Wigner distribution and the synthesis problem: a Hilbert space approach

This paper is concerned with mixed time-frequency filtering using Wigner distribution (WD). The resulting function of such operation is not necessarily a WD function. In order to reconstruct a time domain signal a synthesis procedure is required. Using functional analysis terminology, a fairly complete description of the range of Wigner distribution (viewed as a nonlinear operator W:L² (R)→𝒲⊂L²(R² )) is given, several important properties of the synthesis operator are investigated, and implications of these results for signal processing are discussed