Sigma-Delta (ΣΔ) quantization and finite frames

The K-level Sigma-Delta (Sigma Delta) scheme with step size 6 is introduced as a technique for quantizing finite frame expansions for R-d. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that Sigma Delta quantization of a unit-norm finite frame expansion in R-d achieves approximation error parallel to x - (x) over bar parallel to <= (delta d)(2N) (sigma(F,p) + 1) where N is the frame size, and the frame variation sigma(F, p) is a quantity which reflects the dependence of the E A scheme on the frame. Here parallel to center dot parallel to is the d-dimensional Euclidean 2-norm. Lower bounds and refined upper bounds are derived for certain specific cases. As a direct consequence of these error bounds one is able to bound the mean squared error (MSE) by an order of 1/N-2. When dealing with sufficiently redundant frame expansions, this represents a significant improvement over classical pulse-code modulation (PCM) quantization, which only has MSE of order 1 IN under certain nonrigorous statistical assumptions. Sigma Delta also achieves the optimal MSE order for PCM with consistent reconstruction.