In the frequency domain, the problems of bandwidth equalization and phase matching are orthogonal, since they are concerned with the amplitude and argument of a complex signal.
The steps I applied to equalize the bandwidth were: Fourier transform the training windows, S1 and S2, on the time-axis, and calculate their spectra, and , where
As well as calculating a spectral operator, , I calculated the average difference in phase between the training windows for each frequency, .
Although the phase and bandwidth are orthogonal, in this approach it does matter the order in which the corrections are applied. This is due to the fact I calculate the average phase as the phase of the average value, as opposed to the average of the phases. With my approach it is correct to apply the bandwidth correction before the phase correction, although this is unlikely to be very important.
The results of the frequency domain cross-equalization are displayed in Figure 3. The differences due to the fluid movements are clearly visible, and the signal-to-noise level is high.