
On Statistical Properties of
DeltaSigmaModulated Sequences?
Eugene A. Sandlery, Dmitri A. Gusevz, Gregory Y. Milman, and Mikhail L. Podolsky
Abstract. Some previously unexplored statistical properties of deltasigmamodulated sequences are important when output of deltasigma modulation is used to estimate values of the original elements. Using a deltasigmamodulated sequence, one can compute unbiased sample estimates of averages of consecutive input elements. 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 gn, 1; : : : ; N , such that the expected values of gn are equal to the values of the corresponding inputs of deltasigma modulation) and consistent. Deltasigma modulation may also be described as onedimensional error diffusion (a technique for digital halftoning). However, deltasigma 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 twodimensional case for the purpose of development of a practical digital halftoning algorithm.
IEEE EDICS number: SP 3.8.2
Correspondence should be sent to:
Dmitri A. Gusev
Indiana University
Computer Science Department
Lindley Hall
Bloomington, IN 474054101
email: [email protected]
phone: (812) 8553703
Keywords. deltasigma modulation, delta modulation, estimate, digital signal processing, statistics, sequence, halftoning, error diffusion, image processing.
? Most of this work was done at Moscow Institute of Radioengineering, Electronics, and Automation (MIREA) in 1992.
The work performed at Indiana University was supported by NSF Grant CCR 9402780.
y (19441995) Associate Professor, Department of Cybernetics, Moscow State Institute of Radioengineering, Electronics,
and Automation (University of Technology).
z Graduate student, Indiana University, Computer Science Department, Lindley Hall, Bloomington, IN 474054101
([email protected]).
Chief EngineerResearcher, VIDAR Ltd., 11 Yauzskaya st., bldg 12, rm 14, Moscow, Russia ([email protected]).
 Yaffa, Israel.