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Speaker: Prof. Victor DeBrunner, Department of Electrical and Computer Engineering, Florida State University
Title: Sparseness in Filtering, Signal Representations, and Sampling
Abstract:
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 has also 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 talk, I introduce the Hirschman Uncertainty Principles that are based not on energies and variances, 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. 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 δ[n]. 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. In any case, the expansion is considered to be “sparse.”
I will present a very simple experiment, which indicates that the Hirschman optimal transform (HOT) is superior to the DFT and DCT in terms of its ability to separate or resolve two limiting cases of localization in frequency, viz. pure tones and additive white noise. I will also show how non-parametric spectral estimation can be performed efficiently with the developed HOT. It is superior to the use of the DFT. I will present examples of sparse FIR filter design that have advantages in both implementation and performance. Finally, I will show some results in compressive sampling/sensing. I will conclude with some past example practices, and some future potential ideas.
Please RSVP at: https://meetings.vtools.ieee.org/m/32633
