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IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis

Covariant (α,β), time-frequency, and (a,b) representations

Franz Hlawatsch, T. Twaroch

Published Dec 23, 2002• 9 citations
Publisher

Abstract

We present a general theory of bilinear/quadratic signal representations that are covariant to time-frequency (TF) displacement operators D/sub /spl alpha/,/spl beta//. Covariant representations are developed in the (/spl alpha/,/spl beta/) domain and in the TF domain. In the conjugate case, covariant representations can also be formulated in an eigenfunction parameter domain ((a,b) domain). The relations connecting the (/spl alpha/,/spl beta/), TF, and (a,b) domains are epitomized by the displacement function and the localization function. For conjugate operators, (/spl alpha/,/spl beta/) covariance and (a,b) covariance are equivalent.

DOI
10.1109/tfsa.1996.550086

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We present a general theory of bilinear/quadratic signal representations that are covariant to time-frequency (TF) displacement operators D/sub /spl alpha/,/spl beta//. Covariant representations are developed in the (/spl alpha/,/spl beta/) domain and in the TF domain. In the conjugate case, covariant representations can also be formulated in an eigenfunction parameter domain ((a,b) domain). The relations connecting the (/spl alpha/,/spl beta/), TF, and (a,b) domains are epitomized by the displacement function and the localization function. For conjugate operators, (/spl alpha/,/spl beta/) covariance and (a,b) covariance are equivalent.
Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the "covariant" transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual.
We propose classes of quadratic time-frequency distributions that retain the inner structure of Cohen's class. Each of these classes is based on a pair of conjugate unitary operators producing time-frequency displacements. The classes satisfy covariance and marginal properties corresponding to these operators. For each class, we define a central member generalizing the Wigner distribution and the Q-distribution, and we specify a transformation by which the class can be derived from Cohen's class.
The displacement operator is related to the displaced parity operator through a two dimensional \nFourier transform. Both operators are important operators in phase space \nand the trace of both with respect to the density operator gives the Wigner functions \n(displaced parity operator) and Weyl functions (displacement operator). The generalisation \nof the parity-displacement operator relationship considered here is called \nthe bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional \ndisplacement operators lead to the novel concept of bi-fractional coherent states. \nThe generalisation from Fourier transform to fractional Fourier transform can be \napplied to other phase space functions. The case of the Wigner-Weyl function is considered \nand a generalisation is given, which is called the bi-fractional Wigner functions, \nH(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to \ngive the bi-fractional Q−functions and bi-fractional P−functions respectively. The \ngeneralisation is likewise applied to the Moyal star product and Berezin formalism for \nproducts of non-commutating operators. These are called the bi-fractional Moyal star \nproduct and bi-fractional Berezin formalism. \nFinally, analysis, applications and implications of these bi-fractional transforms \nto the Heisenberg uncertainty principle, photon statistics and future applications are \ndiscussed.
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.>