One-bit sigma-delta quantization with exponential accuracy
Article first published online: 10 JUL 2003
Copyright © 2003 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 56, Issue 11, pages 1608–1630, November 2003
How to Cite
Güntürk, C. S. (2003), One-bit sigma-delta quantization with exponential accuracy. Comm. Pure Appl. Math., 56: 1608–1630. doi: 10.1002/cpa.3044
- Issue published online: 5 SEP 2003
- Article first published online: 10 JUL 2003
- Manuscript Received: SEP 2002
- Courant Institute
- National Science Foundation Grant. Grant Number: DMS-0219072
One-bit quantization is a method of representing bandlimited signals by ±1 sequences that are computed from regularly spaced samples of these signals; as the sampling density λ → ∞, convolving these one-bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. However, unlike fine-resolution quantization, the accuracy of one-bit quantization is not well-understood. A natural error lower bound that decreases like 2−λ can easily be given using information theoretic arguments. Yet, no one-bit quantization algorithm was known with an error decay estimate even close to exponential decay. In this paper, we construct an infinite family of one-bit sigma-delta quantization schemes that achieves this goal. In particular, using this family, we prove that the error signal for π-bandlimited signals is at most O(2−.07λ). © 2003 Wiley Periodicals, Inc.