PAPER Design of Steerable Linear and Planar Array Geometry with Non-uniform Spacing for Side-Lobe Reduction

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1 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY PAPER Design of Steerable Linear and Planar Array Geometry with on-uniform Spacing for Side-Lobe Reduction Ji-Hoon BAE a), Kyung-Tae KIM, and Cheol-Sig PYO, onmembers SUMMARY In this paper, we present a noble pattern synthesis method of linear and planar array antennas, with non-uniform spacing, for simultaneous reduction of their side-lobe level and pattern distortion during beam steering In the case of linear array, the Gauss-ewton method is applied to adjust the positions of elements, providing an optimal linear array in the sense of side-lobe level and pattern distortion In the case of planar array, the concept of thinned array combined with non-uniformly spaced array is applied to obtain an optimal two dimensional (2-D) planar array structure under some constraints The optimized non-uniformly spaced linear array is extended to the 2-D planar array structure, and it is used as an initial planar array geometry ext, we further modify the initial 2-D planar array geometry with the aid of thinned array theory in order to reduce the maximum side-lobe level This is implemented by a genetic algorithm under some constraints, minimizing the maximum side-lobe level of the 2-D planar array It is shown that the proposed method can significantly reduce the pattern distortion as well as the side-lobe level, although the beam direction is scanned key words: antenna array pattern synthesis, non-uniform spacing, optimization technique, side-lobe reduction 1 Introduction In the field of antenna array pattern synthesis, a large number of pattern synthesis techniques have been studied and developed over more than 60 years In general, these techniques can be classified into two categories: one method optimizes the excitation of each element of the uniform array and the other adjusts the elements positions with uniform excitation, resulting in a non-uniform array geometry The excitation includes the amplitudes and phases of array elements Traditional synthesis methods, such as Fourier transform, Talyor, Woodward-Lawson and Dolph-Chebyshev [1], belong to the first category The advantage of these methods is that they can provide standard design rules for uniformly spaced array (USA) pattern synthesis However, these traditional methods are restricted to only a pattern synthesis of arrays with uniformly spaced and isotropic antenna elements Orchard et al [2] presented a power pattern synthesis procedure of linear arrays with uniform element spacing This method is based on Schelkunoff s polynomial representation of the array pattern and it enables us to synthesize an arbitrary beam pattern in the main lobe region as Manuscript received December 12, 2003 Manuscript revised July 12, 2004 The authors are with ETRI, 161 Gajeong-dong, Yuseong-gu, Daejeon, , Korea The author is with the Dept of EECS, Yeungnam University, Daedogng, Kyongsan, Kyungbuk, , Korea a) baejh@etrirekr well as in the side-lobe region by optimally placing roots of the polynomial expression The Orchard-Elliott synthesis method, which yields shaped beams with a non-symmetrical complex distribution, has been extended and improved in [3], [4] to provide a pure-real distribution of the excitation Adifferent approach exploits an adaptive array theory in the synthesis array pattern Olen and Compton [5] developed a numerical pattern synthesis algorithm based on the adaptive array theory Unlike the classical methods, the algorithm can be used for both USA and non-uniformly spaced array (USA) and can provide an arbitrary side-lobe level (SLL) While Olen and Compton s method considers only the SLL, the new algorithm presented in [6] can achieve the desired main-lobe shape as well as low SLL, simultaneously It should be noted however, that most of the methods in the first category inevitably have one disadvantage, namely, an amplitude tapering The amplitude tapering of the excitation may require a complicated feed system and also increases the main beamwidth In addition, if the amplitude tapering is large, then the mutual coupling effectsmay cause appreciable changes in the small antenna current [7] In the second category, Unz [8] originally analyzed a linear array for arbitrarily distributed elements From the initial concept of Unz, Harrington [9] developed a method for reducing the first SLL of a linear array with non-uniform element spacing This synthesis method is based on the Fourier transform formula and can reduce the inner sidelobes (nearby side-lobes from the main beam) to about 2/ times the field intensity of the main lobe, where is the number of elements of linear array Hodjat and Hovanessian [10] suggested an iterative method to adjust positions of a non-uniformly spaced linear array (USLA), which provides symmetrical non-uniform arrangements with respect to the array center In addition, to design several non-uniformly spaced planar arrays (USPAs), they extended their USLA structures to 2-D planar array structures Other array pattern synthesis approaches for USA have been studied and presented in [11] [16] Another major improvement for the second category is the use of thinned array theory It is illustrated in [17] that thinning an array means turning off some elements in a uniformly spaced or periodic array to create a desired amplitude density across the aperture Thinned arrays have been investigated for several decades in many array antenna fields since Skolnik et al [18] applied dynamic programming to the design of thinned array Recently, derivative-free optimization methods, such as simulated annealing and genetic Copyright c 2005 The Institute of Electronics, Information and Communication Engineers

2 346 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY 2005 algorithm, have drawn great attention in this area Simulated annealing (SA) was derived from the physical characteristic of spin glasses and is performed by the mechanism of an annealing or cooling schedule [19] While SA uses a single agent in search for an optimal solution, genetic algorithm (GA) has a multi-agent system GA is composed of the natural evolution mechanism of SA, random search mechanism, and biological mechanism SA was used to design the thinned arrays with low SLL [20] In [21], J elements are optimally placed on a K-linear lattice with uniform spacing under some design constraints to synthesize a desired linear array pattern, where the size of J is less than that of K Ultimately, optimally thinning some elements located on the uniformly spaced linear grid can cause the array to have a suitable non-uniform spacing with low SLL,which corresponds to the USLA theory above In the case of 2-D planar arrays, thinning some elements may change the element density, which can give an amplitude variation to the arrays where the amplitude at each element is presented by one bit [17] Therefore, it is possible to achieve low SLL for the planar array if some elements are optimally turned off, resulting in the effectofthe quantizedamplitude taper ote that in this secondary synthesis field, all the amplitudes and phases of the excited array elements have uniform values or fixed values during optimization procedure The attractive aspect of the USPA is that a radiation pattern with low SLL can be determined by only the proper array structure, maintaining uniform excitations of given array antennas The maximum SLL (MSLL) of an array radiation pattern in the second category can be reduced by adjusting the inter-element spacing appropriately, namely, by applying the USA theory However, it should be noted that the USA geometry may cause the outer SLL to increase especially when the main beam direction is scanned [9] Because of this phenomenon, the undesirable large side-lobes which are greater than the first SLL can be seen within the visible region of 90 θ 90, when the USA geometry is exploited Therefore, in this study, we consider the outer SLL as well as the inner SLL, resulting in a simultaneous reduction of MSLL and pattern distortion of the steerable linear and planar array On the basis of USA theory, we propose linear and planar array pattern synthesis methods for side-lobe reduction The purpose of this paper is to find an optimal USA structure, maintaining a low SLL, without pattern distortion during beam steering In our synthesis method for an optimal USLA with low SLL, first, the non-optimized and non-uniform element positions are calculated by the Fourier transform based formula, and they are used as initial element positions This smart initial guess can facilitate a fast convergence of the iterative optimization process ext, we optimize the inter-element spacing from the initial element positions using the Gauss-ewton method, which is one of the derivative-based optimization techniques The resulting USLA method can reduce both the inner side-lobes and the large outer side-lobes, simultaneously We also propose our design scheme for an optimal USPA with low SLL Our approach makes use of the thinned array theory combined with our proposed USLA technique To generate a 2-D planar grid of non-uniform spacing, the resulting optimized USLA is extended to a 2- D rectangular array lattice This USPA is used as an initial array geometry ext, the GA is applied to implement the thinned array theory in order to adjust the arrangements of the initial USPA The resulting USPA, under some constraints, can accomplish low SLL without pattern distortion, although the beam direction is steered This paper is organized as follows In Sects 2 and 4, we formulate the problem of interest for a USLA and USPA In Sects 3 and 5, we describe in detail the pattern synthesis methods of linear and planar array with nonuniform spacing In Sect 6, we show some simulation examples Finally, we draw our conclusions in Sect 7 2 Problem Formulation for on-uniform Linear Array For an odd number of elements, if isotropic array elements are uniformly distributed along the x-axis and are assumed to be symmetric about the array center, the radiation field pattern over the set of angles θ 1,θ 2, θ L can be described as follows: p 1D (θ i ) = 1 1 exp[ jκndx(sin θ i sin θ 0 )] = 2 n=0 cos[κndx(sin θ i sin θ 0 )] + 1 where is the number of element antennas, κ is the free space propagation constant, dx is the inter-element spacing, θ 0 is the maximum radiation angle, and M is given by ( 1)/2 As shown in Fig 1, if the uniform array element positions are perturbed by the fractional change, e x n,theresulting element spacing can be expressed as follows: (1) d x n = (n + e x n)dx [9] (2) Then, the normalized pattern of the USLA is given by p 1D nu (θ i ) = 2 Fig 1 cos[κ(n + en)dx(sin x θ i sin θ 0 )] Structure of non-uniform linear antenna array

3 BAE et al: DESIG OF STEERABLE LIEAR AD PLAAR ARRAY GEOMETRY (3) In the following section, a pattern synthesis method to find the most suitable element positions is derived from the basic formula of Eq (3) 3 on-uniform Linear Array Pattern Synthesis In this section, the Gauss-ewton algorithm is used to extract the optimal parameter, en x from Eq (3) This algorithm is based on the derivative-based optimization technique and implements the minimization of a cost function that is expressed as the sum of error squares [19] If we expand the cosine term of Eq (3) and assume a very small en, x Eq (3) can be transformed into the following form: p 1D nu (θ i ) p 1D (θ i ) 2 en x sin[κdx(sin θ i sin θ 0 )n] κdx(sin θ i sin θ 0 ) (4) Equation (4) can be further simplified into the following form: g(z i ) = p 1D (z i ) p 1D nu (z i ) [ ] = en x 2(zi z 0 )sin(n (z i z 0 )) where z i = κdxsin θ i, z 0 = κdxsin θ 0,andp 1D (z i )isthe normalized pattern of the uniform linear array When the cost function is defined as C 1 = 1 2 i=1 i=1 L L [g(z i ) g des (z i )] 2 = 1 2 ε 2 (z i ), the Gauss-ewton algorithm can be used to minimize this cost function, where g(z i ) means the difference between the radiation pattern with non-uniform spacing and the original radiation pattern with uniform spacing, and g des (z i ) represents the desired difference pattern between the original pattern and the desired reference pattern with low SLL In our case, to obtain the desired reference pattern, several inner side-lobes including the first side-lobe are set up to a predefined low SLL, and the other SLLs are fixed to each value slightly smaller than the outer SLLs of the original pattern At the same time, the main lobe of the desired pattern is identical with that of the original USLA pattern In order to estimate the predetermined low SLL in an optimal way, we performed many simulations varying the predefined SLL, and found that the predetermined SLL between 25 db and 30 db can guarantee convergence of the proposed algorithm in most cases as wellas sufficiently low SLL without pattern distortion In addition, ε(z i ) is a function of the adjustable variable, e x nif we apply the Talyor series expansion to ε(z i ), then the following equation holds: ε (k) (z i ) e x(k) n [e x(k+1) n e x(k) n ] = ε (k+1) (z i ) ε (k) (z i ), (5) i = 1, 2,, L, n = 1, 2,, M (6) where k denotes an iteration number The matrix form of Eq (6) is given by Ω (k+1) =Ω (k) + J (E (k+1) E (k) ) (7) where Ω=[ε(z 1 ),ε(z 2 ),,ε(z L )] T, E = [e1 x, ex 2,, ex M ]T and J is a Jacobian matrix, given by ε(z 1 ) ε(z 2 ) ε(z L ) T J = = e1 x ε(z 1 ) e2 x e1 x ε(z 2 ) e2 x ε(z 1 ) ε(z 2 ) e x M e x M 2(z 1 z 0 ) sin(z 1 z 0 ) e1 x ε(z L ) e2 x ε(z L ) e x M 2(z 1 z 0 ) sin(2(z 1 z 0 )) 2(z 1 z 0 ) sin(m (z 1 z 0 )) 2(z 2 z 0 ) sin(z 2 z 0 ) 2(z 2 z 0 ) sin(2(z 2 z 0 )) 2(z 2 z 0 ) sin(m (z 2 z 0 )) 2(z L z 0 ) sin(z L z 0 ) 2(z L z 0 ) sin(2(z L z 0 )) 2(z L z 0 ) sin(m (z L z 0 )) Therefore, the desired small perturbation vector, E (k+1) can be obtained by minimizing the following nonlinear leastsquares (LS) criterion: 1 C 2 = arg min Ω (k) + J (E (k+1) E (k) ) 2 (8) E (k) 2 where denotes the Euclidean norm following LS solution: C 2 E (k) = 0 yields the E (k+1) = E (k) [(J T J) 1 J T ]Ω (k) (9) For fast and stable convergence, Eq (9) can be represented in a slightly modified form as follows: E (k+1) = E (k) η[(δ 2 I + J T J) 1 J T ]Ω (k) (10) where I is an L M identity matrix, η = η 0 exp( r k)isan iteration gain, and η 0 and r are all fixed constants A small quantity δ 2 is added to each diagonal element of the Jacobian matrix to prevent it from being ill conditioned [6] For fast and stable convergence of Eq (10), it is essential to assign proper values to the η 0 and r For small value of r, the convergence of Eq (10) may be guaranteed, but its rate of convergence becomes too slow In contrast, for large value of r, the convergence may not be guaranteed due to the fluctuation phenomenon of the convergence curve around the optimal solution From many simulation results, we found that η 0 between 12 and 16, and r between 02 and 025 can provide a suitable performance for our algorithm The initial small distance vector E (0) can be obtained by the formula described in [9] to prevent the algorithm from reaching a local minimum and provide a quick convergence to the global minimum The procedure for the pattern synthesis of the USLA using the proposed algorithm is summarized as follows: Step 1: The initial array element positions are calculated by the following formula based on Fourier coefficients,

4 348 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY 2005 e x(0) n π = n(p 1D (z) p 1D nu (z))s a (n z)dz, π 0 n = 1, 2,, M (11) where S a (n z) = sin(n z) n z Step 2: To improve the radiation pattern of the steerable linear array, new array element positions are obtained by Eq (10) Step 3: Iterate the Step 2 until the following relative error (RE) is less than a predefined small quantity, γ RE = 1 2 L p 1D(k+1) nu i=1 (θ i ) p 1D(k) nu (θ i ) 2 <γ (12) 4 Problem Formulation for on-uniform Planar Array To carry out the design of an optimal planar array, with nonuniform spacing, we use the optimized linear array geometry introduced in Sect 3 as an initial USPA The resulting linear array is extended to a 2-D rectangular array lattice along the row and column directions respectively; according to the non-uniformly distributed positions of the optimized linear array elements Thus, the initial USPA pattern can be described as follows: p 2D nu (u,v) = 1 2 cos(κ(n + en)dx x (u u 2 0 )) cos(κ(m + e y m)dy (v v 0 )) + 1 m=1 = 1 2 cos(κd x 2 n (u u 0 )) cos(κdm y (v v 0 )) + 1 (13) m=1 where u = sin θ cos φ, v = sin θ sin φ, u 0 = sin θ 0 cos φ 0, v 0 = sin θ 0 sin φ 0, d x n = (n + e x n)dx, d y m = (m + e y m)dy, dy=dx, and e y m = e x n Furthermore, this rectangular array geometry can be modified to achieve further reduction of the MSLL In the following section, a pattern synthesis method to find the most suitable planar array structure for the maximum sidelobe reduction is derived from the formula of Eq (13) 5 on-uniform Planar Array Pattern Synthesis In this section, to accomplish lower SLL from the initial USPA of Eq (13), the GA and thinned array concept are exploited to modify the initial USPA structure Consider the linear array pattern of Eq (1) The linear array pattern in Eq (1) is similar to the Fourier series expression for an arbitrary real-valued function in that the array pattern can be expressed as the sum of cosine terms If we define an array frequency, ω n = 2πn dxsin θ, the lowest array frequency can be associated with the center array element, and higher order array frequencies with the outer array elements [12] In addition, when an arbitrary real-valued function is composed of slowly as well as rapidly varying functions, the higher frequencies may determine the higher variations of the function Therefore, SLL of the linear array may be more sensitive to the adjustment of components associated with the high order array frequencies, which physically correspond to the outer array elements far from the array center A concept of the array frequency of the linear array can be extended to a 2-D planar array problem It is shown in [10] that the elimination of some elements of the four corner of a rectangular array, with non-uniform spacing, can give a circular radiation pattern and provide greater reduction of the SLL In this study, to achieve lower SLL from the initial USPA, we find the unnecessary elements around the outer regions of the 2-D rectangular array using a GA, and then they are removed from that array geometry GA is a stochastic search procedure modeled on the Darwinian concepts of natural selection and evolution [22] It is the highest merit of the GA to provide a global optimal solution for complex electromagnetic (EM) problems The basic process, general concepts and applications of the GA in EM problems have been presented in [17], [21] [24] In our case, to formulate a pattern synthesis method from the initial USPA, we start with Fig 2 Fig 2 shows the initial, non-uniformly spaced rectangular array arrangement As shown in Fig 2, each array element is symmetrically positioned along the non-uniformly spaced rectangular grid with respect to x-axis and y-axis Due to this symmetry, we consider merely the array elements in the first quadrant, not whole spaces, for the optimization procedure In addition, among the elements belonging to the first quadrant, only the outer elements contained in (A), (B), and (C) of Region I are optimized through the GA The outer elements within the Regions, II, III, and IV adopt the same geometry as in Region I Therefore, the number of parameters that must be optimized in the GA can be significantly reduced The USPA pattern of Eq (13) can be written as follows: nu = 1 R 2 cos(κd x 2 n (u u 0 )) + 1 p 2D Q 2 cos(κdm y (v v 0 )) + 1 m= m=q+1 cos(κd x n (u u 0 )) } Q m=1 n=r+1 R { W a mn cos(κdm y (v v 0 )) m=q+1 Wm0 a cos(κdy m (v v 0 )) { W b mn cos(κdm y (v v 0 ))

5 BAE et al: DESIG OF STEERABLE LIEAR AD PLAAR ARRAY GEOMETRY 349 Fig 2 USPA geometry for the formulation of planar array pattern synthesis cos(κd x n (u u 0 )) } m=q+1 n=r+1 n=r+1 W0n b cos(κdx n (u u 0 )) W c mn cos(κd y m (v v 0 )) cos(κd x n (u u 0 )) ] (14) where (Wmn a,wa m0 ), (Wb mn,wb 0n ), and Wc mn are the amplitude weights of elements = { 1or0} corresponding to region (A), (B) and (C), respectively, and M, Q, andr are defined in Fig 2 We assume that Wm0 a = Wb 0n =1 In addition, we further assume that Wmn b has the same symmetrical arrangement with Wmn a This can lead to a further reduction of the computational complexity during the optimization process Wm a represents the element status as on, whereas Wmn=0 a represents the element status as off Wmn c is also expressed in the same manner as Wmn a Values for the parameters of the GA can be represented by a binary string or real-valued string In this paper, we adopt a binary string because Wmn, a Wmn,andW b mn c have discrete values, namely, 1 or 0 When the main lobe region of the USPA pattern is U L u U R and V L v V R, the cost function, F, to evaluate the fitness value of given individuals is defined as follows: { F = max 20 log( p 2D nu (u,v) } ) (15) SLL in side-lobe regions subjected to { } 1 u UL and U R u 1for 1 v 1 1 v V L and V R v 1forU L u U R The procedure for the pattern synthesis of the USPA using the GA is summarized as follows: Step 1: The optimized linear array in Sect 3 is extended to a 2-D rectangular array to obtain an initial USPA geometry (ote that a uniformly spaced planar array (USPA), extended by a uniformly spaced linear array (USLA) instead of the optimized USLA, can also be used as an initial planar array in Step 1 for the design of an optimal USPA) Step 2: Randomly generate an initial population for W mn = [W a mn W c mn] which represent a chromosome consisting of binary string Step 3: Calculate the MSLL using Eq (15) Step 4: Rank chromosomes from best to worst, according to their fitness values obtained by Step 3, and discard the bottom 50% Step 5: Create new offspring settings from the selected top 50% using the crossover operator Step 6: The best individual from the present generation is saved, but it will not take part in Step 7 of the mutation process Step 7: Mutate the new offspring based on the probability of mutation Step 8: Iterate Step 2 Step 7 until there is no improvement about the best fitness value F best during K successive generations as follows:

6 350 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY 2005 k+k C = k (F (k+1) best F (k) best ) = 0 during K successive generations (16) Meanwhile, to finish the optimization procedure, the desired low SLL can be also used to the Step 8 Although we cannot expect the lowest MSLL over a given array size in advance, we can roughly set a desired low SLL before the optimization is implemented Therefore, if the best fitness value F best satisfies the desired low SLL in the process of the optimization, the algorithm is stopped In addition, although the best fitness value F best does not satisfy the predetermined low SLL, the algorithm is terminated when the condition in Eq (16) is achieved From many simulation results, in order to guarantee both a robust performance for the MSLL reduction and computational efficiency, approximately, the value of K over five times the array size (=2M) was needed for the pattern synthesis algorithm 6 Simulation Results In this section, we will show a few examples for the purpose of demonstrating the performance of the linear and planar array pattern synthesis methods in Sects Linear Array As the first example, suppose we have a 13-element linear array of isotropic elements spaced every half-wavelength First, this uniformly spaced linear array (USLA) is synthesized to reduce the SLL using the Fourier transform based formula described in [9] The resulting USLA has the array pattern shown in Fig 3 In comparison with the USLA, an amount of about 5 db reduction of the first SLL is achieved with the USLA However, when the main beam of USLA is steered to 30, some large outer side-lobes greater than the first sidelobe are observed within the visible region as shown in Case 1 of Fig 4 Furthermore, their levels in the vicinity of an angle of 60 are also higher than the first SLL of the USLA in Case 3 of Fig 4 In order to reduce the undesirable large SLLs shown above, we now apply the proposed algorithm to pattern synthesis of the scannable USLA with beam steering For the pattern synthesis algorithm, we set the iteration gain to η = 15exp( 0225 k), the maximum allowable error, γ to 10 4,andδ 2 to 0001, respectively The range of π z π with 004π steps corresponding to the observation angle ( 90 θ 90 ) was used for the optimization, resulting in L=51 Considering convergence time and performances, approximately, L was chosen as 4 times the array size for the optimization For the desired reference pattern, the first and second side-lobes with respect to the main lobe positioned at θ 0 = 30 were set up to 25 db and the other SLLs were reduced to 05 times the outer SLLs of the original pattern The resulting radiation pattern is shown in Case 2 of Fig 4 As shown in this figure, the proposed algorithm causes the undesirable large side-lobes near 60 to be significantly reduced They have even smaller levels than the first SLLs of Case 1 and Case 3 As a second example, we consider an equally excited 17-element linear array with the same uniform spacing and scanning angle, as in the previous example For this example, we set the iteration gain to η = 145 exp( 025 k), the maximum allowable error, γ to10 4,andδ 2 to 0001, respectively In case of the 17-element USLA, the range of π z π with 003π increments was considered for this optimization, resulting in L=67 The desired reference pattern was composed of three inner side-lobes of 27 db and the other outer side-lobes which were 05 times the outer ones of the original 17-element USLA The resulting optimized positions of the USLA are given in Table 1 and the beam pattern for the positions of Table 1 is plotted in Fig 5 Case 1 represents the beam pattern of the USLA with the Fourier transform based formula, Case 2 represents the optimized USLA with the proposed technique, and Case 3 that of the USLA, respectively It is shown in Fig 5 that the Fig 3 Radiation pattern of non-uniform linear array using the Fourier transform based formula: the maximum radiation angle is 0 Fig 4 Comarison of the optimized and the non-optimized radiation patterns for the 13-element linear array when the maximum radiation angle is steered to 30

7 BAE et al: DESIG OF STEERABLE LIEAR AD PLAAR ARRAY GEOMETRY 351 Table 1 Position of the optimized non-uniform array elements Case 2: linear array geometry ( = 13) + non-uniform spacing (Fourier transform based method) Case 3: linear array geometry ( = 13) + non-uniform spacing (proposed method) Case 4: linear array geometry ( = 17) + uniform spacing Case 5: linear array geometry ( = 17) + non-uniform spacing (Fourier transform based method) Case 6: linear array geometry ( = 17) + non-uniform spacing (proposed method) From Table 2, the optimized beam patterns of Case 3 and Case 6, with the proposed method, can maintain low SLLs for the scanning range of 30 θ 0 30, while the non-optimized beam patterns of Case 2 and Case 5 can not In addition, we observe that the main-lobe beamwidth broadening in the optimized arrays is very small compared to that of the USLA As shown in Table 2, it should be pointed out that the optimized USLA geometry with the proposed method can provide low SLLs without pattern distortion over the wide scan angles 62 Planar Array As design examples of an optimal planar array, we consider two planar arrays, namely, USPA and USPA The optimized USPA and USPA structure, using the planar array pattern synthesis method in Sect 5, are presented and are also compared 621 Optimized Planar Array from the Uniformly Spaced Planar Array Fig 5 Comarison of the optimized and the non-optimized radiation patterns for the 17-element linear array when the maximum radiation angle is steered to 30 proposed method can reduce not only the first SLL, butalso pattern distortions in the outer side-lobes, although the main beam direction is steered to 30 The MSLLs and 3 db main-lobe beamwidths of several linear array arrangements are compared and summarized in Table 2, when the main beam direction is steered to the five different angles These linear array arrangements are explained as follows: Case 1: linear array geometry ( = 13) + uniform spacing In our planar array pattern synthesis method, it is necessary to determine a design parameter, R or Q, infig2,inadvance It determines the search region of the GA for the optimal solution In our case, the size of R is equal to that of Q to maintain a symmetric array structure Fig 6 shows the relative amount of MSLL reduction (RMSLLR) for several USPAs, when the ratio of R to M (R/M) is varied over several array sizes The RMSLLR is defined as follows: RMSLLR [db] = MSLL of uniform planar array (ie about 13 db) MSLL of optimized planar array The result of Fig 6 is based on the proposed technique when the USPA is used as an initial array geometry in Step 1, rather than the USPA Each initial USPA was obtained by expanding each USLA with a half-wavelength spacing to each 2-D rectangular lattice As shown in Fig 6, RMSLLRs of all the US- PAs, with various array sizes, are respectively significantly increased as the R/M decreases However, there is no noticeable increase of RMSLLRs, although the R/M decreases to a value smaller than 06 It should be pointed out that, as R/M decreases to a value less than 06, the number of adjustable array elements during the GA optimization increases At the same time, there is no further reduction of MSLL when the R/M is less than 06 In contrast, for R/M values larger than about 06, the MSLL of the designed USPA gradually increases, although the computational complexity can

8 352 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY 2005 Table 2 Comparison of several linear array arrangements be reduced Therefore, it is reasonable to choose R/M as 06, since the proposed method with R/M 06 is efficient in the context of MSLL and computational complexity The value of R/M 06 will also be used to determine the proper boundary of the search region in the next USPA case example Although the R/M =06 is a proper choice in terms of MSLL and computation time, as shown in Fig 7, the optimized USPA with R/M =06 yields an amount of about 3 db reduction of relative power level at boresight, compared to the initial USPA case This is because, through the thinning of the array geometry, the number of array elements in the designed USPA with the proposed method and R/M =06 is about 70 75%, compared to the 100% initial filled USPA geometry For example, we consider a USPA for the optimization of an array geometry A USPA extended by a 13-element USLA is used as an initial array structure The MSLL of the initial USPA is 1308 db In order to design an optimal USPA with low SLL from the initial USPA, the GA parameters were determined as follows: population size for the planar array was set to three times the length of each chromosome, a probability Case 1: RMSLLR vs R/M with the initial USPA Case 2: RMSLLR vs R/M with the initial USPA Case 3: RMSLLR vs R/M with the initial USPA Case 4: RMSLLR vs R/M with the initial USPA Case 5: RMSLLR vs R/M with the initial USPA Case 6: RMSLLR vs R/M with the initial USPA Fig 6 Relative amount of MS LL reduction for several USPAs of crossover to 08 and that of mutation to 0025 for the USPA The main lobe region of 018 u 018 and 018 v 018 corresponding to (θ 0,φ 0 ) = (0, 0 )was

9 BAE et al: DESIG OF STEERABLE LIEAR AD PLAAR ARRAY GEOMETRY 353 Case 1: Relative power vs R/M with the initial USPA Case 2: Relative power vs R/M with the initial USPA Case 3: Relative power vs R/M with the initial USPA Case 4: Relative power vs R/M with the initial USPA Case 5: Relative power vs R/M with the initial USPA Case 6: Relative power vs R/M with the initial USPA Fig 7 R/M versus relative power at boresight (a) Optimized USPA structure considered for the optimization K in step 8 was chosen as 70 to terminate the iterative generations Fig 8 shows the optimized USPA with the MSLL of 1973 db for the value of R = Q = 3 Fig 8(c) shows the side view of the planar array pattern when the main beam is scanned to θ 0 = 25 and φ 0 = 0 In comparison with the initial USPA, the optimized USPA structure can provide more reduction of the MSLL than the initial USPA structure ote that generally there is no occurrence of pattern distortion in the USPA case while the main beam direction is steered, as in [17] (b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sin φ 622 Optimized Planar Array from the on-uniformly Spaced Planar Array To generate an optimal USPA geometry in terms of MSLL and pattern distortion, the optimized 17-element USLA designed by the proposed method in Sect 61 is extended to a rectangular array with non-uniform spacing ext, we further modify the initial USPA to achieve an optimal planar array geometry As stated above, in order to determine the genetic search boundary under the design of the optimal USPA structure, we apply the same result of R/M 06 obtained in Sect 621 to the initial USPA, resulting in R = Q = 5forthe17 17 USPA Population size for each planar array is three times the length of each chromosome A probability of crossover is set to 082 and that of mutation to 002 for the USPA The main lobe region of 036 u 063 and 013 v 013 corresponding to (θ 0,φ 0 ) = (30, 0 ) was considered for the calculation of the cost function F in Eq (14) K was selected as 90 to terminate the optimization Fig 9(a) shows the optimized USPA obtained from the initial USPA Fig 9(b) is a 2-D planar array pattern as a function of u = sin θ cos φ and v = sin θ sin φ for the array lattice given in Fig 9(a) Fig 9(c) is the side view of the array pattern when the main (c) Side view of the radiation pattern when the main beam is scanned to θ 0 = 25 and φ 0 = 0 Fig 8 Optimized USPA for the initial USPA beam is scanned to θ 0 = 25 and φ 0 = 0 The MSLL of 2286 db is achieved in all the side-lobe regions except the main-lobe region for the optimized USPA Meanwhile, Fig 10 shows the non-optimized USPA structure with MSLL of about 174 db and its associated beam patterns This USPA was generated from a 17-element USLA which has been obtained by using the Fourier transform based formula of Eq (11) When the main beam angle

10 354 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY 2005 (a) Optimized USPA structure (a) on-optimized USPA structure (b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sin φ (b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sin φ (c) Side view of the radiation pattern when the main beam is scanned to θ 0 = 25 and φ 0 = 0 Fig 9 Optimized USPA geometry for the initial USPA (c) Side view of the radiation pattern when the main beam is scanned to θ 0 = 25 and φ 0 = 0 Fig 10 on-optimized USPA geometry for the initial USPA is scanned to the same direction as in Fig 9(c), the array structure can not maintain a low SLL in the visible region, leading to a pattern distortion as shown in Fig 10(c) In contrast, the optimized USPA structure can provide low SLL without pattern distortions, although the main beam direction is steered ext, Fig 11(a) shows the optimized USPA geometry when a USPA is used as an initial array structure In this case, the MSLL of 2094 db in all the side-lobe regions is accomplished for the value of R = Q = 5 Comparing the optimized USPA in Fig 9 with the optimized USPA in Fig 11, we observe that the optimized USPA structure can achieve more reduction of MSLL without pattern distortions than the optimized USPA geometry In addition, we observe

11 BAE et al: DESIG OF STEERABLE LIEAR AD PLAAR ARRAY GEOMETRY 355 (a) Radiation pattern in the plane of φ = 0 (a) Optimized USPA structure (b) Radiation pattern in the plane of φ = 90 (b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sin φ (c) Steered radiation pattern in the plane of φ = 0 Fig 12 Comparison of normalized radiation patterns when the optimized USPA was obtained from the initial USPA (c) Side view of the radiation pattern when the main beam is scanned to θ 0 = 25 and φ 0 = 0 Fig 11 Optimized USPA geometry for the initial USPA that the two classes of the optimized arrays, namely USPA and USPA, are quite different in shape and the optimized USPA structure rather than the optimized USPA geometry is very similar to a circular array shape From the above described results of Sects 621 and 622, the MSLLs and 3 db main-lobe beamwidths of several planar array for the five different main beam directions are compared and summarized in Table 3 These planar array arrangements are explained as follows: Case 1: Planar array geometry (17 17) with uniform spacing Case 2: on-optimized planar array geometry (17 17) with non-uniform spacing (Fourier transform based formula) Case 3: Optimized planar array geometry with uniform spacing from initial USPA Case 4: Optimized planar array geometry with nonuniform spacing from initial USPA Case 5: Optimized planar array geometry with nonuniform spacing from initial USPA In Case 4, to obtain an initial USPA, the

12 356 IEICE TRAS COMMU, VOLE88 B, O1 JAUARY 2005 Table 3 Comparison of several planar array arrangements non-optimized 17-element USLA was extended to the 2- D USPA In contrast, the optimized USLA was used to construct an initial USPA, in Case 5 From Table 3, we observe that the optimized beam patterns except Case 2 maintain a low SLL, for the scanning range of 30 θ 0 30, without beam pattern distortion Comparing the results of Case 4 and Case 5, we observe that the proper initial array structure is quite important to obtain efficient and reliable performances In addition, beam broadening of Case 4 is more noticeable than any other cases ext, the optimized USPA structure of Case 5 shows a lower SLL than the optimized USPA of Case 3 Meanwhile, when the whole array without boundary condition (R/M 06) was thinned, the same result was obtained like Case 5, after 89 generation However, the result of Case 5 was driven after only 21 generation The result of considering the whole array without the boundary condition also cost much more time than Case 5 Contrary to the 1-D linear array case, as the number of elements are increased in 2-D planar array, the computational complexity for the optimization can be significantly increased and a fast convergence may not be guaranteed Therefore, it is very important to determine the proper boundary condition for the 2-D array case, although the global optimization technique is applied As a result, it can be indicated that the optimized USPA geometry under proper constraints is superior to the other array structures from the viewpoint of MSLL and pattern distortion Finally, we also simulated the array performance using a commercial full-wave analysis software tool (CST Microwave Studio 42) based on FDTD algorithm, when an initial USPA with R = Q = 3 was considered for the construction of an optimal USPA A wire antenna operating at 2 GHz was used as a radiating element The radius of this element is 46 mm and the physical length, 614 mm The antennas were positioned at the optimized USPA lattice, such that the boresight of the wire antenna was under the direction of z-axis The PML (Perfect Matched Layer) which operates like free space is used as an open boundary Fig 12(a) and Fig 12(b) show the normalized radiation patterns in the plane of φ = 0 and φ = 90, respectively In addition, Fig 12(c) represents the scanned radiation pattern in the plane of φ = 0 when the main beam direction is positioned at θ 0 = 30 and φ 0 = 0 While the result of Case 1 was obtained by using the proposed method, that of Case 2 was acquired by using the full-wave analysis software From the result of Fig 12, we observe that, although there is a slight difference of MSLL between Case 1 and Case 2, the pattern shapes of the two cases are very similar each other, maintaining low SLLs without pattern distortion 7 Conclusion In this paper, noble design schemes for the optimal USLA and USPA in the context of MSLL reduction, has been presented using optimization techniques In the USLA case, the Gauss-ewton method was applied to optimally adjust positions of the initial non-uniform array elements which were obtained by the Fourier transform based formula The results show that the optimized scannable USLA structure designed by the proposed synthesis method can reduce outer SLL as well as inner SLL simultaneously without pattern distortion In the USPA case, on the basis of the proposed USLA, the thinned array theory combined with the genetic algorithm was applied to the design of an optimal USPA geometry, in terms of maximum side-lobe reduction First, the initial USPA is obtained by expanding the above-optimized USLA to the 2-D rectangular array lattice along the row and column directions, respectively ext, some elements in the outer regions of the array are optimally turned off from the initial USPA During the process of genetic optimization, the genetic search boundary for the design of the optimal USPA was experimentally derived considering requirements, MSLL and computational complexity The results show that the optimized USPA ge-

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thinned arrays using genetic algorithms, Proc IEEE OCEAS 94, vol2, pp , 1994 [22] JM Johnson and R-S Yahya, Genetic algorithms in engineering electromagnetics, IEEE Antennas Propag Mag, vol39, no4, pp7 21, Aug 1997 [23] R-S Yahya and E Michielssen, Electromagnetic Optimization by Genetic Algorithms, John Wiley & Sons, 1999 [24] DF Li and ZL Gong, Design of hexagonal planar arrays using genetic algorithms for performance improvement, nd International Conference on Microwave and Millimeter Wave Technology Proceedings, pp , 2000 Ji-Hoon Bae received the BS degree in electronic engineering from Kyungpook ational University in 2000, and the MS degree in electrical and computer engineering from Pohang University of Science and Technology (POSTECH) in 2002 Since 2002, he has been with Electronics and Telecommunications Research Institute (ETRI) as a researcher His research interests include array antenna, radar target imaging, and RFID system Kyung-Tae Kim received the BS, MTS, and PhD degrees in electrical engineering from the Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Kyungbuk, Korea, in 1994, 1996, and 1999, respectively During March 1999 and March 2001, he was with the Electromagnetics Technology Laboratory, POSTECH, as a Research Fellow During April 2001 and February 2002, he was a Research Assistant Professor, Electrical and Computer Engineering Division, POSTECH He joined the Faculty of the Department of Electrical Engineering and Computer Science, Yeungnam University, Kyongsan, Kyungbuk His primary research interests include radar target recognition and imaging, array signal processing, spectral estimation, pattern recognition, neural networks, and RCS measurement and prediction Cheol-Sig Pyo received the BS degree in electronic engineering from Yonsei University in 1991, the MS degree in electrical engineering from Korea Advanced institute of Science and Technology (KAIST) in 1999 Since 1991, he has been a senior engineer at Electronics and Telecommunications Research Institute (ETRI) At present, he is a team leader of RFID technology research team in ETRI His research interests include antenna, radio system, and RFID system