KERNEL SYNTHESIS FOR GENERALIZED TIME-FREQUENCY DISTRIBUTIONS USING THE METHOD OF ALTERNATING PROJECTIONS ONTO CONVEX-SETS

Cohen's generalized time-frequency distribution (GTFR) requires the ch oice of a two-dimensional kernel. The kernel directly affects many per formance attributes of the GTFR such as time resolution, frequency res olution, realness, and conformity to time and frequency marginals. A n umber of different kernels may suffice for a given performance constra int (high-frequency resolution, for example). Interestingly, most sets of kernels satisfying commonly used performance constraints are conve x. In this paper, we describe a method whereby kernels can be designed that satisfy two or more of these constraints. If there exists a none mpty intersection among the constraint sets, then the theory of altern ating projection onto convex sets (POCS) guarantees convergence to a k ernel that satisfies all of the constraints. If the constraints can be partitioned into two sets, each with a nonempty intersection, then PO CS guarantees convergence to a kernel that satisfies the inconsistent constraints with minimum mean-square error. We apply kernels synthesiz ed using POCS to the generation of some example GTFR's, and compare th eir performance to the spectrogram, Wigner distribution, and cone kern el GTFR.