This question might be a bit stupid, anyway, i'll risk it, since i want to get better understanding of this subject.
Let's consider random signal x(t), and let's say that we know that it is ergodic by mean, and by autocorrelation (this implies that it is wide sense stationary).
This means that time averaged mean of this signal is equal to it's statistic averaged mean, same goes for autocorrelation.
So we have the following:
$\mu_X=E(x)=\frac{1}{T} \int_T x(t)dt=\langle x(t)\rangle \\ R_{xx}(\tau)=E[x(t)x(t+\tau)]= \lim_{T->\infty} \frac{1}{T} \int_T x(t)x(t+\tau)dt=\langle x(t)x(t+\tau)\rangle$
Then, as i found in one of the books i have, it says following:
$\langle x(t) \rangle$- corresponds to DC level of given signal
$\langle x(t) \rangle ^2$ - corresponds to normalized power of DC component
$\langle x^2(t) \rangle$ - corresponds to total average normalized power
$\overline{\sigma_x^2}= \langle x^2(t) \rangle - \langle x(t) \rangle^2$-corresponds to average normalized power in AC component of the signal
$\overline{\sigma_x}$-corresponds to rms of AC component
My question is, is there any way to prove this, or to gain intuitive understanding why this is true. Any help appreciated!