Foundations and Trends in Signal Processing
Quantization of filter bank frame expansions through moving horizon optimization
IEEE Transactions on Signal Processing
Performance of sigma-delta quantizations in finite frames
IEEE Transactions on Information Theory
Duals of Weighted Exponential Systems
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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The K-level Sigma-Delta (ΣΔ) scheme with step size δ is introduced as a technique for quantizing finite frame expansions for Rd. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that ΣΔ quantization of a unit-norm finite frame expansion in Rd achieves approximation error where N is the frame size, and the frame variation σ(F,p) is a quantity which reflects the dependence of the ΣΔ scheme on the frame. Here ||·|| 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/N2. 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/N under certain nonrigorous statistical assumptions. ΣΔ also achieves the optimal MSE order for PCM with consistent reconstruction.