8.2 Discrete Wigner Transform

Amongst the myriad of time-frequency distributions, the Wigner distribution stands alone in satisfying many desirable mathematical properties [OFW99]. The continuous and aperiodic Wigner function is (i) a quadratic functional, (ii) is covariant to time shifts and frequency shifts, (iii) satisfies Moyal's formula, (iv) is compatible with modulations, (v) is compatible with filtering, and (vi) is real.

While there has been much success in the development of a theory for continuous distributions, attempts to extend the definition of the Wigner distribution to the discrete case have not been completely successful. An ideal approach for computing a discrete distribution would preserve the properties present in the continuous distribution. Depending on whether the function is continuous and periodic there are four different types of Fourier transform and hence arguably four kinds of Wigner transform.

In our applications we have a wave function $\psi$ on the (infinite) real line. If we discretize it this corresponds to the discrete and aperiodic case. It is proved in [OFW99] and in [OW98] that in this case there exists no definition of the discrete Wigner transform which satisfies all the desirable properties mentioned above. The problem of defining a ``good" discrete Wigner transform is further discussed in [RPS98]. We refer the reader to these literature references for an exact formulation concerning these ``no go'' theorems. Good solutions only exist for distributions (and operators) which are periodic both in $x$ and $k$. While in our application case periodicity of the Wigner distribution in $k$ is not too unreasonable, periodicity in $x$ can not be assumed if a bias is applied.

In the literature on Wigner function method simulation authors propose various definitions of a discrete Wigner transform. Starting from the discrete Schrödinger equation they derive different discretizations of the Wigner equation. Each of these definitions has its shortcomings. And the cited results suggests that the search for a ``good'' definition of discrete Wigner transform is in vain.