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Positive time-frequency distribution functions [for formulating signal theory]

Deep Sea Research Part B Oceanographic Literature ReviewJan 1, 19850 citations

Properties of the positive time-frequency distribution functions

Leon W. Cohen

Joint positive distributions of time and frequency which satisfy the correct marginals exist and can be readily obtained. In this presentation we generalize the continuous case previously considered to the discrete time case and the two dimensional situation.

IEEE International Conference on Acoustics Speech and Signal ProcessingMar 23, 20052 citations

Integral transforms covariant to unitary operators and their implications for joint signal representations

A.M. Sayeed, Douglas L. Jones

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.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.

IEEE Transactions on Signal ProcessingJun 1, 199633 citations

Positivity of time-frequency distribution functions

Augustus J. E. M. Janssen

Signal ProcessingApr 1, 198815 citations

Joint Representations in Quantum Mechanics and Signal Processing Theory: Why a Probability Function of Time and Frequency is Disallowed,

David Lowe

Abstract : A discussion is presented which attempts to clarify several misconceptions regarding the nature of distribution functions constructed as simultaneous functions of time and frequency. The problems associated with trying to interpret such densities as probability distributions are pointed out and the close correspondence between signal processing theory and quantum mechanics is exploited in order to address the problems raised. On the basis of the arguments presented, a preferred joint distribution is constructed, its properties outlined and its relevance to the representation of complex time-varying signals is discussed.Abstract : A discussion is presented which attempts to clarify several misconceptions regarding the nature of distribution functions constructed as simultaneous functions of time and frequency. The problems associated with trying to interpret such densities as probability distributions are pointed out and the close correspondence between signal processing theory and quantum mechanics is exploited in order to address the problems raised. On the basis of the arguments presented, a preferred joint distribution is constructed, its properties outlined and its relevance to the representation of complex time-varying signals is discussed.

Dec 1, 19864 citations

Renormalized theory of the time-dependent pair distribution function. I. General formulation

Giuseppe Tenti, Frederick R. W. McCourt

Journal of Statistical PhysicsJul 1, 19763 citations

Distributions in signal theory

Abstract

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