The design of discrete Gabor expansion algorithms and an efficient realization of the Gerchberg-Papoulis algorithm in the Zak space.
Item
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Title
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The design of discrete Gabor expansion algorithms and an efficient realization of the Gerchberg-Papoulis algorithm in the Zak space.
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Identifier
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AAI9605576
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identifier
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9605576
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Creator
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Brodzik, Andrzej K.
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Contributor
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Adviser: Richard Tolimieri
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Date
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1995
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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 | Mathematics
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Abstract
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The effectiveness of any successful signal processing method ultimately depends on the availability of efficient and stable algorithms. Since in general, Gabor systems are not orthogonal, standard Hilbert space inner product methods do not apply. Moreover, since Gabor systems may not form a basis, oversampling in time-frequency is necessary for the existence of arbitrary signal expansions. In this work the finite Zak transform is established as a fundamental tool for studying critically sampled and oversampled Gabor systems and for designing algorithms for computing Gabor coefficients for discrete and periodic signals, by developing a deeper understanding of the relationship between finite Zak transform zero sets and properties of Gabor systems. The role of the finite Zak transform is analogous to that played by the Fourier transform in replacing complex convolution computations by simple pointwise multiplication. In this new setting properties of Gabor systems such as their spanning space and dimension can be determined by simple operations on functions in Zak space. This relationship impacts on questions of existence, parameterization and computation of Gabor expansions and leads to: (1) simple criteria for completeness of a Gabor system based on an explicit description in Zak space of the linear span of a Gabor system and (2) design of algorithms both at critical sampling and oversampling whose main computational task is implemented by 2-D finite Fourier transform offering the advantage of efficient, accurate and flexible code.;Another example of successful application of the finite Zak transform to signal processing problems presented in this work is the Zak space formulation of the Gerchberg-Papoulis algorithm. The iterative procedure of Gerchberg-Papoulis for restoration of band-limited signals from partial data involves transition between time domain and frequency domain at each iteration, rendering the algorithm computationally expensive. This work shows that the computational cost of the algorithm can be significantly reduced by performing the computations in a joined time-frequency space, using the finite Zak transform calculus as a tool. The intimate relationship between the Fourier transform and the Zak transform is exploited to replace the Fourier transform computations by pointwise multiplications. The new approach accomplishes reduction in multiplicative complexity from {dollar}2Nlog\sb2N{dollar} to {dollar}2N{dollar} per iteration.
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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.
