Question regarding AC power of ergodic process

Question

We know Ergodic process is the subset of Weakly stationary process which permits us to substitute time average for ensemble Average

My teacher said If $X(t)$ is Ergodic random process then following relationships holds true and we even solved many questions based on these relations in which either graph of ACF(autocorelation function) or expression of ACF or PSD (fourier transform of ACF) is given and we're asked to find AC power,total power,Dc power etc.

but i don't know how to derive these expressions by exploiting property of ergodicity in mean and Auto-corelation . $\lim_{\tau \to\infty}R_{XX}(\tau)=\text{d.c power}$

and $R_{XX}(0)$ = $\text{Total power}$

so, $R_{XX}(0)-R_{X}(\infty)=$$\text{A C power}$

any help or referance material will be appreciated.thank you

Answer 1

Considering real valued WSS processes whose auto-correlation and auto-covariance functions are defined as $$ r_x(\tau) = E[ x(t)x(t+\tau) ] $$

$$ c_x(\tau) = E[ (x(t)-\mu_x)(x(t+\tau)-\mu_x) ] = r_x(\tau) - \mu_x^2 $$

Then the following are just basic observations.

• The total power of a WSS random process is given by $$ P_x = E[ x^2(t) ] = E[ x(t)x(t) ] = r_x(0) $$

• For a nonperiodic, WSS, RP to be ergodic in the mean, a necessary and sufficient condition is such that the auto covariance function $c_x(\tau)$ goes to zero as $\tau$ goes to $\infty$ : $$ \lim_{\tau \to \infty} c_x(\tau) = 0$$ Since auto correlation sequence and autocovariance sequences are related by $$ r_x(\tau) = c_x(\tau) + \mu_x^2 $$

then the value of ACF as $\tau$ goes infinity yields the DC power as $$ \lim_{\tau \to \infty} r_x(\tau) =\lim_{\tau \to \infty} c_x(\tau) + \mu_x^2 = 0 + \mu_x^2 $$

hence $r_x(\infty)$ is the DC power for an ergodic process.

• The difference between the total power and the DC power is the AC power.

For a zero-mean RP, the total power and the AC power are the same.