A novel translation and modulation invariant discrete-discrete uncertainty measure

The quantification of signal localization simultaneously in time and in frequency is fundamental to a variety of signal processing applications where time-frequency analysis is to be performed on nonstationary signals. In this paper, we develop novel joint localization measures defined on equivalence classes of finitely supported discrete-time signals. These measures bear strong analogies to the well-known continuous-time Heisenberg-Weyl inequality. In particular, they are invariant to signal translations and modulations and admit an intuitive interpretation in terms of the temporal and spectral variance of the signal energy. The new measures are used to design optimal wavelet quadrature mirror filter banks that exhibit improved localization relative to the Haar and Daubechies analysis filters.