Estimating from outputs of oversampled delta-sigma modulation

Abstract

Oversampling a delta-sigma-modulated sequence, one can compute unbiased sample estimates of averages of consecutive input elements for a wide variety of inputs. We prove that these estimates are most efficient in their class (that is, variances of sample means are minimum in the class of random binary sequences g n n = 1,…, N, such that the expected values of g n are equal to the values of the corresponding inputs of delta-sigma modulation) and consistent. Delta-sigma modulation may also be described as one-dimensional error diffusion (a technique for digital halftoning). However, delta-sigma modulation is not a practical digital halftoning algorithm, because human vision averages small luminance deviations in two dimensions. We pose an open problem that invites the reader to extend our approach to the two-dimensional case for the purpose of development of a practical digital halftoning algorithm.