Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

In Analog-to-digital (A/D) conversion on oversampled bandlimited functions, integration of samples in blocks before down-sampling, known as decimation, has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. In particular, when coupled with the Σ∆ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. Here, a similar result is proved for finite unitarily generated frames. Specifically, under certain constraints on the generator, decimation merely increases the reconstruction error estimate by a factor of π/2, independent of the block size. On the other hand, efficient encodings of samples are made possible from decimation, and thus the error decays exponentially with respect to total bits used for data storage. Moreover, the decimation on finite frames has the multiplicative structure that allows the process to be broken down into successive iterations of decimation with smaller blocks, which opens up the possibility for parallel computation and signal transmission through multiple nodes.