Sigma-Delta Quantization of Geometrically Uniform Finite Frames

We consider sigma-delta quantization of cyclic geometrically uniform (CGU) finite frames, family of frames containing finite harmonic frames (both in C^M and R^M). For first- and second-order sigma-delta quantizers, we establish that the reconstruction minimum squares error (MSE) behaves as 1/r² where r denotes the frame redundancy. This result is shown to be true both under the quantization model used in J.J. Benedetto et al. (2004) as well as under the widely used additive white quantization noise assumption. For the widely used L-th order noise shaping filter G(z) = (1 - z^-1)^L, we show that the MSE behaves as 1/r² irrespectively of the filter order L. More importantly, we prove also that in the case of tight and normalized CGU frame, when the frame length is too large compared to the filter order, the reconstruction MSE can decay as faster as O(1/r^2L+1).