One-bit sigma-delta quantization with exponential accuracy

Abstract

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.