Advertisement

Operator basis for analytic signal construction

  • Frederick W. King
Article

Abstract

The operator basis for one-dimensional analytical signal construction is considered. Why the Hilbert transform plays a fundamental role is clarified. Extension to the case of multidimensional signals is considered. Possible ramifications for the definition of a fractional analytic signal are discussed.

Keywords

Hilbert transform Analytic signal Multidimensional analytic signal Fractional analytic signal 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cusmariu A. (2002). Fractional analytic signals. Signal Processing 82, 267–272MATHCrossRefGoogle Scholar
  2. Hogan J.A., Lakey J.D. (2005). Non-translational-invariance in principal shift-invariant spaces. In: Begehr H.G.W., Gilbert R.P., Muldoon M.E., Wong M.W. (eds) Advances in analysis. New Jersey, World Scientific, pp. 471–485Google Scholar
  3. Lohmann A.W., Mendlovic D., Zalevsky Z. (1996a). Fractional Hilbert transform. Optics Letters 21, 281–283CrossRefGoogle Scholar
  4. Lohmann A.W., Ojeda-Castañeda J., Diaz-Santana L. (1996b). Fractional Hilbert transform: Optical implementation for 1D objects. Optical Memormy & Neural Networks 5, 131–135Google Scholar
  5. Loughlin P.J. (1998). Do bounded signals have bounded amplitudes?. Multidimensional Systems and Signal Processing 9, 419–424MATHCrossRefGoogle Scholar
  6. McLean W., Elliott D. (1988). On the p-norm of the truncated Hilbert transform. Bulletin of the Australian Mathematical Society 38, 413–420MATHMathSciNetGoogle Scholar
  7. Meyer Y., Coifman R. (1997). Wavelets Calderón-Zygmund and multilinear operators. Cambridge, Cambridge University PressMATHGoogle Scholar
  8. Pandey J.N. (1996). The Hilbert transform of Schwartz distributions and applications. New York, John Wiley & SonsMATHGoogle Scholar
  9. Pei, S.-C., & Yeh, M.-H. (1998). Discrete fractional Hilbert transform. In: Proceedings of the 1998 IEEE International Symposium on Circuits and Systems, ISCAS ’98 (Vol. 4, pp. 506–509).Google Scholar
  10. Pei S.-C., Yeh M.-H. (2000). Discrete fractional Hilbert transform. IEEE Transactions on Circuits Systems II Analog Digital Signal Processing 47, 1307–1311CrossRefGoogle Scholar
  11. Stein E.M. (1970). Singular integrals and differentiability properties of functions. Princeton, Princeton University PressMATHGoogle Scholar
  12. Vakman D. (1996). On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for defining amplitude and frequency. IEEE Transactions on Signal Processing, 44, 791–797CrossRefGoogle Scholar
  13. Vakman D. (1997). Analytic waves. International Journal of Theoretical Physics 36, 227–247MATHCrossRefMathSciNetGoogle Scholar
  14. Vakman D., Vaĭnshteĭn L.A. (1977). Amplitude, phase, frequency – fundamental concepts of oscillation theory. Soviet Physics Uspekhi 20, 1002–1016CrossRefGoogle Scholar
  15. Zayed A.I. (1998). Hilbert transform associated with the fractional Fourier transform. IEEE Signal Processing Letters 5, 206–208CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of Wisconsin-Eau ClaireEau ClaireUSA

Personalised recommendations