Why coherence function calculation needs averaging of different segments of the signal? I am trying to understand coherence function calculation. I am working with a non-linear system without any noise. Coherence function is defined as:

I understand that coherence is 1 for N=1, but I don't understand why coherence is different with segmentation? As far as understand with segmentation without any noise in system, each segment should give same values and them coherence will go to 1 again. I think I am missing something. I am quite new to this subject.
Could anyone help me with this? Is there any good reference related to this where it has proved that why coherence is different with segmentation?
I have gone through the following questions:
Optimal segment length for coherence estimation
and Magnitude-squared Coherence calculation inconsistence 
but I still it is not clear to me.
Thanks,
Swati
You can see how it falls out if you work through the simple case of N=2 (I've elided the 1/N for brevity)
\begin{equation}
\frac{(Y_{1}X_{1}^{*}+Y_{2}X_{2}^{*})(X_{1}Y_{1}^{*}+X_{2}Y_{2}^{*})}
{(X_{1}X_{1}^{*}+X_{2}X_{2}^{*})(Y_{1}Y_{1}^{*}+Y_{2}Y_{2}^{*})}
\end{equation}
It should be pretty obvious that there are now some cross products. Multiplying through:
\begin{equation}
\frac
{
X_{1}X_{1}^{*}Y_{1}Y_{1}^{*} + X_{1}^{*}X_{2}Y_{1}Y_{2}^{*} + X_{1}X_{2}^{*}Y_{2}Y_{1}^{*} + X_{2}X_{2}^{*}Y_{2}Y_{2}^{*}
}
{
X_{1}X_{1}^{*}Y_{1}Y_{1}^{*} + X_{1}X_{1}^{*}Y_{2}Y_{2}^{*} + X_{2}X_{2}^{*}Y_{1}Y_{1}^{*} + X_{2}X_{2}^{*}Y_{2}Y_{2}^{*}
}
\end{equation}
The second and third terms of the numerator are no longer the same as the second and third terms of the denominator.