A RRAY F ACTOR O PTIMIZATION OF AN A CTIVE P LANAR P HASED A RRAY USING E VOLUTIONARY A LGORITHM

International Journal of Antennas (JANT) Vol.2, No.3, July 2016
DOI: 10.5121/jant.2016.2301 1
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE
PLANAR PHASED ARRAY USING EVOLUTIONARY
ALGORITHM
Samaresh Bhattacharjee, Ashish Kumar, Manish Naja
Aryabhatta Research Institute of observational sciences (ARIES),
Nainital-263001, Uttarakhand, India
ABSTRACT
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization
problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular
choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation
of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of
nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective
approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a
49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In
the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery
is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry
out the work presented in this paper.
KEYWORDS
Active planar phased array, array factor (AF), particle swarm optimization (PSO), radar, transmit receive
module (TRM)
1. INTRODUCTION
With rapid development and advancement in the technologies, the trend of incorporating an
‘active’ phased array configuration in majority of wireless, mobile, space and radar
communication system is increasing day by day [1, 2]. Studies revealed that an active phased
array configuration offers the collective advantages of high reliability, improved efficiency, low
loss and higher sensitivity over a passive configuration [3]. An active phased array system
contains several antenna elements assembled with individual solid state transmit receive module
(TRM) to meet the required power aperture product. The system uses the abilities of active array
to electronically steer the array beam instantaneously in any direction by controlling in-build
phase shifters of TRMs [4].
Failure of single or multiple TRMs in such systems is a common occurrence which makes an
antenna element become non-radiative. In the event of the failure, the far field array factor (AF)
pattern of the system degrades seriously in terms of increased side lobe levels (SLLs),
displacement of nulls and reduction in the null depths. To get rid of such degradations, repair/
replacement of the faulty modules could be best choice, as it always gives the best recovery of the
original AF pattern. However, such repair/ replacement in a space based or ground based systems
commissioned with hundreds of TRMs will be expensive in terms of time and budget. Under such
circumstances, without undergoing any repair/ replacement, evolutionary algorithms (EAs) could

2
be used as an alternative to recover the optimized pattern closest to the original by perturbing the
input amplitudes and/ or phases of input excitations of remaining functional TRMs.
EAs are inspired by the biological models of evolution and natural selection [5, 6]. To solve a
particular problem using EAs, a situation is created in which potential solutions can evolve. The
situation is outlined by the parameters of the problem that pushes the algorithm towards optimal
solutions. Most commonly used EAs are genetic algorithm (GA) [7], particle swarm optimization
(PSO) [8, 9], cuckoo search (CS) [10], and artificial bee colony (ABC) [11]. Several studies have
been reported on optimization of AF pattern by amplitude and/ or phase perturbation using EAs
for linear array [12 – 15]. The present work deals with recovery of the AF pattern of an active
planar phased array by recalculating a new set of amplitudes of input excitations of working
elements using PSO. Being a stochastic technique and based on the movement and intelligence
of swarms, PSO has gained an immense popularity in the field of electromagnetics [16]. The
reason behind selecting PSO is that PSO algorithm has the advantages of its simplicity and less
computational complexity over the other EAs. The computational time taken by PSO to arrive at
the desired solution is also less as compared to other EAs [17 – 19].
This paper presents multiple cases on the recovery of the AF pattern of a 49 elements active
planar phased array operating in very high frequency (VHF) band. Active array in VHF band has
wide application in the areas of wind profiling and weather radar [20]. In the presented cases, the
number of non-working elements is increased to a maximum 20% of the total elements. The
recovered pattern in each case is evaluated numerically by comparing the levels of first side lobes
(FSLs) and the depth of the first nulls (FNs) with the corresponding values in the original AF
pattern.
2. ARRAY CONCEPT
The AF pattern of an active planar phased array consisting of M x N identical elements placed on
XY plane (Fig. 1) scanning to a direction (θo, φo) may be expressed as [4]:
1 1
[ ( ) ( )]
0 0
( , )
M N
j m T T n T Tx xs y ys
mn
m n
AF A eθ ϕ
− −
− + −
= =
= ∑ ∑
(1)
where, dx and dy = inter-element spacing along X and Y direction respectively; Tx = (2π/λ) dx
cos(αx); Ty = (2π/λ) dy cos(αy); Txs = (2π/λ) dx cos(αxs ) ; Tys = (2π/λ) dy cos(αys); cos(αx) =
sin(θ)cos(φ); cos(αy) = sin(θ) sin(φ); cos(αxs) = sin(θo)cos(φo); cos(αys) = sin(θo) sin(φo); and Amn =
Amplitude of input excitation of the mnth
element.
Fig. 1. A planar phased array

3
The phasing of mnth
element, ψmn for beam steering is calculated as mTxs + nTys and the far field
radiation pattern of the array is:
ff(θ,φ) = AF(θ,φ) x EP(θ,φ) (2)
where, EP(θ,φ) is the Element Pattern (EP) of an antenna element at (θ,φ) plane.
3. PSO
EAs are stochastic optimization techniques popularly being used for real life problem solving
tasks in wide and diverse range of application areas like science, engineering, medical and
economics [5, 6] and [21]. Among all EAs inspired from biological process, the PSO technique is
simple, effective and capable of handling tricky multidimensional nonlinear optimization
problems across the fields [16 – 19]. The PSO was originally introduced in 1995 by Russell
Eberhart, an electrical engineer, and James Kennedy, a social psychologist [8, 9]. Here it is
necessary to mention that the present work is not intended to carry an extensive review on PSO,
and therefore only the main steps of the algorithm are discussed.
The PSO is a population based search strategy which is applied in an N dimensional search space
containing set of points, referred as particles. These set of particles moves in the search space
with velocities that are dynamically adjusted according to their historical performance as well as
the experience and knowledge of its neighbours. During movement, at each iteration, the particles
update their velocities according to the best positions already found by themselves as personnel
best (Pbest) and global best (Gbest) [16].
{ 1 1 2 2
, ,
1
( ) ( )
. . .
best n n best n n
n n
inertia
cognitive personal component social neighbourhood component
P X G X
v w v c rand c rand
t t+
− −   
= + +   
∆ ∆      1 4 4 44 2 4 4 4 43 1 4 4 44 2 4 4 4 43
(3)
where, vn+1 and vn are the velocity vectors of the particle in the present and previous iterations
respectively; Xn is the current position of the particle; w is the inertial weight which varies as the
algorithm progresses and determines the weight by which the particles current velocity depends
on its previous velocity; c1 is the cognitive coefficient controlling the pull to the Pbest position; c2
is the social rate coefficient controlling the pull to the Gbest position; rand1 and rand2 are the
random numbers uniformly distributed in (0, 1); ∆t is the time difference between the two
successive positions of the particles and is most often taken as unity. The next position, Xn+1 of
particle in the search space is calculated as:
1 1
.n n n
X X v t+ +
= + ∆
(4)
The choice of different parameters and the constants in above two equations can be decided
according to the work described in [22– 25]. During the process of algorithm, the Pbest and Gbest
are decided according to the criteria set by a fitness function (FF) [16]. A FF plays a vital role in
controlling the performance of the algorithm. The selection of FF depends on the type and
complexities present in the identified problem.
4. METHODOLOGY OF THE WORK
In the present work, a planar phased array having a quasi-circular aperture is considered as the
test array. The diagram of the array shown in Fig. 2 is populated with 49 elements, each

4
represented by a square box labeled with a fixed numerical number for easy identification of
element location. Elements of the array are arranged as equilateral triangular grid in 9 rows and
15 columns. During the design of the array geometry, it is attempted to make the periphery of the
aperture close to circular to maintain angular symmetry in the AF pattern. A configuration of a
planar array having circular or near to circular aperture with triangular grid gives a grating lobe
free AF pattern at maximum possible off-zenith angle with comparatively low FSLs without any
amplitude tapering. Triangular grid also helps to reduce the number of elements demanded to fill
a certain aperture [26]. These inherent advantages make the configuration a popular choice in
applications like wireless, space borne synthetic aperture radar (SAR), weather radar, wind
profilers [20], [27-29].
Fig. 2. The test array
EP of the array elements is considered as isotropic and initial amplitude of input excitations are
taken as unity. The inter-element spacing is kept 0.7 λ which places the rows and columns at 0.88
m and 1.02 m apart, respectively. Regardless of aperture or grid structure, the AF equation
described in equation (1) takes individual element location as inputs. The same equation is used
for calculating the AF of the test array. Effect of mutual coupling among elements is not
considered during the analysis and discussion.
After finalizing the structure of the test array, a suitable FF is developed and expressed as
equation (5). There could be many possibilities of the FF depending on the factors decided to
achieve the optimization. The present FF is developed after a number of trials to bring the
solution close to the optima by minimizing it till last iteration. The weights (a1, a2, a3 and a4) in
the FF equation are taken as unity.
1 1 2 2 3 3 4 4FF a F a F a F a F= + + + (5)
where,
2
1 [ (0,0) (0,0)]pso originalF AF AF= −
2
2 _ _[ ]FSL pso FSL originalF AF AF= −
2
3 _ _[ ]FN pso FN origF AF AF= −
90
2
4
90
[ ( , ) ( , )]pso originalF AF 0 AF 0
θ
θ θ
+ °
=− °
= −∑
Balanis says that the characteristic of an AF pattern can be broadly determined by the three
levels: main beam, FSLs and FNs [30]. F1, F2 and F3 of FF uses these three levels of optimized
and original patterns for deciding movement of particles in problem space. To test the
effectiveness of the PSO algorithm for the present problem with the FF, a software tool is
developed. The coding of the tool is done based on the array concepts and the PSO algorithm.

5
The tool takes the values of all the PSO parameters and the location of non-working element as
user inputs. Research revealed that PSO can achieve faster convergence towards optima with less
oscillation by careful selection of its parameters [22 – 25]. After a detailed study, following
values of the PSO parameters are fed into the developed tool and kept fixed in all the presented
cases.
No of particles = 200, No of iterations =100,
Cognitive component (c1) = 2, and Social component (c2) =1
Different schemes on variation of ‘w’ of equation (3) are tested, and finally for faster convergence
‘w’ is linearly decreased over 100 iterations from 0.9 to 0.4. The same tool can be used for
optimization of any configurations of planar phased array by modifying the elements co-
ordinates.
In the code, Amn of equation (1) is written as (1+ µmn), where, µmn is introduced to represent the
perturbation required in initial amplitude of the input excitations of working elements for
recovery. Each iteration produces a new set of µmn which is algebraically added with unity to
generate a new set of amplitudes for calculating the AF in present iteration required by all four
entities of FF. The perturbations are restricted within a window of ±0.3, so the final amplitude
factor, Amn is restricted within a boundary between 0.7 and 1.3, which is equal to ± 30%
perturbation of initial amplitude. The window in amplitude factor can be linked with the tuning of
power amplifier modules in TRMs. In practical application, depending on the type and quality of
a power amplifier module, the set window may vary. Based on the methodologies, the results
obtained after single run for each case are presented and discussed in the next section.
5. RESULTS AND DISCUSSION
The original AF pattern of the test array in normalized form without any element failure at zenith
(θo, = 0°, φo = 0°) is depicted in Fig. 3. From the figure, it is noted that the levels of FSLs are -
19.08 dB down whereas the depths of FNs are -38.84 dB down from the peak level of the main
beam which is at 0 dB after normalization. Throughout the work, the recovery of the pattern at
zenith is only presented. No beam steering to other elevation (θo) or azimuth (φo) plane is
introduced during the pattern optimization. After recovery in each case, the original AF, AF with
fault and PSO optimized AF patterns are presented in a single figure for visual comparison and
performance evaluation.
Fig. 3. Original AF pattern of the test array

6
In the next couple of paragraphs detailed evaluations on the AF pattern recovery in four different
cases using the developed methodologies are discussed. Work starts with the recovery of the
original pattern when a single element is made non-working. Thereafter, number of non-working
elements are increased in subsequent cases to three, five and ten, respectively. The locations of
the non-working elements are picked up randomly.
Case-1: In the first case the center element at location 1 is made non-working. The presence of a
non-working element at the said location increases the levels of both FSLs and FNs by 1.64 dB
and 7.16 dB, respectively. After creation of the fault, now the tool runs with set PSO parameters
to recover the original pattern. At the end of 100th
iteration with 200 particles, the tool produces
an optimized AF pattern close to the original one shown in Fig. 4. The tool has recovered the
pattern by generating a new set of amplitudes of input excitations of 48 working elements and
brings down the levels of FSLs and FNs to -19.09 dB and -38.87 dB, respectively resulting very
close matching with the corresponding values in the original pattern.
Fig. 4. AF patterns for case-1
Case-2: Work advances with three elements failure at locations 5, 2 and 28 selected randomly as
user inputs. With these three non-working elements, the levels of FSLs increase to -16.52 dB. But
in opposite of the case-1, the depths of FNs increase to -49.67 dB with one degree shift towards
the main beam. Again the tool runs to recover the original pattern by amplitude perturbation of
remaining 46 working elements. Both fault and PSO optimized patterns with original one are
depicted in Fig. 5.
Fig. 5. AF pattern for case-2

7
The figure illustrates that the optimized pattern nicely follows the original pattern at every
location of θ with hardly any variation in the levels. The shifting of FNs after faults is also
rectified by placing them exactly at their original locations.
Case-3: Till now the tool has shown satisfactory performances in recovering the optimized
pattern closest to the original after the occurrence of faults. But to imitate a more complex
situation in an active phased array having hundreds of TRM, the tool is tested further with 10%
non-working elements i.e. 5 of 49 elements. The location numbers of non-working elements are
randomly taken as 37, 12, 30, 1 and 18. Remaining 44 working elements with a void at said
locations produces a pattern with phenomena entirely opposite to the occurrences in the last two
cases. FSLs of the fault pattern remain at very close to its original levels with one degree shift
towards the main beam, whereas the second side lobes (SSLs) increase from -20.87 dB to -15.9
dB. These increased SSLs may pick up a strong interference signal from any external source and
may raise the probability of occurrence of ambiguous detection [4] and [30]. Fig. 6 plots the AF
pattern with faults at the said locations. The plot also depicts that the optimized pattern achieved
after the final iteration is made highly resemble the original pattern. After optimization, the
maximum levels of SLLs are brought down to -20.84 dB close to the original value -20.87 dB. In
optimized pattern, the shifts in FSLs are rectified and original levels are also maintained.
Case-4: To add much more complexity, the number of non-working elements is increased further
to ten which is equal to 20 % of the total elements. The position numbers of the fault elements are
again randomly selected as 49, 42, 13, 5, 10, 32, 31, 19, 26 and 17. Fig. 7 presents the AF pattern
before and after faults. The fault pattern is deformed to a typical formation. The lobes after
second nulls are merged together and pop up to maximum -15.98 dB at ±30o
. This merging
eliminates the third and fourth nulls in both sides of the main beam. In opposite, on occurrence of
merging, FSLs shift downward by approximately 10 dB. This downward shifting of FSLs could
be better at the cost of the deformation, but the popping up of second and third side lobes could
damage the entire application by inviting strong interferences into the system.

8
Once again the tool runs in the quest of searching a new set of amplitudes for the remaining 39
working elements for correcting the failed pattern. The optimized pattern after recovery is
presented in Fig. 7. The pattern illustrates that the process of recovery is successfully handled by
the tool to a great extent. FSLs are pulled up to the values of the original pattern. The region after
FSL in both sides of the main lobe is restructured at every point and brought near to the original
values.
The convergence profiles of 200 particles obtained by minimizing the FF are presented in Fig. 8.
The figure illustrates that the profiles of convergence are nicely damped and reached closest to
optima at around 70th
iteration. Out of 200 particles, the 81th
particle emerges as the global best to
produce the best optimized AF pattern.
Fig. 8. Convergence profiles of the FF for case-4
The final amplitudes of input excitations of 39 working elements of case-4 are displayed in Fig.
9. The values of excitations are represented by a grey scale whose values vary between 0.6 and
1.3. The figure depicts the final excitations searched by the tool within the set window. In the
figure, the non-working elements are represented by only number without any shaded box.

9
Fig. 9. Final amplitude distribution for case-4
A comparative summary on the values of the critical parameters of AF patterns in original, fault
and recovered conditions discussed in above four cases is presented in Table-1.
Table-1. Comparative results
Cases Parameters Original After fault
After
recovery
Case-1 FSLs -19.08 dB -17.44 dB -19.09 dB
FNs -38.84 dB -31.68 dB -38.87 dB
FNBW 28o
26 o
28 o
Case-2 FSLs -19.08 dB -16.52 dB -19.08 dB
FNs -38.84 dB -49.67 dB -38.85 dB
FNBW 28o
26 o
28 o
Case-3 FSLs
FNs
FNBW
-19.08 dB
-38.84 dB
28o
-18.28 dB
-39.48 dB
26 o
-19.15 dB
-38.16 dB
28 o
Case-4 FSLs -19.08 dB -27.73 dB -19.18 dB
FNs -38.84 dB -32.65 dB -36.87 dB
FNBW 28o
30 o
28 o
The degree of recovery of each case is further examined and compared using linear correlation
coefficient (LCC), median absolute deviation correlation coefficient (MADCC), linear least
square regression (LLSR) and root mean square error (RMSE) statistical methods [31]. The
results after examination are tabulated in Table-2.

10
Table-2. Statistical comparison
Cases
LCC MADCC LLSR RMSE (%)
O-F O-P O-F O-P O-F O-P O-F O-P
Case-1 0.9965 0.9999 0.8564 0.9991 0.9931 0.9999 1.9029 0.0395
Case-2 0.9966 0.9999 0.8838 0.9990 0.9933 0.9999 1.8526 0.0272
Case-3 0.9821 0.9997 -0.0683 0.9679 0.9646 0.9995 4.2718 0.4674
Case-4 0.9883 0.9992 0.7615 0.9643 0.9768 0.9985 4.1567 1.0870
The first two methods estimate the correlation coefficients, LLSR calculates the linear regression
and RMSE gives the errors before and after recovery. In the table, O-F represents estimation
between original and fault patterns, whereas O-P signifies the relation between original and PSO
optimized patterns. The values in Table-1 and Table-2 illustrate that the recovery process after
faults works satisfactorily.
6. CONCLUSION
The recovery of the AF pattern in case of element failure by perturbing the amplitudes of input
excitations is discussed. The FF with set PSO parameters was able to produce close optimization
even when 20 % fault elements were present in the array. The numerical results and the statistical
data presented for all the cases shows that the developed methodology worked well in obtaining
the optimized AF pattern with acceptable values of SLLs, FNs and its depths. In present era when
any solid state based active array system is equipped with embedded technology where the
algorithm can be stored and processed, the developed methodology can easily be adopted for
recovering AF pattern of any large and complex planar phased array. However, the present work
shows that the performance of the recovery degrades with the number of fault elements present in
an array. Therefore for large number of faults, the replacement of TRMs could be the right
choice. In future, AF optimization using phase or combination of phase and amplitude
perturbation will be attempted as the extension of the present work.
ACKNOWLEDGEMENTS
The authors are thankful to Dr. D. V. Phanikumar for going through the manuscript critically and
suggesting improvements.
REFERENCES
[1] A. K. Golshayan, S. V. D. Schoot, and P. V. Genderen, “Active Phased Array Radar (APAR),” Proc.
of the 5th International Conference on Radar Systems, Brest (F), 1999.
[2] J. Ronghong, Y. Sheng, L. Xianling, and G. Junping, "Research on planar antenna arrays," Antenna
Technology: "Small Antennas, Novel EM Structures and Materials, and Applications" (iWAT),
International Workshop on, pp.140-142, 4-6 Mar. 2014
[3] A. K. Agrawal and E. L. Holzman, "Active phased array design for high reliability," IEEE Trans.
Aerosp. Electron. Syst., vol.35, no.4, pp.1204, 1211, Oct. 1999.
[4] M. Skolnik, Radar Handbook, 3rd ed., McGraw-Hill Education, 2008.
[5] T. Bäck and H. P. Schwefel, “An overview of evolutionary algorithms for parameter optimization,”
Evol. Comput., vol.1, no.1, pp. 1–24, Spring 1993.
[6] K. R. Krishnanand, S. K. Nayak, K. B. Panigrahi, and P. K. Rout, "Comparative study of five bio-
inspired evolutionary optimization techniques," Nature & Biologically Inspired Computing, NaBIC,
World Congress on, pp. 1231-1236, 9-11 Dec. 2009.
[7] J. F. Frenzel, "Genetic algorithms," Potentials, IEEE, vol.12, no.3, pp.21-24, Oct. 1993.

11
[8] R. Eberhart and J. Kennedy, "A new optimizer using particle swarm theory," Proc. 6th Int. Symp.
Micromachine and Human Science, Nagoya, Japan, 4-6 Oct. 1995, pp.39-43.
[9] J. Kennedy, R. Eberhart, "Particle swarm optimization," Proc. IEEE Int. Conf. Neural Netw., Perth,
Australia, Nov/Dec 1995, vol. IV, pp.1942-1948.
[10] R. Rajabioun, "Cuckoo optimization algorithm," Applied Soft Computing, vol. 11, no. 8, pp. 5508-
5518, Dec. 2011.
[11] D. Karaboga and B. Basturk, “A powerful and efficient algorithm for numerical function
optimization: artificial bee colony (ABC) algorithm,” J Glob Optim, vol. 39, no. 3, pp.459-471, Oct.
2007.
[12] B. Basu and A. Nandi, “Evolutionary Algorithms in Optimization of Linear Antenna Array,” in LAP
Lambert Academic Publishing, May 2014.
[13] Y. Cengiz and H. Tokat, "Linear antenna array design with use of genetic, memetic and tabu search
optimization algorithms," Progress In Electromagnetics Research C, vol. 1, pp. 63-72, 2008.
[14] H. M. Elkamchouchi, M. M. Hassan, "Array pattern synthesis approach using a genetic algorithm,"
Microwaves, Antennas & Propagation, IET, vol.8, no.14, pp.1236-1240, Nov. 2014.
[15] V. Zuniga, A. T. Erdogan, T. Arslan, "Adaptive radiation pattern optimization for antenna arrays by
phase perturbations using particle swarm optimization," Adaptive Hardware and Systems (AHS),
NASA/ESA Conference on, pp.209-214, 15-18 June 2010.
[16] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans.
Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004.
[17] D.W. Boeringer and D.H.Werner, “A comparison of particle swarm optimization and genetic
algorithms for a phased array synthesis problem,” Antennas and Propagation Society International
Symposium, 2003. IEEE, vol. 1, pp.181–184, 22-27 June 2003.
[18] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for
phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004.
[19] J.R Perez and J. Basterrechea, "Comparison of Different Heuristic Optimization Methods for Near-
Field Antenna Measurements," IEEE Trans. Antennas Propag., vol.55, no.3, pp.549-555, Mar. 2007.
[20] A. Kumar, S Bhattacharya, M. Naja and P. Kumar, “Front-End Digital Signal Processing Scheme
For 206.5 MHz Atmospheric Radar Application,” Int. Jr. of Adv. Tech., vol.2, pp. 71-81, Jan 2011.
[21] Coello Coello and A. Carlos, "Evolutionary multi-objective optimization: a historical view of the
field," Computational Intelligence Magazine, IEEE, vol.1, no.1, pp.28-36, Feb. 2006.
[22] M. Jiang, Y.P. Luo, and S.Y. Yang, “Stochastic convergence analysis and parameter selection of the
standard particle swarm optimization algorithm,” Information Processing Letters, vol. 102, no. 1, pp.
8-16, 15 Apr. 2007.
[23] R. C. Eberhart and Y. Shi, “Comparing inertia weights and constriction factors in particle swarm
optimization,” Evolutionary Computation, 2000. Proceedings of the 2000 Congress on, vol. 1, pp.
84-88, July 2000.
[24] I.C. Trelea, “The Particle Swarm Optimization Algorithm: Convergence Analysis and Parameter
Selection,” Inf. Process. Lett., vol. 85, no. 6, pp. 317-325, Mar. 2003.
[25] M. Clerc and J. Kennedy. The particle swarm: explosion stability, and convergence in a multi-
dimensional complex space, Evolutionary Computation, IEEE Transactions on, vol.6, no.1, pp.58-
73, Feb. 2002.
[26] E. Sharp, "A triangular arrangement of planar-array elements that reduces the number needed," IEEE
Trans. Antennas Propag., vol. AP-9, no.2, pp.126-129, Mar. 1961.
[27] T. Kozu, T. Kawanishi, H. Kuroiwa, M. Kojima, K. Oikawa, H. Kumagai, K. Okamoto, M.
Okumura, H. Nakatsuka, and K. Nishikawa, "Development of precipitation radar onboard the
Tropical Rainfall Measuring Mission Satellite," IEEE Trans. Geosci. Remote Sens., vol.39, no.1,
pp.102-116, Jan. 2001.
[28] R. L. Jordan, B. L. Huneycutt, and M. Werner, "The SIR-C/X-SAR Synthetic Aperture Radar
system," IEEE Trans. Geosci. Remote Sensing, vol.33, no.4, pp.829-839, Jul. 1995.
[29] D. J. McLaughlin, E. Knapp, Y. Wang, and V. Chandrasekar, "Distributed weather radar using X-
band active arrays," Aerospace and Electronic Systems Magazine, IEEE, vol.24, no.7, pp.21,26, Jul.
2009.
[30] C. A. Balanis, "Antenna theory analysis and design," 2nd ed., John Willey and Son's Inc., New York,
1997.
[31] D. Yadolah, “The Concise Encyclopedia of Statistics,” 1st ed. New York, USA: Springer-Verlag,
2008.

A RRAY F ACTOR O PTIMIZATION OF AN A CTIVE P LANAR P HASED A RRAY USING E VOLUTIONARY A LGORITHM

More Related Content

What's hot (14)

Similar to A RRAY F ACTOR O PTIMIZATION OF AN A CTIVE P LANAR P HASED A RRAY USING E VOLUTIONARY A LGORITHM (20)

Recently uploaded (20)

A RRAY F ACTOR O PTIMIZATION OF AN A CTIVE P LANAR P HASED A RRAY USING E VOLUTIONARY A LGORITHM