This question is related to one posted a couple years ago at (Why the spectral coherence is unity for all frequencies between single-frequency time series and itself).
Suppose two signals have common frequency, $\omega_0$, and a fixed phase difference, $\phi_0$:
$$x(t)=\sin(2\pi \omega_0 t)$$ $$y(t)=\sin(2\pi \omega_0(t-\phi_0))$$
where $t$ varies over a set of discrete time samples. The coherence, in general, is defined as
$$Coh(\omega)=\frac{|P_{xy}(\omega)|^2}{P_{xx}(\omega)P_{yy}(\omega)},$$
where the numerator is the magnitude squared of the co-spectral density, and the denominator is the product of the corresponding power spectral densities.
If $\omega=\omega_0$ in the example above, then $Coh(\omega_0)=1.$
The mathematician in me is troubled by the fact that the coherence is even defined for any other frequency, $\omega \ne \omega_0$, since the definition results in an undefined quantity, $\frac{0}{0}$.
When I plot the coherence function for the above using a Matplotlib tool such as cohere(x,y), the function appears to equal one for all $\omega$. I gather, from the post at the above URL, that this stems from the numeric estimation tool being used.
But from a theoretical point of view, is $Coh(\omega)$ even defined in my example above, when $\omega \ne \omega_0$? Or is it just a convention to set it equal to 1?
