Generalized Gaussian quadrature applied to an inverse problem in antenna theory

We discuss the linear inverse problem of determining the current distribution (weighting function) for a line source antenna from its directivity pattern. The relationship between the two is a finite Fourier transform and we use its singular functions to expand the directivity pattern. These singular functions are the prolate spheroidal wavefunctions (PSWFs). We point out that the truncation of this expansion, advocated by Rhodes to control the superdirective ratio, is a way of regularizing the inverse problem. We discuss the problem of trying to improve the resolution by narrowing the main lobe of the directivity pattern. Given that the line source is a mathematical abstraction, in order to make a physical device with the given directivity pattern one needs to discretize the line source. To do this we show that the PSWFs 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 for the equivalent discrete array. We show how superdirective arrays fit into the framework discussed in this paper and finally we look at a couple of discrete arrays for which the beam pattern has a narrowed main lobe.