Get documents about "Side Lobe Reduction Of A Planar Array Antenna By Complex Weight Control Using SQP Algorithm And Tchebychev Method
Shared by: ijcsiseditor
Categories
Tags
IJCSIS, call for paper, journal computer science, research, google scholar, IEEE, Scirus, download, ArXiV, library, information security, internet, peer review, scribd, docstoc, cornell university, archive, Journal of Computing, DOAJ, Open Access, June 2011, Volume 9, No. 6, Impact Factor, engineering, international, proQuest, computing, computer, technology
-
Stats
- views:
- 280
- posted:
- 7/5/2011
- language:
- English
- pages:
- 6
Document Sample
JOURNAL OF LTEX CLASS FILES, VOL. 9, NO. 6, JUNE 2011
A 1
Side Lobe Reduction of a planar array antenna by
complex weight control using SQP algorithm and
Tchebychev method
A. Hammami,R. Ghayoula,and A. Gharsallah
e ´
Unit´ de recherche : Circuits et systmes electroniques HF
e
Facult´ des Sciences de Tunis,
Campus Universitaire Tunis EL-manar, 2092,
Tunisie
Email: hammami.a.fst@gmail.com
Abstract—In this paper, we propose an efficient hybrid method algorithm [11], particle swarm optimization algorithm [19] .
based on the sequential quadratic programming (SQP) algorithm In this study, we propose a synthesis method for an uniform
and Dolph-Tchebychev for the pattern synthesis of planar an- planar array antennas based on SQP and Tchebychev algo-
tenna arrays with prescribed pattern nulls in the interference
direction and minimum side lobe levels SLL by controlling only rithms by controlling both the amplitude and phase (complex
the phase of each array element. The SQP algorithm is the weights control).
most widely used to solve nonlinear optimization problems. It The paper is organized as follows; theoretical description is
consists of transforming the nonlinear problem to sequence of presented in section II. Section III shows the optimization
quadratic subproblems by using a quadratic approximation of process. Section VI shows simulations and results and finally,
the lagrangian function. In order to illustrate the performance
of the proposed method, several examples of complex excited section VI makes conclusions.
planar array patterns with one-half wavelength spaced isotropic
elements to place the main beam in the direction of the useful
signal while reduce the side lobe level were investigated. II. T HEORETICAL DESCRIPTION
Index Terms—Planar antenna arrays, Synthesis method, null Consider a planar array composed by M × N equi-spaced
steering, sequential quadratic programming isotropic antenna elements arranged in a regular rectangular
array in the x-y plane, with inter-elements spacing d = dx =
I. I NTRODUCTION dy = λ/2 as shown in figure 2.
A RRAYS antennas are used to steer radiated power to-
wards a desired angular sector and to suppress or to
reduce side lobe level. The choice of the array antenna param-
The planar array antenna can be seen as a linear array of
linear arrays. The total field of a M × N elements is the
multiplication of the element radiation E0 in the coordinates
eters such us number elements, geometrical arrangement, and by the the sum of the normalized excitations currents on the
amplitudes and phases excitation depend on the angular pattern array elements [15][16].
that must be achieved. A linear array allows beam steering in
one dimension. however, a planar array has two dimensions of The perturbed array pattern can be written as
control, permitting a narrow pencil beam to be produced and
the control of the shape in both directions. Several antenna M N
array pattern synthesis antenna have been developed to steer Etot = |E0 . amn ej(ψmn +αmn ) | (1)
the main beam at the desired direction and to reduce the side m=1 n=1
lobe level [1]-[17]. with amn is the amplitude of the excitation of the element
Although the large number of elements in planar arrays, (m, n), αmn represents the phase excitation of the element
which translates into Although the large number of unknown (m, n),Moreover, ψmn is the phase of element (m, n) relative
parameters for the optimization algorithms,Some of these to the (1, 1)element.
works are interested in planar array antennas pattern synthesis. ψmn is then given by
In the few years, various techniques has been used to synthesis
a planar array antenna radion pattern. For the arbitrarily shaped
coverages, the synthesis pattern techniques can be classified ψkl = k(m − 1)dx sinθcosϕ + k(n − 1)dy sinθsinϕ) (2)
into categories [17]. Mathematical optimization deterministic
techniques are based on local gradient optimization method-
ologies such as Newton type algorithm, sequential quadratic
programming (SQP)[2] algorithm... Mathematical optimiza- with dx and dy are inter-element distance along the x-axis
tion stochastic techniques are global optimization methodolo- and inter-element distance along the y-axis respectively.
gies like genetic algorithm [17][23], Differential evolution The planar array radiation pattern a becomes
284 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
JOURNAL OF LTEX CLASS FILES, VOL. 9, NO. 6, JUNE 2011
A 2
The complex weight wmn can be written as the following
expression:
wmn = wm .wn (8)
wmn = am .ejαm .bn .ejαn (9)
2D Array Geometry Plot
Where wm and wn are the complex weight for the linear array
in the x-direction and the complex weight for the linear array
in the y-direction, respectively.
In an adaptive algorithm, several methods have been used
to compute amplitude weights amn such as Hamming,
Blackman,Dolph-Tchebychev, Hanning...[27]
Y−axis
A. Hanning
The coefficients of a Hanning are computed from the
following equation:
n
ω(n) = 0.5(1 − cos(2π )), 0 ≤ n ≤ N (10)
N
B. Hamming
The coefficients of a Hamming window are determined by
X−axis the following equation:
n
ω(n) = 0.54 − 0.46cos(2π ), 0 ≤ n ≤ N (11)
N
Fig. 1. Geometry of a M × N array of isotropic sources for field analysis.
C. Blackman
The equation for computing the Blackman coefficients is :
M N n n
ω(n) = 0.42 − 0.5cos(2π ) + 0.08cos(4π ), 0 ≤ n ≤ N
Etot = |E0 . amn ej(k(m−1)dx ux +k(n−1)dy uy +αmn ) | N N
m=1 n=1
(12)
(3)
with ux = sinθcosϕ and uy = sinθsinϕ)
D. Dolph-Tchebychev
Assuming that the current amplitude is amn = am × an and
the phase excitation can be decomposed in progressive parts The Dolph-Chebyshev weights are defined by
for the row and column elements of the array. So αmn can be cos[N cos−1 [βcos(πk/N )]]n
written as ω(k) = (−1)k ), 0 ≤ k ≤ N −1
cosh[N cosh−1 (β)]
αmn = αm + αn (4) (13)
Where
As a consequence of expressions 3 and 4, the planar array β = cos[1/N cosh−1 (10α )]
antenna radiation pattern may be written as the product of a The α parameter determines the level of the sidelobe attenu-
element pattern and two linear array factors like expression ation.
(9). The width of main-lobe 2ωc may be computed as follows:
1
Etot = E0 .AFm .AFn (5) ωc = 2cos−1 ( ) (14)
x0
Where
cosh−1 ( r )
1
where AFm and AFn are the array factor for the linear array x0 = N −1
in the x-direction and the array factor of the linear array in
the y-direction, respectively. III. O PTIMIZATION PROCESS
M
Our objective is to steer the main beam in the direction
AFm = ej(k(m−1)dx sinθcosϕ+αm ) (6) of desired signal and to reduce or to suppress interfering
m=1
signals from prescribed directions while receiving desired sig-
N nal from a chosen direction by complex weights (amplitudes
AFn = ej(k(m−1)dy sinθsinϕ+αn ) (7) and phases) control. The form of optimization problem is
n=1 expressed in mathematical terms as:
285 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
JOURNAL OF LTEX CLASS FILES, VOL. 9, NO. 6, JUNE 2011
A 3
where
minimise −fθ0 ,ϕ0 (α) λ1,1 , λ1,2 ... λ1,n
λ2,1 , λ2,2 ... λ2,n
subject to fθi ,ϕj (α) = δij i = 1, . . . , me λ= . .. . (21)
with
j = 1, . . . , ne .
. . .
.
λm,1 , λm,2 ... λm,n
k = me + 1, . . . , m
fθk ,ϕl (α) ≤ δkl with
l = ne + 1, . . . , n
is the matrix of the Lagrange multiplier. The solution to the
m = 1, . . . , M QP sub-problem, described on equation (20) produces a search
−2π ≤ αmn ≤ 2π with
n = 1, . . . , N direction vector dk , which is used to form a new iterate αk+1 .
(15)
αk+1 = αk + µp dk (22)
Where
where, µp ∈]0, 1] is a suitable step length parameter.
M N
j(k(m−1)dx ux +k(n−1)dy uy +αmn ) 2
fθ,ϕ (α) = | amn e |
m=1 n=1
IV. S IMULATIONS AND R ESULTS
(16) To illustrate the effectiveness of the proposed of theses
is the matrix of the objective function, fθi ,ϕj (17) is the presented techniques, we consider three planar array antennas
matrix of equality constraints and fθk ,ϕl (18) is the matrix of of sizes 12 × 12, 16 × 16 and 20 × 20 isotropic elements with
inequality constraints. equi-spaced inter-element is half wavelength.
fθ1 ,ϕ1 fθ2 ,ϕ1 . . . fθme ,ϕ1
fθ1 ,ϕ2 fθ2 ,ϕ2 . . . fθme ,ϕ2
fθi ,ϕj = . .. . (17)
.
. . .
.
fθ1 ,ϕne fθ2 ,ϕne ... fθme ,ϕne
fθme +1 ,ϕne +1 fθme +2 ,ϕne +1 ... fθm ,ϕne +1
fθme +1 ,ϕne +2 fθme +2 ,ϕne +2 ... fθm ,ϕne +2
fθk ,ϕl = . .. .
.
. . .
.
fθme +1 ,ϕn fθme +2 ,ϕn ... fθm ,ϕn
(18)
(θ0 , ϕ0 ) is the direction of the main lobe, (me , ne ) is the
equality constraints matrix size and (mi , ni ) is the inequality
constraints matrix size. The most widely used algorithm
to solve the problem (15) is the Sequential Quadratic
Programming (SQP) [3]-[4]. The basic idea of this method
is based on linearisations of the constraints and a quadratic
model of the objective function.
The quadratic programming problem solved at each iteration
of SQP can be defined as
minimise − fθ0 ,ϕ0 (αkl )T d + 2 dT Mkl d
1
Fig. 2. Three dimensional planar antenna array Pattern with main beam
imposed at (ϕ = 50◦ , θ = 50◦ ).
i = 1, . . . , me
subject to fθi ,ϕj (αkl )T d + fθi ,ϕj (αkl ) = δij
j = 1, . . . , ne
For these three examples, a sequential quadratic
programming SQP has been used to optimize the phase
k = me + 1, . . . , m
fθk ,ϕl (αkl )T d + fθk ,ϕl (αkl ) ≤ δkl
l = ne + 1, . . . excitation of the planar array antenna in order to point the
,n
main beam in the desired direction, and Tchebychev method
d ∈ RM ∗N has been used to optimize the amplitude excitation of planar
(19) array antenna in order to achieve a minimum peak sidelobe
where Mkl is the BFGS (Broyden, Fletcher, Goldfarb, and level with narrower beam width. The results obtained by
Shanno) approximation of the Hessian 2 L(αkl , λkl ).
α Tchebychev method were compared with Hanning, Blackman
The lagrangian function L is defined as and Hamming methods.
m n
L(α, λ) = −fθ0 ,ϕ0 (α) + λkl (fθk ,ϕl (α) − δkl ) (20) Figure 2 shows a 3-D surface plot of the radiation pattern
k=1 l=1 of the planar array antenna with a main beam imposed at
(ϕ = 50◦ , θ = 50◦ ) synthesized by SQP and Tchebychev
method silde lobe levels SSL is 50dB.
286 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
JOURNAL OF LTEX CLASS FILES, VOL. 9, NO. 6, JUNE 2011
A 4
0
Uniform Amplitude 0
Dolph−Tchebychev Uniform Amplitude
−10 blackman
−20 Hanning
Hamming
−20
−40
−30
Amplitue(dB)
Amplitue(dB)
−60 −40
−50
−80
−60
−100 −70
−80
−120
−90
−150 −100 −50 0 50 100 150
φ(deg)
−140
−150 −100 −50 0 50 100 150
φ(deg)
Fig. 5. Planar array antenna radiation pattern obtained by uniform amplitude
weights ,the Blackman amplitude weights and Hamming amplitude weights
Fig. 3. Planar array antenna radiation pattern with main beam at (ϕ = for N = 12 × 12 elements.
50◦ , θ = 50◦ ) obtained by uniform amplitude weights ,the Tchebychev
amplitude weights and Hanning amplitude weights for N = 12×12 elements.
0
0 Uniform Amplitude
Uniform Amplitude blackman
Dolph−Tchebychev −2 Hamming X: 46
−1 Y: −2.897
Hanning
−4
Amplitue(dB)
−2 Mainlobe width= 12°
Amplitue(dB)
Mainlobe width= 19°
−3 Mainlobe width= 24°
X: 30
Y: −3.682
−6
−4 Mainlobe width= 12°
−8
−5 Mainlobe width= 18°
Mainlobe width= 20° −10
−6
20 30 40 50 60
−7 φ(deg)
26 28 30 32 34 36 38 40 42 44 46 48 50 52 54
φ(deg)
Fig. 6. the −3dB beam-width planar array antenna radiation pattern obtained
Fig. 4. the −3dB beam-width for planar antenna array radiation pattern by uniform amplitude weights ,the Blackman amplitude weights and Hamming
obtained by uniform amplitude weights ,the Tchebychev amplitude weights amplitude weights for N = 12 × 12 elements.
and Hanning amplitude weights for N = 12 × 12 elements.
0
Figures 3and 5 present the array factors for different value Uniform Amplitude
of ϕ where φ ∈] − ππ] at θ0 = 50◦ and the −3dB beam- −1 Dolph−Tchebychev
width obtained by uniform amplitude weights ,the Tchebychev, −2 Hanning
Hanning, Blackman ant Hamming amplitudes weights for −3
Amplitue(dB)
N = 12 × 12 elements and the −3dB beam-width of the −4 Mainlobe width= 9°
radiation pattern obtained these methods. Figures 4and 6 show −5 Mainlobe width= 13°
the beam-width of theses radiations patterns.
−6 Mainlobe width= 16°
The second test treats with a square array with 16 × 16 ele-
−7
ments. Figures 7 and 9, show the radiation pattern synthesized
by SQP algorithm and Tchebychev, Hanning, Blackman and −8
Hamming methods at θ = 50◦ . The Tchebychev method can −9
cut down the side lobe level less than −80dB with −3dB −10
26 28 30 32 34 36 38 40 42 44 46 4849
main-width is 16. φ(deg)
In the third example, the array elements are placed on regu-
lar grid of size 20 × 20 with interelement spaced λ/2. Figures Fig. 7. Planar array antenna radiation pattern cut at with main beam at
(ϕ = 50◦ , θ = 50◦ ) obtained by uniform amplitude weights ,the Tchebychev
11and 13 show repectively the radiation pattern of planar array amplitude weights and Hanning amplitude weights for N = 16×16 elements.
antenna of 20 × 20 elements at θ = 50◦ ) synthesized by SQP
287 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
JOURNAL OF LTEX CLASS FILES, VOL. 9, NO. 6, JUNE 2011
A 5
algorithm and Tchebychev, Hanning, Blackman and Hamming
0
Uniform Amplitude methods.
Dolph−Tchebychev
−20 Hanning
0
−40 Uniform Amplitude
−10
Dolph−Tchebychev
−60 Hanning
Amplitue(dB)
−30
−80
−50
Amplitue(dB)
−100
−70
−120
−90
−140
−110
−160
−150 −100 −50 0 50 100 150
φ(deg) −130
−150
Fig. 8. The −3dB beam-width planar array antenna radiation pattern −150 −100 −50 0 50 100 150
obtained by uniform amplitude weights ,the Tchebychev amplitude weights φ(deg)
and Hanning amplitude weights for N = 16 × 16 elements.
Fig. 11. Planar array antenna radiation pattern cut at θ = 50◦ obtained by
uniform amplitude weights ,the Tchebychev amplitude weights and Hanning
0 amplitude weights for N = 20 × 20 elements.
Uniform Amplitude
−10 Blackman
−20
−30
0
−40
Uniform Amplitude
−50 −1
Amplitue(dB)
Dolph−Tchebychev
−60
Hanning
−70 −2
−80
Amplitue(dB)
−3
−90
−100 −4 Mainlobe width= 7°
−110
−5 Mainlobe width= 11°
−120
−130
Mainlobe width= 13°
−150 −100 −50 0 50 100 150 −6
φ(deg)
−7
Fig. 9. Planar antenna array pattern obtained by uniform amplitude weights
,the Blackman amplitude weights and Hamming amplitude weights for N = −8
30 32 34 36 38 40 42 44 46 48 50
16 × 16 elements. φ(deg)
Fig. 12. the −3dB beam-width of Planar array antenna radiation pattern
0 obtained by uniform amplitude weights ,the Tchebychev amplitude weights
Uniform Amplitude and Hanning amplitude weights for N = 20 × 20 elements.
−1 Blackman
Hamming
−2 The cut of the radiation pattern plot in figures 2-13 obtained
by using sequential quadratic algorithm SQP and Tchebychev
Amplitue(dB)
−3
demonstrates the ability of this method to steer main beam
Mainlobe width= 9°
−4 in desired direction and to reduce the side lobe level in the
Mainlobe width= 13°
−5 Mainlobe width= 17° direction of interfering signal by controlling only the phase
excitation of each array elements.
−6
By comparing the results shown in figures 2-13 we deduced
−7 that the method based on SQP algorithm and Tchebychev is
more adequate than others methods described above in order
−8
28 30 32 34 36 38 40
φ(deg)
42 44 46 48 50 52 to reduce the side lobe level with an acceptable main-beam
width.
Fig. 10. the −3dB beam-width of planar antenna array pattern obtained by
uniform amplitude weights ,the Blackman amplitude weights and Hamming
amplitude weights for N = 16 × 16 elements.
288 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
JOURNAL OF LTEX CLASS FILES, VOL. 9, NO. 6, JUNE 2011
A 6
Communications, Computing and Control Applications March 3–5, 2011
0
Uniform Amplitude Hammamet Tunisia
−10 Blackman [2] Hammami, A.; Ghayoula, R.; Gharsallah, “A.Planar array antenna pattern
−20 nulling based on sequential quadratic programming (SQP) algorithm
Systems,”Signals and Devices (SSD), 2011 8th International Multi-
−30
Conference on Digital Object Identifier
−40 [3] M. Mouhamadou, P. Vaudon,“Complex Weight Control of Array Pattern
Nulling,” International Journal of RF and Microwave Computer-Aided
Amplitue(dB)
−50
Engineering, pp. 304–310, 2007.
−60 [4] M. Mouhamadou, P. Vaudon, “Smart Antenna a Array Patterns Synthesis:
−70 Null Steering and multi-user Bealforming by Phase control,”Progress In
Electromagnetics Research, Vol. 60, pp. 95–106, 2006.
−80
[5] R.L. Haupt, “Phase-only adaptive nulling with a genetic algorithm,”IEEE
−90 Trans Antennas Propag 45 (1997), 1009-1015
−100
u
[6] K. G¨ ney and A. Akdagli, Null steering of linear antenna arrays using a
modified tabu search algorithm, Prog Electromagn Res 33 (2001), 167-
−110 182.
−120 a
[7] N. Karaboga, K. Gu¨ ney, and A. Akdagli, Null steering of linear antenna
−150 −100 −50 0 50 100 150 arrays with use of modified touring ant colony optimization algorithm,
φ(deg)
Wiley Periodicals; Int J RF Microwave Computer-Aided Eng 12 (2002),
375-383.
Fig. 13. Planar antenna array pattern cut at θ = 50◦ obtained by uniform [8] R.Ghayoula, N.Fadlallah, A.Gharsallah, M.Rammal, IET Microw. An-
amplitude weights ,the Blackman amplitude weights and Hamming amplitude tennas Propag., Phase-Only Adaptive Nulling with Neural Networks for
weights for N = 20 × 20 elements. Antenna Array Synthesis, Vol. 3, pp. 154–163, 2009.
[9] R.Ghayoula, N.Fadlallah, A.Gharsallah, M.Rammal, Neural Network
Synthesis Beamforming Model for One-Dimensional Antenna Arrays ,
The Mediterranean Journal of Electronics and Communications, ISSN:
0 1744-2400, Vol. 4, No. 1, pp.126–131, 2008.
Uniform Amplitude
[10] D.I. Abu-Al-Nadi and M.J. Mismar, Genetically evolved phase-
−1 Blackman aggregation technique for linear arrays control, Prog Electromagn Res
Hamming 43 (2003), 287-304.
[11] S. Yang, Y.B. Gan, and A. Qing, Antenna-array pattern nulling using a
−2
differential evolution algorithm, Wiley Periodicals; Int J RF Microwave
Amplitue(dB)
Computer- Aided Eng 14 (2004), 5763.
−3 [12] T.H. Ismail, D.I. Abu-AL-Nadi, and M.J. Mismar, Phase-only control
for antenna pattern synthesis of linear arrays using LevenbergMarquart
−4 Mainlobe width= 7° algorithm, Electromagnetics 24 (2004), 555-564.
[13] M.M. Khodier and C.G. Christodoulou, Linear array geometry synthesis
Mainlobe width= 10° with minimum sidelobe level and null control using particle swarm
−5 optimization, IEEE Trans Antennas Propag 53 (2005), 2674-2679.
Mainlobe width= 14° [14] Spellucci, P., Numerische Verfahren der Nichtlinearen Optimierung,
−6 a
Birkh¨ user, 1993.
[15] Hubregt J. Visser, Array and Phased Array Antenna Basics,John Wiley&
Sons Ltd, England, 2005.
−7
29 31 33 35 37 39 41 43 45 47 49 5152 [16] R. PONCE HEREDIA, Contribution la modlisation du rayonnemnet
φ(deg) d’antennes conformes pour des applications C.E.M. en aronautique,
´
FACULTE des SCIENCES et TECHNIQUES, Universite de Limoges,
PhD Thesis,2008.
Fig. 14. the −3dB beam-width planar antenna array pattern obtained by
[17] Frank J. Villegas, Parallel Genetic-Algorithm Optimization of Shaped
uniform amplitude weights ,the Blackman amplitude weights and Hamming
Beam Coverage Areas Using 2-D Phased Arrays, in IEEE Transations
amplitude weights for N = 20 × 20 elements.
on antennas and Propagation.), VOL. 55, NO. 6, JUNE 2007.
[18] E. Aksoy, E. Afacan, Planar antenna pattern nulling using differential
evolution algorithm, International Journal of Electronics and Communi-
¨
cations. (AEU) 63 (2009) 116 122.
V. C ONCLUSION
[19] S. Chamaani., S. AbdullahMirtaheri, Planar UWB monopole antenna op-
A complex weight method used to optimize the planar timization to enhance time-domain characteristics usingPSO,International
¨
Journal of Electronics and Communications. (AEU) 64 (2010) 351359.
array with height precision based on the sequential quadratic [20] Einarsson O. Optimization of planar arrays. IEEE Trans Antennas
algorithm SQP and Dolph-Tchebychev method to steer the Propagat 1979;27:86-92.
main beam at desired direction and to reduce the maximum [21] Laxpati SR. Planar array synthesis with prescribed pattern nulls. IEEE
Trans Antennas Propagat 1982;30:1176-83.
side lobe level. This hybrid method exploit the advantages of [22] DeFord JF, Gandhi OP. Phase-only synthesis for minimum peak sidelobe
both methods:the facility and the reduced response time of the patterns for linear and planar arrays. IEEE Trans Antennas Propagat
SQP algorithm to point the main beam easily at any direction 1988;36:191-201.
[23] Kumar BP, Branner GR. Generalized analytical technique for the synthe-
desired and the important attenuation of the side lobe level by sis of unequally spaced arrays with linear, planar, cylindrical or spherical
Dolph-Tchebychev method. geometry. IEEE Trans Antennas Propagat 2005;53:621-34.
Simulation results showed that the proposed method is efficient [24] Haupt RL. Thinned arrays using genetic algorithms. IEEE Trans Anten-
nas Propagat 1994;42:993-9.
for synthesizing a planar array antenna pattern ; indeed, it is [25] Yang S, Nie Z. Time modulated planar arrays with square lattices and
capable to reduce the side lobe level less than −80dB. circular boundaries. Int J Numer Model 2005;18:469-80.
[26] Alicano CR, Rosales DC, Rodriguez CB, Mendoza MP. Differential
evolution algorithm applied to sidelobe level reduction on a planar array.
R EFERENCES Int J Electron Commun (AEU) 2007;61:286-90.
[27] Oppenheim, A.V., and R.W. Schafer, Discrete-Time Signal Processing,
[1] Hammami, A.; Ghayoula, R.; Gharsallah, “Antenna Array Synthesis Prentice-Hall, 1989, pp. 447–448.
with Chebyshev-Genetic Algorithm Method,”International Conference on
289 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
Related docs
Other docs by ijcsiseditor
Digital Images Encryption in Spatial Domain Based on Singular Value Decomposition and Cellular Automata
Views: 0 | Downloads: 0
Agent Behavior in Multiagent Systems: Issues and Challenges in Design, Development and Implementation
Views: 1 | Downloads: 0
Optimizing Cost, Delay, Packet Loss and Network Load in AODV Routing Protocols
Views: 2 | Downloads: 0
