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Sampling Theorems for Nonbandlimited Signals

  • P. P. Vaidyanathan
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In recent years many of the results for bandlimited sampling have been extended to the case of nonbandlimited signals. These recent extensions have been found to be useful in digital signal processing applications such as image interpolation, equalization of communication channels, and multiresolution computation. In this chapter we give a brief overview of some of these ideas.

Keywords

Filter Bank Finite Impulse Response Sampling Theorem Finite Impulse Response Filter Synthesis Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. Aldroubi, Oblique projections in atomic spacesProc. Amer. Math. Soc124:2051–2060, 1996MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Aldroubi and K. Grochenig, Nonuniform sampling and reconstruction in shift invariant spacesSIAM Review43:585–620, 2001MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A. Aldroubi and M. Unser, Oblique projections in discrete signal subspaces ofl 2 and the wavelet transform, Proc. SPIE. vol. 2303, Wavelet applications in signal and image processing, II, 36–45, San Diego, CA, 1994Google Scholar
  4. [4]
    I. Daubechies, Ten lectures on waveletsSIAM, CBMS seriesApril 1992Google Scholar
  5. [5]
    I. Djokovic and P. P. Vaidyanathan, Generalized sampling theorems in multiresolution subspacesIEEE Trans. Signal Proc45: 583–599, March 1997CrossRefGoogle Scholar
  6. [6]
    A. J. Jerri, The Shannon Sampling Theorem - its various extensions and applications: a tutorial reviewProc. IEEE65:1565–1596, Nov. 1977CrossRefMATHGoogle Scholar
  7. [7]
    S. Mallat, A theory for multiresolution signal decomposition: the wavelet representationIEEE Trans. Patt. Recog. and Mach. Intell.11:674–693, July 1989CrossRefMATHGoogle Scholar
  8. [8]
    S. MallatA Wavelet Tour of Signal ProcessingAcademic Press, New York, 1998MATHGoogle Scholar
  9. [9]
    H. Nyquist, Certain topics in telegraph transmission theoryAIEE Trans. 47:617–644, Jan. 1928Google Scholar
  10. [10]
    A. V. Oppenheim, A. S. Willsky, and I. T. YoungSignals and SystemsPrentice-Hall, Inc., Englewood Cliffs, NJ, 1983Google Scholar
  11. [11]
    A. Papoulis, Generalized sampling expansionsIEEE Trans. Circuits and Systems24:652–654, Nov. 1977MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    I. J. SchoenbergCardinal Spline InterpolationSIAM, Philadelphia, PA, 1973CrossRefMATHGoogle Scholar
  13. [13]
    C. E. Shannon, Communications in the presence of noiseProc. IRE37:10–21, Jan. 1949MathSciNetCrossRefGoogle Scholar
  14. [14]
    L. Tong, G. Xu, and T. Kailath, Blind identification and equalization based on second order statistics: a time domain approachIEEE Trans. Info. Theory40:340–349, March 1994CrossRefGoogle Scholar
  15. [15]
    J. R. Treichler, I. Fijalkow, and C. R. Johnson, Jr., Fractionally spaced equalizers: how long should they be?IEEE Signal Processing Magazine13:65–81, May 1996.CrossRefGoogle Scholar
  16. [16]
    J. Tuqan and P. P. Vaidyanathan, Oversampling PCM techniques and optimum noise shapers for quantizing a class of non bandlimited signalsIEEE Trans. Signal Proc47:389–407, Feb. 1999CrossRefGoogle Scholar
  17. [17]
    M. Unser, A. Aldroubi, and M. Eden, Fast B-spline transforms for continuous image representation and interpolationIEEE Trans. Patt. Recog. and Mach. Intell.13:277–285, March 1991.CrossRefGoogle Scholar
  18. [18]
    M. Unser, A. Aldroubi, and M. Eden, B-spline signal processing: Part I-TheoryIEEE Trans. Signal Proc41:821–833, Feb. 1993.CrossRefMATHGoogle Scholar
  19. [19]
    M. Unser, A. Aldroubi, and M. Eden, B-spline signal processing: Part II-Efficient design and applicationsIEEE Trans. Signal Proc41:834–848, Feb. 1993.CrossRefMATHGoogle Scholar
  20. [20]
    M.Unser,Sampling-50yearsafterShannonProc.of the IEEE88:569–587, April 2000.CrossRefGoogle Scholar
  21. [21]
    M. Unser and J. Zerubia, A generalized sampling theory without bandlimiting constraintsIEEE Trans. Circuits and Systems, II45:959–969, Aug. 1998.CrossRefMATHGoogle Scholar
  22. [22]
    P. P. Vaidyanathan and S-M. Phoong, Reconstruction of sequences from nonuniform samplesProc. IEEE Int. Symp. Circuits and Sys.Seattle, April-May, 1995.Google Scholar
  23. [23]
    P. P. Vaidyanathan and I. Djokovic, Wavelet transformsThe Circuits and Filters Handbookedited by W. K. Chen, CRC Press, Inc., Boca Raton, FL, 134–219, 1995.Google Scholar
  24. [24]
    P. P. Vaidyanathan, Generalizations of the sampling theorem: seven decades after NyquistIEEE Trans. Circuits and Systems-I48:1094–1109, Sept. 2001.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    B. Vrcelj and P. P. Vaidyanathan, Efficient implementation of all digital interpolationIEEE Trans. Image Proc10:1639–1646, Nov. 2001.CrossRefMATHGoogle Scholar
  26. [26]
    B. Vrcelj and P. P. Vaidyanathan, MIMO biorthogonal partners and applicationsIEEE Trans. Signal Proc50:528–542, March 2002.CrossRefGoogle Scholar
  27. [27]
    B. Vrceljand and P. P. Vaidyanathan, Results on vector biorthogonal partnersProc. Int. Conf. Acoust., Speech, and Sig. ProcVI:3637–3640, Salt Lake City, May 2001Google Scholar
  28. [28]
    P. P. Vaidyanathan and B. Vrcelj, On sampling theorems for non bandlimited signalsProc Int. Conf. Acoust., Speech, and Sig. ProcVI:3897–3900, Salt Lake City, May 2001.Google Scholar
  29. [29]
    X-G. Xia, New precoding for intersymbol interference cancellation using nonmaximally decimated multirate filter banks with ideal FIR equalizersIEEE Trans. Signal Proc45:2431–2441, Oct. 1997.CrossRefGoogle Scholar
  30. [30]
    P. P. Vaidyanathan, Multirate systems and filter banks, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1993.Google Scholar
  31. [31]
    P. P. Vaidyanathan and B. Vrcelj, Biorthogonal parters and applicationsIEEE Trans. Signal Proc49:1013–1027, May 2001.MathSciNetGoogle Scholar
  32. [32]
    G. G. Walter, A sampling theorem for wavelet subspacesIEEE Trans. Info. Theory38:881–884, March 1992.CrossRefMATHGoogle Scholar
  33. [33]

Copyright information

© Springer Science+Business Media New York 2004

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  • P. P. Vaidyanathan

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