Note on quantization for signals with bounded (r + 1)st derivative

G. W. Wasilkowski

Research output: Contribution to journalArticlepeer-review


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 Bhr+1 (c + 1 - 2r + 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 2r+2 for initial values of r, and is 2r+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 languageEnglish
Pages (from-to)278-289
Number of pages12
JournalJournal of Complexity
Issue number3
StatePublished - Sep 1990

ASJC Scopus subject areas

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


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