The Wigner distribution function has been found to be useful in the field of signal analysis, where it provides a way to define a time-dependent frequency spectrum (Claasen and Mecklenbräuker, 1980). The notion that a frequency distribution can vary with time is quite intuitive: consider our usual concept of music as a temporal sequence of notes. But it encounters precisely the same problem with respect to the Fourier uncertainty principle that the notion of a position-dependent momentum distribution does with respect to the quantum-mechanical uncertainty principle. Thus, the Wigner distribution may be employed for the same purpose as in quantum mechanics: as a rigorous description which has a simple interpretation in the ``classical'' regime (in this case, for signals whose frequency spectrum changes slowly).
The relevance of this body of work to the present discussion is that digital signal analysis employs discretely sampled signals which are fully analogous to the discrete models discussed in Section 6.1. Many of the mathematical properties (and difficulties) of the discrete Wigner distribution discussed there have already been explored in the context of signal analysis. The purpose of this Appendix is to delineate the parallels between the signal analysis work and the work reviewed in the body of the present paper.
In the signal-analysis problem, one has a function which has been sampled with an interval T so that only the values are known for integral n. The sampled signal corresponds to a Schrödinger wavefunction defined on a spatially discrete basis. The autocorrelation sequence, (or a statistical average of this quantity, Oppenheim and Schafer, 1975, ch. 8), corresponds to the density matrix. The Wigner distribution function , where n represents the time (and corresponds to j) and represents the frequency (and corresponds to p), is obtained from the autocorrelation sequence by a transformation similar to (4.42).
The initial work on the discrete Wigner distribution by Claasen and Mecklenbräuker (1980) used precisely the definition (4.42) (but regarded as a continuous variable). They observed that only one-half of the autocorrelation information is employed in this definition, as illustrated in Fig. 21, and noted that, as a consequence, is periodic with a period of rather than . (The corresponding expression in the present work is .) In a later work, Claasen and Mecklenbräuker (1983) investigated the consequences of modifying the definition of the discrete Wigner distribution by modifying the kernel of the transformation (4.42) to be something more elaborate than just an exponential function. In particular, they weighted the exponential by a factor which is very similar to that which appears in (7.122), used by Mains and Haddad (1988a, 1988c) to weight the potential kernel. Poletti (1988) has further developed this analysis.
If the details of the physical system which produced the signal are unknown, as is usually the case in signal analysis, the analog of the Liouville equation is also unknown. Thus, in this context it is natural to try to resolve the problems of the discrete Wigner function by modifying the expression by which it is defined. This approach is complementary (and quite possibly equivalent) to that explored in Sec. 7 for modifying the Liouville equation.