# Integral over power spectral density

The wikipedia entry on PSD has one confusing line:

Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process)

But would the total power correspond to the uncentralized second moment, so variance + mean2? In particular, there are stationary processes with a nonzero mean. Is the integral over the PSD still just the variance?

Summation or integration of the spectral components yields the total power (for a physical process)

yes, that's how "power density" is defined ...

or variance (in a statistical process)

For any bound density, that's true.

But would the total power correspond to the uncentralized second moment, so variance + mean²?

Yes, but if you have a process with a non-zero mean, that would imply that the spectral power density at $$f=0$$ diverges (a "Dirac delta $$\delta(f)$$" with "infinite height"), and that would imply the PSD isn't actually a function (it's still a distribution / test function, but seriously, this is just getting into math).

If you will, just imagine your "mean" is something at very low frequencies ($$f\in[-\epsilon;\epsilon]$$), but not concentrated to exactly $$f=0$$.

Still works out the same, for anything technical (because nothing will ever experience whatever process they're observing for eternity, so who's to say whether something happens at $$0\,\text{Hz}$$ or $$10^{-24}\,\text{Hz}$$?).

Honestly, the wording of the article isn't the greatest.