Discrete Analogs of Wigner Transforms and Weyl Transforms

Abstract

We first introduce the discrete Fourier–Wigner transform and the discrete Wigner transform acting on functions in $L^2({\mathbb Z})$. We prove that properties of the standard Wigner transform of functions in $L^2({\mathbb R}^n)$ such as the Moyal identity, the inversion formula, time-frequency marginal conditions, and the resolution formula hold for the Wigner transforms of functions in $L^2({\mathbb Z})$. Using the discrete Wigner transform, we define the discrete Weyl transform corresponding to a suitable symbol on ${\mathbb Z}\times {\mathbb S}^1$. We give a necessary and sufficient condition for the self-adjointness of the discrete Weyl transform. Moreover, we give a necessary and sufficient condition for a discrete Weyl transform to be a Hilbert–Schmidt operator. Then we show how we can reconstruct the symbol from its corresponding Weyl transform. We prove that the product of two Weyl transforms is again a Weyl transform and an explicit formula for the symbol of the product of two Weyl transforms is given. This result gives a necessary and sufficient condition for the Weyl transform to be in the trace class.