The topic of this course is a new approach to the foundation of linear signal processing (SP), termed algebraic signal processing theory (ASP), that was developed by the instructor and his collaborators. Linear signal processing is built around the fundamental concepts of signals, filters, spectrum, z-transform, Fourier transforms, and many others. ASP generalizes these concepts, and thus linear SP, to provide a unifying approach to many existing SP methods and to enable the derivation of many new ones. This is made possible through the connection between linear SP and abstract algebra that ASP reveals and explores.
Specifically, the course will provide a new look at standard time SP but then introduce various other forms of SP including the non-standard space SP in one and higher dimensions, separable and nonseparable. In each case there will appropriate z-transforms, Fourier transforms, etc.These different SP methods will be derived as instantiations of the general, axiomatic ASP theory. ASP provides detailed insights into existing and novel transforms, their properties, and their fast algorithms, and naturally connects to linear statistical SP (Gauss-Markov random fields and Karhunen-Loeve transforms) as will be discussed in detail. Other topics will include uncertainty relations, sampling theorems, and shift-variant SP.
The course is mathematical in nature. The mathematics needed for ASP will be introduced in the course and learned through concrete exercises. In particular, no background in abstract algebra is needed. The instructor's emphasis regarding math is "hands-on" rather than "abstract."
The course is targeted for graduate students who are interested in enhancing their understanding of the foundation of SP. For any questions please contact the instructor.
Prerequisites: PhD standing, 18-396, one graduate level signal processing course, and Matrix Algebra or by instructor's consent.