Celebratio Mathematica — Daubechies

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I. Daubechies and R. Saab: “A deterministic analysis of decimation for Sigma-Delta quantization of bandlimited functions,” IEEE Sig. Proc. Lett. 22 : 11 (November 2015), pp. 2093–2096. ArXiv 1506.00179 article

We study Sigma-Delta (\( \Sigma\Delta \)) 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 \( \Sigma\Delta \) 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 \( r \)-th order \( \Sigma\Delta \) 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 \( \Sigma\Delta \) scheme.