Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

  • Kung-Ching LinEmail author


In analog-to-digital 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. In this study, similar results are proved for finite unitarily generated frames. We introduce a process called alternative decimation on finite frames that is compatible with first- and second-order \(\Sigma \Delta \) quantization. In both cases, alternative decimation results in exponential error decay with respect to the bit usage.


Decimation Sigma-Delta quantization Unitarily generated frames 

Mathematics Subject Classification




The author greatly acknowledges the support from ARO Grant W911NF-17-1-0014, and John Benedetto for the thoughtful advice and insights. Further, the author appreciates the constructive analysis and suggestions of the referees.


  1. 1.
    Aldroubi, A., Davis, J., Krishtal, I.: Exact reconstruction of spatially undersampled signals in evolutionary systems. arXiv preprint arXiv:1312.3203 (2013)
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benedetto, J.J., Powell, A.M., Yilmaz, O.: Sigma-delta quantization and finite frames. IEEE Trans. Inf. Theory 52(5), 1990–2005 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, J., Lammers, M., Powell, A.M., Yılmaz, Ö.: Sobolev duals in frame theory and sigma-delta quantization. J. Fourier Anal. Appl. 16(3), 365–381 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candy, J.: Decimation for sigma delta modulation. IEEE Trans. Commun. 34, 72–76 (1986)CrossRefGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Chou, E., Güntürk, C.S., Krahmer, F., Saab, R., Yılmaz, Ö.: Noise-Shaping Quantization Methods for Frame-Based and Compressive Sampling Systems, pp. 157–184. Springer, Berlin (2015)zbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chou, W., Wong, P.W., Gray, R.M.: Multistage sigma-delta modulation. IEEE Trans. Inf. Theory 35(4), 784–796 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eldar, Y.C., Bolcskei, H.: Geometrically uniform frames. IEEE Trans. Inf. Theory 49(4), 993–1006 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Forney, G.D.: Geometrically uniform codes. IEEE Trans. Inf. Theory 37(5), 1241–1260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goyal, V.K., Kovačević, J., Kelner, J.A.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10(3), 203–233 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    Güntürk, C.S.: One-bit sigma-delta quantization with exponential accuracy. Commun. Pure Appl. Math. 56(11), 1608–1630 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Inose, H., Yasuda, Y.: A unity bit coding method by negative feedback. Proc. IEEE 51, 1524–1535 (1963)CrossRefGoogle Scholar
  21. 21.
    Lu, Y.M., Vetterli, M.: Distributed spatio-temporal sampling of diffusion fields from sparse instantaneous sources. In: Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2009 3rd IEEE International Workshop on, pp. 205–208 (2009)Google Scholar
  22. 22.
    Lu, Y.M., Vetterli, M.: Spatial super-resolution of a diffusion field by temporal oversampling in sensor networks. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, no. LCAV-CONF-2009-009, pp. 2249–2252 (2009)Google Scholar
  23. 23.
    Tewksbury, S., Hallock, R.W.: Oversampled, linear predictive and noise-shaping coders of order n> 1. IEEE Trans. Circuits Syst. 25(7), 436–447 (1978)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Norbert Wiener Center, Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations