A signal-dependent time-frequency representation: fast algorithm for optimal kernel design

A time-frequency representation based on an optimal, signal-dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time-frequency distributions: that the best kernel and distribution depend on the signal to be analyzed. The optimization formulation for the signal-dependent kernel results in a linear program with a unique feature: a tree structure that summarizes a set of constraints on the kernel. The authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest. For a kernel with Q variables, the running time of the algorithm is O(Q log Q), which is several orders of magnitude less than any other known method for solving this class of linear program. This efficiency enables the computation of the signal-dependent, optimal-kernel time-frequency representation at a cost that is on the same order as a fixed-kernel distribution. An important property of the optimal kernel is that it takes on essentially only the values of 1 and 0