# Joint Angle-Amplitude Estimation for Multiple Signals with L-Structured Arrays Using Bioinspired Computing

Authors: Fawad Zaman
Journal: Wireless Communications and Mobile Computing
 (2017)

---

## Abstract

The aim of this work is to estimate jointly the elevation and azimuth angles along with the amplitudes of multiple signals impinging on 1-L- and 2-L-shape arrays. An efficient mechanism based on hybrid Bioinspired techniques is proposed for this purpose. The global search optimizers such as Differential Evolution (DE) and Particle Swarm optimization (PSO) are hybridized with a local search optimizer called pattern search (PS). Approximation theory in Mean Square Error sense is exploited to develop a fitness function of the problem. The unknown parameters of multiple signals transmitted by far-field sources are estimated with the strength of hybrid DE-PS and PSO-PS. The effectiveness of the proposed techniques is tested in terms of estimation accuracy, proximity effect, convergence, and computational complexity.


---

## Body

## 1. Introduction

Parameter estimation such as Direction of Arrival (DOA) of electromagnetic signals for multiple sources is one of the vital areas of research in array signal processing from the last few decades. The DOA estimation ensues in adaptive beamforming to place the nulls in the direction of jammers or unwanted signals, while placing the main beam in the desired direction [1–3]. Therefore, various schemes are investigated for DOA estimation in literature but mostly limited to one-dimensional (1D) DOA estimation of sources impinging on linear array [4–6]. The 1D DOA estimation is comparatively straight forward because it only involves the angle of elevation as function of DOA. However, 2D DOA estimation is relatively complicated as it involves both elevation and azimuth angles [7–11]. 2D DOA estimation has engaged the interest of research community in the field of radar, sonar, wireless communication systems, and so forth [12]. The estimation failure, pair matching between elevation and azimuth angles, and computational complexity are the major problems observed in 2D DOA estimation. The different array configurations have been proposed to address the 2D DOA estimation problem [13–15] but lack the requisite resources to resolve the aforementioned problems. In [16], Propagator Method (PM) with parallel shape array is used to overcome the computational complexity of [15], but the estimation failure and pair matching problems still exist. In [17], the PM is applied with L-shape array to surmount the drawbacks of [16] but requires a large number of snapshots and sensors.In the present scientific society era, the significance of Evolutionary Computing Techniques (ECT) that include Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE) cannot be vilipended. These techniques are not only easy to implement but also have the significant ability of hybridization with other heuristic and nonheuristic techniques. Therefore, the significance of these techniques is realized in different varying nature of applications [18–22].In this paper, DE and PSO are hybridized with PS to jointly estimate the amplitude and 2D DOA of far-field sources impinging on 1-L- and 2-L-shape arrays. In this hybridization process, DE and PSO act as global search optimizers, while PS is used as rapid local search optimizer. The best individual results of DE and PSO are given to PS for further tuning. The performance criterion is devised on the basis of Mean Square Error (MSE) that is applied as an objective evaluation function. This fitness function is derived from Maximum Likelihood Principle (MLP) [21]. The interest behind this objective function is ease in concept, ease in implementation, and robustness against noise and requires single snapshot. The performance of hybrid (DE-PS) and hybrid (PSO-PS) is first compared with DE and PSO techniques. Moreover, the results are also compared with the existing PM, which uses parallel [16] and L-shape arrays [17].The rest of the paper is organized as follows. Section2 formulates the problem, while Section 3 describes the proposed methodology structure. In Section 4 results and discussion are provided, while Section 5 is dedicated for conclusion and future work directions.


## 2. Data Model

In this section, data model is developed forP independent sources impinging on 1-L- and 2-L-shape arrays from far field. The 1-L-shape array consists of two subarrays, which are placed along x-axis and z-axis. The 2-L-shape array is similar to 1-L-shape array except that it has an extra subarray along y-axis. The number of antenna elements in each subarray is M
-
1, while the reference element is common for all subarrays as shown in Figures 1 and 2. The distance “d” between two consecutive elements in each subarray is λ
/
2. All the impinging signals are considered to be narrow band with known frequency (
ω
0
) by having different amplitudes s
i, elevation angles (
α
i
), and azimuth angles (
β
i
), where i
=
1,2
,
…
,
P.Figure 1
1-L-shape array having 2 subarrays.Figure 2
2-L-shape array having 3 subarrays.(a) 1-L-Shape Array. The output at lth sensor of z-axis subarray due to P sources is given as(1)
y
z
l
1
L
=
∑
i
=
1
P
a
z
l
α
i
s
i
+
n
z
l
.In (1), y
z
l
(
1L) means the response of lth element placed in z-axis subarray of 1-L-shape array, while a
z
l
α
i
=
exp
⁡
-
j
2
π
l
d
cos
⁡
α
i
/
λ and, for d
=
λ
/
2  and  l
=
0,1
,
2
,
…
,
M
-
1, the output of complete subarray can be given as (2)
y
0
y
1
⋮
y
M
-
1
z
=
1
⋯
1
e
-
j
π
cos
⁡
α
1
⋯
e
-
j
π
cos
⁡
α
p
⋮
⋯
⋮
e
-
j
M
-
1
π
cos
⁡
α
1
⋯
e
-
j
M
-
1
π
cos
⁡
α
P
s
1
s
2
⋮
s
P
+
n
0
n
1
⋮
n
M
-
1
z
.Generally, it can be written as(3)
y
z
=
A
z
α
s
+
n
z
.In (3), A
z is called steering matrix which contains the elevation angles of the received signals, while n
z is an Additive White Gaussian Noise (AWGN) added at the output of each sensor along z-axis. In the same way, the response of x-axis subarray for lth sensor can be given as (4)
y
x
l
1
L
=
∑
i
=
1
P
a
x
l
α
i
,
β
i
s
i
+
n
x
l
.In (4), a
x
l
(
α
i
,
β
i
)
=
exp
⁡
-
j
2
π
l
d
cos
⁡
α
i
cos
⁡
β
i
/
λ and, for d
=
λ
/
2 and l
=
0,1
,
2
,
…
,
M
-
1, we have(5)
y
0
y
1
⋮
y
M
-
1
x
=
1
⋯
1
e
-
j
π
sin
⁡
α
1
cos
⁡
β
1
⋯
e
-
j
π
sin
⁡
α
P
cos
⁡
β
P
⋮
⋯
⋮
e
-
j
M
-
1
π
sin
⁡
α
1
cos
⁡
β
1
⋯
e
-
j
M
-
1
π
sin
⁡
α
P
cos
⁡
β
P
s
1
s
2
⋮
s
P
+
n
0
n
1
⋮
n
M
-
1
x
.Generally,(6)
y
x
=
A
x
α
,
β
s
+
n
x
.(b) 2-L-Shape Array. In case of 2-L-shape array, the output of x-axis and z-axis subarrays is exactly similar as discussed for 1-L-shape array; however, the output of y-axis subarray at lth sensor can be modeled as(7)
y
y
l
2
L
=
∑
i
=
1
P
a
y
l
α
i
,
β
i
s
i
+
n
y
l
.In (7), a
y
l
(
α
i
,
β
i
)
=
exp
⁡
-
j
2
π
l
d
sin
⁡
α
i
sin
⁡
β
i
/
λ, and (8)
y
0
y
1
⋮
y
M
-
1
y
=
1
⋯
1
e
-
j
π
sin
⁡
α
1
sin
⁡
β
1
⋯
e
-
j
π
sin
⁡
α
P
sin
⁡
β
P
⋮
⋯
⋮
e
-
j
M
-
1
π
sin
⁡
α
1
sin
⁡
β
1
⋯
e
-
j
M
-
1
π
sin
⁡
α
P
sin
⁡
β
P
s
1
s
2
⋮
s
P
+
n
0
n
1
⋮
n
M
-
1
y
,
(9)
y
y
=
A
y
α
,
β
s
+
n
y
.From (2), (5), and (8), it is clear that the unknown signal parameters are amplitudes (
s
i
), elevation angles (
α
i
), and azimuth angles (
β
i
) to be estimated for i
=
1,2
,
…
,
P.


## 3. Proposed Techniques

In this section, we have discussed the brief introduction, flow chart, parameter setting, and pseudocode of PS, PSO, and DE.


### 3.1. Pattern Search

Pattern Search (PS) which is also called direct search method belongs to a family of numerical optimization techniques. It can be used for linearly constrained and bounded optimization problem, which does not require the gradient of the problem. PS works on a group of points called a pattern. If improvement does not occur in the objective function at the current iteration, the pattern is refined and the process is repeated [23].The PS acts even well when it is used as a local search optimizer with any other global optimizer technique such as GA, PSO, and DE. In the present work, PS is used as a local search optimizer with DE and PSO. The best particle achieved through DE and PSO is given as starting point to PS for further tuning. For PS, we have used a MATLAB built-in optimization tool box for which the parameters settings are given in Table1.Table 1
Parameter setting for PS.
Parameters
Setting
Starting point
The best particle achieved through PSO or DE
Poll method
GPS positive basis 2 N
Polling order
Consecutive
Maximum iteration
800
Function evaluation
15000
Mesh size
01
Expansion factor
2.0
Contraction factor
0.5
Penalty factor
100
Bind tolerance
10-07
Mesh tolerance
10-09
X tolerance
10-09


### 3.2. Particle Swarm Optimization (PSO)

The idea of PSO was first introduced by Kennedy and Eberhart being inspired from the group of birds flocking for food in random manner [24]. It is an Evolutionary Computing Technique (ECT) based on iterative process in order to obtain the best optimization. The idea to search for food is heuristic because all the birds know their distance from the food but do not actually know the location. They find their food by sharing their search information in cooperative behavior in contrast to GA by crossover and kids production method. The PSO has acquired applications almost in every field of engineering [25, 26]. In the current work, PSO is used as global optimizer and the results of PSO are further provided to PS for substantial improvement. For both L-shape arrays, the implementation of PSO is quite similar with some instance of divergence. The generic flow diagram for PSO is shown in Figure 3(a), whereas the steps involved in the implementation of PSO are characterized in the form of pseudocodes.Figure 3Step 1 (initialization). Like other ECT, the first step is to produce randomly N number of particles. The length of each particle depends on the number of the unknowns in the problem. In the present case, the length of each particle is 3
×
P, where P is the total number of sources. The first P genes in each particle represent elevation angles and the next P genes represent azimuth angles, while the last P genes correspond to the amplitudes of sources. The particles in case of 1-L-shape array are produced as(10)
C
=
α
1,1
α
1,2
⋯
α
1
,
P
β
1
,
P
+
1
β
1
,
P
+
2
⋯
β
1,2
P
s
1,2
P
+
1
s
1,2
P
+
2
⋯
s
1,3
P
α
2,1
α
2,2
⋯
α
2
,
P
β
2
,
P
+
1
β
2
,
P
+
2
⋯
β
2,2
P
s
2,2
P
+
1
s
2,2
P
+
2
⋯
s
2,3
P
⋮
⋮
⋮
α
N
,
1
α
N
,
2
⋯
α
N
,
P
β
N
,
P
+
1
β
N
,
P
+
2
⋯
β
N
,
2
P
s
N
,
2
P
+
1
s
N
,
2
P
+
2
⋯
s
N
,
3
P
.In the same way, N number of random particles for 2-L-shape array can be generated. The major difference between the particles of 1-L- and 2-L-shape array is the range of elevation angle. For 1-L-shape array, the genes of elevation angles are generated from 0 to π
/
2, while, for 2-L-shape array, the genes are produced for the range of elevation angles from 0 to π. The range of amplitudes and azimuth angles are the same for both L-shape arrays. The lower and upper ranges of α, β, and s are defined as(11)
α
i
,
k
1
L
∈
R
:
0
≤
α
i
,
k
≤
π
2
α
i
,
k
2
L
∈
R
:
0
≤
α
i
,
k
≤
π
β
i
,
P
+
k
∈
R
:
0
≤
β
i
,
P
+
k
≤
2
π
s
i
,
2
P
+
k
∈
R
:
L
b
≤
s
i
,
2
P
+
k
≤
U
b
for
,
i
=
1,2
,
…
,
N
,
k
=
1,2
,
…
,
P
,where L
b and U
b represent the lower and upper bounds of signals amplitude.Step 2 (fitness function). MSE is used as fitness evaluation function for both L-shape arrays which can be given as(12)
F
F
i
=
1
1
+
ζ
i
,where ζ
(
i
) defines the difference function for ith particle. For 1-L-shape array it can be given as(13)
ζ
=
1
2
M
ζ
1
+
ζ
2
.For 2-L-shape array it can be defined as(14)
ζ
=
1
3
M
ζ
1
+
ζ
2
+
ζ
3
.In (13) and (14), (15)
ζ
1
i
=
∑
l
=
0
M
-
1
y
z
l
-
y
^
z
l
i
2
,
ζ
2
i
=
∑
l
=
0
M
-
1
y
x
l
-
y
^
x
l
i
2
,
ζ
3
i
=
∑
l
=
0
M
-
1
y
y
l
-
y
^
y
l
i
2
.In (15), y
z
l
,
y
x
l
,
y
y
l are defined in (1), (2), and (5) respectively, while y
^
z
l
i
,
y
^
x
l
i
,
y
^
y
l
i are defined as(16)
y
^
z
l
i
=
∑
k
=
1
P
exp
⁡
-
j
π
l
cos
⁡
c
k
i
·
c
2
P
+
k
i
,
y
^
x
l
i
=
∑
k
=
1
P
exp
⁡
-
j
π
l
sin
⁡
c
k
i
cos
⁡
c
P
+
k
i
·
c
2
P
+
k
i
,
y
^
y
l
i
=
∑
k
=
1
P
exp
⁡
-
j
π
l
sin
⁡
c
k
i
sin
⁡
c
P
+
k
i
·
c
2
P
+
k
i
,where c
i is defined in (10). The particle as global best g
b is stored that has maximum fitness function, while marking each c
i as local best l
i for this step where i
=
1,2
,
…
,
N.Step 3 (termination criterion). If the desired fitness function is achieved, stop and go to Step 7; otherwise go to Step 4
a.Step 4a (updating particle velocity). To update the velocity of each particle generated for both L-shape arrays, we used the following relation:(17)
v
i
k
n
=
v
i
k
n
-
1
+
ϕ
1
1
-
γ
l
i
k
-
c
i
k
n
-
1
+
ϕ
2
γ
g
b
-
c
i
k
n
-
1
,where ϕ
1
=
ϕ
2
=
1 while 0.1
≤
γ
≤
0.9. Initially, the value of γ is taken as 0.1 and then gradually increases from 0.1 to 0.9. Therefore, in the beginning more weight factor or ratio is given to local intelligence, whereas at the end more importance is provided to collective intelligence. Hence, the velocity is doubly bounded(18)
-
v
m
a
x
≤
v
i
k
≤
v
m
a
x
,
If
v
i
k
n
>
v
m
a
x
,
v
i
k
n
=
v
m
a
x
,
if
v
i
k
n
<
-
v
m
a
x
,
v
i
k
n
=
-
v
m
a
x
.Step 4b (updating position). To update the position of particles, the following relation is used: (19)
c
i
k
n
=
c
i
k
n
-
1
+
v
i
k
n
.Step 5a (choosing local best particle). The previous local best l
(
n
) is replaced with c
(
n
), if fitness of c
n
> fitness of l
(
n
).Step 5b (choosing global best particle). Also replace g
b with c
(
n
) as new global best if fitness of c
(
n
)
>
g
b
.Step 6 (termination). The termination criteria are based on the achievement of the following goals:(i)
The required fitness function is achieved.
(ii)
The desired MSE has been reached which is10
-
7.
(iii)
The total number of iterations has been completed.Step 7 (hybridization). In this step, the best values of amplitude and DOA got for both L-shape arrays are given as starting point to PS for further refinement. The parameter setting for PS is provided in Table 1.Step 8 (storage). Store all the results for later in discussion and comparison.


### 3.3. Differential Evolution (DE)

The Differential Evolution (DE) was first introduced by Stone and Price in 1996 [27] and belongs to the family of nature inspired techniques. It is simple and easy to cope, perform well in the presence of local minima, and has more randomness as compared to GA and PSO. The flow diagram for DE is shown in Figure 3(b), while the algorithm steps in the form of pseudocode are given.Step 1 (initialization). The initialization step is similar to the one developed for PSO in (10).Step 2 (updating). In this step, all chromosomes (particles) from 1 to N of the current generation “ge” are updated. Suppose that we select ith chromosome, that is, c
k
i
,
g
e from (7), where i
=
1,2
,
…
,
N and k
=
1,2
,
…
,
3
×
P and “ge” represents the particular generation. Now the goal is to find the chromosome of next generation, that is, c
i
,
g
e
+
1, by using the following steps.(a) Mutation. In this step, pick up any three different numbers (chromosomes) from 1 to N, that is, (
n
1
,
n
2
,
n
3
), under the following conditions: (20)
1
≤
n
1
,
n
2
,
n
3
≤
N
,where(21)
n
i
≠
n
k
∀
i
,
k
=
1,2
,
3
,
n
i
≠
i
∀
i
=
1,2
,
3
;now,(22)
d
i
,
g
e
=
c
n
1
,
g
e
+
F
c
n
2
,
g
e
-
c
n
3
,
g
e
,where “F” is a constant whose values usually lie in the range 0.5 to 1.(b) Crossover. The crossover can be performed as(23)
o
k
i
,
g
e
=
d
k
i
,
g
e
if rand
≤
C
R
or
k
=
k
r
a
n
d
c
k
i
,
g
e
o
w
,where 0.5
≤
C
R
≤
1 and k
r
a
n
d is between 1 and 3
∗
P chosen at random.(c) Selection Operation. The selection operation for the chromosome of next generation is performed as(24)
c
i
,
g
e
+
1
=
o
i
,
g
e
if err
o
i
,
g
e
<
err
⁡
c
i
,
g
e
c
i
,
g
e
o
w
,where e
r
r
(
c
i
,
g
e
) and e
r
r
(
o
i
,
g
e
) are defined in (15). Repeat this for all chromosomes.(I)
e
r
r
(
c
i
,
g
e
+
1
)
<
ε (where ε is a very small positive number).
(II)
Total number of generations has completed.Else, go back to Step 2.Step 3 (termination). The termination criterion of DE is based on the following results achieved.Step 4 (hybridization). In this step, the best results achieved through DE are given to PS for further refinement.Step 5 (storage). Store all the results for later in discussion and comparison.


## 3.1. Pattern Search

Pattern Search (PS) which is also called direct search method belongs to a family of numerical optimization techniques. It can be used for linearly constrained and bounded optimization problem, which does not require the gradient of the problem. PS works on a group of points called a pattern. If improvement does not occur in the objective function at the current iteration, the pattern is refined and the process is repeated [23].The PS acts even well when it is used as a local search optimizer with any other global optimizer technique such as GA, PSO, and DE. In the present work, PS is used as a local search optimizer with DE and PSO. The best particle achieved through DE and PSO is given as starting point to PS for further tuning. For PS, we have used a MATLAB built-in optimization tool box for which the parameters settings are given in Table1.Table 1
Parameter setting for PS.
Parameters
Setting
Starting point
The best particle achieved through PSO or DE
Poll method
GPS positive basis 2 N
Polling order
Consecutive
Maximum iteration
800
Function evaluation
15000
Mesh size
01
Expansion factor
2.0
Contraction factor
0.5
Penalty factor
100
Bind tolerance
10-07
Mesh tolerance
10-09
X tolerance
10-09


## 3.2. Particle Swarm Optimization (PSO)

The idea of PSO was first introduced by Kennedy and Eberhart being inspired from the group of birds flocking for food in random manner [24]. It is an Evolutionary Computing Technique (ECT) based on iterative process in order to obtain the best optimization. The idea to search for food is heuristic because all the birds know their distance from the food but do not actually know the location. They find their food by sharing their search information in cooperative behavior in contrast to GA by crossover and kids production method. The PSO has acquired applications almost in every field of engineering [25, 26]. In the current work, PSO is used as global optimizer and the results of PSO are further provided to PS for substantial improvement. For both L-shape arrays, the implementation of PSO is quite similar with some instance of divergence. The generic flow diagram for PSO is shown in Figure 3(a), whereas the steps involved in the implementation of PSO are characterized in the form of pseudocodes.Figure 3Step 1 (initialization). Like other ECT, the first step is to produce randomly N number of particles. The length of each particle depends on the number of the unknowns in the problem. In the present case, the length of each particle is 3
×
P, where P is the total number of sources. The first P genes in each particle represent elevation angles and the next P genes represent azimuth angles, while the last P genes correspond to the amplitudes of sources. The particles in case of 1-L-shape array are produced as(10)
C
=
α
1,1
α
1,2
⋯
α
1
,
P
β
1
,
P
+
1
β
1
,
P
+
2
⋯
β
1,2
P
s
1,2
P
+
1
s
1,2
P
+
2
⋯
s
1,3
P
α
2,1
α
2,2
⋯
α
2
,
P
β
2
,
P
+
1
β
2
,
P
+
2
⋯
β
2,2
P
s
2,2
P
+
1
s
2,2
P
+
2
⋯
s
2,3
P
⋮
⋮
⋮
α
N
,
1
α
N
,
2
⋯
α
N
,
P
β
N
,
P
+
1
β
N
,
P
+
2
⋯
β
N
,
2
P
s
N
,
2
P
+
1
s
N
,
2
P
+
2
⋯
s
N
,
3
P
.In the same way, N number of random particles for 2-L-shape array can be generated. The major difference between the particles of 1-L- and 2-L-shape array is the range of elevation angle. For 1-L-shape array, the genes of elevation angles are generated from 0 to π
/
2, while, for 2-L-shape array, the genes are produced for the range of elevation angles from 0 to π. The range of amplitudes and azimuth angles are the same for both L-shape arrays. The lower and upper ranges of α, β, and s are defined as(11)
α
i
,
k
1
L
∈
R
:
0
≤
α
i
,
k
≤
π
2
α
i
,
k
2
L
∈
R
:
0
≤
α
i
,
k
≤
π
β
i
,
P
+
k
∈
R
:
0
≤
β
i
,
P
+
k
≤
2
π
s
i
,
2
P
+
k
∈
R
:
L
b
≤
s
i
,
2
P
+
k
≤
U
b
for
,
i
=
1,2
,
…
,
N
,
k
=
1,2
,
…
,
P
,where L
b and U
b represent the lower and upper bounds of signals amplitude.Step 2 (fitness function). MSE is used as fitness evaluation function for both L-shape arrays which can be given as(12)
F
F
i
=
1
1
+
ζ
i
,where ζ
(
i
) defines the difference function for ith particle. For 1-L-shape array it can be given as(13)
ζ
=
1
2
M
ζ
1
+
ζ
2
.For 2-L-shape array it can be defined as(14)
ζ
=
1
3
M
ζ
1
+
ζ
2
+
ζ
3
.In (13) and (14), (15)
ζ
1
i
=
∑
l
=
0
M
-
1
y
z
l
-
y
^
z
l
i
2
,
ζ
2
i
=
∑
l
=
0
M
-
1
y
x
l
-
y
^
x
l
i
2
,
ζ
3
i
=
∑
l
=
0
M
-
1
y
y
l
-
y
^
y
l
i
2
.In (15), y
z
l
,
y
x
l
,
y
y
l are defined in (1), (2), and (5) respectively, while y
^
z
l
i
,
y
^
x
l
i
,
y
^
y
l
i are defined as(16)
y
^
z
l
i
=
∑
k
=
1
P
exp
⁡
-
j
π
l
cos
⁡
c
k
i
·
c
2
P
+
k
i
,
y
^
x
l
i
=
∑
k
=
1
P
exp
⁡
-
j
π
l
sin
⁡
c
k
i
cos
⁡
c
P
+
k
i
·
c
2
P
+
k
i
,
y
^
y
l
i
=
∑
k
=
1
P
exp
⁡
-
j
π
l
sin
⁡
c
k
i
sin
⁡
c
P
+
k
i
·
c
2
P
+
k
i
,where c
i is defined in (10). The particle as global best g
b is stored that has maximum fitness function, while marking each c
i as local best l
i for this step where i
=
1,2
,
…
,
N.Step 3 (termination criterion). If the desired fitness function is achieved, stop and go to Step 7; otherwise go to Step 4
a.Step 4a (updating particle velocity). To update the velocity of each particle generated for both L-shape arrays, we used the following relation:(17)
v
i
k
n
=
v
i
k
n
-
1
+
ϕ
1
1
-
γ
l
i
k
-
c
i
k
n
-
1
+
ϕ
2
γ
g
b
-
c
i
k
n
-
1
,where ϕ
1
=
ϕ
2
=
1 while 0.1
≤
γ
≤
0.9. Initially, the value of γ is taken as 0.1 and then gradually increases from 0.1 to 0.9. Therefore, in the beginning more weight factor or ratio is given to local intelligence, whereas at the end more importance is provided to collective intelligence. Hence, the velocity is doubly bounded(18)
-
v
m
a
x
≤
v
i
k
≤
v
m
a
x
,
If
v
i
k
n
>
v
m
a
x
,
v
i
k
n
=
v
m
a
x
,
if
v
i
k
n
<
-
v
m
a
x
,
v
i
k
n
=
-
v
m
a
x
.Step 4b (updating position). To update the position of particles, the following relation is used: (19)
c
i
k
n
=
c
i
k
n
-
1
+
v
i
k
n
.Step 5a (choosing local best particle). The previous local best l
(
n
) is replaced with c
(
n
), if fitness of c
n
> fitness of l
(
n
).Step 5b (choosing global best particle). Also replace g
b with c
(
n
) as new global best if fitness of c
(
n
)
>
g
b
.Step 6 (termination). The termination criteria are based on the achievement of the following goals:(i)
The required fitness function is achieved.
(ii)
The desired MSE has been reached which is10
-
7.
(iii)
The total number of iterations has been completed.Step 7 (hybridization). In this step, the best values of amplitude and DOA got for both L-shape arrays are given as starting point to PS for further refinement. The parameter setting for PS is provided in Table 1.Step 8 (storage). Store all the results for later in discussion and comparison.


## 3.3. Differential Evolution (DE)

The Differential Evolution (DE) was first introduced by Stone and Price in 1996 [27] and belongs to the family of nature inspired techniques. It is simple and easy to cope, perform well in the presence of local minima, and has more randomness as compared to GA and PSO. The flow diagram for DE is shown in Figure 3(b), while the algorithm steps in the form of pseudocode are given.Step 1 (initialization). The initialization step is similar to the one developed for PSO in (10).Step 2 (updating). In this step, all chromosomes (particles) from 1 to N of the current generation “ge” are updated. Suppose that we select ith chromosome, that is, c
k
i
,
g
e from (7), where i
=
1,2
,
…
,
N and k
=
1,2
,
…
,
3
×
P and “ge” represents the particular generation. Now the goal is to find the chromosome of next generation, that is, c
i
,
g
e
+
1, by using the following steps.(a) Mutation. In this step, pick up any three different numbers (chromosomes) from 1 to N, that is, (
n
1
,
n
2
,
n
3
), under the following conditions: (20)
1
≤
n
1
,
n
2
,
n
3
≤
N
,where(21)
n
i
≠
n
k
∀
i
,
k
=
1,2
,
3
,
n
i
≠
i
∀
i
=
1,2
,
3
;now,(22)
d
i
,
g
e
=
c
n
1
,
g
e
+
F
c
n
2
,
g
e
-
c
n
3
,
g
e
,where “F” is a constant whose values usually lie in the range 0.5 to 1.(b) Crossover. The crossover can be performed as(23)
o
k
i
,
g
e
=
d
k
i
,
g
e
if rand
≤
C
R
or
k
=
k
r
a
n
d
c
k
i
,
g
e
o
w
,where 0.5
≤
C
R
≤
1 and k
r
a
n
d is between 1 and 3
∗
P chosen at random.(c) Selection Operation. The selection operation for the chromosome of next generation is performed as(24)
c
i
,
g
e
+
1
=
o
i
,
g
e
if err
o
i
,
g
e
<
err
⁡
c
i
,
g
e
c
i
,
g
e
o
w
,where e
r
r
(
c
i
,
g
e
) and e
r
r
(
o
i
,
g
e
) are defined in (15). Repeat this for all chromosomes.(I)
e
r
r
(
c
i
,
g
e
+
1
)
<
ε (where ε is a very small positive number).
(II)
Total number of generations has completed.Else, go back to Step 2.Step 3 (termination). The termination criterion of DE is based on the following results achieved.Step 4 (hybridization). In this step, the best results achieved through DE are given to PS for further refinement.Step 5 (storage). Store all the results for later in discussion and comparison.


## 4. Results and Discussion

In the first part of simulations, various results are compared to evaluate the estimation accuracy and reliability of PSO, PSO-PS, DE, and DE-PS for the joint estimation of amplitudes and DOA (elevation and azimuth) of far filed sources impinging on 1-L- and 2-L-shape arrays. In the second part of simulation, the comparison is carried out with PM, which has used parallel shape array [16] and L-shape arrays [17]. We have used 60 particles and 60 generations for PSO and DE, respectively. Each result is averaged over 100 independent runs.Case 1.
In this case, the estimation accuracy of PSO, DE, PSO-PS, and DE-PS is examined for 1-L- and 2-L-shape arrays without having any noise in the system. Two sources are considered which have amplitudes{
s
1
=
0.5
,
s
2
=
2
} and DOA α
1
=
30
°
,
β
1
=
110
°
,
α
2
=
70
°
,
β
2
=
210
°
. The 1-L-shape array consists of 5 sensors; that is, 2 sensors are placed along both the subarrays, that is, along x-axis and along z-axis, while the reference sensor is common for them. As provided in Table 2, one can clearly observe that the estimation accuracy of DE and PSO has been increased when they are hybridized with PS. The accuracy of PSO alone is less than DE but as the PSO is hybridized with PS, the estimation accuracy of PSO becomes even better than DE. However, the more accurate scheme is DE hybridized with PS.Table 2
Estimation accuracy of 1-L-shape array for 2 sources.
Scheme
s
1
s
2
α
1
°
α
2
°
β
1
°
β
2
°
Desired values
0.5000
2.0000
30.0000
70.0000
110.0000
210.0000
PSO
0.4951
2.0050
30.0061
70.0060
109.0038
210.0061
DE
0.5022
1.9977
30.0044
69.0058
110.0044
209.9955
PSO-PS
0.5018
2.0019
29.9967
70.0036
110.0033
210.0033
DE-PS
0.4991
1.9988
30.0015
69.0082
110.0015
209.9984In Table3, the estimation accuracy of the same four schemes is provided for 2-L-shape array. The 2-L-shape array consists of 4 sensors; that is, each subarray consists of 1 sensor while the reference sensor is common for them. As listed in Table 3, again the hybrid DE-PS approach created fairly accurate results as compared to the other three schemes. Similarly, the other hybrid approach (PSO-PS) produced the second best accurate results.Table 3
Estimation accuracy of 2-L shape array for 2 sources.
Scheme
s
1
s
2
α
1
°
α
2
°
β
1
°
β
2
°
Desired values
0.5000
2.0000
30.0000
70.0000
110.0000
210.0000
PSO
0.5036
1.9963
29.9950
70.0049
110.0050
210.0050
DE
0.4964
2.0014
30.0034
69.9966
110.0034
209.9965
PSO-PS
0.4998
1.9991
30.0022
70.0021
109.9978
209.9979
DE-PS
0.5001
2.0001
29.9998
70.0001
110.0002
210.0001Now by comparing Tables2 and 3, it can be deduced very easily that each scheme not only produced better estimation accuracy in case of 2-L-shape array but also requires less number of sensors as compared to 1-L-shape array.Case 2.
In this subsection, the estimation accuracy is discussed for 3 sources impinging on L-shape arrays. The 1-L- and 2-L-shape arrays are composed of 13 and 7 sensors, respectively. The desired values of amplitudes{
s
1
=
2
,
s
2
=
4
,
s
3
=
6
} and DOA are α
1
=
60
°
,
β
1
=
15
°
,
α
2
=
25
°
,
β
2
=
85
°
,
α
3
=
40
°
,
β
3
=
170
°
. The estimation accuracy of DE and PSO increases when they are hybridized with PS as depicted in Tables 4 and 5.
The best estimation accuracy is given by DE-PS for both L-shape arrays, while the second best result is given by the hybrid PSO-PS technique. Overall, 2-L-shape array produced better estimation accuracy as compared to 1-L-shape array for all techniques.Table 4
Estimation accuracy for 3 sources using 1-L-shape array.
Scheme
s
1
s
2
s
3
α
1
°
α
2
°
α
3
°
β
1
°
β
2
°
β
3
°
Desired values
2.0000
4.0000
6.0000
60.0000
25.0000
40.0000
15.0000
85.0000
170.0000
PSO
2.1789
3.8209
5.8210
60.3843
24.6158
39.6156
15.3847
84.6157
169.6153
DE
1.9028
3.9026
6.0971
58.8028
25.1974
40.1973
15.1979
85.1972
169.8022
PSO-PS
2.0191
4.0192
6.0193
59.9264
24.9261
40.0738
14.9261
84.9263
170.0737
DE-PS
1.9931
3.9932
5.9932
60.0268
24.9733
39.9731
15.0269
85.0260
170.0272Table 5
Estimation accuracy for 3 sources using 2-L-shape array.
Scheme
s
1
s
2
s
3
α
1
°
α
2
°
α
3
°
β
1
°
β
2
°
β
3
°
Desired values
2.0000
4.0000
6.0000
60.0000
25.0000
40.0000
15.0000
85.0000
170.0000
PSO
1.8447
3.8445
6.1556
60.1791
25.1790
39.8207
15.1791
84.8206
170.1792
DE
2.0481
4.0480
6.0483
60.0977
24.9024
40.0979
14.9020
85.0983
169.9020
PSO-PS
2.0137
3.9862
6.0137
60.0328
25.0331
40.0330
15.0327
84.9670
170.0329
DE-PS
1.9980
4.0021
5.9980
59.9923
25.0077
39.9925
14.9925
85.0076
169.9926Case 3.
In this case, 4 sources are considered that have desired amplitude and DOA values are{
s
1
=
1
,
s
2
=
3
,
s
3
=
5
,
s
4
=
7
}, α
1
=
30
°
,
β
1
=
40
°
,
α
2
=
50
°
,
β
2
=
65
°
,
α
3
=
85
°
,
β
3
=
2
55
°
,
α
4
=
70
°
,
β
4
=
31
5
°. However, the 1-L- and 2-L-shape arrays consist of 15 and 10 sensors, respectively. The estimation accuracy of PSO and DE degraded due to increase of sources. Again, the best result is produced by DE-PS. Moreover, each scheme produced better results using 2-L-shape array with less number of antenna sensors as compared to 1-L-shape array as listed in Tables 6 and 7.Table 6
Estimation accuracy for 4 sources using 1-L-shape array.
Scheme
s
1
s
2
s
3
s
4
α
1
°
α
2
°
α
3
°
α
4
°
β
1
°
β
2
°
β
3
°
β
4
°
Desired values
1.0000
3.0000
5.0000
7.0000
30.0000
50.0000
85.0000
70.0000
40.0000
65.0000
255.0000
315.0000
PSO
0.5211
2.5212
5.4792
7.4789
31.3274
51.3276
86.3282
69.6723
41.3276
66.3278
256.4006
316.4102
DE
1.2988
2.7009
5.2987
7.2990
29.0324
50.9678
85.9681
70.9679
40.9678
64.0320
256.0673
316.0874
PSO-PS
0.8143
2.8140
5.1858
7.1858
30.5741
50.5745
84.4252
70.5746
39.4252
65.5748
255.6775
315.8776
DE-PS
1.0989
2.9908
4.9010
7.0990
30.2468
50.2469
85.2471
69.7530
40.2470
65.2473
254.7431
315.3571Table 7
Estimation accuracy for 4 sources using 2-L-shape.
Scheme
s
1
s
2
s
3
s
4
α
1
°
α
2
°
α
3
°
α
4
°
β
1
°
β
2
°
β
3
°
β
4
°
Desired values
1.0000
3.0000
5.0000
7.0000
30.0000
50.0000
85.0000
70.0000
40.0000
65.0000
255.0000
315.0000
PSO
0.7202
3.2799
5.2797
7.2711
31.1284
51.1285
86.1286
71.1283
41.1282
66.1284
256.1385
316.1369
DE
1.0988
3.0990
5.0989
6.9010
30.6656
49.3375
85.6654
70.6655
40.6656
65.6659
254.2342
315.6766
PSO-PS
1.0477
3.0479
4.9522
7.0480
30.2941
50.2942
84.7056
69.7060
40.2941
65.2946
255.2944
315.3041
DE-PS
1.0109
3.0111
5.0100
7.0112
30.0869
50.0870
85.0867
70.0869
40.0870
64.9125
254.9125
314.9082Case 4.
In this case, the convergence of each scheme is investigated for 2, 3, and 4 sources. For convergence, the number of times particular schemes able to get the desired goal is analyzed. The number of sensors, values of amplitudes, and DOA are kept the same for this simulation as in Case1. As shown in Figure 4, the convergence of PSO and DE has increased when they are hybridized with PS for all number of sources in case of 1-L-shape array. Moreover, the DE-PS scheme got fairly good convergence rate compared to the other schemes. The second best convergence rate is achieved by the hybrid PSO-PS approach.
Similarly, for 2-L-shape array, the DE-PS scheme got best convergence, while the second best convergence rate is achieved by PSO-PS as shown in Figure5. From comparison of Figures 4 and 5, one can easily verify that each scheme produced better convergence for 2-L-shape array as compared to 1-L-shape array.
It is evident from Figures4 and 5 that DE-PS and PSO-PS schemes have performed better for both L-shape arrays as compared to the other schemes. So further investigation will be carried out to analyze the performance of DE-PS and PSO-PS schemes.Figure 4
Performance analysis of convergence versus number of sources using 1-L-shape array.Figure 5
Performance analysis of convergence versus number of sources using 2-L-shape array.Case 5.
In this subsection, the proximity effects of elevation and azimuth angles are discussed using DE-PS and PSO-PS schemes for both L-shape arrays. This experiment is performed for three sources. In Table8, the proximity of elevation angles for fixed amplitudes and azimuth angles is discussed. Due to proximity, the estimation accuracy and convergence rate of DE-PS and PSO-PS techniques degraded for both L-shape arrays but still the DE-PS technique is robust enough to produce fairly accurate results especially in case of 2-L-shape array.
In Table9, the proximity of azimuth angles is discussed for fixed values of amplitudes and elevation. Again the DE-PS technique acted well for the proximity of azimuth angles. It has shown good estimation accuracy and convergence rate especially for both L-shape arrays. Overall, the results of DE-PS with 2-L-shape array are better as compared to 1-L-shape array and PSO-PS with both L-shape arrays.Table 8
Proximity effect of elevation angle fors
1
=
1
,
s
2
=
3
,
s
3
=
5  and  β
1
=
50
°
,
β
2
=
160
°
,
β
3
=
215
°.
Scheme
α
1
°
α
2
°
α
3
°
% convergence
Desired values
30.0000
80.0000
50.0000
100
DE-PS (1-L)
30.3843
80.3842
50.3844
90
PSO-PS (1-L)
30.5132
80.5674
50.6312
88
DE-PS (2-L)
30.1791
80.1790
50.1793
94
PSO-PS (2-L)
30.3671
80.4101
50.3982
92
Desired values
30.0000
65.0000
75.0000
100
DE-PS (1-L)
30.3846
65.9832
75.9834
84
PSO-PS (1-L)
30.5133
66.4378
76.6549
81
DE-PS (2-L)
30.1792
65.4301
75.4302
92
PSO-PS (2-L)
30.5134
66.1367
76.2136
88
Desired values
30.0000
40.0000
50.0000
100
DE-PS (1-L)
31.3965
41.4011
51.4013
70
PSO-PS (1-L)
31.8709
41.9654
51.9987
64
DE-PS (2-L)
30.7692
40.7694
50.7690
88
PSO-PS (2-L)
31.3245
40.6788
50.9899
82
Desired values
30.0000
35.0000
40.0000
100
DE-PS (1-L)
32.3417
37.3518
42.3519
64
PSO-PS (1-L)
34.1303
39.4321
43.9987
56
DE-PS (2-L)
31.1105
36.1107
41.1105
82
PSO-PS (2-L)
32.7989
37.0121
43.5467
76Table 9
Proximity effect of azimuth angles fors
1
=
1
,
s
2
=
3
,
s
3
=
5  and  α
1
=
30
°
,
α
2
=
50
°
,
α
3
=
85
°.
Scheme
β
1
°
β
2
°
β
3
°
% convergence
Desired values
15.0000
80.0000
230.0000
100
DE-PS (1-L)
15.3841
80.3840
230.3845
90
PSO-PS (1-L)
15.8865
80.7985
230.8891
88
DE-PS (2-L)
15.1790
80.1791
230.1793
94
PSO-PS (2-L)
15.4989
80.5110
230.5478
91
Desired values
15.0000
80.0000
70.0000
100
DE-PS (1-L)
15.3844
80.9830
70.9832
83
PSO-PS (1-L)
15.5988
81.7896
71.9989
76
DE-PS (2-L)
15.1792
80.4301
70.4302
91
PSO-PS (2-L)
15.4989
81.1564
71.1211
87
Desired values
60.0000
70.0000
80.0000
100
DE-PS (1-L)
61.3966
71.4012
81.4014
72
PSO-PS (1-L)
62.1456
62.5641
82.4538
65
DE-PS (2-L)
60.7694
70.7696
80.7692
86
PSO-PS (2-L)
61.5476
71.7873
81.6549
81
Desired values
60.0000
65.0000
70.0000
100
DE-PS (1-L)
62.3419
67.3519
72.3523
66
PSO-PS (1-L)
64.8979
70.1653
74.9019
57
DE-PS (2-L)
61.1103
66.1105
71.1104
80
PSO-PS (2-L)
62.9109
68.9469
73.9381
74Case 6.
In this second part of simulations, the comparison of DE-PS using 2-L-shape array is carried out with PSO-PS using 2-L-shape array, PM using parallel shape array [16], and L-shape arrays [17]. Tables 10 and 11 listed the mean, variance, and standard deviations for PM using parallel shape array and L-shape array, respectively [16, 17], while Tables 12 and 13 provided the same calculation for PSO-PS and DE-PS techniques, respectively, using 2-L-shape array. The PM uses 11 sensors for both parallel array and L-shape arrays configuration [16, 17], while for the PSO-PS and DE-PS schemes only 4 sensors are used in 2-L-shape array. The range of elevation angle is varied from 70° to 90° for fixed azimuth angle of 50°. This range of elevation angle has practical importance in wireless communication systems. The readers are encouraged to see [28] for further study. As obvious from Table 9, the PM method with parallel shape array has failed to produce accurate results as soon as the elevation angle is getting close to 90° but the same PM method using 1-L-shape array configuration has got accurate results for the same range of elevation angle as listed in Table 10. However, at the same time, if we look at Tables 11 and 12, then one can observe that the DE-PS and PSO-PS schemes produced even better results for the same range of elevation angles by using 2-L-shape array configuration with less number of sensors. Overall the results of DE-PS are better.Table 10
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuthβ
=
50
° by using PM with parallel shape array [16].
α
Mean ofα
^
Variance ofα
Standard deviation ofα
72°
72.0681
0.7021
0.8379
76°
74.9583
0.9895
0.9947
79°
77.8421
1.4082
1.1867
82°
80.4967
2.3482
1.5323
86°
83.3760
4.7215
2.1729
89°
84.7062
9.4567
3.0752Table 11
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuthβ
=
50
° by using PM with 1-L-shape array [17].
α
Mean ofα
^
Variance ofα
Standard deviation ofα
72°
72.3528
0.0151
0.1228
76°
76.2237
0.0111
0.1053
79°
79.0999
0.0068
0.0825
82°
82.0998
0.0037
0.0608
86°
86.0566
0.000632
0.0251
89°
89.0146
0.000371
0.0192Table 12
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuthβ
=
50
° by using PSO-PS with 2-L-shape array.
α
Mean ofα
^
Variance ofα
Standard deviation ofα
72°
72.1354
0.002363
8.76
×
10
-
03
76°
76.0003
0.003416
9.49
×
10
-
03
79°
78.0075
0.008934
9.99
×
10
-
03
82°
82.0046
0.006984
9.95
×
10
-
03
86°
86.0097
0.007821
9.89
×
10
-
03
89°
89.0078
0.008947
8.69
×
10
-
03Table 13
Means, variances, and standard deviations at 10 dB noise for different elevation angles and fixed azimuthβ
=
50
° by using DE-PS with 2-L-shape array.
α
Mean ofα
^
Variance ofα
Standard deviation ofα
72°
71.9998
0.00002415
4.91
×
10
-
03
76°
76.0003
0.00001871
4.32
×
10
-
03
79°
78.9996
0.00006854
8.27
×
10
-
03
82°
81.9998
0.00003278
5.72
×
10
-
03
86°
86.0006
0.00006132
7.83
×
10
-
03
89°
89.0008
0.00001371
3.70
×
10
-
03Case 7.
In this case, the Root Mean Square Error (RMSE) of DE-PS is compared with PSO-PS and PM using parallel shape array [16] and L-shape arrays [17]. Only one source is considered, which has elevation and azimuth angles 40° and 65°, respectively. The range of signal-to-noise ratio (SNR) is taken from 5 dB to 30 dB. The DE-PS technique maintains minimum RMSE for all values of SNR as depicted in Figure 6. The second best RMSE is maintained by PSO-PS using 2-L-shape arrays.Figure 6
Performance analysis of Root Mean Square Error versus SNR.Case 8.
In this subcase, the computational complexity of DE-PS is compared with PM using L-shape array [16]. The PM required total O
3
×
M
×
T
×
P computations, where M, P, and T represent total number of sensors, sources, and snapshots, respectively [16, 17]. The main computations in DE-PS for 2-L-shape array involved the number of computations (multiplication) in fitness function O
N
2
3
+
16
×
P plus the multiplications required for mutation and cross-over which are approximately 2
(
16
×
N
2
) where N is the total number of particles. Moreover, the computations required for PS are approximately (
16
×
P
). Therefore, the computational complexity of DE-PS for 2-L-shape array in terms of multiplication is almost O
N
2
(
3
+
16
×
P
)
+
2
(
16
×
N
2
)
+
(
16
×
P
). In the same way computations involved in DE-PS with 1-L-shape array are O
N
2
(
3
+
20
×
P
)
+
(
20
×
P
). The DE-PS for 2-L-shape array is computationally expensive as compared to DE-PS with 1-L-shape array, while it is less expensive as compared to PM.Case 9.
In Table14, some general properties are listed for PM with parallel shape array, L-shape arrays, and DE-PS technique using 2-L-shape array. As presented in Table 8, the PM using I-L-shape array has an estimation failure problem in the range of elevation angles from 0° to 20°. The second drawback with 1-L-shape and parallel shape arrays is their limitation for elevation angles beyond 90°. Moreover, the parallel shape array has the pair matching problem. These limitations can be overcome by using PM with 2-L-shape arrays but it requires extra sensors and a large number of snapshots. It requires at least 200 snapshots that obviously increase the computational burdens. Therefore, the DE-PS technique removes the shortcomings of PM (1-L-shape and parallel shape) by using less number of sensors. The DE-PS technique also provides the estimation of source amplitude, which is very important parameter for accurate estimation of remote source. Moreover, it provides the appreciable decrease in the computational cost by requiring only a single snapshot to converge.Table 14
Comparison among 2-L-shape array, parallel shape array [16], and L-shape arrays [17].
Property
Parallel shape array
1-L-shape array
2-L-shape array
2-L-shape array
Scheme used
PM
PM
PM
DE-PS
Range of elevation and azimuth angles
(0°, 90°) and (0°, 360°)
(0°, 90°) and (0°, 360°)
(0°, 180°) and (0°, 360°)
(0°, 180°) and (0°, 360°)
Number of sensors
15
11
10
4
Number of snapshots
200
200
200
1
Pair matching
Required
Not required
Not required
Not required
Failure estimation
From 70° to 90°
From 0° to 20°
No failure
No failure
Amplitude estimation
Cannot estimate
Cannot estimate
Cannot estimate
Yes, can estimate


## 5. Conclusion and Future Work Directions

In this work, an efficient and robust approach has been proposed for joint estimation of amplitude and DOA of far-field sources using L-shape arrays. In this scheme, DE and PSO have been used as global optimizers, which were further assisted by PS as a rapid local search optimizer. We have demonstrated that the hybrid DE-PS approach performed considerably well as compared to the PSO, PSO-PS, and DE techniques. It has been also shown that all the schemes performed well using 2-L-shape array as compared to 1-L-shape array. Moreover, the DE-PS scheme using 2-L-shape array performed well as compared to PM using parallel shape array, as well as L-shape array. The simulation results are in complete conformation of adopted approach.In future, one can check the same approach for sidelobe reduction and beam steering in adaptive beamforming



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*Source: 9428196-2017-01-15.xml*