Equivalence of Generalized Joint Signal Representations of Arbitrary Variables

Abstract

Joint signal representations (JSRs) of arbitrary variables generalize time-frequency representations (TFRs) to a much broader class of nonstationary signal characteristics. Two main distributional approaches to JSRs of arbitrary variables have been proposed by Cohen and Baraniuk. Cohen's method is a direct extension of his original formulation of TFRs, and Baraniuk's approach is based on a group theoretic formulation; both use the powerful concept of associating variables with operators. One of the main results of the paper is that despite their apparent differences, the two approaches to generalized JSRs are completely equivalent. Remarkably, the JSRs of the two methods are simply related via <i>axis warping</i> transformations, with the broad implication that JSRs with radically different covariance properties can be generated efficiently from JSRs of Cohen's method via simple pre- and post-processing. The development in this paper, illustrated with examples, also illuminates other related issues in the theory of generalized JSRs. In particular, we derive an explicit relationship between the Hermitian operators in Cohen's method and the unitary operators in Baraniuk's approach, thereby establishing the relationship between the two types of operator correspondences.