Kernel Synthesis for Generalized Time-Frequency Distributions Using the Method of Projections Onto Convex Sets

The kernel in Cohen's generalized time-frequency representation (GTFR) requires is chosen in accordance to certain desired per-formance attributes. Properties of the kernel are typically expressed as constraints. We es-tablish that many commonly used constraints are convex in the sense that all allowable ker-nels satisfying a given constraint form a convex set. Thus, for a given set of constraints, the kernel can be designed by alternately project-ing among these sets. If there exists a non-empty intersection among the constraint sets, then the theory of projection onto convex sets (POCS) guarantees convergence to a point in the intersection. If the constraints can be par-titioned into two sets, each with a nonempty intersection, then POCS guarantees conver-gence to a kernel that satisfies the inconsistent constraints with minimum mean square error.