Linear and Planar Antenna Array Nulling based on Schelkunoff

Polynomial and Genetic Algorithm

Abubakar Hamza, Khurram Karim Qureshi, Sharif I. Sheikh and Hussein Attia

King Fahd University of Petroleum and Minerals

Dhahran 31261, Saudi Arabia

Abstract— In many aircraft and electronic warfare sys- Z

tems, a lot of obtrusion comes from unwanted sources, which
may be noise or interfering signal. In this work, a null or
multiple nulls are placed at predefined direction(s) where we
have the interference while maintaining the maximum at a
specific location. This is achieved through an algorithm em-
ploying the Schelkunoff nulling theory. The genetic algorithm
is then used to optimize the array excitations. The proposed
technique reduces the number of unknown array coefficients
and hence the processing time is considerably decreased
without compromising other radiation pattern specifications.
.
WN
. . . . . .
W1 W2 W3
. WN
Y
Index Terms— Antenna arrays, Schelkunoff, Genetic algo- d
rithm, pattern nulling.
Fig. 1: Linear antenna array configuration.
I. I NTRODUCTION
For many applications that include aircraft radars, satel-
problem is reduced. The GA is a technique that is based
lite communication systems and electronic warfare sys-
on the mechanism of natural genetics and natural selection,
tems, the required radiation characteristics may not be
it is used in this context to optimize a fewer number of
achieved using a single antenna element. However, these
array elements for the search of both the required nulls
required radiation characteristics can be made possible
and the maxima in the specified directions. This modifies
by means of an arrangement of some specific radiating
the population of individual chromosomes (excitations),
components that are evenly or non-uniformly distributed,
each with an associated fitness function into a new gen-
depending on the structural position of the situation [1]–
eration of the population using the Darwinian principle of
[5]. The obtrusion to the radiation pattern of the array,
reproduction and survival of the fittest, and analogues of
coming from some external sources and radiating at same
naturally occurring genetic operations such as crossover
angle(s) of interest, causes interference or even damage to
and mutation [11]. In conclusion, this novel technique
the receiver. In order to avoid this problem, an algorithm
reduces the processing time significantly compared with
is introduced which employs the Schelkunoff polynomial
the conventional all parameter optimization techniques. In
technique that introduces null(s) at such particular direc-
circumstances where the time taken to deflect the radiated
tion(s) within the array factor (AF) and simultaneously
beam is a concern, particularly in the military warfare
keeping the maximum radiation at a chosen angle θmax .
system, the proposed technique will surely be an excellent
The synthesis of the antenna array in the available literature
candidate.
are based on analytic null placement [6]–[7], as well as
different optimization algorithms methods [1]–[5]. Differ- II. L INEAR A RRAY S YNTHESIS
ent optimization algorithms have been proposed for the In this section, a linearly arranged antenna array as
synthesis of antenna arrays, that include genetic algorithm shown in Fig. 1, with an N +1 identical radiating elements
(GA) [1]–[2], the firefly algorithm [8], the particle swarm and a separation of d between any two adjacent elements
optimization [3], the memetic algorithm [4], the tabu is considered. The AF is given by [10]:
search algorithm [5], and the spatial optimization of the
AF [9]. 
N

In this paper, GA is used to perform the optimization AF = an Z n (1)

of a linear and planar antenna array excitations, and the n=0

Schelkunoff nulling technique [10] is imposed into the where Z = ej(βd cos θ) with β as the wave number for
excitations of the antenna such that the complexity of the free space (β = 2π/λ) and λ is the free space wavelength,

978-1-7281-1120-9/20/$31.00 © 2020 IEEE 112 RWS 2020

the angle θ is taken with respect to a plane normal to the
array and the set an are the elements excitations which z
may be complex. If there is an interference at some specific dy P(r, θ, φ)
directions, then there may be a need to place null(s) within . . . . . . . . . . . .
the array in order to get rid of that unwanted signal, dz θ
and within the shortest period of time. Performing the . . . . . . . . . . . .
optimization directly on all the array excitations will take
a longer time such that the spatial situation could have
been changed. Assuming M number of zeros are needed
. . . . . . . . . . . .
at some desired locations θ1 ,θ2 ,. . . θm , then by applying
Schelkunoff theory [10] to (1), one gets:
. . . . . . . . . . . . Y
. . . . .φ . . . . . . .

N
an Z = aN (Z − Z1 )(Z − Z2 )...(Z − ZM )
n

n=0
[(Z − Z M+1 )(Z − Z M+2 )(Z − ZN )]
(2)
X
Fig. 2: Planar antenna array configuration in yz plane.
Multiplying the square brackets in (2), the AF will now
be in the form:
along z-axis forming an yz plane. Planar arrays in general
2
AF = f0 (y) + f1 (y)Z + f2 (y)Z + ...f N-1 (y)Z N-1
+Z N comprise M × N elements where they are in the form of
(3) odd×odd, odd×even, even×odd and even×even number
The set [y] = [y0 , y1 , y2 , y3 ...y N-M-1 ] are the constants to of elements. Schelkunoff procedure is enforced into each
be determined, where y has a distinguishable function for of the AF and the desired objectives are achieved. For M
each of fn (y) for a specified n. For an odd number of elements placed along the y-axis, the AF will be
elements, 2N + 1 elements will be considered and 2N for

M
an even number of elements. Considering an AF of 13 AFy = ay ej(m−1)(kdy sin θ sin φ+βy ) (5)
elements (N = 6) with multiple zeros needed at θ1 and θ2 , m=1
the final relation to perform the optimization will then be Similarly, For N elements placed along z-axis, the AF is
in the form: given by:
AF = (Z  − Z1 )(Z  − Z2 )[Z 3 + x1 Z 2 + x2 Z  + x3 ] (4)

N

The parameters now to be optimized will only be x1 , x2 AFz = az ej(n−1)(kdz cos θ+βz ) (6)
n=1
and x3 instead of all the parameters of the 13 elements in
the array, and finally the relation between sets y and x is So, the AF of a two dimensional planar array arranged in
summarized in table 1 the yz plane is given as:

TABLE I: RELATION BETWEEN THE SET [y] AND AF =

THE OPTIMIZED PARAMETERS [x]

M 
N
y0 p1 p2 p3 ay ej(m−1)(kdy sin θ sin φ+βy ) az ej(n−1)(kdz cos θ+βz )
y1 p1 p2 x2 − (p1 + p2 )x3 m=1 n=1
y2 x3 − (p1 + p2 )x2 + p1 p2 x1 (7)
y3 x2 − (p1 + p2 )x1 + p1 p2
y4 x1 − (p1 + p2 ) where ay and az are the excitation coefficient of each
y5 1 element along y and z axes, respectively, dy and dz are
Where p1 = Z  1 + Z 2 = Z’1 and p2 = Z 1 Z 2 = Z  2 . the spacing between the elements along y and z axes
respectively. βy and βz are the progressive phase shift be-
tween the elements along y and z axes. For simplicity and
III. P LANAR A RRAY S YNTHESIS assuming the excitations coefficients of the elements of the
In some applications, especially that of satellite commu- array in y-direction are proportional to those in z-direction,
nications and radar systems, the antennas are required to without loss of generality, ayz = ay az . Therefore, for
be in the form of planar arrays. In this section, a planar uniform amplitude and progressive phase excitation, ayz
array is considered that is arranged in the yz plane as = a0 .
shown in Fig. 2. The total AF is based on the product The GA proceeds to generate a new population, which
of two linear syntheses; one along y-axis and the other is a new generation from the old population of the antenna

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excitations by a process of crossover and mutation. In
this process, the fittest of the reduced element excitations
survive and is able to produce offspring/children(desired
excitations). The three main steps in the algorithm are
selection, crossover and mutation.

IV. RESULTS AND DISCUSSION

In order to signify the originality of the proposed
technique, linear and planar array cases are considered, GA
optimization is performed using the standard all-parameter
approach and compared to the presented optimization
approach based on a reduced number of parameters. For
the linear array, an example of 10 and 11 element array is Fig. 3: Radiation pattern of conventional GA and proposed
considered. The separation between the element is chosen technique of a 10 elements linear antenna array.
as d = λ/2, and nulls are needed at 40o and 110o with
the main beam chosen at 90o . Tables II and III show
the optimized excitation coefficients of the standard all-
parameter technique and that of the proposed method of
the linear array, respectively. Fig. 3 shows the radiation
pattern of optimized excitation coefficients of the standard
all parameter technique and that of the of the proposed
method of the 10 and 11 elements linear array, respectively.
The run-time of the conventional GA technique for a linear
array of 10 elements is 22.22 minutes compared with 1.31
minutes for the proposed reduced-parameters method, the
processing time is reduced significantly by about 94%
using a core i7 CPU machine with 8 GB RAM.

TABLE II: Ten elements linear array design (conventional Fig. 4: Radiation pattern of conventional GA and proposed
GA technique) technique of an 11 elements linear antenna array.

Excitations Design specifications

w1 = 1.0000 Maxima is achieved at 90o
w2 = 1.0000 Nulls are at 40o (−128dB) and at 110o (−110dB) standard-all parameter technique and that of the proposed
w3 = 0.7000 SLL = −22dB
w4 = 0.4000 Optimization Time = 22.22 minutes
method of the planar array, respectively. The run-time of
w5 = 0.6000 the conventional GA technique for the planar array of 6×9
elements is 14.6 minutes compared with 1.5 minutes for
the proposed reduced-parameters method, which is reduced
significantly by about 90%.
TABLE III: Ten elements linear array design (proposed
These results confirm that the processing time is sig-
technique)
nificantly reduced which is crucial in many applications
set x set y set w (excitations) especially in the electronic warfare systems where time is a
x6 = 0.04760 y4 = 1.00000 w5 = 0.00390
x5 = 0.20000 y3 = −1.8402 w4 = 0.00640 prior factor. Moreover, better results have been obtained for
x4 = −0.1009 y2 = 1.09360 w3 = 0.00770 the side-lobe level using the proposed technique compared
x3 = −1.0990 y1 = −0.2313 w2 = 0.00820 with the all-parameters technique as shown in Fig. 3 and
x2 = −0.5864 y0 = 0.01410 w1 = 0.01000
4. Another advantage of the proposed method is that it
provides the exact locations of the nulls which reduces
Similarly, for the planar array, even × odd and even × signal interference from neighboring radiating systems.
even cases are considered with element separation also
chosen as d = λ/2. For the even × even case, an 8 × 8 V. CONCLUSION
array is considered with nulls needed at 70o and 130o A new technique employing the Schelkunoff nulling
with the main beam chosen at θ = 90o and Φ = 0o . theory incorporated with GA is developed. This new
Similarly, an even × odd case is considered as 6 × 9 technique reduces the complexity of the problem and also
elements, nulls needed at 55o and 125o with the main beam the processing time for optimizing the elements excita-
chosen at θ = 90o and Φ = 0o . Fig. 5 and 6 shows the tions for both linear and planar antenna arrays. Linear
radiation pattern of optimized excitation coefficients of the array examples of 10 and 11 elements are presented for

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Fig. 6: Radiation pattern of conventional GA and proposed

technique of 8 × 8 planar antenna array.

the conventional and the proposed techniques. Moreover,

planar array examples of 6 × 9 and 8 × 8 elements are
considered, which demonstrate the significant reduction in
the processing time. Furthermore, an additional advantage
in the suggested method is achieving the exact nulling
locations which help to reduce any possible interference
from external sources.
ACKNOWLEDGMENT
The authors wish to acknowledge the assistance and
support provided by King Fahd University of Petroleum
& Minerals (KFUPM) .
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