IPASJ International Journal of Electronics & Communication (IIJEC)
Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm
Email: editoriijec@ipasj.org
ISSN 2321-5984
A Publisher for Research Motivatin........
Volume 2, Issue 3, March 2014
Parametric study on the model of linear antenna
arrays for cosecant squared pattern
Priyanka Behera1, D.S.N. Das2 and A.K. Behera3
3
1&2
Centurion university, Bhubaneswar
Templecity institute of technology, Bhubaneswar
Abstract
This work describes the synthesis of cosecant squared pattern (CSP) from a linear array antenna which is placed along positive
x-axis with a uniform spacing of 0.6 times the operating wavelength. The amplitudes of the current excitations are taken as the
optimizing parameters. The soft-computing techniques are used to optimize the element excitations. The cosecant squared
shaped beam pattern is synthesized with a linear array of 30 elements for the angular width of 30 0 and the reduced side lobe
level (SLL) as well as the ripple level using a soft computing tool. The effectiveness of the optimization algorithm, i.e. Artificial
Bee Colony (ABC) algorithm has been employed to synthesize non-uniformly excited, non-uniformly spaced, linear arrays.
Keywords: cosecant squared pattern (CSP), Artificial Bee Colony (ABC), side lobe level (SLL)
1. Introduction
Many analytical techniques such as Fourier transform method, Dolph–Chebyshev and Taylor series methods are
introduced [1], to synthesis the antenna arrays with the desired radiation pattern. But these analytical techniques in
synthesizing shaped beam pattern are not efficient enough to provide optimum pattern. So in order to get the best result
the stochastic methods [2, 3] are necessary. For a symmetrical array synthesis a Genetic Algorithm (GA) method is
presented in [4] for the optimization of the weights of the array elements with element spacing of 0.6λ; where λ is the
wavelength of the operating signal. The Cosecant Squared Pattern (CSP) is useful for an aircraft system as it can provides
high gain near the horizon [5] where the range is higher but low gain at high-elevation angles where range is lower. As
a result when it moves from the horizon to towards the higher elevation, the corresponding aircraft receiver gets almost
constant power at different elevation angles although its range is changing. A cosecant squared radiation pattern can be
achieved by using multiple feeders and a reflector antenna. Also antennas with cosecant squared radiation pattern are
used for costal and air surveillance radars because of its efficient scanning ability in the space. A number of
optimization methods such as particle swarm optimization [6], genetic algorithm [7, 8], simulate annealing [9] etc.
have been applied for the synthesis of antenna array. In [10], a weighted alternating reverse projection (WARP) method
is applied to synthesize a 12 element linear array with a spacing of 0.5  and a angular range of 350 which also
contains the ripple level of about 0.5 dB. Although, Orchard et al. [11] introduced a method to synthesize the power
pattern from linear array but the solution is not the optimum one because, the reduced ripple results in shrinkage of the
shaped region and increases the complexity of the array excitation. In this paper a cosecant squared pattern (CSP) in
the far field power pattern of the array is generated using Artificial Bee Colony (ABC) algorithm with optimizing the non
uniform excitation coefficients and static phase shifts of linear array elements. The obtained CSP has the much fewer
ripples in the shaped region and also have a lower side lobe level (SLL).
2. LINEAR ARRAY ANTENNA
A configuration of multiple antenna elements arranged along a straight line is known as linear array. A linear array
having N-isotropic elements placed along x-axis. Generally suitable values of three parameters as amplitude, phase and
the inter element spacing are considered to design an antenna array for the desired radiation pattern. The radiation
pattern of an antenna array is the sum of the contribution of all the signals radiated by the individual array elements
and can be obtained by the array factor expression as given in (1), [1].
N
i( k xn s in  n )
(1)
AF 

U ne
n 1
Where, U n is the normalized excitation coefficient of the n-th element, k = 2  , the wave number;  , the angle
made by the line joining the observing point and the origin with the z-axis; xn and n be the position and static phase
of the n-th element respectively. Hence from (1), it is observed that the desired power pattern from a linear array can be
Volume 2, Issue 3, March 2014
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IPASJ International Journal of Electronics & Communication (IIJEC)
A Publisher for Research Motivatin........
Volume 2, Issue 3, March 2014
Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm
Email: editoriijec@ipasj.org
ISSN 2321-5984
synthesized with the suitable values of the normalized amplitude coefficients, U n and static phase shift n of the array
elements. In this paper, the effectiveness of DE algorithm is applied to select the suitable values of U n and n for
n=1,2,….N to generate the desired pattern.
3. ARTIFICIAL BEE COLONY ALGORITHM
In 2005, [1] the ABC algorithm is developed using the foraging behavior of real honey bees which can be used to solve
the multidimensional and multimodal optimization problem. In this algorithm, two different types of artificial bees
present in the colony, that are - employed and unemployed bees. In addition of these two types of bees food source also
plays an important role in the foraging system. Unemployed bees are of two types scout bees and onlooker bees.
Employed bees are employed at a particular food source which they are currently exploiting and they also recruit the
onlooker bees; onlooker bees wait in the hive and gather the information about the food sources such as nectar amounts
from the employed bees. These information sharing takes place in the dancing area by a dance known as waggle dance.
Since onlooker bees watch many dances before choosing a food source so the probability of choosing a profitable food
source by onlookers are very high. The job of scouts is to randomly search the new food sources near the hive. First half
of the colony contains the employed bees and another half contains the onlookers. Since there is only one employed bee
for each food source therefore the number of employed bees and food sources are equal. When the food source of an
employed bee is fully exploited by the employed and onlooker bees, she becomes a scout. The possible solution of the
optimization problem is represented by the position of a food source whereas the quality (fitness) of the solution is
represented by the nectar amount of a food source. ABC initializes with a randomly distributed initial population (at
G=0) of Nfood food source positions or number of solutions, where Nfood denotes size of the population. Each solution Si
(i=1, 2,…, Nfood) is a D-dimensional vector where D is the number of optimization parameters. After initialization, all
the employed, onlooker and scout bees begin to search for food sources within maximum number of Cmax cycles.
3.1 Employed bees’ stage
For finding a new food source, an employed bee modifies the position in her memory and tests the nectar amount
(fitness value) of the new source (new solution). Like real bees, the artificial bees also use the comparison process for
finding a new food source; however, they do not use any information in comparison. An employed bee randomly selects
a food source position and the old food source position will be replaced by the new food source position in the employed
bee’s memory if the new source has better nectar amount than that of the previous one. Otherwise the position of the
previous food source will retain. In other words, a greedy selection mechanism is employed as the selection operation
between the old and the candidate one. The position of the new food source v i in the neighborhood of its current source
xi is found by each employed bee by using the following equation-
vij  xij   ij ( xij  x kj )
(2)
where k{1,2,...,N food } and j {1,2,...., D} are randomly chosen indexes and also k must be different from i.  ij is a
random number between [-1, 1] and it controls the production of a neighbor food source around xij .
3.2 Onlooker bees’ stage
After the completion of the search process, all the employed bees will return to their hive and share the information of
the nectar and position of the food source with the onlookers on the dance area. Each onlooker bee now selects one of
the food sources with a probability related to its nectar amount obtained from the employed bees. Like employed bees,
greedy selection is also applied for onlooker bees. The onlookers select the food source with the probability, Pi , given as
follows(3)
fitn e s s i
Pi 
N fo o d

fitn e s s n
n 1
where fitness i represents the fitness value of the ith solution which is proportional to the nectar amount of the food
source in the position i and Nfood denotes the number of food sources which is equal to the number of employed bees.
Volume 2, Issue 3, March 2014
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IPASJ International Journal of Electronics & Communication (IIJEC)
Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm
Email: editoriijec@ipasj.org
ISSN 2321-5984
A Publisher for Research Motivatin........
Volume 2, Issue 3, March 2014
3.3 Scout bees’ stage
If within the predetermined number of cycles called ‘limit’, the fitness value of a food source does not improve then it
will be abandoned and replaced by a new one. The employed bee associated with that food source becomes scout. This
new position is randomly generated by the scout bee as belowj
j
j
s ij  s min
 rand [0,1] ( s max
 s min
)
where
(4)
j
j
s min
and s max are lower and upper bounds of parameter j, respectively. All of the above three steps are
repeated up to a predetermined, Cmax, number of cycles, or until a threshold value is achieved. In this work ABC
parameters are chosen as follows- number of food source (Nfood) =17, number of parameters to be optimizated=60,
limit=270 and maximum number of cycle (Cmax) =4560.
4. NUMERICAL ANALYSIS
To demonostrat the proposed work, linear antenna array of 30 eIement is considered with equal element spacing of
0.6  . It is desired that the radiated power pattern from the array follow the cosecant square function for at least 300
angular widths and the maximum value of SLL is ≤ -18.37dB. For the desired pattern, the normalized amplitude
coefficients and the static phase shifts of the array elements are taken as the optimizing parameters of the ABC
algorithm. The cost function used for the optimizing algorithm is as given in (5).
f cos t  W1 S112  W2 S 2 22   sum
(5)
Where, 1  SLLmax  SLLd ,  2  Rmax  Rd ; SLLmax and Rmax are the maximum value of SLL and ripple measured
at each iteration of the searching process whereas SLLd and Rd are their corresponding desired values. S1 and S2 are
the Heaviside step function of the corresponding terms and  sum is the sum of the difference between the sampled
values of the array factor and actual cosecant square values at each point within the desired angular width. The
optimized pattern using ABC is shown in Fig. 1 and the corresponding optimum values of Un and n are shown in
table-1. In Fig. 1, the actual cosecant square plot for the desired range is also shown. The ABC optimized CSP
approximately follows the actual plot for the angular width of 300 with the range (96-126) degrees. The maximum
value of the side lobe level is also reduced to ─18.32 dB and almost a negligible ripple in the desired pattern. However
as seen from table-1, the dynamic range ratio (DRR) of amplitude is quite higher (76.92). But when the elements with
lowest and largest amplitude coefficients (6th and 22nd elements) are discarded the corresponding values of DRR
becomes 57.59 and the resultant pattern is not affected for that. If further two elements are discarded (1st and 20th
elements) are turned off then the DRR is reduced to almost half, i.e 38.74 retaining the pattern unchanged.
10
ABC
CSC
norm aliz ed power pattern in dB
0
-10
-20
-30
-40
-50
-60
20
40
60
80
100
120
140
160
180
angle
Figure 1 ABC optimized Cosecant squared pattern
Volume 2, Issue 3, March 2014
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IPASJ International Journal of Electronics & Communication (IIJEC)
A Publisher for Research Motivatin........
Volume 2, Issue 3, March 2014
Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm
Email: editoriijec@ipasj.org
ISSN 2321-5984
Table 1: Normalized amplitude and phase
Element no
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Normalized
0.0171 0.0933 0.0388 0.0809 0.1855 0.0130 0.0860 0.0250 0.0628 0.1348 0.0233 0.2100 0.2580 0.1728 0.2908
amplitudes, (Un)
Phase shift,  n
2.518 2.453 2.941 2.845 3.109 3.528 4.198 4.681 5.112 5.452 6.060 0.337 0.985 1.851 .8720
(degree)
Element no
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Normalized
0.1189 0.3212 0.8611 0.9193 0.9849 0.9686 1.0000 0.9334 0.7764 0.7802 0.5852 0.5433 0.3418 0.1771 0.1181
amplitude, (Un)
Phase
shift,
0.835 1.085 1.520 2.139 2.285 4.633 1.437 1.676 1.391 5.921 0.440 5.436 1.118 1.002 5.975
 n (degree)
5. CONCLUSION
In this paper, the effectiveness of artificial bee colony algorithm is successfully applied for synthesizing uniformly
spaced 30 elements, linear antenna arrays. The suitable element excitations and the static phase shifts are determined
by the algorithm those reduce the side lobe level of the array to satisfactorily low values. The obtained pattern is also
compared to the cosec square plot in the desired region. The synthesized pattern of the antenna is much more helpful at
the sea bank for effective communication.
References
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[10]Haddadi, A. Ghorbani, A.:‘Cosecant-squared pattern synthesis using a weighted alternating reverse projection
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AUTHOR
Priyanka Behera received the B.Tech degree in Electronics and communication engineering from
Maharaja Institute of Technology in 2012 and M.Tech in Communication System from CUTM in 2014
Volume 2, Issue 3, March 2014
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