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Mathematical Foundations of Signal Processing. 2. The Role of Group Theory
MASSACHUSETTS INST OF TECH LEXINGTON LINCOLN LAB
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Several aspects of group theory that prove useful for various signal processing applications are presented. Chapter I begins with a discussion of signal processing activities and goals at an abstract level, and continues with a look at the mathematical underpinnings of this subject. There follows a list of specific mathematical results that seem to be of greatest relevance to signal processing. Chapter II surveys the role played by infinite groups in modeling signals and filters. Here substantial use is made of the associated harmonic analysis in the abelian case the dual group serves as the natural frequency domain. Chapter III presents a fairly detailed review of the representation theory of finite groups, through the Plancherel formula. The essential idea here is to then use those special unitary transforms which are also group transforms for digital signal compression and decorrelation, and the associated group filters as fast suboptimal Wiener or other filters. Initial evidence suggests that nonabelian group filters can improve on the standard DFTFFT methods without significant increase in computational complexity. Chapter IV contains a summary of the main points and conclusions, and suggests some directions for further research, particularly on the use of finite nonabelian group transforms and filters.
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