Oversampling and Coarse Quantization for Signals

CSIC Building (#406), Seminar Room 4122.
Directions: home.cscamm.umd.edu/directions

Unified Principle of Sigma-Delta as a Quantization Technique of Overcomplete Expansions

Dr. Thao Nguyen

Electrical Engineering at City College, CUNY

Abstract:   As with the general theme of this workshop, the goal is to refine approximations of signals by coarsely quantized expansions over an overcomplete generating family of signal vectors. In this talk, we present the most abstract idea that can be extracted from the method of Sigma-Delta modulation to perform the quantization operation. The first principle implicitly introduced by Sigma-Delta modulation is to perform a change of generating family, in the same sense as "change of basis". The new family is constructed by taking each individual vector of the original family and subtracting to it a linear combination of the other generating vectors, to obtain a residual vector of reduced norm. The second principle is to quantize the expansion coefficients of the input signal with respect to the original generating family, while observing the expansion coefficients of the quantization error signal with respect to the generating family of residual vectors. Given the small norm of these vectors, the primary concern is to ensure that the latter coefficients remain bounded with reasonable bounds. Sigma-Delta modulation, the way it is commonly known, results from these abstract principles when the quantization control algorithm is restricted to be causal (one-pass) and time-invariant. This automatically implies the use of a dynamical system. Based on these abstract principles, we give an overview of the various aspects of Sigma-Delta modulation (often ignoring each other) under a unified signal-processing framework. We will show from a top-down presentation what is the position of the various research directions in Sigma-Delta modulation with respect to each other. We will include research in the context of finite and infinite dimensional signal spaces, one-dimensional and multi-dimensional signal index spaces, lowpass, bandpass and multi-channel signal expansions. With respect to the second principle, we perform a unified classification of all existing dynamical system architectures of Sigma-Delta modulation.
 
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