PSDs and Parseval's theorem OK, I have been banging my head for quite a while trying to make sense of this simple equation:



The $1/T$ is what is causing me all the grief.  I assume that the square of the Fourier transform of the signal $X$ is a finite function.  So won't the $T$ in the denominator mean that the PSD, $S_x$, will always tend toward zero?  If $X$ does increase to infinity with time, can someone explain this to me?  I am thinking that $X$ is a measure of the frequency content of the random time signal, and that shouldn't change appreciably with the time taken to measure it.
That equation is indeed simple, but it's also wrong. So your doubts are completely justified. What is probably meant is something like

$$S_X(f)=\lim_{T\rightarrow\infty}\frac{1}{T}\left|X_T(f)\right|^2\tag{1}$$

with

$$X_T(f)=\int_{-T/2}^{T/2}x(t)e^{-j2\pi f t}dt\tag{2}$$

This definition of the power spectrum is used for functions $x(t)$ with finite power (but infinite energy) for which the Fourier integral does not exist.