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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.
Shift Covariant TimeFrequency Distributions of Discrete Signals
 IEEE Trans. on Signal Processing
, 1997
"... Many commonly used timefrequency distributions are members of the Cohen class. This class is defined for continuous signals and since timefrequency distributions in the Cohen class are quadratic, the formulation for discrete signals is not straightforward. The Cohen class can be derived as the cla ..."
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Cited by 14 (5 self)
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Many commonly used timefrequency distributions are members of the Cohen class. This class is defined for continuous signals and since timefrequency distributions in the Cohen class are quadratic, the formulation for discrete signals is not straightforward. The Cohen class can be derived as the class of all quadratic timefrequency distributions that are covariant to time shifts and frequency shifts. In this paper we extend this method to three types of discrete signals to derive what we will call the discrete Cohen classes. The properties of the discrete Cohen classes differ from those of the original Cohen class. To illustrate these properties we also provide explicit relationships between the classical Wigner distribution and the discrete Cohen classes. I. Introduction I N signal analysis there are four types of signals commonly used. These four types are based on whether the signal is continuous or discrete, and whether the signal is aperiodic or periodic. The four signal types ...
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...
DataDriven TimeFrequency and TimeScale Detectors
 Proc. SPIE’s 42 nd Meeting
, 1997
"... In many practical signal detection problems, the detectors have to designed from training data. Due to limited training data, which is usually the case, it is imperative to exploit some inherent signal structure for reliable detector design. The signals of interest in a variety of applications manif ..."
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Cited by 4 (1 self)
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In many practical signal detection problems, the detectors have to designed from training data. Due to limited training data, which is usually the case, it is imperative to exploit some inherent signal structure for reliable detector design. The signals of interest in a variety of applications manifest such structure in the form of nuisance parameters. However, datadriven design of detectors by exploiting nuisance parameters is virtually impossible in general due to two major difficulties: identifying the appropriate nuisance parameters, and estimating the corresponding detector statistics. We address this problem by using recent results that relate joint signal representations (JSRs), such as timefrequency and timescale representations, to quadratic detectors for a wide variety of nuisance parameters. We propose a general datadriven framework that: 1) identifies the appropriate nuisance parameters from an arbitrarily chosen finite set, and 2) estimates the secondorder statistics ...
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 theo ..."
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Cited by 4 (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 3 (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.
Virtues and Vices of Quartic TimeFrequency Distributions
 in IEEE Trans. on Signal Processing
, 2000
"... We present results concerning three different types of quartic (fourth order) timefrequency distributions. First, we present new results on the recently introduced local ambiguity function, and show that it provides more reliable estimates of instantaneous chirp rate than the Wigner distribution. S ..."
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Cited by 2 (1 self)
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We present results concerning three different types of quartic (fourth order) timefrequency distributions. First, we present new results on the recently introduced local ambiguity function, and show that it provides more reliable estimates of instantaneous chirp rate than the Wigner distribution. Second, we introduce the class of quartic, shiftcovariant, timefrequency distributions, and investigate distributions that localize quadratic chirps. Finally, we present a shift covariant distribution of time and chirprate. I. Introduction T HE notion of a timefrequency distribution (TFD) [1], [2], [3] is inherently a concept that is not well defined [4]. A frequency is something that is measured over a period of time (e.g. how many times does the heart beat in a minute), and we would like to specify this frequency description at an instant of time (e.g. how fast is the heart beating right now). Nevertheless, TFD's have proven to be useful in many applications [5]. TFD's have been defin...
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 1 (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.