An eigen space approach for the synthesis of linear and circular arrays

Abstract

An eigen space non-iterative technique for the synthesis of both linear and circular arrays is introduced in this paper. The technique is applicable in case of equispaced linear arrays with equal phase and symmetric excitation coefficients and circular arrays with quadrant symmetry with number of elements divisible by 4. The technique is based on formulating the synthesis process as a least-squared optimization problem. Some design examples are presented to illustrate the potential and the quality of the considered technique.L INTRODUCTION The problem of antenna array synthesis is to find the antenna configuration in addition to the determination of the elements' excitation to achieve the desired radiation patterm. The classical techniques of linear arrays synthesis include the Wood-Ward sampling method [1I where the desired radiation pattern is sampled at specified points to achieve the synthesized pattern and the Fourier method [11 in which a given pattem is approximated by a Fourier partial sum. Other techniques are also exist such as the iterative sampling method [21 and iterative optimization method [3]. This paper introduces a new approach to both linear and circular arrays synthesis. The proposed method is a least-squares optimization problem and is basically based on the computation of the array excitation as the eigenvector that corresponds to the smallest eigenvalue of an appropriate real, symmetric and positive-definite matrix. This technique follows that used in the design of linear-phased finite impulse response (FIR) digital filters [4]. IIL APPROACH AND PROBLEM FORMULATIONStarting with a desired radiation pattem D(o) and the synthesized array factor AF(O), the approach is to minimize the error functional given by: E = Ja(6)ID(0) -AF(0)I2d9 (1) where a(o) is a positive weighting factors that adjust the relative accuracy in the different sub-regions. The resulting error is small in an average sense and not at particular locations on the antenna radiation pattern. Equation (1) can be rewritten in quadratic matrix form as follows:(2) where a and at are a real vector and its transpose that represent the synthesized excitation coefficient and p is a real, symmetric and positive -definite matrix whose elements depend on the desired pattern. Thus, following the Rayleigh principle [61, the eigenvector that corresponds to the least eigenvalue of matrix p minimizes E, then the elements of this eigen vector yield the required excitation coefficients.a-Droblem formulation for non-uniform linear arrays with symmetric feeding.Considering a linear array of N equally spaced elements with non-uniform symmetric excitations, the array factor for even and odd number of elements can be expressed as [1]: 0-7803-8883