Abstract In this paper, we proposed an evolutionary algorithm for joint estimation of amplitude and DOA of signals impinging from far field on a uniform linear array. We used Differential Evolution algorithm with Mean Square Error as a fitness evaluation function. This fitness function defines an error between actual and estimated signal and is derived from Maximum Likelihood Principle. It does not require any permutations to link it with the angles estimated in the previous snapshot. It requires only single snapshot to converge and produce fairly good results even in the presence of low Signal to Noise Ratio. The usefulness and competence of proposed algorithm is tested on the basis of large number of Monte-Carlo simulations and its statistical analysis.

Index Terms Direction of Arrival, Differential Evolution, Mean Square Error, Signal to Noise Ratio.

I. INTRODUCTIONThe adaptive array system consists of sensors which are proficient to steer the main beam in any desired direction while placing a suitable null in the direction of interference or jammer [1]. In this regard, Direction of Arrival estimation of signals arriving on a sensor array is one of the important steps to put up an adaptive receiver. DOA estimation has found a wide range of applications in radar, sonar, mobile communication etc. [2]. Meta-heuristic techniques such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE) etc. have been the focusing area of research since last few decades. These meta-heuristic techniques are successfully widely applied to different problems in array signal processing, wireless communication and medical imaging [3], [4]. Among these techniques, DE got special attention because of its strong ability of avoiding getting stuck in local minima and ease in implementation. Similar to GA, DE also uses three operators i.e. mutation, crossover and selection to search the solution space. Among these operators, mutation has the central role in the performance of DE algorithm and basically, the kind of DE strategies to be constituted depends upon mutation variants. Every evolutionary technique applied to any problem, requires a fitness function which is problem specific. One can find in literature a number of fitness functions used for different problems e.g. Maximum Likelihood principle, Auto

Correlation function etc. [5], [6]. Mean Square Error (MSE) which defines an error between actual and estimated signal, is one of the easy and optimum fitness functions. In [7], Genetic Algorithm, Pattern Search, Simulated Annealing, Genetic Algorithm hybridized with pattern Search and Simulated Annealing have been successfully applied for joint estimation of amplitude and DOA of sources by taking MSE as a fitness function. Similarly, particle Swarm optimization (PSO) with a fitness function as MSE has also been successfully applied for joint estimation of amplitude and DOA of signals impinging on a uniform linear array [8]. By seeing the strength of DE and MSE as a fitness function, we have applied this to joint estimate the amplitude and DOA of signals impinging on a ULA which according to the best of authors knowledge, no one has done yet.

In this paper, we proposed a method based on evolutionary technique for joint estimation of amplitude and DOA of signals impinging from far field on a ULA. We exploited the strength of DE with MSE as a fitness function. This fitness function defines an error between actual and estimated signals and is derived from Maximum Likelihood Principle. This fitness function is capable of automatically pairing the DOA estimated in previous snapshot to current estimation. It avoids any ambiguity between the angles that are supplement to each other and requires only a single snapshot to converge. Moreover, it produced fairly good results even in the presence of low signal to noise ratio. The usefulness and competence of proposed algorithm is tested on the basis of Monte-Carlo simulations and its statistical analysis.

The remaining paper is organized as follows: In section 2, we discussed the problem formulation. Proposed Methodology is discussed in section 3. Section 4 and 5 are dedicated for results and future work direction respectively.

II. PROBLEM FORMULATION

Consider a uniform linear array (ULA) composed of Lantenna elements with equal inter-element spacing d as shown in fig.1. Assume that P sources impinging from different directions on sensor array from far field where P L and each signal having amplitude ia . The signals are

Amplitude and Direction of Arrival Estimation using Differential Evolution

Y. Ali Sheikh1, F. Zaman2, I. M. Qureshi3 and M. Atique-ur-Rehman4 1,3,4Department of Electrical Engineering, Air University

Islamabad, Pakistan 2Department of Electronics Engineering, International Islamic University

Islamabad, Pakistan yawarali@mail.au.edu.pk, fawadphdee31@iiu.edu.pk, imqureshi@mail.au.edu.pk,

atiq@mail.au.edu.pk

978-1-4673-4450-0/12/$31.00 2012 IEEE

narrow band and have known frequency .o

The appropriate output of the array is given as follows:

( )= +B a x (1)

1( )

Pi i

i

=

= +b a (2)

The vectors and matrices in Eq. (1) and Eq. (2) are defined as follow,

1 2 3[ , , ,..., ]TLx x x x=x (3) 1 2 3[ , , ,..., ]TPa a a a=a (4) 1 2 3[ , , ,..., ]TL = (5)

1 2( ) [ ( ) ( )... ( )]TP =B b b b (6) where,

(7)

and 2 ( / ) cosi id = for 1, 2,...,i P= . ( )b denotes the steering vector while is the additive white noise added at the output of array. As shown in equation (2), the unknown parameters are the amplitudes and the angles of arrival .iSo, the problem in hand is to estimate jointly the amplitudes i.e. 1 2, ,... Pa a a and DOA of sources i.e. 1 2, ,..., P from the received data by using intelligent hybrid technique.

III. PROPOSED METHODOLOGY Differential Evolution is an evolutionary stochastic technique, population based algorithm like Genetic Algorithm (GA) or Particle Swarm Optimization (PSO). It was introduced by Stone and Price in 1996 [9]. The algorithm steps are given below in the form of Pseudo code steps.

Step 1 Initialization: Let L and H be respective lower and upper limits of the chromosomes. Let number of chromosomes in a generation= N while number of genes in any chromosome= D.

! " # $ where, i = number of chromosomes for 1 i N and j = number of genes for 1 j D

Step 2: Update all chromosomes of present generation g from 1 to N. Suppose we pick up ith chromosome %&'.a) Mutation: Choose any three numbers from 1to N i.e. ( ( )) all different and not equal to i. * + ,+( $+)where, F is a constant usually between 0.5 to 1. b) Crossover: - .*

/0 ! 1 2345 567+849: ;

where, CR=0.5 to 1and 567+8 is between 1 and D chosen at random. c) Selection Operation: ? @-/00- A 049: Bwhere, 0 CD $(is the Mean Square Error. Repeat chromosomes.

Step 3: If FC%'?D A G , where is a very small positive number, stop else if number of generation reached, stop else go back to step 2.

IV. RESULTS AND DISCUSSION In this section, we have discussed the performance of DE

algorithm in terms of accuracy and reliability for joint estimation of amplitude and DOA of far field sources. We have used a uniform linear array which consists of L antenna elements. The distance between the two consecutive elements d is kept half the wavelength of received signal i.e. H9I. Throughout the simulations, we have used number of Generation (NOG) = 60, Cross over rate (CR) = 0.7, Number of chromosomes (N) = 50 and the constant (F) = 0.7 which is problem dependent. MSE is used as a fitness evaluation function. Different cases are discussed on the basis of different number of sources impinging on ULA. All the values of DOA and signal to noise ratio (SNR) are taken in degrees and in dB respectively. The value of SNR is ranging from 30dB to 5 dB. For all of our simulations, we have used the MATLAB version R2009b and each result is averaged over twenty independent runs.

L

Fig.1. Linear Array having L elements

Case 1: In this case, we have discussed the performance of DE algorithm for two sources. For this, we have used a ULA consists of four elements. The amplitude of the signals are denoted by J and J( while the angles are represented by Kand K(. The values of the amplitudes and angles are J , J( LM, K MN, K( OMNrespectively as shown in table I. As one can observe from table I, that even at very low values of SNR, the DE algorithm performed well for joint estimation of amplitude and DOA of two sources. The error between actual and estimated values is given in table II.

TABLE I AMPLITUDE AND DOA ESTIMATION OF 2 SOURCES

SNR A1 A2 1o 2oNo Noise 1.00 1.50 114.99 145.01 30 dB 1.03 1.53 114.97 144.96 25 dB 0.99 1.49 115.06 144.94 20 dB 0.98 1.49 115.09 145.08 15 dB 0.98 1.48 114.88 144.88 10 dB 1.02 1.48 115.14 145.16 5 dB 1.04 1.46 115.16 144.80

TABLE II ESTIMATION ERROR OF 2 SOURCES

SNR A1 A2 1o 2o

No Noise +0.00 +0.00 - 0.01 +0.01 30 dB +0.03 +0.03 - 0.03 - 0.04 25 dB - 0.01 - 0.01 +0.06 - 0.06 20 dB - 0.02 - 0.01 +0.09 +0.08 15 dB - 0.02 - 0.02 - 0.12 - 0.12 10 dB +0.02 - 0.02 +0.14 +0.16 5 dB +0.04 - 0.04 +0.16 - 0.20

Case II: In this case, we have discussed the performance of DE algorithm for three sources. For this we have taken six sensors in the ULA. The values of the amplitudes and angles are J LM, J( I, J) ILM and K PQN, K( RQN, K) IQN respectively. With the increase of unknowns, we faced few local minima due to which the performancedegraded slightly. As one can see from table III that even in the presence of local minima and low values of SNR, the DE algorithm produced fairly good results for joint estimation of amplitude and DOA of three sources. The error between actual and estimated values is given in table IV.

TABLE III AMPLITUDE AND DOA ESTIMATION OF 3 SOURCES

TABLE IV ESTIMATION ERROR OF 3 SOURCES

Case III: In this case, we have exploited the performance of DE algorithm for four sources and ULA of eight sensors. The values of amplitudes and angles are J LM, J( I, J) ILM, JS P, and K PQN, K( RQN, K) IQN, KS RQNrespectively. In this case, we have eight unknowns i.e. four amplitudes and four DOAs. Again we faced few strong local minima but DE algorithm is quite efficient to perform well even in the presence of theses strong local minima. As shown in table V, the results are also fairly good even at very low values of SNR. The error between actual and estimated values is given in table VI.

TABLE V AMPLITUDE AND DOA ESTIMATION OF 4 SOURCES

SNR A1 A2 A3 A4 1o 2o 3o 4oNo Noise 1.49 1.99 2.51 3.00 29.99 59.99 120.01 159.99 30 dB 1.48 1.99 2.52 2.99 30.02 60.01 120.01 159.99 25 dB 1.47 1.98 2.53 3.03 29.96 59.99 120.03 159.8 20 dB 1.54 1.97 2.46 3.05 30.05 59.97 120.04 159.97 15 dB 1.56 1.96 2.45 2.95 29.94 60.04 120.05 159.95 10 dB 1.43 1.93 2.56 2.94 29.92 59.95 120.06 159.93 5 dB 1.42 1.91 2.43 3.08 30.12 59.94 120.07 159.90

TABLE VI ESTIMATION ERROR OF 4 SOURCES

SNR A1 A2 A3 A4 1o 2o 3o 4oNo Noise - 0.01 - 0.01 +0.01 0.00 - 0.01 - 0.01 +0.01 - 0.01

30 dB - 0.02 - 0.01 +0.02 - 0.01 +0.02 +0.01 +0.01 - 0.01

25 dB - 0.03 - 0.02 +0.03 +0.03 - 0.04 - 0.01 +0.03 - 0.20

20 dB +0.04 - 0.03 - 0.04 +0.05 +0.05 - 0.03 +0.04 - 0.0315 dB +0.06 - 0.04 - 0.05 - 0.05 - 0.06 +0.04 +0.05 - 0.05

10 dB - 0.07 - 0.07 +0.06 - 0.06 - 0.08 - 0.05 +0.06 - 0.07 5 dB - 0.08 - 0.09 - 0.07 +0.08 +0.12 - 0.06 +0.07 - 0.10

In this section, we evaluate the MSE against noise for two, three and four sources. Fig. 2 shows that the MSE decreases in all cases with increase of SNR. As shown in Fig. 2, we have a best curve in case of two sources which is quite obvious as in case of three and four sources we have faced few local minima due to which the performance of DE degraded slightly and hence increased the MSE.

SNR A1 A2 A3 1o 2o 3o

NoNoise 1.49 1.99 2.50 30.02 60.01 119.98 30 dB 1.48 2.02 2.49 29.96 59.99 120.06 25 dB 1.47 1.98 2.51 29.94 59.96 119.93 20 dB 1.54 1.96 2.47 30.09 60.09 120.07 15 dB 1.57 1.92 2.58 29.89 59.91 120.12 10 dB 1.41 2.10 2.41 30.12 60.12 119.88 5 dB 1.39 1.89 2.38 29.85 59.86 120.14

SNR A1 A2 A3 1o 2o 3o

No Noise - 0.01 - 0.01 0.00 +0.02 +0.01 - 0.02 30 dB - 0.02 +0.02 - 0.01 - 0.04 - 0.01 +0.06 25 dB - 0.03 - 0.02 +0.01 - 0.06 - 0.04 - 0.07 20 dB +0.04 - 0.04 - 0.03 +0.09 +0.09 +0.07 15 dB +0.07 - 0.08 +0.08 - 0.11 - 0.09 +0.12 10 dB - 0.09 +0.10 - 0.09 +0.12 +0.12 - 0.12 5 dB - 0.11 - 0.11 - 0.12 - 0.15 - 0.14 +0.14

Fig.2 MSE Vs SNR

Conclusion and Future work directions In this work, an Evolutionary Algorithm i.e. DE has been

proposed for joint estimation of amplitude and DOA of signal impinging on ULA. The proposed algorithm is quite capable and efficient to correctly estimate the amplitude and DOA. It uses MSE as a fitness evaluation function which is optimum and requires only single snap-shot to converge. The DE algorithm fails when the number of sensors in the array is less than the number of sources as it becomes an under-determined problem. In future, we will use the DE algorithm for joint estimation of amplitude, range and DOA of near field sources.

References [1] Z. U. Khan, A. Naveed, I. M. Qureshi and F. zaman, Independent Null steering by decoupling complex weights, IEICE Electronic

Express, Vol. 8, pp. 1008- 1013, 2011. [2] H. Karim and M. Viberg, Two decades of array processing research:

The parametric approach, IEEE Signal Processing Magazine, Vol. 13, No. 4, pp. 6794, July 1996.

[3] B. Addad, S. Amari, and J.-J. Lesage, Genetic algorithms for delays evaluation in networked automation systems, Engineering

Applications of Artificial Intelligence, Vol. 24, pp.485-490, Elsevier,2011.

[4] U. Maulik, Analysis of gene microarray data in a soft computing framework," Engineering Applications of Artificial Intelligence,

Elsevier, Signal Process, Vol. 24, pp. 485-490, 2011. [5] Z. Jiankui, H. zishu and L. benyong, Maximum Likelihood DOA Estimation Using Particle Swarm Optimization Algorithm, Using

Particle Swarm Optimization Algorithm, Proc IEEE, 2006. [6] B. Errasti1, D. Escot , D. Poyatos, I. Montiel, Performance analysis of the Particle Swarm Optimization algorithm when applied to direction of arrival estimation, ICEAA, Sept. 2009, pp.447-450. [7] F. zaman, I. M. Qureshi, A. Naveed, J. A. khan, and R. M. A. Zahoor, Amplitude and Directional of Arrival estimation: comparison

between different techniques, Progress in Electromagnetics Research B, vol. 39, 319-335, 2012.

[8] F. 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, Accepted for publication in Research Journal of Applied Sciences, Engineering and Technology2012.

[9] R. Storn, K. Price, K. Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, 11, (1997) pp. 341-359.

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