Quantization noise shaping on arbitrary frame expansions

Quantization noise shaping is commonly used in oversampled A/D and D/A converters. This paper considers quantization noise shaping for arbitrary frame expansions of signals based on generalizing the view of first order noise shaping as a compensation of the quantization error through a projection. Two levels of generalization are developed, one a special case of the other, and two different cost models are proposed to evaluate the quantizer structures. Within our framework, the implementation of the quantizer and reconstruction are computationally straightforward. The computational complexity is in the initial determination of frame vector ordering, which is part of the quantizer design and carried out off-line. We show that in the case of frame representation corresponding to uniform oversampling, the natural ordering implied by sequential time sampling is optimal. Furthermore, for general finite frame expansions, the problem of optimal ordering corresponds to known problems in graph theory.