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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 10, 2011
Optimum Design of Linear Antenna Arrays Using a Hybrid MoM/GA Algorithm A. H. Hussein, H. H. Abdullah, A. M. Salem, Member, IEEE, S. Khamis, and M. Nasr
Abstract—Synthesis of arbitrarily shaped radiation patterns using linear antenna arrays has a significant importance in many applications. Many attempts based on analytical schemes are exerted for this purpose. However, these analytical methods are developed for specific problems, and usually synthesis of the radiation pattern is subject to only one restriction. On the other hand, optimization algorithms are utilized for more general problems. However, these algorithms require more computational time. In this letter, a new hybrid technique for synthesizing arbitrary-shaped radiation pattern using a linear array is developed. The algorithm is based on a combination between the method of moments (MoM) and the genetic algorithm (GA). The proposed algorithm is applied to synthesis of both symmetric and asymmetric radiation pattern distributions with minimum number of elements. Excellent agreement is obtained by comparison to other analytical and optimization techniques. Index Terms—Genetic algorithm (GA), linear array, method of moments (MoM).
I. INTRODUCTION
T
HE REDUCTION of the number of antenna array elements has a significant importance in different applications such as radar systems, satellite communications, mobile communications, and the GEO atmospheric sounder (GAS) instrument [1]. Many research efforts attempted to reduce the number of elements by introducing nonuniform spacing between the antenna array elements [2]–[8]. A noniterative algorithm based on the matrix pencil method (MPM) was introduced in [2]–[4]. However, the MPM introduces an ill-conditioned matrix that needs special treatments such as the use of the singular value decomposition method (SVD). Furthermore, the application of this method to synthesize shaped-beam patterns may encounter a problem such that the estimated poles usually do not lie on the unit circle. This results in unrealized imaginary parts of the element positions [2]. This problem is partially solved using the forward–backword matrix pencil method (FBMPM) [3]. Therefore, the synthesis problem still needs more effort to overcome the aforementioned obstacles. On the other hand, new evolutionary algorithms based on the optimization techniques are used successfully to solve typically
Manuscript received October 06, 2011; revised October 13, 2011; accepted October 24, 2011. Date of publication October 31, 2011; date of current version November 14, 2011. A. H. Hussein, S. Khamis, and M. Nasr are with the Faculty of Engineering, Tanta University, Tanta 1600, Egypt (e-mail:
[email protected]). H. H. Abdullah and A. M. Salem are with the Electronics Research Institute, Giza 12622, Egypt (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2011.2174189
complicated radiation pattern synthesis problems. These algorithms include vector tabu search [8], simulated annealing [9], genetic [10], particle swarm optimization [11], and differential evolution algorithms [12]. The common step of these optimization techniques is based on finding the solution of many unknowns such as the excitation amplitudes, the phases, and the location of each element. These optimization techniques provide good results, but at the expense of the computational time, where thousands of iterations may be required to obtain an optimum solution [8]. In this letter, a new algorithm based on a combination between the method of moments (MoM) [13] and the genetic algorithm (GA) [14], [15] is introduced. The proposed algorithm provides a number of elements reduction using either uniform or nonuniform element spacing. The MoM provides a deterministic solution for the excitation coefficients. On the other hand, the GA is used to estimate the optimum element locations to obtain the required radiation pattern within a minimum tolerance. Unlike the MPM, the MoM provides a well-conditioned matrix that does not require any specific treatment to be solved. In this case, the developed matrix is solved using any of the conventional methods such as Gaussian elimination methods or conjugate gradient methods [16]. Furthermore, the MoM solution provides excitation coefficients with reduced dynamic range ratio (DRR) [17]. On the other hand, GA is based on a random search method that is robust and capable of solving complicated and nonlinear search problems. GA is also characterized by not being limited by restrictive assumptions about the search space [6]. The proposed algorithm is directly applied to synthesis of both pencil-beam patterns and shaped-beam patterns through a few tens or hundreds of iterations. II. PROBLEM FORMULATION In this section, the number of elements reduction for linear antenna arrays with uniform and nonuniform element spacing is presented. The array factor of a linear antenna array consisting of isotropic elements located at along the -axis as shown in Fig. 1 is given by (1) where is the synthesized array factor, is the desired array factor, is the excitation coefficient of the th is the free-space wavenumber [18]. Let element, and denote the adjacent element spacing, and the position of the first element is at the origin . Then, the array elements positions are given by
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(2)
HUSSEIN et al.: OPTIMUM DESIGN OF LINEAR ANTENNA ARRAYS USING HYBRID MoM/GA ALGORITHM
and the elements of the vector
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are given by (9)
The excitation coefficients linear system of (3), where trix , and
are determined by solving the are the elements of the ma.
III. ESTIMATION OF THE MINIMUM NUMBER ANTENNA ELEMENTS Fig. 1. Geometry of a linear array of
elements positioned along the -axis.
The unknown excitation coefficients of (1) are presented in a matrix form using the MoM.The MoM refers to a family of numerical methods for solving integral equations [13]. The MoM transforms the problem to the form (3) The synthesized excitation coefficients are determined by solving (3) for . The first step in the MoM is to assign a basis function that discretizes an integral equation to form a polynomial with unknown coefficients. In our case, we do not need this step as we already have this polynomial represented in (1). The second step is to introduce a testing procedure to generate a system of linear equations that could be solved for the unknown coefficients according to (3). Applying the testing procedure to (1) yields (4) where the inner product in (4) is defined as
OF
In this section, the estimation of the minimum number of elements is introduced. The number of antenna elements decreases as the element spacing increases to maintain approximately the same aperture. To avoid the appearance of grating lobes, the maximum element spacing should not exceed the operating wavelength [18], [19]. Thus, the maximum element spacing for some tolerance is chosen to be . To maintain the same aperture size for an antenna array consisting of equispaced elements, one can write (11) Thus, the minimum number of antenna elements culated as
is cal-
(12) Hence, the initial value of should be chosen to be . If the required accuracy is not met, the number of elements should be increased by one. The process is repeated until the required accuracy is met. IV. ESTIMATION OF THE ARRAY ELEMENT SPACING USING THE GENETIC ALGORITHM
(5) Based on the formulation of (1), it can be shown that an appropriate choice for the weighting functions is
The GA is applied to estimate the optimum element position that introduces minimum least mean square error (LMSE) between the desired and the synthesized patterns. The Cost Function (CF) to be minimized is given by
(6) Substituting (1), (5), and (6) in (4) and rewriting (4) results in
(7) Equation (7) is a set of equations in unknowns. and are the indices of the equation number and the coefficient number, respectively. By casting (7) in the form of (3), the elements of the matrix are given by (8)
(13) where is the number of samples that should be large enough to cover the variations in the desired pattern. and are the desired and synthesized half-power beamwidths, respectively. In this sequence, it is assumed that there is no error in the HPBW. For a specific number of elements , the GA algorithm searches for the optimum that provides a zero error in the HPBW and at the same time introduces a minimum LMSE over the whole pattern. If this condition is not satisfied, the number of elements is increased by one, and the process is repeated until reaching the optimum and the minimum . The GA optimization tool in MATLAB is used to estimate the optimum element spacing within a preassigned range . The corresponding excitation coefficients
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 10, 2011
Fig. 2. Synthesized radiation pattern with 12 uniformly spaced elements comChebyshev array and [2, Fig. 1(b)]. pared to the 20-element
Fig. 3. Synthesized radiation pattern with 12 nonuniformly spaced elements Chebyshev array and [2, Fig. 1(b)]. compared to the 20-element
are determined directly using the MoM procedure. The initial value of is set to , and the GA optimization process is performed until the minimum value of the LMSE is reached.
TABLE I EXCITATION COEFFICIENTS, ELEMENT SPACING, AND HPBWS OF THE SYNTHESIZED 12-ELEMENT ARRAYS, COMPARED TO THE 20-ELEMENT CHEBYSHEV ARRAY
V. RESULTS AND DISCUSSIONS In this section, the proposed algorithm is verified for different types of array factors. The first example is the synthesis of a symmetric Chebyshev pattern. The second example is the synthesis of an asymmetric arbitrary-shaped pattern. Although the proposed algorithm includes an iterative part in the genetic algorithm to determine the spacing between elements, the computation time of this part is not large. In most cases, the required computational time is found to be about 125 ms per iteration. The required number of iterations to satisfy the required cost function lies between 51–400 iterations. These computations are performed with the MATLAB package over a processor of 2-GHz speed and 3 GB of RAM. A. Synthesis of a
Broadside Chebyshev Pattern
A 20-element broadside Chebyshev array with a half-wavelength spacing between elements and sidelobe level 30 dB [20] is used for comparison. By applying (12), the minimum number of elements required for array synthesis is found to be . A satisfactory approximation of the desired pattern is synthesized with 12 elements of uniform element spacing . A good agreement is obtained when it is compared to both the analytical Chebyshev pattern and the MPM as shown in Fig. 2. The optimum spacing between the elements are obtained using only 51 GA iterations. In this case, a 40% reduction in the number of array elements is obtained. The excitation coefficients, the element spacing, and the HPBW of the synthesized pattern are listed in Table I compared to the analytical Chebyshev array. Exact HPBW is obtained. The same example is repeated with nonuniform element spacing. The nonuniformity provides another degree of freedom for the GA to obtain more accurate results as shown in Fig. 3. In this case, an excellent agreement is obtained compared to both the analytical Chebyshev pattern and the MPM. However, this excellent agreement occurs at the expense of the increase in the number of iterations from 51 in the uniform case to 94 in the nonuniform case. Table I shows the corresponding element positions and excitation coefficients for both the uniform and the nonuniform
spacing compared to the analytical Chebyshev solution. Even though the uniform element spacing synthesis provides good accuracy , the nonuniform element spacing synthesis provides better accuracy compared to the MPM. The LMSE approaches 5.6548e-6 using the MoM/GA with the nonuniform spacing, while it reaches 7.8964e-6 using the MPM [2]. In the point of view of the aperture size, the nonuniform element spacing with the high accuracy provides the same aperture size of as in the MPM method, while the uniform element spacing results in a smaller aperture size of . The MPM algorithm has an SLL and a DRR of 29.0437 dB and 3.725, respectively. From Table I, it is noted that the proposed algorithm has a very close SLL and DRR as in the MPM algorithm. B. Synthesis of Asymmetric Shaped-Beam Pattern Another example is the synthesis of an asymmetric arbitrary pattern synthesized by Marcano et al. [6] with 16 equispaced elements. Marcano’s algorithm is based on the GA in estimating the excitation coefficients with half-wavelength element spacing. Our proposed algorithm considers both the
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methods such as Marcano’s GA-based algorithm, MPM, and FBMPM. In MPM and FBMPM, the computation of the roots that correspond to the elements positions and the computation of the excitation coefficient require the more complicated SVD to solve the ill-conditioned matrix problem. On the other hand, the MoM uses a simple matrix inversion algorithm that is simply implemented. Also, the MoM does not suffer from the existence of the complex spacing values that are unrealizable in the case of the appearance of high imaginary parts as in the MPM and the FBMPM algorithms. REFERENCES
Fig. 4. Synthesized radiation pattern with 13 nonuniformly spaced elements compared to the asymmetric shaped-beam patterns in [3] and [6].
TABLE II EXCITATION COEFFICIENTS OF THE SYNTHESIZED ARRAY OF FIG. 4
element spacing and the excitation coefficients. The use of the MoM provides a fast analytical solution for the excitation coefficients. For this reason, our proposed algorithm is not time-consuming. The desired pattern is synthesized with only 13 elements through 400 GA iterations. In this case, an 18.8% reduction in the number of array elements is obtained. The algorithm starts by assigning according to (12). By testing the LMSE, it is found that 13 elements are enough to get excellent accuracy compared to both Marcano’s algorithm and the FBMPM as shown in Fig. 4. The excitation coefficients of the synthesized array are listed in Table II. It should be noted that both the proposed MoM/GA algorithm and the FBMPM provide the same number of elements reduction, but our MoM/GA algorithm provides more accurate results than the FBMPM. The LMSE for the MoM/GA and FBMPM approaches 4.493e-6 and 12.257e-6, respectively. VI. CONCLUSION A new efficient algorithm based on a combination between the method of moments and the genetic algorithm (MoM/GA) is presented. The proposed algorithm provides a robust synthesis tool for both pencil-beam and shaped power patterns. The MoM is used to estimate the excitation coefficients of the antenna array, while the GA is used to estimate the optimum element spacing. The algorithm is efficiently used to minimize the number of the required array elements for the tested radiation patterns. It is also characterized by fast convergence, high accuracy, and less computational complexity compared to other
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