A Deterministic Analysis of Decimation for Sigma-Delta Quantization of Bandlimited Functions

We study Sigma-Delta ( ΣΔ) quantization of oversampled bandlimited functions. We prove that digitally integrating blocks of bits and then down-sampling, a process known as decimation, can efficiently encode the associated ΣΔ bit-stream. It allows a large reduction in the bit-rate while still permitting good approximation of the underlying bandlimited function via an appropriate reconstruction kernel. Specifically, in the case of stable rth order ΣΔ schemes we show that the reconstruction error decays exponentially in the bit-rate. For example, this result applies to the 1-bit, greedy, first-order ΣΔ scheme.