Dictionary:Hilbert-transform technique

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A technique for determining the phase of a minimum-phase function from its power spectrum, used in computing a deconvolution operator. Given the power spectrum P(f) and that the wavelet is minimum phase, the wavelet's frequency-domain representation W(f) is


.


The amplitude A(f) is the square root of the power spectrum. Taking the logarithm of both sides splits the function into real and imaginary parts:


.


To be minimum phase, the function must be analytic in the lower half-plane. Then the Hilbert transform can be used to find the phase γ(f) from lnP(f)/2:


.


Since the amplitude and phase are known, the Fourier transform can be computed and the time-domain expression for W(t) determined. [1].


References

  1. Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.


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