## Abstract

We prove that signals with bounded (r + 1)st derivative can be quantized using a uniform c-level quantizer with the sample quantization error bounded by Bh^{r+1} (c + 1 - 2^{r + 1}). Here B is the bound on ∥s^{(r+1)}∥_{∞} and h is the sampling interval. Next, for this quantization we study the optimal choice of c that minimizes the worst case error in reconstructing s by a piecewise-polynomial function subject to a constraint that the number (log c)/h of bits per second used to represent the signal is held fixed. This optimal level is 2^{r+2} for initial values of r, and is 2^{r+3} for large r. Finally, we discuss a problem of representing a signal using a finite amount of memory. We prove that this quantization is almost optimal among all representations. (Here by optimal we mean a representation that allows one to reconstruct signals with the worst case error as small as possible.) This, in particular, provides an answer to the question of optimal trade-off between sampling and quantization in signal processing.

Original language | English |
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Pages (from-to) | 278-289 |

Number of pages | 12 |

Journal | Journal of Complexity |

Volume | 6 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1990 |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Mathematics (all)
- Control and Optimization
- Applied Mathematics