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Title:Shear Madness: Signal-Dependent and Metaplectic Time-Frequency Representations
Author(s):Baraniuk, Richard Gordon
Doctoral Committee Chair(s):Jones, Douglas L.
Department / Program:Electrical Engineering
Discipline:Electrical Engineering
Degree Granting Institution:University of Illinois at Urbana-Champaign
Engineering, Electronics and Electrical
Abstract:Time-frequency representations are multidimensional transformations that indicate the joint time-frequency content of a signal. Representations such as the wavelet transform, the short-time Fourier transform, and the Wigner distribution have proven to be powerful tools for signal analysis and processing; however, current techniques are not without their drawbacks. This thesis presents two new approaches to time-frequency analysis that attempt to overcome two of the primary limitations inherent in current techniques.
The lack of a single time-frequency representation that is "best" for all applications has resulted in a proliferation of representations, each corresponding to a different, fixed mapping from signals to the time-frequency plane. A major drawback of all fixed mappings is that, for each mapping, the resulting representation is satisfactory only for a limited class of signals. To counter this hindrance, we derive two new time frequency representations that adapt to each signal and thus perform well for a large class of signals. To find the "best" representation for a given signal, the design of each signal-dependent time-frequency representation is formulated as an optimization problem.
The recent development of the wavelet transform has rekindled tremendous interest in proportional bandwidth, or "constant-Q," time-frequency analysis. In many applications, the time-scale analysis performed by the wavelet transform could be more appropriate than the constant-bandwidth analysis performed by representations such as the short-time Fourier transform, because it more closely matches the underlying physical mechanisms of some signals. However, the wavelet transform is ill-suited for the analysis of signals not exhibiting constant-Q structure. Using concepts from group representation theory, we propose the metaplectic transform, a transform that allows great freedom in the time-frequency resolution tradeoff and, hence, permits better matching of the transform to the signal characteristics. The metaplectic transform unites the conventional wavelet and short-time Fourier transforms under a common framework and provides a systematic method for designing new representations with resolution tradeoffs that are useful for certain types of signals. Using this framework, we construct two new classes of orthonormal bases for signals. A distinctive feature of these bases is that they are composed of linear-FM "chirp" functions.
Issue Date:1992
Description:251 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992.
Other Identifier(s):(UMI)AAI9305460
Date Available in IDEALS:2014-12-16
Date Deposited:1992

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