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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 ...
Applications of Operator Theory to TimeFrequency Analysis and Classification
 IEEE Transactions on Signal Processing
, 1997
"... An entirely new set of criteria for the design of kernels (generating functions) for timefrequency representations (TFRs) is presented. These criteria aim only to produce kernels (and thus, TFRs) which will enable more accurate classification. We refer to these kernels, which are optimized to discr ..."
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Cited by 11 (2 self)
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An entirely new set of criteria for the design of kernels (generating functions) for timefrequency representations (TFRs) is presented. These criteria aim only to produce kernels (and thus, TFRs) which will enable more accurate classification. We refer to these kernels, which are optimized to discriminate among several classes of signals, as signal class dependent kernels, or simply class dependent kernels. The genesis of the class dependent kernel is to be found in the area of operator theory, which we use to establish a direct link between a discretetime, discretefrequency TFR and its corresponding discrete signal. We see that many similarities, but also some important differences, exist between the results of the continuoustime operator approach and our discrete one. The differences between the continuous representations and discrete ones may not be the simple sampling relationship which has often been assumed. From this work, we obtain a very concise, matrixbased expression ...
Understanding Discrete Rotations
, 1997
"... The concept of rotations in continuoustime, continuousfrequency is extended to discretetime, discretefrequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field ..."
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Cited by 10 (0 self)
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The concept of rotations in continuoustime, continuousfrequency is extended to discretetime, discretefrequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field. Use of this definition ensures that discrete rotations will share many of the same mathematical properties as continuous ones. A formula is given for the number of possible rotations of a primelength signal, and an example is provided to illustrate what such rotations look like. In addition, by studying a 90 degree rotation, we formulate an algorithm to compute a primelength discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps. This algorithm provides a further connection between the DFT and the discrete Wigner distribution based on group theory. 1. INTRODUCTION The Wigner distribution satisfies many desirable properties, among them bei...
A function of time, frequency, lag, and doppler
 IEEE Trans. on Signal Processing
, 1999
"... Abstract—In signal processing, four functions of one variable are commonly used. They are the signal in time, the spectrum, the autocorrelation function of the signal, and the autocorrelation function of the spectrum. The variables of these functions are denoted, respectively, as time, frequency, ..."
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Cited by 3 (2 self)
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Abstract—In signal processing, four functions of one variable are commonly used. They are the signal in time, the spectrum, the autocorrelation function of the signal, and the autocorrelation function of the spectrum. The variables of these functions are denoted, respectively, as time, frequency, lag, and doppler. In time–frequency analysis, these functions of one variable are extended to quadratic functions of two variables. In this paper, we investigate a method for creating quartic functions of three of these variables as well as a quartic function of all four variables. These quartic functions provide a meaningful representation of the signal that goes beyond the wellknown quadratic functions. The quartic functions are applied to the design of signaladaptive kernels for the Cohen class and shown to provide improvements over previous methods. I.
ClassDependent, Discrete TimeFrequency Distributions via Operator Theory
 Proc. ICASSP 97
, 1997
"... ABSTRACT We propose a property for kernel design which results in distributions for each of two classes of signals which maximally separates their energies in the timefrequency plane. Such maximally separated distributions may result in improved classification because the signal representation is ..."
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Cited by 3 (1 self)
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ABSTRACT We propose a property for kernel design which results in distributions for each of two classes of signals which maximally separates their energies in the timefrequency plane. Such maximally separated distributions may result in improved classification because the signal representation is optimized to accentuate the differences in signal classes. This is not the case with other timefrequency kernels which are optimized based upon some criteria unrelated to the classification task. Using our operator theory formulation for timefrequency representations, our "maximal separation" criteria takes on a very easily solved form. Analysis of the solution in both the timefrequency and ambiguity planes is given along with an example on discrete signals.
Distributions In The Discrete Cohen Classes
"... The Cohen class of timefrequency distributions for continuous signals has recently been to extended to discrete signals using both an axiomatic approach and an operator theory approach. In this paper, we investigate the formulation of several classical timefrequency distributions (Wigner, Rihaczek, ..."
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Cited by 1 (1 self)
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The Cohen class of timefrequency distributions for continuous signals has recently been to extended to discrete signals using both an axiomatic approach and an operator theory approach. In this paper, we investigate the formulation of several classical timefrequency distributions (Wigner, Rihaczek, MargenauHill, Page, Levin, BornJordan, spectrogram) in the discrete Cohen classes. The main result of this paper concludes that there does not exist a formulation of the Wigner distribution in all of the discrete Cohen classes. 1. INTRODUCTION There are four types of signals often used in signal processing, and to analyze these signals, there are four types of Fourier transforms. In Table 1 we list the four types of signals along with their properties and the appropriate Fourier transform. Since the Fourier transform is linear, the discrete Fourier transforms are samples of the continuous Fourier transform under the appropriate sampling conditions. The Cohen class of timefrequency distr...
A Function of Time, Frequency, Lag, and Doppler
, 1998
"... In signal processing, four functions of one variable are commonly used. They are the signal in time, the spectrum, the autocorrelation function of the signal, and the autocorrelation function of the spectrum. The variables of these functions are denoted, respectively, as time, frequency, lag, and ..."
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Cited by 1 (0 self)
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In signal processing, four functions of one variable are commonly used. They are the signal in time, the spectrum, the autocorrelation function of the signal, and the autocorrelation function of the spectrum. The variables of these functions are denoted, respectively, as time, frequency, lag, and doppler. In timefrequency analysis, these functions of one variable are extended to quadratic functions of two variables. In this paper, we investigate a method for creating quartic functions of three of these variables and also a quartic function of all four variables. These quartic functions provide a meaningful representation of the signal that goes beyond the well known quadratic functions. The quartic functions are applied to the design of signaladaptive kernels for the Cohen class and shown to provide improvements over previous methods. Corresponding Author Jeffrey C. O'Neill Laboratoire de Physique Ecole Normale Sup'erieure 46 All'ee d'Italie 69364 Lyon Cedex 07 FRANCE Tel: (+33) 4 ...