The aim of this work is to estimate jointly the elevation and azimuth angles along with the amplitudes of multiple signals impinging on 1L and 2Lshape arrays. An efficient mechanism based on hybrid Bioinspired techniques is proposed for this purpose. The global search optimizers such as Differential Evolution (DE) and Particle Swarm optimization (PSO) are hybridized with a local search optimizer called pattern search (PS). Approximation theory in Mean Square Error sense is exploited to develop a fitness function of the problem. The unknown parameters of multiple signals transmitted by farfield sources are estimated with the strength of hybrid DEPS and PSOPS. The effectiveness of the proposed techniques is tested in terms of estimation accuracy, proximity effect, convergence, and computational complexity.
Parameter estimation such as Direction of Arrival (DOA) of electromagnetic signals for multiple sources is one of the vital areas of research in array signal processing from the last few decades. The DOA estimation ensues in adaptive beamforming to place the nulls in the direction of jammers or unwanted signals, while placing the main beam in the desired direction [
In the present scientific society era, the significance of Evolutionary Computing Techniques (ECT) that include Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE) cannot be vilipended. These techniques are not only easy to implement but also have the significant ability of hybridization with other heuristic and nonheuristic techniques. Therefore, the significance of these techniques is realized in different varying nature of applications [
In this paper, DE and PSO are hybridized with PS to jointly estimate the amplitude and 2D DOA of farfield sources impinging on 1L and 2Lshape arrays. In this hybridization process, DE and PSO act as global search optimizers, while PS is used as rapid local search optimizer. The best individual results of DE and PSO are given to PS for further tuning. The performance criterion is devised on the basis of Mean Square Error (MSE) that is applied as an objective evaluation function. This fitness function is derived from Maximum Likelihood Principle (MLP) [
The rest of the paper is organized as follows. Section
In this section, data model is developed for
1Lshape array having 2 subarrays.
2Lshape array having 3 subarrays.
In this section, we have discussed the brief introduction, flow chart, parameter setting, and pseudocode of PS, PSO, and DE.
Pattern Search (PS) which is also called direct search method belongs to a family of numerical optimization techniques. It can be used for linearly constrained and bounded optimization problem, which does not require the gradient of the problem. PS works on a group of points called a pattern. If improvement does not occur in the objective function at the current iteration, the pattern is refined and the process is repeated [
The PS acts even well when it is used as a local search optimizer with any other global optimizer technique such as GA, PSO, and DE. In the present work, PS is used as a local search optimizer with DE and PSO. The best particle achieved through DE and PSO is given as starting point to PS for further tuning. For PS, we have used a MATLAB builtin optimization tool box for which the parameters settings are given in Table
Parameter setting for PS.
Parameters  Setting 

Starting point  The best particle achieved through PSO or DE 
Poll method  GPS positive basis 2 N 
Polling order  Consecutive 
Maximum iteration  800 
Function evaluation  15000 
Mesh size  01 
Expansion factor  2.0 
Contraction factor  0.5 
Penalty factor  100 
Bind tolerance  1007 
Mesh tolerance  1009 
X tolerance  1009 
The idea of PSO was first introduced by Kennedy and Eberhart being inspired from the group of birds flocking for food in random manner [
The required fitness function is achieved.
The desired MSE has been reached which is
The total number of iterations has been completed.
The Differential Evolution (DE) was first introduced by Stone and Price in 1996 [
Total number of generations has completed.
In the first part of simulations, various results are compared to evaluate the estimation accuracy and reliability of PSO, PSOPS, DE, and DEPS for the joint estimation of amplitudes and DOA (elevation and azimuth) of far filed sources impinging on 1L and 2Lshape arrays. In the second part of simulation, the comparison is carried out with PM, which has used parallel shape array [
In this case, the estimation accuracy of PSO, DE, PSOPS, and DEPS is examined for 1L and 2Lshape arrays without having any noise in the system. Two sources are considered which have amplitudes
Estimation accuracy of 1Lshape array for 2 sources.
Scheme 







Desired values  0.5000  2.0000  30.0000  70.0000  110.0000  210.0000 
PSO  0.4951  2.0050  30.0061  70.0060  109.0038  210.0061 
DE  0.5022  1.9977  30.0044  69.0058  110.0044  209.9955 
PSOPS  0.5018  2.0019  29.9967  70.0036  110.0033  210.0033 
DEPS  0.4991  1.9988  30.0015  69.0082  110.0015  209.9984 
In Table
Estimation accuracy of 2L shape array for 2 sources.
Scheme 







Desired values  0.5000  2.0000  30.0000  70.0000  110.0000  210.0000 
PSO  0.5036  1.9963  29.9950  70.0049  110.0050  210.0050 
DE  0.4964  2.0014  30.0034  69.9966  110.0034  209.9965 
PSOPS  0.4998  1.9991  30.0022  70.0021  109.9978  209.9979 
DEPS  0.5001  2.0001  29.9998  70.0001  110.0002  210.0001 
Now by comparing Tables
In this subsection, the estimation accuracy is discussed for 3 sources impinging on Lshape arrays. The 1L and 2Lshape arrays are composed of 13 and 7 sensors, respectively. The desired values of amplitudes
The best estimation accuracy is given by DEPS for both Lshape arrays, while the second best result is given by the hybrid PSOPS technique. Overall, 2Lshape array produced better estimation accuracy as compared to 1Lshape array for all techniques.
Estimation accuracy for 3 sources using 1Lshape array.
Scheme 










Desired values  2.0000  4.0000  6.0000  60.0000  25.0000  40.0000  15.0000  85.0000  170.0000 
PSO  2.1789  3.8209  5.8210  60.3843  24.6158  39.6156  15.3847  84.6157  169.6153 
DE  1.9028  3.9026  6.0971  58.8028  25.1974  40.1973  15.1979  85.1972  169.8022 
PSOPS  2.0191  4.0192  6.0193  59.9264  24.9261  40.0738  14.9261  84.9263  170.0737 
DEPS  1.9931  3.9932  5.9932  60.0268  24.9733  39.9731  15.0269  85.0260  170.0272 
Estimation accuracy for 3 sources using 2Lshape array.
Scheme 










Desired values  2.0000  4.0000  6.0000  60.0000  25.0000  40.0000  15.0000  85.0000  170.0000 
PSO  1.8447  3.8445  6.1556  60.1791  25.1790  39.8207  15.1791  84.8206  170.1792 
DE  2.0481  4.0480  6.0483  60.0977  24.9024  40.0979  14.9020  85.0983  169.9020 
PSOPS  2.0137  3.9862  6.0137  60.0328  25.0331  40.0330  15.0327  84.9670  170.0329 
DEPS  1.9980  4.0021  5.9980  59.9923  25.0077  39.9925  14.9925  85.0076  169.9926 
In this case, 4 sources are considered that have desired amplitude and DOA values are
Estimation accuracy for 4 sources using 1Lshape array.
Scheme 













Desired values  1.0000  3.0000  5.0000  7.0000  30.0000  50.0000  85.0000  70.0000  40.0000  65.0000  255.0000  315.0000 
PSO  0.5211  2.5212  5.4792  7.4789  31.3274  51.3276  86.3282  69.6723  41.3276  66.3278  256.4006  316.4102 
DE  1.2988  2.7009  5.2987  7.2990  29.0324  50.9678  85.9681  70.9679  40.9678  64.0320  256.0673  316.0874 
PSOPS  0.8143  2.8140  5.1858  7.1858  30.5741  50.5745  84.4252  70.5746  39.4252  65.5748  255.6775  315.8776 
DEPS  1.0989  2.9908  4.9010  7.0990  30.2468  50.2469  85.2471  69.7530  40.2470  65.2473  254.7431  315.3571 
Estimation accuracy for 4 sources using 2Lshape.
Scheme 













Desired values  1.0000  3.0000  5.0000  7.0000  30.0000  50.0000  85.0000  70.0000  40.0000  65.0000  255.0000  315.0000 
PSO  0.7202  3.2799  5.2797  7.2711  31.1284  51.1285  86.1286  71.1283  41.1282  66.1284  256.1385  316.1369 
DE  1.0988  3.0990  5.0989  6.9010  30.6656  49.3375  85.6654  70.6655  40.6656  65.6659  254.2342  315.6766 
PSOPS  1.0477  3.0479  4.9522  7.0480  30.2941  50.2942  84.7056  69.7060  40.2941  65.2946  255.2944  315.3041 
DEPS  1.0109  3.0111  5.0100  7.0112  30.0869  50.0870  85.0867  70.0869  40.0870  64.9125  254.9125  314.9082 
In this case, the convergence of each scheme is investigated for 2, 3, and 4 sources. For convergence, the number of times particular schemes able to get the desired goal is analyzed. The number of sensors, values of amplitudes, and DOA are kept the same for this simulation as in Case
Similarly, for 2Lshape array, the DEPS scheme got best convergence, while the second best convergence rate is achieved by PSOPS as shown in Figure
It is evident from Figures
Performance analysis of convergence versus number of sources using 1Lshape array.
Performance analysis of convergence versus number of sources using 2Lshape array.
In this subsection, the proximity effects of elevation and azimuth angles are discussed using DEPS and PSOPS schemes for both Lshape arrays. This experiment is performed for three sources. In Table
In Table
Proximity effect of elevation angle for
Scheme 



% convergence 

Desired values  30.0000  80.0000  50.0000  100 
DEPS (1L)  30.3843  80.3842  50.3844  90 
PSOPS (1L)  30.5132  80.5674  50.6312  88 
DEPS (2L)  30.1791  80.1790  50.1793  94 
PSOPS (2L)  30.3671  80.4101  50.3982  92 


Desired values  30.0000  65.0000  75.0000  100 
DEPS (1L)  30.3846  65.9832  75.9834  84 
PSOPS (1L)  30.5133  66.4378  76.6549  81 
DEPS (2L)  30.1792  65.4301  75.4302  92 
PSOPS (2L)  30.5134  66.1367  76.2136  88 


Desired values  30.0000  40.0000  50.0000  100 
DEPS (1L)  31.3965  41.4011  51.4013  70 
PSOPS (1L)  31.8709  41.9654  51.9987  64 
DEPS (2L)  30.7692  40.7694  50.7690  88 
PSOPS (2L)  31.3245  40.6788  50.9899  82 


Desired values  30.0000  35.0000  40.0000  100 
DEPS (1L)  32.3417  37.3518  42.3519  64 
PSOPS (1L)  34.1303  39.4321  43.9987  56 
DEPS (2L)  31.1105  36.1107  41.1105  82 
PSOPS (2L)  32.7989  37.0121  43.5467  76 
Proximity effect of azimuth angles for
Scheme 



% convergence 

Desired values  15.0000  80.0000  230.0000  100 
DEPS (1L)  15.3841  80.3840  230.3845  90 
PSOPS (1L)  15.8865  80.7985  230.8891  88 
DEPS (2L)  15.1790  80.1791  230.1793  94 
PSOPS (2L)  15.4989  80.5110  230.5478  91 


Desired values  15.0000  80.0000  70.0000  100 
DEPS (1L)  15.3844  80.9830  70.9832  83 
PSOPS (1L)  15.5988  81.7896  71.9989  76 
DEPS (2L)  15.1792  80.4301  70.4302  91 
PSOPS (2L)  15.4989  81.1564  71.1211  87 


Desired values  60.0000  70.0000  80.0000  100 
DEPS (1L)  61.3966  71.4012  81.4014  72 
PSOPS (1L)  62.1456  62.5641  82.4538  65 
DEPS (2L)  60.7694  70.7696  80.7692  86 
PSOPS (2L)  61.5476  71.7873  81.6549  81 


Desired values  60.0000  65.0000  70.0000  100 
DEPS (1L)  62.3419  67.3519  72.3523  66 
PSOPS (1L)  64.8979  70.1653  74.9019  57 
DEPS (2L)  61.1103  66.1105  71.1104  80 
PSOPS (2L)  62.9109  68.9469  73.9381  74 
In this second part of simulations, the comparison of DEPS using 2Lshape array is carried out with PSOPS using 2Lshape array, PM using parallel shape array [
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuth

Mean of 
Variance of 
Standard deviation of 

72°  72.0681  0.7021  0.8379 
76°  74.9583  0.9895  0.9947 
79°  77.8421  1.4082  1.1867 
82°  80.4967  2.3482  1.5323 
86°  83.3760  4.7215  2.1729 
89°  84.7062  9.4567  3.0752 
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuth

Mean of 
Variance of 
Standard deviation of 

72°  72.3528  0.0151  0.1228 
76°  76.2237  0.0111  0.1053 
79°  79.0999  0.0068  0.0825 
82°  82.0998  0.0037  0.0608 
86°  86.0566  0.000632  0.0251 
89°  89.0146  0.000371  0.0192 
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuth

Mean of 
Variance of 
Standard deviation of 

72°  72.1354  0.002363 

76°  76.0003  0.003416 

79°  78.0075  0.008934 

82°  82.0046  0.006984 

86°  86.0097  0.007821 

89°  89.0078  0.008947 

Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuth

Mean of 
Variance of 
Standard deviation of 

72°  71.9998  0.00002415 

76°  76.0003  0.00001871 

79°  78.9996  0.00006854 

82°  81.9998  0.00003278 

86°  86.0006  0.00006132 

89°  89.0008  0.00001371 

In this case, the Root Mean Square Error (RMSE) of DEPS is compared with PSOPS and PM using parallel shape array [
Performance analysis of Root Mean Square Error versus SNR.
In this subcase, the computational complexity of DEPS is compared with PM using Lshape array [
In Table
Comparison among 2Lshape array, parallel shape array [
Property  Parallel shape array  1Lshape array  2Lshape array  2Lshape array 

Scheme used  PM  PM  PM  DEPS 
Range of elevation and azimuth angles  (0°, 90°) and (0°, 360°)  (0°, 90°) and (0°, 360°)  (0°, 180°) and (0°, 360°)  (0°, 180°) and (0°, 360°) 
Number of sensors  15  11  10  4 
Number of snapshots  200  200  200  1 
Pair matching  Required  Not required  Not required  Not required 
Failure estimation  From 70° to 90°  From 0° to 20°  No failure  No failure 
Amplitude estimation  Cannot estimate  Cannot estimate  Cannot estimate  Yes, can estimate 
In this work, an efficient and robust approach has been proposed for joint estimation of amplitude and DOA of farfield sources using Lshape arrays. In this scheme, DE and PSO have been used as global optimizers, which were further assisted by PS as a rapid local search optimizer. We have demonstrated that the hybrid DEPS approach performed considerably well as compared to the PSO, PSOPS, and DE techniques. It has been also shown that all the schemes performed well using 2Lshape array as compared to 1Lshape array. Moreover, the DEPS scheme using 2Lshape array performed well as compared to PM using parallel shape array, as well as Lshape array. The simulation results are in complete conformation of adopted approach.
In future, one can check the same approach for sidelobe reduction and beam steering in adaptive beamforming
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by Higher Education Commission (HEC) of Pakistan under Research Grant no. 21323/SRGP/R&D/HEC/2014.