Entropy-based uncertainty measures for L2(Rn), ℓ2(Z), and ℓ2(Z/NZ) with a Hirschman optimal transform for ℓ2(Z/NZ)

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 literature, the term compactness also has been used to refer to this same notion of joint localization, in the sense of a signal representation that is efficient simultaneously in time (or space) and frequency. In this paper, we consider Hirschman uncertainty principles 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 concentration with respect to the Hirschman uncertainty principle is realized by translated and modulated Gaussian functions. In the two discrete 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 interesting results. First, in the finitely supported case, the Hirschman-optimal functions coincide with the so-called "picket fence" functions 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 reconciled with continuous-time Gaussians through a certain limiting process. While a different limiting process can be used to reconcile 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 experiment that indicates that the Hirschman optimal transform (HOT) is superior to - the discrete Fourier transform (DFT) and discrete cosine 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 the infinitely supported continuous-time case and the finitely supported discrete-time case.