A SIGNAL-DEPENDENT TIME-FREQUENCY REPRESENTATION:
  FAST ALGORITHM FOR OPTIMAL KERNEL DESIGN


Richard G. Baraniuk 

  Department of Electrical and Computer Engineering
  Rice University
  Houston, TX 77251-1892

Douglas L. Jones 
  Department of Electrical and Computer Engineering
  University of Illinois
  Urbana, IL 61801


A time-frequency representation based on an optimal, signal-dependent
kernel has been proposed recently 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.  In this paper, we
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 logQ),
which is several orders of magnitude less than any other known method
for solving this class of linear programs.  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  1  and  0.