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International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 316962, 9 pages

http://dx.doi.org/10.1155/2015/316962

## An Improved Antenna Array Pattern Synthesis Method Using Fast Fourier Transforms

^{1}Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China^{2}University of Chinese Academy of Sciences, Beijing 100190, China

Received 21 April 2014; Revised 18 August 2014; Accepted 22 August 2014

Academic Editor: Michelangelo Villano

Copyright © 2015 Xucun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An improved antenna array pattern synthesis method using fast Fourier transform is proposed, which can be effectively applied to the synthesis of large planar arrays with periodic structure. Theoretical and simulative analyses show that the original FFT method has a low convergence rate and the converged solution can hardly fully meet the requirements of the desired pattern. A scaling factor is introduced to the original method. By choosing a proper value for the scaling factor, the convergence rate can be greatly improved and the final solution is able to fully meet the expectations. Simulation results are given to demonstrate the effectiveness of the proposed algorithm.

#### 1. Introduction

In order to solve complex antenna pattern synthesis problems, various methods using various optimization algorithms have been developed. In [1], a quadratic program is formed for arbitrary array pattern synthesis. In [2], a convex optimization problem [3] is formulated for pattern synthesis subject to arbitrary upper bounds. For certain cases, the convex programming problem can be reduced to a linear programming problem [4]. In [5] an effective hybrid optimization method is proposed for footprint pattern synthesis of very large planar antenna arrays [6]. The methods mentioned above all adopt conventional optimization algorithms [7]. Global optimization algorithms such as genetic algorithms [8–10] and particle swarm optimization algorithms [11–13] have also been successfully applied in pattern synthesis problems. To approximate the desired pattern for an array, we need to discretize the angular space. The bigger the elements number is, the greater the required discrete density needs to be. Usually, the computational complexity would grow greatly as the elements number increases. As a consequence, normal synthesis techniques using local or global optimization algorithms are usually not suitable for large planar arrays.

As we know, fast Fourier transforms (FFT) are able to quickly compute the radiation pattern of an array with periodic structure. Once the number of FFT points is specified, the computation time is barely affected by the element number. In [14], an FFT method suitable for large planar arrays is proposed. The operation is very straightforward, which mainly involves direct and inverse fast Fourier transforms. In [15], a modified iterative FFT technique is proposed for leaky-wave antenna pattern synthesis. In [16], FFT is used for the pattern synthesis of nonuniform antenna arrays. The iterative FFT method is very efficient as shown in [14], and many examples are presented, but why the method is effective has not been explained.

In this paper, both theoretical and simulative analyses of the FFT method are presented. It is found that, though effective, it is a slow-convergence method and can hardly converge to the optimum solution. Based on the analyses, we introduce a scaling factor and the performance can be greatly improved.

#### 2. Original FFT Method for Pattern Synthesis

First, we establish the relationship between a certain point in the FFT result and the corresponding angle of the radiation pattern. Consider a planar array with elements arranging in a rectangular grid and spacing and between rows and columns. Assume the element pattern is isotropic. The array factor is given bywhere is the complex excitation of the th element and is the wavelength. If the coordinates are used, the array pattern can be written asPerforming points 2D inverse fast Fourier transform (IFFT) on the excitations, we getIf we want to represent the array factor using , then the coordinates and are related bywhere and are integers, making sure both sides of the equations have the same value range. Suppose that both and are even numbers. DefineCombining (4) and (5) and considering the value ranges of and , we haveSo, the relation between the array factor and the IFFT of the array excitation isFinally, the visible space is given byNote that the indices change in (5) is in fact the fftshift operation in Matlab.

Once the corresponding relationship is established, the procedure of the FFT method for pattern synthesis is given as follows.(1)Specify dimension of the array , the initial excitation , the FFT points , and the maximum iteration times .(2)Perform IFFT on the excitation of the th iteration and obtain the array factor .(3)Extract the amplitude and phase of .(4)Compare with the desired pattern . If the computed pattern fully meets the requirements or the maximum iteration time is reached, terminate the procedure; otherwise go to the next step.(5)Obtain the new pattern by replacing the undesired with as follows: where is a set containing all the points where the pattern is undesired. For example, if is a point within the side lobe region and , then it means that the side lobe level exceeds the desired level and that .(6)Perform points 2-D FFT on and choose the first points as the initial excitation for the next iteration . Constraints can be easily made for amplitude-only or phase-only synthesis.(7)Go to step .

The procedure is also illustrated in Figure 1.