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dc.contributor.authorSayeed, Akbar M.
Jones, Douglas L.
dc.creatorSayeed, Akbar M.
Jones, Douglas L. 2007-10-31T01:04:18Z 2007-10-31T01:04:18Z 1996-06-01 2004-11-07
dc.identifier.citation A. M. Sayeed and D. L. Jones, "Integral Transforms Covariant to Unitary Operators and their Implications for Joint Signal Representations," IEEE Transactions on Signal Processing, vol. 44, 1996.
dc.description Journal Paper
dc.description.abstract 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.
dc.language.iso eng
dc.subjecttime frequency
dc.subject.otherTime Frequency and Spectral Analysis
dc.title Integral Transforms Covariant to Unitary Operators and their Implications for Joint Signal Representations
dc.type Journal article
dc.citation.bibtexName article
dc.citation.journalTitle IEEE Transactions on Signal Processing 2004-11-08
dc.contributor.orgDigital Signal Processing (
dc.subject.keywordtime frequency
dc.citation.volumeNumber 44
dc.type.dcmi Text
dc.type.dcmi Text

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