Power spectral density interpretation

Question

After reading this question: PSD (Power spectral density) explanation I am still a little confused as to what extra information the PSD gives us over simply taking the magnitude of the fourier transform.

So my understanding is that the PSD is equal to:

$$|X(\omega)|^2$$

but what does that tell me that simply

$$|X(\omega)|$$

does not. I am assuming that squaring the fourier transform's magnitude somehow makes the number more physically (or mathematically) relevant? So how do I interpret the values of the power spectral density, especially when the signal is not a physical one and hence has no actual energy (say for example a financial signal or an image)?

Also, does the power spectral density contain information about the probability of the frequencies in the signal? As in is there a way to use the power spectral density to make a statement about the significance level of making a claim about a harmonic frequency in a stochastic signal? So something kind of like checking the significance of the coefficients from an AR model?

Answer 1

it doesn't tell you new information, but because the abs value function is non-analytic (i.e. not all of its derivatives are continuous), and the magnitude-squared is, then the latter can be manipulated mathematically in ways that the former cannot. one important property of the power spectrum is that it is the Fourier transform of the autocorrelation function in the time domain.

the PSD gives you the energy of each frequency component for a deterministic signal and the expectation value of the energy for each frequency component for a random (or "stochastic" if you like jargon) signal.