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Consider we have two time series and for each of them , we know the phase (Phase1 and Phase2). We want to compute phase difference between these two time series. We define the phase difference as (Phase1 - Phase2) and we wrap the results to the range of [-pi,pi]. Based on this definition of phase difference, how can we say which signal is leading and which signal is lagging? Is it something we can determine only based on the sign of the phase difference?

For instance, if phase difference is positive (between 0 and +pi), can we say that signal 1 is leading?

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  • $\begingroup$ yes, you can, according to your definition. Not really sure what the problem with that is?` $\endgroup$
    – Marcus Müller
    Commented Dec 28, 2021 at 23:55
  • $\begingroup$ Please clarify if you are trying to confirm the definition of "phase lead" and "phase lag" or if you are really asking if that means one signal or the other occurs first in time. $\endgroup$
    – Dan Boschen
    Commented Dec 29, 2021 at 13:54

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If you say a signal has a certain phase, I assume you are referring to two sine waves, i.e.

$$x_1(t) = \sin(\omega t + \phi_1) \\ x_2(t) = \sin(\omega t + \phi_2)$$

how can we say which signal is leading and which signal is lagging?

In this case you can't. Both signals are infinitely long and have no starting or end point. The question of which one is "leading" or "lagging" is non-sensical.

You can do it heuristically by something like "I pick the two peaks that are closer together and declare the left one to be leading" but there is no difference between "cosine leads sine by 90 degrees" and "sine leads cosine by 270 degrees". It makes no difference one way or the other.

Things are different if the signal are finite or not sine waves, but then you have to specify in more details what exactly they are.

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  • $\begingroup$ "Phase lead" and "Phase lag" is commonly defined (calculushowto.com/phase-lag-definition) . You could certainly argue that a 90 degree phase lag is the same as a 270 degree phase lead. The question for me is if the OP is really referring to phase or time delay (or confusing the two). $\endgroup$
    – Dan Boschen
    Commented Dec 29, 2021 at 13:52

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