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How do we calculate the AC and DC power of random process $X(t)$ , provided we have $R_x (\tau)$, and $S_x(f)$ ?

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Assuming that the process is wide-sense-stationary, the total power in a random process with autocorrelation function $R(\tau)$ and power spectral density $S(f) = \mathcal F\{R(\tau)\}$ is given by $$\text{Total Power} = \int_{-\infty}^\infty S(f) \,\mathrm df = R(0)$$ while the "DC power" (also equal to the squared mean of the process) is given by $$\text{DC Power} = \lim_{\tau \to \infty} R(\tau).$$ The DC power is also given by the magnitude or amplitude of the impulse (if any) at the origin in $S(f)$. Note that $S(0)> 0$ is not enough to claim that the process has DC power -- for example, white noise has no DC power --; we need to have an impulse (a.k.a Dirac delta) at the origin in $S(f)$ for the process to have DC power.

What the AC power might be is left as an exercise for the reader to figure out.

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