The Zak transform and a new approach to waveform design.
Item
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Title
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The Zak transform and a new approach to waveform design.
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Identifier
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AAI9820532
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identifier
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9820532
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Creator
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Gladkova, Irina Vladimirovna.
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Contributor
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Adviser: Jose Moura
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Date
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1998
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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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Mathematics
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Abstract
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The ambiguity function {dollar}A\sb{lcub}f{rcub}(\tau,\nu){dollar} of a transmitted signal {dollar}f(t){dollar} measures the uncertainty with which the returning echo distinguishes, simultaneously, both ranges and velocities of a target system. For purposes of certain applications discussed in the engineering literature, {dollar}A\sb{lcub}f{rcub}(\tau,\nu){dollar} is desired to be thumbtack, i.e. a function whose absolute value has a graph with a strong peak at the origin, over a broad shallow base. The ambiguity function can be computed directly from the Zak transform {dollar}Z\sb{lcub}f{rcub}(x,y){dollar} of the signal {dollar}f(t){dollar} and so we propose to design a waveform in the Zak domain. A theorem that shows what properties the Zak transform of a signal should have for the ambiguity function to be thumbtack is derived and several new constructions based on this result are obtained. Furthermore, a simple method for the construction of generalized pulse train signals, whose Zak transforms are appropriately chosen trigonometric two-dimensional Chebyshev polynomials, is suggested, and applications to a multipath problem are considered. A multipath signal contains multiple returns of a transmitted signal, each is delayed and possibly dilated. Detecting multipath signals using a standard engineering approach is prohibitively difficult because it is based on a multidimensional minimization problem. One way to avoid this optimization problem is to use a signal with a thumbtack ambiguity function. Then we suggest another way to overcome the difficulties of minimization which may be useful when there is a need to work with known, but not specially designed, signals. Our approach is based on simplifications of the minimization problem through increasing the dimension of the problem. Then, after the structure of our solution is studied, the continuous case is considered and a simple algorithm of a standard high frequency cutoff Fourier transform technique gives a 'smoothed out' solution of the original problem.
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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.
