7.1 Introduction

In this chapter we consider how the rules and pairs technique can be applied to relate aperture distributions to antenna beam patterns, particularly for antennas made up of linear arrays of similar elements. We start by showing there is a Fourier transform relationship between linear aperture distributions and beam patterns. This relationship holds in general, including for continuous distributions, but we subsequently restrict our study here to multielement arrays, which are in effect discretely sampled apertures. In the case of uniform, or regular (evenly spaced), linear arrays, the aperture distribution is of the form of a comb function, which has a rep function as its transform. For this kind of array, the rules and pairs method works well and is easy to apply for suitable problems; examples are given where, in one case, a simple beam is required (with further study of variations with low side-lobe patterns) and in another case a beam covering a sector at a uniform gain level is generated.

If the array elements are not uniformly distributed, then the convenience of the comb/rep transform is not applicable and a more general least squares error solution is required. As the Fourier transform is a least squares error solution also, this general approach, if applied in the uniform array case, would result in the same solution, if not quite so directly achieved. This approach still requires Fourier transforms and is presented in Section 7.4. Although we cannot use the Fourier transform of the aperture in the...