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

Shah, Fiaz Hussain; Raja, Muhammad Asif Zahoor; Al-Turjman, Fadi; Zaman, Fawad; Yang, Xiaodong

doi:https://doi.org/10.1155/2022/4851364

Wireless Communications and Mobile Computing

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Research Article | Open Access

Volume 2022 | Article ID 4851364 | https://doi.org/10.1155/2022/4851364

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

Fiaz Hussain Shah,¹Muhammad Asif Zahoor Raja,²Fadi Al-Turjman,³Fawad Zaman,⁴and Xiaodong Yang¹

Academic Editor: Muhammad Asghar Khan

Received03 Oct 2021

Revised21 Nov 2021

Accepted06 Apr 2022

Published30 Apr 2022

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.

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 Section 2. 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 with antennas have the angle of elevation between 0 and and azimuth between 0 and for far field sources, while the angle between th antenna element for between 0 and -1 as shown in Figure 1.

Each electric field signal of ^th antenna in UCA from ^th sources is written as where denotes the frequency, be the ^th amplitude and be ^th propagation direction at instance . The relation for in polar coordinates is given as

Then,

Equation (1) becomes for and ; we get

In case of response, of ^th antenna of UCA with noise is given as

The response for single snapshot is given as

In a matrix form, the response of UCA can be written as follows:

In a vector form, equation (8) is given as

Here, is a steering matrix for the source signals, a matrix for amplitude and noise signal is denoted by .

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 Figure 2.

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

The fitness function is developed for 2D-DOA estimation of plane waver form sources impinging on circulate array compose of antenna elements by the proficiency of approximation theory in mean square error sense as follows: where

Here, is the desired response or signal in case of single snapshot as given equation (7), while is an approximate signal of . Now, one has to find the appropriate weights as such that ; then accordingly, .

3.2. Optimization of 2D-DOA Parameters

To find the unknown adjustable weights , 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].

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].

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 Tables 1 and 2, respectively.

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 with sources and antenna elements on UCL is taken with settings of elevation and azimuth angles as follows: while the values of amplitude are . The fitness function for case 1 with is formulated as follows:

Case 2. In this case, 2D-DOA estimation problem with sources and antenna elements on UCL is taken with settings of elevation and azimuth angles in degree or radian as follows: while the values of amplitude are . Using equation (14) for , the fitness function for case 2 is constructed.

Case 3. In this scenario, 2D-DOA estimation problem with sources and antenna elements on UCL is taken with settings of elevation and azimuth angles as follows: while the values of amplitude are . Using equation (7) for , 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 Figures 5 and 6 in case of 2 and 3 far-field sources impinging on UCL.

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 Tables 3–5, respectively.

While in case of different noise levels, results of proposed computing paradigm are presented in Tables 6–8 for sources , 3, and 4, respectively. The values of fitness 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.

One 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 Figure 7 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.

The analysis is further conducted with the increase in the number of antenna elements in UCA, i.e., value of . The results of convergence analysis of 2D-DOA estimation for , 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 , but the performance of hybridized approach GA-PS is better from the rest.

Robustness 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 Figures 8–10 for sources , 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.

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 Tables 13–15 for sources , 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.

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 sources , 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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported in part by the Fundamental Research Funds for the Central Universities (No. JB180205).

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Copyright © 2022 Fiaz Hussain Shah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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