Source coding with quantized redundant expansions: Accuracy and reconstruction

Signal representations based on low resolution quantization of redundant expansions is an interesting source coding paradigm, the most important practical case of which is oversampled A/D conversion. Signal reconstruction from quantized coefficients of a redundant expansion and accuracy of representations of this kind are problems which are still not well understood and these are studied in this paper in finite dimensional spaces. It has been previously proven that accuracy of signal representations based on quantized redundant expansions, measured as the squared Euclidean norm of reconstruction error, cannot be better than O(1/r(2)), where r is expansion redundancy. We give some general conditions under which the 1/r(2) accuracy can be attained. We also suggest a form of structure for overcomplete families which facilitates reconstruction and which enables efficient encoding of quantized coefficients with a logarithmic increase of the bit-rate in redundancy.