View metadata, citation and similar papers at core.ac.

uk brought to you by CORE

provided by Elsevier - Publisher Connector

Journal of King Saud University – Computer and Information Sciences (2013) 25, 117–126

King Saud University

Journal of King Saud University –
Computer and Information Sciences
www.ksu.edu.sa
www.sciencedirect.com

ORIGINAL ARTICLE

A genetic algorithm for the level control of nulls and side

lobes in linear antenna arrays
Bipul Goswami, Durbadal Mandal *

Department of Electronics and Communication Engineering, National Institute of Technology Durgapur, West Bengal 713 209, India

Received 3 January 2012; revised 4 June 2012; accepted 26 June 2012

Available online 4 July 2012

KEYWORDS Abstract The design problem of imposing deeper nulls in the interference direction of uniform lin-
Side lobe level; ear antenna arrays under the constraints of a reduced side lobe level (SLL) and a fixed first null
Deeper nulls; beam width (FNBW) is modeled as a simple optimization problem. The real-coded genetic algo-
Real coded genetic algo- rithm (RGA) is used to determine an optimal set of current excitation weights of the antenna ele-
rithm; ments and the optimum inter-element spacing that satisfies the design goal. Three design examples
Linear antenna array; are presented to illustrate the use of the RGA, and the optimization goal in each example is easily
First null beam width achieved. The numerical results demonstrate the effectiveness of the proposed method.
ª 2013 Production and hosting by Elsevier B.V. on behalf of King Saud University.

1. Introduction ual elements (Ballanis, 1997; Elliott, 2003). Many communica-

tion applications require a highly directional antenna. Array
An antenna array is composed of an assembly of radiating ele- antennas have higher gain and directivity than an individual
ments in an electrical or geometrical configuration. In most radiating element. A linear array consists of elements placed
cases, the elements are identical. The total field of the antenna in a straight line with a uniform spacing between the elements
array is found by vector addition of the fields radiated by each (Haupt and Werner, 2007). The goal of antenna array geome-
individual element. Five controls in an antenna array can be try synthesis is to determine the physical layout of the array
used to shape the pattern properly: the geometrical configura- that produces a radiation pattern that is closest to the desired
tion (linear, circular, rectangular, spherical) of the overall ar- pattern.
ray, the spacing between the elements, the excitation The increasing amount of electromagnetic pollution has
amplitude of the individual elements, the excitation phase of prompted the study of array pattern nulling techniques. These
the individual elements, and the relative pattern of the individ- techniques are important in radar, sonar and communication
systems to minimize degradation of the signal to noise ratio
due to undesired interference (Haupt and Werner, 2007).
* Corresponding author. Tel.: +91 9474119721. Much current research on antenna arrays (Haupt, 1997; Steys-
E-mail address: durbadal.bittu@gmail.com (D. Mandal). kal et al., 1986; Yang et al., 2004; Mandal et al., 2010; Guney
Peer review under responsibility of King Saud University. and Akdagli, 2001) is focused on using robust and easily
adapted optimization techniques to improve the nulling per-
formance. Classical gradient-based optimization methods are
not suitable for improving the nulling performance of linear
Production and hosting by Elsevier antenna arrays for several reasons, including the following:

1319-1578 ª 2013 Production and hosting by Elsevier B.V. on behalf of King Saud University.
http://dx.doi.org/10.1016/j.jksuci.2012.06.001
118 B. Goswami, D. Mandal

(i) the methods are highly sensitive to the starting points when excitation for each antenna to steer the main beam in specific
the number of variables, and hence the size of the solution directions.
space, increases, (ii) they frequently converge to local optimum The goal of this paper is to introduce deeper null/nulls in
solutions, diverge or arrive at the same suboptimal solution, the interference directions and to suppress the relative SLLs
(iii) they require a continuous and differentiable objective with respect to the main beam with the constraint of a fixed
function (gradient search methods), (iv) they require piecewise first null beam width (FNBW) for a symmetric linear antenna
linear cost approximation (linear programming), and (v) they array of isotropic elements. This is done by designing the rel-
have problems with convergence and algorithm complexity ative spacing between the elements with a non-uniform excita-
(non-linear programming). Thus, evolutionary optimization tion over the array aperture. An evolutionary technique, the
methods have been employed for the optimal design of deeper RGA (Haupt and Werner, 2007; Haupt, 1995; Holland,
nulls. Different evolutionary optimization algorithms, such as 1975), is used to obtain the desired pattern of the array.
fuzzy logic (Mukherjee and Kar, 2012; Anooj, 2012; De and Several aspects of the RGA are different from other search
Sil, 2012), the bees algorithm (Fahmy, 2012), the genetic algo- techniques. First, the algorithm is a multi-path technique that
rithm (GA) (Haupt and Werner, 2007), and particle swarm searches many peaks in parallel and hence decreases the possi-
optimization (PSO) (Mandal et al., 2012), have been widely bility of local minimum trapping. Secondly, the RGA only
used in the development of design methods that are capable needs to evaluate the objective function (fitness) to guide its
of satisfying constraints that would otherwise be unattainable. search. Hence, there is no need to compute derivatives or other
Of these algorithms, GA is a promising global optimization auxiliary functions, so the RGA can also minimize the non-
method for the design of antenna arrays. derivable objective function. Finally, the RGA explores the
Several methods for the synthesis of array antenna patterns search space where the probability of finding improved perfor-
with prescribed nulls are reviewed below. A method for null mance is high.
control and the effects on radiation patterns is discussed in A broadside uniform linear array with uniform spacing is
Steyskal et al. (1986). An approach of null control using considered. The array is symmetric with respect to the origin
PSO, where single or multiple wide nulls are generated by opti- with equal spacing between any two consecutive elements.
mum perturbations of the elements’ current amplitude weights The phase difference between any two elements is fixed at zero.
to create symmetric nulls about the main beam, is discussed in The RGA adjusts the excitation coefficients and location of the
Mandal et al. (2010). An approach for the pattern synthesis of elements from the array center to impose deeper nulls in the
linear antenna arrays with broad nulls is described in Guney interference directions. A cost function is defined that keeps
and Akdagli (2001). In Yang et al. (2002), a differential evolu- the nulls and side lobes at lower levels.
tion algorithm is used to optimize the static-mode coefficients The remainder of the paper is arranged as follows. In Sec-
and the durations of the time pulses, leading to a significant tion 2, the general design equations for a non-uniformly ex-
reduction of the sideband level. A binary coded genetic algo- cited and unequally spaced linear antenna array are stated.
rithm is used in Haupt (1995, 1975) and Yan and Lu (1997) A brief introduction to the Genetic Algorithm is presented in
to reduce the sidelobe level of a linear array by excitation coef- Section 3, and the numerical simulation results are presented
ficient tapering. The spacing is assumed to be equal to half of in Section 4. The paper concludes with a summary of the work
the wavelength throughout the array aperture. The study in Section 5.
shows good sidelobe performance (approximately 33 dB)
for a 30 element array. The radiation pattern of linear arrays
with large numbers of elements (20–100) is improved using a 2. Design equation
GA in Ares-Pena et al. (1999). The sidelobes for 20 and 100
element arrays are reduced to 20 dB and 30 dB, respec- A broadside linear antenna array (Ballanis, 1997; Elliott, 2003)
tively. A decimal GA technique to taper the amplitude of the of 2M isotropic radiators, as shown in Fig. 1, is considered.
array excitation to achieve reduced side lobe and null steering Each element is excited with a non-uniform current. The array
in single or multiple beam antenna arrays is proposed in elements are assumed to be uncoupled and equally spaced
Abdolee et al. (2007). In Son and Park (2007), a low-profile along the z-axis, and the center of the array is located at the
phased array antenna with a low sidelobe was designed and origin. The array is symmetric in both geometry and excitation
fabricated using a GA. The sidelobe level was suppressed by with respect to the center.
only 6.5 dB after optimization. An approach for sidelobe The radiation characteristics of antennas are most impor-
reduction in a linear antenna array using a GA is proposed tant in the far field (Fraunhofer) region. An array consisting
in Recioui et al. (2008), Das et al. (2010). In Das et al. of identical and identically oriented elements has a far field
(2010), the sidelobes for symmetric linear antenna arrays are radiation pattern that can be expressed as the product of the
reduced without significantly sacrificing the first null beam- element pattern and a factor that is widely referred to as the
width, and non-uniform excitations and optimal uniform spac- array factor. Each array has its own array factor. The array
ing are proposed generate the desired result. Optimal values factor, in general, is a function of the number of elements, their
are found using the real-coded genetic algorithm (RGA). An geometrical arrangement, their relative magnitudes, their rela-
approach to determine an optimum set of weights for antenna tive phases, and their relative spacings. Because the array fac-
elements to reduce the maximum side lobe level (SLL) in a tor does not depend on the directional characteristics of the
concentric circular antenna array (CCAA) with the constraint radiating elements, it can be formulated by replacing the actual
of a fixed beamwidth is proposed in Mandal et al. (2009), elements with isotropic (point) sources. For the array in Fig. 1,
Mondal et al. (2010). In (Cafsi et al. (2011), a method of the array factor, AFðI; u; dÞ Ballanis, 1997; Elliott, 2003 in the
adaptive beamforming is described for a phased antenna array azimuth plane (x–y plane) with symmetric amplitude distribu-
using a GA. The algorithm can determine the values of phase tions (Ballanis, 1997) may be written as (1):
A genetic algorithm for the level control of nulls and side lobes in linear antenna arrays 119

lobes in the original pattern, Qk is the side lobe level in dB

generated by the individual population at some peak point
hk , and d is the desired value of the side lobe level in dB.
HðkÞ is defined as (3):
1; ðQk  dÞ > 0
HðkÞ ¼ ð3Þ
0; ðQk  dÞ 6 0
The side lobes whose peaks exceed the threshold d must be
suppressed, so HðkÞ is adopted in the ‘‘cost function’’ expres-
sion. FNBW denotes the first null beamwidth, which is the
angular width between the first nulls on either side of the main
beam. The third term in (2) is introduced to keep FNBW of the
optimized pattern the same as in the initial pattern (the pattern
for In ¼ 1 and d ¼ k=2). In (2), the two beamwidths
FNBWcomputed and FNBWðIn ¼ 1Þ refer to the computed first
null beamwidth in radian for the non-uniform excitation for
the optimal spacing case and for the uniform excitation
ðIn ¼ 1Þ with uniform inter-element spacing ðd ¼ k=2Þ case,
respectively. The actual value of FNBW for a uniform linear
array can be calculated by (4):

Figure 1 Geometry of a 2M element symmetric linear antenna 2k

array along the z axis. hn ¼ ð4Þ
Nd
where N (= 2M) is the total number of elements in the array.
X
M    C1 ; C2 , and C3 are weighting coefficients to control the relative
2n  1
AFðI; u; dÞ ¼ 2 In cos kd cosðhÞ þ un ð1Þ importance of each term of (2). Because the primary aim is to
2
n¼1 achieve a deeper null, the value of C1 is higher than the values
where h denotes the zenith angle measured from the broadside of C2 and C3 . In the first term of (2), both the numerator and
direction of the array, In and un are the current excitation denominator are absolute values. A smaller value of the cost
amplitude and the excitation phase of the nth array element, function means that the array factor values at predefined posi-
respectively, d is the spacing between two consecutive elements tions are lower. Consequently, RGA controls the amplitude
and k ¼ 2p=k are the wave numbers, where k is the signal excitations and the inter-element spacing to minimize the cost
wave-length. In this paper, un is fixed at zero. The array ele- function.
ments are numbered from 1 to M from the origin in a symmet-
ric array, where the total number of elements is 2M.
After defining the far-field radiation pattern, the next step 3. Evolutionary optimization technique: genetic algorithm
in the design process is to formulate the objective function that
is to be minimized. The objective function is defined using the Genetic algorithms are a family of computational models in-
array factor in such a way that the objective of the optimiza- spired by evolution (Haupt and Werner, 2007; Haupt, 1995;
tion is satisfied. For the optimization problem of the null Holland, 1975). GAs can be used to find approximate solu-
placement in the far field pattern of the array, the array factor tions to search problems through the application of the princi-
value at the particular null position must be less than the ref- ples of evolutionary biology. GA uses biologically inspired
erence pattern. Similarly, for the side lobe reduction problem, techniques, such as genetic inheritance, natural selection,
the array factor values at the side lobe peaks must be less than mutation, and sexual reproduction (recombination or cross-
the reference pattern. To satisfy these objectives, the array fac- over). The GA was first introduced in 1975 by Prof. Holland
tor is included in the cost function expression. The objective (1975). Real-coded GA (RGA) uses floating-point number rep-
function ‘‘cost function’’ (CF) to be minimized with the resentations for the real variables and thus is free of binary
RGA to introduce the deeper null and reduce the relative encoding and decoding. Hence, it is faster than binary GA.
SLL is given in (2): The algorithm performs the following steps:
Qm 
  XK
i¼1 AFðnulli Þ (1) Randomly or heuristically generates an initial popula-
CF ¼ C1  þ C2  HðkÞ  ðQk  dÞ
jAFmax j k¼1 tion within the variable constraint range.
  (2) Computes and saves the fitness for each individual in the
þ C3  FNBWcomputed  FNBWðIn ¼ 1Þ ð2Þ
current population.
where m is the maximum number of positions where the null (3) Defines the selection probability for each individual so
can be imposed. In this paper, the value of ‘m’ is considered that it is proportional to its fitness.
to be one and two. AFðnulli Þ is the value of the array factor (4) Generates the next population by probabilistically
at the particular null position, and AFmax is the maximum va- selecting the individuals from the previous current pop-
lue of the array factor. The second term in (2) is summed to ulation to produce offspring via genetic operators.
reduce the SLL to a desired level. K denotes the number of side (5) Repeats step 2 until a satisfactory solution is obtained.
120 B. Goswami, D. Mandal

to be symmetric with respect to the center of the array. The

chromosomes correspond to the current excitation weights
and the inter-element spacing of the antenna elements. Because
of symmetry, each chromosome consists of M + 1 number of
genes, where M is the number of antenna elements on either
side of the array center. Here, the 1st to Mth genes represent
the current excitation weights of the antenna elements, and
the ðM þ 1Þth gene represents the inter-element spacing. For
example, chromosome one W1 can be represented by (5):
W1 ¼ ½W11 ; W12 ; . . . W1M ; W1ðMþ1Þ  ð5Þ
where W11 ; W12 ; . . . W1M are the antenna element weights or
genes, and W1ðMþ1Þ is the inter-element spacing. Each of these
current excitation weights and the inter-element spacing has
upper and lower limits. The random set of chromosomes can
easily be constructed using the following relation represented
by (6):
Wn ¼ ðu1  u2 Þ  
r þ u2 ; u2 < Wn 6 u1 ð6Þ
where u1 and u2 are the maximum and minimum limit values of
r is a real random vector between
the weights, respectively, and 
zero and one. All of the current excitation weights are re-
stricted to lie between 0 and 1, and the inter-element spacing
is restricted to lie between k=2 and k.

4. Numerical simulation results

Figure 2 The GA flow for determining the optimized excitation Linear antenna arrays composed of 12, 16, and 20 isotropic
amplitude and optimum location of array elements. radiating elements, with an inter-element spacing of k/2, are
considered for reference. RGA is applied to obtain deeper
nulls and to reduce the SLLs. RGA was executed with 500 iter-
GA consists of a data structure of individuals called the ations, and the population size was fixed at 120. For the RGA,
population. Individuals are also called chromosomes. Each the mutation probability was set to 0.05, and uniform cross-
chromosome is evaluated by a function known as a fitness over was used. The RGA algorithm is initialized using random
function or a cost function, which is usually the fitness func- values of the excitation (0 < In < 1) and the spacing between
tion or the objective function of the corresponding optimiza- the elements (k=2 6 d < k). The nulling performances are im-
tion problem. proved for predefined nulls of the radiation pattern. Similarly,
The working principle of a GA is explained briefly in Fig. 2 nulls are imposed at predefined peak positions. The program
based on the problem addressed in this paper. was written in Matlab and run in MATLAB version
The important parameters of the GA are as follows: 7.8.0(R2009a) on a 3.00 GHz core (TM) 2 duo processor with
2 GB RAM.
 Selection – this is based on the fitness criterion to choose The initial values of the maximum side lobe level (SLL) and
which chromosome from a population will go onto the FNBW for a uniform amplitude ðIn ¼ 1Þ and uniform
reproduce. spacing (k=2 between adjacent elements) for all of the array
 Reproduction – the propagation of individuals from one structures (linear arrays with 12, 16 and 20 elements) are given
generation to the next. in Table 1.
 Crossover – this operator exchanges genetic material, which Figs. 3–5 show the generation of deeper nulls over the 3rd
are the features of the optimization problem. Single point null. For the 12, 16, and 20-element arrays, the nulls have im-
crossover is used here. proved up to 79.54 dB, 80 dB, and 98.51 dB from the ini-
 Mutation – the modification of chromosomes in single indi- tial values of 51.90 dB, 50.60 dB, and 77.20 dB,
viduals. Mutation does not permit the algorithm to get respectively. Table 2 shows the resulting amplitude excitation
stuck at a local minimum.

Stopping criteria – The iteration stops when the maximum Table 1 SLL and FNBW for uniform excitation (In ¼ 1) of
number of cycles is reached. The grand minimum CF and its linear array sets with an inter-element spacing of k=2.
corresponding chromosome string or the desired solution are Set no. Total number SLL (dB) FNBW (degrees)
finally obtained. of elements (2M)
The desired pattern is generated by jointly optimizing the
I 12 13.06 19.10
amplitude distributions and the inter-element spacing with II 16 13.14 14.40
the fixed first null beam width. In this paper, both the ampli- III 20 13.19 11.52
tude and the inter-element spacing distributions are assumed
A genetic algorithm for the level control of nulls and side lobes in linear antenna arrays 121

0
In:1 & d:lamda/2
GA Optimized Pattern

Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 3 Best array pattern found by RGA for the 12-element array case with an improved null at the 3rd null; i.e., h = 60 and
h = 120.

0
In=1 & d=lamda/2
GA Optimized Pattern
Side lobe level(dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 4 Best array pattern found by RGA for the 16-element array case with an improved null at the 3rd null; i.e., h = 68 and
h = 112.

0
In:1 & d:lamda/2
GA Optimized pattern
Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 5 Best array pattern found by RGA for the 20-element array case with an improved null at the 3rd null; i.e., h = 72.5 and
h = 107.5.

distribution, optimal inter-element spacing, initial depth and inter-element spacing ðd ¼ k=2Þ. Table 2A shows the SLL
final null depth over the 3rd null position obtained by optimiz- and FNBW of the optimized pattern for a null imposed at
ing the cost function using RGA. In this case, the weightings of the 3rd null position.
the array elements I1 ; I2 ; . . . IM are normalized using max Figs. 6–8 show the generation of nulls at the 3rd peak for
ðIM Þ ¼ 1, and the inter-element spacings d are normalized by 12, 16 and 20 element structures, respectively. For 12, 16,
k=2. Figs. 3–5 also depict the substantial reductions in the and 20 elements, the nulls have improved up to 123.5 dB,
maximum peak of the SLL with non-uniform current excita- 83.17 dB, and 92.00 dB from the initial peak values of
tion weights and optimal inter-element spacing, compared to 19.56 dB, 20.10 dB, and 20.35 dB, respectively. Figs. 6–
the uniform current excitation weights and uniform 8 also depict the substantial reductions in the maximum peak
122 B. Goswami, D. Mandal

Table 2 Current excitation weights and initial and final null depths for a non-uniformly excited linear array with optimal inter-
element spacing (d) for one null imposed in the 3rd null position.
No. of elements (I1 ; I2 ; . . . IM ); d normalized with respect to k=2 Initial depth (dB) Final depth (dB)
(for In ¼ 1 and d ¼ k=2) (optimized In and d)
12 0.84511 0.6556 0.8444 0.71671 0.47139 51.90 79.54
0.40992; 1.1601
16 0.6013 0.5029 0.4866 0.4084 0.2438 0.1575 0.0173 50.60 88.29
0.0718; 1.1248
20 0.5478 0.7969 0.5051 0.5722 0.6221 0.6894 0.5206 77.20 98.51
0.4061 0.3769 0.1785; 1.2099

spacing ðd ¼ k=2Þ. For example, in the optimized pattern,

Table 2A SLL and FNBW for a non-uniformly excited linear the SLL is reduced by more than 3 dB from 13.06 dB to
array with optimal inter-element spacing (d) for one null 16.96 dB for the 12 element array. For the 16 element array
imposed in the 3rd null position. with the optimized pattern, the SLL is reduced by more than
4 dB from 13.14 dB to 17.46 dB. For the 20 element array
No. of elements SLL final (dB) FNBW final (degrees)
with the optimized pattern, the SLL is reduced by more than
12 16.76 19.10 1 dB from 13.19 dB to 14.22 dB. The improved values are
16 14.51 14.40
shown in Tables 3 and 3A.
20 15.50 11.52
Figs. 9–11 show the generation of nulls at the 2nd and 3rd
peaks for 12, 16 and 20 element structures, respectively. For
of the SLL with the non-uniform current excitation weights 12, 16, and 20 elements, the pair of nulls has improved up to
and the optimal inter-element spacing compared to the uni- (62.97 dB, 85.10 dB), (83.87 dB, 69.09 dB), and
form current excitation weights and a uniform inter-element (71 dB, 78.92 dB) from the initial peak values of

0
In=1 & d=lamda/2
GA Optimized pattern
Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 6 Best array pattern found by RGA for the 12-element array case with a null introduced at the 3rd peak; i.e., h = 54.50 and
h = 125.50.

0
In:1 & d:lamda/2
GA Optimized pattern
Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180

Angle of arival (degrees)

Figure 7 Best array pattern found by RGA for the 16-element array case with a null introduced at the 3rd peak; i.e., h = 64.4 and
h = 115.6.
A genetic algorithm for the level control of nulls and side lobes in linear antenna arrays 123

0
In:1 & d:lamda/2
GA Optimized pattern

Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 8 Best array pattern found by RGA for the 20-element array case with a null introduced at the 3rd peak; i.e., h = 69.70 and
h = 110.30.

Table 3 Current excitation weights and initial and final null depths for a non-uniformly excited linear array with optimal inter-
element spacing (d) for one null imposed in the 3rd peak position.
No. of elements (I1 ; I2 ; . . . IM ); d normalized with respect to k=2 Initial depth (dB) Final depth (dB)
(for In ¼ 1 and d ¼ k=2) (optimized In and d)
12 0.60797 0.5196 0.40068 0.51144 0.3882 19.56 123.5
0.35568; 1.1273
16 0.5744 0.59995 0.45826 0.4777 0.39889 0.37541 20.10 89.17
0.14133 0.3515; 3.1903
20 0.8496 0.5783 0.8708 0.5555 0.5727 0.7696 0.6757 20.35 92.00
0.9098 0.3100 0.0361; 1.3892

with the non-uniform current excitation weights and optimal

Table 3A SLL and FNBW for a non-uniformly excited linear inter-element spacing compared to the uniform current
array with optimal inter-element spacing (d) for one null excitation weights and uniform inter-element spacing
imposed in the 3rd peak position. ðd ¼ k=2Þ. For example, the SLL is reduced by more than
No. of elements SLL final (dB) FNBW final (degrees) 3 dB from 13.06 dB to 16.17 dB for the 12 element array.
For the 16 element array, the SLL is reduced by more than
12 16.96 19.08
6 dB from 13.14 dB to 20.00 dB. For the 20 element array,
16 17.46 17.10
20 14.22 10.00
the SLL is reduced by more than 2 dB from 13.19 dB to
15.85 dB. The improved values are shown in Tables 4 and
4A.
(17.22 dB, 19.56 dB), (17.49 dB, 20.10 dB), and Figs. 12–14 show the generation of nulls at the 2nd and 3rd
(17.61 dB, 20.40 dB), respectively. Figs. 9–11 also depict nulls for 12, 16 and 20 element structures, respectively. For 12,
the substantial reductions in the maximum peak of the SLL 16, 20 elements, the pair of nulls has improved up to

0
In=1 & d=lamda/2
GA Optimized pattern
Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 9 Best array pattern found by RGA for the 12-element array case with nulls introduced at the 2nd (h ¼ 65:7 ; 114:3 ) and 3rd
(h ¼ 54:50 ; 125:50 ) peaks.
124 B. Goswami, D. Mandal

0
In:1 & d:lamda/2
GA Optimized pattern

Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 10 Best array pattern found by RGA for the 16-element array case with nulls introduced at the 2nd (h ¼ 72 ; 108 ) and 3rd
(h ¼ 64:4 ; 115:6 ) peaks.

0
In=1 & d=lamda/2
GA Optimized pattern
Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 11 Best array pattern found by RGA for the 20-element array case with nulls introduced at the 2nd (h ¼ 75:6 ; 104:4 ) and 3rd
(h ¼ 69:7 ; 110:3 ) peaks.

Table 4 Current excitation weights, initial peak depth and final null depth for a non-uniformly excited linear array with optimal inter-
element spacing (d) for nulls imposed in the 2nd and 3rd peak positions.
No. of elements (I1 ; I2 ; . . . IM ); d normalized with respect to k=2 Initial depth (dB) Final depth (dB)
(for In ¼ 1 and d ¼ k=2) (optimized In and d)
(2nd; 3rd) (2nd; 3rd)
12 0.6876 0.7969 0.7023 0.67534 0.2092 0.1201; 1.2525 17.22; 19.56 62.97; 85.10
16 0.6049 0.5893 0.5405 0.5850 0.3186 0.3795 0.4347 17.49; 20.10 83.87; 69.09
0.2377; 1.2011
20 0.6163 0.5076 0.7290 0.4194 0.7148 0.2845 0.8719 17.61; 20.40 71.00; 78.92
0.3889 0.2575 0.5260; 1.0442

(82.75 dB, 94.66 dB), (122.30 dB, 123.00 dB), and

(68.60 dB, 86.36 dB) from initial values of (55.83 dB,
Table 4A SLL and FNBW for a non-uniformly excited linear 53.93 dB), (45.35 dB, -50.60 dB), and (56.62 dB,
array with optimal inter-element spacing (d) for nulls imposed 77.2 dB), respectively. Figs. 12–14 also depict the substantial
at the 2nd and 3rd peak positions. reductions in the maximum peak of the SLL with the non-uni-
No. of elements SLL final (dB) FNBW final (degrees)
form current excitation weights and optimal inter-element
spacing compared to the uniform current excitation weights
12 16.17 21.22 and uniform inter-element spacing ðd ¼ k=2Þ. For example,
16 20.00 14.40
the SLL is reduced by more than 7 dB from 13.06 dB to
20 15.85 11.52
20.53 dB for the 12 element array. For the 16 element array,
A genetic algorithm for the level control of nulls and side lobes in linear antenna arrays 125

0
In:1 & d:lamda/2
GA Optimization

Side lobe level (dB)

-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 12 Best array pattern found by RGA for the 12-element array case with improved nulls at the 2nd (h ¼ 70:6 ; 109:4 ) and 3rd
(h ¼ 60 ; 120 ) nulls.

0
In:1 & d:lamda/2
Optimized pattern
Side lobe level (dB)

-20

-40

-60

-80

-100
0 20 40 60 80 100 120 140 160 180
Angle of arival (degrees)

Figure 13 Best array pattern found by RGA for the 16-element array case with improved nulls at the 2nd (h ¼ 75:6 ; 104:4 ) and 3rd
(h ¼ 68 ; 112 ) nulls.

0
In=1 & d=lamda/2
Side lobe level (dB)

GA Optimized pattern
-20

-40

-60

-80
0 20 40 60 80 100 120 140 160 180

Angle of arival (degrees)

Figure 14 Best array pattern found by RGA for the 20-element array case with improved nulls at the 2nd (h ¼ 78:5 ; 101:5 ) and 3rd
(h ¼ 72:5 ; 107:5 ) nulls.

Table 5 Current excitation weights and initial and final null depths for a non-uniformly excited linear array with optimal inter-
element spacing (d) for nulls imposed in the 2nd and 3rd null positions.
No. of elements (I1 ; I2 ; . . . IM ); d normalized with respect to k=2 Initial depth (dB) Final depth (dB)
(for In ¼ 1 and d ¼ k=2) (optimized In and d)
(2nd; 3rd) (2nd; 3rd)
12 0.5974 0.5762 0.5642 0.4591 0.2516 0.2240; 1.2422 55.83; 53.93 82.75; 94.66
16 0.6198 0.5511 0.3251 0.4087 0.1548 0.2899 0.0937 0.1639; 45.35; 50.60 122.30; 123.00
1.7686
20 0.6163 0.5076 0.7290 0.4194 0.7148 0.2845 0.8719 0.3889 56.62; 77.20 68.60; 86.36
0.2575 0.5260; 1.0442
126 B. Goswami, D. Mandal

11th Mediterranean Microwave Symposium (MMS 2011), Hamm-

Table 5A SLL and FNBW for a non-uniformly excited linear amet, Category number FP1146H-ART; Code 87531, 8–10
array with optimal inter-element spacing (d) for nulls imposed September.
at the 2nd and 3rd null positions. Das, S., Bhattacharjee, S., Mandal, D., Bhattacharjee, A.K., 2010.
No. of elements SLL final (dB) FNBW final (degrees) Optimal sidelobe reduction of symmetric linear antenna array using
genetic algorithm. In: IEEE INDICON 2010, Dec 17–19, Kolkata,
12 20.53 19.88 India.
16 15.05 14.40 De, I., Sil, J., 2012. Entropy based fuzzy classification of images on
20 13.27 11.62 quality assessment. J. King Saud Univ. Comput. Inf. Sci. 24, 165–
173.
Elliott, R.S., 2003. Antenna Theory and Design, revised ed. John
Wiley, New Jersey.
the SLL is reduced by approximately 2 dB from 13.14 dB to Fahmy, A.A., 2012. Using the bees algorithm to select the optimal
15.05 dB. For the 20 element array, the SLL is reduced by speed parameters for wind turbine generators. J. King Saud Univ.
less than 1 dB from 13.19 dB to 13.27 dB. The improved Comput. Inf. Sci. 24, 17–26.
values are shown in Tables 5 and 5A. Guney, K., Akdagli, A., 2001. Tabu search algorithm for the optimal
pattern synthesis of linear antenna arrays with broad nulls. J.
5. Conclusions Marm. Pure Appl. Sci. 17 (2), 61–70.
Haupt, R.L., 1995. An introduction to genetic algorithm for electro-
magnetic. IEEE Antennas Propag. Mag. 37 (2), 7–15.
This paper describes the design of a non-uniformly excited Haupt, R.L., 1997. Phase-only adaptive nulling with a genetic
symmetric linear antenna array with optimized non-uniform algorithm. IEEE Trans. Antennas Propag. 45 (6), 1009–1015.
spacings between the elements using the optimization tech- Haupt, R.L., Werner, D.H., 2007. Genetic Algorithms in Electromag-
niques of a RGA. The simulated results reveal that optimizing netics. IEEE Press Wiley-Interscience.
the excitation values of the elements with the optimal inter-ele- Holland, J.H., 1975. Adaptation in Natural and Artificial Systems.
ment spacings of the array antennas can impose deeper nulls in University Michigan Press, Ann Arbor.
Mandal, D., Ghoshal, S.P., Bhattacharjee, A.K., 2009. Determination
the interference direction and reduce the SLL for a given num-
of the optimal design of three-ring concentric circular antenna
ber of array elements with respect to the corresponding uni- array using evolutionary optimization techniques. Int. J. Recent
formly excited linear array with an inter-element spacing of Trends Eng. 2 (5), 110–115.
k=2. For instance, an optimized 16 element linear array anten- Mandal, D., Yallaparagada, N.T., Ghoshal, S.P., Bhattacharjee, A.K.,
na imposes nulls with values of 83.87 dB and 69.09 dB at 2010. Wide null control of linear antenna arrays using particle
the second and third peaks, respectively, from initial peak val- swarm optimization. In: IEEE INDICON 2010, Dec 17–19,
ues of 17.49 dB and 20.10 dB at second and third peaks, Kolkata, India.
respectively. The SLL was also reduced by 6.84 dB. The paper Mandal, S., Ghoshal, S.P., Kar, R., Mandal, D., 2012. Design of
makes three main contributions: (i) In almost all design config- optimal linear phase fir high pass filter using craziness based
urations, the null depth improves by approximately 80 dB. particle swarm optimization technique. J. King Saud Univ.
Comput. Inf. Sci. 24, 83–92.
(ii) The maximum SLL is also reduced in all cases. (iii) The
Mondal, D., Chandra, A., Ghoshal, S.P., Bhattacharjee, A.K., 2010.
FNBW of the initial and final radiation pattern remains Side lobe reduction of a concentric circular antenna array using
approximately the same. It is worth noting that although the genetic algorithm. Serb. J. Electr. Eng. 7 (2), 141–148.
proposed algorithm is implemented to constrain the synthesis Mukherjee, S., Kar, S., 2012. Application of fuzzy mathematics and
of a linear array with isotropic elements, it is not limited to this grey systems in education. J. King Saud Univ. Comput. Inf. Sci. 24,
case. The proposed algorithm can easily be implemented in 157–163.
non-isotropic element antenna arrays with different geometries Recioui, A., Azrar, A., Bentarzi, H., Dehmas, M., Chalal, M., 2008.
to design various array patterns. Synthesis of linear arrays with sidelobe level reduction constraint
using genetic algorithms. Int. J. Microwave Opt. Technol. 3 (5),
524–530.
References Son, S.H., Park, U.H., 2007. Sidelobe reduction of low-profile array
antenna using a genetic algorithm. ETRI J. 29 (1), 95–98.
Abdolee, R., Ali, M.T., Rahman, T.A., 2007. Decimal genetics Steyskal, H., Shore, R.A., Haupt, R.L., 1986. Methods for null control
algorithms for null steering and sidelobe cancellation in switch and their effects on the radiation pattern. IEEE Trans. Antennas
beam smart antenna system. Int. J. Comput. Sci. Secur. 1 (3), 19– Propag. 34 (3), 404–409.
26. Yan, K.K., Lu, Y., 1997. Sidelobe reduction in array-pattern synthesis
Anooj, P.K., 2012. Clinical decision support system: risk level using genetic algorithm. IEEE Trans. Antennas Propag. 45, 1117–
prediction of heart disease using weighted fuzzy rules. J. King 1122.
Saud Univ. Comput. Inf. Sci. 24, 27–40. Yang, S., Gan, Y.B., Qing, A., 2002. Sideband suppression in time-
Ares-Pena, F.J., Rodriguez-Gonzalez, J.A., Villanueva-Lopez, E., modulated linear arrays by the differential evolution algorithm.
Rengarajan, S.R., 1999. Genetic algorithms in the design and IEEE Antennas Wirel. Propag. Lett. 1, 173–175.
optimization of antenna array patterns. IEEE Trans. Antennas Yang, S., Gan, Y.B., Qing, A., 2004. Antenna-array pattern nulling
Propag. 47 (3), 506–510. using a differential evolution algorithm. Int. J. RF Microwave
Ballanis, C.A., 1997. Antenna Theory Analysis and Design, second ed. Comput. Aided Eng. 14 (1), 57–63.
John Willey and Son’s Inc, New York.
El Cafsi, M.A., Ghayoula, R., Trabelsi, Gharsallah, H., 2011. Phased
uniform linear antenna array synthesis using genetic algorithm. In: