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High Precision Numerical Implementation of Bandlimited Signal Extrapolation Using Prolate Spheroidal Wave Functions

Conference paper

Abstract

An efficient and reliable yet simple method to extrapolate bandlimited signals up to an arbitrarily high range of frequencies is proposed. The orthogonal properties of linear prolate spheroidal wave functions (PSWFs) are exploited to form an orthogonal basis set needed for synthesis. A significant step in the process is the higher order piecewise polynomial approximation of the overlap integral required for obtaining the expansion coefficients accurately with very high precision. A PSWFs set having a fixed Slepian frequency is utilized for performing extrapolation. Numerical results of extrapolation of some standard test signals using our algorithm are presented, compared, discussed, and some interesting inferences are made.

Keywords

Bandlimited signals High-precision numerical integration Linear prolate spheroidal wave functions Overlap integral Signal extrapolation Slepian series 

Notes

Acknowledgment

The authors thank Jose Gonzalez-Cueto and William Phillips for their helpful suggestions on the material. Partial support by Applied Science in Photonics and Innovative Research in Engineering (ASPIRE), a program under Collaborative Research and Training Experience (CREATE) program funded by Natural Sciences and Engineering Research Council (NSERC) of Canada, is also acknowledged.

References

  1. 1.
    M. Abramowitz, I. Stegen, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Addison-Wesley, New York, 1972)MATHGoogle Scholar
  2. 2.
    D. Slepian, H.O. Pollack, Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I. Bell Syst. Tech. J. 40, 43–63 (1961)CrossRefMATHGoogle Scholar
  3. 3.
    D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Techn. J. 43, 3009–3057 (1962)CrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Slepian, Some asymptotic expansions for prolate spheroidal wave functions. J. Math. Phys. 44, 99–140 (1965)MATHMathSciNetGoogle Scholar
  5. 5.
    D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25, 379–393 (1983)Google Scholar
  6. 6.
    K. Khare, N. George, Sampling theory approach to prolate spheroidal wave functions, J. Phys. A: Math. Gen. 36, 10011–10021 (2003)Google Scholar
  7. 7.
    P. Kirby, Calculation of spheroidal wave functions, Comput. Phys. Comm. 175(7), 465–472 (2006)Google Scholar
  8. 8.
    I.C. Moore, M. Cada, Prolate spheroidal wave functions, an introduction to the Slepian series and its properties. Appl. Comput. Harmon. Anal. 16, 208–230 (2004)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    H. Xiao, V. Rokhlin, N. Yarvin, Prolate spheroidal wave functions, quadrature and interpolation, Inverse Prob. 17, 805–838 (2001)Google Scholar
  10. 10.
    S. Senay, L.F. Chaparro, A. Akan, Sampling and reconstruction of non-bandlimited signals using Slepian functions, in EUSIPCO 2008, Lousanne, 25–29 Aug 2008Google Scholar
  11. 11.
    S. Senay, L.F. Chaparro, L. Durak, Reconstruction of non-uniformly sampled time-limited signals using prolate spheroidal wave functions. Sig. Process. 89, 2585–2595 (2009)CrossRefMATHGoogle Scholar
  12. 12.
    S. Senay, J. Oh, L.F. Chaparro, Regularized signal reconstruction for level-crossing sampling using Slepian functions. Sig. Process. 92, 1157–1165 (2012)CrossRefGoogle Scholar
  13. 13.
    M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations. J. Math. Phys. 12, 1772–1783 (1971)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000)Google Scholar
  15. 15.
    H. Zhao, R.Y. Wang, D.P. Song, D.P. Wu, An extrapolation algorithm for M-bandlimited signals. IEEE Signal Process. Lett. 18(12), 745–748 (2011)CrossRefGoogle Scholar
  16. 16.
    H. Zhao et al., Extrapolation of discrete bandlimited signals in linear canonical transform domain. Signal Process. (2013). http://dx.doi.org/10.1016/j.sigpro.2013.06.001
  17. 17.
    J. Shi, X. Sha, Q. Zhang, N. Zhang, Extrapolation of bandlimited signals in linear canonical transform domain, IEEE Trans. Signal Process. 60(3), 1502–1508 (2012)Google Scholar
  18. 18.
    H. Zhao, Q.W. Ran, J. Ma, L.Y. Tan, Generalized prolate spheroidal wave functions associated with linear canonical transform. IEEE Trans. Signal Process. 58(6), 3032–3041 (2010)CrossRefMathSciNetGoogle Scholar
  19. 19.
    R. Gerchberg, Super-resolution through error energy reduction. Opt. Acta. 12(9), 709–720 (1974)CrossRefGoogle Scholar
  20. 20.
    A. Papoulis, A new algorithm in spectral analysis and band-limited extrapolation. IEEE Trans. Circuit Syst. CAS-22(9), 735–742 (1975)Google Scholar
  21. 21.
    L. Gosse, Effective band-limited extrapolation relying on Slepian series and ℓ1 regularization. Comput. Math. Appl. 60, 1259–1279 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    E.J. Candès, Compressive sampling, in Proceedings of the International Congress of Mathematicians, Madrid, 2006Google Scholar
  23. 23.
    E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris, Ser. I 346, 589–592 (2008)Google Scholar
  24. 24.
    M. Cada, Private communication, Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, B3H 4R2, Canada, (2012)Google Scholar
  25. 25.
    C. Flammer, Spheroidal Wave Functions (Stanford Univ. Press, Stanford, 1956)Google Scholar
  26. 26.
    V. Rokhlin, H. Xiao, Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit. Appl. Comput. Harmon. Anal. 22, 105–123 (2007)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    A. Kaw, E.E. Kalu, D. Nguyen, Numerical Methods with Applications, 2nd ed. (2011) (Abridged) Google Scholar
  28. 28.
    A. Devasia, M. Cada, Extrapolation of bandlimited signals using Slepian functions, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering and Computer Science 2013, WCECS 2013, 23–25 October 2013, San Francisco, pp. 492–497Google Scholar
  29. 29.
    A. Devasia, M. Cada, Bandlimited Signal Extrapolation Using Prolate Spheroidal Wave Functions. IAENG Int. J. Comput. Sci. 40(4), 291–300 (2013) Google Scholar
  30. 30.
    I. Kauppinen, K. Roth, Audio signal extrapolation-theory and applications, in Proceedings of (DAFx-02), Hamburg, 26–28 Sept 2002Google Scholar
  31. 31.
    A. Kaup, K. Meisinger, T. Aach, Frequency selective signal extrapolation with applications to error concealment in image communication. Int. J. Electron. Commun. 59, 147–156 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringDalhousie UniversityHalifaxCanada

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