On the equivalence of generalized joint signal representations

Joint time-frequency representations have proven very useful for analyzing signals in terms of time-frequency content. In an attempt to tailor joint representations to a richer class of signals, two approaches (Cohen´s (1991) and Baraniuk´s (see Proc. ICASSP 1993, p.III-320, 1993)) to obtaining joint representations of arbitrary variables have been proposed. Baraniuk´s generalization appears broader than Cohen´s, since the latter can be recovered from the former as a special case. One of the main results of this paper is that, despite being apparently quite different, the two approaches to generalized joint representations are exactly equivalent. We explicitly characterize the mapping which relates the representations of the two methods, and also determine the corresponding relationship between the operators of the two methods. A practical implication of the results is that one can avoid the group transforms in Baraniuk´s approach, which may not be computationally efficient, by replacing them with Fourier transforms in Cohen´s method