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10
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.
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 5 (0 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 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.
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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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 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.
Range Bias Modeling for Hyperbolic Frequency Modulated Waveforms in Target Tracking
"... Abstract—Hyperbolic frequency modulated (HFM) waveforms offer detection performance that is “Doppler insensitive”— relatively unaffected by target rangerate in matched filtering. They have been applied in radar; but more they are particularly useful in sonar where the wideband Doppler insensitivity ..."
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Abstract—Hyperbolic frequency modulated (HFM) waveforms offer detection performance that is “Doppler insensitive”— relatively unaffected by target rangerate in matched filtering. They have been applied in radar; but more they are particularly useful in sonar where the wideband Doppler insensitivity is especially prized. However, Doppler insensitivity does come hand in hand with a range bias, which would significantly degrade the tracking performance. In this paper, we model the range bias as a function of rangerate and system parameters, and then utilize that to calibrate the measurement equation in very precise target tracking. By doing so, the tracking performance can be improved, particularly for fast maneuvering targets. Index Terms—Hyperbolic frequency modulation (HFM), linear frequency modulation (LFM), wideband, narrowband, ambiguity function, Doppler tolerant waveform, bias compensation, sonar, radar, tracking, (extended) Kalman filter. I.
ESTIMATING MAXIMA OF GENERALIZED CROSS AMBIGUITY FUNCTIONS, AND UNCERTAINTY PRINCIPLES
"... Abstract. In certain signal processing problems, it is customary to estimate parameters in distorted signals by approximating what is termed a cross ambiguity function and estimating where it attains its maximum modulus. To unify and generalize these procedures, we consider a generalized form of the ..."
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Abstract. In certain signal processing problems, it is customary to estimate parameters in distorted signals by approximating what is termed a cross ambiguity function and estimating where it attains its maximum modulus. To unify and generalize these procedures, we consider a generalized form of the cross ambiguity function and give error bounds for estimating the parameters, showing that these bounds are lower if we maximize the real part rather than the modulus. We also reveal a connection between these bounds and certain uncertainty principles, which leads to a new type of uncertainty principle. 1.
unknown title
, 2008
"... An extension of the class of unitary time–warping operators to discrete–time sequences E ..."
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An extension of the class of unitary time–warping operators to discrete–time sequences E