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Given the signals

\begin{align} x(t) &= A_1\sin(\omega t)\\ y(t) &= A_2\sin(\omega t) \end{align}

Do $A_1$ and $A_2$ play a role in the $C_{xy}$ coherence estimator? MATLAB's mscohere.m shows no differences for $A_1=A_2$ and $A_1\neq A_2$ at all.

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From the definition of the (magnitude-squared) coherence

$$C_{xy}(f)=\frac{|G_{xy}(f)|^2}{G_{xx}(f)G_{yy}(f)}$$

with the cross-spectral density $G_{xy}(f)$, and the power spectra $G_{xx}(f)$ and $G_{yy}(f)$, respectively, it is clear that scaling of $x(t)$ or $y(t)$ does not change the value of $C_{xy}(f)$, because the scaling constants appear in the numerator as well as in the denominator and cancel out.

The coherence is a normalized quantity (and hence scaling-invariant), and it always satisfies $0\le C_{xy}(f)\le 1$.

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  • $\begingroup$ Thank you. I know it may sounds a naive question to make, however in a very popular paper [IF 29.2] that is clearly stated as in "Coherence mixes the effects of amplitude and phase in the interrelations between two signals". $\endgroup$
    – IDKreally
    Feb 1 '17 at 20:32
  • $\begingroup$ @IDKreally And now you do know really. :D $\endgroup$
    – Peter K.
    Feb 2 '17 at 0:11

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