2.5. Instantaneous Frequency, Phase

To calculate the instantaneous phase, the analytic signal is created by joining the original data with its Hilbert transform [24]. The analytic signal zx(t), which is a complex signal of time t can be created as follows:

where x(t) is the real signal, and H{x(t)} is the Hilbert transform of the real signal, which is defined as follows:

where P.V. indicates Cauchy’s principal value. The instantaneous phase of the signal can be found from the complex analytic signal as follows: