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 choice of a two-dimensional kernel. The kernel directly affects many performance attributes of the GTFR such as time resolution, frequency resolution, realness, and conformity to time and frequency marginals. A number of different kernels may suffice for a given performance constraint (high-frequency resolution, for example). Interestingly, most sets of kernels satisfying commonly used performance constraints are convex. We describe a method whereby kernels can be designed that satisfy two or more of these constraints. If there exists a nonempty intersection among the constraint sets, then the theory of alternating projection onto convex sets (POCS) guarantees, convergence to a kernel that satisfies all of the constraints. If the constraints can be partitioned into two sets, each with a nonempty intersection, then POCS guarantees convergence to a kernel that satisfies the inconsistent constraints with minimum mean-square error. We apply kernels synthesized using POCS to the generation of some example GTFRs, and compare their performance to the spectrogram, Wigner distribution, and cone kernel GTFR