# demonstration using parseval

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)