Coarse quantization for random interleaved sampling of bandlimited signals^∗,∗∗

¹ Department of Mathematics, Vanderbilt University, Nashville, 37240 TN, USA
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² School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, EH9 3 JL Edinburgh, UK
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³ Department of Mathematics, Michigan State University, East Lansing, 48824 MI, USA
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⁴ Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver B.C., V6T 1Z2 Canada
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Received: 30 September 2009

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 + T_n} _k ∈ Z with offsets $\mathrm{\{}{\mathit{T}}_{\mathit{n}}{\mathrm{\}}}_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\subset }\mathrm{[}\mathrm{0}\mathit{,T}\mathrm{]}$. If the offsets T_n 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 $\mathrm{\left|}\mathit{f}\mathrm{\right(}\mathit{t}\mathrm{)}\mathrm{-}\begin{array}{}{}_{\mathrm{􏽥}}\\ \mathit{f}\end{array}\mathrm{(}\mathit{t}\mathrm{\left)}\mathrm{\right|}$is at most of order N^-1log N.