Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1460–1490 | Cite as

Sigma Delta Quantization with Harmonic Frames and Partial Fourier Ensembles

  • Rongrong WangEmail author


Sigma Delta (\(\Sigma \Delta \)) quantization, a quantization method first surfaced in the 1960s, has now been widely adopted in various digital products such as cameras, cell phones, radars, etc. The method features a great robustness with respect to quantization noises through sampling an input signal at a Super-Nyquist rate. Compressed sensing (CS) is a frugal acquisition method that utilizes the sparsity structure of an objective signal to reduce the number of samples required for a lossless acquisition. By deeming this reduced number as an effective dimensionality of the set of sparse signals, one can define a relative oversampling/subsampling rate as the ratio between the actual sampling rate and the effective dimensionality. When recording these “compressed” analog measurements via Sigma Delta quantization, a natural question arises: will the signal reconstruction error previously shown to decay polynomially as the increase of the vanilla oversampling rate for the case of band-limited functions, now be decaying polynomially as that of the relative oversampling rate? Answering this question is one of the main goals in this direction. The study of quantization in CS has so far been limited to proving error convergence results for Gaussian and sub-Gaussian sensing matrices, as the number of bits and/or the number of samples grow to infinity. In this paper, we provide a first result for the more realistic Fourier sensing matrices. The main idea is to randomly permute the Fourier samples before feeding them into the quantizer. We show that the random permutation can effectively increase the low frequency power of the measurements, thus enhance the quality of \(\Sigma \Delta \) quantization.


Compressed sensing Sigma delta quantization Partial fourier matrix 

Mathematics Subject Classification




The author would like to thank Rayan Saab, Özgur Yılmaz and Joe-Mei Feng for valuable discussion about the topic. R.  Wang was funded in part by an NSERC Collaborative Research and Development Grant DNOISE II (22R07504).


  1. 1.
    Ai, A., Lapanowski, A., Plan, Y., Vershynin, R.: One-bit compressed sensing with non-gaussian measurements. Linear Algebra Appl. 441, 222–239 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baraniuk, R., Foucart, S., Needell, D., Plan, Y., Wootters, M.: Exponential decay of reconstruction error from binary measurements of sparse signals. arXiv preprint arXiv:1407.8246 (2014)
  3. 3.
    Benedetto, J.J., Powell, A.M., Yılmaz, Ö.: Sigma-delta (\(\Sigma \Delta \)) 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.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.J.: An improved estimate in the restricted isometry problem. In: Geometric Aspects of Functional Analysis. Springer, Berlin, pp. 65–70 (2014)Google Scholar
  7. 7.
    Cai, T., Zhang, A.: Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inf. Theory 60, 122–132 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candes, J.E.: The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique 346(9), 589–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheraghchi, M., Guruswami, V., Velingker, A.: Restricted isometry of fourier matrices and list decodability of random linear codes. SIAM J. Comput. 42(5), 1888–1914 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chou, E., Güntürk, C.S., Krahmer, F., Saab, R., Yılmaz, Ö.: Noise-shaping quantization methods for frame-based and compressive sampling systems. arXiv preprint arXiv:1502.05807 (2015)
  12. 12.
    Chou, E., Güntürk: Distributed noise-shaping quantization: I. beta duals of finite frames and near-optimal quantization of random measurements. arXiv preprint arXiv:1405.4628, (2015)
  13. 13.
    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
  14. 14.
    Deift, P., Güntürk, C.S., Krahmer, F.: An optimal family of exponentially accurate one-bit sigma-delta quantization schemes. Commun. Pure Appl. Math. 64(7), 883–919 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Feng, J., Krahmer, F.: An RIP-based approach to \(\Sigma \Delta \) quantization for compressed sensing. IEEE Signal Process. Lett. 21(11), 1351–1355 (2014)CrossRefGoogle Scholar
  16. 16.
    Foucart, S.: Stability and robustness of \(\ell _1\)-minimizations with Weibull matrices and redundant dictionaries. Linear Algebra Appl. 441, 4–21 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Güntürk, C.S.: One-bit sigma-delta quantization with exponential accuracy. Commun. Pure Appl. Math. 56(11), 1608–1630 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Güntürk, C.S., Lammers, M., Powell, A.M., Saab, R., Yılmaz, Ö.: Sobolev duals for random frames and sigma-delta quantization of compressed sensing measurements. Found. Comput. Math. 13(1), 1–36 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Haviv, I., Regev, O.: The restricted isometry property of subsampled fourier matrices. In: Klartag, B., Milman, E. (eds.) Geometric Aspects of Functional Analysis, pp. 163–179. Springer, Berlin (2017)CrossRefGoogle Scholar
  20. 20.
    Huynh, T.: Accurate quantization in redundant systems: From frames to compressive sampling and phase retrieval. Ph.D. Thesis, New York University (2016)Google Scholar
  21. 21.
    Inose, H., Yasuda, Y.: A unity bit coding method by negative feedback. Proc. IEEE 51(11), 1524–1535 (1963)CrossRefGoogle Scholar
  22. 22.
    Iwen, M., Saab, R.: Near-optimal encoding for sigma-delta quantization of finite frame expansions. J. Fourier Anal. Appl. 19(6), 1255–1273 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jacques, L., Laska, J.N., Boufounos, P.T., Baraniuk, R.G.: Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors. IEEE Trans. Inf. Theory 59(4), 2082–2102 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Knudson, K., Saab, R., Ward, R.: One-bit compressive sensing with norm estimation. arXiv preprint arXiv:1404.6853 (2014)
  25. 25.
    Krahmer, F., Saab, R., Ward, R.: Root-exponential accuracy for coarse quantization of finite frame expansions. IEEE Trans. Inf. Theory 58(2), 1069–1079 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Krahmer, F., Saab, R., Yılmaz, Ö.: Sigma-Delta quantization of sub-Gaussian frame expansions and its application to compressed sensing. Inf. Inference 3(1), 40–58 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lammers, M., Powell, A.M., Yılmaz, Ö.: Alternative dual frames for digital-to-analog conversion in sigma-delta quantization. Adv. Comput. Math. 32, 73–102 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mackey, L., Jordan, M., Chen, R., Farrell, B., Tropp, J.: Matrix concentration inequalities via the method of exchangeable pairs. Ann. Probab. 42(3), 906–945 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maricic, D.: Image sensors employing oversampling sigma-delta analog-to-digital conversion with high dynamic range and low power. University of Rochester, Diss (2011)Google Scholar
  30. 30.
    Pe na, V., Giné, E. (eds.): Decoupling: From Dependence to Independence. Springer, Berlin (2012)Google Scholar
  31. 31.
    Plan, Y., Vershynin, R.: One-bit compressed sensing by linear programming. Commun. Pure Appl. Math. 66(8), 1275–1297 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Plan, Y., Vershynin, R.: Robust 1-bit compressed sensing and sparse logistic regression: a convex programming approach. IEEE Trans. Inf. Theory 59(1), 482–494 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Powell, A.M., Saab, R., Yılmaz, Ö.: Quantization and finite frames. In: Casazza, P.G., Kutyniok, G. (eds.) Finite Frames, ANHA, pp. 267–302. Birkhäuser, Boston (2013)CrossRefGoogle Scholar
  34. 34.
    Rudelson, M., Vershynin, R.: On sparse reconstruction from fourier and gaussian measurements. Commun. Pure Appl. Math. 61(8), 1025–1045 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Saab, R., Wang, R., Yılmaz, Ö.: Quantization of compressive samples with stable and robust recovery. preprint arXiv:1504.00087 (2015)
  36. 36.
    Saab, R., Wang, R., Yılmaz, Ö: From compressed sensing to compressed bit-streams: practical encoders, tractable decoders. IEEE Trans. Inf. Theory (to appear) (2017)Google Scholar
  37. 37.
    Vershynin, R.: A simple decoupling inequality in probability theory. Preprint (2011)Google Scholar

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Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA

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