Uncertainty, information and time-frequency distributions *

The well-known uncertaintly principle is often invoked in signal processing. It is also often considered to have the same implications in signal analysis as does the uncertainty principle in quantum mechanics. The uncertainty principle is often incorrectly interpreted to mean that one cannot locate the time-frequency coordinates of a signal with arbitrarily good precision, since, in quantum mechanics, one can not determine the position and momentum of a particle with arbitrarily good precision. Rényi information of the third order is used to provide an information measure on time-frequency distributions. The results suggest that even though this new measure tracks time-bandwidth results for two Gabor logons separated in time and/or frequency, the information measure is more general and provides a quantitative assessment of the number of resolvable components in a time frequency representation. As such the information measure may be useful as a tool in the design and evaluation of timefrequency distributions. 1. TIME-FREQUENCY UNCERTAINTY Time frequency uncertainty is defined in various ways with slightly different results. Ca*This research was supported in part by grants from the Rackham School of Graduate Studies and the Office of Naval Research, ONR contracts no. N00014-89J-1723 and N00014-90-J-1654 bor's 10 definition will be used for the purposes of this paper. It is: