Classes of smoothed Weyl symbols

Iem, Byeong-Gwan; Papandreou-Suppappola, Antonia; Boudreaux-Bartels, G.F.

doi:10.1109/97.847364

Cited by 6 publications

(6 citation statements)

References 17 publications

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“…1) Quadratic Form Property: Using the TF smoothed versions of the WS, we can obtain an alternative expression for the quadratic form property in (7) using QTFRs from Cohen's class [43]. Any Cohen's class TF shift-covariant QTFR with a signal independent kernel can be expressed as AF (27) where is a 2-D kernel characterizing the QTFR [1]- [3].…”

Section: A Tf Shift Covariant Classmentioning

confidence: 99%

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Wideband Weyl symbols for dispersive time-varying processing of systems and random signals

Iem

Papandreou-Suppappola

Boudreaux-Bartels

2002

IEEE Trans. Signal Process.

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We extend the narrowband Weyl symbol (WS) and the wideband P 0-Weyl symbol (P 0 WS) for dispersive time-frequency (TF) analysis of nonstationary random processes and time-varying systems. We obtain the new TF symbols using unitary transformations on the WS and the P 0 WS. 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.

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Section: A Tf Shift Covariant Classmentioning

confidence: 99%

“…For the 2-D inner product in (28) to be equated with the 1-D inner product in (7), we have shown that the following relationship must hold between the symbol kernel and the corresponding QTFR kernel that is nonzero for [43], [46]:…”

Section: (; ) Is Equal To Cohen's Class Qtfr Kernel 9 (;)mentioning

confidence: 99%

Wideband Weyl symbols for dispersive time-varying processing of systems and random signals

Iem

Papandreou-Suppappola

Boudreaux-Bartels

2002

IEEE Trans. Signal Process.

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“…This spectrum can be rewritten as It belongs to the class of type I spectra and will here be referred to as type I Page spectrum. The underlying TF operator symbol is the "Page symbol" [74] The type I Page spectrum satisfies the property of causality. As the corresponding type II spectrum, we define the new type II Page spectrum by using the Page symbol as the TF operator symbol in (14), i.e.,…”

Section: ) Generalized Wigner-ville Spectrum and Generalized Evolutimentioning

confidence: 99%

“…Consider a linear operator with kernel [69], [70]. A fairly wide class of linear TF representations of is given by the TF shift covariant TF operator symbols 1 [71]- [74] (…”

Section: A Tf Operator Symbolsmentioning

confidence: 99%

“…He arrived at the (type I) Levin spectrum which is equal to the real part of the Rihaczek spectrum . The underlying TF operator symbol is the "Levin symbol" [74] Using the same TF operator symbol in (14) yields the new type II Levin spectrum 4) Type I and Type II Physical Spectrum: Motivated by the idea of measuring the mean energy about a TF analysis point via an inner product, the (type I) physical spectrum was defined as the expectation of the spectrogram of [8], [101] Here, , where is an analysis window localized about the origin of the TF plane and normalized such that . The type I physical spectrum can be rewritten as…”

Section: ) Type I and Type Ii Levin Spectrum: Levin [3] Augmentedmentioning

confidence: 99%

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Nonstationary spectral analysis based on time-frequency operator symbols and underspread approximations

Matz

Hlawatsch

2006

IEEE Trans. Inform. Theory

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Abstract-We present a unified framework for time-varying or time-frequency (TF) spectra of nonstationary random processes in terms of TF operator symbols. We provide axiomatic definitions and TF operator symbol formulations for two broad classes of TF spectra, one of which is new. These classes contain all major existing TF spectra such as the Wigner-Ville, evolutionary, instantaneous power, and physical spectrum. Our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high-lag TF correlations (so-called underspread processes). We demonstrate that for underspread processes all TF spectra yield effectively identical results and satisfy several desirable properties at least approximately. We also show that Gabor frames provide approximate Karhunen-Loève (KL) functions of underspread processes and TF spectra provide a corresponding approximate KL spectrum. Finally, we formulate simple approximate input-output relations for the TF spectra of underspread processes that are passed through underspread linear timevarying systems. All approximations are substantiated mathematically by upper bounds on the associated approximation errors. Our results establish a TF calculus for the second-order analysis and time-varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes.Index Terms-Evolutionary spectrum, Gabor expansion, instantaneous power spectrum, Karhunen-Loève (KL) expansion, nonstationary random processes, nonstationary spectral analysis, time-frequency (TF) analysis, time-varying systems, Wigner-Ville spectrum.

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Time-Frequency Processing of Time-Varying Signals with Nonlinear Group Delay

Papandreou-Suppappola¹

2003

Wavelets and Signal Processing

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Classes of smoothed Weyl symbols

Cited by 6 publications

References 17 publications

Wideband Weyl symbols for dispersive time-varying processing of systems and random signals

Wideband Weyl symbols for dispersive time-varying processing of systems and random signals

Nonstationary spectral analysis based on time-frequency operator symbols and underspread approximations

Time-Frequency Processing of Time-Varying Signals with Nonlinear Group Delay

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