A hybrid method for the design of oversampled uniform DFT filter banks

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doi:10.1016/j.sig

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Signal Processing 86 (2006) 1355–1364

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A hybrid method for the design of oversampled uniform DFTfilter banks

Ka Fai Cedric Yiua,�, Nedelko Grbicb, Sven Nordholmc,1, Kok Lay Teod

aDepartment of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong KongbDepartment of Signal Processing, Blekinge Institute of Technology, SE-372 25 Ronneby, Sweden

cWATRI, Western Australian Telecommunications Research Institute, AustraliadDepartment of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845 Australia

Received 27 January 2003; received in revised form 15 June 2004

Available online 28 September 2005

Abstract

Subband adaptive filters have been proposed to speed up the convergence and to lower the computational complexity of

time domain adaptive filters. However, subband processing causes signal degradations due to aliasing effects and

amplitude distortions. This problem is unavoidable due to further filtering operations in subbands. In this paper, the

problem of aliasing effect and amplitude distortion is studied. The prototype filter design problem is formulated as a multi-

criteria optimization problem and all the Pareto optima are sought. Since the problem is highly nonlinear and nonsmooth,

a new hybrid optimization method is proposed. Different prototype filters are used and their performances are compared.

Moreover, the effect of the number of subbands, the oversampling factors and the length of prototype filter are also

studied. We find that prototype filters designed via Kaiser or Dolph–Chebyshev window provide the best overall

performance. Also, there is a critical oversampling factor beyond which the improvement in performance is not justified.

Finally, if the length of the prototype filter increases with the number of subbands, an increase in the subband level will not

deteriorate the performance.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Subband adaptive filter; Aliasing effect; Amplitude distortion; Simulated annealing; Pareto optimum

1. Introduction

Adaptive filtering in subbands is an attractivealternative to the full-band scheme in manyapplications to achieve faster convergence andlower computational cost. In a typical subband

e front matter r 2005 Elsevier B.V. All rights reserved

pro.2005.02.023

ing author. Tel.: +852 22415956;

6535.

ss: cedric@hkucc.hku.hk (K.F.C. Yiu).

ure between The University of Western Australia

versity of Technology, Perth, Australia

adaptive filter, the filter input is first partitionedinto a set of subband signals through an ana-lysis filter bank. These subband signals are thendecimated to a lower rate and passed through a setof independent or partially independent adaptivefilters that operate at the decimated rate. Theoutputs from these filters are subsequently com-bined using a synthesis filter bank to reconstructthe full-band output. The DFT multirate filterbanks are commonly used for efficient rea-lization of the analysis and synthesis filter banks[1,2].

.

ARTICLE IN PRESSK.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–13641356

However, the analysis of a signal into a subbandrepresentation and the synthesis back into itsoriginal full-band form has several difficulties.Noticeably, subband filtering introduces signaldegradations which include signal distortions andaliasing effects [3]. It is well known that a filter bankcan be designed alias-free and perfectly recon-structed when certain conditions are met by theanalysis and synthesis filters. However, any filteringoperation in the subbands may cause a possiblephase and amplitude change and thereby alteringthe perfect reconstruction property. There are trade-offs in controlling both the aliasing effect and thedistortion level. Non-critical decimation has beensuggested in [4] to improve the overall performanceof the filter banks. Depending on the level ofoversampling, the cost of computation also in-creases significantly.

In general, the filter bank design problem is amulti-criteria decision problem, where the criteriaare the level of distortion and the level of aliasingeffect. A very sharp prototype filter will decrease thealiasing effect and distortion, but the length ofthe filter is usually prohibitively long. Depending onthe computation complexity, the length of theprototype filter is usually limited. Within this limit,the optimal filter is sought. If a simple least-squarestechnique is used to minimize both criteria together,there is no direct control over each individualcriterion. During the design process, it is thereforenot possible to specify the performance of the filterbank in advance. Methods have been proposed tominimize both criteria simultaneously, such as [5].However, more flexible design of the prototype filterhas not been considered, and individual criterion isnot controlled directly.

The performance of the filter bank depends on thechoice of prototype filter, the length of it, thenumber of subbands, and the oversampling factor.Here, we study the optimal designs for differentcombinations of parameters. A multi-criteria for-mulation is employed to trade off the aliasing effectagainst the distortion level. In order to allow for theworst scenario, the maximum norm is applied.Consequently, the filter design problem becomeshighly nonlinear and both the cost function andconstraint are not differentiable. In order to tacklethis problem, the L1 exact penalty function is firstapplied to transform the constrained problem intoan equivalent unconstrained problem. A new hybridmethod is then proposed to solve the resultanthighly nonlinear optimization problem. The hybrid

method combines the simulated annealing (whichhas the advantage of escaping from local minima)and the simplex search method (which is a no-derivative search method to locate local minima) toachieve fast convergence to the global mini-mum. One main desirable property of the proposedhybrid descent method is that the convergence ismonotonic.

This approach is versatile in the way that aspecific performance of the filter bank can beimposed in advance. The aliasing and distortionlevel can be controlled easily and the correspondingoptimal weights can be found. In this way, all thePareto optima can be sought. In assessing theperformance, different prototype filter designs arestudied. These include the window method with theHamming window, Kaiser window and Dolph–Chebyshev window, and the minimax method. Weshow that Kaiser and Dolph–Chebyshev windowgive the best overall performance with or withoutoversampling. Finally, the effects of the oversam-pling factor, the number of subbands and the lengthof the prototype filter are investigated.

2. The uniform DFT modulated filter bank

In a typical analysis–synthesis DFT filter bank,two sets of filters form a uniform DFT analysisfilter bank and synthesis filter bank. Assume thesame prototype filter is applied for both ana-lysis and synthesis, the subband filters are relatedto the prototype filter, h0ðnÞ, by means of modu-lation as

HkðzÞ ¼ H0ðzW kK Þ ¼

X1n¼�1

h0ðnÞðzW kK Þ�n

¼ hTfðzW kK Þ; k ¼ 0; . . . ;K � 1, ð1Þ

where W K ¼ e�j2p=K ; h ¼ ½hð0Þ; . . . ; hðL� 1Þ�T andfðzÞ ¼ ½1; z�1; . . . ; z�ðL�1Þ�T. Each subband signal isdecimated by a factor D. An implementation ofsuch a filter bank is depicted in Fig. 1. A typicalanalysis operation can be summarized in Fig. 2.Using the subband signal definitions according toFig. 1, we can describe the signal path through thefilter-bank realization. Each branch signal, V kðzÞ, issimply a filtered input signal defined as

VkðzÞ ¼ HkðzÞX ðzÞ ¼ H0ðzW kK Þ;X ðzÞ,

k ¼ 0; . . . ;K � 1. ð2Þ

The decimators cause a summation of repeatedand expanded spectrum of the input signal

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Fig. 1. Direct form realization of an analysis and synthesis filter-bank.

Filter D

D

Filter D

x(n) x0(n)

x1(n)

xK-1(n)

Filter

Filter Spectrum

Filter Spectrum

Filter Spectrum

fs

f

fs

f

sf

f

Fig. 2. A typical analysis operation.

K.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–1364 1357

according to

X kðzÞ ¼1

D

XD�1l¼0

V kðz1=DW l

DÞ

¼1

D

XD�1l¼0

H0ðz1=DW k

K W lDÞX ðz

1=DW lDÞ,

k ¼ 0; . . . ;K � 1, ð3Þ

where W D ¼ e�j2p=D. The interpolators have acompressing effect according to

UkðzÞ ¼ X kðzDÞ ¼

1

D

XD�1l¼0

H0ðzW kK W l

DÞX ðzW lDÞ,

k ¼ 0; . . . ;K � 1. ð4Þ

Finally, the signal will be synthesized by thereconstruction filters and we can state input–output

relationship as

bX ðzÞ ¼ XK�1k¼0

F kðzÞUkðzÞ

¼1

D

XD�1l¼0

X ðzW lDÞXK�1k¼0

H0ðzW kK W l

DÞH�0ðzW k

K Þ,

ð5Þ

where the superscript * denotes the conjugate. Thisexpression can be rewritten as

bX ðzÞ ¼XD�1l¼0

AlðzÞX ðzW lDÞ, (6)

where

AlðzÞ ¼1

D

XK�1k¼0

H0ðzW kK W l

DÞH�0ðzW k

K Þ. (7)

ARTICLE IN PRESSK.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–13641358

If AlðzÞ ¼ 0 for l ¼ 1; 2; . . . ;D� 1, and A0ðzÞ ¼

az�b, for any a; b where aa0, we get a per-fect reconstruction filter-bank. However, anyfiltering operation in the subbands may cause apossible phase and amplitude change and there-by altering the perfect reconstruction property.Our main objective is to find the prototypefilter coefficients h to minimize both the alias-ing power and the amplitude distortion defi-ned as

AP ¼ maxo

XD�1l¼1

jAlðe�joÞj

!, (8)

AD ¼ maxoð1� jA0ðe

�joÞjÞ. (9)

The aliasing effect is best understood by Figs. 3and 4 where a critical sampling clearly create severealiasing effect due to the transition region of theprototype filter. When the oversampling increases,the lines of aliasing will gradually move further toreduce the aliasing effect. In optimizing the proto-type filter, simply minimizing a sum of both

Fig. 3. The cause of the aliasing

measures may result in performance skewingtoward one extreme. There is no easy way tointroduce any scaling factor to adjust such unevenperformances. Because there are more than oneobjective in the design of the filter-bank, it isbasically a multi-criteria design problem [6,7]. Whendifferent scaling factors are applied to the criteria inthe design process, a solution set can be derived inwhich all solutions are efficient, or Pareto optima.In the present context, the set of weights h0 is aPareto optimum if and only if there does not exist aset of weights h such that

APðhÞpAPðh0Þ; ADðhÞpADðh

0Þ (10)

with strict inequality to at least one of the criteria.In order to solve for the Pareto optima, oneof the criteria can be formulated as a constraintinstead so that it becomes a nonlinear programmingproblem. An additional advantage of using thisformulation is that the constraint can be adjustedfreely to select the desired filter from the set ofPareto optima.

effect in critical sampling.

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Fig. 4. The aliasing effect is reduced after over-sampling ð2xÞ.

K.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–1364 1359

3. Prototype filter design

A typical nonrecursive causal prototype filter canbe defined by the transfer function

HðzÞ ¼XL�1n¼0

hðnÞz�n. (11)

There are several ways to design this type of filter.One method is to use a window function. The filtercoefficients hðnÞ is given by the Window method as

hðnÞ ¼sinð2pf cðn� ðL� 1Þ=2ÞÞ

pðn� ðL� 1Þ=2ÞwðnÞ, (12)

where wðnÞ is a window function. For a given numberof subbands, M, and a given decimation/interpolationfactor, D, and for a certain length of the prototypefilter, L, we need to design the cut-off frequency0of co 1

2and the corresponding window function.

A simple popular window function is the Ham-ming window, defined as

wðnÞ ¼a� ð1� aÞ cos

2pn

L� 1for n ¼ 0; 1; . . . ;L� 1;

0 otherwise;

8<:(13)

where a ¼ 0:54. There is no additional parameterfor this window function.

In case of the Kaiser window, there is anadditional design parameter a. Also, it can controlthe ripple ratio and the main-lobe width. Thiswindow is given by

wðnÞ ¼

I0 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L�12

� �2� n� L�1

2

� �2q� �Io a

ffiffiffiffiffiffiffiffiffiffiffiffiffiL�12

� �2q� � , (14)

where IoðxÞ is the zeroth-order modified Besselfunction of the first kind.

Another window which can vary the ripple ratioand main-lobe width is the Dolph–Chebyshevwindow defined as

wðnÞ ¼1

L

1

rþ 2

XðL�1Þ=2i¼1

TL�1ðx0cosip=LÞ cosð2npi=LÞ

" #,

(15)

where r is the ripple ratio as a fraction, and

x0 ¼ cosh1

L� 1cosh�1

1

r

� �. (16)

ARTICLE IN PRESSK.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–13641360

Function TkðxÞ is the kth-order Chebyshev poly-nomial assoicated with the Chebyshev approxima-tion for recursive filters and is given by

TkðxÞ ¼cosðk cos�1xÞ for jxjp1;

coshðk cosh�1xÞ for jxj41:

((17)

The additional design parameter is therefore r forthis window.

Apart from using the window method, it is alsopossible to use the minimax technique instead. Oncethe cut-off frequency f c and the stop-band fre-quency f s is fixed, the minimax optimizationproblem can be solved quickly via the Remezexchange algorithm. Therefore, the two designparameters for the minimax method are f c and f s.

In order to control the aliasing effect and theamplitude distortion separately, the final optimaldesign problem can be formulated into an equiva-lent optimization problem ðP1Þ as

ming

APðhðgÞÞ (18)

subject to

ADðhðgÞÞpad , (19)

where ad is a pre-defined tolerance for the amplitudedistortion. The level of distortion can now becontrolled freely. Note that g is one-dimensionalfor the Hamming window and two-dimensional forthe other windows and the minimax method. ðP1Þ isa nonsmooth constrained optimization problem. Inorder to simplify the problem, the L1 exact penaltyfunction is applied to convert it into an equivalentunconstrained minimization problem ðP2Þ as

ming

f ðgÞ � APðhðgÞÞ þM maxðADðhðgÞÞ � ad ; 0Þ,

(20)

where M is a large number. In this way, constraintviolations are penalized by a weighted L1 term.

4. Algorithms

Since ðP2Þ is a non-differentiable and highlynonlinear minimization problem, gradient-basedapproaches cannot adequately be used. Becausethe dimension of the problem is relatively low, amore realistic approach is to apply the simplexsearch technique to locate the minima via systematicmoves in the solution space [8]. However, this stilldoes not guarantee to reach the global minimumbecause of the nonlinearity. Consequently, a hybrid

technique comprises the simulated annealing algo-rithm and the simplex search technique is proposed.

Theoretically, the global minimum of f ðgÞ couldbe sought by using the simulated annealing algo-rithm alone. However, its convergence rate isusually very slow. On the other hand, the simplexsearch algorithm is much more efficient in conver-ging to a stationary point. Thus, by combining thesimulated annealing algorithm with the simplexsearch algorithm, we obtain an efficient hybriddescent method, which is formally stated in thefollowing:

A Hybrid Descent Algorithm

1.

2.

3.

4.

In Step 3 of the algorithm, the simulated annealingiterations composes of three key steps, namely thegeneration of the next trial point in the solutionspace via random perturbations, a choice of aprobability distribution to govern the acceptance ofuphill steps, and an annealing schedule.

In this paper, following [9,10], the Boltzmannprobability distribution is used. The annealingschedule is determined by the parameters b, thecooling speed; Nc, the number of cooling steps; N,the number of random perturbations for eachtemperature; and the initial temperature, T. Typicalchoices of these parameters can be found in [11].The simulated algorithm algorithm can be imple-mented as follows:

Initiation: Select b;Nc, N, and initial T. Evaluatef ðgðkÞ0Þ.

Cooling

(a)

(i) i ¼ randomf1; 2g.(ii) Depending on the outcome of i, re-generate

randomly either one element of g, or thewhole vector of g. This gives ~g.

(iii) Calculate D ¼ f ð~gÞ � f ðgÞ. If Do0 orrandom½0; 1�oT expð�D=TÞ, then g ¼ ~g.

ARTICLE IN PRESSK.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–1364 1361

(iv) Set j :¼ j þ 1 and return to (i) until N

perturbations are executed.

until Nc cooling steps are executed.

5. Evaluation results

In the multi-criteria formulation, either thealiasing effect or the amplitude distortion can beimposed as a constraint. Since it is easier to assessthe impact of the amplitude distortion for theapplication, it is therefore better to treat it as aconstraint.

The first example is to assess different designof prototype filter and its implication to the alias-ing effect for different oversampling factors. Theamplitude distortion ad is first restricted to be�20 dB and the result is depicted in Fig. 5. It isobserved that by increasing the oversampling factor,the minimized aliasing effect is decreasing. How-ever, the decrease is very slow for the Hammingwindow and is diminishing for the Remez method.Only the Kaiser and Dolph–Chebyshev windowmaintains a fast decrease in the minimized aliasingeffect with the increase of the oversampling factor.

0.1 0.2 0.3 0.4 0.5-180

-160

-140

-120

-100

-80

-60

-40

Tot

al a

liasi

ng p

ower

(dB

)

Downsampling ratio M/D

Hamming

Remez

Dolph Chebyshev

Kaise

M=32, L=5M, Amp

Fig. 5. Comparison for diff

In this way, the Kaiser and Dolph–Chebyshevwindow out-performed other methods for all over-sampling factor. We then varied the amplitudedistortion ad requirement to �30 dB and re-opti-mized the aliasing effect for different oversamplingfactors. The result is shown in Fig. 6. We observe avery similar pattern of performance. Also, weobserve a trade-off between the amplitude distor-tion and the optimized aliasing effect, which isexpected due to the multi-criteria nature of theproblem. The choice of ad depends largely on theapplication requirements.

In order to understand the influence of theprototype filter length and the number of sub-bands, we apply the Kaiser window as a demon-stration. As the number of subbands increases,the passband gets narrower. Thus, it is hardto maintain the low distortion level unless thelength of the filter increases to allow for a narrowertransition region. If the filter length is propor-tional to the number of subbands, the resultis shown in Fig. 7. It is interesting to see that asthe number of subbands increases, there is auniform improvement in the minimized aliasingeffect for all oversampling levels. The improvementis more significant for sampling factors near to

0.6 0.7 0.8 0.9 1

(critical downsampling=1)

r

litude distortion= 20

erent design methods.

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-180

-160

-140

-120

-100

-80

-60

-40

-20M=32, L=5M, Amplitude distortion= 30

Tot

al a

liasi

ng p

ower

(dB

)

Downsampling ratio M/D (critical downsampling=1)

Hamming

Remez Dolph Chebyshev

Kaiser

Fig. 6. Comparison for different design methods.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-180

-160

-140

-120

-100

-80

-60

-40

M=32M=64M=128

Prototype filter designed via Kaiser window with L=5M

Downsampling ratio M/D (critical downsampling=1)

Tot

al a

liasi

ng p

ower

(dB

)

Fig. 7. Comparison for different number of subbands.

K.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–13641362

critical one. The final designs of prototype filtersare shown in Fig. 8. All designs have a similarlevel of minimized aliasing effect; however, they

all have different transition widths and stopbandripples. As the maximum of the aliasing effectusually occurs near to the transition region, the

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-300

-250

-200

-150

-100

-50

0

50

Frequency (Hz)

Mag

nitu

de (

dB)

M=32, D=32, L=5M, amplitude distortion=-20

Hamming

Remez

Kaiser Dolph Chebyshev

Fig. 8. Comparison for different designs with critical sampling.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-250

-200

-150

-100

-50

0

Frequency (Hz)

Mag

nitu

de (

dB)

M=32, D=15, L=5M, amplitude distortion=-20

Hamming

Remez

Dolph Chebyshev Kaiser

Fig. 9. Comparison for different designs with over-sampling.

K.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–1364 1363

transition width and the stopband ripple clearlycompensates for each other. When oversamplingfactor about 0.5 is applied, the result is given in

Fig. 9. From the figure, the transition widths arevery similar for different designs. However, thereare significant differences in the stopband ripples.

ARTICLE IN PRESSK.F.C. Yiu et al. / Signal Processing 86 (2006) 1355–13641364

This account for the differences in the aliasing effectsuppression.

6. Conclusions

In this paper, the problem of filter bank design hasbeen studied. The design problem has been formu-lated as a multi-criteria optimization problem. Usingthe L1 exact penalty function, a new hybrid method isproposed to tackle the problem. Consequently, all thePareto optima can be solved and studied. Differentdesigns of the prototype filter have been investigated.It turns out that Kaiser and Dolph–Chebyshevwindow give the best performance with the lowestaliasing effect for a fixed amplitude distortion level. Ifthe length of the prototype filter is proportional to thenumber of subbands, the optimal aliasing effectgenerally improves with the number of subbands.The improvement is more significant for samplingfactors close to the critical one.

Acknowledgements

The third author would like to thank ARC, theAustralian Research Council for supporting thisresearch under grant DP0451111. The fourth authorwould also like to thank ARC for supporting thisresearch.

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