Oversampling and Coarse Quantization for Signals

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Sigma-delta Quantization of Finite Frames - Part 1

Dr. Ozgür Yılmaz

Department Of Mathematics at University of British Columbia

Abstract:   A basic problem in signal processing, when analyzing a given signal of interest, is to obtain a digital representation that is suitable for storage, transmission, and recovery. A reasonable approach is to first decompose the signal as a sum of appropriate harmonics, where each harmonic has a real (or complex) coefficient. Next, one "quantizes" the coefficients, i.e., one replaces each coefficient by an element of a given finite set (e.g.,{-1,1}). We explore quantization of atomic decompositions in a Hilbert space. When the decomposition is in the form of a basis expansion, the optimal quantizer for a fixed quantization alphabet is given by PCM. However, when the expansion is redundant, the problem of finding the optimal quantizer (or even a good quantizer) is nontrivial. We investigate this problem in the case of finite frames. In particular, we show that first-order sigma-delta quantizers perform better than the traditional PCM algorithms whenever the redundancy of the frame is sufficiently high. We present refined error estimates for the first-order sigma-delta quantizers. Furthermore, we address the problem of optimal quantization for harmonic frames in Euclidean space. This is joint work with J. Benedetto and A. Powell.