# Power contained in a random process $X(t)$

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

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.