Shear Madness: Signal-Dependent and Metaplectic Time-Frequency Representations
Baraniuk, Richard Gordon
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https://hdl.handle.net/2142/71974
Description
Title
Shear Madness: Signal-Dependent and Metaplectic Time-Frequency Representations
Author(s)
Baraniuk, Richard Gordon
Issue Date
1992
Doctoral Committee Chair(s)
Jones, Douglas L.
Department of Study
Electrical Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
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.
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