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Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6403-6409 6403

www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

Genetic Algorithm based Optimization of Uniform

Circular Array

Vinod Kumar

Department of ECE
Guru Jambheshwar University of Science & Technology

Hissar, India

vinodspec@yahoo.co.in

Sanjeev Kumar Dhull

Department of ECE
Guru Jambheshwar University of Science & Technology

Hissar, India

sanjeevdhull2011@yahoo.com

Abstract-Signal estimation at the antenna is a major challenge of

the antenna array structure because the received signals have

different directions. Therefore, in this paper, a Genetic

Algorithm (GA) is applied to the uniform circular array for the

optimization of array structure in regard to its geometry. On the

optimized array structure, four different algorithms (Estimation
of Signal Parameter via Rotational Invariance Technique –

ESPRIT, First Order Forward Prediction - FOFP, Beamscan,

and Multiple Signal Classification - MUSIC) have been

implemented in order to estimate the signal direction accurately

with quick estimation time. The accuracy has been calculated

with Root Mean Square Error (RMSE) indices. From the

experimental analysis, it has been found that the performance of

the ESPRIT algorithm is better than the others in terms of
accuracy and estimation time.

Keywords-computational complexity; FOFP; genetic algorithm;

RMSE; uniform circular array

I. INTRODUCTION

When an incoming signal wave is detected and measured at
a sensor array, it is processed and information is extracted from
it (e.g.: Direction Of Arrival - DOA). The algorithms that are
used to find DOA in the antenna array are generally used in
wireless communication systems to increase the system
capacity and the throughput of the wireless network. The
antenna array consists of several antennas for finding the
direction of incident waves and suppressing interference
signals. The estimation problem is of significant importance in
the array signal processing. Parameter estimation is an area
where research is conducted for performance improvement. A
major topic of interest is the accurate discovery of the signal
direction. Sensor array field processing is an area for research
focused in accurate signal estimation. As improvements in the
geometry of the antenna array occur, the DOA estimation
approach will become an important part of smart antenna
systems. The antenna array collects data from all the elements
and then combines them for spatial information and detects the
incoming signal direction based on signal processing
algorithms.

The main aim of the antenna array is to find the signal
directions from signals that come from different directions. The
estimation of the signal may be obtained with the help of the
array’s geometry and DOA algorithms. DOA estimation is an

interesting field and many DOA algorithms were proposed
including spatial spectrum estimation. The first method based
on the spatial spectrum was proposed in [1]. This method is
used for estimating the direction of the incoming signal by
computing and observing the spatial spectrum. After that, a
local maxima point is decided. Author in [2] presented a new
method based on Minimum Variance Distortionless Response
(MVDR). This MVDR is used to conduct spectrum estimation
with the help of Maximum Likelihood (ML) method [3] which
maximizes the log-likelihood function among a set of arrays. In
[4] a subspace based algorithm was proposed which scans all
the angles between the signal subspace and noise subspace.
Author in [5] improves the solution ability of the MUSIC
algorithm and gives good estimation results by introducing a
variant of MUSIC algorithm. Authors in [6] propose a new
algorithm named ESPRIT which has higher computation
efficiency. Authors in [7-8] proposed the weighted subspace
fitting method for better estimation results. Authors in [9]
proposed a new method for the signal estimation in two
dimensions. Authors in [10] improved the MUSIC algorithm.
Authors in [11] gave an improvement in the static performance
of the MUSIC algorithm. Authors in [12] proposed a new
scheme for the increase of resolution ability of the MUSIC
algorithm. Authors in [13] gave advancement in the ESPRIT
algorithm for two dimensions. Table I gives the detailed
description of the existing DOA algorithms. These algorithms
are based on spectral search and subspace methods.

TABLE I. DIFFERENT DOA ALGORITHMS

Algorithm Comments

Bartlett [1]
Estimate DOA by computing spatial spectrum and

finalizing local maxima. The noise signal is excluded.

MVDR [2] Spectrum estimation with maximum likelihood method.

MUSIC [4] Scans all the angles from signal and noise subspace.

ESPRIT [6]
Increased computation efficiency with the help of two

identical sub arrays.

Weighted subspace

fitting [7-8]
Unified approach for signal estimation.

MUSIC [14]
Discussion on maximum likelihood and Cramor Rao

bound

MUSIC [9] A method for signal detection on two dimensions.

MUSIC [11] Static performance of MUSIC algorithm.

MUSIC [12] Improved the resolution ability of MUSIC algorithm

Root MUSIC [5]
Does not find the spectral peaks, it solves the rooting

problem of a polynomial with good resolution ability.

Corresponding author: Vinod Kumar

Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6403-6409 6404

www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

II. ESTIMATION OF SIGNAL PARAMETERS VIA ROTATIONAL
INVARIANCE TECHNIQUES

Authors in [6] proposed an algorithm named Estimation of
Signal Parameter via Rotational Invariance Technique
(ESPRIT) for DOA estimation. In ESPRIT, array doublets are
used and formed by N/2 pairs which further form a
displacement vector, where N is the number of array elements.
The starting two elements of the doublet are separated and
grouped to make two N/2 sub arrays. The vectors x and y are
the data vectors corresponding to each of the sub arrays. The
output of the sub arrays x and y can be expressed as:

1
( )

[ ] [ ] ( ) [ ]
r

x
k i k i k

x n s n a v nθ
−

= +∑ (1)

1
( )2 sin

[ ] [ ] ( ) [ ]
r

yj
k i k i k

y n s n e a v n
πδ θ θ

−

= +∑ (2)

where similar notation has been used and δ is the displacement
magnitude in wavelengths. The estimated angle by ESPRIT
algorithm is relative to the displacement vector. The output of
sub arrays, x and y, in matrix form is given as:

( )x
n n nx As v= + (3)

( )y
n n ny A s vϕ= + (4)

The matrix φ is a diagonal r×r matrix where the diagonal
elements are:

0 1 1{exp( 2 sin ),exp( 2 sin ),....exp( 2 sin )}rj j jπδ θ πδ θ πδ θ −

The phase delay may be represented by the complex
exponentials between the r signals and the doublet pair. The
data vectors may be concatenated from sub arrays to make a
single 2N-2 data vector like:

n
n b n

x
z A S

 
= = 
 

(5)

( )

( )
,

x
n

b n y
n

vA
A V

A vϕ

  
= =   
    

(6)

The columns of Ab occupy the signal subspace of the new
array. Let Vs be the column matrix depending upon the signal
subspace and zn, Ab. Vs is related with r×r transformation T and
can be written as:

s bV A T= (7)

and can be portioned as follows:

s
y

E AT
V

E A Tϕ
   

= =   
  

(8)

From this step, A will be equal to Ex, Ey, and have the same
range. The rank r of matrix Exy is:

[ ]xy x yE E E= (9)

To find the r×2r rank r matrix having null space of Exy to
form the matrix F is written as:

[ ]x y x x y y x yE E F E F E F ATF A TFϕ= + = + (10)

Assuming Ψ is:

1
[ ]x yF Fψ

−= − (11)

by reshuffling the above equations, we get:

x yE Eψ = (12)

Now, by substituting we get:

1 1
AT A T AT T A T Tψ ϕ ψ ϕ ψ ϕ− −= ⇒ = ⇒ = (13)

The given equations mean that the eigenvalues of Ψ are the
same as the diagonal elements of φ. Once the eigenvalues λ of
φ have been calculated, the angle of arrival is calculated as:

2 sin kj
k e

πδ θλ = (14)

arg( )
arcsin

k
k

λ
θ

πδ
 

=  
 

(15)

If A is the full rank matrix, then the eigenvalues of the
matrix Ψ are the diagonal elements of φ and the eigenvectors of
Ψ are the columns of T. Practically, the signal subspace is not
known exactly, the only estimate is taken from sample
covariance matrix Rxx or from a sub space tracking algorithm.
Therefore, Exψ=Εy will not be exactly satisfied and we will
have to resort to a least square solution to compute Ψ. The least
square process assumes that the columns in Ex are known
exactly whereas the data in Ey are noisy. If the assumption that
Ex and Ey are equally noisy is made, then the total least square
criteria is used to solve to above problem with better results.

III. OPTIMIZATION USING GENETIC ALGORITHM

The Genetic Algorithm (GA) is a population search
algorithm inspired by the evolutionary process of natural
selection [15]. GA is a simple algorithm that has been proven
to be very powerful and widely applicable. The algorithm
arrives at a solution by performing operations on a population
of solutions referred to as chromosomes over several iterations,
which are referred to as generations. The operations used are
borrowed from genetic concepts such as mutation,
reproduction, fitness, etc. The algorithm uses fitness evaluation
to rank solutions and randomness in creating new generations
of solutions in order to promote diversity. Authors in [16] used
the GA to optimize the relative position in an antenna system.
Authors in [17] used the GA in a linear array with infinite SNR
in order to extract the amplitude and DOA of various signals.
Authors in [18] introduced the use of GA in ML based
parameter and spatial spectrum estimation. Authors in [19]
introduced a new refined GA for accurate and reliable DOA
estimation with the help of a sensor array based on the ML
function. Authors in [20] introduced a new genetic based
estimation method with the help of fourth order cumulant. A
new genetic firefly hybrid algorithm based on the neural
network was designed in [21] for finding the best position in

Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6403-6409 6405

www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

the data cube. A new algorithm was designed in[22] for finding
the best path in all the clusters based on the generalized
travelling salesman problem. This algorithm was based on the
minimum cost tour within a clustered set of cities. The
application of GA on beam forming of correlated and
uncorrelated static sources was discussed. When the target is
moving in CDMA systems by knowing the spreading code, the
LMS algorithm received the signal at zero rates. The SNR was
as low as -9dB for uncorrelated static sources but when the
signals were correlated, the LMS gave unstable beam forming.
Hence, GA gives good results for all sources which are static
with good error rate [23]. Authors in [24] proposed a multiple
source localization method for DOA estimation at each sensor
mode. The GA based proposed algorithm was compared with
conventional sequential search techniques. The computational
load and inter node communication burden was reduced, so the
algorithm was considered suitable for use in IT based
applications having large numbers of sensor nodes and sources.
The GA is most commonly used as an optimization algorithm
and has efficiently solved various optimization problems across
many disciplines in proficient manners. Alternating projection
maximum likelihood, stimulated annealing grid based search,
and data based grid search are used in the GA. The GA has
widely been used in the DOA estimation by using the spatial
spectrum estimation. The ML technique gives an optimal
solution when compared to other methods. Based on the
likelihood function, the ML is superior to other techniques [3,
14, 25]. A new optimized array structure for DOA estimation
was designed which can estimate both azimuth and elevation
angles. The array is designed with the help of the GA that has
as a fitness function the optimization of the array structure. The
array aperture size increases when the number of elements in
the array is increased. After that, various DOA algorithms have
been applied to the proposed structure and simulation results
were obtained. The proposed array was compared with other
arrays in terms of RMSE of azimuth and elevation angles.

A. Structure of the New Optimized Array

Based on the GA, the proposed design array structure is
given in Figures 1 and 2. In this structure, the population size is
taken as 100 with 1000 iterations and a randomly generated
initial array structure of 10 elements on the XY plane. The
signal generated from various directions is received at each
array element and combined to produce the output.

Fig. 1. Optimum configuration with number of iterations=1.

The optimized new array structure is formed and the total
spacing between array elements is 314.1593 for 10 array
elements and 157.0796 for 20 array elements. Total spacing
reduces when the number of array elements increases along
with the complexity of the structure. The formed structure is
the optimized structure that can be used to detect the signal in
two dimensions. The optimized structure is designed with the
help of GA after several iterations.

Fig. 2. The purposed array structure of 20 elements.

B. Steps Involved in Array Optimization

Figure 3 gives the flowchart of the used algorithm. The
steps involved in the optimization of the array are:

Input: Config (PQ, distance, Cost Function), Population,
Crossover 90%, and Mutation 10%, iterations
for Number of Iterations
for population
Selection Cost( Config) ==> MSE ==> Small MSE 10%
Crossover= offsprings are created in hope of producing

better Config
Mutation =Changing the configuration PQ in hope of

producing better Config
Population for next iteration
end
end

Fig. 3. Flowchart of the MATLAB code.

0 1 2 3 4 5 6 7 8 9 10

Array Spacing

10
Total Spacing Between Array's Elements (mm) = 188.7047, Iteration = 1

0 125 250 375 500

Array Spacing

125

250

375

500

Total Spacing Between Array Elements= 157.0796 mm

Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6403-6409 6406

www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

C. Execution Time of DOA Estimation Algorithms

In terms of execution time (the time taken by the algorithm
to estimate the signal), the ESPRIT algorithm requires the least
time when compared to the other algorithms, as can be seen in
Figure 4.

Fig. 4. Execution time of algorithms on the proposed configuration.

In the optimized array, we have applied simulations of
various DOA algorithms in order to compare their performance
in terms of execution time. The simulation results are obtained
with different signal directions at various angles. Figures 4-9
give the simulation results of the different DOA algorithms on
the optimized array structure. Figure 4 gives the simulation
results of the different algorithms in terms of execution time. It
is evident that the ESPRIT algorithm shows better results than
the other algorithms.

Figures 5 and 6 show the simulation results of the ESPRIT
and beam-forming algorithm at different angles (60°, 80°, 90°,
100°, 120°). The angles are accurately estimated through the
ESPRIT algorithm but the estimation through beam-forming is
poor. Also, in Figures 7-9 (estimation through MUSIC, FOFA,
and Capon algorithms) the estimation is accurate and clear. The
simulation results of the discussed algorithms depend on the
number of sources, the number of antenna elements, the
spacing between the elements, and the SNR. When the number
of sources is bigger than the antenna elements, it is difficult to
estimate the signal directions. When there are more of antenna
elements, the obtained results are better. The spacing between
the elements is taken as 0.5λ.

Fig. 5. ESPRIT histogram corresponding to 5 signals directions.

Fig. 6. Beam-forming spatial spectrum corresponding to 5 signals.

Fig. 7. MUSIC spatial spectrum corresponding to 5 signals.

Fig. 8. FOFP spatial spectrum corresponding to 5 signals.

D. Comparison with Existing Arrays

The proposed array is compared with Circular, Rectangular,
Concentric, and IASA arrays in terms of RMSE of azimuth and
elevation angles. The results of the GA based array have been
observed to be better when compared with the other arrays.
Figures 9-21 give the detailed comparison and show that the
RMSE of the proposed design offers better results. Figures 10-
11 depict the azimuth and elevation angle comparison with
RMSE of the proposed GA based array. The circular array is
compared with the GA based array in Figure 10. The result
shows that the RMSE of the GA based array is lower for both
azimuth and elevation angles.

Execution time of Algorithms

ESPRIT Beam Forming Music FOFP
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 20 40 60 80 100 120 140 160 180

angle°

10
ESPRIT Histogram

0 20 40 60 80 100 120 140 160 180

angle°

25
beamforming spatial spectrum

0 20 40 60 80 100 120 140 160 180

angle°

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000
Music algorithm spatial spectrum

0 20 40 60 80 100 120 140 160 180

angle°

100

120

140
first-order forward prediction algorithm spatial spectrum

Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6403-6409 6407

www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

Figures 12 and 13 show the azimuth and elevation angle
comparison with RMSE of the proposed GA based array with
the rectangular array. The result shows that the RMSE of the
GA based array is lower for both angles. Figures 14 and 15
show the comparison results of azimuth and elevation angle
with RMSE of the proposed GA based array with the
concentric array. The result shows that the RMSE of the GA
based array is lower for both angles. Figures 16 and 17
illustrate the comparison of azimuth and elevation angle with
RMSE of the proposed GA based array with the IASA array.
The result shows that the RMSE of the GA based array is lower
for both angles.

Fig. 9. Capon algorithm spatial spectrum corresponding to 5 signals.

Fig. 10. RMSE of the azimuth angles of circular and GA based arrays.

Fig. 11. RMSE of the elevation angle of circular and GA based arrays.

Fig. 12. RMSE of the azimuth angle of rectangular and GA based arrays.

Fig. 13. RMSE of the elevation angle of rectangular and GA based arrays.

Fig. 14. RMSE of the azimuth angle of concentric and GA based arrays.

Fig. 15. RMSE of the elevation angle of concentric and GA based arrays.

0 20 40 60 80 100 120 140 160 180

angle°

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8
Capon algorithm spatial spectrum

-10 -5 0 5 10 15 20 25 30

SNR

0.01

0.02

0.03

0.04

0.05

0.06
RMSE v s SNR of azimuth angle

Circular Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.005

0.01

0.015

0.02

0.025

0.03
RMSE v s SNR of Elev ation angle

Circular Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1
RMSE v s SNR of azimuth angle

Rectangular Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.005

0.01

0.015

0.02

0.025

0.03
RMSE v s SNR of Elev ation angle

Rectangular Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.01

0.02

0.03

0.04

0.05

0.06
RMSE v s SNR of azimuth angle

Cocentric Ring Array

GA Based Array

Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6403-6409 6408

www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

Fig. 16. RMSE of the azimuth angle of IASA and GA based arrays.

Fig. 17. RMSE of the elevation angle of IASA and GA based arrays.

Fig. 18. RMSE error of the azimuth angle of circular, rectangular,
concentric, and IASA arrays.

Figures 18 and 19 show the azimuth and elevation angle
comparison with RMSE of the four arrays. The result shows
that the RMSE of the IASA array is lower for both azimuth and
elevation angles. Figures 20 and 21 show the azimuth and
elevation angle comparison with RMSE of the proposed GA
based array with all the considered existing arrays. The result
shows that the RMSE of the GA based array is lower for both
azimuth and elevation angles. This type of GA based array is
generally used in the communication field for the estimation of
the signal and radar and sonar field for signal estimation. The
only limitation of this type of optimized GA based array is that
it basically estimates the signal direction of static targets.

Concluding, this optimized array structure detects the signal
accurately in terms of RMSE.

Fig. 19. RMSE of the elevation angles of circular, rectangular, concentric,
and IASA arrays.

Fig. 20. RMSE error of the azimuth angle of circular, rectangular,
concentric, IASA, and GA based arrays.

Fig. 21. RMSE error of the elevation angle of circular, rectangular,
concentric, IASA, and GA based arrays.

IV. CONCLUSIONS

The DOA of signal estimation is a major concern for the
communication technologist. The proposed geometry for DOA
estimation aims to meet the estimation accuracy requirements.
The simulation results of the proposed configuration show that
the GA based array with ESPRIT algorithm has better
estimation accuracy among all four proposed DOA estimation

-10 -5 0 5 10 15 20 25 30

SNR

0.01

0.02

0.03

0.04

0.05

0.06
RMSE v s SNR of azimuth angle

IASA Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04
RMSE v s SNR of Elev ation angle

IASA Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.01

0.02

0.03

0.04

0.05

0.06
RMSE v s SNR of azimuth angle

Circular Array

Rectangular Array

Cocentric Ring Array

IASA Array

-10 -5 0 5 10 15 20 25 30

SNR

0.005

0.01

0.015

0.02

0.025

0.03
RMSE v s SNR of Elev ation angle

Circular Array

Rectangular Array

Cocentric Ring Array

IASA Array

-10 -5 0 5 10 15 20 25 30

SNR

0.01

0.02

0.03

0.04

0.05

0.06
RMSE v s SNR of azimuth angle

Circular Array

Rectangular Array

Cocentric Ring Array

IASA Array

GA Based Array

-10 -5 0 5 10 15 20 25 30

SNR

0.005

0.01

0.015

0.02

0.025

0.03
RMSE v s SNR of Elev ation angle

Circular Array

Rectangular Array

Cocentric Ring Array

IASA Array

GA Based Array

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www.etasr.com Kumar & Dhull: Genetic Algorithm based Optimization of Uniform Circular Array

algorithms. The results can be further improved with the help
of some other optimized algorithms in terms of number of
elements and SNR. Other array configurations can also be
taken into consideration for the estimation of the signal angles.

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AUTHORS PROFILE

Vinod Kumar is presently working in the department of Electronics &

Communication Engineering, Guru Jambheshwar University of Science &
Technology, Hisar, India. He is pursuing his Ph.D. from the same department.

His research area of speclization is array signal processing.

Sanjeev Kumar Dhull received his Bachelor of Engineering (ECE) from
Mangalore University, Mangalore, in 1996, his M.Tech. from Punjab

University Chandigarh, in 2004 and his Ph.D. from Guru Jambheshwar
University of Science and Technology (GJUS&T), Hisar, in 2013. He joined

GJUS&T as an Assistant Professor in 2006 and became an Associate
Professor in 2013 and Professor in 2018. His research interests include

adaptive signal processing and speech processing.