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

Alexander M. Powell; Jared Tanner; Yang Wang; Özgür Yılmaz

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 3, page 605-618
  • ISSN: 0764-583X

Abstract

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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 + Tn} k ∈ Z with offsets { 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 | f ( t ) - f ˜ ( t ) | is at most of order N-1log N.

How to cite

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Powell, Alexander M., et al. "Coarse quantization for random interleaved sampling of bandlimited signals∗∗∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.3 (2012): 605-618. <http://eudml.org/doc/222103>.

@article{Powell2012,
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 + Tn\} k ∈ Z with offsets \hbox\{$\\{T_n\\}_\{n=1\}^N\subset [0,T]$\}. 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\{$|f(t) - \widetilde\{f\}(t)|$\}is at most of order N-1log N.},
author = {Powell, Alexander M., Tanner, Jared, Wang, Yang, Yılmaz, Özgür},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Analog-to-digital conversion; bandlimited signals; interleaved sampling; random sampling; sampling expansions; Sigma-Delta quantization; analog-to-digital conversion; sigma-delta quantization},
language = {eng},
month = {1},
number = {3},
pages = {605-618},
publisher = {EDP Sciences},
title = {Coarse quantization for random interleaved sampling of bandlimited signals∗∗∗},
url = {http://eudml.org/doc/222103},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Powell, Alexander M.
AU - Tanner, Jared
AU - Wang, Yang
AU - Yılmaz, Özgür
TI - Coarse quantization for random interleaved sampling of bandlimited signals∗∗∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/1//
PB - EDP Sciences
VL - 46
IS - 3
SP - 605
EP - 618
AB - 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 + Tn} k ∈ Z with offsets \hbox{$\{T_n\}_{n=1}^N\subset [0,T]$}. 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{$|f(t) - \widetilde{f}(t)|$}is at most of order N-1log N.
LA - eng
KW - Analog-to-digital conversion; bandlimited signals; interleaved sampling; random sampling; sampling expansions; Sigma-Delta quantization; analog-to-digital conversion; sigma-delta quantization
UR - http://eudml.org/doc/222103
ER -

References

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  1. R.F. Bass and K. Gröchenig, Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal.36 (2005) 773–795.  Zbl1096.94008
  2. J.J. Benedetto, A.M. Powell and Ö. Yılmaz, Sigma-delta (ΣΔ) quantization and finite frames. IEEE Trans. Inf. Theory52 (2006) 1990–2005.  Zbl1285.94014
  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.  Zbl1058.94004
  4. H.A. David and H.N. Nagarja, Order Statistics, 3th edition. John Wiley & Sons, Hoboken, NJ (2003).  
  5. L. Devroye, Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Probab.9 (1981) 860–867.  Zbl0465.60038
  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 Mathematik65. Birkhäuser, Basel (1984) 355–362.  
  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.  Zbl1032.94502
  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.  Zbl1165.94314
  10. F. Natterer, Efficient evaluation of oversampled functions. J. Comput. Appl. Math.14 (1986) 303–309.  Zbl0632.65142
  11. R.A. Niland, Optimum oversampling. J. Acoust. Soc. Amer.86 (1989) 1805–1812.  
  12. E. Slud, Entropy and maximal spacings for random partitions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete41 (1977/78) 341–352.  Zbl0353.60019
  13. T. Strohmer and J. Tanner, Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling. SIAM J. Numer. Anal.44 (2006) 1073–1094.  Zbl1118.42010
  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.  URIhttp://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.  Zbl1027.94515
  17. A.I. Zayed, Advances in Shannon’s sampling theory. CRC Press, Boca Raton (1993).  Zbl0868.94011

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