Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 320489 16 pageshttpdxdoiorg1011552013320489

Research ArticleEfficient and Accurate Optimal Linear Phase FIR Filter DesignUsing Opposition-Based Harmony Search Algorithm

S K Saha1 R Dutta2 R Choudhury2 R Kar1 D Mandal1 and S P Ghoshal3

1 Department of ECE NIT Durgapur Durgapur 713209 India2Department of ECE BCET Durgapur India3 Department of EE NIT Durgapur Durgapur 713209 India

Correspondence should be addressed to D Mandal durbadalbittugmailcom

Received 11 March 2013 Accepted 29 April 2013

Academic Editors C Grimm and D A Zeze

Copyright copy 2013 S K Saha et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper opposition-based harmony search has been applied for the optimal design of linear phase FIR filters RGA PSO andDE have also been adopted for the sake of comparison The original harmony search algorithm is chosen as the parent one andopposition-based approach is applied During the initialization randomly generated population of solutions is chosen oppositesolutions are also considered and the fitter one is selected as a priori guess In harmonymemory each such solution passes throughmemory consideration rule pitch adjustment rule and then opposition-based reinitialization generation jumping which gives theoptimum result corresponding to the least error fitness in multidimensional search space of FIR filter design Incorporation ofdifferent control parameters in the basic HS algorithm results in the balancing of exploration and exploitation of search space Lowpass high pass band pass and band stop FIR filters are designed with the proposed OHS and other aforementioned algorithmsindividually for comparative optimization performance A comparison of simulation results reveals the optimization efficacy of theOHS over the other optimization techniques for the solution of the multimodal nondifferentiable nonlinear and constrained FIRfilter design problems

1 Introduction

Digital filter is essentially a system or network that improvesthe quality of a signal andor extracts information fromsignals or separates two or more signals which are previ-ously combined The linear time invariant (LTI) system andthe filter are synonymous and are often used to performspectral shaping or frequency selective filtering The natureof this filtering action is determined by the frequencyresponse characteristics which depend on the choice ofsystem parameters that is the coefficients of the differenceequations Thus by proper selection of the coefficients onecan design frequency selective filters that pass signals withfrequency components in some bands while attenuate signalscontaining frequency components in other frequency bands[1 2] There are different techniques for the design of FIRfilters such as window method and frequency samplingmethod All thesemethods are based on approximation to thefrequency characteristics of ideal filters The design method

is based on the requirements of ripples in the passband andthe stopband the stop band attenuation and the transitionwidth In the window method ideal impulse response ismultiplied with a window function There are various kindsof window functions (Butterworth Chebshev Kaiser etc)These windows limit the infinite length impulse response ofideal filter into a finite window to design an actual response[3ndash5] But the major drawback of windowing methods isthat it does not allow sufficient control of the frequencyresponse in the various frequency bands and other filterparameters such as transition width and it tends to processrelatively long filter lengths The designer always has tocompromise on one or other design specifications [6] Theconventional gradient-based optimization method [7] andother classical optimization algorithms [3 4] are not sufficientto optimize multimodal and nonuniform objective functionsof FIR filters and the objective function cannot convergeto the global minimum solution So evolutionary methodshave been implemented in the design of optimal digital filters

2 The Scientific World Journal

with better control of filter parameters and achievement ofthe highest stop band attenuation and the lowest stop bandripples

Different evolutionary optimization techniques arereported in the literatures When considering globaloptimization methods for digital filter design GA [8ndash11]seems to have attracted a considerable attention Althoughstandard GA (also known as real-coded GA (RGA)) showsa good performance for finding the promising regionsof the search space they are inefficient in determiningthe global optimum and prone to revisiting the samesuboptimal solution In order to overcome the problemassociated with RGA orthogonal genetic algorithm (OGA)[12] hybrid Taguchi GA (TGA) [13] have been proposedTabu search [14] Simulated Annealing (SA) [15 16] BeeColony algorithm (BCA) [17] differential evolution (DE)[18 19] seeker optimization algorithm [20] particle swarmoptimization (PSO) [21ndash23] opposition-based BAT (OBAT)algorithm [24] some variants of PSO like PSOwithQuantumInfusion (PSO-QI) [25 26] adaptive inertia weight PSO[27] chaotic mutation PSO (CMPSO) [28 29] Novel PSO(NPSO) [30] Gravitational search algorithm (GSA) [31]seeker optimization algorithm (SOA) [32] some hybridalgorithms like DE-PSO [33] have also been used for thefilter design problems with varying degree of comparativeoptimization effectiveness

Most of the above algorithms show the problems of fixingalgorithmrsquos control parameters premature convergence stag-nation and revisiting of the same solution over and againIn order to overcome these problems in this paper a noveloptimization algorithm called opposition-based harmonysearch (OHS) is employed for the FIR filter design

Tizhoosh introduced the concept of opposition-basedlearning (OBL) in [34] This notion has been applied toaccelerate the reinforcement learning [35ndash37] and the backpropagation learning [38] in neural networks The main ideabehind OBL is the simultaneous consideration of an estimateand its corresponding opposite estimate (ie guess andopposite guess) in order to achieve a better approximationfor the current candidate solution In the recent literature theconcept of opposite numbers has been utilized to speed up theconvergence rate of an optimization algorithm for exampleopposition-based differential evolution (ODE) [39] Can thisidea of opposite number OBL be incorporated during theharmonymemory (HM) initialization and also for generatingthe new harmony vectors during the process of HS In thispaper OBL has been utilized to accelerate the convergencerate of the HS Hence our proposed approach has been calledas opposition-based HS (OHS) OHS uses opposite numbersduringHM initialization and also for generating the newHMduring the evolutionary process of HS

This paper describes the comparative optimal designsof linear phase low pass (LP) high pass (HP) band pass(BP) and band stop (BS) FIR digital filters using otheraforementioned algorithms and the proposed OHS approachindividually The OHS does not prematurely restrict thesearching space A comparison of optimal designs revealsbetter optimization efficacy of the proposed algorithm overthe other optimization techniques for the solution of the

multimodal nondifferentiable highly nonlinear and con-strained FIR filter design problems

The rest of the paper is arranged as follows In Section 2the FIR filter design problem is formulated Section 3 brieflydiscusses the evolutionary approaches employed for theFIR filter designs Section 4 describes the simulation resultsobtained by employingPMRGA PSODE andOHS FinallySection 5 concludes the paper

2 FIR Filter Design

Digital filters are classified as finite impulse response (FIR)or infinite impulse response (IIR) filter depending uponwhether the response of the filter is dependent on only thepresent and past inputs or on the present and past inputs aswell as previous outputs respectively

An FIR filter has a system function of the form given inthe following

119867(119911) = ℎ (0) + ℎ (1) 119911minus1

+ sdot sdot sdot + ℎ (119873) 119911minus119873 (1)

or

119867(119911) =

119873

sum

119899=0

ℎ (119899) 119911minus119899

(2)

where ℎ(119899) is called impulse responseThe difference equation representation is

119910 (119899) = ℎ (0) 119909 (119899) + ℎ (1) 119909 (119899 minus 1) + sdot sdot sdot + ℎ (119873) 119909 (119909 minus 119873)

(3)

The order of the filter is 119873 while the length of the filter(which is equal to the number of coefficients) is 119873 + 1 TheFIR filter structures are always stable and can be designedto have linear phase response The impulse responses ℎ(119899)

are to be determined in the design process and the valuesof ℎ(119899) will determine the type of the filter for example LPHP BP BS and so forth The choice of filters is based onfour broad criteria The filters should provide the followingas little distortion as possible to the signal flat pass band andexhibit high attenuation characteristicswith as low as possiblestop band ripples

Other desirable characteristics include short filter lengthshort frequency transition beyond the cut-off point andthe ability to manipulate the attenuation in the stop bandIn many filtering applications minus3 dB frequency 119891

minus3 dB hasbecome a recognizable parameter for defining the cut-offfrequency 119891

119888

(the frequency at which the magnitude attainsan absolute value of 05) The consequence of using the 3 dBmeasure is that it varies with filter length since the sharpnessof the transition width is a function of the filter orderAdditionally as the filter order increases the transition widthdecreases and119891

minus3dB approaches119891119888 asymptotically [40 41] Inany filter design problem some of these parameters are fixedwhile others are optimized

In this paper the OHS is applied in order to obtain theactual filter response as close as possible to the ideal filterresponse The designed FIR filter with ℎ(119899) individuals or

The Scientific World Journal 3

particlessolutions is positive even symmetric and of evenorder The length of ℎ(119899) is 119873 + 1 that is the number ofcoefficients is also 119873 + 1 In each iteration these solutionsare updated Fitness values of updated solutions are calculatedusing the new coefficients and the new error fitness functionThe solution obtained after a certain number of iterations orafter the error fitness below a certain limit is considered tobe the optimal result The error is used to evaluate the errorfitness function of the solution It takes the error between themagnitudes of frequency responses of the ideal and the actualfilters An ideal filter has amagnitude of one on the pass bandand a magnitude of zero on the stop band The error fitnessfunction is minimized using the evolutionary algorithmsRGA PSO DE and OHS individually The individuals thathave lower error fitness values represent the better filter thatis the filter with better frequency response

The frequency response of the FIR digital filter can becalculated as

119867(119890119895119908119896) =

119873

sum

119899=0

ℎ (119899) 119890minus119895119908119896119899 (4)

where 120596119896

= 2120587119896119873 and 119867(119890119895119908119896) or 119867(119908

119896

) is the Fouriertransform complex vector The frequency is sampled with 119873

points in [0 120587] One has the following

119867119889

(120596) = [119867119889

(1205961

) 119867119889

(1205962

) 119867119889

(1205963

) 119867119889

(120596119873

)]119879

119867119894

(120596) = [119867119894

(1205961

) 119867119894

(1205962

) 119867119894

(1205963

) 119867119894

(120596119873

)]119879

(5)

where119867119894

represents themagnitude response of the ideal filterand for LP HP BP and BS it is given respectively as

119867119894

(120596119896

) = 1 for 0 le 120596 le 120596

119888

0 otherwise

119867119894

(120596119896

) = 0 for 0 le 120596 le 120596

119888

1 otherwise

119867119894

(120596119896

) = 1 for 120596

119901119897

le 120596 le 120596119901ℎ

0 otherwise

119867119894

(120596119896

) = 0 for 120596

119901119897

le 120596 le 120596119901ℎ

1 otherwise

(6)

119867119889

(120596119896

) represents the approximate actual filter to be design-ed and the119873 is the number of samples

Different kinds of fitness functions have been used indifferent literatures as given in the following [18 19 21]

Error = max

119873

sum

119894=1

[1003816100381610038161003816

1003816100381610038161003816119867119889 (120596119896)1003816100381610038161003816 minus

1003816100381610038161003816119867119894 (120596119896)1003816100381610038161003816

1003816100381610038161003816] (7)

Error =

119873

sum

119894=1

[1003816100381610038161003816119867119889 (120596119896)

1003816100381610038161003816 minus1003816100381610038161003816119867119894 (120596119896)

1003816100381610038161003816]2

12

(8)

An error function given by the following equation is theapproximate error function used in popular Parks-McClellan(PM) algorithm for digital filter design [3]

119864 (120596) = 119866 (120596) [119867119889

(120596119896

) minus 119867119894

(120596119896

)] (9)

where 119866(120596) is the weighting function used to provide differ-ent weights for the approximate errors in different frequencybands

Themajor drawback of the PM algorithm is that the ratioof 120575119901

120575119904

is fixed In order to improve the flexibility in theerror function to be minimized so that the desired levels of120575119901

and 120575119904

may be individually specified the error functiongiven in the following equation has been considered as fitnessfunction in [23 26]

1198691

= max120596le120596119901

(|119864 (120596)| minus 120575119901

) +max120596ge120596119904

(|119864 (120596)| minus 120575119904

) (10)

where 120575119901

and 120575119904

are the ripples in the pass band and the stopband respectively and 120596

119901

and 120596119904

are the pass band and stopband normalized edge frequencies respectively

In this paper a novel error fitness function has beenadopted in order to achieve higher stop band attenuation andto have moderate control on the transition width The errorfitness function used in this paper is given in (11) Using thefollowing equation it is found that the proposed filter designapproach results in considerable improvement over the PMand other optimization techniques

1198692

= sum abs [abs (|119867 (120596)| minus 1)minus120575119901

]+sum[abs (|119867 (120596)| minus 120575119904

)]

(11)

For the first term of (11) 120596 isin pass band including a por-tion of the transition band and for the second term of (11)120596 isin stop band including the rest portion of the transitionband The portions of the transition band chosen depend onpass band edge and stop band edge frequencies

The error fitness function given in (11) represents thegeneralized fitness function to beminimized using the evolu-tionary algorithms individually Each algorithm individuallytries to minimize this error and thus improves the filterperformance Since the coefficients of the linear phase FIRfilter are matched the dimension of the problem is thushalved By only determining half of the coefficients the FIRfilter can be designedThis greatly reduces the computationalburdens of the algorithms applied to the design of linearphase FIR filters

3 Optimization Techniques Employed

Evolutionary algorithms stand upon some common char-acteristics like stochastic adaptive and learning in orderto produce intelligent optimization schemes Such schemeshave the potential to adapt to their ever-changing dynamicenvironment through the previously acquired knowledgeTizhoosh introduced the concept of opposition-based learn-ing (OBL) in [34] In this paper OBL has been utilizedto accelerate the convergence rate of the HS Hence ourproposed approach has been called as opposition-basedharmony search (OHS) OHS uses opposite numbers duringHM initialization and also for generating the new harmonymemory (HM) during the evolutionary process of HS Theother algorithms RGA PSO and DE considered in this paperare well known and not discussed here

4 The Scientific World Journal

0 01 02 03 04 05 06 07 08 09 1

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Frequency

Mag

nitu

de (d

B)

PMRGAPSO

DEOHS

Figure 1 dB plots for the FIR LP filter of order 20

31 A Brief Description of HS Algorithm In the basic HSalgorithm each solution is called a harmony It is representedby an 119899-dimension real vector An initial randomly generatedpopulation of harmony vectors is stored in an HM Then anew candidate harmony is generated from all the solutionsin the HM by adopting a memory consideration rule a pitchadjustment rule and a random reinitialization Finally theHM is updated by comparing the new candidate harmonyvector and the worst harmony vector in the HM The worstharmony vector is replaced by the new candidate vector ifit is better than the worst harmony vector in the HM Theabove process is repeated until a certain termination criterionis met Thus the basic HS algorithm consists of three basicphases These are initialization improvisation of a harmonyvector and updating the HM Sequentially these phases aredescribed below

311 Initialization of the Problem and the Parameters of theHS Algorithm In general a global optimization problem canbe enumerated as follows min 119891(119909) st 119909

119895

isin [119901119886119903119886min119895

119901119886119903119886max119895

] 119895 = 1 2 119899 where 119891(119909) is the objectivefunction 119883 = [119909

1

1199092

119909119899

] is the set of design variablesand 119899 is the number of design variables Here 119901119886119903119886

min119895

119901119886119903119886

max119895

are the lower and upper bounds for the designvariable 119909

119895

respectively The parameters of the HS algorithmare the harmonymemory size (HMS) (the number of solutionvectors in HM) the harmony memory consideration rate(HMCR) the pitch adjusting rate (PAR) the distance band-width (BW) and the number of improvisations (NI)NI is thesame as the total number of fitness function calls (NFFCs) Itmay be set as a stopping criterion

312 Initialization of the HM The HM consists of HMSharmony vectors Let 119883

119895

= [119909119895

1

119909119895

2

119909119895

119899

] represent the119895th harmony vector which is randomly generated within the

0

0

01 02 03 04 05 06 07 08 09 1

14

12

1

08

06

04

02

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 2 Normalized plots for the FIR LP filter of order 20

0 005 01 015 02 025 03 035 04 045

12

115

105

11

1

095

09

085

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 3 Normalized pass band ripple plots for the FIR LP filter oforder 20

parameter limits [119901119886119903119886min119895

119901119886119903119886max119895

] Then the HM matrixis filled with the HMS harmony vectors as in the following

HM =

[[[[[

[

1199091

1

1199091

2

sdot sdot sdot 1199091

119899

1199092

1

1199092

2

sdot sdot sdot 1199092

119899

sdot sdot sdot

119909119867119872119878

1

119909119867119872119878

2

sdot sdot sdot 119909119867119872119878

119899

]]]]]

]

(12)

313 Improvisation of a New Harmony A new harmony vec-tor119883new

= (119909new1

119909new2

sdot sdot sdot 119909new119899

) is generated (called improvi-sation) by applying three rules namely (i) a memory consid-eration (ii) a pitch adjustment and (iii) a random selectionFirst of all a uniform random number 119903

1

is generated in the

The Scientific World Journal 5

Table 1 Control parameters of RGA PSO DE and OHS

RGA PSO DE OHSPopulation size = 120 iterationcycles = 600 crossover rate = 1crossover = two point crossovermutation rate = 001 mutation =Gaussian mutation and selection =roulette

Population size = 120 iterationcycles = 600 119862

1

= 1198622

=

205 Vmin119894

= 001 Vmax119894

=

10 119908max = 10 and119908min = 04

Population size = 120iteration cycles = 600

119862119903

= 03 119865 = 05

Population size = 120iteration cycles = 600

HMCR = 06 PARmin = 0PARmax = 09 BWmin =0000001 and BWmax = 1

Table 2 Optimized coefficients of the FIR LP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0020644508012550 0025116793352393 0027005399982491 0020509263646238

ℎ(2) = ℎ(20) 0048721413185106 0047219259300299 0047266866797926 0032132175858325

ℎ(3) = ℎ(19) 0005868601564964 0003546242723169 0005320204222841 minus0013117080938103

ℎ(4) = ℎ(18) minus0040966865300227 minus0040094047283599 minus0038982294859373 minus0053943547883771

ℎ(5) = ℎ(17) minus0000863506780022 minus0000520432067214 minus0003452235386096 minus0012003255218944

ℎ(6) = ℎ(16) 0059796031265565 0060907207778672 0057946858872171 0048106434177123

ℎ(7) = ℎ(15) minus0001408842862974 minus0001759240756773 minus0002051400593964 minus0007969148591832

ℎ(8) = ℎ(14) minus0103117834700311 minus0103613994946693 minus0102715267629915 minus0113260035845183

ℎ(9) = ℎ(13) minus0000440644382089 0000627623037422 0001692937801793 minus0007125766240178

ℎ(10) = ℎ(12) 0317600651261946 0318119036548684 0319795676258768 0310972208736947

ℎ(11) 0500018538901556 0500018538901556 0500018538901556 0500018538901555

Table 3 Optimized coefficients of the FIR HP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0021731353326545 0025559145974814 0029041921147266 0027958875232026

ℎ(2) = ℎ(20) minus0048131602227058 minus0047413653181042 minus0045873202416582 minus0046219133874889

ℎ(3) = ℎ(19) 0006298189918824 0005135430273491 0002950561225606 0005908752612340

ℎ(4) = ℎ(18) 0041895345956760 0039988099089174 0041311799862169 0041684671886351

ℎ(5) = ℎ(17) 0000879943669486 0001405996354021 minus0000283997158910 0000484971329494

ℎ(6) = ℎ(16) minus0059027866591514 minus0060283192968605 minus0060002355552046 minus0058130379060402

ℎ(7) = ℎ(15) minus0000013559660394 0000768613197325 minus0003921102337490 0003431061619144

ℎ(8) = ℎ(14) 0104257677520726 0105120739785348 0106119151142982 0105123375775675

ℎ(9) = ℎ(13) 0003823743541217 0001471927911810 minus0000565063060302 minus0007107362649439

ℎ(10) = ℎ(12) minus0316631427282300 minus0315471590838371 minus0320083906578923 minus0316061867104696

ℎ(11) 0499468012025621 0499981461098444 0499981461098444 0499981461098444

Table 4 Optimized coefficients of the FIR BP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0028502857888104 0024910907374264 0024315759656957 0026978129356347

ℎ(2) = ℎ(20) minus0001893868108392 0000092972958187 0002788616388318 0003014521009408

ℎ(3) = ℎ(19) minus0076189026154460 minus0074535581888545 minus0075883731240533 minus0075573145593866

ℎ(4) = ℎ(18) 0000994123920259 minus0000579129089510 minus0003313368788351 minus0005667273096237

ℎ(5) = ℎ(17) 0053196793860741 0058322287561503 0056134992376798 0054764314739432

ℎ(6) = ℎ(16) minus0000639149080848 minus0000187613541059 0000257826174027 minus0001285546925313

ℎ(7) = ℎ(15) 0100057194730152 0093164875388599 0090912344142406 0093422699573072

ℎ(8) = ℎ(14) 0001409980793664 0001012723950710 0002199187065772 0003282759787987

ℎ(9) = ℎ(13) minus0299380312728113 minus0296866917983546 minus0300934749358008 minus0299362912756530

ℎ(10) = ℎ(12) minus0000752480372393 minus0000392232750468 minus0001799401229551 minus0004731509416213

ℎ(11) 0400369877077545 0400369877077545 0400369877077545 0400369877077545

6 The Scientific World Journal

Table 5 Optimized coefficients of the FIR BS filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0008765244188382 0005065078955931 0005738163937772 0011156843719480

ℎ(2) = ℎ(20) 0054796923249762 0054496716662981 0053905628215447 0052282877968109

ℎ(3) = ℎ(19) 0001796419983890 0005809988516188 0002902448937586 0009942391820387

ℎ(4) = ℎ(18) 0048911654246731 0051144048751957 0049349878942931 0047219028225649

ℎ(5) = ℎ(17) minus0054718457691943 minus0050663949788261 minus0050884656047053 minus0049531151275131

ℎ(6) = ℎ(16) minus0060963142228236 minus0062741465298722 minus0063088550820316 minus0064382546432969

ℎ(7) = ℎ(15) 0004293459264617 minus0000062718416445 0004089341810059 minus0000512890612780

ℎ(8) = ℎ(14) minus0065342448643273 minus0068916923681426 minus0068023108311494 minus0067902719326839

ℎ(9) = ℎ(13) 0300682045893488 0297478557865240 0299063386928411 0296806655163297

ℎ(10) = ℎ(12) 0069036675664641 0074390206250426 0071701365941159 0074321537635153

ℎ(11) 0499582536276171 0499582536276171 0500000357523254 0500000357523254

055 06 065 07 075 08 085 09 095 1

008

007

006

005

004

003

002

001

0

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 4 Normalized stop band ripple plots for the FIR LP filter oforder 20

Table 6 Comparison of stop band attenuations for different typesof FIR filters each of order 20 using different algorithms

Filter type Maximum stop band attenuation (dB)PM RGA PSO DE OHS

LP 2354 2611 2803 2953 3516HP 2355 2525 2810 2916 3386BP 2238 3080 3203 3258 3476BS 2165 2973 3056 3096 3245

range [0 1] If 1199031

is less thanHMCR the decision variable119909new119895

is generated by the memory consideration otherwise 119909new119895

isobtained by a random selection (ie random reinitializationbetween the search bounds) In the memory consideration119909new119895

is selected from any harmony vector 119894 in [1 2 119867119872119878]Secondly each decision variable 119909

new119895

will undergo a pitchadjustment with a probability of PAR if it is updated by the

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 5 dB plots for the FIR HP filter of order 20

memory consideration The pitch adjustment rule is given asfollows

119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 (13)

where 1199033

is a uniform random number between 0 and 1

314 Updating of HM After a new harmony vector 119883new119895

isgenerated the HMwill be updated by the survival of the fittervector between119883

new and the worst harmony vector119883worst inthe HM That is 119883new will replace 119883

worst and become a newmember of the HM if the fitness value of 119883new is better thanthe fitness value of119883worst

The computational procedure of the basic HS algorithmcan be summarized as shown in Algorithm 1

32 The Improved Harmony Search (IHS) Algorithm Thebasic HS algorithm uses fixed values for PAR and BWparameters The IHS algorithm proposed by Mahdavi et al

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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2 The Scientific World Journal

with better control of filter parameters and achievement ofthe highest stop band attenuation and the lowest stop bandripples

Different evolutionary optimization techniques arereported in the literatures When considering globaloptimization methods for digital filter design GA [8ndash11]seems to have attracted a considerable attention Althoughstandard GA (also known as real-coded GA (RGA)) showsa good performance for finding the promising regionsof the search space they are inefficient in determiningthe global optimum and prone to revisiting the samesuboptimal solution In order to overcome the problemassociated with RGA orthogonal genetic algorithm (OGA)[12] hybrid Taguchi GA (TGA) [13] have been proposedTabu search [14] Simulated Annealing (SA) [15 16] BeeColony algorithm (BCA) [17] differential evolution (DE)[18 19] seeker optimization algorithm [20] particle swarmoptimization (PSO) [21ndash23] opposition-based BAT (OBAT)algorithm [24] some variants of PSO like PSOwithQuantumInfusion (PSO-QI) [25 26] adaptive inertia weight PSO[27] chaotic mutation PSO (CMPSO) [28 29] Novel PSO(NPSO) [30] Gravitational search algorithm (GSA) [31]seeker optimization algorithm (SOA) [32] some hybridalgorithms like DE-PSO [33] have also been used for thefilter design problems with varying degree of comparativeoptimization effectiveness

Most of the above algorithms show the problems of fixingalgorithmrsquos control parameters premature convergence stag-nation and revisiting of the same solution over and againIn order to overcome these problems in this paper a noveloptimization algorithm called opposition-based harmonysearch (OHS) is employed for the FIR filter design

Tizhoosh introduced the concept of opposition-basedlearning (OBL) in [34] This notion has been applied toaccelerate the reinforcement learning [35ndash37] and the backpropagation learning [38] in neural networks The main ideabehind OBL is the simultaneous consideration of an estimateand its corresponding opposite estimate (ie guess andopposite guess) in order to achieve a better approximationfor the current candidate solution In the recent literature theconcept of opposite numbers has been utilized to speed up theconvergence rate of an optimization algorithm for exampleopposition-based differential evolution (ODE) [39] Can thisidea of opposite number OBL be incorporated during theharmonymemory (HM) initialization and also for generatingthe new harmony vectors during the process of HS In thispaper OBL has been utilized to accelerate the convergencerate of the HS Hence our proposed approach has been calledas opposition-based HS (OHS) OHS uses opposite numbersduringHM initialization and also for generating the newHMduring the evolutionary process of HS

This paper describes the comparative optimal designsof linear phase low pass (LP) high pass (HP) band pass(BP) and band stop (BS) FIR digital filters using otheraforementioned algorithms and the proposed OHS approachindividually The OHS does not prematurely restrict thesearching space A comparison of optimal designs revealsbetter optimization efficacy of the proposed algorithm overthe other optimization techniques for the solution of the

multimodal nondifferentiable highly nonlinear and con-strained FIR filter design problems

The rest of the paper is arranged as follows In Section 2the FIR filter design problem is formulated Section 3 brieflydiscusses the evolutionary approaches employed for theFIR filter designs Section 4 describes the simulation resultsobtained by employingPMRGA PSODE andOHS FinallySection 5 concludes the paper

2 FIR Filter Design

Digital filters are classified as finite impulse response (FIR)or infinite impulse response (IIR) filter depending uponwhether the response of the filter is dependent on only thepresent and past inputs or on the present and past inputs aswell as previous outputs respectively

An FIR filter has a system function of the form given inthe following

119867(119911) = ℎ (0) + ℎ (1) 119911minus1

+ sdot sdot sdot + ℎ (119873) 119911minus119873 (1)

or

119867(119911) =

119873

sum

119899=0

ℎ (119899) 119911minus119899

(2)

where ℎ(119899) is called impulse responseThe difference equation representation is

119910 (119899) = ℎ (0) 119909 (119899) + ℎ (1) 119909 (119899 minus 1) + sdot sdot sdot + ℎ (119873) 119909 (119909 minus 119873)

(3)

The order of the filter is 119873 while the length of the filter(which is equal to the number of coefficients) is 119873 + 1 TheFIR filter structures are always stable and can be designedto have linear phase response The impulse responses ℎ(119899)

are to be determined in the design process and the valuesof ℎ(119899) will determine the type of the filter for example LPHP BP BS and so forth The choice of filters is based onfour broad criteria The filters should provide the followingas little distortion as possible to the signal flat pass band andexhibit high attenuation characteristicswith as low as possiblestop band ripples

Other desirable characteristics include short filter lengthshort frequency transition beyond the cut-off point andthe ability to manipulate the attenuation in the stop bandIn many filtering applications minus3 dB frequency 119891

minus3 dB hasbecome a recognizable parameter for defining the cut-offfrequency 119891

119888

(the frequency at which the magnitude attainsan absolute value of 05) The consequence of using the 3 dBmeasure is that it varies with filter length since the sharpnessof the transition width is a function of the filter orderAdditionally as the filter order increases the transition widthdecreases and119891

minus3dB approaches119891119888 asymptotically [40 41] Inany filter design problem some of these parameters are fixedwhile others are optimized

In this paper the OHS is applied in order to obtain theactual filter response as close as possible to the ideal filterresponse The designed FIR filter with ℎ(119899) individuals or

The Scientific World Journal 3

particlessolutions is positive even symmetric and of evenorder The length of ℎ(119899) is 119873 + 1 that is the number ofcoefficients is also 119873 + 1 In each iteration these solutionsare updated Fitness values of updated solutions are calculatedusing the new coefficients and the new error fitness functionThe solution obtained after a certain number of iterations orafter the error fitness below a certain limit is considered tobe the optimal result The error is used to evaluate the errorfitness function of the solution It takes the error between themagnitudes of frequency responses of the ideal and the actualfilters An ideal filter has amagnitude of one on the pass bandand a magnitude of zero on the stop band The error fitnessfunction is minimized using the evolutionary algorithmsRGA PSO DE and OHS individually The individuals thathave lower error fitness values represent the better filter thatis the filter with better frequency response

The frequency response of the FIR digital filter can becalculated as

119867(119890119895119908119896) =

119873

sum

119899=0

ℎ (119899) 119890minus119895119908119896119899 (4)

where 120596119896

= 2120587119896119873 and 119867(119890119895119908119896) or 119867(119908

119896

) is the Fouriertransform complex vector The frequency is sampled with 119873

points in [0 120587] One has the following

119867119889

(120596) = [119867119889

(1205961

) 119867119889

(1205962

) 119867119889

(1205963

) 119867119889

(120596119873

)]119879

119867119894

(120596) = [119867119894

(1205961

) 119867119894

(1205962

) 119867119894

(1205963

) 119867119894

(120596119873

)]119879

(5)

where119867119894

represents themagnitude response of the ideal filterand for LP HP BP and BS it is given respectively as

119867119894

(120596119896

) = 1 for 0 le 120596 le 120596

119888

0 otherwise

119867119894

(120596119896

) = 0 for 0 le 120596 le 120596

119888

1 otherwise

119867119894

(120596119896

) = 1 for 120596

119901119897

le 120596 le 120596119901ℎ

0 otherwise

119867119894

(120596119896

) = 0 for 120596

119901119897

le 120596 le 120596119901ℎ

1 otherwise

(6)

119867119889

(120596119896

) represents the approximate actual filter to be design-ed and the119873 is the number of samples

Different kinds of fitness functions have been used indifferent literatures as given in the following [18 19 21]

Error = max

119873

sum

119894=1

[1003816100381610038161003816

1003816100381610038161003816119867119889 (120596119896)1003816100381610038161003816 minus

1003816100381610038161003816119867119894 (120596119896)1003816100381610038161003816

1003816100381610038161003816] (7)

Error =

119873

sum

119894=1

[1003816100381610038161003816119867119889 (120596119896)

1003816100381610038161003816 minus1003816100381610038161003816119867119894 (120596119896)

1003816100381610038161003816]2

12

(8)

An error function given by the following equation is theapproximate error function used in popular Parks-McClellan(PM) algorithm for digital filter design [3]

119864 (120596) = 119866 (120596) [119867119889

(120596119896

) minus 119867119894

(120596119896

)] (9)

where 119866(120596) is the weighting function used to provide differ-ent weights for the approximate errors in different frequencybands

Themajor drawback of the PM algorithm is that the ratioof 120575119901

120575119904

is fixed In order to improve the flexibility in theerror function to be minimized so that the desired levels of120575119901

and 120575119904

may be individually specified the error functiongiven in the following equation has been considered as fitnessfunction in [23 26]

1198691

= max120596le120596119901

(|119864 (120596)| minus 120575119901

) +max120596ge120596119904

(|119864 (120596)| minus 120575119904

) (10)

where 120575119901

and 120575119904

are the ripples in the pass band and the stopband respectively and 120596

119901

and 120596119904

are the pass band and stopband normalized edge frequencies respectively

In this paper a novel error fitness function has beenadopted in order to achieve higher stop band attenuation andto have moderate control on the transition width The errorfitness function used in this paper is given in (11) Using thefollowing equation it is found that the proposed filter designapproach results in considerable improvement over the PMand other optimization techniques

1198692

= sum abs [abs (|119867 (120596)| minus 1)minus120575119901

]+sum[abs (|119867 (120596)| minus 120575119904

)]

(11)

For the first term of (11) 120596 isin pass band including a por-tion of the transition band and for the second term of (11)120596 isin stop band including the rest portion of the transitionband The portions of the transition band chosen depend onpass band edge and stop band edge frequencies

The error fitness function given in (11) represents thegeneralized fitness function to beminimized using the evolu-tionary algorithms individually Each algorithm individuallytries to minimize this error and thus improves the filterperformance Since the coefficients of the linear phase FIRfilter are matched the dimension of the problem is thushalved By only determining half of the coefficients the FIRfilter can be designedThis greatly reduces the computationalburdens of the algorithms applied to the design of linearphase FIR filters

3 Optimization Techniques Employed

Evolutionary algorithms stand upon some common char-acteristics like stochastic adaptive and learning in orderto produce intelligent optimization schemes Such schemeshave the potential to adapt to their ever-changing dynamicenvironment through the previously acquired knowledgeTizhoosh introduced the concept of opposition-based learn-ing (OBL) in [34] In this paper OBL has been utilizedto accelerate the convergence rate of the HS Hence ourproposed approach has been called as opposition-basedharmony search (OHS) OHS uses opposite numbers duringHM initialization and also for generating the new harmonymemory (HM) during the evolutionary process of HS Theother algorithms RGA PSO and DE considered in this paperare well known and not discussed here

4 The Scientific World Journal

0 01 02 03 04 05 06 07 08 09 1

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Frequency

Mag

nitu

de (d

B)

PMRGAPSO

DEOHS

Figure 1 dB plots for the FIR LP filter of order 20

31 A Brief Description of HS Algorithm In the basic HSalgorithm each solution is called a harmony It is representedby an 119899-dimension real vector An initial randomly generatedpopulation of harmony vectors is stored in an HM Then anew candidate harmony is generated from all the solutionsin the HM by adopting a memory consideration rule a pitchadjustment rule and a random reinitialization Finally theHM is updated by comparing the new candidate harmonyvector and the worst harmony vector in the HM The worstharmony vector is replaced by the new candidate vector ifit is better than the worst harmony vector in the HM Theabove process is repeated until a certain termination criterionis met Thus the basic HS algorithm consists of three basicphases These are initialization improvisation of a harmonyvector and updating the HM Sequentially these phases aredescribed below

311 Initialization of the Problem and the Parameters of theHS Algorithm In general a global optimization problem canbe enumerated as follows min 119891(119909) st 119909

119895

isin [119901119886119903119886min119895

119901119886119903119886max119895

] 119895 = 1 2 119899 where 119891(119909) is the objectivefunction 119883 = [119909

1

1199092

119909119899

] is the set of design variablesand 119899 is the number of design variables Here 119901119886119903119886

min119895

119901119886119903119886

max119895

are the lower and upper bounds for the designvariable 119909

119895

respectively The parameters of the HS algorithmare the harmonymemory size (HMS) (the number of solutionvectors in HM) the harmony memory consideration rate(HMCR) the pitch adjusting rate (PAR) the distance band-width (BW) and the number of improvisations (NI)NI is thesame as the total number of fitness function calls (NFFCs) Itmay be set as a stopping criterion

312 Initialization of the HM The HM consists of HMSharmony vectors Let 119883

119895

= [119909119895

1

119909119895

2

119909119895

119899

] represent the119895th harmony vector which is randomly generated within the

0

0

01 02 03 04 05 06 07 08 09 1

14

12

1

08

06

04

02

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 2 Normalized plots for the FIR LP filter of order 20

0 005 01 015 02 025 03 035 04 045

12

115

105

11

1

095

09

085

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 3 Normalized pass band ripple plots for the FIR LP filter oforder 20

parameter limits [119901119886119903119886min119895

119901119886119903119886max119895

] Then the HM matrixis filled with the HMS harmony vectors as in the following

HM =

[[[[[

[

1199091

1

1199091

2

sdot sdot sdot 1199091

119899

1199092

1

1199092

2

sdot sdot sdot 1199092

119899

sdot sdot sdot

119909119867119872119878

1

119909119867119872119878

2

sdot sdot sdot 119909119867119872119878

119899

]]]]]

]

(12)

313 Improvisation of a New Harmony A new harmony vec-tor119883new

= (119909new1

119909new2

sdot sdot sdot 119909new119899

) is generated (called improvi-sation) by applying three rules namely (i) a memory consid-eration (ii) a pitch adjustment and (iii) a random selectionFirst of all a uniform random number 119903

1

is generated in the

The Scientific World Journal 5

Table 1 Control parameters of RGA PSO DE and OHS

RGA PSO DE OHSPopulation size = 120 iterationcycles = 600 crossover rate = 1crossover = two point crossovermutation rate = 001 mutation =Gaussian mutation and selection =roulette

Population size = 120 iterationcycles = 600 119862

1

= 1198622

=

205 Vmin119894

= 001 Vmax119894

=

10 119908max = 10 and119908min = 04

Population size = 120iteration cycles = 600

119862119903

= 03 119865 = 05

Population size = 120iteration cycles = 600

HMCR = 06 PARmin = 0PARmax = 09 BWmin =0000001 and BWmax = 1

Table 2 Optimized coefficients of the FIR LP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0020644508012550 0025116793352393 0027005399982491 0020509263646238

ℎ(2) = ℎ(20) 0048721413185106 0047219259300299 0047266866797926 0032132175858325

ℎ(3) = ℎ(19) 0005868601564964 0003546242723169 0005320204222841 minus0013117080938103

ℎ(4) = ℎ(18) minus0040966865300227 minus0040094047283599 minus0038982294859373 minus0053943547883771

ℎ(5) = ℎ(17) minus0000863506780022 minus0000520432067214 minus0003452235386096 minus0012003255218944

ℎ(6) = ℎ(16) 0059796031265565 0060907207778672 0057946858872171 0048106434177123

ℎ(7) = ℎ(15) minus0001408842862974 minus0001759240756773 minus0002051400593964 minus0007969148591832

ℎ(8) = ℎ(14) minus0103117834700311 minus0103613994946693 minus0102715267629915 minus0113260035845183

ℎ(9) = ℎ(13) minus0000440644382089 0000627623037422 0001692937801793 minus0007125766240178

ℎ(10) = ℎ(12) 0317600651261946 0318119036548684 0319795676258768 0310972208736947

ℎ(11) 0500018538901556 0500018538901556 0500018538901556 0500018538901555

Table 3 Optimized coefficients of the FIR HP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0021731353326545 0025559145974814 0029041921147266 0027958875232026

ℎ(2) = ℎ(20) minus0048131602227058 minus0047413653181042 minus0045873202416582 minus0046219133874889

ℎ(3) = ℎ(19) 0006298189918824 0005135430273491 0002950561225606 0005908752612340

ℎ(4) = ℎ(18) 0041895345956760 0039988099089174 0041311799862169 0041684671886351

ℎ(5) = ℎ(17) 0000879943669486 0001405996354021 minus0000283997158910 0000484971329494

ℎ(6) = ℎ(16) minus0059027866591514 minus0060283192968605 minus0060002355552046 minus0058130379060402

ℎ(7) = ℎ(15) minus0000013559660394 0000768613197325 minus0003921102337490 0003431061619144

ℎ(8) = ℎ(14) 0104257677520726 0105120739785348 0106119151142982 0105123375775675

ℎ(9) = ℎ(13) 0003823743541217 0001471927911810 minus0000565063060302 minus0007107362649439

ℎ(10) = ℎ(12) minus0316631427282300 minus0315471590838371 minus0320083906578923 minus0316061867104696

ℎ(11) 0499468012025621 0499981461098444 0499981461098444 0499981461098444

Table 4 Optimized coefficients of the FIR BP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0028502857888104 0024910907374264 0024315759656957 0026978129356347

ℎ(2) = ℎ(20) minus0001893868108392 0000092972958187 0002788616388318 0003014521009408

ℎ(3) = ℎ(19) minus0076189026154460 minus0074535581888545 minus0075883731240533 minus0075573145593866

ℎ(4) = ℎ(18) 0000994123920259 minus0000579129089510 minus0003313368788351 minus0005667273096237

ℎ(5) = ℎ(17) 0053196793860741 0058322287561503 0056134992376798 0054764314739432

ℎ(6) = ℎ(16) minus0000639149080848 minus0000187613541059 0000257826174027 minus0001285546925313

ℎ(7) = ℎ(15) 0100057194730152 0093164875388599 0090912344142406 0093422699573072

ℎ(8) = ℎ(14) 0001409980793664 0001012723950710 0002199187065772 0003282759787987

ℎ(9) = ℎ(13) minus0299380312728113 minus0296866917983546 minus0300934749358008 minus0299362912756530

ℎ(10) = ℎ(12) minus0000752480372393 minus0000392232750468 minus0001799401229551 minus0004731509416213

ℎ(11) 0400369877077545 0400369877077545 0400369877077545 0400369877077545

6 The Scientific World Journal

Table 5 Optimized coefficients of the FIR BS filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0008765244188382 0005065078955931 0005738163937772 0011156843719480

ℎ(2) = ℎ(20) 0054796923249762 0054496716662981 0053905628215447 0052282877968109

ℎ(3) = ℎ(19) 0001796419983890 0005809988516188 0002902448937586 0009942391820387

ℎ(4) = ℎ(18) 0048911654246731 0051144048751957 0049349878942931 0047219028225649

ℎ(5) = ℎ(17) minus0054718457691943 minus0050663949788261 minus0050884656047053 minus0049531151275131

ℎ(6) = ℎ(16) minus0060963142228236 minus0062741465298722 minus0063088550820316 minus0064382546432969

ℎ(7) = ℎ(15) 0004293459264617 minus0000062718416445 0004089341810059 minus0000512890612780

ℎ(8) = ℎ(14) minus0065342448643273 minus0068916923681426 minus0068023108311494 minus0067902719326839

ℎ(9) = ℎ(13) 0300682045893488 0297478557865240 0299063386928411 0296806655163297

ℎ(10) = ℎ(12) 0069036675664641 0074390206250426 0071701365941159 0074321537635153

ℎ(11) 0499582536276171 0499582536276171 0500000357523254 0500000357523254

055 06 065 07 075 08 085 09 095 1

008

007

006

005

004

003

002

001

0

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 4 Normalized stop band ripple plots for the FIR LP filter oforder 20

Table 6 Comparison of stop band attenuations for different typesof FIR filters each of order 20 using different algorithms

Filter type Maximum stop band attenuation (dB)PM RGA PSO DE OHS

LP 2354 2611 2803 2953 3516HP 2355 2525 2810 2916 3386BP 2238 3080 3203 3258 3476BS 2165 2973 3056 3096 3245

range [0 1] If 1199031

is less thanHMCR the decision variable119909new119895

is generated by the memory consideration otherwise 119909new119895

isobtained by a random selection (ie random reinitializationbetween the search bounds) In the memory consideration119909new119895

is selected from any harmony vector 119894 in [1 2 119867119872119878]Secondly each decision variable 119909

new119895

will undergo a pitchadjustment with a probability of PAR if it is updated by the

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 5 dB plots for the FIR HP filter of order 20

memory consideration The pitch adjustment rule is given asfollows

119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 (13)

where 1199033

is a uniform random number between 0 and 1

314 Updating of HM After a new harmony vector 119883new119895

isgenerated the HMwill be updated by the survival of the fittervector between119883

new and the worst harmony vector119883worst inthe HM That is 119883new will replace 119883

worst and become a newmember of the HM if the fitness value of 119883new is better thanthe fitness value of119883worst

The computational procedure of the basic HS algorithmcan be summarized as shown in Algorithm 1

32 The Improved Harmony Search (IHS) Algorithm Thebasic HS algorithm uses fixed values for PAR and BWparameters The IHS algorithm proposed by Mahdavi et al

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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The Scientific World Journal 3

particlessolutions is positive even symmetric and of evenorder The length of ℎ(119899) is 119873 + 1 that is the number ofcoefficients is also 119873 + 1 In each iteration these solutionsare updated Fitness values of updated solutions are calculatedusing the new coefficients and the new error fitness functionThe solution obtained after a certain number of iterations orafter the error fitness below a certain limit is considered tobe the optimal result The error is used to evaluate the errorfitness function of the solution It takes the error between themagnitudes of frequency responses of the ideal and the actualfilters An ideal filter has amagnitude of one on the pass bandand a magnitude of zero on the stop band The error fitnessfunction is minimized using the evolutionary algorithmsRGA PSO DE and OHS individually The individuals thathave lower error fitness values represent the better filter thatis the filter with better frequency response

The frequency response of the FIR digital filter can becalculated as

119867(119890119895119908119896) =

119873

sum

119899=0

ℎ (119899) 119890minus119895119908119896119899 (4)

where 120596119896

= 2120587119896119873 and 119867(119890119895119908119896) or 119867(119908

119896

) is the Fouriertransform complex vector The frequency is sampled with 119873

points in [0 120587] One has the following

119867119889

(120596) = [119867119889

(1205961

) 119867119889

(1205962

) 119867119889

(1205963

) 119867119889

(120596119873

)]119879

119867119894

(120596) = [119867119894

(1205961

) 119867119894

(1205962

) 119867119894

(1205963

) 119867119894

(120596119873

)]119879

(5)

where119867119894

represents themagnitude response of the ideal filterand for LP HP BP and BS it is given respectively as

119867119894

(120596119896

) = 1 for 0 le 120596 le 120596

119888

0 otherwise

119867119894

(120596119896

) = 0 for 0 le 120596 le 120596

119888

1 otherwise

119867119894

(120596119896

) = 1 for 120596

119901119897

le 120596 le 120596119901ℎ

0 otherwise

119867119894

(120596119896

) = 0 for 120596

119901119897

le 120596 le 120596119901ℎ

1 otherwise

(6)

119867119889

(120596119896

) represents the approximate actual filter to be design-ed and the119873 is the number of samples

Different kinds of fitness functions have been used indifferent literatures as given in the following [18 19 21]

Error = max

119873

sum

119894=1

[1003816100381610038161003816

1003816100381610038161003816119867119889 (120596119896)1003816100381610038161003816 minus

1003816100381610038161003816119867119894 (120596119896)1003816100381610038161003816

1003816100381610038161003816] (7)

Error =

119873

sum

119894=1

[1003816100381610038161003816119867119889 (120596119896)

1003816100381610038161003816 minus1003816100381610038161003816119867119894 (120596119896)

1003816100381610038161003816]2

12

(8)

An error function given by the following equation is theapproximate error function used in popular Parks-McClellan(PM) algorithm for digital filter design [3]

119864 (120596) = 119866 (120596) [119867119889

(120596119896

) minus 119867119894

(120596119896

)] (9)

where 119866(120596) is the weighting function used to provide differ-ent weights for the approximate errors in different frequencybands

Themajor drawback of the PM algorithm is that the ratioof 120575119901

120575119904

is fixed In order to improve the flexibility in theerror function to be minimized so that the desired levels of120575119901

and 120575119904

may be individually specified the error functiongiven in the following equation has been considered as fitnessfunction in [23 26]

1198691

= max120596le120596119901

(|119864 (120596)| minus 120575119901

) +max120596ge120596119904

(|119864 (120596)| minus 120575119904

) (10)

where 120575119901

and 120575119904

are the ripples in the pass band and the stopband respectively and 120596

119901

and 120596119904

are the pass band and stopband normalized edge frequencies respectively

In this paper a novel error fitness function has beenadopted in order to achieve higher stop band attenuation andto have moderate control on the transition width The errorfitness function used in this paper is given in (11) Using thefollowing equation it is found that the proposed filter designapproach results in considerable improvement over the PMand other optimization techniques

1198692

= sum abs [abs (|119867 (120596)| minus 1)minus120575119901

]+sum[abs (|119867 (120596)| minus 120575119904

)]

(11)

For the first term of (11) 120596 isin pass band including a por-tion of the transition band and for the second term of (11)120596 isin stop band including the rest portion of the transitionband The portions of the transition band chosen depend onpass band edge and stop band edge frequencies

The error fitness function given in (11) represents thegeneralized fitness function to beminimized using the evolu-tionary algorithms individually Each algorithm individuallytries to minimize this error and thus improves the filterperformance Since the coefficients of the linear phase FIRfilter are matched the dimension of the problem is thushalved By only determining half of the coefficients the FIRfilter can be designedThis greatly reduces the computationalburdens of the algorithms applied to the design of linearphase FIR filters

3 Optimization Techniques Employed

Evolutionary algorithms stand upon some common char-acteristics like stochastic adaptive and learning in orderto produce intelligent optimization schemes Such schemeshave the potential to adapt to their ever-changing dynamicenvironment through the previously acquired knowledgeTizhoosh introduced the concept of opposition-based learn-ing (OBL) in [34] In this paper OBL has been utilizedto accelerate the convergence rate of the HS Hence ourproposed approach has been called as opposition-basedharmony search (OHS) OHS uses opposite numbers duringHM initialization and also for generating the new harmonymemory (HM) during the evolutionary process of HS Theother algorithms RGA PSO and DE considered in this paperare well known and not discussed here

4 The Scientific World Journal

0 01 02 03 04 05 06 07 08 09 1

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Frequency

Mag

nitu

de (d

B)

PMRGAPSO

DEOHS

Figure 1 dB plots for the FIR LP filter of order 20

31 A Brief Description of HS Algorithm In the basic HSalgorithm each solution is called a harmony It is representedby an 119899-dimension real vector An initial randomly generatedpopulation of harmony vectors is stored in an HM Then anew candidate harmony is generated from all the solutionsin the HM by adopting a memory consideration rule a pitchadjustment rule and a random reinitialization Finally theHM is updated by comparing the new candidate harmonyvector and the worst harmony vector in the HM The worstharmony vector is replaced by the new candidate vector ifit is better than the worst harmony vector in the HM Theabove process is repeated until a certain termination criterionis met Thus the basic HS algorithm consists of three basicphases These are initialization improvisation of a harmonyvector and updating the HM Sequentially these phases aredescribed below

311 Initialization of the Problem and the Parameters of theHS Algorithm In general a global optimization problem canbe enumerated as follows min 119891(119909) st 119909

119895

isin [119901119886119903119886min119895

119901119886119903119886max119895

] 119895 = 1 2 119899 where 119891(119909) is the objectivefunction 119883 = [119909

1

1199092

119909119899

] is the set of design variablesand 119899 is the number of design variables Here 119901119886119903119886

min119895

119901119886119903119886

max119895

are the lower and upper bounds for the designvariable 119909

119895

respectively The parameters of the HS algorithmare the harmonymemory size (HMS) (the number of solutionvectors in HM) the harmony memory consideration rate(HMCR) the pitch adjusting rate (PAR) the distance band-width (BW) and the number of improvisations (NI)NI is thesame as the total number of fitness function calls (NFFCs) Itmay be set as a stopping criterion

312 Initialization of the HM The HM consists of HMSharmony vectors Let 119883

119895

= [119909119895

1

119909119895

2

119909119895

119899

] represent the119895th harmony vector which is randomly generated within the

0

0

01 02 03 04 05 06 07 08 09 1

14

12

1

08

06

04

02

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 2 Normalized plots for the FIR LP filter of order 20

0 005 01 015 02 025 03 035 04 045

12

115

105

11

1

095

09

085

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 3 Normalized pass band ripple plots for the FIR LP filter oforder 20

parameter limits [119901119886119903119886min119895

119901119886119903119886max119895

] Then the HM matrixis filled with the HMS harmony vectors as in the following

HM =

[[[[[

[

1199091

1

1199091

2

sdot sdot sdot 1199091

119899

1199092

1

1199092

2

sdot sdot sdot 1199092

119899

sdot sdot sdot

119909119867119872119878

1

119909119867119872119878

2

sdot sdot sdot 119909119867119872119878

119899

]]]]]

]

(12)

313 Improvisation of a New Harmony A new harmony vec-tor119883new

= (119909new1

119909new2

sdot sdot sdot 119909new119899

) is generated (called improvi-sation) by applying three rules namely (i) a memory consid-eration (ii) a pitch adjustment and (iii) a random selectionFirst of all a uniform random number 119903

1

is generated in the

The Scientific World Journal 5

Table 1 Control parameters of RGA PSO DE and OHS

RGA PSO DE OHSPopulation size = 120 iterationcycles = 600 crossover rate = 1crossover = two point crossovermutation rate = 001 mutation =Gaussian mutation and selection =roulette

Population size = 120 iterationcycles = 600 119862

1

= 1198622

=

205 Vmin119894

= 001 Vmax119894

=

10 119908max = 10 and119908min = 04

Population size = 120iteration cycles = 600

119862119903

= 03 119865 = 05

Population size = 120iteration cycles = 600

HMCR = 06 PARmin = 0PARmax = 09 BWmin =0000001 and BWmax = 1

Table 2 Optimized coefficients of the FIR LP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0020644508012550 0025116793352393 0027005399982491 0020509263646238

ℎ(2) = ℎ(20) 0048721413185106 0047219259300299 0047266866797926 0032132175858325

ℎ(3) = ℎ(19) 0005868601564964 0003546242723169 0005320204222841 minus0013117080938103

ℎ(4) = ℎ(18) minus0040966865300227 minus0040094047283599 minus0038982294859373 minus0053943547883771

ℎ(5) = ℎ(17) minus0000863506780022 minus0000520432067214 minus0003452235386096 minus0012003255218944

ℎ(6) = ℎ(16) 0059796031265565 0060907207778672 0057946858872171 0048106434177123

ℎ(7) = ℎ(15) minus0001408842862974 minus0001759240756773 minus0002051400593964 minus0007969148591832

ℎ(8) = ℎ(14) minus0103117834700311 minus0103613994946693 minus0102715267629915 minus0113260035845183

ℎ(9) = ℎ(13) minus0000440644382089 0000627623037422 0001692937801793 minus0007125766240178

ℎ(10) = ℎ(12) 0317600651261946 0318119036548684 0319795676258768 0310972208736947

ℎ(11) 0500018538901556 0500018538901556 0500018538901556 0500018538901555

Table 3 Optimized coefficients of the FIR HP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0021731353326545 0025559145974814 0029041921147266 0027958875232026

ℎ(2) = ℎ(20) minus0048131602227058 minus0047413653181042 minus0045873202416582 minus0046219133874889

ℎ(3) = ℎ(19) 0006298189918824 0005135430273491 0002950561225606 0005908752612340

ℎ(4) = ℎ(18) 0041895345956760 0039988099089174 0041311799862169 0041684671886351

ℎ(5) = ℎ(17) 0000879943669486 0001405996354021 minus0000283997158910 0000484971329494

ℎ(6) = ℎ(16) minus0059027866591514 minus0060283192968605 minus0060002355552046 minus0058130379060402

ℎ(7) = ℎ(15) minus0000013559660394 0000768613197325 minus0003921102337490 0003431061619144

ℎ(8) = ℎ(14) 0104257677520726 0105120739785348 0106119151142982 0105123375775675

ℎ(9) = ℎ(13) 0003823743541217 0001471927911810 minus0000565063060302 minus0007107362649439

ℎ(10) = ℎ(12) minus0316631427282300 minus0315471590838371 minus0320083906578923 minus0316061867104696

ℎ(11) 0499468012025621 0499981461098444 0499981461098444 0499981461098444

Table 4 Optimized coefficients of the FIR BP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0028502857888104 0024910907374264 0024315759656957 0026978129356347

ℎ(2) = ℎ(20) minus0001893868108392 0000092972958187 0002788616388318 0003014521009408

ℎ(3) = ℎ(19) minus0076189026154460 minus0074535581888545 minus0075883731240533 minus0075573145593866

ℎ(4) = ℎ(18) 0000994123920259 minus0000579129089510 minus0003313368788351 minus0005667273096237

ℎ(5) = ℎ(17) 0053196793860741 0058322287561503 0056134992376798 0054764314739432

ℎ(6) = ℎ(16) minus0000639149080848 minus0000187613541059 0000257826174027 minus0001285546925313

ℎ(7) = ℎ(15) 0100057194730152 0093164875388599 0090912344142406 0093422699573072

ℎ(8) = ℎ(14) 0001409980793664 0001012723950710 0002199187065772 0003282759787987

ℎ(9) = ℎ(13) minus0299380312728113 minus0296866917983546 minus0300934749358008 minus0299362912756530

ℎ(10) = ℎ(12) minus0000752480372393 minus0000392232750468 minus0001799401229551 minus0004731509416213

ℎ(11) 0400369877077545 0400369877077545 0400369877077545 0400369877077545

6 The Scientific World Journal

Table 5 Optimized coefficients of the FIR BS filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0008765244188382 0005065078955931 0005738163937772 0011156843719480

ℎ(2) = ℎ(20) 0054796923249762 0054496716662981 0053905628215447 0052282877968109

ℎ(3) = ℎ(19) 0001796419983890 0005809988516188 0002902448937586 0009942391820387

ℎ(4) = ℎ(18) 0048911654246731 0051144048751957 0049349878942931 0047219028225649

ℎ(5) = ℎ(17) minus0054718457691943 minus0050663949788261 minus0050884656047053 minus0049531151275131

ℎ(6) = ℎ(16) minus0060963142228236 minus0062741465298722 minus0063088550820316 minus0064382546432969

ℎ(7) = ℎ(15) 0004293459264617 minus0000062718416445 0004089341810059 minus0000512890612780

ℎ(8) = ℎ(14) minus0065342448643273 minus0068916923681426 minus0068023108311494 minus0067902719326839

ℎ(9) = ℎ(13) 0300682045893488 0297478557865240 0299063386928411 0296806655163297

ℎ(10) = ℎ(12) 0069036675664641 0074390206250426 0071701365941159 0074321537635153

ℎ(11) 0499582536276171 0499582536276171 0500000357523254 0500000357523254

055 06 065 07 075 08 085 09 095 1

008

007

006

005

004

003

002

001

0

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 4 Normalized stop band ripple plots for the FIR LP filter oforder 20

Table 6 Comparison of stop band attenuations for different typesof FIR filters each of order 20 using different algorithms

Filter type Maximum stop band attenuation (dB)PM RGA PSO DE OHS

LP 2354 2611 2803 2953 3516HP 2355 2525 2810 2916 3386BP 2238 3080 3203 3258 3476BS 2165 2973 3056 3096 3245

range [0 1] If 1199031

is less thanHMCR the decision variable119909new119895

is generated by the memory consideration otherwise 119909new119895

isobtained by a random selection (ie random reinitializationbetween the search bounds) In the memory consideration119909new119895

is selected from any harmony vector 119894 in [1 2 119867119872119878]Secondly each decision variable 119909

new119895

will undergo a pitchadjustment with a probability of PAR if it is updated by the

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 5 dB plots for the FIR HP filter of order 20

memory consideration The pitch adjustment rule is given asfollows

119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 (13)

where 1199033

is a uniform random number between 0 and 1

314 Updating of HM After a new harmony vector 119883new119895

isgenerated the HMwill be updated by the survival of the fittervector between119883

new and the worst harmony vector119883worst inthe HM That is 119883new will replace 119883

worst and become a newmember of the HM if the fitness value of 119883new is better thanthe fitness value of119883worst

The computational procedure of the basic HS algorithmcan be summarized as shown in Algorithm 1

32 The Improved Harmony Search (IHS) Algorithm Thebasic HS algorithm uses fixed values for PAR and BWparameters The IHS algorithm proposed by Mahdavi et al

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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4 The Scientific World Journal

0 01 02 03 04 05 06 07 08 09 1

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Frequency

Mag

nitu

de (d

B)

PMRGAPSO

DEOHS

Figure 1 dB plots for the FIR LP filter of order 20

31 A Brief Description of HS Algorithm In the basic HSalgorithm each solution is called a harmony It is representedby an 119899-dimension real vector An initial randomly generatedpopulation of harmony vectors is stored in an HM Then anew candidate harmony is generated from all the solutionsin the HM by adopting a memory consideration rule a pitchadjustment rule and a random reinitialization Finally theHM is updated by comparing the new candidate harmonyvector and the worst harmony vector in the HM The worstharmony vector is replaced by the new candidate vector ifit is better than the worst harmony vector in the HM Theabove process is repeated until a certain termination criterionis met Thus the basic HS algorithm consists of three basicphases These are initialization improvisation of a harmonyvector and updating the HM Sequentially these phases aredescribed below

311 Initialization of the Problem and the Parameters of theHS Algorithm In general a global optimization problem canbe enumerated as follows min 119891(119909) st 119909

119895

isin [119901119886119903119886min119895

119901119886119903119886max119895

] 119895 = 1 2 119899 where 119891(119909) is the objectivefunction 119883 = [119909

1

1199092

119909119899

] is the set of design variablesand 119899 is the number of design variables Here 119901119886119903119886

min119895

119901119886119903119886

max119895

are the lower and upper bounds for the designvariable 119909

119895

respectively The parameters of the HS algorithmare the harmonymemory size (HMS) (the number of solutionvectors in HM) the harmony memory consideration rate(HMCR) the pitch adjusting rate (PAR) the distance band-width (BW) and the number of improvisations (NI)NI is thesame as the total number of fitness function calls (NFFCs) Itmay be set as a stopping criterion

312 Initialization of the HM The HM consists of HMSharmony vectors Let 119883

119895

= [119909119895

1

119909119895

2

119909119895

119899

] represent the119895th harmony vector which is randomly generated within the

0

0

01 02 03 04 05 06 07 08 09 1

14

12

1

08

06

04

02

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 2 Normalized plots for the FIR LP filter of order 20

0 005 01 015 02 025 03 035 04 045

12

115

105

11

1

095

09

085

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 3 Normalized pass band ripple plots for the FIR LP filter oforder 20

parameter limits [119901119886119903119886min119895

119901119886119903119886max119895

] Then the HM matrixis filled with the HMS harmony vectors as in the following

HM =

[[[[[

[

1199091

1

1199091

2

sdot sdot sdot 1199091

119899

1199092

1

1199092

2

sdot sdot sdot 1199092

119899

sdot sdot sdot

119909119867119872119878

1

119909119867119872119878

2

sdot sdot sdot 119909119867119872119878

119899

]]]]]

]

(12)

313 Improvisation of a New Harmony A new harmony vec-tor119883new

= (119909new1

119909new2

sdot sdot sdot 119909new119899

) is generated (called improvi-sation) by applying three rules namely (i) a memory consid-eration (ii) a pitch adjustment and (iii) a random selectionFirst of all a uniform random number 119903

1

is generated in the

The Scientific World Journal 5

Table 1 Control parameters of RGA PSO DE and OHS

RGA PSO DE OHSPopulation size = 120 iterationcycles = 600 crossover rate = 1crossover = two point crossovermutation rate = 001 mutation =Gaussian mutation and selection =roulette

Population size = 120 iterationcycles = 600 119862

1

= 1198622

=

205 Vmin119894

= 001 Vmax119894

=

10 119908max = 10 and119908min = 04

Population size = 120iteration cycles = 600

119862119903

= 03 119865 = 05

Population size = 120iteration cycles = 600

HMCR = 06 PARmin = 0PARmax = 09 BWmin =0000001 and BWmax = 1

Table 2 Optimized coefficients of the FIR LP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0020644508012550 0025116793352393 0027005399982491 0020509263646238

ℎ(2) = ℎ(20) 0048721413185106 0047219259300299 0047266866797926 0032132175858325

ℎ(3) = ℎ(19) 0005868601564964 0003546242723169 0005320204222841 minus0013117080938103

ℎ(4) = ℎ(18) minus0040966865300227 minus0040094047283599 minus0038982294859373 minus0053943547883771

ℎ(5) = ℎ(17) minus0000863506780022 minus0000520432067214 minus0003452235386096 minus0012003255218944

ℎ(6) = ℎ(16) 0059796031265565 0060907207778672 0057946858872171 0048106434177123

ℎ(7) = ℎ(15) minus0001408842862974 minus0001759240756773 minus0002051400593964 minus0007969148591832

ℎ(8) = ℎ(14) minus0103117834700311 minus0103613994946693 minus0102715267629915 minus0113260035845183

ℎ(9) = ℎ(13) minus0000440644382089 0000627623037422 0001692937801793 minus0007125766240178

ℎ(10) = ℎ(12) 0317600651261946 0318119036548684 0319795676258768 0310972208736947

ℎ(11) 0500018538901556 0500018538901556 0500018538901556 0500018538901555

Table 3 Optimized coefficients of the FIR HP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0021731353326545 0025559145974814 0029041921147266 0027958875232026

ℎ(2) = ℎ(20) minus0048131602227058 minus0047413653181042 minus0045873202416582 minus0046219133874889

ℎ(3) = ℎ(19) 0006298189918824 0005135430273491 0002950561225606 0005908752612340

ℎ(4) = ℎ(18) 0041895345956760 0039988099089174 0041311799862169 0041684671886351

ℎ(5) = ℎ(17) 0000879943669486 0001405996354021 minus0000283997158910 0000484971329494

ℎ(6) = ℎ(16) minus0059027866591514 minus0060283192968605 minus0060002355552046 minus0058130379060402

ℎ(7) = ℎ(15) minus0000013559660394 0000768613197325 minus0003921102337490 0003431061619144

ℎ(8) = ℎ(14) 0104257677520726 0105120739785348 0106119151142982 0105123375775675

ℎ(9) = ℎ(13) 0003823743541217 0001471927911810 minus0000565063060302 minus0007107362649439

ℎ(10) = ℎ(12) minus0316631427282300 minus0315471590838371 minus0320083906578923 minus0316061867104696

ℎ(11) 0499468012025621 0499981461098444 0499981461098444 0499981461098444

Table 4 Optimized coefficients of the FIR BP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0028502857888104 0024910907374264 0024315759656957 0026978129356347

ℎ(2) = ℎ(20) minus0001893868108392 0000092972958187 0002788616388318 0003014521009408

ℎ(3) = ℎ(19) minus0076189026154460 minus0074535581888545 minus0075883731240533 minus0075573145593866

ℎ(4) = ℎ(18) 0000994123920259 minus0000579129089510 minus0003313368788351 minus0005667273096237

ℎ(5) = ℎ(17) 0053196793860741 0058322287561503 0056134992376798 0054764314739432

ℎ(6) = ℎ(16) minus0000639149080848 minus0000187613541059 0000257826174027 minus0001285546925313

ℎ(7) = ℎ(15) 0100057194730152 0093164875388599 0090912344142406 0093422699573072

ℎ(8) = ℎ(14) 0001409980793664 0001012723950710 0002199187065772 0003282759787987

ℎ(9) = ℎ(13) minus0299380312728113 minus0296866917983546 minus0300934749358008 minus0299362912756530

ℎ(10) = ℎ(12) minus0000752480372393 minus0000392232750468 minus0001799401229551 minus0004731509416213

ℎ(11) 0400369877077545 0400369877077545 0400369877077545 0400369877077545

6 The Scientific World Journal

Table 5 Optimized coefficients of the FIR BS filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0008765244188382 0005065078955931 0005738163937772 0011156843719480

ℎ(2) = ℎ(20) 0054796923249762 0054496716662981 0053905628215447 0052282877968109

ℎ(3) = ℎ(19) 0001796419983890 0005809988516188 0002902448937586 0009942391820387

ℎ(4) = ℎ(18) 0048911654246731 0051144048751957 0049349878942931 0047219028225649

ℎ(5) = ℎ(17) minus0054718457691943 minus0050663949788261 minus0050884656047053 minus0049531151275131

ℎ(6) = ℎ(16) minus0060963142228236 minus0062741465298722 minus0063088550820316 minus0064382546432969

ℎ(7) = ℎ(15) 0004293459264617 minus0000062718416445 0004089341810059 minus0000512890612780

ℎ(8) = ℎ(14) minus0065342448643273 minus0068916923681426 minus0068023108311494 minus0067902719326839

ℎ(9) = ℎ(13) 0300682045893488 0297478557865240 0299063386928411 0296806655163297

ℎ(10) = ℎ(12) 0069036675664641 0074390206250426 0071701365941159 0074321537635153

ℎ(11) 0499582536276171 0499582536276171 0500000357523254 0500000357523254

055 06 065 07 075 08 085 09 095 1

008

007

006

005

004

003

002

001

0

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 4 Normalized stop band ripple plots for the FIR LP filter oforder 20

Table 6 Comparison of stop band attenuations for different typesof FIR filters each of order 20 using different algorithms

Filter type Maximum stop band attenuation (dB)PM RGA PSO DE OHS

LP 2354 2611 2803 2953 3516HP 2355 2525 2810 2916 3386BP 2238 3080 3203 3258 3476BS 2165 2973 3056 3096 3245

range [0 1] If 1199031

is less thanHMCR the decision variable119909new119895

is generated by the memory consideration otherwise 119909new119895

isobtained by a random selection (ie random reinitializationbetween the search bounds) In the memory consideration119909new119895

is selected from any harmony vector 119894 in [1 2 119867119872119878]Secondly each decision variable 119909

new119895

will undergo a pitchadjustment with a probability of PAR if it is updated by the

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 5 dB plots for the FIR HP filter of order 20

memory consideration The pitch adjustment rule is given asfollows

119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 (13)

where 1199033

is a uniform random number between 0 and 1

314 Updating of HM After a new harmony vector 119883new119895

isgenerated the HMwill be updated by the survival of the fittervector between119883

new and the worst harmony vector119883worst inthe HM That is 119883new will replace 119883

worst and become a newmember of the HM if the fitness value of 119883new is better thanthe fitness value of119883worst

The computational procedure of the basic HS algorithmcan be summarized as shown in Algorithm 1

32 The Improved Harmony Search (IHS) Algorithm Thebasic HS algorithm uses fixed values for PAR and BWparameters The IHS algorithm proposed by Mahdavi et al

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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The Scientific World Journal 5

Table 1 Control parameters of RGA PSO DE and OHS

RGA PSO DE OHSPopulation size = 120 iterationcycles = 600 crossover rate = 1crossover = two point crossovermutation rate = 001 mutation =Gaussian mutation and selection =roulette

Population size = 120 iterationcycles = 600 119862

1

= 1198622

=

205 Vmin119894

= 001 Vmax119894

=

10 119908max = 10 and119908min = 04

Population size = 120iteration cycles = 600

119862119903

= 03 119865 = 05

Population size = 120iteration cycles = 600

HMCR = 06 PARmin = 0PARmax = 09 BWmin =0000001 and BWmax = 1

Table 2 Optimized coefficients of the FIR LP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0020644508012550 0025116793352393 0027005399982491 0020509263646238

ℎ(2) = ℎ(20) 0048721413185106 0047219259300299 0047266866797926 0032132175858325

ℎ(3) = ℎ(19) 0005868601564964 0003546242723169 0005320204222841 minus0013117080938103

ℎ(4) = ℎ(18) minus0040966865300227 minus0040094047283599 minus0038982294859373 minus0053943547883771

ℎ(5) = ℎ(17) minus0000863506780022 minus0000520432067214 minus0003452235386096 minus0012003255218944

ℎ(6) = ℎ(16) 0059796031265565 0060907207778672 0057946858872171 0048106434177123

ℎ(7) = ℎ(15) minus0001408842862974 minus0001759240756773 minus0002051400593964 minus0007969148591832

ℎ(8) = ℎ(14) minus0103117834700311 minus0103613994946693 minus0102715267629915 minus0113260035845183

ℎ(9) = ℎ(13) minus0000440644382089 0000627623037422 0001692937801793 minus0007125766240178

ℎ(10) = ℎ(12) 0317600651261946 0318119036548684 0319795676258768 0310972208736947

ℎ(11) 0500018538901556 0500018538901556 0500018538901556 0500018538901555

Table 3 Optimized coefficients of the FIR HP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0021731353326545 0025559145974814 0029041921147266 0027958875232026

ℎ(2) = ℎ(20) minus0048131602227058 minus0047413653181042 minus0045873202416582 minus0046219133874889

ℎ(3) = ℎ(19) 0006298189918824 0005135430273491 0002950561225606 0005908752612340

ℎ(4) = ℎ(18) 0041895345956760 0039988099089174 0041311799862169 0041684671886351

ℎ(5) = ℎ(17) 0000879943669486 0001405996354021 minus0000283997158910 0000484971329494

ℎ(6) = ℎ(16) minus0059027866591514 minus0060283192968605 minus0060002355552046 minus0058130379060402

ℎ(7) = ℎ(15) minus0000013559660394 0000768613197325 minus0003921102337490 0003431061619144

ℎ(8) = ℎ(14) 0104257677520726 0105120739785348 0106119151142982 0105123375775675

ℎ(9) = ℎ(13) 0003823743541217 0001471927911810 minus0000565063060302 minus0007107362649439

ℎ(10) = ℎ(12) minus0316631427282300 minus0315471590838371 minus0320083906578923 minus0316061867104696

ℎ(11) 0499468012025621 0499981461098444 0499981461098444 0499981461098444

Table 4 Optimized coefficients of the FIR BP filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0028502857888104 0024910907374264 0024315759656957 0026978129356347

ℎ(2) = ℎ(20) minus0001893868108392 0000092972958187 0002788616388318 0003014521009408

ℎ(3) = ℎ(19) minus0076189026154460 minus0074535581888545 minus0075883731240533 minus0075573145593866

ℎ(4) = ℎ(18) 0000994123920259 minus0000579129089510 minus0003313368788351 minus0005667273096237

ℎ(5) = ℎ(17) 0053196793860741 0058322287561503 0056134992376798 0054764314739432

ℎ(6) = ℎ(16) minus0000639149080848 minus0000187613541059 0000257826174027 minus0001285546925313

ℎ(7) = ℎ(15) 0100057194730152 0093164875388599 0090912344142406 0093422699573072

ℎ(8) = ℎ(14) 0001409980793664 0001012723950710 0002199187065772 0003282759787987

ℎ(9) = ℎ(13) minus0299380312728113 minus0296866917983546 minus0300934749358008 minus0299362912756530

ℎ(10) = ℎ(12) minus0000752480372393 minus0000392232750468 minus0001799401229551 minus0004731509416213

ℎ(11) 0400369877077545 0400369877077545 0400369877077545 0400369877077545

6 The Scientific World Journal

Table 5 Optimized coefficients of the FIR BS filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0008765244188382 0005065078955931 0005738163937772 0011156843719480

ℎ(2) = ℎ(20) 0054796923249762 0054496716662981 0053905628215447 0052282877968109

ℎ(3) = ℎ(19) 0001796419983890 0005809988516188 0002902448937586 0009942391820387

ℎ(4) = ℎ(18) 0048911654246731 0051144048751957 0049349878942931 0047219028225649

ℎ(5) = ℎ(17) minus0054718457691943 minus0050663949788261 minus0050884656047053 minus0049531151275131

ℎ(6) = ℎ(16) minus0060963142228236 minus0062741465298722 minus0063088550820316 minus0064382546432969

ℎ(7) = ℎ(15) 0004293459264617 minus0000062718416445 0004089341810059 minus0000512890612780

ℎ(8) = ℎ(14) minus0065342448643273 minus0068916923681426 minus0068023108311494 minus0067902719326839

ℎ(9) = ℎ(13) 0300682045893488 0297478557865240 0299063386928411 0296806655163297

ℎ(10) = ℎ(12) 0069036675664641 0074390206250426 0071701365941159 0074321537635153

ℎ(11) 0499582536276171 0499582536276171 0500000357523254 0500000357523254

055 06 065 07 075 08 085 09 095 1

008

007

006

005

004

003

002

001

0

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 4 Normalized stop band ripple plots for the FIR LP filter oforder 20

Table 6 Comparison of stop band attenuations for different typesof FIR filters each of order 20 using different algorithms

Filter type Maximum stop band attenuation (dB)PM RGA PSO DE OHS

LP 2354 2611 2803 2953 3516HP 2355 2525 2810 2916 3386BP 2238 3080 3203 3258 3476BS 2165 2973 3056 3096 3245

range [0 1] If 1199031

is less thanHMCR the decision variable119909new119895

is generated by the memory consideration otherwise 119909new119895

isobtained by a random selection (ie random reinitializationbetween the search bounds) In the memory consideration119909new119895

is selected from any harmony vector 119894 in [1 2 119867119872119878]Secondly each decision variable 119909

new119895

will undergo a pitchadjustment with a probability of PAR if it is updated by the

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 5 dB plots for the FIR HP filter of order 20

memory consideration The pitch adjustment rule is given asfollows

119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 (13)

where 1199033

is a uniform random number between 0 and 1

314 Updating of HM After a new harmony vector 119883new119895

isgenerated the HMwill be updated by the survival of the fittervector between119883

new and the worst harmony vector119883worst inthe HM That is 119883new will replace 119883

worst and become a newmember of the HM if the fitness value of 119883new is better thanthe fitness value of119883worst

The computational procedure of the basic HS algorithmcan be summarized as shown in Algorithm 1

32 The Improved Harmony Search (IHS) Algorithm Thebasic HS algorithm uses fixed values for PAR and BWparameters The IHS algorithm proposed by Mahdavi et al

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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6 The Scientific World Journal

Table 5 Optimized coefficients of the FIR BS filter of order 20

ℎ(119873) RGA PSO DE OHSℎ(1) = ℎ(21) 0008765244188382 0005065078955931 0005738163937772 0011156843719480

ℎ(2) = ℎ(20) 0054796923249762 0054496716662981 0053905628215447 0052282877968109

ℎ(3) = ℎ(19) 0001796419983890 0005809988516188 0002902448937586 0009942391820387

ℎ(4) = ℎ(18) 0048911654246731 0051144048751957 0049349878942931 0047219028225649

ℎ(5) = ℎ(17) minus0054718457691943 minus0050663949788261 minus0050884656047053 minus0049531151275131

ℎ(6) = ℎ(16) minus0060963142228236 minus0062741465298722 minus0063088550820316 minus0064382546432969

ℎ(7) = ℎ(15) 0004293459264617 minus0000062718416445 0004089341810059 minus0000512890612780

ℎ(8) = ℎ(14) minus0065342448643273 minus0068916923681426 minus0068023108311494 minus0067902719326839

ℎ(9) = ℎ(13) 0300682045893488 0297478557865240 0299063386928411 0296806655163297

ℎ(10) = ℎ(12) 0069036675664641 0074390206250426 0071701365941159 0074321537635153

ℎ(11) 0499582536276171 0499582536276171 0500000357523254 0500000357523254

055 06 065 07 075 08 085 09 095 1

008

007

006

005

004

003

002

001

0

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 4 Normalized stop band ripple plots for the FIR LP filter oforder 20

Table 6 Comparison of stop band attenuations for different typesof FIR filters each of order 20 using different algorithms

Filter type Maximum stop band attenuation (dB)PM RGA PSO DE OHS

LP 2354 2611 2803 2953 3516HP 2355 2525 2810 2916 3386BP 2238 3080 3203 3258 3476BS 2165 2973 3056 3096 3245

range [0 1] If 1199031

is less thanHMCR the decision variable119909new119895

is generated by the memory consideration otherwise 119909new119895

isobtained by a random selection (ie random reinitializationbetween the search bounds) In the memory consideration119909new119895

is selected from any harmony vector 119894 in [1 2 119867119872119878]Secondly each decision variable 119909

new119895

will undergo a pitchadjustment with a probability of PAR if it is updated by the

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 5 dB plots for the FIR HP filter of order 20

memory consideration The pitch adjustment rule is given asfollows

119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 (13)

where 1199033

is a uniform random number between 0 and 1

314 Updating of HM After a new harmony vector 119883new119895

isgenerated the HMwill be updated by the survival of the fittervector between119883

new and the worst harmony vector119883worst inthe HM That is 119883new will replace 119883

worst and become a newmember of the HM if the fitness value of 119883new is better thanthe fitness value of119883worst

The computational procedure of the basic HS algorithmcan be summarized as shown in Algorithm 1

32 The Improved Harmony Search (IHS) Algorithm Thebasic HS algorithm uses fixed values for PAR and BWparameters The IHS algorithm proposed by Mahdavi et al

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

The Scientific World Journal 7

Table 7 Other comparative results of performance parameters of all algorithms for the FIR LP filter of order 20

Algorithm FIR LP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006651 0066282 00838 mdashRGA 004949 0025620 00853 57174PSO 003967 0019052 00869 33286DE 003339 0076192 00908 39543OHS 001746 0045708 00994 38321

Table 8 Other comparative results of performance parameters of all algorithms for the FIR HP filter of order 20

Algorithm FIR HP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles (s)

PM 006645 006637 00839 mdashRGA 005461 002860 00864 53667PSO 003935 001916 00867 304358DE 003483 001611 00878 393745OHS 002027 001651 01004 379341

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 6 Normalized plots for the FIR HP filter of order 20

[42] applies the same memory consideration pitch adjust-ment and random selection as the basic HS algorithm butdynamically updates the values of PAR and BW as in (14) and(15) respectively

119875119860119877 (119892119899) = 119875119860119877min

+119875119860119877

maxminus 119875119860119877

min

119873119868times 119892119899 (14)

119861119882(119892119899) = 119861119882max

times 119890((ln((119861119882min

)(119861119882

max))119873119868)times119892119899)

(15)

In (14) 119875119860119877(119892119899) is the pitch adjustment rate in the cur-rent generation (119892119899) 119875119860119877

min and 119875119860119877max are the minimum

and the maximum adjustment rates respectively In (15)119861119882(119892119899) is the distance bandwidth at generation (119892119899) 119861119882min

and119861119882max are theminimum and themaximumbandwidths

respectively

33 Opposition-Based Learning A Concept Evolutionaryoptimizationmethods start with some initial solutions (initialpopulation) and try to improve them toward some optimalsolution(s) The process of searching terminates when somepredefined criteria are satisfied In the absence of a prioriinformation about the solution we usually start with randomguesses The computation time among others is related tothe distance of these initial guesses from the optimal solutionWe can improve our chance of starting with a closer (fitter)solution by simultaneously checking the opposite solution[34] By doing this the fitter one (guess or opposite guess)can be chosen as an initial solution In fact according tothe theory of probability 50 of the time a guess is furtherfrom the solution than its opposite guess [36] Thereforestarting with the closer of the two guesses (as judged by itsfitness) has the potential to accelerate convergenceThe sameapproach can be applied not only to initial solutions but alsocontinuously to each solution in the current population [36]

331 Definition of Opposite Number Let 119909 isin [119906119887 119897119887] be areal number The opposite number is defined as

= 119906119887 + 119897119887 minus 119909 (16)

Similarly this definition can be extended to higherdimensions [34] as stated in the next subsection

332 Definition of Opposite Point Let 119883 = (1199091

1199092

119909119899

)

be a point in 119899-dimensional space where (1199091

1199092

119909119899

) isin 119877

and 119909119894

isin [119906119887119894

119897119887119894

] for all 119894 isin 1 2 119899 The opposite point = (

1

2

119899

) is completely defined by its componentsas

119894

= 119906119887119894

+ 119897119887119894

minus 119909119894

(17)

Now by employing the opposite point definition theopposition-based optimization is defined in the followingsubsection

8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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8 The Scientific World Journal

Table 9 Other comparative results of performance parameters of all algorithms for the FIR BP filter of order 20

Algorithm FIR BP filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 007609 0076017 00875 mdashRGA 002885 0016855 00945 63827PSO 002504 0015893 01009 46850DE 002350 0015125 00987 49832OHS 001828 001408 00988 47156

Table 10 Other comparative results of performance parameters of the FIR BS filter of order 20 for all algorithms

Algorithm FIR BS filter of order 20Maximum average stop band ripple (normalized) Transition width (normalized) Execution time for 100 cycles

PM 008273 008268 00905 mdashRGA 003262 002322 00959 62846PSO 002966 002092 00936 48777DE 002832 002161 00981 50005OHS 002385 001906 01069 49943

Table 11 Statistical parameters of FIR LP filters for different algorithms

AlgorithmFIR LP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00664 006616 384119890 minus 8 0000196 2354 23572 0000696 0026382RGA 01142 011224 511119890 minus 6 0002261 2611 33056 205999 4538712PSO 01230 011714 155119890 minus 5 0003939 2803 35588 186021 4313015DE 01360 012152 656119890 minus 5 0008099 2953 36784 1330738 3647929OHS 01400 012195 173119890 minus 5 0013171 3516 37014 1804384 1343274

Table 12 Statistical parameters of FIR HP filters for different algorithms

AlgorithmFIR HP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00663 006612 216119890 minus 8 0000147 2355 23560 000012 0010954RGA 01170 011262 664119890 minus 6 0002577 2525 32110 2059505 4538177PSO 01249 011820 312119890 minus 5 0005590 281 35396 1609082 4011337DE 01360 012060 0000137 0011693 2916 37058 183217 4280385OHS 01400 012228 0000191 0013819 3386 35728 1474056 1214107

Table 13 Statistical parameters of FIR BP filters for different algorithms

AlgorithmFIR BP filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 00763 007610 2119890 minus 8 0000141 2238 2242 0007267 0085245RGA 01670 0145167 0000448 0021175 3080 35941667 7752181 2784278PSO 01460 0141367 211119890 minus 5 0004597 3203 366 1040607 3225844DE 01520 0142533 0000101 0010062 3258 3686 7605033 2757722OHS 01530 0144567 0000126 0011227 3476 37215 3209158 1791412

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

The Scientific World Journal 9

Table 14 Statistical parameters of FIR BS filters for different algorithms

AlgorithmFIR BS filter of order 20

Pass band ripple (normalized) Stop band attenuation (dB)Maximum Mean Variance Standard deviation Maximum Mean Variance Standard deviation

PM 0083 008272 154119890 minus 7 0000392 2165 21653333 222119890 minus 5 0004714RGA 0118 01037 0000285 0016869 2973 32924 4148024 203667PSO 0125 010554 0000221 0014868 3056 33858 4566376 2136908DE 0115 010024 0000322 0017937 3096 33552 4405096 2098832OHS 0140 009952 000123 0035072 3245 34564 2914104 1707075

Table 15 Comparison of OHS-based results with other reported results

Model Parameter

Filter type Order Maximum stop bandattenuation (dB)

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Transitionwidth

Oliveira et al [15] Band pass 30 lt33 dB NRlowast NRlowast gt01Karaboga andCetinkaya [18] Low pass 20 NRlowast gt008 gt009 gt016

Liu et al [19] Low Pass 20 NRlowast 004 gt007 gt006Najjarzadeh andAyatollahi [21]

Low pass 33 lt29 dB NRlowast NRlowast NRlowast

Band pass 33 lt25 dB NRlowast NRlowast NRlowast

Ababneh andBataineh [23] Low pass 30 lt30 dB (Approx) 015 0031 005

Sarangi et al [26] Low pass 20 lt27 dB gt01 gt006 gt015Band pass 20 lt8 dB gt02 gt005 gt007

Mondal et al [30] High pass 20 3403 0129 002392 00825Luitel andVenayagamoorthy[33]

Low pass 20 lt27 dB 0291 0270 gt013

OHS

Low pass 20 3516 0140 001746 00994High pass 20 3386 0140 002027 01004Band pass 20 3476 0153 001828 00988Band stop 20 3245 0140 002385 01069

lowastNR means not reported in the referred literature

Step 1 Set the parameters HMS HMCR PAR BW NI and 119899Step 2 Initialize the HM and calculate the objective function value for each harmony vectorStep 3 Improvise the HM filled with new harmony119883

new vectors as followsfor (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119909new119895

= 119909119886

119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877) then119909new119895

= 119909new119895

plusmn 1199033

times 119861119882 1199031

1199032

1199033

isin [0 1]

end ifelse

119909new119895

= 119901119886119903119886min119895

+ 119903 times (119901119886119903119886max119895

minus 119901119886119903119886min119895

) 119903 isin [0 1]

end ifend for

Step 4 Update the HM as119883worst= 119883

new if119891(119883new

) lt 119891(119883worst

)Step 5 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 3

Algorithm 1 HS Algorithm

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

12

115

105

11

1

095

09

085

Mag

nitu

de (n

orm

aliz

ed)

Frequency

PMRGAPSO

DEOHS

055 06 065 07 075 08 085 09 095 1

Figure 7 Normalized pass band ripple plots for the FIR HP filter oforder 20

0 005 01 015 02 025 03 035 04 045

Frequency

008

006

004

002

0

Mag

nitu

de (n

orm

aliz

ed)

014

012

01

PMRGAPSO

DEOHS

Figure 8 Normalized stop band ripple plots for the FIR HP filter oforder 20

333 Opposition-Based Optimization Let 119883 = (1199091

1199092

119909119899

) be a point in 119899-dimensional space (ie a candidatesolution) Assume 119891 = (sdot) is a fitness function which is usedtomeasure the candidatersquos fitness According to the definitionof the opposite point = (

1

2

119899

) is the opposite of119883 = (119909

1

1199092

119909119899

) Now if 119891() ge 119891(119883) then point 119883 canbe replaced with otherwise we continue with 119883 Hencethe point and its opposite point are evaluated simultaneouslyin order to continue with the fitter one

34 Opposition-Based Harmony Search (OHS) AlgorithmSimilar to all population-based optimization algorithms two

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

10

Mag

nitu

de (d

B)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 9 dB plots for the FIR BP filter of order 20

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 10 Normalized plots for the FIR BP filter of order 20

main steps are distinguishable for HS namely HM initial-ization and producing new HM by adopting the principleof HS In the present work the strategy of the OBL [34] isincorporated in those two stepsTheoriginalHS is chosen as aparent algorithm and opposition-based ideas are embeddedin it with an intention to exhibit accelerated convergenceprofile Corresponding pseudo code for the proposed OHSapproach can be summarized as shown in Algorithm 2

4 Results and Discussions

This section presents the simulations performed inMATLAB75 for the design of LP HP BP and BS FIR filters Eachfilter order (119873) is taken as 20 which results in the number

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

International Journal of

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RoboticsJournal of

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Control Scienceand Engineering

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Electrical and Computer Engineering

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 11

03 035 04 045 05 055 06 065 07

04

05

06

07

08

09

1

11

12

13

Frequency

Mag

nitu

de (n

orm

aliz

ed)

PMRGAPSO

DEOHS

Figure 11 Normalized pass band ripple plots for the FIR BP filter oforder 20

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 12 Normalized stop band ripple plots for the FIR BP filter oforder 20

of coefficients as 21 The sampling frequency is taken to be119891119904

= 1Hz The number of frequency samples is 128 Eachalgorithm is run for 50 times to obtain its best results Table 1shows the best chosen control parameters for RGA PSO DEand OHS respectively

The parameters of the filters to be designed using anyalgorithm are as follows pass band ripple (120575

119901

) = 01stop band ripple (120575

119904

) = 001 For the LP filter passband (normalized) edge frequency (120596

119901

) = 045 stop band(normalized) edge frequency (120596

119904

) = 055 transition width =01 For the HP filter stop band (normalized) edge frequency(120596119904

) = 045 pass band (normalized) edge frequency (120596119901

) =

055 transition width = 01 For the BP filter lower stop

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

20

minus100

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 13 dB plots for the FIR BS filter of order 20

band (normalized) edge frequency (120596119904119897

) = 025 lower passband (normalized) edge frequency (120596

119901119897

) = 035 upper passband (normalized) edge frequency (120596

119901ℎ

) = 065 upper stopband (normalized) edge frequency (120596

119904ℎ

) = 075 transitionwidth = 01 For the BS filter lower pass band (normalized)edge frequency (120596

119901119897

) = 025 lower stop band (normalized)edge frequency (120596

119904119897

) = 035 upper stop band (normalized)edge frequency (120596

119904ℎ

) = 075 upper pass band (normalized)edge frequency (120596

119901ℎ

) = 085 transition width = 01 Tables 23 4 and 5 show the optimized filter coefficients obtained forLP HP BP and BS FIR filters respectively using RGA PSODE and OHS individually

Table 6 shows the highest stop band attenuations for allfour types of filters using OHS as 3516 dB (for LP filter)3386 dB (for HP filter) 3476 dB (for BP filter) and 3245 dB(for BS filter) as compared to those of PM RGA PSO andDE Tables 7 8 9 and 10 show the comparative results ofperformance parameters in terms of maximum and averagestop band ripple (normalized) transition width (normalized)for LP HP BP and BS filters using PM RGA PSO DE andOHS respectively It is also noticed that for almost samelevel of transition width and stop band ripple OHS resultsin the best stop band attenuation among all algorithms for alltypes of filters Tables 11 12 13 and 14 summarize maximummean variance and standard deviation for pass band ripple(normalized) and stop band attenuations in dB for LP HP BPand BS filters using all concerned algorithms

In Table 15 OHS-based results are compared with otherreported results Oliveira et al [15] have designed 30th-orderBP filter with stop band attenuation and transition width of33 dB and 01 respectively A 20th-order LP filter has beendesigned by Karaboga and Cetinkaya [18] with transitionwidth pass band and stop band ripples of 016 008 and009 respectively Liu et al [19] also reported for 20th-orderFIR filter with transition width pass band and stop bandripples of 006 004 and 007 respectively Najjarzadeh and

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

12 The Scientific World Journal

Step 1 Set the parameters119867119872119878119867119872119862119877 119901119886119903119886min

119901119886119903119886max

119861119882min

119861119882max and 119873119868

Step 2 Initialize the HM with1198830119894119895

Step 3 Opposition based HM initializationfor (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

1198741198830119894119895

= 119901119886119903119886min119895

+ 119901119886119903119886max119895

minus 1198830119894119895

1198741198830

Opposite of initial X0end forend for End of Opposition-based HM initializationSelect HMS fittest individuals from set of 119883

0119894119895

1198741198830119894119895

as initial HM HM beingthe matrix of fittest X vectors

Step 4 Improvise a new harmony119883new as follows

Update 119875119860119877(119892119899) by (14) and 119861119882(119892119899) using (15)for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

if (1199031

lt 119867119872119862119877) then119883

new119894119895

= 119883119886

119894119895

119886 isin (1 2 119867119872119878)

if (1199032

lt 119875119860119877 (119892119899)) then119883

new119894119895

= 119883new119894119895

plusmn 1199033

times 119861119882(119892119899) 1199031

1199032

1199033

isin [0 1]

end ifelse

119883new119894119895

= 119901119886119903119886min119894119895

+ 119903 times (119901119886119903119886max119894119895

minus 119901119886119903119886min119894119895

) 119903 isin [0 1]

end ifend for

end forStep 5 Update the HM as119883worst

= 119883new if119891(119883

new) lt 119891(119883

worst)

Step 6 Opposition based generation jumpingif (119903119886119899119889

2

lt 119869119903

) 1199031198861198991198892

isin [0 1] 119869119903

Jumping rate

for (119894 = 0 119894 lt 119867119872119878 119894 + +)

for (119895 = 0 119895 lt 119899 119895 + +)

119874119883119894119895

= min119892119899119895

+max119892119899119895

minus 119883119894119895

min119892119899119895

minimum value of the jth variable in the current generation (gn) max119892119899

119895

maximum value of the jth variable in the current generation (gn)Select 119867119872119878 fittest HM from the set of 119883

119894119895

119874119883119894119895

as current HMend for

end forend if End of opposition-based generation jumping

Step 7 If119873119868 is completed return the best harmony vector 119883best in the HM otherwise go back to Step 4

Algorithm 2 OHS Algorithm

Ayatollahi [21] have designed LP and BP filters of order 33with approximate values of stop band attenuation 29 dB and25 dB respectively A 30th-order FIR filter has been designedby Ababneh and Bataineh [23] with stop band attenuationtransition width pass band and stop band ripples of 30 dB005 015 and 0031 respectively Sarangi et al [26] havedesigned 20th-order FIR filters with stop band attenuationtransitionwidth pass band and stop band ripples for LP filterof values 27 dB 015 01 and 006 respectively For the sameorder BP these values are respectively as follows 8 dB 00702 and 005 Mondal et al have reported the 20th-order HPfilter [30] with stop band attenuation transition width passband and stop band ripples of 3403 dB 00825 0129 and

002392 respectively Luitel and Venayagamoorthy also havedesigned 20th-order LP filter with stop band attenuationtransition width pass band and stop band ripples of 27 dB013 0291 and 0270 respectively as reported in [33]

In this paper OHS-based design is applied to 20th-orderLP HP BP and BS filters maximum stop band attenuationsof 3516 dB 3386 dB 3476 dB and 3245 dB maximum passband ripples of 0140 0140 0153 and 0140 Maximumstop band ripples of 001746 002027 001828 and 002385Transition width values of 00994 01004 00988 and 01069are achieved respectively with LP HP BP and BS filtersThe above-mentioned values can be verified from the resultspresented in Table 15 Thus it is observed from Table 15 that

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 13

0

14

12

1

08

06

04

02

Mag

nitu

de (n

orm

aliz

ed)

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

Figure 14 Normalized plots for the FIR BS filter of order 20

0 01 02 03 04 05 06 07 08 09 1

Frequency

PMRGAPSO

DEOHS

115

105

11

1

095

09

085

08

Mag

nitu

de (n

orm

aliz

ed)

Figure 15 Normalized pass band ripple plot for the FIR BS filter oforder 20

the stop band attenuations in all cases for the 20th-orderfilters using OHS are much better than the other reportedresults

Figures 1ndash4 show the magnitude responses of the 20th-order LP filter in various forms using PM RGA PSO DEand OHS respectively The magnitude responses in dB areplotted in Figure 1 for the same The normalized magnituderesponses are shown in Figure 2 Figure 3 shows normalizedpass band ripple plots Figure 4 shows normalized stop bandripple plots Figures 5 6 7 8 9 10 11 12 13 14 15 and16 show magnitude responses in dB normalized magnituderesponses normalized pass band ripples and normalizedstop band ripples for HP BP and BS filters respectivelyFrom the above figures and tables it is observed that OHS

035 04 045 05 055 06 065 07 075

Frequency

008

007

006

005

004

003

002

001

0

Mag

nitu

de (n

orm

aliz

ed)

009

PMRGAPSO

DEOHS

Figure 16 Normalized stop band ripple plot for the FIR BS filter oforder 20

0 50 100 150 200 250 300 350 400 450 500

4

45

5

55

6

65

7

75

8

85

Iteration cycles

Erro

r fitn

ess

Figure 17 Convergence profile for RGA in case of FIR HP filter oforder 20

results in bettermagnitude responses (dB) better normalizedmagnitude responses and better normalized stop band rippleplots for all filters as compared to those of PM RGA PSOand DE

41 Comparative Effectiveness and Convergence Profiles ofRGA PSO DE and OHS Algorithms In order to comparethe algorithms in terms of the error fitness values theconvergences of error fitness values obtained for the HPfilter of order 20 for RGA PSO DE and OHS are plottedas shown in Figures 17 18 19 and 20 Similar plots havealso been obtained for the LP BP and BS filters of the sameorder which are not shown here OHS converges to muchlower error fitness value as compared to RGA PSO andDE which yield suboptimal higher values of error fitnessvalues RGA converges to the minimum error fitness value

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 The Scientific World Journal

0 100 200 300 400 500 6002

3

4

5

6

7

8

9

10

Iteration cycles

Erro

r fitn

ess

Figure 18 Convergence profile for PSO in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

0 100 200 300 400 500 600

Iteration cycles

Erro

r fitn

ess

Figure 19 Convergence profile for DE in case of FIR HP filter oforder 20

1

2

3

4

5

6

7

8

9

Erro

r fitn

ess

0 100 200 300 400 500 600

Iteration cycles

Figure 20 Convergence profile for OHS in case of FIR HP filter oforder 20

of 4088 in 18676116 s PSO converges to the minimum errorfitness value of 2575 in 172266628 s DE converges to theminimum error fitness value of 1803 in 173641545 s whereasOHS converges to the minimum error fitness value of 1248in 1896705 s

For all types of filters OHS converges to the leastminimum error fitness values in finding the optimum filtercoefficients in moderately less execution times whichmay beverified from Tables 7 8 9 and 10 With a view to the abovefact it may be inferred that the performance of OHS is thebest among all algorithms All optimization programs wererun in MATLAB 75 version on core (TM) 2 duo processor300GHz with 2GB RAM

5 Conclusion

In this paper a novel opposition-based harmony search(OHS) algorithm is applied to the solution of the constrainedmultimodal FIR filter design problem yielding optimal filtercoefficients Comparison of the results of PM RGA PSODEand OHS algorithm has been made It is revealed that OHShas the ability to converge to the best quality near optimalsolution and possesses the best convergence characteristicsin moderately less execution times among the algorithmsThe simulation results clearly indicate thatOHSdemonstratesbetter performance in terms of magnitude response mini-mum stop band ripple and maximum stop band attenuationwith a very little deterioration in the transition width ThusOHS may be used as a good optimizer for obtaining theoptimal filter coefficients in any practical digital filter designproblem of digital signal processing systems

References

[1] T W Parks and C S Burrus Digital Filter Design John Wileyamp Sons New York NY USA 1987

[2] E C Ifeachor and B W Jervis Digital Signal Processing APractical Approach Pearson Education Upper Saddle River NJUSA 2002

[3] T W Parks and J H McClellan ldquoChebyshev approximation fornon recursive digital filters with linear phaserdquo IEEE Transac-tions on Circuits Theory vol 19 no 2 pp 189ndash194 1972

[4] L R Rabiner ldquoApproximate design relationships for low passFIR digital filtersrdquo IEEE Transactions on Audio and Electroa-coustics vol 21 no 5 pp 456ndash460 1973

[5] L Litwin ldquoFIR and IIR digital filtersrdquo IEEE Potentials vol 19no 4 pp 28ndash31 2000

[6] J H McClellan T W Parks and L R Rabiner ldquoA computerprogram for designing optimumFIR linear phase digital filtersrdquoIEEE Transactions on Audio and Electroacoustics vol 21 no 6pp 506ndash526 1973

[7] T Ciloglu ldquoAn efficient local search method guided by gradientinformation for discrete coefficient FIR filter designrdquo SignalProcessing vol 82 no 10 pp 1337ndash1350 2002

[8] A Lee M Ahmadi G A Jullien W C Miller and R SLashkari ldquoDigital filter design using genetic algorithmrdquo inProceedings of the IEEE Symposium on Advances in DigitalFiltering and Signal Processing pp 34ndash38 Victoria Canada1998

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 15

[9] S U Ahmad and A Antoniou ldquoA genetic algorithm approachfor fractional delay FIR filtersrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo06)pp 2517ndash2520 May 2006

[10] N E Mastorakis I F Gonos and M N S Swamy ldquoDesignof two-dimensional recursive filters using genetic algorithmsrdquoIEEE Transactions on Circuits and Systems I vol 50 no 5 pp634ndash639 2003

[11] H C Lu and S T Tzeng ldquoDesign of arbitrary FIR log filters bygenetic algorithm approachrdquo Signal Processing vol 80 no 3 pp497ndash505 2000

[12] S U Ahmad andA Andreas ldquoCascade-formmultiplier less FIRfilter design using orthogonal genetic algorithmrdquo in Proceedingsof the IEEE International Symposium on Signal Processingand Information Technology pp 932ndash937 Vancouver CanadaAugust 2006

[13] W Tang and T Shen ldquoOptimal design of FRM-based FIR filtersby using hybrid taguchi genetic algorithmrdquo in Proceedings ofthe 1st International Conference on Green Circuits and Systems(ICGCS rsquo10) pp 392ndash397 June 2010

[14] D Karaboga D H Horrocks N Karaboga and A KalinlildquoDesigning digital FIR filters using tabu search algorithmrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo97) vol 4 pp 2236ndash2239 June 1997

[15] H A Oliveira Jr A Petraglia and M R Petraglia ldquoFrequencydomain FIR filter design using fuzzy adaptive simulated anneal-ingrdquo Circuits Systems and Signal Processing vol 28 no 6 pp899ndash911 2009

[16] S Chen ldquoIIR model identification using batch-recursive adap-tive simulated annealing algorithmrdquo in Proceedings of the 6thAnnual Chinese Automation and Computer Science Conferencepp 151ndash155 September 2000

[17] N Karaboga ldquoA new design method based on artificial beecolony algorithm for digital IIR filtersrdquo Journal of the FranklinInstitute vol 346 no 4 pp 328ndash348 2009

[18] N Karaboga and B Cetinkaya ldquoDesign of digital FIR filtersusing differential evolution algorithmrdquo Circuits Systems andSignal Processing vol 25 no 5 pp 649ndash660 2006

[19] G Liu Y X Li and G He ldquoDesign of digital FIR filters usingdifferential evolution algorithm based on reserved generdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash7 Barcelona Spain July 2010

[20] S K Saha R Kar D Mandal and S P Ghoshal ldquoSeekeroptimization algorithm application to the design of linearphase FIR filterrdquo IET Signal Processing vol 6 no 8 pp 763ndash7712012

[21] M Najjarzadeh and A Ayatollahi ldquoFIR digital filters designparticle swarm optimization utilizing LMS andminimax strate-giesrdquo in Proceedings of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT rsquo08) pp129ndash132 Sarajevo Bosnia and Herzegovina December 2008

[22] D J Krusienski and W K Jenkins ldquoA modified particle swarmoptimization algorithm for adaptive filteringrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo06) pp 137ndash140 May 2006

[23] J I Ababneh andMH Bataineh ldquoLinear phase FIRfilter designusing particle swarm optimization and genetic algorithmsrdquoDigital Signal Processing vol 18 no 4 pp 657ndash668 2008

[24] S K Saha R Kar D Mandal and S P Ghoshal ldquoA new designmethod using opposition-based BAT algorithm for IIR systemidentification problemrdquo International Journal of Bio-InspiredComputation vol 5 no 2 pp 99ndash132 2013

[25] B Luitel and G K Venayagamoorthy ldquoParticle swarm opti-mization with quantum infusion for system identificationrdquoEngineering Applications of Artificial Intelligence vol 23 no 5pp 635ndash649 2010

[26] A Sarangi R K Mahapatra and S P Panigrahi ldquoDEPSOand PSO-QI in digital filter designrdquo Expert Systems withApplications vol 38 no 9 pp 10966ndash10973 2011

[27] X Yu J Liu and H Li ldquoAn adaptive inertia weight particleswarm optimization algorithm for IIR digital filterrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo09) vol 1 pp 114ndash118November 2009

[28] D Jia Y Jiao and J Zhang ldquoSatisfactory design of IIR digitalfilter based on chaotic mutation particle swarm optimizationrdquoinProceedings of the 3rd International Conference onGenetic andEvolutionary Computing (WGEC rsquo09) pp 48ndash51 October 2009

[29] Y Gao Y Li and H Qian ldquoThe design of IIR digital filterbased on chaos particle swarm optimization algorithmrdquo inProceedings of the 2nd International Conference on Genetic andEvolutionary Computing (WGEC rsquo08) pp 303ndash306 September2008

[30] S Mondal D Chakraborty S P Ghoshal A Majumdar R Karand D Mandal ldquoNovel particle swarm optimization for highpass FIR filter designrdquo in Proceedings of the IEEE Symposiumon Humanities Science and Engineering Research (SHUSER rsquo12)p 413 418 Kuala Lumpur Malaysia June 2012

[31] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFiltermodeling using gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 24 no 1 pp 117ndash1222011

[32] C Dai W Chen and Y Zhu ldquoSeeker optimization algorithmfor digital IIR filter designrdquo IEEE Transactions on IndustrialElectronics vol 57 no 5 pp 1710ndash1718 2010

[33] B Luitel and G K Venayagamoorthy ldquoDifferential evolutionparticle swarmoptimization for digital filter designrdquo inProceed-ings of the IEEE Congress on Evolutionary Computation (CECrsquo08) pp 3954ndash3961 June 2008

[34] H R Tizhoosh ldquoOpposition-based learning a new scheme formachine intelligencerdquo in Proceedings of the International Con-ference on Computational Intelligence for Modelling Control andAutomation 2005 and International Conference on IntelligentAgents Web Technologies and Internet Commerce vol 1 pp695ndash701 Vienna Austria November 2005

[35] M Shokri H R Tizhoosh and M Kamel ldquoOpposition-basedQ(120582) algorithmrdquo inProceedings of the International Joint Confer-ence on Neural Networks (IJCNN rsquo06) pp 254ndash261 VancouverCanada July 2006

[36] H R Tizhoosh ldquoReinforcement learning based on actions andopposite actionsrdquo in Proceedings of the ICGST InternationalConference on Artificial Intelligence and Machine LearningCairo Egypt December 2005

[37] H R Tizhoosh ldquoOpposition-based reinforcement learningrdquoJournal of Advanced Computational Intelligence and IntelligentInformatics vol 10 pp 578ndash585 2006

[38] M Ventresca and H R Tizhoosh ldquoImproving the convergenceof backpropagation by opposite transfer functionsrdquo in Proceed-ings of the International Joint Conference on Neural Networks(IJCNN rsquo06) pp 4777ndash4784 Vancouver Canada July 2006

[39] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 The Scientific World Journal

[40] Z Lin ldquoAn introduction to time-frequency signal analysisrdquoSensor Review vol 17 pp 46ndash53 1997

[41] A V Oppenheim R W Schafer and J R Buck Discrete TimeSignal Processing Prentice Hall New York NY USA 2ndedition 1999

[42] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of