# Multiple Target Localization with Bistatic Radar Using Heuristic Computational Intelligence Techniques

Authors: Fawad Zaman; Ijaz Mansoor Qureshi; Ata Ur Rehman; Shujaat Ali Khan Tanoli
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

We assume Bistatic Phase Multiple Input Multiple Output radar having passive Centrosymmetric Cross Shape Sensor Array (CSCA) on its receiver. Let the transmitter of this Bistatic radar send coherent signals using a subarray that gives a fairly wide beam with a large solid angle so as to cover up any potential relevant target in the near field. We developed Heuristic Computational Intelligence (HCI) based techniques to jointly estimate the range, amplitude, and elevation and azimuth angles of these multiple targets impinging on the CSCA. In this connection, first the global search optimizers, that is,are developed separately Particle Swarm Optimization (PSO) and Differential Evolution (DE) are developed separately, and, to enhance the performances further, both of them are hybridized with a local search optimizer called Active Set Algorithm (ASA). Initially, the performance of PSO, DE, PSO hybridized with ASA, and DE hybridized with ASA are compared with each other and then with some traditional techniques available in literature using root mean square error (RMSE) as figure of merit.


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## Body

## 1. Introduction

It is easy to deal with the far field targets because all the waves emitted by them are assumed to be plane waves and, by incorporating such assumption, one can characterize the signal model only by the Direction of Arrival (DOA) of the targets [1, 2]. On the other hand, it is comparatively difficult and challenging to localize the targets located close to the sensors array, that is, in near field. In this case, the plane wave assumption is no longer applicable and one has to deal with spherical waves. Thus, to localize the near field targets, the range information must also be incorporated into the signal model along with DOA [3, 4]. Near field targets localization, especially 3D (range, elevation angle, and azimuth angle), plays significant role in radar, cognitive radio networks, and array signal processing, since it is a preliminary step for adaptive beamformer to guide the main beam in preferred direction and simultaneously manage the nulls in the direction of jammers [5–7].Several algorithms are available to address the issue of 3D near field source localization problem. In [8–10] different schemes are discussed but they not only are computationally expensive but also have the problem of pair matching between elevation and azimuth angles. To solve the pair matching problem, a two-stage separated steering vector-based algorithm is proposed in [11] but it has higher MSE and is computationally expensive as it requires more than 400 snapshots to achieve the results. It also fails to estimate the amplitude of sources which is also sometimes an important parameter to be estimated. Clearly the goal is to develop a scheme which must be able to jointly estimate the amplitude, range, elevation angle, and azimuth angle, should provide an improved MSE, and finally should be free of pair matching problem.In order to achieve the desired goals Heuristic Computational Intelligence (HCI) techniques are available. In the last two decades, these techniques were being widely applied to a mixture of problems ranging from handy applications in industry and commerce to leading scientific research [12–14]. These techniques include Differential Evolution (DE), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA). Besides the other useful properties, one of the most significant aspects of these techniques is their property of hybridization with any other heuristic or nonheuristic techniques. Due to this property, their reliability and applicability increase even more. In [15], an efficient scheme based on GA hybridized with Pattern Search (PS) is developed for the joint estimation of amplitude and 2D DOA of far field sources impinging on L shape arrays (1-L and 2-L shape arrays). Similarly, another useful algorithm is designed in [16] which is based on GA hybridized with Interior Point Algorithm (IPA) to jointly estimate the amplitude, range, and elevation angle of near field sources impinging on Uniform Linear Array (ULA).In this paper, we assume a scenario of Bistatic Phase Multiple Input Multiple Output (MIMO) radar with colocated antennas which is capable of sending multiple beams with diverse wave forms on diverse targets. But, before getting into the mode of tracking targets by sending high gain sharp beams on the targets, it is required to estimate their spherical coordinates, that is, range and elevation and azimuth angles(r,θ,ϕ). To achieve this, the transmitter sends a coherent signal beam using any subarray of the transmitter such that a beam has a fairly wide solid angle so as to cover up any potential target in the space. The receiver in the first mode of this scenario is a CSCA of passive sensors consisting of two X and Y subarrays. The reflected signals from different targets will impinge on these X-Y arrays. Since we are interested in the ranges of the targets apart from 2D DOA, we treat the targets as near field in order to localize them completely. To jointly estimate the amplitude, range, and elevation and azimuth angles of the targets, we employ(a)
PSO,(b)
DE,(c)
PSO hybridized with Active Set Algorithm (ASA),(d)
DE hybridized with ASA.In both hybrid schemes, the best individual results achieved through DE and PSO are given as starting point to ASA for further tuning. The performance criterion is made on the basis of mean square error (MSE) which is used as fitness function derived from maximum likelihood principle (MLP) [16]. This fitness function is able to automatically pair the amplitude, range, and elevation and azimuth angles of a specific target. The other properties of this fitness function are its ease in concept, ease in implementation, robustness against noise, and requirement of single snapshot. The performances of PSO, DE, Particle Swarm Optimization hybridized with Active Set Algorithm (PSO-ASA), and Differential Evolution hybridized with Active Set Algorithm (DE-ASA) are not only compared with each other but also with some traditional techniques available in literature [10, 11] using RMSE as figure of merit.The rest of the paper is organized as follows: Section2 evaluates the problem formulation, while the proposed HCI techniques are discussed in Section 3. In the same way, results and discussion are carried out in Section 4, while Section 5 is dedicated to conclusion and future work directions.


## 2. Data Model for Near Field Targets

In this section, we developed a data model forK near field targets impinging on CSCA placed on the receiver of Bistatic radar. The CSCA is composed of two symmetric sub-ULAs placed along x-axis and y-axis, respectively, as shown in Figure 1. Each ULA carries 2×P passive sensors, while the reference sensor is common for them. For K<4P+1, the data model at mth and nth sensor in the x-axis and y-axis subarray, respectively, can be represented as(1)wm,0=∑k=1Kskejψxkm+ηm,0,(2)w0,n=∑k=1Kskejψykn+η0,n.In (1) and (2), ψxk(m) is the propagation delay for kth target between (0,0)th and (m,0)th sensors, while ψyk(n) is the propagation delay for kth target between (0,0)th and (0,n)th sensors. For kth near field target, the signal phases ψxk(m) and ψyk(n) are parameterized mathematically as intermediate parameters in the form given as(3)ψxkm=mαxk+m2βxk,ψykn=nαyk+n2βyk.In (3), αxk, βxk, αyk, and βyk are the function of kth source amplitude (s), elevation angle (θ), azimuth angle (ϕ), and range (r). Mathematically, they can be expressed as (4)αxk=-2πdsin⁡θkcos⁡ϕkλ,βxk=πd21-sin2⁡θkcos2⁡ϕkλrk,αyk=-2πdsin⁡θksin⁡ϕkλ,βyk=πd21-sin2⁡θksin2⁡ϕkλrk.In (1) and (2), ηm,0 and η0,n are Additive White Gaussian Noise (AWGN) added at mth and nth sensors, respectively. In vector form, the signal model can be represented as(5)w=As+η,where(6)s=s1,s2,s3,…,sKT,w=w-P,0,w-P+1,0,…,w-1,0,w1,0,w2,0,…,wP-1,0,wP,0,w0,0,w0,-P,w0,-P+1,…,w0,-1,w0,1,w0,2,…,w0,P-1,w0,PT,η=η-P,0,η-P+1,0,…,η-1,0,η1,0,η2,0,…,ηP-1,0,ηP,0,η0,0,η0,-P,η0,-P+1,…,η0,-1,η0,1,η0,2,…,η0,P-1,η0,PT,A=aαx1,βx1,αy1,βy1,aαx2,βx2,αy2,βy2,…,aαxK,βxK,αyK,βyK,where(7)aαxk,βxk,αyk,βyk=ej-Pαxk+-P2βxk,ej-P+1αxk+-P+12βxk,…,ej-αxk+βxk,ejαxk+βxk,…,ejP-1αxk+P-12βxk,ejPαxk+P2βxk,1,ej-Pαyk+-P2βyk,ej-P+1αyk+-P+12βyk,…,ej-αyk+βyk,ejαyk+βyk,…,ejP-1αyk+P-12βyk,ejpαyk+p2βykT.Now clearly the problem in hand is to accurately and jointly estimate the unknown parameters (amplitudes, ranges, and elevation and azimuth angles) of the reflected signals from targets.Figure 1
Schematic diagram for Bistatic radar.


## 3. Heuristic Computational Intelligence

Heuristic Computational Intelligence (HCI) is a subfield of artificial intelligence that can be used for continuous and combinatorial optimization problems. The entire HCI algorithms are global optimization based methods which have stochastic or meta-heuristic characteristics which can be used for black box or gradient free problems. Basically, these kinds of algorithms make use of the iterative progress in a population and, to achieve the desired goal, the population is selected using parallel processing in random guided manner [17]. The major advantages of HCI over traditional optimization methods include wide range of applicability, conceptual simplicity, knowledge usage capability, hybridization with any other scheme, robustness against dynamic changes, and ability of self-optimization [18]. In literature, several HCI based algorithms are available but we are focusing on DE and PSO.


### 3.1. Particle Swarm Optimization (PSO)

PSO algorithm is inspired from bird flocking and school fishing which was proposed by Kennedy and Eberhart in 1995 [19]. It can handle multimodal, discontinuous, and nonconvex optimization problems because it is a pseudooptimization method [20, 21]. The schematic diagram of PSO is shown in Figure 2, while its execution steps are given below.Figure 2
Generic flow diagram of Hybrid Particle Swarm Optimization.Step 1(initialization). Initialize the swarm randomly, that is, randomly producing Q particles. In this problem, each particle consists of 4∗K genes, where K represents the total number of available targets. Mathematically, it can be written as(8)B=b1,b2,b3,…,bQT,b1b2b3⋮bQ=θ1,1,θ1,2,…,θ1,K,ϕ1,K+1,ϕ1,K+2,…,ϕ1,2∗K,r1,2∗K+1,r1,2∗K+2,…,r1,3∗K,s1,3∗K+1,s1,3∗K+2,…,s1,4∗Kθ2,1,θ2,2,…,θ2,K,ϕ2,K+1,ϕ2,K+2,…,ϕ2,2∗K,r2,2∗K+1,r2,2∗K+2,…,r2,3∗K,s2,3∗K+1,s2,3∗K+2,…,s2,4∗Kθ3,1,θ3,2,…,θ3,K,ϕ3,K+1,ϕ3,K+2,…,ϕ3,2∗K,r3,2∗K+1,r3,2∗K+2,…,r3,3∗K,s3,3∗K+1,s3,3∗K+2,…,s3,4∗K⋮θQ,1,θQ,2,…,θQ,K,ϕQ,K+1,ϕQ,K+2,…,ϕQ,2∗K,rQ,2∗K+1,rQ,2∗K+2,…,rQ,3∗K,sQ,3∗K+1,sQ,3∗K+2,…,sQ,4∗K.The upper and lower bounds of θ, ϕ, r, and s are given as(9)θq,k∈R:0≤θq,k≤π2ϕq,K+k∈R:0≤ϕq,K+k≤2πrq,2K+k∈R:rl≤rq,2K+k≤rusq,3K+k∈R:sl≤sq,3K+k≤sufor,q=1,2,…,Q,k=1,2,…,K,where rl and ru represent the lower and upper limits of ranges, while su and sl are the upper and lower limits of amplitudes.Step 2(fitness function). We have used MSE as a fitness function to compute the fitness of every particle. For qth particle, it can be mathematically written as(10)FFq=11+ξq,where ξ(q) is called MSE which is derived from maximum likelihood principle [16]. The total MSE for qth particle can be mathematically written as (11)ξq=ξ1q+ξ2q,(12)ξ1q=12Px∑m=-PxPxwm,o-w^m,0q2,(13)ξ2q=12Py∑n=-PyPyw0,n-w^0,nq2,where Px and Py are the number of sensors placed along x-axis and y-axis, respectively. In (12) and (13), wm,0 and w0,n are given in (1) and (2), respectively, while, w^m,0q and w^0,nq can be defined as(14)w^m,0q=∑k=1Kb^3K+kqej-m2πdsin⁡b^kqcos⁡b^3K+kq/λ+m2πd21-sin2⁡b^kqcos2⁡b^K+kq/λb^2K+kq,w^0,nq=∑k=1Kb^3K+kqej-n2πdsin⁡b^Kqsin⁡b^K+kq/λ+n2πd21-sin2⁡b^kqsin2⁡b^K+kq/λb^2K+kq.Now, store each particle as local best (l) while the one having maximum fitness function is stored as global best (gb) particle.Step 3a(update particle velocity). We used the following relation to update the velocity of each particle:(15)vqmn=vqmn-1+σ11-γlqm-bqmn-1+σ2γgbm-bqmn-1.In the right hand side of (15), the first term is called inertia or momentum which shows the previous velocity. The second term shows the private thinking and is also called as cognitive term of the particle while the last term explains the collective behavior of the population and is known as a social component. At the beginning, the value of γ=0.1 to ensure that more weightage is given to local intelligence at the beginning; then there is gradual increase in the value of γ towards 0.9 which means that more weightage is given to collective intelligence at the end. In (15), both σ1 and σ2 are positive constants and for the ongoing problem σ1=σ2=1.The velocity is doubly bounded; that is,-vmax⁡≤vqm≤vmax⁡;ifvqmn>vmax⁡,vqmn=vmax⁡;ifvqm(n)<-vmax⁡,vqm(n)=-vmax⁡.vmax and vmin are taken to be 2 and −2, respectively, in order to avoid the premature convergence.Step 3b(update particle position). The position of each particle can be updated as(16)bqmn=bqmn-1+vqmn.Step 4a(choose local best particle). If fitness bq(n)>l(n), replace l with b.Step 4b(choose global best particle). Similarly replace the previous global best gb  particle with bq(n) as a new global best particle if fitness bq(n)>gb.Step 5(termination). Terminate, if any of the following conditions is satisfied; otherwise go to Step 3a:(i)
The predefined total number of iterations/flights is executed for the algorithm.(ii)
The desired MSE is attained which is10-7.Step 6(hybridization). In this step, the best particle achieved through PSO is given to ASA for further improvement.Step 7(storage). Store the results for later on discussion and comparison.


### 3.2. Differential Evolution (DE)

DE was introduced by Storn and Price in 1997 [22] which iteratively searches large spaces of candidate solution that improve the candidate solution with respect to specified measurement of quality. In other words, DE deals with the optimization problem in the following way: first it achieves the candidate solution and then, by using simple formula, it generates new solution by combining with the existing ones. As a result only those candidate solutions will survive which has best fitness or score. The block diagram of DE is shown in Figure 3, while its steps are given as follows.Figure 3
Generic flow diagram for hybrid DE.Step 1(initialization). The first step is similar to the one given for PSO in (8).Step 2(updating). Update all particles of the current generation “ge” from 1 to Q. Let us choose qth particle from (8), that is, bhq,ge, where q=1,2,…,Q and h=1,2,…,4×K, while “ge” represents the particular generation. The main goal of DE is to find out the particles of next generation, that is, cq,ge+1, by adopting the following steps:(a)
Mutation: select any three different numbers (particles) from 1 to Q, that is, (a1,a2,a3), by keeping the following conditions:(17)1≤a1,a2,a3≤Q,whereaq≠nh∀q,h=1,2,3,nq≠q∀q=1,2,3,dq,ge=ba1,ge+Fba2,ge-ba3,ge,where “F” is a constant whose value usually lies in the range 0.5 to 1.(b)
Crossover: the crossover can be performed as(18)okq,ge=dhq,geifrand⁡≤CR orh=hrandbhq,geotherwise,where0.5≤CR≤1 and hrand is between 1 and 4∗K chosen at random.(c)
Selection operation: in this step, select the particle of next generation by using the following criterion of selection:(19)bq,ge+1=oq,geiferr⁡oq,ge<err⁡bq,gebq,geotherwise,where theerr⁡(bq,ge) and err⁡(oq,ge) are defined in (8). Repeat this for all particles.Step 3(stopping criteria). The stopping criteria of DE are based on any of the following conditions being fulfilled:(I)
err⁡(bq,ge+1)<ε, where ε is a very small positive number OR.(II)
The number of iterations is completed.Else go back to Step 2.Step 4(hybridization). In order to further refine the results of DE, we give the best particle to ASA as starting point.


### 3.3. Active Set Algorithm (ASA)

Active set methods are recursive procedures for solving sequence of equality-constrained quadratic subproblems effectively from majority of algorithms based on interior methods and simplex techniques due to the fact that they operate in two phases; first they focus on feasibility, while others focus on optimality. The applications in which their dominance is well established are quadratic programming including portfolio analysis, structural analysis, and optimal control and optimal mesh refinement in ODE or PDE constrained problems, sparse linear programming problems, and box constrained optimization problem [23–25]. Active set quadratic programming is exploited in many state-of-the-art optimization solvers such as QPOPT [26], SQOPT [27], and QPA (part of the GALAHAD library) [28]. Although the global optimization methods (PSO and DE) alone can produced excellent results, their rate of convergence usually decreases with the increase of iteration especially in nonlinear problems. Therefore, to get accurate results, we need an efficient local search optimizer like ASA. The ASA is mainly used in constrained optimization problems whose basic purpose is to transform the problem into an easier solvable problem [29]. The standard working procedure of ASA in the form of pseudocode is given as follows, while its flow diagram in terms of processing blocks is shown in Figure 4.Figure 4
Generic flow diagram of Active Set Algorithm.Generic Pseudocode of Active Set AlgorithmStart. Initialize ASA algorithm with random assignment of parameters.Step   1.
Determine a suitable starting point.Step   2.
Repeat till an optimal condition is achieved.Step  3.
Solve the equality problem defined roughly by ASA.Step   4.
Calculate the Lagrange multipliers for ASA and remove all those constraints having negative Lagrange multipliers.End RepeatThe mathematical formulation of ASA algorithm is given as.The constrained optimization problem can be mathematically written as(20)min⁡fysuch thatl≤y≤u,where l and u represent the lower and upper bounds, respectively, of the desired vectors. In early methods, penalty functions were used to transform the constrained problem into unconstrained one which is beyond or near the constrained boundary. In such a way, the constrained problems were solved by a sequence of parameterized unconstrained optimizations, which is obviously an inefficient way, that are now replaced by methods which focus on the solution of Karush-Kuhn-Tucker (KKT) equations such as ASA. The KKT equations are the essential condition for constrained optimization problem. In order to achieve the global solution point, the KKT equations are the sufficient and essential conditions to be satisfied. The KKT equations can be mathematically written as(21)∇fy∗+∑i=1mλi·ΔGiy∗=0,(22)λi·Giy∗=0,i=1,…,me,where λi≥0, i=me+1,…,m, are called Lagrange multipliers. Equation (21) represents the gradient cancelation of active constraints and the objective function at the solution point where the Lagrange multipliers are necessary to balance the deviation in magnitude of the constrained gradients and objective function.For ASA, we used a MATLAB built-in optimization toolbox where the parameter setting is provided in Table1.Table 1
Parameters setting for ASA.
Parameters
Setting
Starting point
Best particles achieved by DE and PSO
Number of iterations
2000
Number of variables
4
×
K
Fitness limit
10−15
Function tolerance
10−15
Nonlinear constraints tolerance
10−15
Maximum function evaluations
5000
X-tolerance
10−15
Derivative approximate
Finite central difference


## 3.1. Particle Swarm Optimization (PSO)

PSO algorithm is inspired from bird flocking and school fishing which was proposed by Kennedy and Eberhart in 1995 [19]. It can handle multimodal, discontinuous, and nonconvex optimization problems because it is a pseudooptimization method [20, 21]. The schematic diagram of PSO is shown in Figure 2, while its execution steps are given below.Figure 2
Generic flow diagram of Hybrid Particle Swarm Optimization.Step 1(initialization). Initialize the swarm randomly, that is, randomly producing Q particles. In this problem, each particle consists of 4∗K genes, where K represents the total number of available targets. Mathematically, it can be written as(8)B=b1,b2,b3,…,bQT,b1b2b3⋮bQ=θ1,1,θ1,2,…,θ1,K,ϕ1,K+1,ϕ1,K+2,…,ϕ1,2∗K,r1,2∗K+1,r1,2∗K+2,…,r1,3∗K,s1,3∗K+1,s1,3∗K+2,…,s1,4∗Kθ2,1,θ2,2,…,θ2,K,ϕ2,K+1,ϕ2,K+2,…,ϕ2,2∗K,r2,2∗K+1,r2,2∗K+2,…,r2,3∗K,s2,3∗K+1,s2,3∗K+2,…,s2,4∗Kθ3,1,θ3,2,…,θ3,K,ϕ3,K+1,ϕ3,K+2,…,ϕ3,2∗K,r3,2∗K+1,r3,2∗K+2,…,r3,3∗K,s3,3∗K+1,s3,3∗K+2,…,s3,4∗K⋮θQ,1,θQ,2,…,θQ,K,ϕQ,K+1,ϕQ,K+2,…,ϕQ,2∗K,rQ,2∗K+1,rQ,2∗K+2,…,rQ,3∗K,sQ,3∗K+1,sQ,3∗K+2,…,sQ,4∗K.The upper and lower bounds of θ, ϕ, r, and s are given as(9)θq,k∈R:0≤θq,k≤π2ϕq,K+k∈R:0≤ϕq,K+k≤2πrq,2K+k∈R:rl≤rq,2K+k≤rusq,3K+k∈R:sl≤sq,3K+k≤sufor,q=1,2,…,Q,k=1,2,…,K,where rl and ru represent the lower and upper limits of ranges, while su and sl are the upper and lower limits of amplitudes.Step 2(fitness function). We have used MSE as a fitness function to compute the fitness of every particle. For qth particle, it can be mathematically written as(10)FFq=11+ξq,where ξ(q) is called MSE which is derived from maximum likelihood principle [16]. The total MSE for qth particle can be mathematically written as (11)ξq=ξ1q+ξ2q,(12)ξ1q=12Px∑m=-PxPxwm,o-w^m,0q2,(13)ξ2q=12Py∑n=-PyPyw0,n-w^0,nq2,where Px and Py are the number of sensors placed along x-axis and y-axis, respectively. In (12) and (13), wm,0 and w0,n are given in (1) and (2), respectively, while, w^m,0q and w^0,nq can be defined as(14)w^m,0q=∑k=1Kb^3K+kqej-m2πdsin⁡b^kqcos⁡b^3K+kq/λ+m2πd21-sin2⁡b^kqcos2⁡b^K+kq/λb^2K+kq,w^0,nq=∑k=1Kb^3K+kqej-n2πdsin⁡b^Kqsin⁡b^K+kq/λ+n2πd21-sin2⁡b^kqsin2⁡b^K+kq/λb^2K+kq.Now, store each particle as local best (l) while the one having maximum fitness function is stored as global best (gb) particle.Step 3a(update particle velocity). We used the following relation to update the velocity of each particle:(15)vqmn=vqmn-1+σ11-γlqm-bqmn-1+σ2γgbm-bqmn-1.In the right hand side of (15), the first term is called inertia or momentum which shows the previous velocity. The second term shows the private thinking and is also called as cognitive term of the particle while the last term explains the collective behavior of the population and is known as a social component. At the beginning, the value of γ=0.1 to ensure that more weightage is given to local intelligence at the beginning; then there is gradual increase in the value of γ towards 0.9 which means that more weightage is given to collective intelligence at the end. In (15), both σ1 and σ2 are positive constants and for the ongoing problem σ1=σ2=1.The velocity is doubly bounded; that is,-vmax⁡≤vqm≤vmax⁡;ifvqmn>vmax⁡,vqmn=vmax⁡;ifvqm(n)<-vmax⁡,vqm(n)=-vmax⁡.vmax and vmin are taken to be 2 and −2, respectively, in order to avoid the premature convergence.Step 3b(update particle position). The position of each particle can be updated as(16)bqmn=bqmn-1+vqmn.Step 4a(choose local best particle). If fitness bq(n)>l(n), replace l with b.Step 4b(choose global best particle). Similarly replace the previous global best gb  particle with bq(n) as a new global best particle if fitness bq(n)>gb.Step 5(termination). Terminate, if any of the following conditions is satisfied; otherwise go to Step 3a:(i)
The predefined total number of iterations/flights is executed for the algorithm.(ii)
The desired MSE is attained which is10-7.Step 6(hybridization). In this step, the best particle achieved through PSO is given to ASA for further improvement.Step 7(storage). Store the results for later on discussion and comparison.


## 3.2. Differential Evolution (DE)

DE was introduced by Storn and Price in 1997 [22] which iteratively searches large spaces of candidate solution that improve the candidate solution with respect to specified measurement of quality. In other words, DE deals with the optimization problem in the following way: first it achieves the candidate solution and then, by using simple formula, it generates new solution by combining with the existing ones. As a result only those candidate solutions will survive which has best fitness or score. The block diagram of DE is shown in Figure 3, while its steps are given as follows.Figure 3
Generic flow diagram for hybrid DE.Step 1(initialization). The first step is similar to the one given for PSO in (8).Step 2(updating). Update all particles of the current generation “ge” from 1 to Q. Let us choose qth particle from (8), that is, bhq,ge, where q=1,2,…,Q and h=1,2,…,4×K, while “ge” represents the particular generation. The main goal of DE is to find out the particles of next generation, that is, cq,ge+1, by adopting the following steps:(a)
Mutation: select any three different numbers (particles) from 1 to Q, that is, (a1,a2,a3), by keeping the following conditions:(17)1≤a1,a2,a3≤Q,whereaq≠nh∀q,h=1,2,3,nq≠q∀q=1,2,3,dq,ge=ba1,ge+Fba2,ge-ba3,ge,where “F” is a constant whose value usually lies in the range 0.5 to 1.(b)
Crossover: the crossover can be performed as(18)okq,ge=dhq,geifrand⁡≤CR orh=hrandbhq,geotherwise,where0.5≤CR≤1 and hrand is between 1 and 4∗K chosen at random.(c)
Selection operation: in this step, select the particle of next generation by using the following criterion of selection:(19)bq,ge+1=oq,geiferr⁡oq,ge<err⁡bq,gebq,geotherwise,where theerr⁡(bq,ge) and err⁡(oq,ge) are defined in (8). Repeat this for all particles.Step 3(stopping criteria). The stopping criteria of DE are based on any of the following conditions being fulfilled:(I)
err⁡(bq,ge+1)<ε, where ε is a very small positive number OR.(II)
The number of iterations is completed.Else go back to Step 2.Step 4(hybridization). In order to further refine the results of DE, we give the best particle to ASA as starting point.


## 3.3. Active Set Algorithm (ASA)

Active set methods are recursive procedures for solving sequence of equality-constrained quadratic subproblems effectively from majority of algorithms based on interior methods and simplex techniques due to the fact that they operate in two phases; first they focus on feasibility, while others focus on optimality. The applications in which their dominance is well established are quadratic programming including portfolio analysis, structural analysis, and optimal control and optimal mesh refinement in ODE or PDE constrained problems, sparse linear programming problems, and box constrained optimization problem [23–25]. Active set quadratic programming is exploited in many state-of-the-art optimization solvers such as QPOPT [26], SQOPT [27], and QPA (part of the GALAHAD library) [28]. Although the global optimization methods (PSO and DE) alone can produced excellent results, their rate of convergence usually decreases with the increase of iteration especially in nonlinear problems. Therefore, to get accurate results, we need an efficient local search optimizer like ASA. The ASA is mainly used in constrained optimization problems whose basic purpose is to transform the problem into an easier solvable problem [29]. The standard working procedure of ASA in the form of pseudocode is given as follows, while its flow diagram in terms of processing blocks is shown in Figure 4.Figure 4
Generic flow diagram of Active Set Algorithm.Generic Pseudocode of Active Set AlgorithmStart. Initialize ASA algorithm with random assignment of parameters.Step   1.
Determine a suitable starting point.Step   2.
Repeat till an optimal condition is achieved.Step  3.
Solve the equality problem defined roughly by ASA.Step   4.
Calculate the Lagrange multipliers for ASA and remove all those constraints having negative Lagrange multipliers.End RepeatThe mathematical formulation of ASA algorithm is given as.The constrained optimization problem can be mathematically written as(20)min⁡fysuch thatl≤y≤u,where l and u represent the lower and upper bounds, respectively, of the desired vectors. In early methods, penalty functions were used to transform the constrained problem into unconstrained one which is beyond or near the constrained boundary. In such a way, the constrained problems were solved by a sequence of parameterized unconstrained optimizations, which is obviously an inefficient way, that are now replaced by methods which focus on the solution of Karush-Kuhn-Tucker (KKT) equations such as ASA. The KKT equations are the essential condition for constrained optimization problem. In order to achieve the global solution point, the KKT equations are the sufficient and essential conditions to be satisfied. The KKT equations can be mathematically written as(21)∇fy∗+∑i=1mλi·ΔGiy∗=0,(22)λi·Giy∗=0,i=1,…,me,where λi≥0, i=me+1,…,m, are called Lagrange multipliers. Equation (21) represents the gradient cancelation of active constraints and the objective function at the solution point where the Lagrange multipliers are necessary to balance the deviation in magnitude of the constrained gradients and objective function.For ASA, we used a MATLAB built-in optimization toolbox where the parameter setting is provided in Table1.Table 1
Parameters setting for ASA.
Parameters
Setting
Starting point
Best particles achieved by DE and PSO
Number of iterations
2000
Number of variables
4
×
K
Fitness limit
10−15
Function tolerance
10−15
Nonlinear constraints tolerance
10−15
Maximum function evaluations
5000
X-tolerance
10−15
Derivative approximate
Finite central difference


## 4. Results and Discussion

In this section, several simulations are carried out to validate the performance of the proposed techniques based on PSO, DE, PSO-ASA, and DE-ASAs. For theoretical consideration among the optimization techniques, the strength of memetic computing techniques based on PSO-ASA and DE-ASAs is superior to the rest. Comparison of the results of these nonconventional meta-heuristic techniques is presented here in number of graphical and numerical illustrations to establish memetic optimization methodologies in terms of estimation accuracy, reliability, robustness, and convergence.Simulation studies are present here in two parts; in part 1, we compared the performances of PSO, DE, PSO-ASA, and DE-ASA with each other in terms of estimation accuracy and convergence rate for different number of targets. In part 2, the performances of the two best techniques among them are compared with existing traditional algorithms [10, 11] by using MSE as a figure of merit. Every time, the number of sensors in both subarrays is taken to be same. In order to avoid the phase ambiguity, the interelement spacing between the consecutive elements d=λ/4 is required [10]. All the signals reflected back from targets are assumed to be statistically independent and having constant frequency. The received data at the output of each sensor are polluted by zero mean, unit variance AWGN. The values of elevation and azimuth angles are taken in degrees, while the values of ranges are taken in terms of wavelength (λ). All the simulations for PSO and DE are carried out for 1000 independent trials.


### 4.1. Estimation Accuracy

In this subsection, estimation accuracy of PSO, DE, PSO-ASA, and DE-ASA is discussed without adding any noise to the system. In Tables2–4, the estimation accuracy of all schemes is provided for two, three, and four sources, respectively, for different values of elevation angle, azimuth angle, range, and amplitudes. For this, the CSCA consists of 9, 13, and 17 sensors for 2, 3, and 4 targets, respectively. One can see from Tables 2–4 that each scheme produced better estimation accuracy for two targets but their accuracy degraded for three and four targets due to the presence of local minima. From these results, the advantages of hybridization are quite obvious as the estimation accuracy of PSO alone is less as compared to DE alone but its accuracy increases than DE when it is hybridized with ASA. Overall the DE-ASA produced better estimation accuracy as compared to the other three techniques. The second and third best results are given by PSO-ASA and DE alone, respectively.Table 2
Estimation accuracy for 2 targets.
Scheme
θ
1
∘
ϕ
1
∘
r
1
(
λ
)
s
1
θ
2
∘
ϕ
2
∘
r
2
(
λ
)
s
2
Desired values
35.0000
73.0000
1.5000
4.0000
52.0000
105.0000
3.0000
1.0000
PSO
35.0061
73.0060
1.4951
4.0050
52.0060
105.0061
3.1032
1.0051
DE
35.0044
72.0058
1.5022
3.9977
51.0058
104.9955
3.1022
0.9976
PSO-ASA
34.9967
73.0036
1.5018
4.0019
52.0036
105.0033
3.1017
1.0020
DE-ASA
35.0015
72.0082
1.4991
3.9988
51.0082
104.9984
3.1008
0.9988Table 3
Estimation accuracy for 3 targets.
Scheme
θ
1
°
ϕ
1
°
r
1
(
λ
)
s
1
θ
2
°
ϕ
2
°
r
2
(
λ
)
s
2
θ
3
°
ϕ
3
°
r
3
(
λ
)
s
3
Desired values
80.0000
40.0000
7.0000
2.0000
65.0000
160.0000
1.0000
8.0000
25.0000
120.0000
3.0000
6.0000
PSO
80.3843
39.6156
6.8209
2.1789
64.6158
159.6156
1.3847
8.6157
24.6158
120.1789
2.8209
5.8210
DE
79.8028
40.1973
6.9026
1.9028
65.1974
160.1973
1.1979
8.1972
25.1974
119.9028
2.9026
6.0971
PSO-ASA
79.9264
40.0738
7.0192
2.0191
64.9261
160.0738
0.9261
7.9263
24.9261
120.0191
3.0192
6.0193
DE-ASA
80.0268
39.9731
6.9932
1.9931
64.9733
159.9731
1.0269
8.0260
24.9733
119.9931
2.9932
5.9932Table 4
Estimation accuracy for 4 targets.
(a)
Scheme
θ
1
°
ϕ
1
°
r
1
(
λ
)
s
1
θ
2
°
ϕ
2
°
r
2
(
λ
)
s
2
Desired values
40.0000
120.0000
2.5000
3.0000
75.0000
50.0000
4.0000
9.0000
PSO
38.2202
121.8799
2.9797
3.8711
73.1884
48.1285
4.8686
9.9128
DE
41.2388
121.2490
2.8989
3.7010
73.8656
48.8375
4.6654
9.6655
PSO-ASA
40.4177
119.6279
2.6522
3.4805
75.4947
50.3942
4.4056
9.4060
DE-ASA
40.0709
120.0781
2.5791
3.0812
75.0869
50.0870
4.0867
9.0869
(b)
Scheme
θ
3
°
ϕ
3
°
r
3
(
λ
)
s
3
θ
4
°
ϕ
4
°
r
4
(
λ
)
s
4
Desired values
55.00000
160.0000
12.0000
6.0000
30.0000
220.0000
6.0000
4.0000
PSO
56.8282
161.8284
12.6582
5.2837
28.2814
218.1369
6.6182
4.8273
DE
53.8656
161.2659
12.4942
5.4766
28.7766
218.7751
5.6656
4.6659
PSO-ASA
55.4941
160.4658
12.2944
6.4041
30.5041
220.4041
6.32941
4.2946
DE-ASA
55.0870
160.0791
12.0912
6.0908
30.0988
220.0715
6.0870
4.0912In Table5, mean, variance, and standard deviation are carried out for all schemes in terms of RMSE. For this simulation, two targets are considered to be impinging on CSCA having nine sensors. It is apparent that again DE-ASA produced better results as compared to the other algorithms while the second and third best results are once again given by PSO-ASA and DE alone.Table 5
Mean, variance, and standard deviation in terms of RMSE for two targets.
Scheme
Mean
Variance
Standard deviation
PSO
0.23
E
-
02
1.0
E
-
03
31.6
E
-
03
DE
5.45
E
-
03
1.0
E
-
04
10.0
E
-
03
PSO-ASA
3.63
E
-
04
1.0
E
-
05
3.1
E
-
03
DE-ASA
8.19
E
-
06
1.0
E
-
07
0.31
E
-
03


### 4.2. Convergence

In this section, the convergence of each scheme for different number of targets is discussed. By convergence we mean the total number of times a particular scheme got its desire results. As shown in Figure5, the convergence of each scheme decreases with the increase of unknowns (targets) in the problem. However, the convergence of hybrid schemes are less degraded and they maintained fairly good convergence every time. The first and second best schemes among them are DE-ASA and PSO-ASA as they maintained better convergence. The third best convergence is given by DE alone. From the above discussion, the first and second best schemes are DE-ASA and PSO-ASA, respectively, so, to summarize the discussion, we shall be limited to the discussion of these two hybrid schemes only in the upcoming discussion.Figure 5
Convergence versus number of targets.


### 4.3. Proximity Effects

In this subsection, the results are carried out to assess the proximity effects of elevation and azimuth angles for DE-ASA and PSO-ASA. We considered 3 sources impinging on CSCA, composed of 13 sensors. As provided in Tables6 and 7, both of the hybrid schemes produced fairly good estimation accuracy and convergence for closely spaced elevation and azimuth angles. However, the DE-ASA produced better results as compared to PSO-ASA.Table 6
Proximity effect of elevation angles fors1=1,s2=3,s3=5,  r1=1.5λ,r2=3λ,r3=4λ,  and  ϕ1=130°,  ϕ2=70°,ϕ3=160°.
Scheme
θ
1
°
θ
2
°
θ
3
°
%convergence
Desired values
35.0000
75.0000
60.0000
—
PSO-ASA
35.3843
75.3842
60.3844
89
DE-ASA
35.1791
75.1790
60.1793
95
Desired values
30.0000
65.0000
75.0000
—
PSO-ASA
30.3846
65.9832
75.9834
81
DE-ASA
30.1792
65.4301
75.4302
92
Desired values
30.0000
40.0000
50.0000
—
PSO-ASA
31.3965
41.4011
51.4013
68
DE-ASA
30.7692
40.7694
50.7690
87
Desired values
30.0000
35.0000
40.0000
—
PSO-ASA
32.3417
37.3518
42.3519
62
DE-ASA
31.1105
36.1107
41.1105
80Table 7
Proximity effect of azimuth angles fors1=1,s2=3,s3=5,  r1=1.5λ,r2=3λ,r3=4λ , and  θ1=30°,θ2=50°,θ3=85°.
Scheme
∅
1
°
∅
2
°
∅
3
°
%convergence
Desired values
25.0000
80.0000
240.0000
—
PSO-ASA
24.6841
79.6170
239.6254
90
DE-ASA
25.1790
79.8210
240.1793
93
Desired values
50.0000
80.0000
70.0000
—
PSO-ASA
49.6154
80.9832
70.9832
81
DE-ASA
49.8210
80.4305
70.4303
90
Desired values
50.0000
60.0000
70.0000
—
PSO-ASA
48.6031
58.5981
71.4014
70
DE-ASA
50.7694
60.7696
70.7692
85
Desired values
50.0000
55.0000
60.0000
—
PSO-ASA
52.4420
57.3519
57.6468
64
DE-ASA
51.1104
56.1105
58.8967
82


### 4.4. Estimation Accuracy for DOA on Reference Axis

Some of the elevation angles(θ=0∘,90∘) and azimuth angles ϕ=(0∘,90∘,180∘) are considered to be critical angles where the performance of most algorithms degraded. We considered the three targets at (θ1=0∘,ϕ1=90∘,r1=3λ,s1=1), (θ2=90∘,ϕ2=0∘,r2=1λ,s2=2), and (θ3=35∘,ϕ3=180∘,r3=5λ,s3=4). In Figure 6(a), the estimation accuracy of both hybrid schemes for elevation angles is evaluated on reference axis (0∘ and 90∘). It can be seen that both hybrid schemes produced significant error (6∘ to 8∘) especially for 90∘. However, the DE-ASA produced comparatively less errors as compared to PSO-ASA.Figure 6
(a) Elevation angle estimation on reference axis. (b) Elevation angle estimation on reference axis.
(a)
(b)In Figure6(b), the estimation accuracy of both hybrid schemes is evaluated for azimuth angles at 0∘, 90∘, and 180∘. In this case, both schemes are up to the mark and produced negligible error. Once again the DE-ASA produced better results as compared to PSO-ASA.


### 4.5. Comparison with Other Techniques Using Root Mean Square Error (RMSE)

In this subsection, we compared the RMSE of DE-ASA and PSO-ASA with existing nonheuristic techniques [10, 11] in the presence of noise. We considered two sources, (θ1=25∘,ϕ1=55∘,r1=3.5λ,s1=1) and (θ2=60∘,ϕ2=170∘,r2=1λ,s2=3), where the CSCA consist of 17 sensors. In Figures 7(a)–7(c), both hybrid schemes produced lower RMSE as compared to the algorithms described in [10, 11]. The RMSE of both hybrid schemes is lower for the target located near to the array as compared to the target located comparatively away from the array as shown in Figure 7(c). Among all of them, the DE-ASA produced lower RMSE while the second best RMSE is obtained by the PSO-ASA scheme.Figure 7
(a) Root mean square error of elevation angles versus SNR. (b) Root mean square error of azimuth angles versus SNR. (c) Root mean square error of ranges versus SNR. (d) Root mean square error of amplitudes versus SNR.
(a)
(b)
(c)
(d)One of the other features of both hybrid schemes is that they can also be used for the amplitude estimation which is missing in [10, 11]. In Figure 7(d), RMSE is shown against SNR for the amplitude of the targets. Once again the DE-ASA scheme maintained a lower RMSE for both targets.


## 4.1. Estimation Accuracy

In this subsection, estimation accuracy of PSO, DE, PSO-ASA, and DE-ASA is discussed without adding any noise to the system. In Tables2–4, the estimation accuracy of all schemes is provided for two, three, and four sources, respectively, for different values of elevation angle, azimuth angle, range, and amplitudes. For this, the CSCA consists of 9, 13, and 17 sensors for 2, 3, and 4 targets, respectively. One can see from Tables 2–4 that each scheme produced better estimation accuracy for two targets but their accuracy degraded for three and four targets due to the presence of local minima. From these results, the advantages of hybridization are quite obvious as the estimation accuracy of PSO alone is less as compared to DE alone but its accuracy increases than DE when it is hybridized with ASA. Overall the DE-ASA produced better estimation accuracy as compared to the other three techniques. The second and third best results are given by PSO-ASA and DE alone, respectively.Table 2
Estimation accuracy for 2 targets.
Scheme
θ
1
∘
ϕ
1
∘
r
1
(
λ
)
s
1
θ
2
∘
ϕ
2
∘
r
2
(
λ
)
s
2
Desired values
35.0000
73.0000
1.5000
4.0000
52.0000
105.0000
3.0000
1.0000
PSO
35.0061
73.0060
1.4951
4.0050
52.0060
105.0061
3.1032
1.0051
DE
35.0044
72.0058
1.5022
3.9977
51.0058
104.9955
3.1022
0.9976
PSO-ASA
34.9967
73.0036
1.5018
4.0019
52.0036
105.0033
3.1017
1.0020
DE-ASA
35.0015
72.0082
1.4991
3.9988
51.0082
104.9984
3.1008
0.9988Table 3
Estimation accuracy for 3 targets.
Scheme
θ
1
°
ϕ
1
°
r
1
(
λ
)
s
1
θ
2
°
ϕ
2
°
r
2
(
λ
)
s
2
θ
3
°
ϕ
3
°
r
3
(
λ
)
s
3
Desired values
80.0000
40.0000
7.0000
2.0000
65.0000
160.0000
1.0000
8.0000
25.0000
120.0000
3.0000
6.0000
PSO
80.3843
39.6156
6.8209
2.1789
64.6158
159.6156
1.3847
8.6157
24.6158
120.1789
2.8209
5.8210
DE
79.8028
40.1973
6.9026
1.9028
65.1974
160.1973
1.1979
8.1972
25.1974
119.9028
2.9026
6.0971
PSO-ASA
79.9264
40.0738
7.0192
2.0191
64.9261
160.0738
0.9261
7.9263
24.9261
120.0191
3.0192
6.0193
DE-ASA
80.0268
39.9731
6.9932
1.9931
64.9733
159.9731
1.0269
8.0260
24.9733
119.9931
2.9932
5.9932Table 4
Estimation accuracy for 4 targets.
(a)
Scheme
θ
1
°
ϕ
1
°
r
1
(
λ
)
s
1
θ
2
°
ϕ
2
°
r
2
(
λ
)
s
2
Desired values
40.0000
120.0000
2.5000
3.0000
75.0000
50.0000
4.0000
9.0000
PSO
38.2202
121.8799
2.9797
3.8711
73.1884
48.1285
4.8686
9.9128
DE
41.2388
121.2490
2.8989
3.7010
73.8656
48.8375
4.6654
9.6655
PSO-ASA
40.4177
119.6279
2.6522
3.4805
75.4947
50.3942
4.4056
9.4060
DE-ASA
40.0709
120.0781
2.5791
3.0812
75.0869
50.0870
4.0867
9.0869
(b)
Scheme
θ
3
°
ϕ
3
°
r
3
(
λ
)
s
3
θ
4
°
ϕ
4
°
r
4
(
λ
)
s
4
Desired values
55.00000
160.0000
12.0000
6.0000
30.0000
220.0000
6.0000
4.0000
PSO
56.8282
161.8284
12.6582
5.2837
28.2814
218.1369
6.6182
4.8273
DE
53.8656
161.2659
12.4942
5.4766
28.7766
218.7751
5.6656
4.6659
PSO-ASA
55.4941
160.4658
12.2944
6.4041
30.5041
220.4041
6.32941
4.2946
DE-ASA
55.0870
160.0791
12.0912
6.0908
30.0988
220.0715
6.0870
4.0912In Table5, mean, variance, and standard deviation are carried out for all schemes in terms of RMSE. For this simulation, two targets are considered to be impinging on CSCA having nine sensors. It is apparent that again DE-ASA produced better results as compared to the other algorithms while the second and third best results are once again given by PSO-ASA and DE alone.Table 5
Mean, variance, and standard deviation in terms of RMSE for two targets.
Scheme
Mean
Variance
Standard deviation
PSO
0.23
E
-
02
1.0
E
-
03
31.6
E
-
03
DE
5.45
E
-
03
1.0
E
-
04
10.0
E
-
03
PSO-ASA
3.63
E
-
04
1.0
E
-
05
3.1
E
-
03
DE-ASA
8.19
E
-
06
1.0
E
-
07
0.31
E
-
03


## 4.2. Convergence

In this section, the convergence of each scheme for different number of targets is discussed. By convergence we mean the total number of times a particular scheme got its desire results. As shown in Figure5, the convergence of each scheme decreases with the increase of unknowns (targets) in the problem. However, the convergence of hybrid schemes are less degraded and they maintained fairly good convergence every time. The first and second best schemes among them are DE-ASA and PSO-ASA as they maintained better convergence. The third best convergence is given by DE alone. From the above discussion, the first and second best schemes are DE-ASA and PSO-ASA, respectively, so, to summarize the discussion, we shall be limited to the discussion of these two hybrid schemes only in the upcoming discussion.Figure 5
Convergence versus number of targets.


## 4.3. Proximity Effects

In this subsection, the results are carried out to assess the proximity effects of elevation and azimuth angles for DE-ASA and PSO-ASA. We considered 3 sources impinging on CSCA, composed of 13 sensors. As provided in Tables6 and 7, both of the hybrid schemes produced fairly good estimation accuracy and convergence for closely spaced elevation and azimuth angles. However, the DE-ASA produced better results as compared to PSO-ASA.Table 6
Proximity effect of elevation angles fors1=1,s2=3,s3=5,  r1=1.5λ,r2=3λ,r3=4λ,  and  ϕ1=130°,  ϕ2=70°,ϕ3=160°.
Scheme
θ
1
°
θ
2
°
θ
3
°
%convergence
Desired values
35.0000
75.0000
60.0000
—
PSO-ASA
35.3843
75.3842
60.3844
89
DE-ASA
35.1791
75.1790
60.1793
95
Desired values
30.0000
65.0000
75.0000
—
PSO-ASA
30.3846
65.9832
75.9834
81
DE-ASA
30.1792
65.4301
75.4302
92
Desired values
30.0000
40.0000
50.0000
—
PSO-ASA
31.3965
41.4011
51.4013
68
DE-ASA
30.7692
40.7694
50.7690
87
Desired values
30.0000
35.0000
40.0000
—
PSO-ASA
32.3417
37.3518
42.3519
62
DE-ASA
31.1105
36.1107
41.1105
80Table 7
Proximity effect of azimuth angles fors1=1,s2=3,s3=5,  r1=1.5λ,r2=3λ,r3=4λ , and  θ1=30°,θ2=50°,θ3=85°.
Scheme
∅
1
°
∅
2
°
∅
3
°
%convergence
Desired values
25.0000
80.0000
240.0000
—
PSO-ASA
24.6841
79.6170
239.6254
90
DE-ASA
25.1790
79.8210
240.1793
93
Desired values
50.0000
80.0000
70.0000
—
PSO-ASA
49.6154
80.9832
70.9832
81
DE-ASA
49.8210
80.4305
70.4303
90
Desired values
50.0000
60.0000
70.0000
—
PSO-ASA
48.6031
58.5981
71.4014
70
DE-ASA
50.7694
60.7696
70.7692
85
Desired values
50.0000
55.0000
60.0000
—
PSO-ASA
52.4420
57.3519
57.6468
64
DE-ASA
51.1104
56.1105
58.8967
82


## 4.4. Estimation Accuracy for DOA on Reference Axis

Some of the elevation angles(θ=0∘,90∘) and azimuth angles ϕ=(0∘,90∘,180∘) are considered to be critical angles where the performance of most algorithms degraded. We considered the three targets at (θ1=0∘,ϕ1=90∘,r1=3λ,s1=1), (θ2=90∘,ϕ2=0∘,r2=1λ,s2=2), and (θ3=35∘,ϕ3=180∘,r3=5λ,s3=4). In Figure 6(a), the estimation accuracy of both hybrid schemes for elevation angles is evaluated on reference axis (0∘ and 90∘). It can be seen that both hybrid schemes produced significant error (6∘ to 8∘) especially for 90∘. However, the DE-ASA produced comparatively less errors as compared to PSO-ASA.Figure 6
(a) Elevation angle estimation on reference axis. (b) Elevation angle estimation on reference axis.
(a)
(b)In Figure6(b), the estimation accuracy of both hybrid schemes is evaluated for azimuth angles at 0∘, 90∘, and 180∘. In this case, both schemes are up to the mark and produced negligible error. Once again the DE-ASA produced better results as compared to PSO-ASA.


## 4.5. Comparison with Other Techniques Using Root Mean Square Error (RMSE)

In this subsection, we compared the RMSE of DE-ASA and PSO-ASA with existing nonheuristic techniques [10, 11] in the presence of noise. We considered two sources, (θ1=25∘,ϕ1=55∘,r1=3.5λ,s1=1) and (θ2=60∘,ϕ2=170∘,r2=1λ,s2=3), where the CSCA consist of 17 sensors. In Figures 7(a)–7(c), both hybrid schemes produced lower RMSE as compared to the algorithms described in [10, 11]. The RMSE of both hybrid schemes is lower for the target located near to the array as compared to the target located comparatively away from the array as shown in Figure 7(c). Among all of them, the DE-ASA produced lower RMSE while the second best RMSE is obtained by the PSO-ASA scheme.Figure 7
(a) Root mean square error of elevation angles versus SNR. (b) Root mean square error of azimuth angles versus SNR. (c) Root mean square error of ranges versus SNR. (d) Root mean square error of amplitudes versus SNR.
(a)
(b)
(c)
(d)One of the other features of both hybrid schemes is that they can also be used for the amplitude estimation which is missing in [10, 11]. In Figure 7(d), RMSE is shown against SNR for the amplitude of the targets. Once again the DE-ASA scheme maintained a lower RMSE for both targets.


## 5. Conclusion and Future Work Direction

In this work, we developed Heuristic Computational Intelligence techniques to estimate jointly the amplitude, range, and elevation and azimuth angle of multiple targets impinging on the passive CSCA. In these computational techniques, first Particle Swarm Optimization (PSO) and Differential Evolution (DE) were used alone and then, to improve the results further, they were hybridized with ASA. It has been found that the hybrid DE-ASA schemes produced better results as compared to PSO and DE in terms of estimation accuracy and convergence and have also shown better results as compared to nonheuristic techniques in terms of RMSE.Besides all the advantages, the proposed schemes have also few limitations:(i)
All the schemes fail if the number of targets increases to that of the number of sensors in the CSCA.(ii)
The performance of the proposed schemes degraded when the number of sources is more than four.In the future, one can use such kind of techniques in the following areas:(i)
This kind of model for finding range, amplitude, and elevation angle can effectively be used for spectrum sensing in the cognitive radio domain in which the source shall be mainly primary user station. With each snapshot, one can find these updated parameters.(ii)
One can use such kind of algorithms for null steering, sidelobe reductions, and main beam steering.



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*Source: 982967-2015-08-31.xml*