### Abstract

Sigma-delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio λ. It was recently shown that exponential accuracy of the form O(2^{-rλ}) can be achieved by appropriate one-bit sigma-delta modulation schemes. By general information-entropy arguments, r must be less than 1. The current best-known value for r is approximately 0:088. The schemes that were designed to achieve this accuracy employ the "greedy" quantization rule coupled with feedback filters that fall into a class we call "minimally supported." In this paper, we study the discrete minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best-known exponential error decay rate to r ≈ 0.102. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind.

Original language | English (US) |
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Pages (from-to) | 883-919 |

Number of pages | 37 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 64 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2011 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics