I have to do a demonstration. If we do the Parseval identity of the signals $x(t)$, $y(t)$ and $z(t)$ that go from $0$ to $T$ and that are real, we have:
$\int_{0}^{T} x(t)^2dt=\int_{-\infty}^{\infty}|X(f)|^2df$
$\int_{0}^{T} y(t)^2dt=\int_{-\infty}^{\infty}|Y(f)|^2df$
$\int_{0}^{T} z(t)^2dt=\int_{-\infty}^{\infty}|Z(f)|^2df$ (1)
If we sum up the these 3 equations, and we divide both sides by the interval $T$, we obtain:
$\frac{1}{T}\int_{0}^{T} (x(t)^2+y(t)^2+z(t)^2)dt=\frac{1}{T}\int_{-\infty}^{\infty}(|X(f)|^2+|Y(f)|^2+|Z(f)|^2)df$ (2)
Can you confirm me that everything is ok in this demonstration? I am trying to demonstrate this because my advisor says that since the PSD of the signals $x$, $y$, $z$, which are $S_{X}$, $S_{Y}$, $S_{Z}$, respect this relation:
$\int_{-\infty}^{\infty}(S_{X}(f)+S_{Y}(f)+S_{Z}(f))df=\frac{1}{T}\int_{0}^{T} (x(t)^2+y(t)^2+z(t)^2)dt$ (3)
then (and this is the point that is totally wrong for me) he says that $S_{X}+ S_{Y}+S_{Z}$ is the Fourier transform of ( $x(t)^2 +y(t)^2+z(t)^2$):
$S_{X}+ S_{Y}+S_{Z} = \int_{-\infty}^{\infty}( x(t)^2 +y(t)^2+z(t)^2)\exp(-itf2\pi)dt$ (4)
this cannot be possible in my opinion. I already tried to show him the procedure that leads to the PSD of a signal etc. but he still believe (maybe because I am young, not expert and just arrived) that is true what he said. So I am trying to do the above mentioned demonstration, saying that since $|X(f)|^2$ is different from the absolute value of the Fourier transform of $x(t)^2$, it cannot be contemporary true what it is stated in the eq. (4) and the application of the Parseval identity (2).
I beg you to tell me if my procedure is good. I want to convince him with something that is unassailable. (I am almost desperate, I have been trying to convince him that is wrong since 3 month ago)