# High-Resolution Direction of Arrival Estimation of Underwater Multitargets Using Swarming Intelligence of Flower Pollination Heuristics

Authors: Nauman Ahmed; Huigang Wang; Shanshan Tu; Norah A.M. Alsaif; Muhammad Asif Zahoor Raja; Muhammad Kashif; Ammar Armghan; Yasser S. Abdalla; Wasiq Ali; Farman Ali
Journal: Shock and Vibration
 (2022)

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## Abstract

Developing the parameter estimation, particularly direction of arrival (DOA), utilizing the swarming intelligence-based flower pollination algorithm (FPA) is considered an optimistic solution. Therefore, in this paper, the features of FPA are applied for viable DOA in the case of several robust underwater scenarios. Moreover, acoustic waves impinging from the far-field multitarget are evaluated using the different number of hydrophones of uniform linear array (ULA). The measuring parameters like robustness against noise and element quantity, estimation accuracy, computation complexity, various numbers of hydrophones, variability analysis, frequency distribution and cumulative distribution function of root mean square error (RMSE), and resolution ability are applied for analyzing the performance of the proposed model with additive white Gaussian noise (AWGN). For this purpose, particle swarm optimization (PSO), minimum variance distortion-less response (MVDR), multiple signal classification (MUSIC), and estimation of signal parameter via rotational invariance technique (ESPRIT) standard counterparts are employed along with Crammer–Rao bound (CRB) to improve the worth of the proposed setup further. The proposed scheme for estimating the DOA generates efficient outcomes compared to the state-of-the-art algorithms over the Monte Carlo simulations.


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## Body

## 1. Introduction

Swarming intelligence of evolutionary algorithms is a significant development in signal processing for direction of arrival (DOA) estimation of underwater multitargets [1, 2]. For solving such problems, subspace-based methods were used like multiple signal classification (MUSIC) [3]. If used in a constraint environment like assuming incoherent sources, with a specified number of snapshots and high SNR signals, these methods give good results. But due to these limitations, it can only be applied to very few scenarios and problems. Parametric methods like maximum likelihood (ML) [4, 5] are also used to address such issues, but they are computationally complex, which impedes the fertility of application. Previously, most of the published research validated the performance of DOA estimators using asymptotic assumptions, which requires either a high signal-to-noise ratio or a more significant number of samples, which are not specific in many real-life problems [6, 7]. DOA estimator accuracy is dependent on the signal power and the rate of transmission, which are beyond the control of the system designer [8]. Therefore, such systems operate in a low SNR, estimating accurate angular localization, a challenging task [9]. New intelligent optimization algorithms have been proposed recently for estimation of DOA like genetic algorithm (GA) [9], differential evolution (DE) method [8], particle swarm optimization (PSO), seeker optimization algorithm (SOA), sine cosine algorithm (SCA), invasive weed optimization (IWO) [10], and squirrel search algorithm (SSA) [11]. In [12], GA is proposed with accurate and reliable results for the estimation of the parameters of DOA problems, and the performances of the GA, ML, and MUSIC algorithm have been compared with different variants of SNR, computational cost, and the number of snapshots. The detection probability is modeled in [13], through the active sonar equation for probability hypothesis density (PHD) and cardinalized PHD (CPHD). Novel complexity measure is proposed in [14] for improving permutation entropy (PE) and analyzing the time series. Similarly, missing amplitude information and single scale problem in PE are addressed in [15] through refined composite multiscale reverse weighted PE (RCMRWE). A modified version of the GA is applied to the nonlinear and highly nonlinear function to estimate the parameters of DOA as presented in [16]. In [17, 18], PSO algorithm and pattern search algorithm were developed to estimate the parameters of the multimodal function. In [19], the PSO ML estimator shows very healthy and reliable results as compared to conventional parameter estimation techniques for DOA. Using the ant colony optimization (ACO) by extending the pheromone, DOA parameters are estimated in [20], with outstanding results and low computational complexity. In [21], the artificial bee colony (ABC) algorithm is used to achieve higher statistical performance. A high degree of freedom for DOA is studied in [22] using Cuckoo search algorithm. The analytical model was discussed for the proposed approach in terms of fitness function, SNR, and cumulative distributive function. In [23], the authors have proposed adaptive FPA mechanism in order to localize the nodes in wireless sensor networks. Back in 2019, a Squirrel Search Algorithm (SSA) was proposed, which is a novel numerical optimization algorithm. It focuses on the foraging and gliding behavior of flying squirrels to determine their efficient way of locomotion. Gliding is a powerful technique used by small mammals for traveling long distances. The present work mathematically models this behavior to realize the process of optimization. These features may be helpful to improve convergence and reduce the number of iterations of the SSA algorithm to determine the ML DOA estimate [24].In this study, optimization strength of nature-inspired heuristics of flower pollination algorithm (FPA) is exploited for possible DOA estimation in case of different scenarios of the underwater environment using a uniform linear array (ULA) of hydrophones for influencing acoustic waves from far-field multitargets. The high resolution for close space targets is achieved using fewer snapshots viably with FPA by investigating the global minima of the highly nonlinear cost function of ULA with multiple local minima. Performance analysis is conducted for different number of targets employing estimation accuracy, robustness against noise, and number of hydrophones in the presence of additive white Gaussian measurement noise, and comparative studies with MVDR, MUSIC, Root MUSIC, and ESPRIT counterparts along with Crammer–Rao bound analysis reveals the worth of the scheme for estimating DOA parameters, which are further endorsed from the results of Monte Carlo simulations.The rest of the paper is arranged as follows: Section2 defines the mathematical model for ULA. The conventional beamforming (CBF) algorithm, MVDR, MUSIC, Root MUSIC, and ESPRIT are explained with their procedure for the DOA problem in Section 3. Performance analysis of algorithms concerning RMSE is illustrated in Section 4. Finally, Section 5 explains the main contributions of the proposed study.


## 2. Mathematical Model

In this study, the ULA of hydrophones is used for 1D-DOA estimation. So, according to the characteristics of ULA, the impinged plane waves from the far-field region are phase-shifted versions of consecutive hydrophones as explained in Figure1. The angle of arrival [25, 26] can be denoted as(1)θ=θ1,θ2,θ3,…,θD.Here, θD is the associated angle to the Dth acoustic source.(2)zt=z1t,z2t,…,zWt,gθi=1,e−jkdsinθi,e−jk2dsinθi,…,e−jkW−1dsinθiT,Gθ=gθ1,gθ2,…,gθD.Figure 1
DOA estimation model.Here,k=2π/λ, while λ is the wave length. zt is the hydrophone’s output vector with dimension W × 1 and can be known as array response.(3)zt=Gθyt+vt.The steering matrixG of dimension W × D comprises the time delay entities of signals for each hydrophone. Here, vt is additive white Gaussian noise of zero mean with a dimension of W × 1. The covariance matrix [27, 28] is defined as(4)R=EztzHt.The previous equation can be written when a finite number of snapshots are available:(5)R=1K∑n=1KznzHn,where E. and .H are ensemble average and Hermitian operators, respectively. So the correlation matrix [29, 30] can be written as(6)R=GRsGH+σ2ID.where Ry is the correlation associated with signal and σ2ID is the noise correlation matrix.


## 3. DOA Estimation

Generally, DOA estimation algorithms are divided into two categories, i.e., CBF techniques and subspace-based techniques. In this work, the performance analysis has been taken for both the CBF and subspace-based algorithms under varying noise levels for different acoustic sources.


### 3.1. Particle Swarm Optimization

The heuristics of PSO was proposed by Kennedy and Eberhart having motivation from the pool of birds congregating for food in a random manner [31]. The idea of seeking food is a heuristic approach because all the birds have the information of distance but are not familiar with the explicit location of food. They seek the food by exchanging their search information via crossover and kid production method. The PSO is introduced for the pool of applications almost in every walk of engineering [32, 33]. In this work, PSO performs searching via a swarm of particles that updates recursively. To approach the optimal solution, each particle (DOAs) moves in the direction to its previously best (pbest) position and the global best (gbest) position in the swarm.(7)vin+1=ωvin+c1r1θin−θpbest+c2r2θin−θgbest.(8)θin+1=θin+vin+1.Here, θpbest and θgbest can be calculated as(9)θpbesti=argmini=1,…,Kfθni,θgbesti=argminn=1,…,Nfθni.Here,i denotes the particle index, n is the current iteration number, and fθni is the fitness function that can be defined as(10)fθe=zθet−zθat2.And, the parametersω,c1,c2,r1, and r2 are inertia weight, two positive constants, and two random parameters within [0, 1], respectively. The velocity v and positions (DOAs) of particles are updated with equations (9) and (10).


### 3.2. Basis Principle of FPA

The features of pollination scheme, flower reliability, and behavior of the pollinator can be analyzed efficiently by the following principles:(1)
The global pollination scheme consists of cross-pollination and biotic methods, while pollinators perform levy flights with pollen (global optimization)(2)
The local pollination mechanism consists of self-pollination and abiotic methods (local optimization) [34](3)
A switch probabilitypϵ0,1 is designed to control global pollination and local pollination schemesIn the complete pollination process, the local pollination scheme can experience a large quantity of fractionp. Its reason can be environmental factors like wind and physical proximity. Generally, many flowers can grow on a plant, and pollen gametes from each flower can be released in billions. However, here we suppose that every plant has the ability to harvest only one flower and only one pollen gamete can be produced from each flower. Therefore, plant, flower, and pollen gamete are easy to identify for finding the solution to a problem. The above assumption develops the most straightforward way that solution can be equal to a pollen gamete and a flower. In future research, especially for multiobjective optimization problems, different numbers of flowers can be associated with each plant and multiple pollen gametes can be assigned to each flower. A flower-based algorithm, known as flower pollination algorithm (FPA), can be designed from the above principles and arguments. Global pollination and local pollination are two major stages of this algorithm [35]. In the phase of global pollination, insects work as pollinators for carrying flower pollens over long distances because of their capability to move and fly for a more extended range. In this way, fittest reproduction and pollination can be ensured. Flower reliability phenomena and the first rule of global pollination step can be mathematically [36, 37] described as(11)θin+1=θin+Lθin−θbest.Here, polleni is represented by θin at iteration n, while at current iteration, the best value among all values is denoted by θbest. Pollination strength is shown with parameter L, which represents step size. Levy flight mechanism can be used accurately to analyze the property of insects to travel over long distances using many steps. Thus, from Levy distribution, we develop L>0. In the end, the best result can be referred to as the best approximated angle.(12)L=λΓλsinπλ2π.s1+λ−1.The standard gamma function is represented byΓλ with λ=1.5, and s consists of Gaussian distributions U and V described as(13)s=U.V−1/λ,where U∼N0,σ2 and V∼N0,1, and σ2 can be computed as(14)σ2=Γ1+λλΓ1+λ/2.sinπλ/22λ−1/21/λ.Local pollination step and flower reliability phenomena can be written as(15)θin+1=θin+εθjn−θkn.Two arbitrary pollens produced from unlike flowers of the similar plant are denoted withθjn and θkn in the above mathematical model. This behavior shows the reliability of a flower in a limited community. Mathematically, these solutions are selected from the same population with a random walk of ϵ drawn from a uniform distribution in [0, 1]. Many of the flower pollination processes happen at both global and local levels. Practically local flower pollens pollinate the nearby flowers and flower patches or those not so from them. Thus, using this property, a switch or proximity probability ps (Rule 3) is developed to shift among global mutual pollination and complete local pollination. By applying switch probability, ps=0.5 can be used as starting value, and after this, a suitable range of parameters can be developed. The literature shows that ps=0.8 is appropriate for many practical applications.


### 3.3. DOA Estimation Using FPA

The general goal of DOA estimation is a continuous optimization that is used to find theθbest which satisfies(16)fθbest=minθϵSfθ,where SϵRD and fθ comprise the cost values of the corresponding solution θ. The cost function of DOA estimation can be defined as(17)fθ=zeθ−zθa2,where zeθ is the estimated (approximated using optimized parameters) array output and zθa is the actual array out. Therefore, the actual goal of the optimizer is to compute the associated argument for the minimum cost of the cost function as shown in Figure 2. Hence, the population of N individuals will be used to solve the optimization problem having T iterations (trials). The set of D-dimensional vectors (total N vectors) for i iteration can be denoted as(18)θ1i,θ2i,θ3i,…,θNi.Figure 2
Flow chart of FPA.Hence, the best solution at iterationi can be found as(19)θbesti=argminn=1,…,Nfθni.


## 3.1. Particle Swarm Optimization

The heuristics of PSO was proposed by Kennedy and Eberhart having motivation from the pool of birds congregating for food in a random manner [31]. The idea of seeking food is a heuristic approach because all the birds have the information of distance but are not familiar with the explicit location of food. They seek the food by exchanging their search information via crossover and kid production method. The PSO is introduced for the pool of applications almost in every walk of engineering [32, 33]. In this work, PSO performs searching via a swarm of particles that updates recursively. To approach the optimal solution, each particle (DOAs) moves in the direction to its previously best (pbest) position and the global best (gbest) position in the swarm.(7)vin+1=ωvin+c1r1θin−θpbest+c2r2θin−θgbest.(8)θin+1=θin+vin+1.Here, θpbest and θgbest can be calculated as(9)θpbesti=argmini=1,…,Kfθni,θgbesti=argminn=1,…,Nfθni.Here,i denotes the particle index, n is the current iteration number, and fθni is the fitness function that can be defined as(10)fθe=zθet−zθat2.And, the parametersω,c1,c2,r1, and r2 are inertia weight, two positive constants, and two random parameters within [0, 1], respectively. The velocity v and positions (DOAs) of particles are updated with equations (9) and (10).


## 3.2. Basis Principle of FPA

The features of pollination scheme, flower reliability, and behavior of the pollinator can be analyzed efficiently by the following principles:(1)
The global pollination scheme consists of cross-pollination and biotic methods, while pollinators perform levy flights with pollen (global optimization)(2)
The local pollination mechanism consists of self-pollination and abiotic methods (local optimization) [34](3)
A switch probabilitypϵ0,1 is designed to control global pollination and local pollination schemesIn the complete pollination process, the local pollination scheme can experience a large quantity of fractionp. Its reason can be environmental factors like wind and physical proximity. Generally, many flowers can grow on a plant, and pollen gametes from each flower can be released in billions. However, here we suppose that every plant has the ability to harvest only one flower and only one pollen gamete can be produced from each flower. Therefore, plant, flower, and pollen gamete are easy to identify for finding the solution to a problem. The above assumption develops the most straightforward way that solution can be equal to a pollen gamete and a flower. In future research, especially for multiobjective optimization problems, different numbers of flowers can be associated with each plant and multiple pollen gametes can be assigned to each flower. A flower-based algorithm, known as flower pollination algorithm (FPA), can be designed from the above principles and arguments. Global pollination and local pollination are two major stages of this algorithm [35]. In the phase of global pollination, insects work as pollinators for carrying flower pollens over long distances because of their capability to move and fly for a more extended range. In this way, fittest reproduction and pollination can be ensured. Flower reliability phenomena and the first rule of global pollination step can be mathematically [36, 37] described as(11)θin+1=θin+Lθin−θbest.Here, polleni is represented by θin at iteration n, while at current iteration, the best value among all values is denoted by θbest. Pollination strength is shown with parameter L, which represents step size. Levy flight mechanism can be used accurately to analyze the property of insects to travel over long distances using many steps. Thus, from Levy distribution, we develop L>0. In the end, the best result can be referred to as the best approximated angle.(12)L=λΓλsinπλ2π.s1+λ−1.The standard gamma function is represented byΓλ with λ=1.5, and s consists of Gaussian distributions U and V described as(13)s=U.V−1/λ,where U∼N0,σ2 and V∼N0,1, and σ2 can be computed as(14)σ2=Γ1+λλΓ1+λ/2.sinπλ/22λ−1/21/λ.Local pollination step and flower reliability phenomena can be written as(15)θin+1=θin+εθjn−θkn.Two arbitrary pollens produced from unlike flowers of the similar plant are denoted withθjn and θkn in the above mathematical model. This behavior shows the reliability of a flower in a limited community. Mathematically, these solutions are selected from the same population with a random walk of ϵ drawn from a uniform distribution in [0, 1]. Many of the flower pollination processes happen at both global and local levels. Practically local flower pollens pollinate the nearby flowers and flower patches or those not so from them. Thus, using this property, a switch or proximity probability ps (Rule 3) is developed to shift among global mutual pollination and complete local pollination. By applying switch probability, ps=0.5 can be used as starting value, and after this, a suitable range of parameters can be developed. The literature shows that ps=0.8 is appropriate for many practical applications.


## 3.3. DOA Estimation Using FPA

The general goal of DOA estimation is a continuous optimization that is used to find theθbest which satisfies(16)fθbest=minθϵSfθ,where SϵRD and fθ comprise the cost values of the corresponding solution θ. The cost function of DOA estimation can be defined as(17)fθ=zeθ−zθa2,where zeθ is the estimated (approximated using optimized parameters) array output and zθa is the actual array out. Therefore, the actual goal of the optimizer is to compute the associated argument for the minimum cost of the cost function as shown in Figure 2. Hence, the population of N individuals will be used to solve the optimization problem having T iterations (trials). The set of D-dimensional vectors (total N vectors) for i iteration can be denoted as(18)θ1i,θ2i,θ3i,…,θNi.Figure 2
Flow chart of FPA.Hence, the best solution at iterationi can be found as(19)θbesti=argminn=1,…,Nfθni.


## 4. Experimental Results and Comparison

In this section, numerous simulations have been carried out to analyze the performance of the FPA against the state-of-the-art algorithms. The performance has been studied in terms of estimation accuracy, convergence analysis, robustness against noise, and the robustness against the number of hydrophones used in the array. The conditions for FPAs are also depicted in this section. The measures of performance illustrate the comprehensive analysis of FPAs as explained in the following areas.


### 4.1. Convergence Analysis

In this section, we examined the performance of FPA and PSO in terms of convergence. The performance is analyzed for two and three sources with the varying noise level. Figures3 and 4 show that the FPA converged towards the minimum cost as compared to that of PSO.Figure 3
Convergence analysis of FPA for two sources.Figure 4
Convergence analysis of FPA for three sources.


### 4.2. Estimation Accuracy

The estimation accuracy of MVDR, MUSIC, Root MUSIC, ESPRIT, PSO, and FPA is examined here by taking signal sources with different positions and different noise levels. The noise is assumed to be additive white Gaussian with zero means. The statistical measures of the mean and variance have been calculated from 300 independent Monte Carlo simulations as discussed in the tables for each algorithm. It can be seen from Tables1–3 that the best performance is for FPAs in all different noise levels and in all two cases of Monte Carlo simulations (mean, variance).Table 1
Estimation accuracy for two sources.
SNR = 5 dB, snapshots = 20, hydrophones = 8,d = λ/2AlgorithmsFPAESPRITRMUSICMUSICMVDRPSOθ1 = 30Mean30.00529.68026.12531.98032.03633.549Variance0.01625.490347.701.9320.86933.549θ1 = 35Mean34.95335.69335.24632.99333.036278.68Variance0.00812.37333.6611.9600.869476.42Table 2
Estimation accuracy for three sources.
SNR = 5 dB, snapshots = 20, hydrophones = 8,d = λ/2AlgorithmsFPAESPRITRMUSICMUSICMVDRPSOθ1 = 30Mean29.99929.09911.96138.71039.66637.447Variance0.018846.4054854.418755.379246.2022439.3751θ2 = 40Mean39.99839.79238.31041.03341.15347.041Variance0.020514.450847.698653.732244.8432345.3394θ1 = 50Mean50.00750.68648.33042.48342.92051.160Variance0.029514.375819.392451.743140.6736138.6291Table 3
Estimation accuracy for four sources.
SNR = 5 dB, snapshots = 20, sydrophones = 8,d = λ/2AlgorithmsFPAESPRITRMUSICMUSICMVDRPSOθ1 = 10Mean10.7986.754−23.20621.39323.2079.859Variance63.4137.7753.898.597.01224.0θ2 = 20Mean20.24818.98914.24724.71325.58341.079Variance16.318.958.3102.394.31032.8θ3 = 30Mean30.00130.62825.54128.47327.85339.7725Variance0.020421.758169.734793.656089.8585550.2132θ4 = 40Mean39.99541.85237.23831.13729.89347.3674Variance0.027429.167435.161375.331378.6620391.5863


### 4.3. Performance against the Independent Monte Carlo Runs

The performance metrics are analyzed by calculating RMSE that is defined as follows:(20)RMSE=∑K∑i=1D1KDθia−θie2.Here, θia is the actual DOA and θie is the estimated DOA.In this section, the performance has been analyzed against the independent Monte Carlo runs. The simulation results show that the oscillations of the maximum and minimum of RMSE describe the best and worst performance of the algorithm. Hence, the reliability of the FPA outperforms that of the MVDR, MUSIC, RMUSIC, and ESPRIT and PSO algorithm for independent Monte Carlo runs as shown in Figures5 and 6.Figure 5
Performance against independent Monte Carlo runs for two sources with 5 dB of SNR.Figure 6
Performance against independent Monte Carlo runs for three sources with 5 dB of SNR.


### 4.4. Robustness against Noise

In this section, the performance of these algorithms is measured by calculating the RMSE under varying levels of additive white noise. The convergence of RMSE also depicts performance analysis of such algorithms with the different number of signal sources having different DOAs.We have used eight hydrophones, and the number of snapshots is 20 for two sources located at 30° and 35° and for three sources situated at 30°, 34°, and 50°. The result analysis in Figures7 and 8 presents that the RMSE is a function of SNR. As the SNR increases, the RMSE decreases substantially. It can be seen that the FPA is robust enough to produce excellent results even in the presence of low SNR as compared to the other algorithms. CRB has also validated the performances of the algorithms.Figure 7
Performance of RMSE vs SNR for two sources.Figure 8
Performance of RMSE vs SNR for three sources.


### 4.5. Robustness against the Number of Hydrophones

The results in Figures9–12 plotted the RMSE as a function of the number of hydrophones. As the number of hydrophones increases, the directivity also increases and hence RMSE decreases significantly. It can be seen that the FPA produces even for fewer hydrophones than the other algorithms in both cases of 5 and 10 dB of SNR.Figure 9
Robustness against the number of elements for two sources of 5 dB of SNR.Figure 10
Robustness against the number of elements for two sources of 10 dB of SNR.Figure 11
Robustness against the number of elements for three sources of 5 dB of SNR.Figure 12
Robustness against the number of elements for three sources of 10 dB of SNR.


### 4.6. Robustness against Snapshots

Another parameter to be considered is the impact of snapshots on RMSE. In this simulation, the SNR at 5 dB is fixed and the number of snapshots from 5 to 200 is varied with a step size of 5. Figures13 and 14 show that as the number of snapshots increases, the RMSE decreases towards zero. The FPA outperforms the MVDR, MUSIC, RMUSIC, and ESPRIT algorithms. The worst performance of MUSIC and MVDR is due to low SNR, which results in a distorted spatial spectrum.Figure 13
Robustness against snapshots for two sources.Figure 14
Robustness against snapshots for three sources.


### 4.7. Analysis of Empirical Cumulative Distribution Function of RMSE

In this section, the observations of the RMSE are depicted in the order from least to greatest. This analysis corresponds to the survival and failure times of the algorithms over the Monte Carlo runs. It can be seen from Figures15 and 16 that the FPA gives a significant amount of the Monte Carlo runs having the least RMSE. MVDR and MUSIC algorithms depict a colossal number of Monte Carlo trials with the most incredible value of RMSE that leads to the failure of the algorithm for DOA estimation. More specifically, the FPA and ESPRIT algorithms also give a remarkable amount of Monte Carlo trials the least RMSEs compared to PSO, MUSIC, Root MUSIC, and MVDR algorithms. Ultimately, the noteworthy performances are possessed by the FPA.Figure 15
CDF of RMSE for two sources.Figure 16
CDF of RMSE for three sources.


### 4.8. Frequency Distribution of RMSE

In this section, the histogram provides a visual interpretation of RMSE observations by showing the number of RMSE observations that fall within a specified range of RMSE values. This analysis also explains the skewness of the RMSE observations to validate the performance of the algorithms. It can be seen from Figures17 and 18 that most of the occurrences underlie the least values of the RMSE for FPA. Furthermore, the ESPRIT algorithm gives a fair distribution of frequency of RMSE over the Monte Carlo runs compared to PSO, Root MUSIC, MUSIC, and MVDR algorithms.Figure 17
Histogram analysis of RMSE for two sources.Figure 18
Histogram analysis of RMSE for three sources.


### 4.9. Variability Analysis of the RMSE

In this section, the spread-out of the RMSE is analyzed in the five pieces of the information (minimum, first quartile, median, third quartile, and maximum) over the Monte Carlo trials. “Minimum” depicts the minimum value of the RMSE from Monte Carlo observations. In comparison, the first and third quartiles describe 25 and 75 percent of the RMSE observations (Monte Carlo runs). This measure of the spread-out is a comprehensive description of the RMSE distribution to validate the performance of the algorithms. It can be seen from Figures19 and 20 that FPA displays the distribution of RMSE with the least values of the RMSE, which reveals it is outperforming the other algorithms. Furthermore, the ESPRIT algorithm also gives significant performance compared to the PSO, Root MUSIC, MUSIC, and MVDR algorithms.Figure 19
Variability analysis for two sources.Figure 20
Variability analysis for three sources.


### 4.10. The Resolution Ability for Closely Spaced Targets

In this section, the simulation background is estimated for the proposed structure to check the superresolution performance. The probability of resolution is defined as(21)Pr=Probθe−θa≤△θ2,where △θ=θ1−θ2. The resolution ability of both closely spaced sources is shown in Figures 21–24. The performance is analyzed based on different DOA positions. It can be seen that the FPA outperforms the others but the PSO performs significantly too for closely spaced sources. Simulation is carried out by fixing the first source and moving the second source from 35 to 34 degrees. The resolution ability is analyzed for each separation for both sources independently.Figure 21
Probability of resolution for the source at 30 degrees.Figure 22
Probability of resolution for the source at 35 degrees.Figure 23
Probability of resolution for the source at 30 degrees.Figure 24
Probability of resolution for the source at 34 degrees.


### 4.11. Computational Complexity Analysis

In this, the computation loading performance is analyzed using MUSIC, MVDR, and RMUSIC algorithms, which are highly computational practical algorithms due to their spectral search approach to estimate the DOA. Moreover, the ESPRIT algorithm neither includes the extrema search optimization nor the spectral search approach; hence, it results in better computational complexity than the other algorithms as mentioned in Table4. The FPA also outperforms the MUSIC, MVDR, and RMUSIC algorithm but is not better than the ESPRIT algorithm due to its extrema searching approach to optimize the cost function. The extrema searching approach of FPA and PSO to optimize the cost function increases the computational complexity that sometimes restrain the application of swarming intelligent algorithms.Table 4
Computational loading analysis.
AlgorithmComplexityMVDRW2K+6+W+4∗D∗PMUSIC5/3W3+W2K+1+D+W+4∗D∗PRMUSIC11/3W3+W2K+D−1+2W−1ESPRITW2K+2∗W+1+DD+1PSOK∗T∗N∗W∗DFPAD∗T∗K∗N∗W1−ps


## 4.1. Convergence Analysis

In this section, we examined the performance of FPA and PSO in terms of convergence. The performance is analyzed for two and three sources with the varying noise level. Figures3 and 4 show that the FPA converged towards the minimum cost as compared to that of PSO.Figure 3
Convergence analysis of FPA for two sources.Figure 4
Convergence analysis of FPA for three sources.


## 4.2. Estimation Accuracy

The estimation accuracy of MVDR, MUSIC, Root MUSIC, ESPRIT, PSO, and FPA is examined here by taking signal sources with different positions and different noise levels. The noise is assumed to be additive white Gaussian with zero means. The statistical measures of the mean and variance have been calculated from 300 independent Monte Carlo simulations as discussed in the tables for each algorithm. It can be seen from Tables1–3 that the best performance is for FPAs in all different noise levels and in all two cases of Monte Carlo simulations (mean, variance).Table 1
Estimation accuracy for two sources.
SNR = 5 dB, snapshots = 20, hydrophones = 8,d = λ/2AlgorithmsFPAESPRITRMUSICMUSICMVDRPSOθ1 = 30Mean30.00529.68026.12531.98032.03633.549Variance0.01625.490347.701.9320.86933.549θ1 = 35Mean34.95335.69335.24632.99333.036278.68Variance0.00812.37333.6611.9600.869476.42Table 2
Estimation accuracy for three sources.
SNR = 5 dB, snapshots = 20, hydrophones = 8,d = λ/2AlgorithmsFPAESPRITRMUSICMUSICMVDRPSOθ1 = 30Mean29.99929.09911.96138.71039.66637.447Variance0.018846.4054854.418755.379246.2022439.3751θ2 = 40Mean39.99839.79238.31041.03341.15347.041Variance0.020514.450847.698653.732244.8432345.3394θ1 = 50Mean50.00750.68648.33042.48342.92051.160Variance0.029514.375819.392451.743140.6736138.6291Table 3
Estimation accuracy for four sources.
SNR = 5 dB, snapshots = 20, sydrophones = 8,d = λ/2AlgorithmsFPAESPRITRMUSICMUSICMVDRPSOθ1 = 10Mean10.7986.754−23.20621.39323.2079.859Variance63.4137.7753.898.597.01224.0θ2 = 20Mean20.24818.98914.24724.71325.58341.079Variance16.318.958.3102.394.31032.8θ3 = 30Mean30.00130.62825.54128.47327.85339.7725Variance0.020421.758169.734793.656089.8585550.2132θ4 = 40Mean39.99541.85237.23831.13729.89347.3674Variance0.027429.167435.161375.331378.6620391.5863


## 4.3. Performance against the Independent Monte Carlo Runs

The performance metrics are analyzed by calculating RMSE that is defined as follows:(20)RMSE=∑K∑i=1D1KDθia−θie2.Here, θia is the actual DOA and θie is the estimated DOA.In this section, the performance has been analyzed against the independent Monte Carlo runs. The simulation results show that the oscillations of the maximum and minimum of RMSE describe the best and worst performance of the algorithm. Hence, the reliability of the FPA outperforms that of the MVDR, MUSIC, RMUSIC, and ESPRIT and PSO algorithm for independent Monte Carlo runs as shown in Figures5 and 6.Figure 5
Performance against independent Monte Carlo runs for two sources with 5 dB of SNR.Figure 6
Performance against independent Monte Carlo runs for three sources with 5 dB of SNR.


## 4.4. Robustness against Noise

In this section, the performance of these algorithms is measured by calculating the RMSE under varying levels of additive white noise. The convergence of RMSE also depicts performance analysis of such algorithms with the different number of signal sources having different DOAs.We have used eight hydrophones, and the number of snapshots is 20 for two sources located at 30° and 35° and for three sources situated at 30°, 34°, and 50°. The result analysis in Figures7 and 8 presents that the RMSE is a function of SNR. As the SNR increases, the RMSE decreases substantially. It can be seen that the FPA is robust enough to produce excellent results even in the presence of low SNR as compared to the other algorithms. CRB has also validated the performances of the algorithms.Figure 7
Performance of RMSE vs SNR for two sources.Figure 8
Performance of RMSE vs SNR for three sources.


## 4.5. Robustness against the Number of Hydrophones

The results in Figures9–12 plotted the RMSE as a function of the number of hydrophones. As the number of hydrophones increases, the directivity also increases and hence RMSE decreases significantly. It can be seen that the FPA produces even for fewer hydrophones than the other algorithms in both cases of 5 and 10 dB of SNR.Figure 9
Robustness against the number of elements for two sources of 5 dB of SNR.Figure 10
Robustness against the number of elements for two sources of 10 dB of SNR.Figure 11
Robustness against the number of elements for three sources of 5 dB of SNR.Figure 12
Robustness against the number of elements for three sources of 10 dB of SNR.


## 4.6. Robustness against Snapshots

Another parameter to be considered is the impact of snapshots on RMSE. In this simulation, the SNR at 5 dB is fixed and the number of snapshots from 5 to 200 is varied with a step size of 5. Figures13 and 14 show that as the number of snapshots increases, the RMSE decreases towards zero. The FPA outperforms the MVDR, MUSIC, RMUSIC, and ESPRIT algorithms. The worst performance of MUSIC and MVDR is due to low SNR, which results in a distorted spatial spectrum.Figure 13
Robustness against snapshots for two sources.Figure 14
Robustness against snapshots for three sources.


## 4.7. Analysis of Empirical Cumulative Distribution Function of RMSE

In this section, the observations of the RMSE are depicted in the order from least to greatest. This analysis corresponds to the survival and failure times of the algorithms over the Monte Carlo runs. It can be seen from Figures15 and 16 that the FPA gives a significant amount of the Monte Carlo runs having the least RMSE. MVDR and MUSIC algorithms depict a colossal number of Monte Carlo trials with the most incredible value of RMSE that leads to the failure of the algorithm for DOA estimation. More specifically, the FPA and ESPRIT algorithms also give a remarkable amount of Monte Carlo trials the least RMSEs compared to PSO, MUSIC, Root MUSIC, and MVDR algorithms. Ultimately, the noteworthy performances are possessed by the FPA.Figure 15
CDF of RMSE for two sources.Figure 16
CDF of RMSE for three sources.


## 4.8. Frequency Distribution of RMSE

In this section, the histogram provides a visual interpretation of RMSE observations by showing the number of RMSE observations that fall within a specified range of RMSE values. This analysis also explains the skewness of the RMSE observations to validate the performance of the algorithms. It can be seen from Figures17 and 18 that most of the occurrences underlie the least values of the RMSE for FPA. Furthermore, the ESPRIT algorithm gives a fair distribution of frequency of RMSE over the Monte Carlo runs compared to PSO, Root MUSIC, MUSIC, and MVDR algorithms.Figure 17
Histogram analysis of RMSE for two sources.Figure 18
Histogram analysis of RMSE for three sources.


## 4.9. Variability Analysis of the RMSE

In this section, the spread-out of the RMSE is analyzed in the five pieces of the information (minimum, first quartile, median, third quartile, and maximum) over the Monte Carlo trials. “Minimum” depicts the minimum value of the RMSE from Monte Carlo observations. In comparison, the first and third quartiles describe 25 and 75 percent of the RMSE observations (Monte Carlo runs). This measure of the spread-out is a comprehensive description of the RMSE distribution to validate the performance of the algorithms. It can be seen from Figures19 and 20 that FPA displays the distribution of RMSE with the least values of the RMSE, which reveals it is outperforming the other algorithms. Furthermore, the ESPRIT algorithm also gives significant performance compared to the PSO, Root MUSIC, MUSIC, and MVDR algorithms.Figure 19
Variability analysis for two sources.Figure 20
Variability analysis for three sources.


## 4.10. The Resolution Ability for Closely Spaced Targets

In this section, the simulation background is estimated for the proposed structure to check the superresolution performance. The probability of resolution is defined as(21)Pr=Probθe−θa≤△θ2,where △θ=θ1−θ2. The resolution ability of both closely spaced sources is shown in Figures 21–24. The performance is analyzed based on different DOA positions. It can be seen that the FPA outperforms the others but the PSO performs significantly too for closely spaced sources. Simulation is carried out by fixing the first source and moving the second source from 35 to 34 degrees. The resolution ability is analyzed for each separation for both sources independently.Figure 21
Probability of resolution for the source at 30 degrees.Figure 22
Probability of resolution for the source at 35 degrees.Figure 23
Probability of resolution for the source at 30 degrees.Figure 24
Probability of resolution for the source at 34 degrees.


## 4.11. Computational Complexity Analysis

In this, the computation loading performance is analyzed using MUSIC, MVDR, and RMUSIC algorithms, which are highly computational practical algorithms due to their spectral search approach to estimate the DOA. Moreover, the ESPRIT algorithm neither includes the extrema search optimization nor the spectral search approach; hence, it results in better computational complexity than the other algorithms as mentioned in Table4. The FPA also outperforms the MUSIC, MVDR, and RMUSIC algorithm but is not better than the ESPRIT algorithm due to its extrema searching approach to optimize the cost function. The extrema searching approach of FPA and PSO to optimize the cost function increases the computational complexity that sometimes restrain the application of swarming intelligent algorithms.Table 4
Computational loading analysis.
AlgorithmComplexityMVDRW2K+6+W+4∗D∗PMUSIC5/3W3+W2K+1+D+W+4∗D∗PRMUSIC11/3W3+W2K+D−1+2W−1ESPRITW2K+2∗W+1+DD+1PSOK∗T∗N∗W∗DFPAD∗T∗K∗N∗W1−ps


## 5. Conclusion

The simulation results demonstrate that the FPA outperforms the conventional beamforming and conventional subspace-based algorithms in most situations. The performance improvement is more significant when multiple signals are incident at closely spaced angles at a low signal-to-noise ratio and when a small number of snapshots are used to estimate direction of arrival (DOA). Statistical analysis of the RMSE in Monte Carlo trials, that is, ECDF of RMSE, variability analysis of RMSE, frequency distribution of RMSE, and the probability of resolution, witnesses the strength of FPA in the challenging environment of low SNR using less number of snapshots. In the future, the estimation of 2D-DOA of underwater multitargets using 2D arrays should be investigated with modern heuristic algorithms to achieve high accuracy and resolution. Moreover, the proposed flower pollination heuristics look promising to deal with the optimization problems in diversified fields.



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*Source: 5876874-2022-06-30.xml*