Celebratio Mathematica — Daubechies

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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
Martin Cohn	Related
Zoran Dusan Cvetković	Related
James A. Storer	Related

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
Zoran Dusan Cvetković	Related