Skip to main content
SpringerLink
Log in

Reconstruction of Stationary Processes Sampled at Random Times

  • Chapter
Nonuniform Sampling

Abstract

The reconstruction of a deterministic function from a finite or infinite number of its values is an old problem initially studied by I. Newton, E. Waring and J. L. Lagrange. This chapter addresses the problem of reconstructing a stationary process (rather than deterministic function) from observations at known or unknown instants. In the first case, the reconstruction depends explicitly on the known instants. In the second case, the unknown instants are modelled by random variables whose joint distributions allow to define interpolation formulas.

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

Access this chapter

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 229.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. J. Jerri. The Shannon Sampling Theorem: Its Various Extensions and its Applications. IEEE Trans. on Inf. Th. 65:1565–1595, 1977.

    MATH  Google Scholar 

  2. R. J. Marks II. Introduction to Shannon Sampling Theory and Interpolation Theory. Springer-Verlag, 1991.

    Google Scholar 

  3. J. R. Higgins. Sampling Theory in Fourier and Signal Analysis. Foundations. Oxford Sc. Pub., 1996.

    Google Scholar 

  4. H. S. Black. Modulation Theory. Van Nostrand, 1953, pp. 41–58.

    Google Scholar 

  5. A. V. Balakhrishnan. On the Problem of Time Jitter in Sampling. IEEE Trans. on Inf. Th., 226–236, 1962.

    Google Scholar 

  6. A. V. Balakhrishnan. A Note on the Sampling Principle for Continuous Signals. IRE Trans. on IT, IT-3:143–146, 1957.

    Article  Google Scholar 

  7. S. P. Lloyd. A Sampling Theorem for (Wide Sense) Stochastic Processes. Trans. Am. Math. Soc. 92:1–12, 1959.

    MathSciNet  MATH  Google Scholar 

  8. O. A. Z. Leneman. J. B. Lewis. Random Sampling of Random Processes, Mean-Square Comparison of Various Interpolators. IEEE Trans. on Aut. Cont. AC 11:396–403, 1966.

    Article  Google Scholar 

  9. O. A. Z. Leneman. Random Sampling of Random Processes. J. of Franklin Inst., 281(4): 302–314, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  10. O. A. Z. Leneman. Random Sampling of Random Processes: Impulse Processes. Inf. and Control, 9:347–363, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. A. Marvasti. Signal Recovery from Nonuniform Samples and Spectral Analysis on Random Nonuniform Samples. ICASSP, Tokyo, 1986, pp. 1649–1652.

    Google Scholar 

  12. F. A. Marvasti. A Unified Approach to Zero-Crossings and Nonuniform Sampling. Nonuniform, Oak Park IL, 1987, pp. 57–61.

    Google Scholar 

  13. F. A. Marvasti, P. M. Clarkson, M. V Dokic, U. Goenchanart, C. Lui. Reconstruction of Speech Signals with Lost Samples. IEEE Trans, on SP., 2897–2903, 1992.

    Google Scholar 

  14. F. J. Beutler and O. A. Z. Leneman. Random Sampling of Random Processes: Stationary Point Processes. Inf. and Control, 9:325–346, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. S. Shapiro and R. A. Silverman. Alias-Free Sampling of Random Noise. J. Soc. Ind. Appl. Math., 8(2):225–248, 1960.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. J. Beutler. Alias-Free Randomly Timed Sampling of Stochastic Processes. IEEE Trans. on IT, IT-16(2):147–152, 1970.

    Article  MathSciNet  Google Scholar 

  17. E. Masry. Random Sampling and Reconstruction of Spectra. Inf. and Control, 19:275–288, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Masry and M-C. C. Lui. Discrete-Time Estimation of Continuous-Parameter Processes — A New Consistent Estimate. IEEE Trans. on IT, IT-22(3):298–312, 1978.

    Google Scholar 

  19. E. Masry. Poisson Sampling and Spectral Estimation of Continuous-Time Processes. IEEE Trans. on IT, 24(2):173–183, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Parzen. On Spectral Analysis with Missing Observations and Amplitude Modulation. Time Series Analysis Papers, Holden-Day, 1967, pp. 180–189.

    Google Scholar 

  21. B. Lacaze. Stationary Clock Changes on Stationary Processes. Signal Processing, 55:191–205, 1996.

    Article  MATH  Google Scholar 

  22. B. Lacaze. A Note about Stationary Process Random Sampling. Stat. Prob. Lett., 31:133–137, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  23. B. Lacaze. Periodic Bi-sampling of Stationary Processes. Signal Processing, 68:283–293, 1998.

    Article  MATH  Google Scholar 

  24. B. Lacaze. Modeling the HF Channel with Gaussian Random Delays. Signal Processing, 64(2): 215–220, 1998.

    Article  Google Scholar 

  25. B. Lacaze. La Formule d’Échantillonnage et A. L. Cauchy. Traitement du Signal, 15(4):289–295, 1998.

    MATH  Google Scholar 

  26. R. Veldhuis. Restoration of Lost Samples in Digital Signals. Prentice Hall, 1990.

    Google Scholar 

  27. W. Feller. An Introduction to Probability Theory and Its Applications. Wiley, 1950.

    Google Scholar 

  28. J. L. Doob. Stochastic Processes. Wiley, 1953.

    Google Scholar 

  29. E. Lukacs. Characteristic Functions. Griffin, London, 1960.

    MATH  Google Scholar 

  30. H. Cramer and M. R. Leadbetter. Stationary and Related Stochastic Processes. Wiley, 1967.

    Google Scholar 

  31. A. M. Yaglom. Theory of Stationary Random Functions. Prentice-Hall, 1962.

    Google Scholar 

  32. A. Papoulis. Probability, Random Variables and Stochastic Processes. McGraw-Hill, 1991.

    Google Scholar 

  33. B. Y. Levin. Zeros of Entire Functions. Am. Math. Soc., 1964.

    Google Scholar 

  34. N. Levinson. Gap and Density Theorems. Am. Math. Soc. Coll. Pub., 1940.

    Google Scholar 

  35. J. M. H. Elmirghani, R. A. Cryan and F. M. Clayton. Spectral Characterization and Frame Synchronisation of Optical Fiber Digital PPM. El. Let., 28(16): 1482–1483, 1992.

    Article  Google Scholar 

  36. J. M. H. Elmirghani, R. A. Cryan and F. M. Clayton. Spectral Analysis of Time Jitter Effects on the Cyclostationary Format. Signal Processing, 43:269–277, 1995.

    Article  MATH  Google Scholar 

  37. J. L. Yen. On Nonuniform Sampling of Bandwith-Limited Signals. IRE on Circ. Th., CT-3(12): 251–257, 1956.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. TéSA, 2 rue Camichel, 31071, Toulouse Cedex, France

    B. Lacaze

Authors
  1. B. Lacaze
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. King’s College, London, UK

    Farokh Marvasti

  2. Sharif College of Technology, Tehran, Iran

    Farokh Marvasti

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lacaze, B. (2001). Reconstruction of Stationary Processes Sampled at Random Times. In: Marvasti, F. (eds) Nonuniform Sampling. Information Technology: Transmission, Processing, and Storage. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1229-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-1229-5_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5451-2

  • Online ISBN: 978-1-4615-1229-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation