Skip to main content
SpringerLink
Log in
Book cover

Harmonic Analysis and Applications pp 253–287Cite as

Periodic Nonuniform Sampling in Shift-Invariant Spaces

  • Chapter

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

This chapter reviews several ideas that grew out of observations of Djokovic and Vaidyanathan to the effect that a generalized sampling method for bandlimited functions, due to Papoulis, could be carried over in many cases to the spline spaces and other shift-invariant spaces. Papoulis’ method is based on the sampling output of linear, time-invariant systems. Unser and Zerubia formalized Papoulis’ approach in the context of shift-invariant spaces. However, it is not easy to provide useful conditions under which the Unser-Zerubia criterion provides convergent and stable sampling expansions. Here we review several methods for validating the Unser-Zerubia approach for periodic nonuniform sampling, which is a very special case of generalized sampling. The Zak transform plays an important role.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Aldroubi and H. Feichtinger, Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: the L p-theory, Proc. Amer. Math. Soc., 126 (1998), pp. 2677–2686.

    Article  MATH  Google Scholar 

  2. A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Review, 43 (2001), pp. 585–620.

    Article  MATH  Google Scholar 

  3. A. Aldroubi and K. Gröchenig, Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces, J. Fourier Anal. Appl., 6 (2000), pp. 93–103.

    Article  MATH  Google Scholar 

  4. A. Aldroubi and M. Unser, Projection based prefiltering for multiwavelet transforms, IEEE Trans. Signal Proc., 46 (1998), pp. 3088–3092.

    Article  Google Scholar 

  5. X. Aragones, J. L. Gonzalez and A. Rubio, Analysis and Solutions for Switching Noise in Coupling Mixed Signal ICs, Kluwer, Dordrecht, 1999.

    Google Scholar 

  6. N. Atreas, J. J. Benedetto and C. Karanikas, Local sampling for regular wavelet and Gabor expansions, Sampl. Theory Signal Image Process., 2 (2003), pp. 1–24.

    MATH  Google Scholar 

  7. A. J. Bastys, Translation invariance of orthogonal multiresolution analyses of L 2(ℝ), Appl. Comput. Harmon. Anal., 9 (2000), pp. 128–145.

    Article  MATH  Google Scholar 

  8. A. J. Bastys, Orthogonal and biorthogonal scaling functions with good translation invariance characteristic, in: Proc. SampTA, Aveiro (1997), pp. 239–244.

    Google Scholar 

  9. J. J. Benedetto, Frames, sampling, and seizure prediction, in: Advances in Wavelets (Hong Kong, 1997), Springer, Singapore, 1999, pp. 1–25.

    Google Scholar 

  10. J. J. Benedetto, Harmonic Analysis and its Applications, CRC Press, Boca Raton, 1997.

    Google Scholar 

  11. J. J. Benedetto, Frame decompositions, sampling, and uncertainty principle inequalities, in: Wavelets: Mathematics and Applications, CRC Press, Boca Raton, 1994, pp. 247–304.

    Google Scholar 

  12. J. J. Benedetto, Irregular sampling and frames, in: Wavelets: A Tutorial in Theory and Applications, C. K. Chui ed., CRC Press, Boca Raton, 1992, pp. 445–507.

    Google Scholar 

  13. J. J. Benedetto, Spectral Synthesis, Academic Press, New York, 1975.

    Google Scholar 

  14. J. J. Benedetto and P. J. S. G. Ferreira, Introduction, in: Modern Sampling Theory, J. J. Benedetto and P. J. S. G. Ferreira, eds., Birkhäuser, Boston, 2001, pp. 1–26.

    Google Scholar 

  15. J. J. Benedetto and W. Heller, Irregular sampling and the theory of frames. I, Note Mat., 10, pp. 103–125.

    Google Scholar 

  16. J. J. Benedetto and S. Scott, Frames, irregular sampling, and a wavelet auditory model, in: Nonuniform Sampling, Kluwer/Plenum, New York, 2001, pp. 585–617.

    Google Scholar 

  17. J. J. Benedetto and O. Treiber, Wavelet frames: multiresolution analysis and extension principles, in: Wavelet Transforms and Time-Frequency Signal Analysis, L. Debnath, ed., Birkhäuser, Boston, 2001, pp. 3–36.

    Google Scholar 

  18. J. J. Benedetto, Ö. Yilmaz, and A. M. Powell, Sigma-Delta quantization and finite frames, in: Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), Vol. 3, Montreal, Canada, 2004, pp. 937–940.

    Google Scholar 

  19. J. J. Benedetto and G. Zimmermann, Sampling operators and the Poisson summation formula, J. Fourier Anal. Appl., 3 (1997), pp. 505–523.

    Article  MATH  Google Scholar 

  20. W. R. Bennett, Spectra of quantized signals, Bell System Tech. J., 27 (1948), pp. 446–472.

    Google Scholar 

  21. H. Bölcskei, Oversampling in wavelet subspaces, in: Proc. IEEE-SP 1998 Int. Sympos. Time-Frequency Time-Scale Analysis, Pittsburgh, 1998, pp. 489–492.

    Google Scholar 

  22. R. Bracewell, The Fourier Transform and its Applications, McGraw-Hill, New York, 1965.

    MATH  Google Scholar 

  23. Y. Bresler and R. Venkataramani, Sampling theorems for uniform and periodic nonuniform MIMO sampling of multiband signals, IEEE Trans. Signal Process., 51 (2003), pp. 3152–3163.

    Article  Google Scholar 

  24. S. D. Casey and D. F. Walnut, Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms, SIAM Review, 36 (1994), pp. 537–577.

    Article  MATH  Google Scholar 

  25. W. Chen, S. Itoh and J. Shiki, Irregular sampling theorems for wavelet subspaces, IEEE Trans. Inform. Theory, 44 (1998), pp. 1131–1142.

    Article  MATH  Google Scholar 

  26. M. Croft and J. A. Hogan, Wavelet-based signal extrapolation, in: Proc. Fourth Int. Symp. Sig. Proc. and Appl., Gold Coast, Australia, 2 (1996), pp. 752–755.

    Google Scholar 

  27. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

    MATH  Google Scholar 

  28. I. Djokovic and P. P. Vaidyanathan, Generalized sampling theorems in multiresolution subspaces, IEEE Trans. Signal Proc., 45 (1997), pp. 583–599.

    Article  Google Scholar 

  29. C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), pp. 628–666.

    Article  MATH  Google Scholar 

  30. C. Herley and P.-W. Wong, Minimum rate sampling and reconstruction of signals with arbitrary frequency support, IEEE Trans. Inform. Theory, 45 (1999), pp. 1555–1564.

    Article  MATH  Google Scholar 

  31. J. A. Hogan and J. D. Lakey, Sampling and aliasing without translation-invariance, in: Proc. SampTA, Orlando, 2001, pp. 61–66.

    Google Scholar 

  32. J. A. Hogan and J. D. Lakey, Sampling for shift-invariant and wavelet subspaces, in: “Wavelet Applications in Signal and Image Processing VIII” (San Diego, CA, 2000), Proc. SPIE 4119, A. Aldroubi, A. F. Laine, and M. A. Unser, eds., SPIE, Bellingham, WA, 2000, pp. 36–47.

    Google Scholar 

  33. J. A. Hogan and J. D. Lakey, Sampling and oversampling in shift-invariant and multiresolution spaces I: Validation of sampling schemes, Int. J. Wavelets Multiresolut. Inf. Process., 3 (2005), pp. 257–281.

    Article  MATH  Google Scholar 

  34. J. A. Hogan and J. D. Lakey, Sampling and oversampling in shift-invariant and multiresolution spaces II: Estimation and aliasing, in preparation.

    Google Scholar 

  35. N. Jacobson, Basic Algebra, W. H. Freeman, New York, 1985.

    MATH  Google Scholar 

  36. A. J. E. M. Janssen, The Zak transform: A signal transform for sampled time-continuous signals, Philips J. Res., 43 (1998), pp. 23–69.

    Google Scholar 

  37. A. J. E. M. Janssen, The Zak transform and sampling theorems for wavelet subspaces, IEEE Trans. Signal Proc., 41 (1993), pp. 3360–3364.

    Article  MATH  Google Scholar 

  38. A. Papoulis, Generalized sampling expansion, IEEE Trans. Circuits and Systems, 24 (1977), pp. 652–654.

    Article  MATH  Google Scholar 

  39. O. Rioul, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal., 23 (1992), pp. 1544–1576.

    Article  MATH  Google Scholar 

  40. D. Seidner and M. Feder, Noise amplification of periodic nonuniform sampling, IEEE Trans. Signal Proc., 48 (2000), pp. 275–277.

    Article  Google Scholar 

  41. W. Sun and X. Zhou, Reconstruction of functions in spline subspaces from local averages, Proc. Amer. Math. Soc., 131 (2003), pp. 2561–2571.

    Article  MATH  Google Scholar 

  42. M. Unser and J. Zerubia, A generalized sampling theory without bandlimiting constraints, IEEE Trans. Circuits and Systems II, 45 (1998), pp. 959–969.

    Article  MATH  Google Scholar 

  43. P. P. Vaidyanathan, Sampling theorems for non bandlimited signals: theoretical impact and practical applications in: Proc. SampTA, Orlando, 2001, pp. 17–26.

    Google Scholar 

  44. M. Vrhel and A. Aldroubi, Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory, Numer. Funct. Anal. Optim., 15 (1994), pp. 1–21.

    Google Scholar 

  45. G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, 38 (1992), pp. 881–884.

    Article  Google Scholar 

  46. X.-G. Xia and Z. Zhang, On sampling theorem, wavelets, and wavelet transforms, IEEE Trans. Signal Proc., 41 (1993), pp. 3524–3535.

    Article  MATH  Google Scholar 

  47. J. L. Yen, On the nonuniform sampling of bandwidth-limited signals, IRE Trans. Circuit Theory 3 (1956), 251–257.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, 72701, USA

    Jeffrey A. Hogan

  2. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003-8001, USA

    Joseph D. Lakey

Authors
  1. Jeffrey A. Hogan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph D. Lakey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Christopher Heil

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Boston

About this chapter

Cite this chapter

Hogan, J.A., Lakey, J.D. (2006). Periodic Nonuniform Sampling in Shift-Invariant Spaces. In: Heil, C. (eds) Harmonic Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4504-7_12

Download citation

Publish with us

Policies and ethics

Search

Navigation