seeker optimization algorithm for interference suppression of linear antenna arrays by controlling...

Seeker Optimization Algorithm for InterferenceSuppression of Linear Antenna Arrays by ControllingPosition-Only, Phase-Only, and Amplitude-Only

K. Guney,1 S. Basbug2

1 Department of Electrical and Electronics Engineering, Faculty of Engineering, Nuh Naci YazganUniversity, 38040, Kayseri, Turkey2 Department of Industrial Electronics, Vocational College, Nevsehir University, 50300,Nevsehir, Turkey

Received 29 December 2010; accepted 2 April 2011

ABSTRACT: In this article, a stochastic search technique based on seeker optimization

algorithm (SOA) is proposed for null steering of linear antenna arrays by controlling the

position-only, phase-only, and amplitude-only. The SOA is relatively new optimization algo-

rithm based on the concept of simulating the act of humans’ intelligent search with their

memory, experience, and uncertainty reasoning. Several numerical examples of Chebyshev

pattern with the single, multiple, and broad nulls imposed at the directions of interference

are given to illustrate the performance and flexibility of the proposed algorithm. For a com-

parison, the nulling patterns obtained by simulated annealing (SA) and tabu search (TS)

algorithms are also given. Furthermore, the results of SOA are statistically compared with

those of SA and TS algorithms. The statistical results of simulations show that SOA is supe-

rior to the other compared algorithms. VC 2011 Wiley Periodicals, Inc. Int J RF and Microwave

CAE 21:505–518, 2011.

Keywords: antenna array synthesis; pattern nulling; seeker optimization algorithm; phase-only

control; position-only control; amplitude-only control

I. INTRODUCTION

The null steering in antenna array pattern to reject unwanted

interference sources while receiving the desired signal from

a chosen direction has received considerable attention over

the years [1–33]. The increasing pollution of the electro-

magnetic environment has prompted the study of array pat-

tern nulling methods. These methods are very important in

radar, sonar, and many communication systems for maxi-

mizing signal-to-interference ratio. The nulling methods are

generally based on controlling the complex weights (both

the amplitude and the phase), the amplitude-only, the

phase-only, and the position only of the array elements. In-

terference suppression with complex weights can be consid-

ered the most effective methods as it has a larger solution

space than others. However, it is also the most expensive

method considering the cost of the controllers used for

phase shifters and variable attenuators for each array ele-

ment. Moreover, the computation performed to find the val-

ues of element amplitudes and phases takes a long time

when the number of elements in the array increases. The

amplitude-only control uses a set of variable attenuators to

adjust the element amplitudes. The number of attenuators

and the computational time can be halved, if the element

amplitudes have even symmetry around the center of the

array. The phase-only control method can be practiced with

the use of digital phase shifters. The position-only control

needs a mechanical driving system such as servomotors,

which can move the array elements.

In this article, a method based on seeker optimization

algorithm (SOA) [34–40] is presented for linear antenna

array pattern synthesis with prescribed nulls by controlling

the position-only, phase-only, and amplitude-only. The

SOA originally proposed in Ref. [34] is a relatively new ev-

olutionary computation technique based on the concept of

simulating the act of human intelligent searching. In the lit-

erature [34–42], SOA was compared with biogeography-

based optimization (BBO) and different and hybrid versions

of particle swarm optimization (PSO), genetic algorithm

(GA), differential evolution algorithm (DEA), ant colony

Correspondence to: K. Guney; e-mail: [email protected]

VC 2011 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20536Published online 27 July 2011 in Wiley Online Library

(wileyonlinelibrary.com).

505

optimization (ACO), and bacterial foraging (BF) for the

benchmark functions and particular engineering problems.

In Refs. [35, 36], the SOA has been applied to optimal reac-

tive power dispatch on standard IEEE 57- and 118-bus

power systems, and compared with several optimization

algorithms including GA, PSO, and DEA. It is clear from

the results obtained in Refs. [35, 36] that the proposed

approach is superior to the other listed algorithms and can

be efficiently used for optimal reactive power dispatch. An

SOA-based digital filter design method has been presented

in Ref. [37], and the benefits of SOA for designing digital

IIR filters have been studied. It was illustrated in Ref. [37]

that SOA has better, or at least equivalent, global search

ability and convergence speed than GA, four version of

PSOs, and three version of DEAs for most of the chosen

and widely used problems. In Ref. [38], the performance of

SOA was studied with a challenging set of benchmark prob-

lems for function optimization. The comparisons of SOA,

DEA, and three modified PSOs show that SOA has better

global search capability and faster convergence speed for

the most chosen benchmark problems [38]. A hybrid

method combining SOA and sequential quadratic program-

ming (SQP) for solving dynamic economic dispatch (DED)

problem with valve-point effects including generator ramp-

rate limits has been proposed in Ref. [41]. It was shown that

the proposed hybrid SOA-SQP method is giving higher

quality solutions than the reported methods for DED prob-

lem with valve-point effects [41]. The application of the

SOA to tuning the structures and parameters of artificial

neural networks (ANNs) was presented in Ref. [39] as a

new evolutionary method of artificial neural network train-

ing. The results in Ref. [39] showed that SOA can simulta-

neously tune the structures, the weight values, and the

parameter of regularization performance function. More-

over, it was shown that SOA is better than, or at least com-

parable with, DE and two PSO algorithms. It was also illus-

trated that the ANNs with link switches trained by SOA not

only have much less number of links but also have better or

equivalent learning capabilities than ones by back propaga-

tion algorithms [39]. SOA was applied to optimal modeling

of the proton exchange membrane fuel cell by using a fuel

cell test system in Ref. [40]. The simulation results showed

that SOA has better performance than other famous ver-

sions of PSO and DEA algorithms [40]. SOA was used in

Ref. [42] for the solution of the constrained economic load

dispatch problems in different power systems considering

various nonlinear characteristics of generators. The results

of SOA were compared with those of BBO, different and

hybrid versions of PSO, DEA, GA, ACO, BF with Nelder-

Mead algorithm, improved fast evolutionary programming,

and Hopfield model. The comparison results showed that

the SOA has the capability to converge to a better quality

near-optimal solution and possesses better convergence

characteristics and robustness than the other algorithms

[42].

As SOA was compared with BBO and different and

hybrid versions of PSO, GA, DEA, ACO, and BF in the

aforementioned Refs. [34–42], in this article we compared

SOA with simulated annealing (SA) [43] and tabu search

(TS) [44–46] algorithms for the position-only, phase-only,

and amplitude-only pattern nulling of linear antenna arrays.

Both algorithms, SA and TS, were successfully used for

antenna array synthesis in the literature [13, 19, 47–53].

The comparison of stochastic optimization algorithms

needs the appropriate statistical methods for the analysis

of results. In this article, the results of SOA are statisti-

cally compared with those of SA and TS algorithms. A

statistical method named Wilcoxon rank sum test is used

to determine whether the difference between the mean

objective function values obtained by SOA and the other

competitive algorithms is statistically significant.

II. PROBLEM FORMULATION

Let us assume that the array elements are symmetrically

placed and conjugate-symmetrically excited about the center

of a linear array. The far field array factor of such an array

with an even number of isotropic elements (2M) is defined as

FðhÞ ¼ 2XMk¼1

ak cos2pkdk sin hþ uk

� �(1)

where y is the angle from broadside, dk is the distance

between position of the kth element and the array centre,

and ak and uk are amplitude and phase weights of the kthelement, respectively. The main purpose of this work is to

find an optimal set of element excitation or position val-

ues such that obtained pattern achieves the desired nulling

performance. Therefore, the following cost function will

be minimized with the use of SOA.

C ¼X90�

h¼�90�WðhÞ FoðhÞ � FdðhÞj j þ ESLðhÞ½ � (2)

where Fo(y) and Fd(y) are, respectively, the pattern

obtained by using SOA and the desired pattern. W(y) andESL(y) are used to control the null depth level (NDL) and

maximum sidelobe level (MSL), respectively. The values

of W(y) and ESL(y) are selected by experience such that

the cost function is capable of guiding potential solutions

to obtain satisfactory array pattern performance with

desired properties. The factors W(y) and ESL(y) give the

antenna designer greater flexibility and control over the

actual pattern. The trade-off of the relative importance

between null depth and sidelobe level can easily be

obtained by changing the values of these factors.

III. SOA

Like other evolutionary algorithms, SOA has a set of

potential solutions named as search population [34–40].

‘‘Seekers’’ are the individuals of these populations. The

population is randomly divided into K subpopulations to

define a neighborhood for each seeker. This organization

ensures that SOA has a kind of social characteristic for

sharing of information. Every subpopulation has the same

size and all seekers in the same populations constitute a

neighborhood.

506 Guney and Basbug

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011

A. Implementation of SOAIn SOA, a search direction dij(t) and a step length aij(t) arecomputed separately for each seeker i on each dimension j foreach time step t, where aij(t) � 0 and dij(t) [ {�1,0,1}. dij(t) ¼1 means the ith seeker goes towards the positive direction of

the coordinate axis on the dimension j, dij(t) ¼�1 means the

seeker goes towards the negative direction, and dij(t) ¼ 0

means the seeker stays at the current position. For each seeker

i (1 � i � s, s is the population size), the position update on

each dimension j (1� j� D) is given as follows,

xijðtþ 1Þ ¼ xijðtÞ þ aijðtÞdijðtÞ (3)

xij(t) is the position of the ith seeker in the jth dimension at

time step t. In this article, ~xiðtÞ in the solution space is

served as the position, phase, and amplitude values of the

antenna array elements for the position-only, phase-only, and

amplitude-only synthesis examples, respectively. Each sub-

populations use their own information for searching activ-

ities. Therefore, they may easily converge to a local opti-

mum solution. To prevent this unwanted convergency, the

positions of the worst K-1 seekers of each subpopulation are

combined with the best one in each of the other K-1 subpo-

pulations using the following binomial crossover operator:

xknj;worst ¼xlj;bestxknj;worst

if Rj � 0:5else

�(4)

where Rj is a uniformly random real number within [0,1],

xknj;worst is denoted as the jth dimension of the nth worst

position in the kth subpopulation, and xlj,best is the jth dimen-

sion of the best position in the lth subpopulation, n, k, l ¼ 1,

2, …, K�1 and k = l. Thus, each subpopulation shares their

good information with the other subpopulations. This

exchange process among the subpopulations boosts the di-

versity of population. When the iteration number reached the

maximum iteration number tmax, the optimization process is

ended. A flowchart of the SOA utilized in this article is

shown in Figure 1.

B. Search DirectionWhen the objective function cannot be differentiated at

all, a so-called empirical gradient (EG) can be determined

by evaluating the response to increments and decrements

in each coordinate. In this way, the seekers are able to

trace an EG to lead their search. Search direction can sim-

ply be determined by the signum function of a better posi-

tion minus a worse position as the SOA involves the dif-

ference of the values instead of the magnitudes of the EG.

The seekers must take into account several EGs by

evaluating their own current or historical positions or their

neighbors to determine their search direction. Similarly,

SOA models three different behaviors to determine search

direction: egotistic, altruistic and proactive.

Although egotistic behavior is totally pro-self, altruistic

behavior is totally pro-group. Egotistic and altruistic behaviors

are two extreme types of co-operative behavior. Every seeker i,

as a single sophisticated agent, is uniformly egotistic, believing

that he should go toward his historical best position ~pi;bestðtÞ.

Then, an EG from ~xiðtÞ to ~pi;bestðtÞ can be involved where

~xiðtÞ ¼ ½xi1; xi2;…; xiD� is the position of the ith seeker at

time step t. Hence, the seeker i is associated with an empiri-

cal direction vector called as egotistic direction ~di;egoðtÞ:

~di;egoðtÞ ¼ signð~pi;bestðtÞ �~xiðtÞÞ (5)

where the sign(�) is a signum function on each dimension of

the input vector.

The altruistic behavior means that the seekers in a same

neighborhood region co-operate explicitly communicate with

each other and adjust their behaviors in response to others to

achieve the desired goal. Hence, the seekers present totally

pro-group behavior. The population then exhibits a self-

organized aggregation behavior. The positive feedback of

self-organized aggregation behaviors usually takes the form

of attraction toward a given signal source. For a ‘‘black-box’’

problem in which the ideal global minimum value is

unknown, the neighbors’ historical best position ~gbestðtÞ or

the neighbors’ current best position ~lbestðtÞ is used as the

attraction signal source of the self-organized aggregation

behavior. Therefore, each seeker i is associated with two

optional altruistic direction, ~di;alt1ðtÞ and ~di;alt2ðtÞ,

~di;alt1ðtÞ ¼ signð~gbestðtÞ �~xiðtÞÞ (6)

~di;alt2ðtÞ ¼ signð~lbestðtÞ �~xiðtÞÞ (7)

Furthermore, seekers also have a pro-active personal-

ity. Seekers do not simply act in response to their

Figure 1 Flowchart of the SOA.

Seeker Optimization Algorithm for Nulling 507

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

environment; they are able to exhibit goal-directed behav-

ior. In addition, the seekers’ past behaviors can affect and

guide their future behaviors. As a result, seekers may be

pro-active to change their search direction and exhibit

goal-directed behavior according to their past behavior.

Hence, each seeker i is associated with a search direction

vector called as proactiveness direction ~di;proðtÞ:

~di;proðtÞ ¼ signð~xiðt1Þ �~xiðt2ÞÞ (8)

where t1, t2 [ {t, t�1, t�2}, and ~xiðt1Þ is better than

~xiðt2Þ.According to human rational judgment, the actual search

direction of the ith seeker, ~diðtÞ, is based on a compromise

among egotistic, altruistic and proactiveness behaviors,

namely, ~di;egoðtÞ, ~di;alt1ðtÞ, ~di;alt2ðtÞ and ~di;proðtÞ. In the study,

every dimension j of ~diðtÞ is selected applying the follow-

ing proportional selection rule (shown in Fig. 2):

dij ¼0 if rj � p

ð0Þj

þ1 if pð0Þj < rj � p

ð0Þj þ p

ðþ1Þj

�1 if pð0Þj þ p

ðþ1Þj < rj � 1

8>><>>:

(9)

where rj is a uniform random number in [0,1] and pðmÞj (m

[ {0,þ1,�1}) is the percentage of the number of ‘‘m’’from the set f~dij;egoðtÞ; ~dij;alt1ðtÞ; ~dij;alt2ðtÞ; ~dij;proðtÞg on each

dimension j of all the four empirical directions, i.e., pðmÞj ¼

the number of m/4.

C. Step LengthThere is mostly a neighborhood region near an extremum

point in the continuous search space. Therefore, the fitness

values calculated from input variables are proportional to

their distances from the extremum point. Searching activ-

ities should be intensified in regions containing good solu-

tions through focusing search because it can be assumed

that better points are likely to be found in the neighbor-

hood of families of good points. Therefore, from the

standpoint on human searching, one may find the near

optimal solutions in a narrower neighborhood of the point

with lower fitness value and, on the contrary, in a wider

neighborhood of the point with higher fitness value.

Fuzzy systems originate from the desire to describe

complex systems with linguistic descriptions. According

to human focusing searching discussed above, the uncer-

tain reasoning of human searching could be described by

natural linguistic variables and a simple control rule as

‘‘If {fitness value is small} (i.e., the conditional part), then

{step length is short} (i.e., the action part).’’ The under-

standing and linguistic description of human searching

makes fuzzy system a good candidate for simulating

human focusing searching behavior.

Ranges of fitness values frequently differ from one

optimization problem to another. To design a fuzzy sys-

tem to be applicable to a wide range of optimization prob-

lems, the fitness values of all the seekers are descendingly

sorted and turned into the sequence numbers from 1 to sas the inputs of fuzzy reasoning. The linear membership

function is used in the conditional part as the universe of

discourse is a given set of numbers, i.e., 1, 2, …, s. The

expression is presented as follows,

li ¼ lmax �s� Iis� 1

ðlmax � lminÞ (10)

where Ii is the sequence number of ~xiðtÞ after sorting the

fitness values and lmax is the maximum membership

degree value which is equal to or a little less than 1.0.

In this study, the Bell membership function

lðxÞ ¼ e�x2=2d2 is used in the action part. For the conven-

ience, one dimension is considered. Thus, the membership

degree values of the input variables beyond [�3d, 3d] areless than 0.0111 (l (63d) ¼ 0.0111), and the elements

beyond [�3d, 3d] in the universe of discourse can be

neglected for a linguistic atom. Thus, the minimum value

lmin ¼ 0.0111 is set. Moreover, the parameter, ~d, of the

Bell function is determined as follows:

~d ¼ x� absð~xbest �~xrandÞ (11)

where abs(�) returns an output vector such that each ele-

ment of the vector is the absolute value of the correspond-

ing element of the input vector. The parameter x is linearly

decreased from xmax to xmin during a run to decrease the

step length with time step increasing and gradually improve

the search precision. The ~xbest and ~xrand are the best seeker

and a randomly selected seeker from the same subpopula-

tion to which the ith seeker belongs, respectively. Notice

that ~xrand is different from ~xbest, and ~d is shared by all the

seekers in the same subpopulation [34–40].

To introduce the randomicity on each dimension and

improve local search capability, the following equation is

used to change li into a vector ~li.

lij ¼ RANDðli; 1Þ (12)

Then the action part of the fuzzy reasoning (shown in

Fig. 3) gives every dimension j of step length by the fol-

lowing equation:

aij ¼ djffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� lnðlijÞ

q(13)

IV. NUMERICAL RESULTS

To show the performance of the proposed SOA for steer-

ing single, multiple, and broad nulls with the imposed

directions by controlling the position-only, phase-only,

and amplitude-only, 16 examples of a linear array have

Figure 2 The proportional selection rule of search directions.



been performed. For all examples, a 30-dB Chebyshev

pattern given in Figure 4 for 20 equispaced elements with

k/2 interelement spacing is utilized as the initial pattern.

In the first six examples, the pattern nulling is achieved

by controlling only the element positions. In the next four

examples, the SOA is used for interference suppression

with the phase-only control. In the remaining six exam-

ples, the SOA is used for interference suppression with

the amplitude-only control.

It was shown in the literature [34–42] that the perform-

ance of SOA is mostly better than the performance of

BBO and different and hybrid versions of PSO, GA,

DEA, ACO, and BF for the benchmark functions and par-

ticular engineering problems. For this reason, in this pa-

per, the results of SOA are compared with the results of

SA and TS for array pattern nulling problem.

It should be noted that the coordinates of seekers, ~xiðtÞ,represent the position, phase, and amplitude values of the

antenna array elements for the position-only, phase-only, and

amplitude-only examples, respectively, for SOA. The result

with the lowest cost function value after a whole simulation

process is the solution which can be achieved by the algo-

rithm. All simulation codes are programmed by using Cþþprogramming language and tested in Windows environment

by running 30 times for each example. All simulations are

run on the same computer, with configuration of Intel Pen-

tium Core 2 Duo 1.6 GHz CPU and 2048 MB RAM.

In the optimization process, population size is chosen

as 60 members for SOA, SA, and TS. For SA algorithm,

the initial temperature has to be high enough to ensure

that the process will start from a state in which all the

suggested solutions may have a chance to be accepted as

a new solution. Initial temperatures T0 are taken as 5, 20,

and 50 for the position-only, phase-only, and amplitude-

only examples, respectively. The cooling ratio is set to

0.85 to decrease temperature gradually. As the classical

TS [44, 45] uses a solution vector consisting of a string of

bits, a transformation from binary to real numbers should

be used for solving numerical problems. Thus, in this

study, a version of TS proposed by Karaboga et al. in

Ref. [46] is used. For TS algorithm, the values of c1, c2,c3, recency, and frequency factors are chosen as 9000, 3,

3, 1.5, and 2, respectively. These control parameters are

the same as those used successfully in Ref. [13] for array

pattern synthesis. The meanings of the SA and TS control

parameters given above are available in the literature [13,

43, 46]. For SOA, the subpopulation number K is set to 3.

Thus, each subpopulation has 20 seekers. xmax and xmin

are set to 0.9 and 0.1, respectively for all examples. The

maximum iteration number is set to 150.

A. Position-Only ControlIn the first example, the direction of interference yi is cho-sen at the peak of the second sidelobe, which occurs about

Figure 3 The action part of the fuzzy reasoning.

Figure 4 Initial Chebyshev pattern.

Figure 5 Radiation patterns obtained by position-only control

with one imposed null at 15�.

Figure 6 Convergence curves of the nulling pattern achieved

by the SOA, SA, and TS.



15�. The values of the cost function parameters given in

Eq. (2) are selected as follows:

FdðhÞ ¼ 0; for h ¼ hiInitial pattern; elsewhere

�(14)

WðhÞ ¼ 170; for h ¼ hi1; elsewhere

�(15)

and

ESLðhÞ ¼ 30; ifMSL> �28dB

0; elsewhere

�(16)

where the MSL given in eq. (16) represents the MSL of

achieved pattern in the sidelobe region. The resultant pat-

terns that make use of the position-only control by SOA,

SA, and TS are shown in Figure 5. These nulling patterns

are produced by the simulation results with the best cost

function values obtained by running 30 times. The pattern

obtained by SOA closely resembles to the initial Cheby-

shev pattern except for the nulling direction (yi ¼ 15�). Itis seen from Figure 5 that a null depth is obtained at a

level of deeper than �140 dB, and there is no sidelobe

level which exceeds �28.08 dB.

Figure 6 shows the convergence curves of SOA, SA,

and TS. The curves are the average values obtained by

running each algorithm 30 times. It is clear from Figure 6

that SOA leads to better convergence than the others and

150 iterations are needed to find the good solutions.

To examine the flexibility of the SOA, in the second

and third examples, the restrictions are made on the MSL.

The values of ESL(y) are modified as given below while

the values of other design parameters are the same as

those of the first example.

ESLðhÞ ¼ 30; ifMSL> �28:5dB0; elsewhere

�(17)

and


with the restricted MSL having one imposed null at 15� (MSL >

�28.5 dB).


with the restricted MSL having one imposed null at 15� (MSL >

�29 dB).

TABLE I NDL and MSL Values of the Patterns Obtainedby SOA, SA, and TS for Figures 5 and 7-11

SOA SA TS

Figure 5 NDL (dB) �144.10 �81.21 �84.26

MSL (dB) �28.08 �28.09 �28.06

Figure 7 NDL (dB) �115.60 �41.03 �69.22

MSL (dB) �28.53 �28.47 �28.62

Figure 8 NDL (dB) �102.50 �37.69 �40.37

MSL (dB) �29.02 �29.04 �29.09

Figure 9 NDL (dB) at 15� �134.90 �35.33 �36.41

NDL (dB) at 25� �113.10 �76.53 �77.14

MSL (dB) �28.03 �28.27 �28.13

Figure 10 NDL (dB) at 15� �88.02 �34.64 �39.30

NDL (dB) at 25� �96.06 �41.01 �41.08

NDL (dB) at 40� �103.60 �56.14 �48.79

MSL (dB) �28.07 �28.12 �28.03

Figure 11 NDL (dB) �50.56 �43.22 �48.66

MSL (dB) �28.06 �28.26 �28.09


with double imposed null at 15� and 25�.



ESLðhÞ ¼ 30; ifMSL> �29dB

0; elsewhere

�: (18)

The patterns produced by the SOA, SA, and TS for the

values of ESL(y) given in Eqs. (17) and (18) are illus-

trated in Figures 7 and 8, respectively. The NDL and MSL

values of the patterns in Figures 5, 7, and 8 are listed in

Table I. It is apparent from Table I and Figures 5, 7 and 8

that the SOA can obtain the patterns with satisfactory NDL

and MSL. As expected, the MSL performances of the pat-

terns given in Figures 7 and 8 are better than those of the

pattern given in Figure 5. But in response to such an

improvement of the MSL, the null depth performance of

the patterns in Figures 7 and 8 is worse than that of the

patterns in Figure 5. The results obtained here illustrate

that the NDL and MSL of the nulling pattern can be easily

adjusted by using SOA. There is a trade-off between the

NDL and the MSL; usually, performance cannot be

improved significantly for one without sacrificing the other.

The proposed SOA is also capable of synthesizing the

pattern with multiple nulls at any desired directions. For

this purpose, in the fourth and fifth examples, only the

Fd(y) given by Eq. (14) is modified to synthesize the pat-

terns with double nulls imposed at the directions of the

second and fourth peaks from the main beam (yi1 ¼ 15�

and yi2 ¼ 25�), and with triple nulls imposed at the direc-

tions of the second, fourth, and sixth peaks from the main

beam (yi1 ¼ 15�, yi2 ¼ 25�, and yi3 ¼ 40�). The patterns

with multiple nulls are shown in Figures 9 and 10. It is

clear from Figures 9 and 10 that all desired nulls obtained

by SOA are deeper than �88 dB. The results depicted in

Figures 9 and 10 show the ability of the proposed tech-

nique for the array pattern synthesis with multiple nulls

steered in the prescribed interference directions.

It is well known that the broad nulls are needed when

the direction of arrival of the unwanted interference may

vary slightly with time or may not be known exactly, and

where a comparatively sharp null would require continu-

ous steering for obtaining a reasonable value for the sig-

nal-to-noise ratio. In the sixth example, the pattern having

a broad null located at 19� with Dy ¼ 5� is achieved and

is shown in Figure 11. From the figure, an NDL deeper

than �50 dB is obtained over the spatial region of inter-

est. This computer simulation example clearly shows the

capacity of SOA to synthesize the array pattern with

broad null imposed at the direction of interference.

The NDL and MSL values of the patterns obtained by

SOA, SA, and TS are given in Table I for Figures 9–11. It


with triple imposed null at 15�, 25�, and 40�.Figure 11 Radiation patterns obtained by position-only control

with a broad null sector centered 19� with Dy ¼ 5�.

TABLE II Element Positions (dk) in k for Figures 5 and 7-11

Initial Chebyshev

pattern Computed with the SOA

k Figure 4 Figure 5 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11

1 0.25 0.25835 0.26128 0.26111 0.27358 0.27369 0.25064

2 0.75 0.77377 0.77351 0.77363 0.79847 0.78951 0.74862

3 1.25 1.28440 1.27546 1.28222 1.28666 1.27494 1.26454

4 1.75 1.76882 1.76462 1.77510 1.75895 1.75998 1.77846

5 2.25 2.24460 2.25109 2.26830 2.25695 2.26261 2.28907

6 2.75 2.71830 2.71366 2.75054 2.76370 2.76653 2.76083

7 3.25 3.21919 3.21159 3.25340 3.26848 3.27775 3.23548

8 3.75 3.72021 3.73381 3.76317 3.72217 3.75795 3.68193

9 4.25 4.28291 4.28283 4.34347 4.26624 4.28335 4.25626

10 4.75 4.81884 4.83268 4.92320 4.90186 4.97987 4.85265



is clear from Table I that the null depth performances of

patterns obtained by SOA are better than those of SA and

TS algorithms while their MSL values are almost the same.

The element position values calculated by the SOA for

the patterns given in Figures 5 and 7–11 are given in

Table II. It is apparent from Figures 5 and 7–11 that the

patterns are symmetric with respect to the main beam.

This is because the symmetry property of the element

positions around the array center results in a pattern that

is symmetric about the main beam peak. Therefore, when

a null imposed at the one side of the main beam, an

image null occurs at the other side of the main beam.

B. Phase-Only ControlTo show the performance of the proposed method, in the

next four examples, forming nulls in the pattern is

achieved by controlling only the phase of each array ele-

ment. The amplitude weights (ak) of these examples are

the same as those of the initial Chebyshev array.

In the seventh example, the direction of interference yi isselected at the peak of the first sidelobe, which occurs about

�10�. The values of the cost function parameters given in

Eq. (2) are selected as in the first example except ESL(y)value. As there is not a specified restriction on the sidelobe

level, the value of ESL(y) is zero in this example. Figure 12

shows the nulling patterns obtained from the SOA, SA, and

TS. It is seen from this figure that the pattern achieved by

SOA closely resembles to the initial Chebyshev pattern

except for the nulling direction (yi ¼ �10�), and there is an

unavoidable sidelobe level increase in the direction symmet-

ric to nulling direction with respect to the main beam.

In the eighth and ninth examples, only the Fd(y) givenby Eq. (14) is modified to synthesize the patterns with

two nulls imposed at the peaks of the first and the third

sidelobes (yi1 ¼ �10� and yi2 ¼ 20�), and with triple

nulls imposed at the peaks of the first, the third, and the

fifth sidelobes (yi1 ¼ �10�, yi2 ¼ 20�, and yi3 ¼ �33�).The patterns with multiple nulls are illustrated in Figures

13 and 14. It is clear that all desired nulls obtained by

SOA are deeper than �110 dB.

In the tenth example, the patterns having a broad null

located at 26� with Dy ¼ 5� are achieved and are shown in

Figure 15. From the figure, a NDL achieved by SOA deeper

than �62 dB is obtained over the spatial region of interest.

Figure 12 Radiation patterns obtained by phase-only control

with one imposed null at �10�.


with double imposed null at �10� and 20�.


with triple imposed null at �33�, �10�, and 20�.


with a broad null sector centered 26� with Dyi ¼ 5�.



In Table III, the NDL and MSL of the patterns

obtained by using SOA are compared with those of the

patterns obtained by using SA and TS for Figures 12–15.

It is apparent from Table III that the NDLs of patterns

produced by SOA are deeper than those of SA and TS

algorithms while their MSL values are almost the same.

The element phase values calculated by the SOA for

the patterns given in Figures 12–15 are given in Table IV.

As it is seen from Figures 12–15, the patterns are not

symmetric with respect to the main beam. This is a conse-

quence of the odd-symmetry of the element phases around

the array center which, coupled with the even symmetry

of the element amplitudes, results in a pattern that is not

symmetric about the maim beam peak at 0�. It should also

be noted that since the element phases have odd-symmetry

about the center of the array, the number of phase shifters

to be used is 2M, but the number of controllers for the

phase shifters is M for an array with 2M elements.

C. Amplitude-Only ControlIn the remaining six examples, forming nulls in the pattern is

achieved by controlling only the amplitude of each array ele-

ment. For these examples, the maximum iteration number is

set to 200 to find the optimal solutions and the other control pa-

rameters of algorithms are the same with the first 10 examples.

In the eleventh example, the Chebyshev pattern with a

single null imposed at the direction of the second peak

from main beam, which occurs about 14�, is considered.

The values of the cost function parameters given in (2)

are selected as in the first example. Figure 16 shows the

nulling patterns obtained by the SOA, SA, and TS. As it

can be seen from the figure, the NDL of the pattern

obtained by using SOA is deeper than �100 dB.

To show the flexibility of the SOA, in the twelfth

example, W(y) is modified as follows

WðhÞ ¼ 270; for h ¼ 14�

1; elsewhere

�(19)

The nulling pattern with the increased NDL is then

obtained by the SOA, and is compared with the patterns

obtained by SA and TS in Figure 17. The NDL and MSL

of the patterns illustrated in Figures 16 and 17 are given

in Table V. The null depth performance of pattern

obtained by SOA in Figure 17 is better than that of the

pattern in Figure 16. But in response to such an improve-

ment of the NDL, the maximum sidelobe performance of

TABLE III NDL and MSL Values of the PatternsObtained by SOA, SA, and TS for Figure 12–15

SOA SA TS

Figure 12 NDL (dB) �145.40 �65.21 �78.30

MSL (dB) �23.99 �24.15 �24.01

Figure 13 NDL (dB) at �10� �119.80 �66.50 �83.49

NDL (dB) at 20� �112.60 �75.21 �79.18

MSL (dB) �23.96 �23.99 �23.94

Figure 14 NDL (dB) at �33� �117.10 �64.25 �58.23

NDL (dB) at �10� �111,10 �69.18 �85.72

NDL (dB) at 20� �133.40 �73.34 �61.62

MSL (dB) �23.72 �23.08 �23.42

Figure 15 NDL (dB) �62.78 �59.49 �55.51

MSL (dB) �22.13 �20.82 �23.27

Figure 16 Radiation patterns obtained by amplitude-only con-

trol with one imposed null at 14�.

TABLE IV The Element Amplitudes (ak) of the Initial Chebyshev Array and the Element Phases (uk) of Nulling PatternsObtained by SOA Given in Figures 12–15

Initial Chebyshev

pattern Element phases (in degree) computed with the SOA

k Figure 4 Figure 12 Figure 13 Figure 14 Figure 15

61 1.00000 60.78319 +0.13387 61.49891 +0.85905

62 0.97010 61.81287 +0.02192 60.43040 +2.49308

63 0.91243 62.24520 60.39395 +0.01824 63.28681

64 0.83102 62.89041 63.05584 66.94246 62.14004

65 0.73147 62.50538 65.81174 610.34060 62.04314

66 0.62034 60.64605 61.56896 61.40288 66.48599

67 0.50461 +1.57898 +5.92681 +7.24097 610.24862

68 0.39104 +4.20985 +9.26273 +0.15470 +10.21150

69 0.28558 +7.51205 +5.93831 65.40499 +22.26463

610 0.32561 +3.70782 +0.21320 +0.31847 614.84453



the pattern in Figure 17 is worse than that of the pattern

in Figure 16.

Further to inspect the versatility of the SOA on the

pattern synthesis with null steering, in the thirteenth

example, the restriction is made on the MSL. The value

of ESL(y) is modified as in Eq. (18). The resulting pat-

terns with the restricted MSL are shown in Figure 18. The

NDL and MSL of the patterns in Figure 18 are given in

Table V. It is evident from Table V that the NDL of the

pattern obtained by SOA is worse than those of the previ-

ous two examples because the better MSL values mean

the worse null depth performance. The flexible results

obtained in this example and the twelfth example are not

observed for SA and TS under the maximum iteration

condition used in this article. It should be noted that the

better results of SA and TS can be obtained by using

more iteration and population numbers. However, the

more iteration and population numbers mean that the lon-

ger computation time for the optimization process. On the

other hand, many modified versions of SA and TS are

also available in the literature. Achieving better results

can be possible with these modified versions of the algo-

rithms. The results in this article are obtained under stated

conditions.

To test capacity of synthesizing the pattern with multi-

ple nulls at any desired directions for the SOA, in the

fourteenth and fifteenth examples, only the Fd(y) given by

Eq. (14) is modified to synthesize the patterns with double

nulls imposed at the directions of the second and the

fourth peaks from main beam (14� and 26�), and with tri-

ple nulls imposed at the directions of the second, fourth,

and fifth peaks from main beam (14�, 26�, and 33�). Thepatterns with multiple nulls are illustrated in Figures 19

and 20. It can be seen from Figures 19 and 20 that all

desired nulls obtained by SOA are deeper than �95 dB.

As the final example, the pattern having a broad null

located at 30� with Dy ¼ 5� is achieved and is shown in

Figure 21. From the Figure 21, the NDL of the pattern

obtained by using SOA is deeper than �70 dB.

In Table V, the NDL and MSL values of the pattern

achieved by using SOA are compared with those of the

patterns achieved by using SA and TS for Figures 19–21.


trol with a NDL deeper than that of the eleventh example.

TABLE V NDL and MSL Values of the PatternsObtained by SOA, SA, and TS for Figures 16-21

SOA SA TS

Figure 16 NDL (dB) �103.90 �73.08 �66.06

MSL (dB) �28.08 �25.18 �26.71

Figure 17 NDL (dB) �126.00 �61.11 �67.69

MSL (dB) �27.86 �27.16 �27.66

Figure 18 NDL (dB) �96.70 �65.93 �74.37

MSL (dB) �29.86 �24.50 �26.92

Figure 19 NDL (dB) at 14� �95.54 �46.12 �65.83

NDL (dB) at 26� �134.40 �51.24 �36.25

MSL (dB) �27.51 �25.12 �26.52

Figure 20 NDL (dB) at 14� �110.60 �54.26 �63.69

NDL (dB) at 26� �116.30 �58.00 �66.43

NDL (dB) at 33� �100.90 �41.86 �52.33

MSL (dB) �25.03 �23.81 �26.26

Figure 21 NDL (dB) �70.84 �44.54 �61.94

MSL (dB) �27.53 �24.68 �26.12


trol with the restricted MSL having one imposed null at 14�.


trol with double imposed null at 14� and 26�.



It is clear from Table V that the NDL and MSL values of

patterns produced by SOA are better than those of SA and

TS algorithms except the MSL value of TS in Figure 20,

it is slightly better than that of SOA.

The element amplitude values normalized according to

center elements for the patterns in Figures 16–21 are

given in Table VI. It is apparent from Figures 16–21 that

the patterns are symmetric with respect to the main beam.

This is because the symmetry property of the element

amplitudes around the array center results in a pattern that

is symmetric about the main beam. Therefore, when a

null imposed at the one side of the main beam, an image

null occurs at the other side of the main beam.

To demonstrate clearly the performances of SOA, SA,

and TS, in this article, the results of SOA are statistically

compared with those of SA and TS algorithms. The statis-

tical values are obtained by running algorithms 30 times

for each example, and are given in Table VII. In this ta-

ble, the statistical values, ‘‘Min,’’ ‘‘Max,’’ ‘‘Mean,’’ and

‘‘StdDev’’ of each set of data represent the best, the

worst, the average, and the standard deviation of cost

function values of simulations, respectively. A statistical

test called Wilcoxon rank sum test is conducted at the 5%

significance level for independent samples. The P-values

obtained by Wilcoxon rank sum test are listed in

Table VII. These values show that the difference between

the mean objective function values obtained with SOA

and the other competitive algorithms is statistically signifi-

cant. As shown in Table VII, SOA can generate the best

minimum, maximum, and mean values. For a comparison,

the mean calculation times for all examples are given

in Table VII. It is clear from the table that the mean com-

putation time of SOA is mostly shorter than those of SA

and TS.

The practical arrays with anisotropic elements can

also be implemented by using the principle of pattern

multiplication. If the current isotropic array elements are

replaced by anisotropic ones, then the new resultant pat-

tern can be obtained with multiplying the array pattern

by the new anisotropic element pattern. The results

depicted in Figures 5 and 7–21 show that the SOA

proposed in this paper can accurately obtain the nulling

patterns of a linear antenna array by controlling the posi-

tion-only, phase-only, and amplitude-only. From the

NDL and the MSL points of view, the performances of

the patterns are very good. It is clearly shown in this

article that the SOA can be used as an alternate to other

antenna synthesis algorithms.


trol with triple imposed null at 14�, 26�, and 33�.

TABLE VI Element Amplitudes ak Normalized According to Center Elements for Figures 16–21

k

Initial Chebyshev

pattern Computed with the SOA

Figure 4 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21

61 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

62 0.97010 1.01839 0.98429 0.97277 1.01460 0.95528 0.92249

63 0.91243 0.96189 0.95341 0.93784 1.04531 1.04066 0.92061

64 0.83102 0.89057 0.89193 0.86154 0.92733 0.92997 0.82005

65 0.73147 0.81277 0.78554 0.75458 0.76162 0.80210 0.68393

66 0.62034 0.64042 0.64634 0.61937 0.63401 0.55175 0.58139

67 0.50461 0.47836 0.50304 0.47107 0.54350 0.60011 0.52456

68 0.39104 0.37336 0.35698 0.34201 0.40914 0.41470 0.41824

69 0.28558 0.24079 0.22555 0.21792 0.21700 0.22251 0.21509

610 0.32561 0.33132 0.35258 0.27947 0.31041 0.31007 0.18908


trol with a broad null sector centered 30� with Dy ¼ 5�.



V. CONCLUSIONS

In this article, a method based on SOA for the pattern

synthesis of linear antenna arrays with the prescribed nulls

is presented. Nulling of the pattern is achieved by the

position-only, phase-only, and amplitude-only. Numerical

results show that the SOA is capable of synthesizing the

array patterns with single, multiple, and broad nulls

imposed at the directions of interferences. SOA is also

statistically compared with SA and TS algorithms. Statisti-

cal results of all examples show that SOA as a stochastic

algorithm outperforms the other competitive algorithms in

all test instances. The results also show that SOA is a fast

evolutionary algorithm and it can produce flexible solu-

tions for antenna array optimization problems. SOA

method will likely be an attractive alternate in the electro-

magnetics and antennas community.

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TABLE VII The Statistical Results of All Examples for SOA, SA, and TS

Algorithm Min Max Mean StdDev P-value Time (s)

Figure 5 SOA 3.37 5.93 4.23 0.60 – 5.67

SA 8.93 14.22 11.83 1.04 3.02 � 10�11 6.80

TS 4.60 12.00 7.44 1.82 9.92 � 10�11 12.27

Figure 7 SOA 4.02 14.62 5.93 1.86 – 5.80

SA 14.51 20.37 17.61 1.57 3.34 � 10�11 6.07

TS 5.76 17.61 12.65 2.75 1.07 � 10�09 11.33

Figure 8 SOA 6.01 13.14 9.12 1.82 – 5.87

SA 18.25 26.47 23.00 1.98 3.02 � 10�11 5.97

TS 15.05 22.27 19.48 1.80 3.02 � 10�11 12.97

Figure 9 SOA 9.17 20.94 12.25 2.59 – 6.10

SA 26.52 36.32 32.17 2.22 3.02 � 10�11 7.20

TS 23.38 30.87 26.56 2.28 3.02 � 10�11 12.67

Figure 10 SOA 14.84 43.34 28.87 7.45 – 6.17

SA 41.91 55.95 48.34 3.62 3.69 � 10�11 7.03

TS 34.09 45.85 39.16 3.02 4.31 � 10�08 11.50

Figure 11 SOA 25.62 28.83 26.90 0.73 – 5.87

SA 30.84 38.52 34.83 2.12 3.02 � 10�11 6.80

TS 27.79 33.06 30.66 1.23 4.50 � 10�11 12.80

Figure 12 SOA 3.66 6.80 4.38 0.68 – 10.87

SA 9.48 23.06 16.10 3.48 3.02 � 10�11 10.93

TS 6.06 13.71 9.02 1.66 4.08 � 10�11 21.10

Figure 13 SOA 6.01 15.19 8.57 2.08 – 10.90

SA 13.15 28.56 22.44 3.12 3.34 � 10�11 11.03

TS 10.14 18.86 15.01 2.42 4.20 � 10�10 21.17

Figure 14 SOA 10.62 22.36 13.72 2.83 – 10.70

SA 20.15 29.90 25.40 2.69 4.50 � 10�11 10.63

TS 13.55 24.16 19.06 2.90 9.83 � 10�08 21.20

Figure 15 SOA 21.22 27.52 23.45 1.72 – 10.70

SA 24.36 45.76 35.29 4.00 8.15 � 10�11 10.80

TS 21.70 30.81 25.88 2.11 5.61 � 10�05 21.20

Figure 16 SOA 6.53 14.62 10.72 2.16 – 7.83

SA 15.10 34.93 24.83 4.43 3.02 � 10�11 7.50

TS 8.50 19.03 13.50 2.77 1.41 � 10�04 16.53

Figure 17 SOA 6.13 16.09 11.71 2.27 – 7.97

SA 15.21 34.00 23.91 4.56 3.69 � 10�11 8.40

TS 5.02 20.01 14.01 3.23 3.18 � 10�03 14.63

Figure 18 SOA 3.72 15.63 11.93 2.50 – 7.43

SA 18.55 83.94 32.00 12.94 3.02 � 10�11 8.50

TS 10.70 22.88 16.62 2.95 2.20 � 10�07 16.33

Figure 19 SOA 8.27 19.14 15.45 2.65 – 7.93

SA 21.64 42.32 31.07 5.28 3.02 � 10�11 8.43

TS 10.84 32.04 18.03 4.15 1.44 � 10�02 14.63

Figure 20 SOA 11.37 19.62 16.36 2.44 – 7.80

SA 26.45 67.38 41.32 9.35 3.02 � 10�11 7.37

TS 15.97 30.26 21.57 3.19 2.19 � 10�08 14.53

Figure 21 SOA 10.17 25.19 16.45 3.61 – 7.90

SA 31.76 80.14 48.86 13.07 3.02 � 10�11 8.50

TS 14.24 33.82 24.72 3.81 5.97 � 10�09 16.50



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BIOGRAPHIES

Kerim Guney was born in Isparta,

Turkey, on February 28, 1962. He

received the B.S. degree from

Erciyes University, Kayseri, in 1983,

the M.S. degree from Istanbul Tech-

nical University, in 1988, and the

Ph.D. degree from Erciyes Univer-

sity, in 1991, all in Electronic Engi-

neering. From 1991 to 1995 he was an Assistant Professor

at the Engineering Faculty in Erciyes University, and now

is a Professor at the Engineering Faculty in Nuh Naci Yaz-

gan University, where he is working in the areas of optimi-

zation techniques (the genetic, the tabu search, the particle

swarm optimization, the differential evolution, the bacterial

foraging, the plant growth simulation, the ant colony opti-

mization, the bee optimization and the clonal selection

algorithms), fuzzy inference systems, neural networks, their

applications to antennas, the analysis and synthesis of pla-

nar transmission lines, microstrip and horn antennas,

antenna pattern synthesis, and target tracking. He has pub-

lished more than 250 journal and conference papers.

Suad Basbug was born in Nevsehir,

Turkey, in 1975. He received the

B.S. degree from Sakarya University,

Sakarya, in 1999, and the M.S.

degree from Erciyes University, in

2008, both in Electrical and Electron-

ics Engineering. Currently, he is a

Ph.D. student. His current research

activities include antennas, antenna arrays, evolutionary

algorithms, and computational electromagnetics.



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