# Evolutionary Heuristic Computing Paradigm for 2D-DOA Estimation along Circular Array

Authors: Fiaz Hussain Shah; Muhammad Asif Zahoor Raja; Fadi Al-Turjman; Fawad Zaman; Xiaodong Yang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Direction of arrival (DOA) estimation problem has growing interest for the researcher investigating in system identification models arising in the field of digital signal processing, mobile communication, controls, and beamforming. In the presented work, evolutionary heuristic computing paradigm is presented for 2D-DOA estimation of plane waves impinging on uniform circular array. Performance metric of mean squared error is utilized as construction of a fitness function for the system, and the optimization strength of three methodologies, genetic algorithms (GAs), Pattern Search (PS), and integration of GAs with PS (GA-PS) is exploited for 2D-DOA estimation based on elevation as well as azimuth angles. Consistent precision, convergence, stability, and robustness of integrated heuristics of GA-PS are endorsed through outcomes of statistical observations.


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## Body

## 1. Introduction

The use of different antenna structural arrays has growing interest in researchers due to remarkable performance in the domain of direction-of-arrival (DOA) parameter estimation, beamforming, radars, sonars, and seismology. Researchers proposed different DOA estimation procedures including MUSIC [1, 2], ESPRIT [3, 4] spatial smoothing methods [5, 6], subspace smoothing procedures [7], and temporal smoothing approach [8] for uniform linear array (ULA). The two-dimensional (2D) DOA (2D-DOA) estimation one preferred to used two-dimensional arrays based on L-shaped array [9, 10], nested array [11–13], coprime array [14, 15], uniform rectangle array (URA) [16, 17], uniform circular array (UCA) [18, 19], virtual uniform-linear-like array (VULA) [20], and visual array VT-MUSIC algorithm [21]. The transformation procedure also exploited for two-dimensional DOA (2D-DOA) estimation algorithms with relatively low computational requirement including UCA-RB-MUSIC [22], UCA-ESPRIT [23], UCA rank reduction [24], and root-MUSIC approach [25]. Beside these deterministic techniques for 1D and 2D DOA estimation, the stochastic procedures are also adopted for these global search based optimization problems [26–29]. All the existing procedures adopted for system identification of DOA models motivate authors to investigate stochastic optimization mechanism by exploitation of evolutionary heuristics for joint estimation of two-dimensional DOA parameters impinging on circular structural array of far field sources.The stochastic optimization mechanism by exploitation of artificial intelligence techniques has been implemented extensively to address constrained and unconstrained optimization model associated with a variety of linear/nonlinear systems [30–33]. Few prevailing recent applications include Hammerstein nonlinear control autoregressive systems, active noise control system, transport model for soft tissues, nonlinear optics, nonlinear Bratu systems, nonlinear fractional Riccati systems, nonlinear Jeffery-Hamel flow, nonlinear prey-predator, nonlinear thin film flow models, nonlinear FalknarSkan system, nonlinear Troesch problem, nonlinear singular Lane-Emden systems, nonlinear Thomas-Fermi model of atom, piezoelectric model, magneto-hydrodynamics, astrophysics, atomic physics, plasma physics, control, signal processing, energy, bioinformatics, economics, and finance (see references [34–36] and citation therein). These are source of incitements for authors to perform exploration and exploitation in evolutionary computational heuristic paradigm reliable treatment of 2-D DOA estimation of plane waves impinging of UCA.In this paper, stochastic optimization solvers are presented for 2D-DOA estimation impinging on UCA from far field sources. The salient features of the scheme are highlighted as follows:A novel application of evolutionary computational heuristic paradigm is presented for two-dimensional DOA estimation of far field sources involving uniform circular array by exploitation of global search efficacy of genetic algorithms (GAs), pattern search (PS), and integrated strength of GA-PS algorithms.The performance of optimization mechanisms is substantiated by effective implementation of uniform circular array-based DOA estimation problem having different degrees of freedom. The results of integrated solver GA-PS are relatively better from standalone counterparts GA and PS for each scenario of the data model for DOA.Consistent accuracy and convergence of the hybrid optimization scheme GA-PS are endorsed through outcomes of statistical observations for DOA problems with different numbers of decision variables and noise variations.Organization of remaining of the paper is as follows. The data model for two-dimensional DOA estimation problem with uniform circular array geometry is presented in Section2. Optimization methodology of all three algorithms is described in Section 3. The results of simulations through enough graphical and numerical illustrations with necessary interpretations are presented in Section 4. While the conclusions and further work are provided in last Section 5.


## 2. Data Model: 2D-DOA Estimation with UCA

When the data or system model for 2D-DOA estimation of plane waves impinging on UCA is presented here, the UCA withM antennas have the angle of elevation α between 0 and π/2 and azimuth β between 0 and 2π for P far field sources, while the angle γm=2πm/M between mth antenna element for m between 0 and M-1 as shown in Figure 1.Figure 1
UCA geometry for plane waves.Each electric field signal ofmth antenna in UCA from ith sources is written as
(1)Erm,t=aitejωt−di.rm,forrm=xm,ym,zm=rsinβm,rcosβm,0,where ω denotes the frequency, ai be the ith amplitude and di be ith propagation direction at instance t. The relation for di in polar coordinates is given as
(2)di=1csinαicosβi,sinαisinβi,cosαi.Then,(3)di⋅rm=rcsinαicosβi−γm.Equation (1) becomes
(4)Erm,t=aitejωt−r/csinαicosβi−γn=aitejωte−jωr/csinαicosβi−γm,for Ait=aitejωt and k=ω/c=2π/λ; we get
(5)Erm,t=Aite−jkrm−1sinαicosβi−γm.In case of response,xmt=Erm,t of mth antenna of UCA with noise nmt is given as
(6)xmt=∑i=1Paite−jkm−1rsinαicosβi−2m−1π/m+nmt.The response for single snapshot is given as(7)xm=∑i=1Paie−jkm−1rsinαicosβi−2m−1π/m+nm.In a matrix form, the response of UCA can be written as follows:(8)x1x2⋮xM=11⋯1e−jkrsinα1cosβ1−πe−jkrsinα2cosβ2−π⋯e−jkrsinαpcosβp−π⋮⋮⋮⋮−jkrm−1sinα1cosβ1−2πm−1me−jkrm−1sinα1cosβ2−2πm−1m⋯e−jkrsinαpcosβp−2πm−1ma1a2⋮ap+n1n2⋮nm.In a vector form, equation (8) is given as
(9)x=As+n.Here,s is a steering matrix for the source signals, a matrix A for amplitude and noise signal is denoted by n.


## 3. Design Methodology

The design methodology consisted of two parts: a fitness function formulation for 2D-DOA parameter estimation and its optimization with the help of genetic algorithms (GA), pattern search (PS) and integrated approach GA-PS. The generic flow diagram of the proposed methodology is shown in Figure2.Figure 2
Block structure representation of workflow of the system.


### 3.1. Fitness Function of 2D-DOA Estimation of UCA

The fitness function is developed for 2D-DOA estimation of plane waver formP sources impinging on circulate array compose of M antenna elements by the proficiency of approximation theory in mean square error sense as follows:
(10)ε=1M∑m=1Mxm−x∧m2,where
(11)xm=∑i=1Paie−jkm−1rsinαicosβi−2m−1π/m+nm,x^m=∑i=1Paie−jkm−1rsinα∧icosβ∧i−2m−1π/m.Here,xm is the desired response or signal in case of single snapshot as given equation (7), while x^m is an approximate signal of xm. Now, one has to find the appropriate weights as
(12)W=α1,α2,⋯,αP,β1,β2,⋯,βP,such that ε⟶0; then accordingly, x^m⟶xm.


### 3.2. Optimization of 2D-DOA Parameters

To find the unknown adjustable weightsW, 2D-DOA parameter estimation, the standalone optimization strength of GAs, and PS method, as well as hybrid computing heuristics of GA-PS algorithm, are exploited.GA is developed on mathematical modeling of natural genetic mechanism in human genetic system and the first renewed application introduced by Onnen [37] in early seventies of the last century. GAs work through its fundamental operators of selection, crossover, and mutation for reproduction of the new population of candidate solution at each step increment in generations. The generic workflow of GAs operations is illustrated in Figure 3, while further necessary details of processing blocks can be seen in [38, 39]. Many constrained and unconstrained nonlinear optimization problems are effectively addressed with competency of GAs such as optimization in filter designing [40], life prediction of supercapacitors [41], salesman problem [42], multiaccess edge computing [43], and multiobjective optimization [44].Figure 3
Optimization cycle of genetic algorithms.A pattern search algorithm belongs to the class of derivative free algorithm used broadly by the researchers for viable solution of constrained and unconstrained optimization tasks [45, 46]. The generic workflow diagram of PS by means of process block structures is shown in Figure 4, while broad recent applications of PS in different fields of science and engineering include the design of PID controller [47], automotive safety [48], and health monitoring [49].Figure 4
Process flow diagram of PS algorithm.In the presented study, standalone and combine strength of both optimization algorithms based on GAs, PS, and GA-PS are used for 2D-DOA estimation of plane waves. The built-in routines are invoked for both GAs and PS methods using the optimization toolbox of MATLAB software with setting of GAs and PS tools as provided in Tables1 and 2, respectively.Table 1
Settings for gas optimization tools.
IndexSetIndividuals in population360Total number of generations1000Crossover schemeHeuristic routineFraction of crossover0.2Function tolerance10-09Range initialization[0-2π]Selection schemeRoutine of stochastic uniformScaling procedureRankingElitism2 countsMutation schemeRoutine of adaptive feasibleOtherDefaultsTable 2
Settings for PS optimization tools.
IndexSetIterations2000Penalty100Polling schemeGPS basis on 2 NPoll orderingConsecutiveMesh size02Evaluation of fitness200000Expansion parameter2Contraction parameter0.5TolX0TolBind0TolMesh10-09


## 3.1. Fitness Function of 2D-DOA Estimation of UCA

The fitness function is developed for 2D-DOA estimation of plane waver formP sources impinging on circulate array compose of M antenna elements by the proficiency of approximation theory in mean square error sense as follows:
(10)ε=1M∑m=1Mxm−x∧m2,where
(11)xm=∑i=1Paie−jkm−1rsinαicosβi−2m−1π/m+nm,x^m=∑i=1Paie−jkm−1rsinα∧icosβ∧i−2m−1π/m.Here,xm is the desired response or signal in case of single snapshot as given equation (7), while x^m is an approximate signal of xm. Now, one has to find the appropriate weights as
(12)W=α1,α2,⋯,αP,β1,β2,⋯,βP,such that ε⟶0; then accordingly, x^m⟶xm.


## 3.2. Optimization of 2D-DOA Parameters

To find the unknown adjustable weightsW, 2D-DOA parameter estimation, the standalone optimization strength of GAs, and PS method, as well as hybrid computing heuristics of GA-PS algorithm, are exploited.GA is developed on mathematical modeling of natural genetic mechanism in human genetic system and the first renewed application introduced by Onnen [37] in early seventies of the last century. GAs work through its fundamental operators of selection, crossover, and mutation for reproduction of the new population of candidate solution at each step increment in generations. The generic workflow of GAs operations is illustrated in Figure 3, while further necessary details of processing blocks can be seen in [38, 39]. Many constrained and unconstrained nonlinear optimization problems are effectively addressed with competency of GAs such as optimization in filter designing [40], life prediction of supercapacitors [41], salesman problem [42], multiaccess edge computing [43], and multiobjective optimization [44].Figure 3
Optimization cycle of genetic algorithms.A pattern search algorithm belongs to the class of derivative free algorithm used broadly by the researchers for viable solution of constrained and unconstrained optimization tasks [45, 46]. The generic workflow diagram of PS by means of process block structures is shown in Figure 4, while broad recent applications of PS in different fields of science and engineering include the design of PID controller [47], automotive safety [48], and health monitoring [49].Figure 4
Process flow diagram of PS algorithm.In the presented study, standalone and combine strength of both optimization algorithms based on GAs, PS, and GA-PS are used for 2D-DOA estimation of plane waves. The built-in routines are invoked for both GAs and PS methods using the optimization toolbox of MATLAB software with setting of GAs and PS tools as provided in Tables1 and 2, respectively.Table 1
Settings for gas optimization tools.
IndexSetIndividuals in population360Total number of generations1000Crossover schemeHeuristic routineFraction of crossover0.2Function tolerance10-09Range initialization[0-2π]Selection schemeRoutine of stochastic uniformScaling procedureRankingElitism2 countsMutation schemeRoutine of adaptive feasibleOtherDefaultsTable 2
Settings for PS optimization tools.
IndexSetIterations2000Penalty100Polling schemeGPS basis on 2 NPoll orderingConsecutiveMesh size02Evaluation of fitness200000Expansion parameter2Contraction parameter0.5TolX0TolBind0TolMesh10-09


## 4. Simulations and Results

In this section, results with interpretations are presented for abundant experimentation to test, analyze, and compare the outcomes of GAs PS and GA-PS based on the proposed methodologies. Results are presented throughout in this study based on average of 100 independent trials.To evaluate the performance of GAs, PS, and their integrated scheme GA-PS, three case studies are taken based on 2, 3, and 4 plane wave sources impinging of UCA as follows:Case 1.
In the said scenario, 2D-DOA estimation problem withP=2 sources and M=6 antenna elements on UCL is taken with settings of elevation α1 and azimuth β1 angles as follows:
(13)W=α1,α2,β1,β2=20∘,70∘,110∘,150∘degrees,0.3491,1.2217,1.9199,2.6180radians,while the values of amplitude are a=a1,a2=1,3. The fitness function for case 1 with P=2andM=6 is formulated as follows:
(14)ε=16∑m=16xm−x∧m2,xm=∑i=12aie−jkm−1rsinαicosβi−2m−1π/m+nm,x^m=∑i=12aie−jkm−1rsinα∧icosβ∧i−2m−1π/m.Case 2.
In this case, 2D-DOA estimation problem withP=3 sources and M=9 antenna elements on UCL is taken with settings of elevation α1 and azimuth β1 angles in degree or radian as follows:
(15)W=α1,α2,α3,β1,β2,β3=20∘,40∘,50∘,110∘,120∘,125∘degrees,0.3491,0.6980,0.8727,1.9190,2.0944,2.1817radians,while the values of amplitude are a=a1,a2,a3=1,4,7. Using equation (14) for P=3andM=9, the fitness function for case 2 is constructed.Case 3.
In this scenario, 2D-DOA estimation problem withP=4 sources and M=8 antenna elements on UCL is taken with settings of elevation α and azimuth β angles as follows:
(16)W=α1,α2,α3,β1,β2,β3=20∘,40∘,50∘,110∘,120∘,125∘degrees,0.3491,0.6980,0.8727,1.9190,2.0944,2.1817radians,while the values of amplitude are a=a1,a2,a3,a3=1. Using equation (7) for P=4andM=8, the fitness function for case 3 is constructed.Results are determined for 100 independent trials of the algorithms to pinpoint their performance for two, three, and four far field sources. The optimization characteristics of GAs for 2 and 3 source models in terms of learning curves, best individual, fitness of each individual in the population, and stoppage criteria are shown in Figures5 and 6 in case of 2 and 3 far-field sources impinging on UCL.Figure 5
Learning curve with global best individual, fitness of each individual, and stopping conditions of GAs for case 1 of 2D-DOA estimation.Figure 6
Learning curve with the best global individual, fitness of each individual, and stopping conditions of GAs for case 2 of 2D-DOA estimation.The results of all three algorithms GAs, PS, and GA-PS against the true parameters of 2, 3 and 4 far-field sources for noiseless environment are presented in Tables3–5, respectively.Table 3
Comparison of results for 2D-DOA model for 2 far field sources.
IndexDesire and Approximated parametersα1α2β1β2True (deg)20.000070.0000110.0000150.0000True (rad)0.34911.22171.91992.6180GAs (deg)19.996269.9925109.9964149.9946GAs(rad)0.34901.22161.91982.6179PS (deg)19.990569.9868109.9850149.9889PS (rad)0.34891.22151.91962.6178GA-PS (deg)20.002069.9983110.0022150.0004GA-PS (rad)0.34911.22171.91992.6180Table 4
Comparison of results for 2D-DOA model for 3 far field sources.
IndexParametersα1α2α3β1β2β3True (deg)20.000040.000050.0000110.000120.000125.000True (rad)0.34910.69800.87271.91902.09442.1817GAs (deg)19.996239.992549.9791109.6469119.9946124.9850GAs (rad)0.34900.69800.87231.91372.09432.1814PS (deg)19.976739.964949.9475109.4138119.9348124.9290PS (rad)0.34880.69780.87211.91042.09412.1813GA-PS (deg)20.002039.992550.0020109.9506120.0003125.0022GA-PS (rad)0.34910.69800.87271.91902.09442.1817Table 5
Comparison of results for 2D-DOA model for 4 far field sources.
IndexDesire and Approximated parametersElevation angleAzimuth angleα1α2α1α2β1β1β1β1True (deg)20.00054.00062.00035.000105.000133.000125.000166.000True (rad)0.34910.94241.08210.61081.83252.32122.18162.8972Gas (deg)19.93953.96761.96034.996104.966132.978124.979165.974Gas (rad)0.34800.94191.08140.61081.83202.32092.18132.8968PS (deg)19.93353.95061.97734.996104.960132.972124.968165.951PS (rad)0.34790.94161.08170.61081.83192.32082.18112.8964GA-PS (deg)19.99053.97861.99434.996104.983132.984124.991165.986GA-PS (rad)0.34890.94211.08200.61081.83232.32102.18152.8970While in case of different noise levels, results of proposed computing paradigm are presented in Tables6–8 for sources P=2, 3, and 4, respectively. The values of fitness ε and complexity parameters are time consumed, generations/iterations (Gens/iter) executed, and fitness function counts (FCs) by the optimization strategy for finding the decision variables.Table 6
Comparative study 2D-DOA model for 2 far field sources.
IndexNoise (dB)εTimeGens/iterFCGAsNil4.03E−2543.121000360360303.39E−2243.041000360360255.61E−1943.921000360360203.79E−1844.141000360360156.96E−1844.691000360360104.19E−1745.01100036036059.68E−1648.501000360360PSNil7.26E−213.79200023507301.34E−213.80200023507254.79E−193.93200023231206.26E−183.83200023145155.58E−163.85200023468104.63E−153.8520002327751.89E−143.86200023601GA-PSNil1.00E−0046.703000383867301.00E−3446.633000380576256.42E−3147.533000383591201.71E−3047.753000381109156.23E−2848.313000381010102.54E−2648.64300038135951.85E−2652.163000381469Table 7
Comparative study 2D-DOA model for 3 far field sources.
IndexNoise (dB)εTimeGens/iterFCGAsNil1E−2155.11000360360302E−2056.41000360360253E−1956.71000360360205E−1858.31000360360158E−1759.41000360360109E−1660.1100036036059E−1462.31000360360PSNil7.E−155.4200035660301E−155.7200035876254.E−146.4200035897207E−126.7200035876154E−106.7200035879102E−86.820003595652E−66.9200035989GA-PSNil2E−2860.203000386246302E−2861.803000386281255E−2662.303000386732202E−2464.103000387149156E−2465.303000387257102E−2266.00300038748552E−2068.403000387681Table 8
Comparative study 2D-DOA model for 4 far field sources.
IndexNoise (dB)εTimeGens/iterFCGAsNil6.3e−1266.51000360360301.2e−0966.81000360360256.7e−0969.81000360360208.3e−0869.31000360360158.9e−0871.11000360360101.8e−0772.3100036036051.9e−0672.51000360360PSNil3.2e−097.2200035660302.7e−087.7200035876257.3e−087.8200035897202.9e−067.9200035876151.8e−068.6200035879101.9e−048.720003595658.2e−048.9200035989GA-PSNil8.3e−1673.203000386246301.2e−1473.603000386281258.7e−1476.603000386732204.3e−1276.203000387149156.9e−1278.203000387257101.8e−1079.50300038748558.9e−0980.003000388681One may observe that all three methods attained reasonably well levels of estimation accuracy; however, the results of integrated computing heuristics of GA-PS are more precise than those of GAs and PS standalone solvers. The performance of integrated algorithm GA-PS at the expense of relatively more computations is better than that of standalone schemes. Additionally, the increase in the number of sources and the level of noise variances results are deteriorated for each computing algorithm GAs, PS, and GA-PS, but still, the hybrid GA-PS achieved better reasonable precision than that of standalone counterparts.The convergence analysis is also conducted for all three optimization solvers GAs, PS and GA-PS for solving 2D-DOA estimation problems are based on 100 trails, and results are presented in Figure7 and Table 9 for each case study. One may see that percentage convergence of integrated heuristic of GA-PS algorithm is higher from standalone methodologies and performance of each optimization solver degraded with increase in sources from 2 to 4.Figure 7
Bar chart illustration of convergence analysis for sourcesP=2, 3, and 4.Table 9
Convergence analysis of 2D-DOA estimation for 2, 3, and 4 far field sources with fixed antenna elements in UCL.
Source/antennaMethodεConvergence2/4Gas1.0E−2594%PS1.0E−2144%GA-PS1.0E−3198%3/6GAs1.0E−2169%PS1.0E−1531%GA-PS1.0E−28876%4/8GAs1.0E−0619%PS1.0E−0509%GA-PS1.0E−1736%The analysis is further conducted with the increase in the number of antenna elements in UCA, i.e., value ofM. The results of convergence analysis of 2D-DOA estimation for P=2andM=6, 8, and 10 in UCL are presented in Table 10 along with achieved fitness level for all three optimization schemes. Accordingly, the results of convergence analysis of 2D-DOA estimation for 3 and 4 far-field sources with different antenna elements in UCA are presented in Tables 11 and 12, respectively. It is seen that rate of convergence for each algorithm increases with the increase in the value of M, but the performance of hybridized approach GA-PS is better from the rest.Table 10
Convergence analysis of 2D-DOA estimation for 2 far field sources with different antenna elements in UCL.
Antenna UCAMethodεConvergence6GAs1.0E−2595%PS1.0E−2145%GA-PS1.0E−3199%8GAs1.0E−2796%PS1.0E−2252%GA-PS1.0E−3299%10GAs1.0E−2898%PS1.0E−2560%GA-PS1.0E−00100%Table 11
Convergence analysis of 2D-DOA estimation for 3 far field sources with different antenna elements in UCL.
Antenna UCAMethodεConvergence9GAs1.0E−2169%PS1.0E−1529%GA-PS1.0E−2874%11GAs1.0E−2274%PS1.0E−1639%GA-PS1.0E−2982%13GAs1.0E−2277%PS1.0E−1743%GA-PS1.0E−3085%Table 12
Convergence analysis of 2D-DOA estimation for 4 far field sources with different antenna elements in UCL.
Antenna UCASchemeεConvergence (%)8GA1.0E−0620PS1.0E−0310GA-PS1.0E−083510GA1.0E−0726PS1.0E−0418GA-PS1.0E−094512GA1.0E−0730PS1.0E−0525GA-PS1.0E−1055Robustness analysis of all three optimization methodologies is conducted for different values of signal to noise (SNR), i.e., 5 dB, 10 dB, 15 dB, 20 dB, 25 dB, and 30Db for 2D-DOA estimation of 2, 3, and 4 sources. The results of robustness analysis each algorithm for different noise variation are presented in Figures8–10 for sources P=2, 3, and 4, respectively. One may see that for both low and high values of SNR, the performance of hybridized computing solver GA-PS remains better than that of GAs and PS standalone schemes.Figure 8
Comparison of fitness for different SNR levels for sourcesP=2.Figure 9
Comparison of fitness for different SNR levels for sourcesP=3.Figure 10
Comparison of fitness for different SNR levels for sourcesP=4.The complexity analysis in terms of minimum-time, maximum-time, mean-time, minimum-FCs, maximum-FCs, and mean-FCs along with the values of best-fitness, worst-fitness and mean-fitness is conducted for 100 executions of each optimization scheme GAs, PS, and GA-PS for all three 2D-DOA scenarios. Results of complexity operators are listed in Tables13–15 for sources P=2, 3, and 4, respectively. It is seen that computation complexity of PS is superior from GA and GA-PS technique but the performance in terms of accuracy and convergence is better for both GAs and GA-PS methodologies for each case.Table 13
Performance comparison on statistics for 2 sources based on 2D-DOA estimation.
IndexGAsPSGA-PSBest-fitness1.0E−251.0E−231.0E−00Worst-fitness1.0E−211.0E−171.0E−28Mean-fitness1.0E−231.0E−211.0E−30Minimum-FCs3603602054523147Maximum-FCs3603602132423986Mean-FCs3603602077723327Minimum-time42.523.6946.10Maximum-time44.275.0148.02Mean-time43.283.9147.00Table 14
Performance comparison on statistics for 3 sources based on 2D-DOA estimation.
IndexGAsPSGA-PSBest-fitness1.0E−171.0E−12−121.0E−31Worst-fitness1.0E−131.0E−021.0E−18Mean-fitness1.0E−151.0E−061.0E−28Minimum-FCs3603602546725538Maximum-FCs3603602858029614Mean-FCs3603602633926598Minimum-time53.094.7158.20Maximum-time61.556.2567.62Mean-time56.805.0961.88Table 15
Performance comparison on statistics for 4 sources based on 2D-DOA estimation.
IndexGAsPSGA-PSBest-fitness1.0E−121.0E−091.0E−16Worst-fitness1.0E−041.0E−011.0E−05Mean-fitness1.0E−111.0E−041.0E−15Minimum-FCs3603602534725538Maximum-FCs3603602872029614Mean-FCs3603602675926598Minimum-time73.019.1279.11Maximum-time69.557.5176.45Mean-time72.017.8378.71


## 5. Conclusion

Novel applications of evolutionary heuristics are effectively presented for 2D-DOA estimation of plane waves impinging on UCL by exploitation of global search efficacy of GAs, efficiency of PS, and integrated optimization strength of GA-PS. The performance of optimization mechanisms is verified by implementation of UCA-based 2D-DOA estimation having different degrees of freedom, i.e., far field sourcesP=2, 3, and 4. The results of integrated solver GA-PS are relatively better from standalone counterparts GA and PS for each scenario of the data model for DOA. Consistent accuracy, stability, and robustness of the hybrid optimization procedure of GA-PS are established through outcomes of statistical observations for DOA problems with different numbers of decision variables and noise variations but at the cost of relative more computations that that of GAs and PS standalone schemes.In the future, one may investigate the application of presented computing platform on other circular array structures based on concentric circular array, conic circular array, and coprime circular array for better estimation accuracy of DOA parameters. Moreover, the use of fractional evolutionary/swarming techniques looks promising for the estimation of 2D-DOA parameters more viably.



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