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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 ...


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).

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