Generalized Gaussian quadrature applied to an inverse problem in antenna theory: II. The two-dimensional case with circular symmetry

We investigate the linear inverse problem of determining the weighting function for a disc source antenna from its directivity pattern. The relationship between the two is a finite two-dimensional Fourier transform and we use its singular functions to expand the directivity pattern and the weighting function. These singular functions are closely related to the generalized prolate spheroidal functions. The inverse problem is regularized by controlling the superdirective ratio. We show how the weighting function may be discretized so that its support is a set of narrow concentric rings. To do this we prove that the appropriate generalized prolate spheroidal functions form a Tchebycheff system. We then use generalized Gaussian quadrature to carry out the discretization, with the Gaussian nodes and weights corresponding to the radii of the rings and the weights to be attached to the rings. We obtain accurate approximations to directivity patterns corresponding to various weighting functions. For reasons of simplicity and space we restrict the discussion to broadside sources and arrays, broadside being the direction normal to the plane of the source or array. We discuss superdirectivity both for disc sources and concentric ring arrays and we consider the relationship between our work and earlier work of Toraldo di Francia on apodization in optics.