Celebratio Mathematica

Z. Cvetkovic and I. Daubechies: “Single-bit oversampled A/D conversion with exponential accuracy in the bit-rate,” pp. 343–352 in Proceedings: Data compression conference 2000 (Snowbird, UT, 28–30 March 2000). Edited by J. A. Storer and M. Cohn. IEEE Computer Society Press (Los Alamitos, CA), 2000. Also published in IEEE Trans. Inform. Theory 53:11 (2007). incollection

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Z. Cvetkovic, I. Daubechies, and B. F. Logan: “Interpolation of bandlimited functions from quantized irregular samples,” pp. 412–421 in Proceedings: Data compression conference (Snowbird, UT, 2–4 April 2002). Edited by J. A. Storer and M. Cohn. IEEE (Piscataway, NJ), 2002. incollection

The problem of reconstructing a \( \pi \)-bandlimited signal \( f \) from its quantized samples taken at an irregular sequence of points \( t_k \), \( k = \dots, -2 \), \( -1,0 \), \( 1,\dots \) arises in oversampled analog-to-digital conversion. The input signal can be reconstructed from the quantized samples \( f(t_k) \) by estimating samples \( f(n/\lambda) \), \( n = \dots,-2 \), \( -1,0 \), \( 1,\dots \), where \( \lambda \) is the average uniform density of the sequence \( (t_k) \), assumed here to be greater than one, followed by linear low-pass filtering. We study three techniques for estimating samples \( f(n/\lambda) \) from quantized irregular samples \( f(t_k) \), including Lagrangian interpolation, and two other techniques which result in a better overall accuracy of oversampled A/D conversion.

Benjamin F. Logan, Jr.	Related
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Z. Cvetković, I. Daubechies, and B. F. Logan, Jr.: “Single-bit oversampled A/D conversion with exponential accuracy in the bit rate,” IEEE Trans. Inform. Theory 53 : 11 (2007), pp. 3979–3989. Also published in Proceedings: DCC 2000 (2000). MR 2446550 Zbl 1231.94041 article

A scheme for simple oversampled analog-to-digital (A/D) conversion using single-bit quantization is presented. The scheme is based on recording positions of zero-crossings of the input signal added to a deterministic dither function. This information can be represented in a manner such that the bit rate increases only logarithmically with the oversampling factor \( r \). The input band-limited signal can be reconstructed from this information locally with \( O(1/r) \) pointwise error, resulting in an exponentially decaying distortion-rate characteristic. In the course of studying the accuracy of the proposed A/D conversion scheme, some new results are established about reconstruction of band-limited signals from irregular samples using linear combination of functions with fast decay. Schemes for local interpolation of band-limited signals from quantized irregular samples are also proposed.

Benjamin F. Logan, Jr.	Related
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Ingrid Daubechies

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