Null Placements in Non-uniformly Excited Beam

Steered Linear Array using Particle Swarm

Optimization

Prarthana Mukherjee, Ankita Hajra, Sauro Ghosal,

Soumyo Chatterjee

Dept. of Electronics and Communication Engineering

Heritage Institute of Technology

Kolkata, India

m.prarthana@rediffmail.com, soumyo33@yahoo.co.in

Sayan Chatterjee

Dept. of Electronics and Telecommunication Engineering

Jadavpur University

Kolkata, India

sayan1234@gmail.com

Abstract— In this paper, a novel method for null placements

in beam steered linear array with unequal excitation amplitude and uniform inter-element spacing has been presented. The proposed method has been developed using Taylor one

parameter distribution for linear array and particle swarm optimization (PSO). The beam steering has been achieved by

maintaining a constant phase difference between two consecutive elements of the array. In order to place nulls in the desired

direction excitation amplitude perturbation method using PSO has been considered. Search space of PSO has been defined using Taylor one parameter distribution for linear array. Within the

defined search space PSO with the null placement objective. The proposed method has been tested for both single and wide null

objectives. As example 12 elements linear array has been

considered, for both single and wide null placement objective. Results show that, null level in the desired directions lies within

89% to 100% of desired null level of -60 dB, which in turn proves the effectiveness of the proposed method.

Keywords— Beam steered linear array, Taylor one parameter

distribution, null placement, Particle Swarm Optimization.

I. INTRODUCTION

Pattern synthesis of antenna array has been the subject of many studies and investigations [1-9]. One of the major requirements of wireless system is to steer the main beam of an array in a specific direction which in turn enhances spectral efficiency and capability of nullifying the effects of multi path propagation[2]. Conventional methods of beam steering are- dynamic phased array beam steering and adaptive beam steering [2-3]. In dynamic beam steering the gain is maximized towards a particular direction in which intended user or target is present [3]. In adaptive beam forming apart from maximizing the gain in a particular direction, null placements are done in the direction of unwanted or interfering signals [4-6].

Beam steering can be achieved by phase shifting process in which feed to each element is varied such that the received or transmitted signals from all directions are in phase in a particular direction [3]. Conventional analytical methods, like Taylor one parameter method for linear arrays, solves single objective problems effectively as cited in dynamic beam

forming methods [1]. Conventional methods are not capable of handling multiple synthesis criteria like reduced SLL and Null placement in beam steered linear array. Consequently evolutionary algorithms have been used to solve multi objective problems that are also equally effective to solve single objective problems.

Evolutionary algorithms like genetic algorithm (GA) [3], differential algorithm (DE) [4], simulated annealing (SA) [5], particle swarm optimization (PSO) [6] has been widely used to solve different multi objective problems in antenna arrays. The desired objectives were generally achieved by controlling the complex weights (both amplitude and phase), amplitude only, phase only, array element position only and combination of any of the mentioned parameters. Selection of a particular evolutionary method and parameters depends upon the optimization problem in hand. Further performance parameters like implementation simplicity, convergence rate and computational time has to be taken in to consideration. Based on comparative study reported in [7], PSO has been selected.

In the present work, a new approach has been proposed to address the null placement problem in beam steered linear array with uniform inter-element spacing. Beam steering has been achieved by maintaining a uniform phase associated with each array element. Value of phase depends upon the desired main beam position. Excitation amplitude associated with each array element has been considered as optimization parameter. By using Taylor one parameter distribution for a particular linear array configuration excitation amplitude range has been defined within which PSO searches for optimum excitation amplitude to achieve null in the desired direction. Efficiency of the proposed method has been illustrated using 12 elements array of isotropic radiator for both single and wide null placement along with main beam steering in the desired direction.

II. BEAM STEERED LINEAR ARRAY AND TALYOR ONE

PARAMETER DISTRIBUTION

In linear array, elements are lied along a straight line with equal inter element spacing. In Fig.1, the geometry of even numbered (2N) linear array has been depicted. In (1), the equation for normalized array factor for even numbered linear

array has been shown. The inter-element spacing of this even numbered linear array is uniform [1].

Fig. 1. Geometry of the 2N-isotropic element symmetric linear array placed

along the y-axis

( )( )∑=

+

−=

N

n

nkd

naAF

1

cos2

12cos)( βθθ

(1)

In (1), k=2π/λ is the wave number, an is the excitation amplitude of the nth element, d is the uniform inter-element spacing, β is the phase associated with each excitation amplitude value and θ is the elevation angle. If the main beam position is at θs then at main beam position kdcos(θs)+β = 0 or β=-kdcos(θs). Substituting the value of β in (1) the array factor expression gets modified. The modified expression is shown in

( )( )∑=

θ−θ

−=θ

N

1nsn )cos(coskd

2

1n2cosa)(AF

(2)

Conventional Taylor one parameter method has been developed for line source distribution generating monotonically decreasing pattern. To find the excitation amplitude value Taylor one parameter method for linear array has been used. Mathematical details of Taylor one parameter method for linear array are given in [1].

III. LINEAR ARRAY DESIGN USING PSO

PSO is a stochastic optimization technique [8-9] that has

been effectively used to solve multidimensional discontinuous

optimization problems in a variety of fields [6].

In order to explain the application of PSO on linear array

design, synthesis problem of reduced SLL has been

considered. According to PSO nomenclature, swarm of bees

representing possible solution sets, searches for the optimum

location (excitation amplitudes) for which desired SLL is

achieved. Location of each solution set is being defined by the

values of excitation amplitudes. Number of excitation

amplitudes required to define the location is equal to the

number of elements of the array under consideration. All the possible solution sets has the ability to modify its location to

obtain the desired objective. The possible solution sets (swarm

of bees) has no previous knowledge of the search space (field),

so each solution set (bee) searches in random locations.

During the search process each solution set updates its

location and velocity on two pieces of information. The first is

its ability to remember the previous location where it has

found the desired SLL (particle best (pbest)). The second

information is all about the location where SLL close to

desired SLL is found by all the solution sets (bees) of the

swarm (global best (gbest)) at present instant of time. This process of continuous updating of velocity and position

continues until one of the solution set (bee) finds the location

of desired SLL within the defined search space. This location

gives the value of optimum excitation amplitude. Ultimately

all the possible solution sets bees will be drawn to this

optimum location as they will not be able to find any other

better location. The process has been pictorially represented in

Fig.2 [8]. In accordance with the ongoing explanation,

parameeter1 and parameter2 in Fig.2 represents the excitation

amplitude values of an array of two elements.

In present application of PSO, particle’s velocity is

manipulated according to the following equation:

]xg()[randc

]xp()[randcwvv

nn,best2

nn,best1n1n

−+

−+=+ (3)

where vn is the velocity of the particle in the nth

dimension and

xn is the particle’s coordinate in the nth dimension. Parameter

Fig. 2. Particles (1 and 2) are accelerated toward the best solution gbest, and

the location of their own personal best pbest, in a 2-D parameter space

w is the inertial weight that specifies the weight by which the

particle’s current velocity depends on the previous velocity.

rand ( ) is a random function generating a number in the range

[0,1]. In this work c1 is decreased from 2.5 to 0.5 whereas c2 is

increased from 0.5 to 2.5 [9]. This facilitates the global search

over the entire search space during the early part of the

optimization. This also encourages the particles to converge to

global optima at the end of the search. Previous work [9] has shown that in PSO algorithm faster convergence is achieved if

inertia weight w is linearly damped with iterations. The

starting value is taken as 0.9 and decreases linearly to 0.4 at

last iteration. After the time step, the new position of the

particle is given by:

)1t(v)t(x)1t(x ++=+ (4)

Performance of PSO algorithm varies considerably upon

boundary conditions. These boundary conditions are

Parameter2

2

2

1

1

Pa

ram

eter

1

Pbest1 Pbest2

gbest

Original velocity Velocity towards gbest

Velocity towards pbest

Resultant velocity

d

Bro

ad s

ide

dir

ecti

on

x

θ

z

2N

y End fire direction

y

categorized in two groups: restricted and unrestricted. In this

work restricted boundary has been used according to which if

a particle goes outside the allowable solution space, it is

relocated randomly within the search space [10]. Termination

criterion has been set to pre-defined number of iterations.

IV. PROBLEM FORMULATION

In this paper, optimization objective is to place nulls in the desired direction by varying the excitation amplitude of each array element using PSO. In order start the search process, PSO algorithm must know where to search for the optimum excitation amplitude values. This means search space has to be defined properly to obtain the desired objective. In the proposed method, instead of defining the search space randomly, it has been proposed to do so by using Taylor one parameter distribution for discrete source (linear array) [3]. From Taylor one parameter distribution upper and lower limit of search space has been defined which in turn facilitates PSO with the knowledge of where to search and removes the initial randomness of the algorithm.

In PSO, concept of fitness function guides the particles (bees) during their search for optimum position within the search space. Corresponding fitness function for the problem is as given in (5).

( )( )

∑2

1

n

ni s

i

AF

AFlog20-)dB(levelnulledefinedPrf

= θ

θ= (5)

In (5), θn1 and θn2 defines the elevation angles where nulls are to be placed. If it is desired to have null for a wide elevation angle range then, θn1 and θn2 defines the lower and upper limit of that range respectively. AF (θs) is the array factor at the main beam position (θs) and AF (θi) is the calculated array factor value at the desired null positions. For each iteration value, within the defined search space, PSO searches for optimum excitation amplitude resulting in minimum fitness value. Fitness function (5) has been defined in the side lobe region (SLR). The SLR definition is represented in (6).

( )( ) [ ] [ ]180,∪,0;AFmaxSLR FNRFNL∈SLL

θθ=θθ=θθ

(6)

In (6), θFNL and θFNR are left and right first nulls around the main beam position. Termination criterion for the optimization

process is reached when iteration count is equal to predefined

number of iterations. Iteration count of 100 and particle count

of 25 has been used to obtain the desired objective.

Accordingly, at each level, PSO algorithm checks for the

desired fitness value, while updating the position and velocity. The algorithm is developed using MATLAB 7 version on core

(TM) 2 duo processor, 3 GHz with 2 GB RAM.

V. EXAMPLES AND NUMERICAL RESULTS

Idea of the proposed method for solving null placement

problem has been illustrated through some examples. The

proposed method has been applied to a linear array of 12

isotropic elements with uniform spacing of 0.5λ. Moreover,

for this array configuration it is assumed that interfering signal

is present at 63 degree and intended user is at 80 degree.

Hence it is required to place null at 63 degree and main beam

has to be steered at 80 degree. For this problem null level has

been taken as -60 dB. Fig.3 shows the radiation pattern of 12

element array with main beam at 80 degree and null at 63

degree. Corresponding optimization curve is shown in Fig. 4.

Fig.3. Radiation pattern of 12 elements optimized linear array with main beam

at 80 degree and null imposed at 63 degree.

Fig.4. Convergence curve for 12 elements optimized linear array with main

beam at 80 degree and imposed null at 63 degree.

From Fig.4 it is observed that convergence is reached at 46th

iteration. Parameters like directivity (D), half power beam

width (HPBW), first null beam width (FNBW), phase (β) and

excitation amplitude associated with 12 elements optimized

array using PSO with null at 63 degree has been summarized

in Table I.

TABLE I PARAMETRIC VALUES FOR 12 ELEMENTS OPTIMIZED ARRAY WITH MAIN BEAM POSITION OF 80 DEGREE AND NULL AT

63 DEGREE

2N

Different parameters of the optimized array

β D

(dB)

HPBW

FNBW

Excitation

Amplitude

12

(θs =80º) 0.554º 10.4 8.6º 20.1º

2.08, 1.86, 1.3,

1.16, 1.17, 2.84.

In order to ascertain wide null placement capacity of the

proposed method, same linear array configuration of 12

elements with main beam position at 100 degree, inter-element

spacing of 0.5λ and nulls imposed at 24 degree to 35 degree

has been considered. The predefined null level has been kept at

-60 dB. Corresponding radiation pattern and convergence

graph are shown in Fig.5 and Fig.6 respectively.

Fig. 5 Radiation pattern of 12 elements optimized linear array using PSO with

main beam at 100 degree and wide nulls imposed at 24 degree to 35 degree.

Fig. 6 Convergence curve for 12 elements optimized linear array using PSO

with the objective of nulls imposed at 24 degree to 35 degree.

From Fig.5 it is observed that for the defined range of 24

degree to 35 degree value is well below -50 dB. From Fig. 6, it

is observed that convergence has been reached at 51th

iteration

with a value of 678.2. Parametric values associated with the

optimized arrays are tabulated in Table II. For all the array

configurations nulls are placed at 24 degree to 35 degree.

TABLE II PARAMETRIC VALUES FOR DIFFERENT OPTIMIZED

ARRAYS AND MAIN BEAM POSITION WITH IMPOSED NULLS AT 24

DEGREE TO 35 DEGREE

2N

Different parameters of the optimized array

β D

(dB)

HPBW

FNBW

Excitation

Amplitude

12

(θs =100º) -0.546º 10.6 9.4º 21.6º

2.33, 2.32, 1.92,

1.93, 2.25, 1.11.

15

(θs =75º) 0.813º 11.63 7.4º 17.1º

1.16, 2.52, 2.39,

2.01, 2.28, 2.47,

2.01, 1.34.

20

(θs = 150º) -2.721º 12.65 10º 27.2º

1.90, 2.46, 1.66,

2.47, 2.69, 2.85,

2.56, 1.15, 1.01,

1.95.

TABLE III MAXIMUM ARRAY FACTOR VALUE WITHIN 24

DEGREE TO 35 DEGREE

2N= 12

(θs =100º)

2N=15

(θs =75º)

2N=20

(θs = 150º)

AF (Maximum)

(24º to 35º) -52.74 dB -53.28 dB -54.59 dB

From Table II it is observed that with the increase in number

of elements Directivity (D), HPBW and FNBW value

increases. From Table III it is observed that for each array

configuration maximum array factor is around 89% of desired

null level of -60 dB which is quite acceptable.

VI. CONCLUSION

In this paper a novel method has been proposed to solve the

null placement problem in linear array. The method uses the

amplitude perturbation to achieve the desired objective of null

placement. The method has been developed using Taylor one

parameter distribution for linear array and PSO. By using

Taylor one parameter distribution search space for PSO has

been defined. Consequently, initial randomness of PSO has

been removed. Simulated result shows that null level in the

desired direction is within 89% to 100% of the desired null

level of -60 dB. Hence it can be concluded that the proposed

method can handle both multiple and wide null placement

objectives in linear array.

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