Generalized Cross-Correlation, Part II: Discretization of Generalized Cross-Correlation and Progress to Date in Its Implementation

Abstract

The companion article, “Generalised Cross-Correlation Functions, Part I,” introduced a generalization of cross-correlation in which the constant shift used in traditional cross-correlation is replaced by a function of time or space. This allows correlation to be applied globally to the whole domain of interest avoiding the need to compromise spatial resolution with statistical reliability. The development in Part I was entirely in terms of continuous variables. This article examines the issues that arise when Generalized Cross-Correlation is applied to discrete variable situations. Topics discussed include sampling rate requirements, noise rejection, and efficient approximate algorithms, with special attention being paid to the condition number for the system.