One-bit sigma-delta quantization with exponential accuracy

One-bit quantization is a method of representing band-limited signals by +/-1 sequences that are computed from regularly spaced samples of these signals; as the sampling density lambda --> infinity, 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(-lambda) 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 pi-band-limited signals is at most O(2(-.07lambda)). (C) 2003 Wiley Periodicals, Inc.