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https://hdl.handle.net/2142/21007
Description
Title
Statistical time-frequency analysis
Author(s)
Sayeed, Akbar M.
Issue Date
1996
Doctoral Committee Chair(s)
Jones, Douglas L.
Department of Study
Electrical and Computer Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Language
eng
Abstract
Time-frequency representations (TFRs), such as the short-time Fourier transform, the wavelet transform, and the Wigner distribution, represent signal characteristics jointly in time and frequency and provide a versatile set of tools for the analysis and processing of nonstationary signals. Recent developments have generalized the concept of TFRs, and a fairly complete theory has been formulated for arbitrary joint signal representations (JSRs) based on variables more general than time, frequency or scale. However, despite the demonstrated potential of TFRs and generalized JSRs, the applicability of existing methods has been severely limited due to the lack of statistical considerations in their development.
In this thesis, we develop a comprehensive framework for statistically optimal techniques that enables time-frequency methods to be more fully exploited in real situations. The result is a unified formulation encompassing arbitrary classes of JSRs, making it applicable in a broad range of radically different scenarios involving nonstationary stochastic signals, noise, and interference. The backbone of the framework is provided by optimal detection and estimation techniques based on JSRs.
The detection framework allows for optimal detection of a rich class of nonstationary stochastic signals, exhibiting some uncertain nuisance parameters, in the presence of Gaussian noise. At the heart of the framework is a family of unitary operators that is a representation of a Lie group. On one hand, it characterizes a class of JSRs, and on the other, a class of composite hypothesis testing scenarios for which such JSRs constitute canonical detectors. The nuisance signal parameters of the composite hypothesis are precisely the group parameters, and the optimal JSR detectors realize a generalized likelihood ratio test. For example, in the case of TFRs, the nuisance parameters are time and frequency shifts. The result is a unified formulation in terms of arbitrary unitary group representations that can serve as a model for a wide variety of nonstationary scenarios.
The estimation framework allows for optimal estimation of a broad range of nonstationary quadratic signal characteristics in the presence of noise or arbitrary statistical distortions. The quadratic characteristics to be estimated are determined by a JSR, which can be chosen from any arbitrary class which also serves as the class of estimators. We derive JSR estimators that are optimal in a minimum-mean-squared-error sense, and our framework allows for estimation of both average characteristics, such as generalized nonstationary spectra, and realization-based characteristics that reveal the structure of particular realizations of the signal process. As a special case, Cohen's class of TFRs can be used for optimal nonstationary spectral estimation.
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