Hindawi Publishing Corporation
International Journal of Microwave Science and Technology
Volume2012,Article ID256759,8pages
doi:10.1155/2012/256759
Rearch Article
Nonuniformly Spaced Linear Antenna Array Design Using Firefly Algorithm
Mohammad Asif Zaman and Md.Abdul Matin
Department of Electrical and Electronic Engineering,Bangladesh University of Engineering and Technology,Dhaka1000,Bangladesh Correspondence should be addresd to Mohammad Asif Zaman,
Received31October2011;Revid23January2012;Accepted31January2012
Academic Editor:Ramon Gonzalo
Copyright©2012M.A.Zaman and Md.Abdul Matin.This is an open access article distributed under the Creative Commons Attribution Licen,which permits unrestricted u,distribution,and reproduction in any medium,provided the original work is properly cited.
A nonuniformly spaced linear antenna array with broadside radiation characteristics is synthesized usingfirefly algorithm and
particle swarm optimization.The objective of the work is tofind the optimum spacing between the radiating antenna elements which will create a predefined arbitrary radiation pattern.The excitation amplitudes of all the antenna elements are assumed to be constant.The optimum spacing between the array elements are obtained usingfirefly algorithm.The minimum allowed distance between the antenna elements is defined in such a way that mutual coupling between the elements can be ignored.Numerical analysis is performed to calculate the far-field radiation characteristics of the array.Two numerical examples are shown to form two different desired predefined radiation patterns.The performance of thefirefly algorithm and particle swarm optimization is compared in terms of convergence rate and global best solution achieved.The performances of the optimized nonuniformly spaced arrays are analyzed.Finally,contour plots of the radiated power from the optimized array in the horizontal plane and vertical plane in the far-field region are provided.
1.Introduction
Multiple antennas can be arranged in space in various geo-metrical configurations to form an antenna array with highly directive radiation pattern[1,2].The radiation characteris-tics of the antenna array depend on some input paramet-ers.The parameters are the relative magnitude and pha of the excitation current of each radiating element,radiation characteristics of each radiating element,the geometrical configuration of the array,and the paration distance between the array elements[2].An antenna array can be designed to produce almost any arbitrary prescribed pat-tern by controlling the parameters.For this reason,antenna arraysfind application in RADAR and wireless communica-tion systems[3,4].
Most antenna arrays are designed to produce a directive beam at a particular direction and while keeping the sidelobe level(SLL)small to avoid interference with other radiating sources.In most cas,this is achieved by controlling the magnitude and pha of the excitation amplitudes[5,6].In most cas,a relatively simple geometry is considered where the distance between two concutive radiating elements is constant.However,exact control of pha and magnitude of excitation current of array elements requires complex and expensive electronic circuitry[4].In ca of phad arrays, where the direction of the main beam needs to be controlled electronically in real time[7],us of suc
h electronic circuits are unavoidable.However,in many applications,a dynamic control of array radiation pattern is not required.It is highly desirable in such cas to design an antenna array that does not require complex circuitry to control the pha and mag-nitude of excitation currents.Desired radiation characteris-tics can be achieved by proper placement of each individual radiating element in space while keeping the excitation cur-rent constant for all elements.Such arrays are known as non-uniformly spaced antenna arrays.
In linear antenna arrays,the antenna elements are placed along a straight line.Linear array of identical radiating
elements are one of the common type of antenna arrays[4, 6].Design method of linear uniformly spaced antenna arrays are widely covered in literature[5–8].Most of the methods employ a heuristic optimization algorithm tofind the pha and/or magnitude of the excitation currents of the array elements while keeping the paration between the array ele-ments uniform.Particle Swarm Optimization(PSO)[5,7], Genetic Algorithm(GA)[6]and Artificial Bee Colony(ABC) algorithm[8]have been successfully ud to design such arrays.Although,design methods of Nonuniformly Spaced Linear Antenna arrays(NUSLA)exist,they have not received equal attention in the literature.
One of thefirst major articles describing nonuniformly spaced antenna arrays was published in1961[9].The design method employed perturbation methods and concentrated on sidelobe reduction.An iterative method for sidelobe reduction was developed by Hodjat and Hovanessian[10]. Recently,Fourier transform and window techniques have been applied for designing NUSLA[11].The methods are suitable for synthesizing radiation pattern with low SLL,but lacksflexibility to synthesize arbitrary radiation pattern.
Optimization-algorithm-bad design methods of NUSLA allowed engineers to easily synthesize arbitrary radiation pat-ters.The applications of DE and GA to synthesize NUSLA for low SLL have been reported in the literature[12,13].PSO has been successfully ud to design NUSLA with low SLL and nulls in arbitrary positions[14].Ant Colony Optimization (ACO)algorithm has also been ud to synthesize NUSLA with arbitrary radiation pattern[15].
Oraizi and Fallahpour have prented an impressive study of NUSLA in their paper[16].They ud modified GA-bad analysis to optimize the position of the array ele-ments.However,the work does not mention the number of iterations required by GA to converge or any other indication convergence time.GA usually is not fast converging for ante-nna array problems and it is often outperformed by PSO and ABC[7,8].In this paper,we ud a newly developed optimi-zation algorithm
called Firefly Algorithm(FA)to synthesize NUSLA and compare the performance of the algorithm with existing methods.
FA is a heuristic numeric optimization algorithm inspir-ed by the behavior offireflies[17,18].FA has been successful-ly ud in many applications.Recently,FA has been ud by Basu and Mahanti for designing linear antenna arrays[19]. The work concentrated on designing equally spaced linear array with variable excitation current.No work has yet been published on the application of FA for NUSLA design.
In this paper,FA is ud tofind the optimum spacing between the array elements to produce a desired broadside radiation pattern.The design examples with numerical results are provided.In thefirst example,a NUSLA is synthe-sized with minimum possible SLL.In the cond example, a NUSLA is synthesized with low SLL and nulls in arbitrary direction.The paper is organized as follows:array geometry and mathematical formulation of the radiated electricfield radiated by the array is prented in Section2.A brief descri-ption of FA is given in Section3.Numerical results are pro-vided in Section4and concluding remarks are given in Section5
.Figure1:Two-dimensional schematic diagram of the array geome-try.
2.Array Geometry and
Mathematical Formulation
The linear antenna array is assumed to be compod of N identical radiating elements.The radiating elements are taken to be dipoles antenna.The dipoles are positioned sym-metrically around the origin on the x-axis.The arms of the dipoles are parallel to the z-axis.A two-dimensional schema-tic diagram of the array geometry is shown in Figure1.
The total number of elements,N is assumed to be even. The elements are divided into two groups of M elements where N=2M.The elements are numbered as−M,−M+ 1···−1,1,2,...M−1,M.The distances
of elements 1,2,...M from origin is denoted by d1,d2,...d M.Due to symmetry,−1,−2,...−M elements have the same distance values.The symmetry around the origin creates symmetrical radiation pattern,which is often desirable.It is possible to create symmetrical radiation pattern with unsymmetrical positioning of the array elements.However,the symmetry condition reduces computational complexities,as now only position of M elements must be optimized instead of position of N elements.To avoid complexities,symmetrical distribution of the elements around the origin is assumed in this paper.
The radiatedfield of the array depends on the radiation pattern of each array element and relative spacing of the array elements.The far-field radiation pattern of the array is given by[1,3]
FF
θ,φ
=EP
θ,φ
×AF
θ,φ
.(1) Here,EP(·)is the radiation pattern of individual array elements,AF(·)is the array factor,and(θ,φ)are the zenith and azimuth angle of the spherical coordinate system.The coordinate system and the three dimensional geometry are shown in Figure2.
The radiation pattern of each array element is assumed to be
EP
θ,φ
=sinθ.(2) The array factor is given by[16]
AF
θ,φ
=
1M
n=1
cos
k d n cosφ
.(3)
r
θφ
z
y
x (r ,θ,φ)
鸡胸肉炒什么好吃
Array
elements
Figure 2:Coordinate system and three dimensional geometry of the array.
Here,k is wave number =2π/λand λis wavelength of radiation.
Using (1),(2),and (3),the overall radiation pattern is given by
FF θ,φ =sin θ⎡
⎣
1
N M n =1
cos k d n cos φ ⎤
⎦.(4)
In this paper,the objective is to find optimum values of d n to produce a desired far-field radiation pattern.It is noted that for the given geometry and type of radiating elements chon,d n does not a ffect the field variation with θ.So,for optimization process is carried out at θ=90◦plane only.Desired ra
diation pattern must be defined to measure the performance of the array.Since two di fferent designs are to be implemented using the propod method,two di fferent desired patterns are defined as follows:
FF des,1 90◦,φ =⎧⎪
⎨⎪
产品总监
⎩
0dB,−
BW
2<φ<BW 2
,SLL des ,otherwi .
(5)
FF des,2 90◦,φ
=⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
0dB,−
BW 2<φ<BW
2,NLL des ,
φNL <φ<φNH
,SLL des ,
otherwi .
(6)
Here,BW is desired main beamwidth,SLL des is desired sidelobe level,NLL des is desired null level,φNS is desired starting angular position of the null,and φNE is desired ending angular position of the null.
Equation (5)reprents a desired radiation pattern with a prescribed SLL and beamwidth.Equation (6)reprents a
desired radiation pattern with a specific SLL and beamwidth along with a predefined null level at an arbitrary direction.The angular extent of the null region is defined by the angles φNS and φNE .
The deviation of the obtained far-field pattern from the desired pattern is evaluated using the cost function.The cost function is defined as
f cos t
=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
180◦
φ=0◦
FF 90◦,φ −FF des 90◦,φ 2,FF 90◦,φ >FF des 90◦,φ ,0,otherwi .
(7)
Here,the summation is performed over the discrete field points that are numerically calculated.For the two di fferent desired patterns defined in (5)and (6),two di fferent cost functions can be defined.For each ca,the fitness function is defined as
f fitness =−f cos t .
(8)
It can be noted that,when the obtained radiation pattern
matches with the desired pattern,the cost function and the fitness function have zero values.When the patterns do not match,the cost function has positive values and the fit-ness function has negative values.Large deviations result is high positive cost values and high negative fitness values.So,the goal of the optimization process is to minimize the cost function or maximize the fitness function by finding appro-priate values of d n .
3.Overview of Firefly Algorithm
FA is an optimization algorithm inspired by behavior and motion of fireflies.It is a population-bad optimization al-gorithm which us swarm intelligence to converge [17,18].It is similar to other optimi
zation algorithms employing swa-rm intelligence such as PSO and ABC.But FA is found to have superior performance in many cas [19].
FA initially produces a swarm of fireflies located ran-domly in the arch space.The initial distribution is usually produced from a uniform random distribution.The position of each firefly in the arch space reprents a potential solu-tion of the optimization problem.The dimension of the arch space is equal to the number of optimizing parameters in the given problem.The fitness function takes the position of a firefly as input and produces a single numerical output value denoting how good the potential solution is.A fitness value is assigned to each firefly.The FA us a phenome-non known is bioluminescent communication to model the movement of the fireflies through the arch space.The bri-ghtness of each firefly depends on the fitness value of that firefly.Each firefly is attracted by the brightness of other fire-flies and tries to move towards them.The velocity or the pull a firefly towards another firefly depends on the attractive-ness.The attractiveness depends on the relative distance be-tween the fireflies.It can be a function of the brightness of the fireflies as well.A brighter firefly far away may not be as attractive as a less bright firefly that is clor.In each iterative
step,FA computes the brightness and the relative attractive-ness of each firefly.Depending on the values,the positions of the fireflies are updated.After a su fficient amount of iterations,all fireflies conv
erge to the best possible posi-tion on the arch space.
The number of fireflies in the swarm is known as the pop-ulation size,P .The lection of population size depends on the specific optimization problem.However,typically a pop-ulation size of 20to 40is ud for PSO and FA for most appli-cations [14,17].For the current problem,the solution space is M dimensional,where each dimension reprents the posi-tion of an array element.The position of the n th firefly is de-noted by a vector x n where,
x n = x 1n
,x 2n ,x 3n ,...,x m n ,...,x M n
.(9)
Here,n =1,2,3···P and m =1,2,3,...,M .
The arch space is limited in mth dimension by the following inequality:
x m Low
<x m
n <x m High .(10)
The value of the variables x m Low
and x m
High depend on the optimization problem.For the current NUSLA synthesis problem,the variables reprent minimum allowed and maximum allowed paration distance of concutive array elements.Initially,the positions of the fireflies are generated from a uniform distribution using the following equation:
x m n
=
x m Low
+
x m High −x m Low
×rand .(11)
梨子的英文Here,rand is a uniform random variable with values ranging from 0to 1.The value of rand is di fferent for each value of m and n .Equation (11)generates random values from a uniform distribution within the prescribed range defined by (10).The initial distribution does not significantly a ffect the performance of the algorithm.Each time the algorithm is executed,the optimization process starts with a di fferent t of initial points.However,in each ca,the algorithm finds the optimum solution.In ca of multiple possible ts of solutions,the algorithm may converge on di fferent solutions each time.But each of tho solutions will be valid as they all will satisfy the design requirements.The brightness of the n th firefly,I n is given by
I n =f fitness (x n ).
(12)
The attractiveness between the n th and pth firefly,βnp given by [17]
βnp =βo exp
−γr 2np
.(13)
Here,r np is Cartesian distance between x n and x p given
by
r np
=
x n
三年级修改病句
−x p =
M m =1
x m n −x m p 2.
(14)
βo is a constant taken to be 1.γis another constant who
value is related to the dynamic range of the arch space.The position of firefly is updated in each iterative step.If the brightness of pth firefly is larger than the brightness of the n th firefly,then the n th firefly moves towards the pth firefly.The motion is denoted by the following equation:x n ,new =x n ,old +βmn
x p ,old −x n ,old
+α(rand −0.5).
(15)
Here,rand is a random number between 0and 1,taken from a uniform distribution.αis a constant who value depends on the dynamic range of the solution space.
At each iterative step,the brightness and the attractive-ness of each firefly is calculated.The brightness of each firefly is compared with all other fireflies and the positions of the fireflies are updated using (15).After a su fficient number of iterations,all the fireflies converge to the same position in the arch space and the global optimum is achieved.
4.Numerical Simulation and Results
For numerical simulations,a N =20element NUSLA is
considered.As symmetry about the origin is assumed,only the position of M =10elements located on the positive x -axis needs to be optimized.The population size is taken to be 40and maximum number of iterations is limited to 65.The arch space is bounded by defining the minimum and maximum paration between two concutive elements to be 0.35λand 0.9λ,respectively,implying the solution space limiting variables defined in (10)have the following values:
x m
Low =0.35λ,x m
High
=0.9λ.(16)
For m =1,2,...,M .The limiting values in all dimensions of the solution space are assumed to be the same.The values are lected so that mutual coupling between the elements remains negligible [16].The dynamic range of the arch space is 0.9λ−0.35λ=0.55λ.αis taken to be 80%of the dynamic range initially and the value is linearly decread to zero at maximum iteration number.The paramete
r γis taken to be equal to the dynamic range.So,
电脑没声音一键恢复
α
=0.8×
x m High
−x m
Low
,
γ=x m High
−x m
miss的用法Low .(17)
The parameters of the first desired radiation pattern defined by (5)are lected as
BW <13.4◦,SLL des =−23.5dB .
(18)
The values are lected bad on typical beamwidth and SLL values of 20element linear antenna array [16].From multiple trial runs of the optimization algorithm,it was verified that the were the lowest possible values of beamwidth and SLL.
Element number,n12345678910 Position,d n/λFA0.1890.5400.934 1.292 1.726 2.141 2.639 3.173 3.899 4.634 PSO0.2390.857 1.362 1.929 2.576 3.193 3.881 4.762 5.662 6.397
The parameters of the cond desired radiation pattern defined by(6)are lected as
BW<12◦
SLL des=−20dB
NLL des=−40dB
φNS=46◦,126◦
φNE=54◦,134◦.
(19)
Two nulls symmetrical spaced around the main beam (located at90◦)extending from46◦to54◦and126◦to 134◦with null level of–40dB are desired.To achieve this, the requirements on desired SLL must be relaxed compared to the previous ca.For this reason,desired SLL in this ca is t to–20dB compared to previous value of–23dB. This relaxation in SLL constraint allows the possibility of reduction of main beamwidth.For this reason,the maximum main beamwidth in this ca is lected as12◦compared to 13.4◦of the previous ca.
The NUSLA for the two desired radiation patterns is designed using FA algorithm.For comparison,the same designs are performed using PSO algorithm also.The value of the parameters of the PSO algorithm is lected the same as the ones ud in[14].However,the population size of20 was ud in[14].For proper comparison,the population size for the PSO algorithm is taken to be40,which is identical to the population size ud in the FA algorithm.
FORTRAN computer coding(with G95compiler)is ud to implement the FA and PSO algorithm and formulating the far-field pattern.The optimum position of the array elements obtained from FA and PS
O for thefirst desired radiation pattern is shown in Table1.
It can be obrved from Table1that the two algorithms produce different ts of solution values.Using the values, the far-field radiation pattern atθ=90◦plane(xy plane) is calculated.The radiation pattern obtained from the FA is shown in Figure3and the radiation pattern obtained from the PSO algorithm is shown in Figure4.The desired pattern is also highlighted in red in the samefigures.
It can be en that both radiation patterns satisfy most of the design criterion.In[16],the maximum SLL of–22.6dB was achieved with a20element NUSLA of similar geometry. In this paper,using FA algorithm,the SLL value is limited to–23.5dB using the same number of array elements.So the propod FA-bad outperforms the GA-bad method described in[16].For the PSO algorithm,SLL constraint has not been met by the design obtained from PSO in only a small angular region.So,FA outperforms PSO in terms of output as well.However,the performance of the algorithms cannot be compared form thefinal output only.The con-vergence speed of the algorithms can show the difference in performance:the convergence rate obrved from thefitness
−5
−10
−15
−20
−25
−30
−35
−40
−45
−50
020406080100120140160180
Angle,φ(deg)
R
e
l
a
t
i
v
e
p
o
w
e
提高酒店r
(
d
B
)
Figure3:Optimized radiation pattern of the NUSLA atθ=90◦plane obtained from FA algorithm for thefirst design example.
−5
−10
−15
−20
−25
−30
−35
−40
−45
−50
020406080100120140160180
Angle,φ(deg)
R
e
l
a
t
i
v
e
p
o
w
e
r
(
d
B
)
Figure4:Optimized radiation pattern of the NUSLA atθ=90◦plane obtained from PSO algorithm for thefirst design example.
versus iteration plot for the algorithms.This is shown in Figure5.
It can be en that for FA,thefitness function value reaches its maximum possible value zero within20itera-tions.The saturation offitness value implies convergence. However,for PSO,maximum value of–14.317is achieved. It took PSO around50iterations to reach this value.The fact that zerofitness value was not achieved implies that the obtained radiation pattern does not perfectly match with the desired radiation pattern,which is apparent from Figure4. The most noticeable fact obrved from Figure5is the fast converging characteristics of FA compared to PSO indicated
Element number,n
12345678910Position,d n /λ
FA 0.2140.689 1.087 1.504 2.018 2.578 3.045 3.805 4.449 5.106PSO
0.219
0.654
1.125
1.539
2.176
2.666
3.236
3.950
4.478
5.083
−100−200−300−400−500
1020
30
405060
Iteration number
B e s t fit n e s s v a l u e
PSO FA
Figure 5:Fitness value versus iteration number for FA and PSO algorithm for the first design
problem.
−5−10
−15−20−25−30−35−40−45−50
0204060
80100120140160180
Angle,φ(deg)
R e l a t i v e p o w e r (d B )
Figure 6:Optimized radiation pattern of the NUSLA at θ=90◦plane obtained from FA algorithm for the cond design example.
by the sharper slope of the blue curve compared to the red one.
Similar analysis was performed to synthesize NUSLA for the cond desired radiation pattern.The cond desired pattern is characterized by two nulls located symmetrically around the main-lobe region.The optimum position of the array elements obtained from FA and PSO for the first desired radiation pattern is shown in Table 2
.
−5−10
−15−20−25−30−35−40−45
−50
0204060
80100120140160180
Angle,φ(deg)
政审模板R e l a t i v e p o w e r (d B )
Figure 7:Optimized radiation pattern of the NUSLA at θ=90◦plane obtained from PSO algorithm for the cond design example.
It can be obrved from Table 2that the two algorithms produce di fferent ts of solution values.The radiation pattern obtained from the FA and PSO algorithm are shown in Figures 6and 7,respectively.
The desired pattern is also highlighted in red in the same figures.
It can be obrved that the results obtained from FA satisfy the design criterion very well.The results from PSO maintain the desired pattern for the most part,with only a few sidelobes exceeding the limit.This implies that PSO has not converged perfectly.This fact can be further illustrated by obrving the fitness function values.This is shown in Figure 8.Again,the superior performance of FA compared to PSO can be obrved by the sharper slope of the blue curve compared to the red curve.The maximum fitness value achieved for FA is –0.1456,whereas this value is –82.223.FA clearly outperforms PSO in terms of convergence rate and maximum fitness value reached within a limited number of iterations.
Due to the prence of random number parameters in FA and PSO,the results vary on each time the program is executed.However,in all cas,the FA converges faster than PSO.Since,PSO usually outperforms GA for antenna array problems,it is expected that FA will outperform GA as well.ABC usually requires over 100iterations (with a population size of 30to 40)to reach convergence in antenna array problems [7,8].So,it can be concluded that FA is a very fast converging algorithm.
Using the optimized position of the array elements,the normalized radiated power in the far-field zon
e of the antenna array is calculated for the first design problem.Equation (4)along with the fact that power varies at 1/r 2is
