An Application of Hybrid Differential Evolution to 3-D Near field Source localization

Fawad Zaman*, Shafqat Ullah khan**, Kabir Ashraf*** and Ijaz Mansoor Qureshi****

* International Islamic University, Islamabad, Pakistan, ** Isra University, Islamabad, Pakistan, ***CESAT, Islamabad, Pakistan, ****Air University, Islamabad, Pakistan.

Abstract—3-D near field source localization is one of the hot areas of research which has found direct applications in Radar, Sonar and digital communication. In this work, we propose hybrid meta-heuristic based algorithm to estimate jointly and efficiently the range, elevation angle and amplitude of the near field sources impinging on uniform linear array. For this, first the Differential evolution (DE) and Interior Point Algorithm (IPA) are employed independently and then both of them are hybridized with each other to improve the accuracy and convergence rate further. In this hybridization, DE is used as a global optimization method while the IPA is acted as rapid local search optimizer. Mean Square Error is used as an objective evaluation function which requires single snapshot to converge and avoids any ambiguity among the angles that are supplement to each other. The proposed hybrid scheme (DE-IPA) produced better results as compare to DE and IPA alone. Moreover, the DE-IPA is also compared with the other hybrid meta-heuristic technique based on Genetic Algorithm hybridized with Interior point Algorithm (GA-IPA). The comparison is made on the basis of estimation accuracy, robustness against noise, convergence rate and root mean square error. The validity and effectiveness of the proposed hybrid scheme is exploited on the basis of large number of Monte-Carlo simulations.

I. INTRODUCTION

Near field source localization is one of the important and challenging area in radar, sonar, wireless communication etc [1]. It is easy to deal with far field sources as all the impinging signals have planar wave front and the signal is dependent only on Direction of arrival (DOA) [2-3]. A degree of difficulty arises when one have to deal with the parameter estimations of near field sources. In this case, all the impinging signals have spherical wave front which is a function of DOA as well as range of the sources [4-5]. The main problems during estimating these parameters are pair matching and estimation failure.

In order to solve the issue in better way, one should test the evolutionary computing techniques, as in this advance technological era, no one can deny their importance especially in the field of optimization [6-8]. These techniques mainly include Differential evolution (DE), Particle swarm optimization (PSO), Genetic Algorithm (GA), Culture Algorithm (CA) etc. These techniques are easy to manipulate and avoiding getting stuck in the presence of local minima. One of the other most fascinating features of these algorithms is hybridization with any other efficient algorithm such as Active set (AS), Interior point Algorithm (IPA), pattern Search (PS) etc [9-

13]. Due to hybridization, their reliability and effectiveness increases even more. In [13], GA is hybridized with IPA to jointly estimate the 3D parameters (Amplitude, range and elevation angle) of near field sources impinging on uniform linear array (ULA). It has used Mean Square Error (MSE) as a fitness evaluation function. In this paper, the same 3D parameters are jointly estimated but this time DE is hybridized with IPA instead of GA. For this, first the Differential evolution (DE) and Interior Point Algorithm (IPA) are employed independently and then both of them are hybridized with each other to improve the accuracy and convergence rate further. In this hybridization, DE is used as a global optimization method while the IPA is acted as rapid local search optimizer. MSE is used as an objective evaluation function which requires single snapshot to converge. Besides, it avoids ambiguity among the angles that are supplement to each other. The proposed hybrid scheme (DE-IPA) produced better results as compare to DE and IPA alone. Moreover, the DE-IPA is also compared with [13]. The comparison is made on the basis of estimation accuracy, robustness against noise and convergence rate.

The rest of the paper is organized as follow, In section II, we have provided data model for near sources impinging on ULA, whereas the proposed methodology structure is developed in section III. Section IV, is dedicated to simulation and results while section V, gives the conclusion and future work directions.

II. DATA MODEL Consider P sources impinging on ULA from near

field. The sources are assumed to be narrow band and statistically independent from each other. The ULA consists of 2M M x sensors by having the same inter-element spacing as shown in Fig.1. For this, the data model on mth sensor in the ULA for P sources can be represented as,

2exp( ( ))1

Py s j m m nm i i i m

i

(1)

In (1), nm is Additive White Gaussian noise (AWGN)

added at the mth sensor in the ULA. i and i are the function of elevation angle as well as, range of the sources and can be given as,

2sin( )

di i

(2)

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978-1-4799-2319-9/14/$31.00 © 2014 IEEE

22cos ( )

di i

ri

(3)

In vector-Matrix form, (1) can be represented as, y Bs n (4)

where in (4), y, s, n, B can be defined as,

[ , ... ... ]1T

y y yM o Mx x y (5)

1 2[ , , ..., ]PT

s s ss (6)

1[ , ..., , ..., ]M Mx

Tn n no xn (7)

1 2[ , , ..., ]PB b b b (8)

where,2( , ) [exp( ( 1) ( 1) ),

..., exp( ( )), 1, exp( ( )), ...,

2exp( ( ))]

r j M j Mi i i x i x i

j ji i i i

Tj M Mx i x i

b

is the steering vector for ith sources where 1, 2, ...i P .From (1-3), it is clear that the unknown parameters are amplitude ( )s , range ( )r and elevation angle ( ) . So, the goal of this paper is to jointly and efficiently estimate these 3D parameters by using DE-IPA.

Figure 1: schematic diagram for near field sources

III. PROPOSED MEHTODOLOGY

In this section, brief introduction, flow diagram and pseudo code of the proposed hybrid scheme is provided.

Differential Evolution: Differential evolution (DE), is comparatively new algorithm and proved to be the most impressive and powerful algorithm as compared to GA and PSO. It was first proposed by Stone and Price in 1996 [14]. It has already shown fairly good results for non-linear, multi-modal and non-differentiable cost functions. It has consistent and excellent convergence towards global minima and also very handy to the solution of discrete, as well as, constrained problems [15]. Thus, it has got applications in every field in every field of science and technology [16-17]. For more details about DE, the readers are encouraged to see [18-19]. In this current work, DE is first used as global optimizer and to improve the performance, the results are further passed through a local search optimizer. The flow chart of DE is shown in

“Fig.2” while it steps in the form of pseudo code are given as,

STEP 1 INITIALIZATION: In this step, randomly generate Q chromosomes where the length of each chromosome is 3 P . The first P genes in each chromosome represent the amplitudes, the second Pgenes represent ranges while the last P genes correspond to elevation angles as given below,

1,1 1,2 1, 1, 1 1,2 1,2 1 1,3

2,1 2,2 2, 2, 1 2,2 2,2 1 2,3

,1 ,2 , , 1 ,2 ,2 1 ,3

s s s r rP P P P Ps s s r rP P P P P

s s s r rQ Q Q P Q P Q P Q P Q P

D

where

:, ,: 1, 2, ... & 1, 2, ..., ,

:0,2 ,2

s R l s ui k b i k br R r r r i Q k Pi P k l i P k b

Ri P k i P k

Step 2 Updating: In this step, update all the chromosomes from 1 to Q of the present generation ‘ge’. Now, select

qth chromosome ,( )q gehd from matrix D, where ‘ge’ and

‘h’ represents the particular generation and length of chromosome respectively. Basically, in this step, our aim is to find out the chromosome of next generation i.e.

, 1q gee by using mutation, cross over and selection operation as, (a) Mutation: To perform the mutation, choose any three different numbers (chromosomes) from matrix ‘D’according to the following conditions,

1 , ,1 2 3c c c Q

, 1, 2, 3c n q hq h

and 1, 2, 3n q qq

now,

,, ,, 31 2( )c gec ge c geq ge

F f d d d (9)

where 0.5 1F .

b) Crossover: The crossover can be done as , (),, /

q geif rand CR or h hq ge randh

k q geo wh

fr

d

where 0.5 1CR and hrand is chosen randomly

between 1 and 3*P .

c) Selection Operation: By using the following step, select the chromosome of next generation as

, , ,, 1 ( ) ( ), /

q ge q ge q geq ge if err errq ge

o w

r r ddd

Proceedings of 2014 11th International Bhurban Conference on Applied Sciences & Technology (IBCAST)Islamabad, Pakistan, 14th – 18th January, 2014 475

where the ,( )q ge

err d and,( )q ge

err r are defined in above matrix ‘D’. Repeat this for all chromosomes.Step 3 stopping Criteria: The stopping criteria is based on any of the following condition is satisfied, (I) If , 1( ) ,q ge

err d where is a very small

positive number OR (II) Total number of generation has completed. else go back to step 2. Step 4 Hybridization: In this hybridization step, pass the results of DE through IPA for further tuning. IPA is basically a local search optimizer and one can see its detail in [20]. In this current work, we have used a MATLAB built-in optimization tool box for IPA whose parameters settings are provided in Table 1. Table 1. Parameter Settings for IPA

IPA Parameters Setting

Starting Point Best chromosomeachieved by DE

Sub problem algorithm Idl factorization Maximum perturbation 0.1 Minimum perturbation 1e^-8

Scaling Objective & Constraint Hessian BFGS

Derivative type Central difference Penalty factor 100

Maximum function evaluation

50000

Maximum Iteration 2000 X Tolerance 10-15

Figure 2: Flow chart of DE-IPA

IV. RESULTS AND DISCUSSION

In this section, we have carried out several simulations to assess the performance of our proposed hybrid scheme. For this purpose, we have taken 60 generation and 100 chromosomes. Throughout the simulations, the inter-element spacing between the sensors is kept / 4 where is the wavelength of signals impinging on ULA. The proposed hybrid scheme (DE-IPA) is compared with DE , IPA and GA-IPA given in [13]. The values of ranges and DOA are taken in terms of and radians (rad) respectively. All the results are averaged over 100 independent runs. Case 1. Robustness: In this case, the robustness of DE-IPA is checked against noise and compared with IPA, DE and GA-IPA. In this simulation, MSE is taken as a evaluation function. The ULA consists of 8 sensors while the values of amplitude, range and DOA of two sources are 2, 2 , 1.91991 1 1s r rad

4, 5 , 0.52362 2 2s r rad . The values of SNR are taken from 0 to 20 dB. Fig 3, shows the joint MSE of amplitudes, ranges and elevation angles by using IPA, DE, GA-IPA and DE-IPA. One can see, that in all cases, the proposed DE-IPA has maintained fairly a minimum MSE for all values of SNR. The second best curves have been produced by the other hybrid GA-IPA scheme [13].

0 2 4 6 8 10 12 14 16 18 2010-10

10-5

100 MSE vs SNR

[SNR]

MS

E

DEIPAGA-IPA[13]DE-IPA

Figure 3: MSE vs SNRCase 2. Convergence Rate: In this case, the convergence rate of DE-IPA is evaluated and compared with IPA, DE and GA-IPA. From convergence, we mean the total number of times an algorithm achieved its desired results. In Fig 6, bar graph is shown, which gives percentage convergence of each algorithm against the number of sources. With the increase of sources, the convergence rate of each scheme has degraded. One can again see that the DE-IPA has greater convergence rate not only as compared to IPA and DE alone but also from GA-IPA in [13].

2 3 40

20

40

60

80

100

Number of sources

% c

onve

rgen

ce ra

te

IPAGA-IPA[13]DEDE-IPA

Figure 4: Convergence rate vs number of sources

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Case 3. Estimation accuracy: In this case, the estimation accuracy of DE-IPA has compared with IPA, DE and GA-IPA in the presence of 10dB noise. For this the ULA consists of 12 sensors while three sources are considered. The desired values of these three sources are

3, 1 , 0.6981 ,1 1 1s r rad

9, 4 , 2.2689 ,2 2 2s r rad

6, 5 , 1.22173 3 3s r rad .As provide in Table 2, the proposed DE-IPA scheme has produced batter estimation accuracy as compared to IPA, DE and GA-IPA. The second best results is given by GA-IPA. Table 2. Estimation accuracy for three sources

Algorithm 1s ( )1r ( )1 rad 2s

Desired 3.0000 1.0000 0.6981 9.0000

IPA 3.0099 1.0088 0.7070 9.0084

DE 3.0066 1.0067 0.7046 9.0067

GA-IPA 3.0060 1.0061 0.7041 9.0062

DE-IPA 3.0024 1.0021 0.7001 9.0022

TABLE 2. CONTINUED

( )2r ( )2 rad 3s ( )3r ( )3 rad

4.0000 2.2689 6.0000 5.0000 1.2217

4.0082 2.2780 6.0088 5.0092 1.2302

4.0068 2.2760 6.0063 5.0064 1.2284

4.0061 2.2751 6.0062 5.0056 1.2278

4.0023 2.2713 6.0026 5.0022 1.2237

V. CONCLUSION AND FUTURE WORK DIRECTION

In this work, mainly a hybrid DE-IPA schemes has been developed for the joint estimation of 3D parameters of near field sources impinging on ULA. The 3D parameters include amplitude, range and elevation angle. In this hybrid scheme, DE was used as a global optimizer while the IPA has been used as a local search optimizer for further tuning. MSE were used as an objective evaluation function that required single snapshot. From various simulations, it has been shown that the proposed DE-IPA scheme has produced better results as compared to IPA, DE and GA-IPA in terms of robustness, convergence rate and estimation accuracy. The proposed scheme fails when the number of sensors in the ULA is kept less than the number of sources. In future, one can develop such hybrid schemes for side lobe reduction, main beam steering and null steering in the field of adaptive beamforming.

REFERENCES

[1] H. Krim, and M. Viberg, “Two decades of array processing research: The parametric approach,” IEEE Signal Processing Magazine, Vol. 13, No. 4, pp.67–94, July 1996.

[2] R.Schmidt, Multiple emitter location and signal parameter estimation. IEEE Transactions on Antennas and Propagation, 34,pp. 276–280. 1986.

[3] T. Sarkar, Pereira, Using the matrix pencil method to estimate the paprameters of a sum of complex exponentials. IEEE Antennas Propag, 37, pp.48-55, 1995.

[4] Swindlehurst, A., Kailath, T. (1988). Passive direction-of-arrival and range estimation for near-field sources. In Proceedings of the 4th Annual ASSP Workshop on Spectrum Estimation and Modeling, USA, pp.123–128,1989.

[5] E.Grosicki, Abed, K., Y. Hua, A weighted linear prediction method for near-field source localization. IEEE Transactions on Signal Processing, 53, part 1,pp. 3651–3660,2005.

[6] A.U. Rehman, Fawad Zaman Y.A.shiekh, I.M.Qureshi. Null and sidelobes adjustment of damaged array using hybrid computing. IEEE International Conference on Emerging Technologies (ICET), 483-484, 2012.

[7] S. U. Khan, I.M. Qureshi, Fawad Zaman, A.N.Malik, “Null placement and sidelobe suppression in failed array using symmetrical element failure technique and hybrid heuristic computation,” Progress in Electromegnatic Research B, pp, 165-184, vol 52, 2013.

[8] Y.A.Shiekh, Fawad Zaman, I.M.Qureshi, A.U. Rehman, “Amplitude and Direction of Arrival Estimation using Differential Evolution,” IEEE, ICET, pp, 45-49, 2012.

[9] Fawad Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, “An Application of Artificial Intelligence for the Joint Estimation of Amplitude and Two Dimensional Direction of Arrival of far field sources using 2-L shape array,” International Journal of Antennas and Propagation, Article ID 593247, 10 pages, Volume 2013.

[10] Fawad Zaman, Ijaz Mansoor Qureshi, A. Naveed , Junaid Ali Khan and Raja Muhammad Asif Zahoor “Amplitude and Directional of Arrival Estimation: Comparison between different techniques,” Progress in Electromagnetic research-B (PIER-B),Vol. 39, pp.319-335, 2012.

[11] Fawad Zaman, I. M. Qureshi, A.Naveed and Z. U. Khan, “Real Time Direction of Arrival estimation in Noisy Environment using Particle Swarm Optimization with single snapshot,” Research Journal of Engineering and Technology (Maxwell Scientific organization), Vol. 4(13) pp. 1949-1952, 2012.

[12] Fawad Zaman, J. A. Khan, Z.U.Khan, I.M.Qureshi, “An application of hybrid computing to estimate jointly the amplitude and Direction of Arrival with single snapshot,” IEEE, 10th-IBCAST, pp-364-368, Islamabad, Pakistan, 2013.

[13] Fawad Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, “joint estimation of amplitude, direction of arrival and range of near field sources using memetic computing” Progress in Electromagnetic research-C (PIER-C) ,Vol,31, pp. 199-213, 2012.

[14] R. Storn, K. Price. Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global optimization, 11: 341-359, 1996.

[15] K. V. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization. Springer, 2005.

[16] T. Rogalsky, R.W. Derksen and Kocabiyik, “Differential evolution in aerodynamic optimization”, Proc. of 46th annual conf. of Canadian Aeronautics and Space Institute, pp 29-36, 1999.

[17] S. Das and A. Konar, “Design of two dimensional IIR filters with modern search heuristics: a comparative study”, International Journal of Computational Intelligence and Applications, Vol. 6, No.3, pp.329-355, 2006.

[18] J.R. Koza, “Hierarchical Genetic Algorithms Operating on Populations of Computer Programs,” In Proc. of the Eleventh International Joint Conference on Artificial Intelligence, vol.1, pp. 768–774, 1989.

[19] L.J. Fogel, “Autonomous Automata,” Industrial Research, vol.4, pp.14–19, 1962.

[20] A. Forsgren, P. E. Gill and M. H. Wright, “Interior methods for nonlinear Optimization,” SIAM Review 44, 525–597, 2002.

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