Skip to main content
SpringerLink
Log in

Adapted Decimation on Finite Frames for Arbitrary Orders of Sigma-Delta Quantization

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the \(\Sigma \Delta \) quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. We build on our previous result, which extended signal decimation to finite frames, albeit only up to the second order. In this study, we introduce a new scheme called adapted decimation, which yields polynomial reconstruction error decay rate of arbitrary order with respect to the oversampling ratio, and exponential with respect to the bit-rate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aldroubi, A., Davis, J., Krishtal, I.: Exact reconstruction of spatially undersampled signals in evolutionary systems. arXiv preprint arXiv:1312.3203 (2013)

  2. Aldroubi, A., Davis, J., Krishtal, I.: Exact reconstruction of signals in evolutionary systems via spatiotemporal trade-off. J. Fourier Anal. Appl. 21(1), 11–31 (2015)

    Article  MathSciNet  Google Scholar 

  3. Benedetto, J.J., Powell, A.M., Yilmaz, O.: Sigma-delta quantization and finite frames. IEEE Trans. Inf. Theory 52(5), 1990–2005 (2006)

    Article  MathSciNet  Google Scholar 

  4. Blum, J., Lammers, M., Powell, A.M., Yılmaz, O.: Sobolev duals in frame theory and sigma-delta quantization. J. Fourier Anal. Appl. 16(3), 365–381 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bodmann, B.G., Paulsen, V.I., Abdulbaki, S.A.: Smooth frame-path termination for higher order sigma-delta quantization. J. Fourier Anal. Appl. 13(3), 285–307 (2007)

    Article  MathSciNet  Google Scholar 

  6. Candy, J.: Decimation for sigma delta modulation. IEEE Trans. Commun. 34(1), 72–76 (1986)

    Article  Google Scholar 

  7. Chou, E., Güntürk, C.S.: Distributed noise-shaping quantization: Ii. classical frames. In: Excursions in Harmonic Analysis, The February Fourier Talks at the Norbert Wiener Center, vol. 5, pp. 179–198 (2017)

    Chapter  Google Scholar 

  8. Chou, E., Güntürk, C.S., Krahmer, F., Saab, R., Yılmaz, O.: Noise-shaping quantization methods for frame-based and compressive sampling systems, pp. 157–184. Springer, New York (2015)

    Chapter  Google Scholar 

  9. Chou, E., Güntürk, C.S.: Distributed noise-shaping quantization: I. Beta duals of finite frames and near-optimal quantization of random measurements. Constr. Approx. 44(1), 1–22 (2016)

    Article  MathSciNet  Google Scholar 

  10. Chou, W., Wong, P.W., Gray, R.M.: Multistage sigma-delta modulation. IEEE Trans. Inf. Theory 35(4), 784–796 (1989)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Daubechies, I., DeVore, R.A., Gunturk, C.S., Vaishampayan, V.A.: A/d conversion with imperfect quantizers. IEEE Trans. Inf. Theory 52(3), 874–885 (2006)

    Article  MathSciNet  Google Scholar 

  13. Daubechies, I., Saab, R.: A deterministic analysis of decimation for sigma-delta quantization of bandlimited functions. IEEE Signal Process. Lett. 22(11), 2093–2096 (2015)

    Article  Google Scholar 

  14. Deift, P., Krahmer, F., Güntürk, C.S.: An optimal family of exponentially accurate one-bit sigma-delta quantization schemes. Commun. Pure Appl. Math. 64, 883–919 (2011)

    Article  MathSciNet  Google Scholar 

  15. Eldar, Y.C., Bolcskei, H.: Geometrically uniform frames. IEEE Trans. Inf. Theory 49(4), 993–1006 (2003)

    Article  MathSciNet  Google Scholar 

  16. Ferguson, P.F., Ganesan, A., Adams, R.W.: One bit higher order sigma-delta A/D converters. In: IEEE International Symposium on Circuits and Systems, pp. 890–893 (1990)

  17. Forney, G.D.: Geometrically uniform codes. IEEE Trans. Inf. Theory 37(5), 1241–1260 (1991)

    Article  MathSciNet  Google Scholar 

  18. Goyal, V.K., Kovačević, J., Kelner, J.A.: Quantized frame expansions with erasures. Appl. Comput. Harmonic Anal. 10(3), 203–233 (2001)

    Article  MathSciNet  Google Scholar 

  19. Goyal, V.K., Kovacevic, J., Vetterli, M.: Quantized frame expansions as source-channel codes for erasure channels. In: Proceedings DCC’99 Data Compression Conference (Cat. No. PR00096), pp. 326–335 (1999)

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

    Article  MathSciNet  Google Scholar 

  21. Huynh, T., Saab, R.: Fast binary embeddings, and quantized compressed sensing with structured matrices. arXiv preprint arXiv:1801.08639 (2018)

  22. Inose, H., Yasuda, Y.: A unity bit coding method by negative feedback. Proc. IEEE 51, 1524–1535 (1963)

    Article  Google Scholar 

  23. Iwen, M., Saab, R.: Near-optimal encoding for sigma-delta quantization of finite frame expansions. J. Fourier Anal. Appl. 19(6), 1255–1273 (2013)

    Article  MathSciNet  Google Scholar 

  24. Lin, K.-C.: Analysis of decimation on finite frames with sigma-delta quantization. Constr. Approx. 50(3), 507–542 (2019)

    Article  MathSciNet  Google Scholar 

  25. Lu, Y.M, Vetterli, M.: Distributed spatio-temporal sampling of diffusion fields from sparse instantaneous sources. In: 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 205–208 (2009)

  26. Lu, Y.M, Vetterli, M.: Spatial super-resolution of a diffusion field by temporal oversampling in sensor networks. In: Proc. IEEE International Conference on Acoustics, Speech and Signal Processin, no. LCAV-CONF-2009-009, pp. 2249–2252 (2009)

  27. Tewksbury, S., Hallock, R.W.: Oversampled, linear predictive and noise-shaping coders of order \(\text{ n }> 1\). IEEE Trans. Circ. Syst. 25(7), 436–447 (1978)

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank the support from ARO Grant W911NF-17-1-0014, NSF-DMS Grant 1814253, and J. Benedetto for all the thoughtful advice and insights. Further, the author appreciates the constructive analysis and suggestions of the referees.

Author information

Authors and Affiliations

  1. Department of Mathematics, Norbert Wiener Center, University of Maryland, College Park, MD, 20742, USA

    Kung-Ching Lin

Authors
  1. Kung-Ching Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kung-Ching Lin.

Additional information

Communicated by Gitta Kutyniok.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, KC. Adapted Decimation on Finite Frames for Arbitrary Orders of Sigma-Delta Quantization. J Fourier Anal Appl 26, 20 (2020). https://doi.org/10.1007/s00041-020-09728-8

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-020-09728-8

Keywords

Mathematics Subject Classification

Search

Navigation