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Side Lobe Reduction Of A Planar Array Antenna By Complex Weight Control Using SQP Algorithm And Tchebychev Method

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					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
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                                                                                    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:
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                                                                                               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.
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                         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
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                                                                                                           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.

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           A                                                                                                                                                               6



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                        0
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       Amplitue(dB)




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