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18
Measuring timefrequency information content using the Rényi entropies
 IEEE TRANS. ON INFO. THEORY
, 2001
"... 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 ..."
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Cited by 30 (1 self)
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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.
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 17 (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.
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
On Separability, Positivity and Minimum Uncertainty in TimeFrequency Energy Distributions
 IEEE Trans. Signal Proc
, 1998
"... Gaussian signals play a very special role in classical timefrequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty or positivity and separability of WignerVille distributions. We investigate here some of the logical connections which exist between ..."
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Cited by 10 (2 self)
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Gaussian signals play a very special role in classical timefrequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty or positivity and separability of WignerVille distributions. We investigate here some of the logical connections which exist between these different features, and we discuss some examples and counterexamples of their extension to more general joint distributions within Cohen's class and the affine class. 1 Introduction Let us consider Gaussian signals of the form g(t) = C e \Gammafft 2 ; (1) with C 2 C and ff 2 R + . Besides the fact that their Fourier transform is also Gaussian, namely that G(f) j Z +1 \Gamma1 g(t) e \Gammai2ßf t dt = C r ß ff e \Gamma ß 2 f 2 ff ; such signals happen to play a very special role in classical timefrequency analysis, and this is so for at least three different reasons: 1. Minimum uncertainty. They are the only minimizers for the timefrequency uncertainty relation [...
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.
Toward The Use Of The Time–Warping Principle With Discrete–Time Sequences
"... Abstract—This paper establishes a new coherent framework to extend the class of unitary warping operators to the case of discrete–time sequences. Providing some a priori considerations on signals, we show that the class of discrete–time warping operators finds a natural description in linear shift– ..."
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Cited by 5 (4 self)
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Abstract—This paper establishes a new coherent framework to extend the class of unitary warping operators to the case of discrete–time sequences. Providing some a priori considerations on signals, we show that the class of discrete–time warping operators finds a natural description in linear shift– invariant spaces. On such spaces, any discrete–time warping operator can be seen as a non–uniform weighted resampling of the original signal. Then, gathering different results from the non–uniform sampling theory, we propose an efficient iterative algorithm to compute the inverse discrete–time warping operator and we give the conditions under which the warped sequence can be inverted. Numerical examples show that the inversion error is of the order of the numerical round–off limitations after few iterations. Index Terms—Time–frequency, Unitary equivalence, Implementation of time–warping operators, Non–stationary filtering. I.
Wideband Weyl Symbols for Dispersive TimeVarying Processing of Systems and Random Signals
, 2002
"... We extend the narrowband Weyl symbol (WS) and the wideband PHWeyl symbol (PHWS) for dispersive time–frequency (TF) analysis of nonstationary random processes and timevarying systems. We obtain the new TF symbols using unitary transformations on the WS and the PHWS. For example, whereas the WS is m ..."
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Cited by 5 (0 self)
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We extend the narrowband Weyl symbol (WS) and the wideband PHWeyl symbol (PHWS) for dispersive time–frequency (TF) analysis of nonstationary random processes and timevarying systems. We obtain the new TF symbols using unitary transformations on the WS and the PHWS. For example, whereas the WS is matched to systems with constant or linear TF characteristics, the new symbols are better matched to systems with dispersive (nonlinear) TF structures. This results from matching the geometry of the unitary transformation to the specific TF characteristics of a system. We also develop new classes of smoothed Weyl symbols that are covariant to TF shifts or time shift and scaling system transformations. These classes of symbols are also extended via unitary warpings to obtain classes of TF symbols covariant to dispersive shifts. We provide examples of the new symbols and symbol classes, and we list some of their desirable properties. Using simulation examples, we demonstrate the advantage of using TF symbols that are matched to the changes in the TF characteristics of a system or random process. We also provide new TF formulations for matched detection applications.
Chapter 7 Quadratic TimeFrequency Analysis III: The Affine Class and Other Covariant Classes
"... Abstract: Affine timefrequency distributions appeared towards the middle of the 1980s with the emergence of wavelet theory. The affine class is built upon the principle of covariance of the affine group, i.e., contractionsdilations and translations in time. This group provides an interesting alter ..."
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Abstract: Affine timefrequency distributions appeared towards the middle of the 1980s with the emergence of wavelet theory. The affine class is built upon the principle of covariance of the affine group, i.e., contractionsdilations and translations in time. This group provides an interesting alternative to the group of translations in time and in frequency, which forms the basis for the conventional timefrequency distributions of Cohen’s class. More precisely, as the Doppler effect on “broadband ” signals is expressed in terms of contractionsdilations, it is for the analysis of this category of signals that the affine class is particularly destined. The objective of this chapter is to present the various approaches for constructing the affine class and the associated tools devised over the past years. We will demonstrate how the latter supported the introduction of new mathematical concepts in signal processing – group theory, operator theory – as well as of new classes of covariant timefrequency distributions.
Research Article Nonstationary System Analysis Methods for Underwater Acoustic Communications
, 2010
"... Copyright © 2011 Nicolas F. Josso et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The underwater environment can be considered ..."
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Copyright © 2011 Nicolas F. Josso et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The underwater environment can be considered a system with timevarying impulse response, causing timedependent spectral changes to a transmitted acoustic signal. This is the result of the interaction of the signal with the water column and ocean boundaries or the presence of fast moving object scatterers in the ocean. In underwater acoustic communications using mediumtohigh frequencies (0.3–20 kHz), the nonstationary transformation on the transmitted signals can be modeled as multiple timedelay and Dopplerscaling paths. When estimating the channel, a higher processing performance is thus expected if the techniques used employ a matched channel model compared to those that only compensate for wideband effects. Following a matched linear timevarying wideband system representation, we propose two different methods for estimating the underwater acoustic communication environment. The first method follows a canonical timescale channel model and is based on estimating the coefficients of the discrete wideband spreading function. The secondmethod follows a ray systemmodel and is based on extracting timescale features for different ray paths using the matching pursuit decomposition algorithm. Both methods are validated and compared using communication data from actual underwater acoustic communication experiments. 1.