### Abstract

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 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-band-limited signals is at most O (2^{-.07λ}).

Original language | English (US) |
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Pages (from-to) | 1608-1630 |

Number of pages | 23 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 56 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2003 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics