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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 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. 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