Coarse quantization for random interleaved sampling of bandlimited signals

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

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

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

Abstract

top
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 ∈ Zwith offsets { T n } n = 1 N ⊂ [ 0 , T ] T n n = 1 N ⊂ [ 0 ,T ] . If the offsetsTn are chosen independently and uniformly at random from  [0,T]  and if the sample values of fare quantized with a first order Sigma-Delta algorithm, then with high probability the quantization error | f ( t ) - f ˜ ( t ) | | f ( t ) − x10ff65; f ( t ) | is at most of orderN-1log N.

How to cite

top

Powell, Alexander M., et al. "Coarse quantization for random interleaved sampling of bandlimited signals." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.3 (2012): 605-618. <http://eudml.org/doc/273138>.

@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 ∈ Zwith offsets $\lbrace T_n\rbrace _\{n=1\}^N\subset [0,T]$ T n n = 1 N ⊂ [ 0 ,T ] . If the offsetsTn are chosen independently and uniformly at random from  [0,T]  and if the sample values of fare quantized with a first order Sigma-Delta algorithm, then with high probability the quantization error $|f(t) - \widetilde\{f\}(t)|$ | f ( t ) − x10ff65; f ( t ) | is at most of orderN-1log N.},
author = {Powell, Alexander M., Tanner, Jared, Wang, Yang, Yılmaz, Özgür},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {analog-to-digital conversion; bandlimited signals; interleaved sampling; random sampling; sampling expansions; sigma-delta quantization},
language = {eng},
number = {3},
pages = {605-618},
publisher = {EDP-Sciences},
title = {Coarse quantization for random interleaved sampling of bandlimited signals},
url = {http://eudml.org/doc/273138},
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 - Modélisation Mathématique et Analyse Numérique
PY - 2012
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 ∈ Zwith offsets $\lbrace T_n\rbrace _{n=1}^N\subset [0,T]$ T n n = 1 N ⊂ [ 0 ,T ] . If the offsetsTn are chosen independently and uniformly at random from  [0,T]  and if the sample values of fare quantized with a first order Sigma-Delta algorithm, then with high probability the quantization error $|f(t) - \widetilde{f}(t)|$ | f ( t ) − x10ff65; f ( t ) | is at most of orderN-1log N.
LA - eng
KW - analog-to-digital conversion; bandlimited signals; interleaved sampling; random sampling; sampling expansions; sigma-delta quantization
UR - http://eudml.org/doc/273138
ER -

References

top
  1. [1] R.F. Bass and K. Gröchenig, Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal.36 (2005) 773–795. Zbl1096.94008MR2111915
  2. [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. [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. Zbl1058.94004MR2018933
  4. [4] H.A. David and H.N. Nagarja, Order Statistics, 3th edition. John Wiley & Sons, Hoboken, NJ (2003). Zbl0223.62057
  5. [5] L. Devroye, Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Probab.9 (1981) 860–867. Zbl0465.60038MR628878
  6. [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. Zbl0547.41016MR820536
  7. [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. Zbl1032.94502MR2015335
  8. [8] S. Huestis, Optimum kernels for oversampled signals. J. Acoust. Soc. Amer.92 (1992) 1172–1173. 
  9. [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. Zbl1165.94314MR2461245
  10. [10] F. Natterer, Efficient evaluation of oversampled functions. J. Comput. Appl. Math.14 (1986) 303–309. Zbl0632.65142MR831076
  11. [11] R.A. Niland, Optimum oversampling. J. Acoust. Soc. Amer.86 (1989) 1805–1812. MR1020409
  12. [12] E. Slud, Entropy and maximal spacings for random partitions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 41 (1977/78) 341–352. Zbl0353.60019MR488242
  13. [13] T. Strohmer and J. Tanner, Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling. SIAM J. Numer. Anal.44 (2006) 1073–1094. Zbl1118.42010MR2231856
  14. [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. [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. [16] Ö. Yılmaz, Coarse quantization of highly redundant time-frequency representations of square-integrable functions. Appl. Comput. Harmonic Anal.14 (2003) 107–132. Zbl1027.94515MR1981204
  17. [17] A.I. Zayed, Advances in Shannon’s sampling theory. CRC Press, Boca Raton (1993). Zbl0868.94011MR1270907

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.