The Hilbert transformation and its applications

A sampled signal can be decomposed using the Fourier series into a sum of cosine functions. The Hilbert transform (HT) changes the phase of all the components by -90 degrees for positive frequencies and by +90 degrees for negative frequencies.
The Hilbert transform gives rise to numerous applications in seismics, for example:

• in introducing a phase change,
• in measuring an envelope,
• in measuring the instantaneous phase,
• in measuring the instantaneous frequency,
• in transforming a signal to one with minimum phase or zero phase.

A signal and its Hilbert transform can be associated to form a complex signal whose modulus is called an envelope.

We will present some applications of the Hilbert transform , notably:

computing an envelope.
introducing a phase change.
measuring the instantaneous frequency of a sweep.
computing an analytic signal.
studying the polarization of a wave.
separating wave by polarization filters.
studying group and phase velocities.
transforming a signal into a zero phase signal.