Quantitative analysis of EEG signals: Time-frequency methods and ...

I xx (k) =jX(k)j 2 = X(k) X (k) (9)
where denotes complex conjugation. The sample cross spectrum can be similarly
dened by making the product of eq. 9 between two time series x(n) andy(n) as follows
I xy (k) =X(k) Y (k) (10)
where X(k) and Y (k) are the Fourier Transforms of x(t) and y(t) respectively. The
sample cross spectrum gives a measure of the linear correlation between two signals for
the dierent frequencies f k , dened as in eq. 7. Since it is a complex measure, it can be
described by an amplitude and a phase. A useful measure of the amplitude is obtained
with its square value, normalized in the following way:
; 2 xy(k) =
jI xy (k)j 2
I xx (k) I yy (k)
(11)
Equation 11 is known as the squared coherence function, and it will give values
tending to 1 for highly linearly correlated signals and to 0 for non correlated ones.
The counterpart of the coherence is the phase function, dened as
= arctan ;=I xy(k)