Problems of Information Transmission

, Volume 48, Issue 3, pp 217–238 | Cite as

On the hilbert transform of bounded bandlimited signals

Information Theory

Abstract

In this paper we analyze the Hilbert transform and existence of the analytical signal for the space Bπ of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in L2(ℝ) and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in Bπ and that the Hilbert transform is not a bounded operator on Bπ, it is nevertheless possible to define the Hilbert transform for the space Bπ. We use a definition that is based on the H1-BMO(ℝ) duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in Bπ is still bandlimited but not necessarily bounded. With these results we continue the work of [1, 2].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Fink, L.M., Relations Between the Spectrum and Instantaneous Frequency of a Signal, Probl. Peredachi Inf., 1966, vol. 2, no. 4, pp. 26–38 [Probl. Inf. Trans. (Engl. Transl.), 1966, vol. 2, no. 4, pp. 11–21].Google Scholar
  2. 2.
    Korzhik, V.I., The Extended Hilbert Transformation and Its Application in Signal Theory, Probl. Peredachi Inf., 1969, vol. 5, no. 4, pp. 3–18 [Probl. Inf. Trans. (Engl. Transl.), 1969, vol. 5, no. 4, pp. 1–14].Google Scholar
  3. 3.
    Gabor, D., Theory of Communication, J. Inst. Electrical Engineers. Part III: Radio and Communication Engineering, 1946, vol. 93, no. 26, pp. 429–457.Google Scholar
  4. 4.
    Voelcker, H.B., Toward a Unified Theory of Modulation. Part I: Phase-Envelope Relationships, Proc. IEEE, 1966, vol. 54, no. 3, pp. 340–353.CrossRefGoogle Scholar
  5. 5.
    Vakman, D.E., On the Definition of Concepts of Amplitude, Phase and Instantaneous Frequency of a Signal, Radiotekhnika i Elektronika, 1972, vol. 17, no. 5, pp. 972–978 [Radio Eng. Electron. Phys. (Engl. Transl.), 1972, vol. 17, no. 5, pp. 754–759].Google Scholar
  6. 6.
    Vakman, D.E., Do We Know What Are the Instantaneous Frequency and Instantaneous Amplitude of a Signal?, Radiotekhnika i Elektronika, 1976, vol. 21, no. 6, pp. 1275–1282 [Radio Eng. Electron. Phys. (Engl. Transl.), 1976, vol. 21, no. 6, pp. 95–100].MathSciNetGoogle Scholar
  7. 7.
    Boas, R.P., Jr., Some Theorems on Fourier Transforms and Conjugate Trigonometric Integrals, Trans. Amer. Math. Soc., 1936, vol. 40, no. 2, pp. 287–308.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Logan, B.F., Jr., Theory of Analytic Modulation Systems, Bell System Tech. J., 1978, vol. 57, no. 3, pp. 491–576.MathSciNetMATHGoogle Scholar
  9. 9.
    King, F.W., Hilbert Transforms, vol. 2. Cambridge: Cambridge Univ. Press, 2009.CrossRefGoogle Scholar
  10. 10.
    Yang, L. and Zhang, H., The Bedrosian Identity for H p Functions, J. Math. Anal. Appl., 2008, vol. 345, no. 2, pp. 975–984.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Brown, J.L., Jr., Analytic Signals and Product Theorems for Hilbert Transforms, IEEE Trans. Circuits Syst., 1974, vol. 21, no. 6, pp. 790–792.CrossRefGoogle Scholar
  12. 12.
    Grafakos, L., Classical Fourier Analysis, New York: Springer, 2008, 2nd ed.MATHGoogle Scholar
  13. 13.
    Butzer, P.L., Splettstößer, W., and Stens, R.L., The Sampling Theorem and Linear Prediction in Signal Analysis, Jahresber. Deutsch. Math.-Verein., 1988, vol. 90, no. 1, pp. 1–70.MATHGoogle Scholar
  14. 14.
    Mönich, U.J. and Boche, H., Non-equidistant Sampling for Bounded Bandlimited Signals, Signal Processing, 2010, vol. 90, no. 7, pp. 2212–2218.MATHCrossRefGoogle Scholar
  15. 15.
    Boche, H. and Mönich, U.J., There Exists No Globally Uniformly Convergent Reconstruction for the Paley-Wiener Space PW 1 π of Bandlimited Functions Sampled at Nyquist Rate, IEEE Trans. Signal Process., 2008, vol. 56, no. 7, pp. 3170–3179.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zemanian, A.H., Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications, New York: Dover, 1987.MATHGoogle Scholar
  17. 17.
    Pandey, J.N., The Hilbert Transform of Schwartz Distributions, Proc. Amer. Math. Soc., 1983, vol. 89, no. 1, pp. 86–90.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Toland, J.F., A Few Remarks about the Hilbert Transform, J. Funct. Anal. 1997, vol. 145, no. 1, pp. 151–174.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kober, H., A Note on Hilbert’s Operator, Bull. Amer. Math. Soc., 1942, vol. 48, no. 6, pp. 421–427.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Garnett, J.B., Bounded Analytic Functions, New York: Academic, 1981.MATHGoogle Scholar
  21. 21.
    Duffin, R. and Schaeffer, A.C., Some Properties of Functions of Exponential Type, Bull. Amer. Math. Soc., 1938, vol. 44, no. 4, pp. 236–240.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Titchmarsh, E.C., The Theory of Functions, Oxford (London): Oxford Univ. Press, 1939, 2nd ed.MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Lehrstuhl für Theoretische InformationstechnikTechnische Universität MünchenMünchenGermany

Personalised recommendations