MEASURING TIME-FREQUENCY INFORMATION CONTENT USING THE RENYI ENTROPIES

Richard G. Baraniuk (1), Patrick Flandrin (2), 
  Augustus J. E. M. Janssen (3), and Olivier Michel (2)

  (1) Department of Electrical and Computer Engineering
      Rice University
      P.O. Box 1892, Houston, TX  77251-1892, USA
      E-mail: richb@rice.edu

  (2) Laboratoire de Physique (URA 1325 CNRS)
      Ecole Normale Superieure de Lyon
      46 allee d'Italie, 69364 Lyon Cedex 07, FRANCE
      E-mail: flandrin@physique.ens-lyon.fr, omichel@physique.ens-lyon.fr

  (3) Philips Research Laboratories
      Prof. Holstlaan 4
      5656 AA Eindhoven, The Netherlands
      E-mail: janssena@natlab.research.philips.com

Submitted to IEEE Transactions on Information Theory, August 1998

Abstract:

The generalized entropies of Renyi inspire new measures for estimating
signal information and complexity in the time-frequency plane.  When
applied to a time-frequency representation from Cohen's class or the
affine class, the Renyi entropies conform closely to the notion of
complexity that we use when visually inspecting time-frequency images.
These measures possess several additional interesting and useful
properties, such as accounting and cross-component and transformation
invariances, that make them natural for time-frequency analysis.  This
paper comprises a detailed study of the properties and several
potential applications of the Renyi entropies, with emphasis on the
mathematical foundations for quadratic TFRs.  In particular, for the
Wigner distribution we establish that there exist signals for which
the measures are not well defined.