Uni-variable cross-spectral densities

Abstract

We introduce a novel class of planar, scalar sources whose cross-spectral density is a function of a single complex variable. For a typical pair of source points, the modulus and argument of such a variable equal the product of their radial coordinates and the difference of their angular coordinates, respectively. As functions of a single complex variable, the corresponding cross-spectral densities (CSD) are expandable in power series in their convergence domain, and a virtually infinite number of source models can be devised. All such sources are shown to have vortex fields as modes. The closed analytical form of these uni-variable CSDs makes it possible to evaluate in a simple way the significant quantities characterizing the sources and the ones of the fields they radiate. The basic tool leading to the definition of the uni-variable CSDs are the reproducing-kernel Hilbert spaces. As an example, the CSD derived from the so called Szegö kernel is studied in some detail, and its features are derived, together with of those of the radiated field in the far zone.