Discrete Wigner synthesis

An important application of the Wigner distribution (WD) is the synthesis of a discrete-time signal whose WD approximates a specified time-frequency distribution in the minimum mean-square error sense. One approach previously proposed (Yu and Cheng, Proc. ICASSP, 1985, pp. 1037–1040) is based upon the notion of expressing the desired signal in terms of orthonormal functions and applying the resulting induced Wigner distributions for WD synthesis. In this paper, we prove that the induced WDs cannot be orthonormal for any choice of the original orthonormal functions. We also prove that the induced WDs can be orthogonal only for particular original basis functions. We utilize these results to introduce a rigorous approach for discrete WD synthesis. We then illustrate our approach through analytical and computational examples.