# Dictionary:Hilbert-transform technique

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 ln*P*(*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

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