Lower bounds for the error decay incurred by coarse quantization schemes

Several analog-to-digital conversion methods for bandlimited sig-nals used in applications, such as Σ ∆ quantization schemes, employ coarse quantization coupled with oversampling. The standard math-ematical model for the error accrued from such methods measures the performance of a given scheme by the rate at which the asso-ciated reconstruction error decays as a function of the oversampling ratio λ. It was recently shown that exponential accuracy of the form O(2−αλ) can be achieved by appropriate one-bit Sigma-Delta modula-tion schemes. However, the best known achievable rate constants α in this setting differ significantly from the general information theoretic lower bound. In this paper, we provide the first lower bound specific to coarse quantization, thus narrowing the gap between existing up-per and lower bounds. In particular, our results imply a quantitative correspondence between the maximal signal amplitude and the best possible error decay rate. Our method draws from the theory of large deviations.