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43
The scale representation
 IEEE Transactions on Signal Processing
, 1993
"... scaleable automated quality assurance technique for semantic ..."
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Cited by 39 (3 self)
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scaleable automated quality assurance technique for semantic
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 32 (12 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...
Beyond timefrequency analysis: Energy densities in one and many dimensions
, 1998
"... 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 meas ..."
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Cited by 16 (4 self)
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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. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen’s general construction for joint distributions of arbitrary variables.
Measuring timefrequency information content using the Rényi entropies
 IEEE Trans. on Info. Theory
, 2001
"... Abstract—The generalized entropies of Rényi inspire new measures for estimating signal information and complexity in the time–frequency plane. When applied to a time–frequency representation (TFR) from Cohen’s class or the affine class, the Rényi entropies conform closely to the notion of complexity ..."
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Cited by 16 (0 self)
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Abstract—The generalized entropies of Rényi inspire new measures for estimating signal information and complexity in the time–frequency plane. When applied to a time–frequency representation (TFR) from Cohen’s class or the affine class, the Rényi entropies conform closely to the notion of complexity that we use when visually inspecting time–frequency images. These measures possess several additional interesting and useful properties, such as accounting and crosscomponent and transformation invariances, that make them natural for time–frequency analysis. This paper comprises a detailed study of the properties and several potential applications of the Rényi entropies, with emphasis on the mathematical foundations for quadratic TFRs. In particular, for the Wigner distribution, we establish that there exist signals for which the measures are not well defined. Index Terms—Complexity, Rényi entropy, time–frequency analysis, Wigner distribution.
Discrete Frequency Warped Wavelets: Theory and Applications
 IEEE Trans. Signal Processing
, 1998
"... In this paper, we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discretetime by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of ..."
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Cited by 14 (7 self)
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In this paper, we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discretetime by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of orthogonal or biorthogonal warped wavelets by means of rational transfer functions. We show that the discretetime warped wavelets lead to welldefined continuoustime wavelet bases, satisfying a warped form of the twoscale equation. The shape of the wavelets is not invariant by translation. Rather, the "wavelet translates" are obtained from one another by allpass filtering. We show that the phase of the delay element is asymptotically a fractal. A feature of the warped wavelet transform is that the cutoff frequencies of the wavelets may be arbitrarily assigned while preserving a dyadic structure. The new transform provides an arbitrary tiling of the timefrequency plane, which can be designed by selecting as little as a single parameter. This feature is particularly desirable in cochlear and perceptual models of speech and music, where accurate bandwidth selection is an issue. As our examples show, by defining pitchsynchronous wavelets based on warped wavelets, the analysis of transients and denoising of inharmonic pseudoperiodic signals is greatly enhanced.
Warped Wavelet Bases: Unitary Equivalence And Signal Processing
 in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing  ICASSP '93
, 1993
"... The notions of time, frequency, and scale are generalized using concepts from unitary operator theory and applied to timefrequency analysis, in particular the wavelet and shorttime Fourier transform orthonormal bases and Cohen's class of bilinear timefrequency distributions. The result is an infin ..."
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Cited by 12 (4 self)
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The notions of time, frequency, and scale are generalized using concepts from unitary operator theory and applied to timefrequency analysis, in particular the wavelet and shorttime Fourier transform orthonormal bases and Cohen's class of bilinear timefrequency distributions. The result is an infinite number of new signal analysis and processing tools that are implemented simply by prewarping the signal by a unitary transformation, performing standard processing techniques on the warped signal, and then (in some cases) unwarping the resulting output. These unitarily equivalent, warped signal representations are useful for representing signals that are well modeled by neither the constantbandwidth analysis of timefrequency techniques nor the proportionalbandwidth analysis of timescale techniques. 1. INTRODUCTION The concepts of time and frequency are the cornerstones of signal analysis and processing, for they are the basis for fundamental tools such as the Fourier transform and ...
The hyperbolic class of quadratic timefrequency representations  Part II: Subclasses, . . .
 IEEE TRANS. SIGNAL PROCESSING
, 1997
"... Part I of this paper introduced the hyperbolic class (HC) of quadratic/bilinear timefrequency representations (QTFR’s) as a new framework for constant timefrequency analysis. The present Part II defines and studies the following four subclasses of the HC: • The localizedkernel subclass of the HC ..."
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Cited by 12 (3 self)
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Part I of this paper introduced the hyperbolic class (HC) of quadratic/bilinear timefrequency representations (QTFR’s) as a new framework for constant timefrequency analysis. The present Part II defines and studies the following four subclasses of the HC: • The localizedkernel subclass of the HC is related to a timefrequency concentration property of QTFR’s. It is analogous to the localizedkernel subclass of the affine QTFR class. • The affine subclass of the HC (affine HC) consists of all HC QTFR’s that satisfy the conventional timeshift covariance property. It forms the intersection of the HC with the affine QTFR class. • The power subclasses of the HC consist of all HC QTFR’s that satisfy a “power timeshift ” covariance property. They form the intersection of the HC with the recently introduced power classes. • The powerwarp subclass of the HC consists of all HC QTFR’s that satisfy a covariance to powerlaw frequency warpings. It is the HC counterpart of the shiftscale covariant subclass of Cohen’s class. All of these subclasses are characterized by 1D kernel functions. It is shown that the affine HC is contained in both the localizedkernel hyperbolic subclass and the localizedkernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary €Hdistribution by a convolution. We furthermore consider the properties of regularity (invertibility of a QTFR) and unitarity (preservation of inner products, Moyal’s formula) in the HC. The calculus of inverse kernels is developed, and important implications of regularity and unitarity are summarized. The results comprise a general method for leastsquares signal synthesis and new relations for the AltesMarinovichdistribution.
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...
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 is a ..."
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Cited by 10 (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 ...
Joint Distributions of Arbitrary Variables Made Easy
 IEEE Signal Processing Letters
, 1996
"... In this paper, we propose a simple framework for studying certain distributions of variables beyond timefrequency and timescale. When applicable, our results turn the theory of joint distributions of arbitrary variables into an easy exercise of coordinate transformation. While straightforward, the ..."
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Cited by 9 (4 self)
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In this paper, we propose a simple framework for studying certain distributions of variables beyond timefrequency and timescale. When applicable, our results turn the theory of joint distributions of arbitrary variables into an easy exercise of coordinate transformation. While straightforward, the method can generate many distributions previously attainable only by the general construction of Cohen, including time versus inverse frequency, time versus Mellin transform (scale), and time versus chirp distributions. In addition to providing insight into these new signal analysis tools, warpbased distributions have efficient implementations for potential use in applications. This work was supported by the National Science Foundation, grant no. MIP9457438, and by the Office of Naval Research, grant no. N000149510849. 1 Introduction The successful application of joint timefrequency distributions to problems in timevarying spectral analysis has stimulated considerable recent...