Adaptive signal processing with Weyl-Heisenberg expansions.
Item
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Title
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Adaptive signal processing with Weyl-Heisenberg expansions.
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Identifier
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AAI9917663
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identifier
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9917663
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Creator
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Joseph, E. Anthony.
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Contributor
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Adviser: Richard Tolimieri
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Date
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1999
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Language
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English
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Publisher
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City University of New York.
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Subject
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Engineering, Electronics and Electrical
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Abstract
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A sufficiently large class of signals exists whose information content is inadequately described in either the time or the frequency domain. These signals, which include speech and music, are nonstationary in nature. Nonstationary signals are best described in a time-frequency representation. However, a time-frequency signal representation must adhere to the lower bound established by the Heisenberg uncertainty principle. In particular, the powerful and robust Zak transform, which preserves many properties of Fourier analysis, has a peculiar property in the zero theorem imposed by the uncertainty principle. The Zak transform resides at the crossroad of Fourier analysis and Fourier synthesis; it is coarsely sampled at the Nyquist rate; and it is inherent in the structure of the Cooley-Tukey algorithm. The Zak transform is included in theory and application of Cohen's class of bilinear distributions, ambiguity functions, wavelet transforms, and Weyl- Heisenberg expansions. Moreover, it is recognized as the most suitable tool for the study of Weyl-Heisenberg expansions. Both Zak transform and Weyl-Heisenberg expansions are adaptive forms of Fourier representations. A Weyl-Heisenberg expansion is a nonorthogonal series expansion whose basis signals usually constitute a redundant and nonorthogonal set.;The main results of this research are the development of an orthogonal projection algorithm for Weyl-Heisenberg expansions in Zak space and an extension of Zak transform theory to cover non-critical sampling situations. This orthogonal projection algorithm projects Weyl-Heisenberg expansions from a critically sampled subspace onto an undersampled subspace. The projection signal on the undersampled subspace is synthesized with a subset of the original signal's coefficient set; the orthogonality of the projection caused an intrinsic change in the subset values. The orthogonal projection algorithm has potential application in data compression; it is adaptable to a multiresolutional decomposition.;The restriction of the Nyquist rate is removed from the definition of the Zak transform. The result is a generalization which is more consistent with Fourier analysis in terms of undersampling, critical sampling, and oversampling. At the critical sampling rate, its operation is identical to the standard Zak transform. This generalized Zak transform is suitable for Weyl-Heisenberg expansions over any Weyl-Heisenberg system.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
