On the robustness of single-loop sigma-delta modulation

C. Sinan Güntürk, J. C. Lagarias, V. A. Vaishampayan

Research output: Contribution to journalArticlepeer-review

Abstract

Sigma-delta modulation, a widely used method of analog-to-digital (A/D) signal conversion, is known to be robust to hardware imperfections, i.e., bit streams generated by slightly imprecise hardware components can be decoded comparably well. We formulate a model for robustness and give a rigorous analysis for single-loop sigma-delta modulation applied to constant signals (dc inputs) for N time cycles, with an arbitrary (small enough) initial condition u0, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with u0 and the offset error known) is bounded below by 1/96 N-3. We also determine the asymptotically best possible MSE as N → ∞ for perfect decoding when u0 = 0 and u0 = 1/2. The robustness result is the upper bound that a triangular linear filter decoder (with both u0 and the offset error unknown) achieves an MSE of 40/3 N-3. These results estab lish the known result that the O(1/N3) decay of the MSE with N is optimal in the single-loop case, under weaker assumptions than previous analyses, and show that a suitable linear decoder is robust against offset error. These results are obtained using methods from number theory and Fourier analysis.

Original languageEnglish (US)
Pages (from-to)1735-1744
Number of pages10
JournalIEEE Transactions on Information Theory
Volume47
Issue number5
DOIs
StatePublished - Jul 2001

Keywords

  • Dynamical systems
  • Oversampled quantization
  • Quantization
  • Robustness
  • Sigma-delta modulation

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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