Random rounding in redundant representations - art. no. 670103

This paper investigates the performance of randomly dithered first and higher-order sigma-delta quantization applied to the frame coefficients of a vector in a finite-dimensional Hilbert space. We compute the mean square error resulting from linear reconstruction with the quantized frame coefficients. When properly, dithered, this computation simplifies in the same way as under the assumption of the white-noise hypothesis. The results presented here are valid for a uniform mid-tread quantizer operating in the no-overload regime. We estimate the large-redundancy asymptotics of the error for each family of tight frames obtained from regular sampling of a bounded, differentiable path in the Hilbert space. In order to achieve error asymptotics that are comparable to the quantization of oversampled band-limited functions, we require the use of smoothly terminated frame paths.