Measurement of phase difference between two signals using cross correlation vs. fourier transform

Question

I am studying eeg signals with the aim of distinguishing between preictal and interictal epilepsy states based on the eeg signal. I have read some papers and one of the metrics used to distinguish between the states has to do with measuring the phase synchronization between eeg samples from multiple electrodes. In this regard, I thought the most straightforward way to measure phase difference between two signals would be to compute the cross correlation function, and find the time $\tau$ at which the cross correlation function reaches a maximum - call this Method 1. However, in practice what is usually done is to fourier transform the signals and measure phase difference of the complex fourier transform (and possibly average it over a frequency band of interest). Details are explained e.g., here - see the section on phase synchronization. Call this Method 2.

So my question is: what is wrong with Method 1? It seems more intuitive, direct and computationally easier to me. Could someone explain this to me?

EDIT: thinking about this even further, it seems the method employed in practice and described at http://www.scholarpedia.org/article/Measures_of_neuronal_signal_synchrony is actually Method 3 and distinct from Method 2 above. This link shows how it can be implemented in MATLAB. Consider two signals $x(t)$ and a time shifted version of it $x(t+T)$. The Fourier Transforms will be $X(\omega)$ and $X(\omega)e^{j\omega T}$ respectively. Using Method 1 one would get $T$ as measure of phase difference. Using Method 2: $\Delta \phi$ = $\omega T$ and this averaged over $[\omega_1,\omega_2]$ will give phase difference as $T\frac{\omega_1+\omega_2}{2}$.

But what would Method 3 give? Is there a difference between phase difference and phase synchronization of two signals?

Answer 1

Methods 1 and 2 are matematically equivalent, but I have experienced that method 2 is computationally more efficient for high sampling rates and large ranges of delay (notice that method 2 does not require a loop).

Answer 2

To clear this up, there is a distinction between phase difference and phase synchronization. Phase synchronization = is the phase difference constant with time? One crude way to measure phase synchronization would be to partition long signals $x(t)$ and $y(t)$ into many disjoint windows, compute the phase difference in each window using either Method 1 or Mathod 2, and then determine if the phase difference is the same across windows. Method 3 gives a more sophisticated method of measuring phase synchronization where we measure the instantaneous phase of signals and track the difference over time to see if its constant. For completeness, I have come across another paper that talks about yet another way to measure phase synchronization.

Also for $x(t)$ and $x(t+T)$ Methods 1 and 2 give the same answer (discarding the frequency term in method 2) but in general I have a feeling that Method 1 and 2 would give different answer for two arbirary signals $x(t)$ and $y(t)$ and would be very interested if someone could explain this mathematically and elaborate on the relationship if any between the answers obtained from Method 1 vs Method 2, which method is better and why.