Sigma-delta quantization and finite frames

It is shown that sigma-delta (/spl Sigma//spl Delta/) algorithms can be used effectively to quantize finite frame expansions for R/sup d/. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that /spl Sigma//spl Delta/ quantizers outperform the standard PCM schemes.

[1]  I. Daubechies,et al.  Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order , 2003 .

[2]  Vivek K. Goyal,et al.  Quantized Overcomplete Expansions in IRN: Analysis, Synthesis, and Algorithms , 1998, IEEE Trans. Inf. Theory.

[3]  Niels Juul Munch.  Noise reduction in tight Weyl-Heisenberg frames , 1992, IEEE Trans. Inf. Theory.

[4]  Jeffrey C. Lagarias,et al.  On the robustness of single-loop sigma-Delta modulation , 2001, IEEE Trans. Inf. Theory.

[5]  John J. Benedetto,et al.  Wavelet Frames: Multiresolution Analysis and Extension Principles , 2001 .

[6]  G. Temes.  Delta-sigma data converters , 1994 .

[7]  Zoran Cvetkovic.  Resilience properties of redundant expansions under additive noise and quantization , 2003, IEEE Trans. Inf. Theory.

[8]  C. S. Gunturk,et al.  Approximating a bandlimited function using very coarsely quantized data: Improved error estimates in sigma-delta modulation , 2003 .

[9]  R. Schreier,et al.  Delta-sigma data converters : theory, design, and simulation , 1997 .

[10]  Christine Guillemot,et al.  Performance analysis and recursive syndrome decoding of DFT codes for bursty erasure recovery , 2003, IEEE Trans. Signal Process..

[11]  Vivek K Goyal,et al.  Multiple description transform coding: robustness to erasures using tight frame expansions , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[12]  Robert M. Gray,et al.  Quantization noise spectra , 1990, IEEE Trans. Inf. Theory.

[13]  Erwin Janssen,et al.  Super-audio CD: an introduction , 2003 .

[14]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[15]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[16]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[17]  Thomas L. Marzetta,et al.  Systematic design of unitary space-time constellations , 2000, IEEE Trans. Inf. Theory.

[18]  B. Torrésani,et al.  Wavelets: Mathematics and Applications , 1994 .

[19]  W. R. Bennett,et al.  Spectra of quantized signals , 1948, Bell Syst. Tech. J..

[20]  Martin Vetterli,et al.  Deterministic analysis of oversampled A/D conversion and decoding improvement based on consistent estimates , 1994, IEEE Trans. Signal Process..

[21]  Antonio Ortega,et al.  Efficient quantization for overcomplete expansions in RN , 2003, IEEE Trans. Inf. Theory.

[22]  John J. Benedetto,et al.  Finite Normalized Tight Frames , 2003, Adv. Comput. Math..

[23]  Vivek K. Goyal,et al.  Quantized frame expansions as source-channel codes for erasure channels , 1999, Proceedings DCC'99 Data Compression Conference (Cat. No. PR00096).

[24]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[25]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[26]  Gabor C. Temes,et al.  Oversampling delta-sigma data converters : theory, design, and simulation , 1992 .

[27]  Peter G. Casazza,et al.  Equal-Norm Tight Frames with Erasures , 2003, Adv. Comput. Math..

[28]  Ozgur Yilmaz,et al.  Coarse quantization of highly redundant time–frequency representations of square-integrable functions , 2003 .

[29]  Helmut Bölcskei,et al.  Noise reduction in oversampled filter banks using predictive quantization , 2001, IEEE Trans. Inf. Theory.

[30]  Georg Zimmermann,et al.  Normalized Tight Frames in Finite Dimensions , 2001 .

[31]  Gabor C. Temes,et al.  Understanding Delta-Sigma Data Converters , 2004 .

[32]  Vivek K Goyal,et al.  Quantized Frame Expansions with Erasures , 2001 .

[33]  N. J. A. Sloane,et al.  Quantizing Using Lattice Intersections , 2002 .

[34]  Yonina C. Eldar,et al.  Optimal tight frames and quantum measurement , 2002, IEEE Trans. Inf. Theory.