Spectral analysis of the convolution and filtering of non-stationary stochastic processes

Mark, William D.

doi:10.1016/s0022-460x(70)80106-7

Cited by 157 publications

(49 citation statements)

References 11 publications

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“…Many publications have been dedicated to it and several different definitions have been proposed [see, for example, (Eberly & Wódkievicz, 1977;Lampard, 1954;Mark, 1970;Page, 1952;Ponomarenko et al, 2004;Silverman, 1957)]. Nevertheless, for statistically non-stationary processes, W (1) (r, ω; r, ω) is measurable and a physically meaningful quantity.…”

Section: Physical Interpretation Of Correlation Functions In the Spacmentioning

confidence: 99%

Quantum Theory of Coherence and Polarization of Light

Lahiri¹

2012

Advances in Quantum Theory

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Section: Physical Interpretation Of Correlation Functions In the Spacmentioning

confidence: 99%

Quantum Theory of Coherence and Polarization of Light

Lahiri¹

2012

Advances in Quantum Theory

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“…The GWVS is a family of type I spectra whose underlying TF operator symbol is the GWS in (4), i.e., . Two important members of the GWVS family are the Wigner-Ville spectrum [4], [8], [14], [23], [24], [27], [29], [30], [94], [96]- [98] and the Rihaczek spectrum [7], [99], which are obtained for and , respectively. The GWVS can also be written as (16) where denotes the generalized expected ambiguity function to be defined in Section III-A.…”

Section: ) Generalized Wigner-ville Spectrum and Generalized Evolutimentioning

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%

“…While we will here consider only the GWVS and GES, approximate input-output relations hold also for other type I and II spectra due to the approximate equivalence of TF spectra in the underspread case (see Section VI). We note that input-output relations for nonstationary processes have previously been considered in [8], [116], however only for time-invariant systems.…”

Section: B Approximate Kl Eigenvalue Spectrummentioning

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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“…In this section, a short summary of the concept of time-varying spectrum will be given. A more detailed treatment of the subject can be found in a long paper by Mark (1970). …”

Section: Time-varying Power Spectrummentioning

confidence: 99%

Spectral analysis of the convolution and filtering of non-stationary stochastic processes

Cited by 157 publications

References 11 publications

Quantum Theory of Coherence and Polarization of Light

Quantum Theory of Coherence and Polarization of Light

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

Time varying spectral characteristics of strong ground motions; the Imperial Valley, California, earthquake of October 15, 1979

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