Entropy-Based Uncertainty Measures for , , and With a Hirschman Optimal Transform for
Victor DeBrunner, Joseph Havlicek, Tomasz Przebinda, Murad Özaydõn · 2005
The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. In the image processing lit- erature, the term compactness also has been used to refer to this same notion of joint localization, in the sense of a signal represen- tation that is efficient simultaneously in time (or space) and fre- quency. In this paper, we consider Hirschman uncertainty prin- ciples based not on energies and variances directly but rather on entropies computed with respect to normalized energy densities in time and frequency. Unlike the Heisenberg-Weyl measure, this entropic Hirschman notion of joint uncertainty extends naturally from the case of infinitely supported continuous-time signals to the cases of both finitely and infinitely supported discrete-time signals. For the first time, we consider these three cases together and study them relative to one another. In the case of infinitely supported continuous-time signals, we find that, consistent with the energy- based Heisenberg principle, the optimal time-frequency concentra- tion with respect to the Hirschman uncertainty principle is realized by translated and modulated Gaussian functions. In the two dis- crete cases, however, the entropy-based measure yields optimizers that may be generated by applying compositions of operators to the Kronecker delta. Study of the discrete cases yields two inter- esting results. First, in the finitely supported case, the Hirschman- optimal functions coincide with the so-called picket fence func- tions that are also optimal with respect to the joint time-frequency counting measure of Donoho and Stark. Second, the Hirschman optimal functions in the infinitely supported case can be recon- ciled with continuous-time Gaussians through a certain limiting process. While a different limiting process can be used to recon- cile the finitely and infinitely supported discrete cases, there does not appear to be a straightforward limiting process that unifies all three cases: The optimizers from the finitely supported discrete case are decidedly non-Gaussian. We perform a very simple exper- iment that indicates that the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete co- sine transform (DCT) in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz. pure tones and additive white noise. We believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between