How is the PSD the Fourier Transform of the Autocorrelation function?

Question

I am getting very confused with the textbook I am reading through: Modern Digital and Analogue Communication Systems:

3.8.1 Parseval's Theorem says that:

The total power in a power signal is its time average energy.

It then goes onto section 3.8.2 where it says that the Autocorrelation function is:

In terms of a discrete sampled signal if I have a signal and I am calculating the autocorrelation function and I get to tau=0. Is that just the energy of the signal? g(t)*g(t-0)=E_gt

But then the text goes on to say that the PSD is the Fourier Transform of the Autocorrelation Function.

How is that possible? if tau=zero, then that one calculation captures the energy in the signal. I haven't even considered the other taus. Wouldn't I go well above the actual energy in the signal?

Answer 1

If you believe the authors' claim (and you should) that the power spectrum is the Fourier transform of the autocorrelation:

$$S_x(f)=\int_{-\infty}^{\infty}R_x(\tau)e^{-j2\pi f\tau}d\tau\tag{1}$$

then it must also be true that the autocorrelation is the inverse Fourier transform of the power spectrum:

$$R_x(\tau)=\int_{-\infty}^{\infty}S_x(f)e^{j2\pi f\tau}df\tag{2}$$

From $(2)$ you see that

$$R_x(0)=\int_{-\infty}^{\infty}S_x(f)df\tag{3}$$

I.e., the power is the integral of the power spectrum, which makes sense. Furthermore, the power in a certain frequency band can be obtained by integrating the power spectrum over that frequency band, and the result must always be less than or equal to the total power given by $(3)$.