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The finite HeisenbergWeyl group in radar and communications
 EURASIP Journal of Applied Signal Processing
"... We investigate the theory of the finite HeisenbergWeyl group in relation to the development of adaptive radar and to the construction of spreading sequences and errorcorrecting codes in communications. We contend that this group can form the basis for the representation of the radar environment i ..."
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Cited by 33 (3 self)
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We investigate the theory of the finite HeisenbergWeyl group in relation to the development of adaptive radar and to the construction of spreading sequences and errorcorrecting codes in communications. We contend that this group can form the basis for the representation of the radar environment in terms of operators on the space of waveforms. We also demonstrate, following recent developments in the theory of errorcorrecting codes, that the finite HeisenbergWeyl groups provide a unified basis for the construction of useful waveforms/sequences for radar, communications, and the theory of errorcorrecting codes. Copyright © 2006 S. D. Howard 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. 1.
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 ...
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.
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.
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...
Recent developments in the theory of the fractional Fourier and linear canonical transforms
, 2004
"... ..."
Operator Theory Approach to Discrete TimeFrequency Distributions
 in Proc. of the IEEE Int. Symp. on TimeFrequency and TimeScale Analysis
, 1996
"... this document, you agree to all provisions of the copyright laws protecting it. Proceedings IEEESP International Symposium on TimeFrequency and TimeScale Analysis, p. 5214, 1996. ..."
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Cited by 9 (2 self)
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this document, you agree to all provisions of the copyright laws protecting it. Proceedings IEEESP International Symposium on TimeFrequency and TimeScale Analysis, p. 5214, 1996.
New Properties For Discrete, Bilinear TimeFrequency Distributions
 in Proc. of the IEEE Int. Symp. on TimeFrequency and TimeScale Analysis
, 1996
"... The most straightforward discrete implementation of bilinear timefrequency distributions (TFDs) are aliased in frequency, so as a result many different discrete TFDs have been developed. The traditional viewpoint in constructing discrete TFDs is that they should be samples of continuous TFDs. This ..."
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Cited by 8 (4 self)
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The most straightforward discrete implementation of bilinear timefrequency distributions (TFDs) are aliased in frequency, so as a result many different discrete TFDs have been developed. The traditional viewpoint in constructing discrete TFDs is that they should be samples of continuous TFDs. This paper introduces a second viewpoint: that discrete TFDs are inherently different, and thus, while having the same general goal, will not emulate continuous TFDs exactly. This paper develops properties corresponding to the second viewpoint for three types of discrete TFDs. A class of TFDs that satisfies the appropriate properties is also presented for each of the three types of discrete distributions. One of these is the class of aliasfree generalized discrete time timefrequency distributions (AFGDTFDs). It can be shown that the three classes of discrete TFDs presented here include every time and frequency shift covariant, bilinear TFD, so can be considered to be Cohen's classes for discre...
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.