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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: kguney@nny.edu.tr
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.
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International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011
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.
Seeker Optimization Algorithm for Nulling 509
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
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
Figure 7 Radiation patterns obtained by position-only control
with the restricted MSL having one imposed null at 15� (MSL >
�28.5 dB).
Figure 8 Radiation patterns obtained by position-only control
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
Figure 9 Radiation patterns obtained by position-only control
with double imposed null at 15� and 25�.
510 Guney and Basbug
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011
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
Figure 10 Radiation patterns obtained by position-only control
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
Seeker Optimization Algorithm for Nulling 511
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
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�.
Figure 13 Radiation patterns obtained by phase-only control
with double imposed null at �10� and 20�.
Figure 14 Radiation patterns obtained by phase-only control
with triple imposed null at �33�, �10�, and 20�.
Figure 15 Radiation patterns obtained by phase-only control
with a broad null sector centered 26� with Dyi ¼ 5�.
512 Guney and Basbug
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011
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
Seeker Optimization Algorithm for Nulling 513
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
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.
Figure 17 Radiation patterns obtained by amplitude-only con-
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
Figure 18 Radiation patterns obtained by amplitude-only con-
trol with the restricted MSL having one imposed null at 14�.
Figure 19 Radiation patterns obtained by amplitude-only con-
trol with double imposed null at 14� and 26�.
514 Guney and Basbug
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011
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.
Figure 20 Radiation patterns obtained by amplitude-only con-
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
Figure 21 Radiation patterns obtained by amplitude-only con-
trol with a broad null sector centered 30� with Dy ¼ 5�.
Seeker Optimization Algorithm for Nulling 515
International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce
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
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Figure 13 SOA 6.01 15.19 8.57 2.08 – 10.90
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Figure 14 SOA 10.62 22.36 13.72 2.83 – 10.70
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Figure 17 SOA 6.13 16.09 11.71 2.27 – 7.97
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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
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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
516 Guney and Basbug
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011
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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.
518 Guney and Basbug
International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011
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