Element positioning for linear arrays using generalized Gaussian quadrature

We present a novel approach for designing small narrowband linear arrays with unequally spaced elements. Our approach involves discretising a line source with a desirable beam pattern and relies on the fact that beam patterns are bandlimited. We show that, given a nonnegative weighting function for a line source, one can design a linear array of elements whose beam pattern approximates that of the line source to any desired degree of accuracy. We make use of the prolate spheroidal wave functions to analyze line-source beam patterns since they are eigenfunctions of the finite Fourier transform. These functions form a Tchebycheff system, which allows us to calculate exactly integrals involving these functions via a generalized Gaussian quadrature rule. This quadrature rule then gives the appropriate set of element positions and weights. We give results for several weighting functions and we consider both broadside and electronically scanned linear arrays. As an example we show how a broadside array with half-wavelength spacing and Dolph-Tchebycheff weights may be improved upon using the method described.