Equivalence of generalized joint signal representations of arbitrary variables

Joint signal representations (JSR's) of arbitrary variables generalize time-frequency representations (TFR's) to a much broader class of nonstationary signal characteristics. Two main distributional approaches to JSR's of arbitrary variables have been proposed by Cohen and Baraniuk. Cohen's method is a direct extension of his original formulation of TFR's 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 JSR's are completely equivalent. Remarkably the JSR's of the two methods are simply related via axis warping transformations with the broad implication that JSR's with radically different covariance properties can be generated efficiently from JSR's of Cohen's method via simple pre- and post-processing. The development in this paper which is illustrated with examples also illuminates other related issues in the theory of generalized JSR's. 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.