Relationships between two definitions of the discrete Wigner distribution and the continuous Wigner distribution

Abstract

We present a very simple relationship between two widely used discrete-time discrete-frequency Wigner distributions. The first one is obtained through sampling and the second one is obtained from the representation theory of the finite Heisenberg group. This relation shows that the values of one can simply be obtained by permuting the values of the other along the frequency axis, which in turn implies a relationship of the second definition to the samples of the continuous Wigner distribution, and the first definition to group representation theory. In the process, we derive a simplified form for the second definition which is completely analogous to the continuous Wigner distribution, and develop a set of relationships relating this definition to a discrete ambiguity function and auxiliary functions.