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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

General approach for obtaining joint representations in signal analysis and an application to scale

Leon W. Cohen

Published Dec 1, 1991• 26 citations
Publisher

Abstract

A general approach for obtaining joint representations in signal analysis is presented. The method is applied to scale representations. We define a new scale operator and show that it leads to joint representations of the Altes and Marinovic type and we also define joint representations with inverse frequency and show that it leads to Bertrand-Bertrand type distributions. We derive the uncertainty principle for scale and obtain the minimum time-scale uncertainty signal.

DOI
10.1117/12.49816

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A general approach for obtaining joint representations in signal analysis is presented. The method is applied to scale representations. We define a new scale operator and show that it leads to joint representations of the Altes and Marinovic type and we also define joint representations with inverse frequency and show that it leads to Bertrand-Bertrand type distributions. We derive the uncertainty principle for scale and obtain the minimum time-scale uncertainty signal.
We give a simple derivation for the covariance of time and frequency. The simplicity remains when applied to arbitrary variables. We show that the covariance enters into the uncertainty principle in a fundamental way. Also we present a seeming paradox: the uncertainty principle depends only on the marginals, but the marginals carry no information about covariance. Then how is it possible that the uncertainty principle involves the covariance? The resolution of the paradox clarifies a number of issues regarding the existence of well behaved manifestly positive joint distributions of time and frequency.>
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.
The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in [Formula: see text]-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.
Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. The construction is quite general and is also easily extended to the multi-operator case, which generalizes previously derived joint densities such as the time-frequency and time-scale distributions.>
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.