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Birkhäuser Boston eBooks

Uncertainty Principles for Time-Frequency Representations

Karlheinz Gröchenig

Published Jan 1, 2003• 76 citations
Publisher

Abstract

We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a functionfand its Fourier transformf.To generate new uncertainty principles, we interpret the pair(f,f)as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.

DOI
10.1007/978-1-4612-0133-5_2

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We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a functionfand its Fourier transformf.To generate new uncertainty principles, we interpret the pair(f,f)as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.
A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.
We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The focus of this article is to propose two different versions of the Heisenberg uncertainty principle related to the fractional Fourier transform (FrFT). The first version is directly derived using the basic connection between the traditional Fourier transform (FT) and the FrFT, and the second is derived by exploiting time differentiation property and Parseval’s formula for the FrFT. In addition, a weighted uncertainty inequality related to this transformation is investigated. As a simple illustration of the use of the obtained results, we design the fractional characteristic function. An inequality describing the relation between the probability density function and its fractional characteristic function is presented.
Abstract Fourier transform is an integral part of harmonic analysis, information theory and signal processing with wide applications. The transformation between the Fourier pair is governed by an uncertainty relation, most generally captured by the Bandwidth theorem, . The usual approaches to the uncertainty principle employ sophisticated methods which lack physical interpretation of the origin and description of the uncertainty for real systems. In this paper, we have proposed a physically inspired model of the Fourier transform with an aim to describe its underlying dynamics and how it leads to the fundamental uncertainty principle. A signal is modeled as helical spring in 3D space, which is wrapped around a circle with controllable wrapping frequency. This process is formulated to reproduce the Fourier transform. The model is used to describe the origin of uncertainty, and to formulate a new generator-based approach to the Bandwidth theorem.
A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 σX(f)σD(f)≡∥Xf−⟨Xf,f⟩f∥2∥Df−⟨Df,f⟩f∥2⩾1, \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,σX​(f)σD​(f)≡∥Xf−⟨Xf,f⟩f∥2​∥Df−⟨Df,f⟩f∥2​⩾1, where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3
Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.