# Phase locking value (PLV) between $x(t)$ and $x(t-T)$

I have been trying to calculate PLV using the hilbert transform (reference). To test if it even works or not, I calculated PLV between $x(t)$ and $x(t-T)$. i find that it drops to zero very quickly as $T$ becomes nonzero as shown in following figure:

Is this as expected? I would expect PLV between $x(t)$ and $x(t-T)$ to be 1. In above graph shift is a normalized quantity such that $T=\rm{shift}\times\rm{length}(x)$

EDIT: thinking about this some more. Let us first define what PLV is. If we define PLV as the phase locking between instantaneous phases of two signals $x(t)$ and $y(t)$ where instantaneous phase is defined according to the analytic signal, then I am not able to derive a mathematical relationship asserting that $PLV(x(t),x(t-T)) = 1$. But then what is the PLV good for? What kind of transformation could be applied to $x(t)$ to yield a $y(t)$ s.t. $PLV(x,y)=1$?

• Is your time series here real data, or something artificial like a sine wave?
– Josh
Commented Oct 28, 2014 at 21:21
• my time series is real eeg data over 10sec. You can run the experiment on any general $x(t)$ that has a continuous FT. Commented Oct 28, 2014 at 21:24
• I asked because if it were artificial then you would expect the PLV to by phasic, but with real data your pattern looks right.
– Josh
Commented Oct 28, 2014 at 21:25
• I want to make sure I'm understanding: shift is essentially the proportion of the 10s window that you shifted the pattern by? So when you get to 1, you should be computing the PLV of the original time series x with itself?
– Josh
Commented Oct 28, 2014 at 21:29
• yes when T=0 I am computing PLV between $x(t)$ and $x(t)$ which gives 1 as expected. But shouldn't the PLV between $x(t)$ and $x(t-T)$ also be 1? If not, could someone explain why not (mathematically) Commented Oct 28, 2014 at 21:33

## 1 Answer

PLV is good for correlating signals that seem out of phase to our measurement techniques due to the digital nature of our recordings but are in sync because of the electro chemical biological nature of the CNS. If you transform two signals into phase you can determine the relatedness of the two signals.

Phase Locking Value NeuroBytes

The schematic below illustrates how PLV is computed in this implementation. Explanation for each step follows.