The development of a discrete transform for the Wigner distribution and ambiguity function

A theoretical basis for the development of the discrete Wigner distribution and ambiguity function is presented that is based on the temporal and spectral properties of the two-dimensional Wigner kernel function. This two-dimensional description is helpful in visualizing the various functions and shows that the increase in the number of points required to compute the discrete Wigner distribution is a natural consequence of the definition of the kernel, and that it does not contradict the Nyquist sampling theorem. A generalized set of pseudofunctions is introduced that involves applying a two-dimensional weighting function to the Wigner kernel or its two-dimensional Fourier transform, resulting in a Wigner distribution that is weighted or smoothed in time or frequency. Two forms of the Wigner distribution are derived. The first is a pure Wigner distribution without any windowing of the kernel that may be computed over the entire data set. The second involves the general set of pseudo-Wigner distribution that may be calculated over a subset of the data set. The discrete Wigner distribution of the analytic signal is also developed, and it is shown that the number of points may be reduced by a factor of 2, in the same way that the number of points required to represent the discrete analytic signal may be reduced. The discrete ambiguity function is developed along with the discrete Wigner distribution and it is shown that the four two-dimensional functions are interrelated by discrete one- and two-dimensional Fourier transforms.