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IEEE Transactions on Signal Processing

Entropy-based uncertainty measures for L/sup 2/(/spl Ropf//sup n/), /spl lscr//sup 2/(/spl Zopf/), and /spl lscr//sup 2/(/spl Zopf//N/spl Zopf/) with a Hirschman optimal transform for /spl lscr//sup 2/(/spl Zopf//N/spl Zopf/)

Victor DeBrunner, Joseph Havlicek, Tomasz Przebinda, Murad Özaydın

Published Jul 19, 2005• 32 citations
Publisher

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 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.

DOI
10.1109/tsp.2005.850329

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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.
In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Rényi entropies with conjugated indexes. We consider here the more general case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes α and β in the plane (α, β). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).
The uncertainty relation is introduced by using operators which are not necessarily self-adjoint. In general, the uncertainty (lower) limit is different for different functions. Operators are found that define the second moments and the uncertainty limit for their product such that the limit is the same for functions obtained from each other by applying unitary support-size preserving operators, e.g. by shifting in time and frequency. In this case, for all functions of finite energy the limit is related to the minimum support size in the time-frequency domain. Moreover, it is possible to find the discrete-time counterpart of the classical uncertainty relation for continuous-time functions. The operator calculus presents results from signal analysis in a more standard form extensively used in quantum mechanics.
We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state . We prove sharp estimates which critically depend on the purity of and which are saturated in the Gaussian case.
We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state ρ\rho ρ . We prove sharp estimates which critically depend on the purity of ρ\rho ρ and which are saturated in the Gaussian case.
Abstract Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.