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17
Unitary Equivalence: A New Twist On Signal Processing
, 1995
"... Unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transfo ..."
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Cited by 72 (15 self)
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Unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transformation, performing standard processing techniques on the transformed signal, and then (in some cases) transforming the resulting output. The resulting unitarily equivalent systems focus on the critical signal characteristics in large classes of signals and, hence, prove useful for representing and processing signals that are not well matched by current techniques. As specific examples of this procedure, we generalize linear timeinvariant systems, orthonormal basis and frame decompositions, and joint timefrequency and timescale distributions, illustrating the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature. This work...
Optimal Detection Using Bilinear TimeFrequency And TimeScale Representations
 IEEE TRANS. SIGNAL PROCESSING
, 1995
"... Bilinear timefrequency representations (TFRs) and timescale representations (TSRs) are potentially very useful for detecting a nonstationary signal in the presence of nonstationary noise or interference. As quadratic signal representations, they are promising for situations in which the optimal de ..."
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Cited by 35 (13 self)
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Bilinear timefrequency representations (TFRs) and timescale representations (TSRs) are potentially very useful for detecting a nonstationary signal in the presence of nonstationary noise or interference. As quadratic signal representations, they are promising for situations in which the optimal detector is a quadratic function of the observations. All existing timefrequency formulations of quadratic detection either implement classical optimal detectors equivalently in the timefrequency domain, without fully exploiting the structure of the TFR, or attempt to exploit the nonstationary structure of the signal in an ad hoc manner. We identify several important nonstationary composite hypothesis testing scenarios for which TFR/TSRbased detectors provide a "natural" framework; that is, in which TFR/TSRbased detectors are both optimal and exploit the many degrees of freedom available in the TFR/TSR. We also derive explicit expressions for the corresponding optimal TFR/TSR kernels. As p...
A Canonical CovarianceBased Method for Generalized Joint Signal Representations
, 1996
"... Generalized joint signal representations extend the scope of joint timefrequency representations to a richer class of nonstationary signals. Cohen's marginalbased generalized approach is canonical from a distributional viewpoint, whereas, in some other applications, for example, in a signal d ..."
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Cited by 13 (7 self)
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Generalized joint signal representations extend the scope of joint timefrequency representations to a richer class of nonstationary signals. Cohen's marginalbased generalized approach is canonical from a distributional viewpoint, whereas, in some other applications, for example, in a signal detection framework, a covariancebased formulation is needed and/or more attractive. In this note, we present a canonical covariancebased recipe for generating generalized joint signal representations. The method is highlighted by its simple characterization and interpretation, and naturally extends the concept of the corresponding linear representations.
Equivalence Of Generalized Joint Signal Representations Of Arbitrary Variables
 in Proc. IEEE Int. Conf. on Acoust., Speech and Signal Proc.  ICASSP '95
, 1995
"... Joint signal representations (JSRs) of arbitrary variables generalize timefrequency representations (TFRs) to a much broader class of nonstationary signal characteristics. Two main distributional approaches to JSRs of arbitrary variables have been proposed by Cohen and Baraniuk. Cohen's method ..."
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Cited by 12 (6 self)
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Joint signal representations (JSRs) of arbitrary variables generalize timefrequency representations (TFRs) to a much broader class of nonstationary signal characteristics. Two main distributional approaches to JSRs of arbitrary variables have been proposed by Cohen and Baraniuk. Cohen's method is a direct extension of his original formulation of TFRs, and Baraniuk's approach is based on a group theoretic formulation; both use the powerful concept of associating variables with operators. One of the main results of the paper is that despite their apparent differences, the two approaches to generalized JSRs are completely equivalent. Remarkably, the JSRs of the two methods are simply related via axis warping transformations, with the broad implication that JSRs with radically different covariance properties can be generated efficiently from JSRs of Cohen's method via simple pre and postprocessing. The development in this paper, illustrated with examples, also illuminates other related ...
Marginals vs. Covariance in Joint Distribution Theory
 in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing  ICASSP '95
, 1995
"... Recently, Cohen has proposed a method for constructing joint distributions of arbitrary physical quantities, in direct generalization of joint timefrequency representations. In this paper, we investigate the covariance properties of this procedure and caution that in its present form it cannot gene ..."
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Cited by 11 (3 self)
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Recently, Cohen has proposed a method for constructing joint distributions of arbitrary physical quantities, in direct generalization of joint timefrequency representations. In this paper, we investigate the covariance properties of this procedure and caution that in its present form it cannot generate all possible distributions. Using group theory, we extend Cohen's construction to a more general form that can be customized to satisfy specific marginal and covariance requirements. 1. INTRODUCTION Joint distributions of arbitrary variables extend the notion of timefrequency analysis to quantities such as scale, Mellin, chirp rate, and inverse frequency. Two complementary approaches to constructing distributions have been developed. Covariancebased methods [14] concentrate on certain canonical signal transformations that leave the form of the distribution unchanged, while marginalbased methods [5, 6] aim for the property that integrating out one variable leaves the valid density...
Multiple window timevarying spectrum estimation
 in Conf. Info. Sci. and Sys. (CISS
, 1996
"... We overview a new nonparametric method for estimating the timevarying spectrum of a nonstationary random process. Our method extends Thomson’s powerful multiple window spectrum estimation scheme to the timefrequency and timescale planes. Unlike previous extensions of Thomson’s method, we identi ..."
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Cited by 11 (0 self)
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We overview a new nonparametric method for estimating the timevarying spectrum of a nonstationary random process. Our method extends Thomson’s powerful multiple window spectrum estimation scheme to the timefrequency and timescale planes. Unlike previous extensions of Thomson’s method, we identify and utilize optimally concentrated Hermite window and Morse wavelet functions and develop a statistical test for extracting chirping line components. Examples on synthetic and realworld data illustrate the superior performance of the technique. 2
Displacementcovariant timefrequency energy distributions
 in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process.—ICASSP
, 1995
"... Abstract’We present a theory of quadratic timefrequency (TF) energy distributions that satisfy a covariance property and generalized marginal properties. The theory coincides with the characteristic function method of Cohen and Earaniuk in the special case of ‘‘conjugate operators.” 1 ..."
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Cited by 10 (1 self)
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Abstract’We present a theory of quadratic timefrequency (TF) energy distributions that satisfy a covariance property and generalized marginal properties. The theory coincides with the characteristic function method of Cohen and Earaniuk in the special case of ‘‘conjugate operators.” 1
Integral Transforms Covariant To Unitary Operators And Their Implications For Joint Signal Representations
 THE IEEE TRANS. SIGNAL PROCESSING
, 1996
"... 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 par ..."
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Cited by 10 (5 self)
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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 wellknown 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 wellknown 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 presente...
Covariant TimeFrequency Analysis
, 2002
"... We present a theory of linear and bilinear/quadratic timefrequency (TF) representations that satisfy a covariance property with respect to \TF displacement operators." These operators cause TF displacements such as (possibly dispersive) TF shifts, dilations/compressions, etc. Our covariance ..."
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Cited by 8 (1 self)
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We present a theory of linear and bilinear/quadratic timefrequency (TF) representations that satisfy a covariance property with respect to \TF displacement operators." These operators cause TF displacements such as (possibly dispersive) TF shifts, dilations/compressions, etc. Our covariance theory establishes a uni ed framework for important classes of linear TF representations (e.g., shorttime Fourier transform and continuous wavelet transform) as well as bilinear TF representations (e.g., Cohen's class and ane class). It yields a theoretical basis for TF analysis and allows the systematic construction of covariant TF representations.
The power classes  Quadratic timefrequency representations with scale covariance and dispersive timeshift covariance
 IEEE TRANS. SIGNAL PROCESSING
, 1999
"... We consider scalecovariant quadratic time–frequency representations (QTFR’s) specifically suited for the analysis of signals passing through dispersive systems. These QTFR’s satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet tran ..."
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Cited by 7 (1 self)
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We consider scalecovariant quadratic time–frequency representations (QTFR’s) specifically suited for the analysis of signals passing through dispersive systems. These QTFR’s satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet transform and a covariance property with respect to generalized time shifts. We derive an existence/representation theorem that shows the exceptional role of time shifts corresponding to group delay functions that are proportional to powers of frequency. This motivates the definition of the power classes (PC’s) of QTFR’s. The PC’s contain the affine QTFR class as a special case, and thus, they extend the affine class. We show that the PC’s can be defined axiomatically by the two covariance properties they satisfy, or they can be obtained from the affine class through a warping transformation. We discuss signal transformations related to the PC’s, the description of the PC’s by kernel functions, desirable properties and kernel constraints, and specific PC members. Furthermore, we consider three important PC subclasses, one of which contains the Bertrand P_k distributions. Finally, we comment on the discretetime implementation of PC QTFR’s, and we present simulation results that demonstrate the potential advantage of PC QTFR’s.