Distributed Noise-Shaping Quantization: II. Classical Frames

  • Evan Chou
  • C. Sinan GüntürkEmail author
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


This chapter constitutes the second part in a series of papers on distributed noise-shaping quantization. In the first part, the main concept of distributed noise shaping was introduced and the performance of distributed beta encoding coupled with reconstruction via beta duals was analyzed for random frames (Chou and Güntürk, Constr Approx 44(1):1–22, 2016). In this second part, the performance of the same method is analyzed for several classical examples of deterministic frames. Particular consideration is given to Fourier frames and frames used in analog-to-digital conversion. It is shown in all these examples that entropic rate-distortion performance is achievable.


Finite frames quantization A/D conversion noise shaping beta encoding beta duals. 


  1. 1.
    R. Adler, T. Nowicki, G. Świrszcz, C. Tresser, Convex dynamics with constant input. Ergodic Theory Dyn. Syst. 30, 957–972 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J.J. Benedetto, A.M. Powell, Ö. Yılmaz, Sigma-delta quantization and finite frames. IEEE Trans. Inf. Theory 52(5), 1990–2005 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J.J. Benedetto, O. Oktay, A. Tangboondouangjit, Complex sigma-delta quantization algorithms for finite frames, in Radon Transforms, Geometry, and Wavelets. Contemporary Mathematics, vol. 464 (American Mathematical Society, Providence, RI, 2008), pp. 27–49Google Scholar
  4. 4.
    B.G. Bodmann, V.I. Paulsen, Frame paths and error bounds for sigma-delta quantization. Appl. Comput. Harmon. Anal. 22(2), 176–197 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P.G. Casazza, S.J. Dilworth, E. Odell, Th. Schlumprecht, A. Zsk, Coefficient quantization for frames in Banach spaces. J. Math. Anal. Appl. 348(1), 66–86 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    E. Chou, C.S. Güntürk, Distributed noise-shaping quantization: I. Beta duals of finite frames and near-optimal quantization of random measurements. Constr. Approx. 44(1), 1–22 (2016)zbMATHGoogle Scholar
  7. 7.
    I. Daubechies, R. DeVore, 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
  8. 8.
    C.S. Güntürk, Mathematics of analog-to-digital conversion. Comm. Pure Appl. Math. 65(12), 1671–1696 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A.M. Powell, R. Saab, Ö. Yılmaz, Quantization and finite frames, in Finite Frames: Theory and Applications, ed. by G.P. Casazza, G. Kutyniok. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2013), pp. 267–302Google Scholar
  10. 10.
    R. Schreier, G.C. Temes, Understanding Delta-Sigma Data Converters (Wiley/IEEE Press, Chichester, 2004)CrossRefGoogle Scholar
  11. 11.
    Y. Wang, Sigma-delta quantization errors and the traveling salesman problem. Adv. Comput. Math. 28(2), 101–118 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.GoogleNew YorkUSA
  2. 2.Courant Institute, NYUNew YorkUSA

Personalised recommendations