Coarse quantization for random interleaved sampling of bandlimited signals
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 3, pp. 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{{T n } n=1 N [0,T]} { 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(t)-f ˜(t)|} | f ( t ) - 􏽥 f ( t ) | is at most of order N-1log N.

DOI : 10.1051/m2an/2011057
Classification : 41A30, 94A12, 94A20
Mots clés : 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 },
     pages = {605--618},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {3},
     year = {2012},
     doi = {10.1051/m2an/2011057},
     mrnumber = {2877367},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2011057/}
}
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 
PY  - 2012
SP  - 605
EP  - 618
VL  - 46
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2011057/
DO  - 10.1051/m2an/2011057
LA  - en
ID  - M2AN_2012__46_3_605_0
ER  - 
%0 Journal Article
%A Powell, Alexander M.
%A Tanner, Jared
%A Wang, Yang
%A Yılmaz, Özgür
%T Coarse quantization for random interleaved sampling of bandlimited signals
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 605-618
%V 46
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2011057/
%R 10.1051/m2an/2011057
%G en
%F 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 , Tome 46 (2012) no. 3, pp. 605-618. doi : 10.1051/m2an/2011057. http://www.numdam.org/articles/10.1051/m2an/2011057/

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

[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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

Cité par Sources :