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Alternative dual frames for digital-to-analog conversion in sigma–delta quantization

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Abstract

We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 / λ r, and λ > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for \(\mathbb{R}^d\) when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order sigma–delta quantizer to achieve approximation error of order 1/N r for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions.

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References

  1. Aldroubi, A.: Portraits of frames. Proceedings of AMS 123, 357–385 (1995)

    MathSciNet  Google Scholar 

  2. Benedetto, J.J., Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18(2–4), 357–385 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benedetto, J.J., Powell, A.M., Yılmaz, Ö.: Sigma–delta quantization and finite frames. In: Proceedings of the International Conference on Acoustics, Speech and Signal Processing, vol. 3, pp. 937–940. Montreal, QC, Canada (May 2004)

  4. Benedetto, J.J., Powell, A.M., Yılmaz, Ö.: Sigma–delta (ΣΔ) quantization and finite frames. IEEE Trans. Inform. Theory 52(5), 1990–2005 (2006)

    Article  MathSciNet  Google Scholar 

  5. Benedetto, J.J., Powell, A.M., Yılmaz, Ö.: Second order sigma–delta (ΣΔ) quantization of finite frame expansions. Appl. Comput. Harmon. Anal. 20(1), 126–148 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bennett, W.: Spectra of quantized signals. Bell Syst. Tech. J. 27, 446–472 (1949)

    Google Scholar 

  7. Blum, J., Lammers, M.C., Powell, A.M., Yılmaz, Ö.: Sobolev duals in frame theory and sigma–delta quantization (preprint)

  8. Bodmann, B.G., Paulsen, V.I.: Frame paths and error bounds for sigma–delta quantization. Appl. Comput. Harmon. Anal. 22(2), 176–197 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bodmann, B.G., Paulsen, V.I., Abdulbaki, S.: Smooth frame path termination for higher order sigma–delta quantization. J. Fourier Anal. Appl. 13, 285–307 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bölcskei, H., Hlawatsch, F.: Noise reduction in oversampled filter banks using predictive quantization. IEEE Trans. Inform. Theory 47(1), 155–172 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Casazza, P.: The art of frame theory. Taiwanese J. Math. 4(2), 129–210 (2000)

    MATH  MathSciNet  Google Scholar 

  12. Christensen, O., Eldar, Y.: Characterization of oblique dual frame pairs. EURASIP J. Appl. Signal Proc. Frames and overcomplete representations in signal processing, communications, and information theory, pp. 11. Art. ID 92674 (2006)

  13. Cvetković, Z.: Resilience properties of redundant expansions under additive noise quantization. IEEE Trans. Inform. Theory 49(3), 644–656 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Daubechies, I., DeVore, R.: Approximating a bandlimited function from very coarsely quantized data: a family of stable sigma–delta modulators of arbitrary order. Ann. of Math. (2) 158(2), 679–710 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Goyal, V., Kovačević, J., Kelner, J.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10, 203–233 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Goyal, V., Vetterli, M., Thao, N.: Quantized overcomplete expansions in \(\mathbb{R}^n\). IEEE Trans. Inform. Theory 44(1), 16–31 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Güntürk, C.S.: Approximating a bandlimited function using very coarsely quantized data: improved error estimates in sigma–delta modulation. J. Amer. Math. Soc. 17(1), 229–242 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Güntürk, C.S.: One-bit sigma–delta quantization with exponential accuracy. Comm. Pure Appl. Math. 56(11), 1608–1630 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gray, R.: Quantization noise spectra. IEEE Trans. Inform. Theory 36(6), 1220–1244 (1990)

    Article  Google Scholar 

  20. He, N., Kuhlmann, F., Buzo, A.: Multi-loop sigma–delta quantization. IEEE Trans. Inform. Theory 38(3), 1015–1028 (1992)

    Article  MATH  Google Scholar 

  21. Jimenez, D., Wang, L., Wang, Y.: White noise hypothesis for uniform quantization errors. SIAM J. Math. Anal. 38(6), 2042–2056 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kuipers, L., Niederrieter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics, Wiley-Interscience, New York (1974)

    MATH  Google Scholar 

  23. Li, S.: On general frame decompositions. Numer. Funct. Anal. Optim. 16(9–10), 1181–1191 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10(4), 409–431 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Montgomery, H.L.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics, 84, American Mathematical Society, Providence, RI (1994)

    Google Scholar 

  26. Marco, D., Neuhoff, D.: The validity of the additive noise model for uniform scalar quantizers. IEEE Trans. Inform. Theory 51(5), 1739–1755 (2005)

    Article  MathSciNet  Google Scholar 

  27. Pinault, S.C., Lopresti, P.V.: On the behavior of the double-loop sigma–delta modulator. IEEE Trans. Circuits and Syst. II 40(8), 467–479 (1993)

    Article  Google Scholar 

  28. Sripad, A., Snyder, D.: A necessary and sufficient condition for quantization errors to be uniform and white. IEEE Trans. Acoust. Speech Signal Proc. 25(5), 442–448 (1977)

    Article  MATH  Google Scholar 

  29. Thao, N., Vetterli, M.: Deterministic analysis of oversampled A/D conversion and decoding improvement based on consistent estimates. IEEE Trans. Inform. Theory 42(3), 519–531 (1994)

    Google Scholar 

  30. Viswanathan, H., Zamir, R.: On the whiteness of high-resolution quantization errors. IEEE Trans. Inform. Theory, 47(5), 2029–2038 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Yilmaz, O.: Stability analysis for several second-order sigma–delta methods of coarse quantization of bandlimited functions. Constr. Approx. 18(4), 599–623 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zimmermann, G.: Normalized tight frames in finite dimensions. In: Jetter, K., Haussmann, W., Reimer, M. (eds.) Recent Progress in Multivariate Approximation, Birkhäuser (2001)

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina at Wilmington, Wilmington, NC, 28403, USA

    Mark Lammers

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Alexander M. Powell

  3. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

    Özgür Yılmaz

Authors
  1. Mark Lammers
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  2. Alexander M. Powell
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  3. Özgür Yılmaz
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Corresponding author

Correspondence to Özgür Yılmaz.

Additional information

Communicated by Qi Yu Sun.

The first author was supported in part by a Cahill Grant from the University of North Carolina at Wilmington. The second author was supported in part by NSF DMS Grant 0219233, NSF DMS Grant 0504924, and by an Erwin Schrödinger Institute (ESI) Junior Research Fellowship. The third author was supported in part by the Natural Sciences and Engineering Research Council of Canada.

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Lammers, M., Powell, A.M. & Yılmaz, Ö. Alternative dual frames for digital-to-analog conversion in sigma–delta quantization. Adv Comput Math 32, 73 (2010). https://doi.org/10.1007/s10444-008-9088-1

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