ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

Coarse quantization for random interleaved sampling of bandlimited signals∗∗

Alexander M. Powella1, Jared Tannera2, Yang Wanga3 and Özgür Yılmaza4

a1 Department of Mathematics, Vanderbilt University, Nashville, 37240 TN, USA. alexander.m.powell@vanderbilt.edu

a2 School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, EH9 3JL Edinburgh, UK; jared.tanner@ed.ac.uk

a3 Department of Mathematics, Michigan State University, East Lansing, 48824 MI, USA; ywang@math.msu.edu

a4 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver B.C., V6T 1Z2 Canada; oyilmaz@math.ubc.ca

Abstract

The compatibility of unsynchronized interleaved uniform sampling with Sigma-Delta analog-to-digital conversion is investigated. Let f be a bandlimited signal that is sampled on a collection of N interleaved grids  {kT + Tnk ∈ Z with offsets \hbox{$\{T_n\}_{n=1}^N\subset [0,T]$}{Tn}n=1N⊂[0,T] . If the offsets Tn are chosen independently and uniformly at random from  [0,T]  and if the sample values of f are quantized with a first order Sigma-Delta algorithm, then with high probability the quantization error \hbox{$|f(t) - \widetilde{f}(t)|$}|f(t)−􏽥f(t)| is at most of order N-1log N.

(Received September 30 2009)

(Online publication January 11 2012)

Key Words:

  • Analog-to-digital conversion;
  • bandlimited signals;
  • interleaved sampling;
  • random sampling;
  • sampling expansions;
  • Sigma-Delta quantization

Mathematics Subject Classification:

  • 41A30;
  • 94A12;
  • 94A20

Footnotes

  In memory of David Gottlieb-mentor and friend.

∗∗  A. Powell was supported in part by NSF Grant DMS-0811086. This author is grateful to the Academia Sinica Institute of Mathematics (Taipei, Taiwan) and the City University of Hong Kong for their hospitality and support during extended visits. J. Tanner was supported in part by the Leverhulme Trust. Y. Wang was supported in part by NSF Grant DMS-0813750. Ö.Yılmaz supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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