A SIGNAL-DEPENDENT TIME-FREQUENCY REPRESENTATION:
  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


Time-frequency distributions (TFDs), which indicate the energy content
of a signal as a function of both time and frequency, are powerful
tools for time-varying signal analysis.  The lack of a single
distribution that is ``best'' for all applications has resulted in a
proliferation of TFDs, each corresponding to a different, fixed
mapping from signals to the time-frequency plane.  A major drawback of
all fixed mappings is that, for each mapping, the resulting time-
frequency representation is satisfactory only for a limited class of
signals.  In this paper, we introduce a new TFD that adapts to each
signal and so offers good performance for a large class of signals.
The design of the {\it signal-dependent} TFD is formulated in Cohen's
class as an optimization problem and results in a special linear
program.  Given a signal to be analyzed, the solution to the linear
program yields the optimal kernel and, hence, the optimal time-
frequency mapping for that signal.  A fast algorithm has been
developed for solving the linear program, allowing the computation of
the signal-dependent TFD with a time-complexity on the same order as a
fixed-kernel distribution.  Besides this computational efficiency, an
attractive feature of the optimization-based approach is the ease with
which the formulation can be customized to incorporate application-
specific knowledge into the design process.