Processing of defective ring data by “super-interpolators”

Abstract

We develop an analytical-numerical scheme to make up for full or partial sensor failures in a ring array. The technique’s goal is to restore missing information to a recorded signal in the form of a Hermitian covariance matrix for an arbitrary dependent variable. Our analysis breaks the problem down into two: first, by considering the special case of an imperfect array in which the positions and degrees of failure of the flawed sensors are known, and, second, by generalizing that approach to one in which neither piece of information is presumed nor in fact is the knowledge of how many sensors are affected required. An important application of the first part is the retrieval of the perfect signal at equispaced points that have been left un-instrumented for reasons of economy or lack of physical access (a missing sensor becomes equivalent to one with a known null gain). The problem’s more general second part has led to a fundamental relationship for a periodic array’s total number of sensors, the bandwidth of the ideal signal being restored, and the rank of an integral equation developed from one of Fourier series’ dual statements of orthogonality. We call the products of our two-tier technique “super-interpolators” even though neither engages in that activity mathematically.