Coarse quantization for random interleaved sampling of bandlimited signals
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, p. 605-618

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{${\left\{{T}_{n}\right\}}_{n=1}^{N}\subset \left[0,T\right]$} { T n } n = 1 N ⊂ [ 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{$|f\left(t\right)-\stackrel{˜}{f}\left(t\right)|$} | f ( t ) - 􏽥 f ( t ) | is at most of order N-1log N.

DOI : https://doi.org/10.1051/m2an/2011057
Classification:  41A30,  94A12,  94A20
Keywords: analog-to-digital conversion, bandlimited signals, interleaved sampling, random sampling, sampling expansions, sigma-delta quantization
@article{M2AN_2012__46_3_605_0,
author = {Powell, Alexander M. and Tanner, Jared and Wang, Yang and Y\i lmaz, \"Ozg\"ur},
title = {Coarse quantization for random interleaved sampling of bandlimited signals},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {3},
year = {2012},
pages = {605-618},
doi = {10.1051/m2an/2011057},
mrnumber = {2877367},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_3_605_0}
}

Powell, Alexander M.; Tanner, Jared; Wang, Yang; Yılmaz, Özgür. Coarse quantization for random interleaved sampling of bandlimited signals. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, pp. 605-618. doi : 10.1051/m2an/2011057. http://www.numdam.org/item/M2AN_2012__46_3_605_0/

[1] R.F. Bass and K. Gröchenig, Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal. 36 (2005) 773-795. | MR 2111915 | Zbl 1096.94008

[2] J.J. Benedetto, A.M. Powell and Ö. Yılmaz, Sigma-delta (ΣΔ) quantization and finite frames. IEEE Trans. Inf. Theory 52 (2006) 1990-2005. | Zbl 1285.94014

[3] I. Daubechies and R. Devore, Reconstructing a bandlimited function from very coarsely quantized data : A family of stable sigma-delta modulators of arbitrary order. Ann. Math. 158 (2003) 679-710. | MR 2018933 | Zbl 1058.94004

[4] H.A. David and H.N. Nagarja, Order Statistics, 3th edition. John Wiley & Sons, Hoboken, NJ (2003). | Zbl 0223.62057

[5] L. Devroye, Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Probab. 9 (1981) 860-867. | MR 628878 | Zbl 0465.60038

[6] R. Gervais, Q.I. Rahman and G. Schmeisser, A bandlimited function simulating a duration-limited one, in Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983), Internationale Schriftenreihe zur Numerischen Mathematik 65. Birkhäuser, Basel (1984) 355-362. | MR 820536 | Zbl 0547.41016

[7] C.S. Güntürk, Approximating a bandlimited function using very coarsely quantized data : improved error estimates in sigma-delta modulation. J. Amer. Math. Soc. 17 (2004) 229-242. | MR 2015335 | Zbl 1032.94502

[8] S. Huestis, Optimum kernels for oversampled signals. J. Acoust. Soc. Amer. 92 (1992) 1172-1173.

[9] S. Kunis and H. Rauhut, Random sampling of sparse trigonometric polynomials II. orthogonal matching pursuit versus basis pursuit. Found. Comput. Math. 8 (2008) 737-763. | MR 2461245 | Zbl 1165.94314

[10] F. Natterer, Efficient evaluation of oversampled functions. J. Comput. Appl. Math. 14 (1986) 303-309. | MR 831076 | Zbl 0632.65142

[11] R.A. Niland, Optimum oversampling. J. Acoust. Soc. Amer. 86 (1989) 1805-1812. | MR 1020409

[12] E. Slud, Entropy and maximal spacings for random partitions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 41 (1977/78) 341-352. | MR 488242 | Zbl 0353.60019

[13] T. Strohmer and J. Tanner, Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling. SIAM J. Numer. Anal. 44 (2006) 1073-1094. | MR 2231856 | Zbl 1118.42010

[14] C. Vogel and H. Johansson, Time-interleaved analog-to-digital converters : Status and future directions. Proceedings of the 2006 IEEE International Symposium on Circuits and Systems (ISCAS) (2006) 3386-3389.

[15] J. Xu and T. Strohmer, Efficient calibration of time-interleaved adcs via separable nonlinear least squares. Technical Report, Dept. of Mathematics, University of California at Davis. http://www.math.ucdavis.edu/-strotimer/papers/2006/adc.pdf

[16] Ö. Yılmaz, Coarse quantization of highly redundant time-frequency representations of square-integrable functions. Appl. Comput. Harmonic Anal. 14 (2003) 107-132. | MR 1981204 | Zbl 1027.94515

[17] A.I. Zayed, Advances in Shannon's sampling theory. CRC Press, Boca Raton (1993). | MR 1270907 | Zbl 0868.94011