Wigner Distribution decomposition and cross-terms deleted representation

Abstract

In this paper, we represent the Wigner Distribution (WD) of an arbitrary signal, via the Gabor expansion, in terms of a linear combination of elementary WDs, which can be easily partitioned into two subsets: auto WDs and cross WDs. The Gabor coefficients, C m, n , for this decomposition are obtained with a Gaussian-shaped synthesis function. The optimally concentrated auto WDs are non-negative and entirely free of cross-terms; the sum of these auto WDs we call the cross-terms deleted representation (CDR). The sum of the cross WDs is an oscillating function with non-zero energy in general; it can be removed and returned depending on the user's needs. Such a decomposition illustrates and isolates the mechanism of WD negative values and cross-term interference. Moreover, new information is provided to facilitate the design of valid joint time-frequency signal representations and time-varying filters. Also in this paper, analogous, yet more practical, results are shown for the Discrete Wigner Distribution (DWD) for finite or periodic discrete-time signals. Examples are presented to demonstrate the CDR technique and its performance in comparison with other joint time-frequency distributions. It is shown that the CDR has the high energy concentration of the WD without the interference problems that occur in many other approaches. Moreover, because only the Gabor coefficients, C m, n , need be computed on-line, the CDR is suitable for on-line implementation.