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13
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...
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 18 (6 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 ...
A Pedestrian's Approach to Pseudodifferential Operators
, 2005
"... This article, in particular, owes much to my joint work and many stimulating discussions with Chris Heil ..."
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Cited by 18 (3 self)
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This article, in particular, owes much to my joint work and many stimulating discussions with Chris Heil
Optimizing TimeFrequency Kernels for Classification
, 2001
"... In many pattern recognition applications, features are traditionally extracted from standard timefrequency representations (TFRs). This assumes that the implicit smoothing of, say, a spectrogram is appropriate for the classification task. Making such assumptions may degrade classification performa ..."
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Cited by 13 (1 self)
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In many pattern recognition applications, features are traditionally extracted from standard timefrequency representations (TFRs). This assumes that the implicit smoothing of, say, a spectrogram is appropriate for the classification task. Making such assumptions may degrade classification performance. In general, any timefrequency classification technique that uses a singular quadratic TFR (e.g., the spectrogram) as a source of features will never surpass the performance of the same technique using a regular quadratic TFR (e.g., Rihaczek or WignerVille). Any TFR that is not regular is said to be singular. Use of a singular quadratic TFR implicitly discards information without explicitly determining if it is germane to the classification task. We propose smoothing regular quadratic TFRs to retain only that information that is essential for classification. We call the resulting quadratic TFRs classdependent TFRs. This approach makes no a priori assumptions about the amount and type of timefrequency smoothing required for classification. The performance of our approach is demonstrated on simulated and real data. The simulated study indicates that the performance can approach the Bayes optimal classifier. The realworld pilot studies involved helicopter fault diagnosis and radar transmitter identification.
Proof of a conjecture on the supports of Wigner distributions
, 1998
"... In this note we prove that the Wigner distribution of an f 2 L 2 (R n ) cannot be supported by a set of finite measure in R 2n unless f = 0. We prove a corresponding statement for crossambiguity functions. As a strengthening of the conjecture we show that for an f 2 L 2 (R n ) its Wigner d ..."
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Cited by 13 (0 self)
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In this note we prove that the Wigner distribution of an f 2 L 2 (R n ) cannot be supported by a set of finite measure in R 2n unless f = 0. We prove a corresponding statement for crossambiguity functions. As a strengthening of the conjecture we show that for an f 2 L 2 (R n ) its Wigner distribution has a support of measure 0 or 1 in any halfspace of R 2n . Math Subject Classifications: 42B10, 94A12. Keywords and Phrases: uncertainty principle, Wigner distribution, ambiguity function. 1.
DataDriven TimeFrequency Classification Techniques Applied To ToolWear Monitoring
 University of Wisconsin
, 2000
"... In many pattern recognition applications features are traditionally extracted from standard timefrequency representations (e.g. the spectrogram) and input to a classifier. This assumes that the implicit smoothing of, say, a spectrogram is appropriate for the classification task. It is better to beg ..."
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Cited by 11 (3 self)
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In many pattern recognition applications features are traditionally extracted from standard timefrequency representations (e.g. the spectrogram) and input to a classifier. This assumes that the implicit smoothing of, say, a spectrogram is appropriate for the classification task. It is better to begin with no implicit smoothing assumptions and optimize the timefrequency representation for each specific classification task. Here we describe two different approaches to datadriven timefrequency classification techniques, one supervised and one unsupervised. We show that a certain class of quadratic timefrequency representations will always provide best classification performance. Using our techniques we explore the wear process of milling cutters. Our initial experiments give strong evidence to the nonlinear nature of the wear process and the importance of capturing nonstationary information about each flutestrike to accurately understand the wear process.
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 Wigner distribution of a linear signal space
 IEEE Trans. on Signal Processing
, 1993
"... AbstractWe introduce a timefrequency representation of linear signal spaces which we call the Wigner distribution (WO) ofa linear signal space. Similar to the WD of a signal, the WD of a linear signal space describes the space’s energy distribution over the timefrequency plane. The WD of a signa ..."
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Cited by 8 (2 self)
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AbstractWe introduce a timefrequency representation of linear signal spaces which we call the Wigner distribution (WO) ofa linear signal space. Similar to the WD of a signal, the WD of a linear signal space describes the space’s energy distribution over the timefrequency plane. The WD of a signal space can be defined both in a deterministic and in a stochastic framework, and it can be expressed in a simple way in terms of the space’s projection operator and bases. It is shown to satisfy many interesting properties which are often analogous to corresponding properties of the WD of a signal. The results obtained for some specific signal spaces are found to be intuitively satisfactory. Further topics discussed are the crossWD of two signal spaces, a discretetime WD version, and the extension of the WD definition to arbitrary quadratic signal representations. I.
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.
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.