Approximation Theory of Quantization of Redundant Expansions

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}). The problem of how to quantize a given expansion is non-trivial when the expansion is redundant.

In this talk, we consider (redundant) frame expansions, and show that Sigma-Delta modulators provide efficient quantization algorithms in the cases of oversampled bandlimited functions, Gabor frame expansions of square-integrable functions, and finite frame expansions in Euclidean space. In particular, we show that Sigma-Delta algorithms outperform PCM algorithms (the current state-of-the-art). We also address the problem of optimal quantization, and present recent results in the case of finite frames.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).