# 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

${\displaystyle W(f)=A(f)e^{i\gamma (f)}=\vert P(f)\vert ^{1/2}e^{i\gamma (f)}}$.

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:

${\displaystyle \ln \left[W(f)\right]=(1/2)\ln \left[P(f)\right]+i\gamma (f)}$.

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:

${\displaystyle \gamma (f)=(1/2)\ln \left[P(f)\right]*{\frac {1}{\pi f}}}$.

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.