A general approach for obtaining joint representations for arbitrary physical quantities is presented. The characteristic function operator method of Moyal (1949) and Ville (1960) and generalized by Cohen (1966, 1976) and Scully and Cohen (1987) for the time-frequency case is developed for arbitrary variables.
We compare the methods of Cohen (1971) and of Baraniuk and Jones (see Proc. IEEE ICASSP-93, p.320-323, 1993) for obtaining joint distributions for arbitrary variables. We show that the two methods produce identical results for variables whose associated operators are obtained via a unitary transformation of the time and frequency operators. In addition, we generalize this result and show that all pairs of variables connected by a unitary transformation have joint distributions that are identical in functional form. This result shows how pairs of variables can be grouped into classes whose joint distributions are functionally equivalent and, therefore, share equivalent properties.>
We discuss some issues relevant to time-frequency analysis. Among them are, time-frequency spatial/spatial-frequency representations, generalized marginals and generalized characteristic functions. We derive representations that jointly involve the four variables, time, frequency, space, and spatial frequency. We apply these representations to wave packet motion. For the study on generalized marginals we show that there is an inherent difference between the standard methods of transforming variables and in time- frequency analysis. We use the method of generalized characteristic function operator method to generalize Taylor's theorem.
We discuss the issues that arise in developing quasi-distributions for arbitrary operators in contrast to the usual case of time and frequency. The arbitrary operator case has many mathematical and physical challenges that have not been solved. We also discuss the connection with differential equations and pseudo-differential operators. In regard to differential equations we argue that the proper generalization of the constant coefficient case is not the variable coefficients case but an equation where the coefficients are kept constant and the differential operator is replaced by a Hermitian operator.
Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen's (see Englewood Cliffs, NJ: Prentice-Hall, 1995) general construction for joint distributions of arbitrary variables.