Ergodic process; WSS random signals through LTI

Ergodic stochastic process

A stationary random process is ergodic if its time averages, taken over one long sample function, equal its ensemble averages, taken across all sample functions at one instant:

$$\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}x(t)\,dt=E[X(t)]=m_X,\qquad \lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}x(t)x(t+\tau)\,dt=R_X(\tau)$$

Ergodicity allows the mean (DC value), power and autocorrelation of noise to be measured from a single long record. An ergodic process must be stationary, but a stationary process need not be ergodic.

Passage of WSS random signals through an LTI system

Let a WSS process $X(t)$ (mean $m_X$, autocorrelation $R_X(\tau)$, PSD $S_X(f)$) be applied to a stable LTI system with impulse response $h(t)$ and $H(f)$:

$$Y(t)=\int_{-\infty}^{\infty}h(\alpha)X(t-\alpha)\,d\alpha$$

  X(t) (WSS)   +--------------+   Y(t)
 ------------->| h(t) , H(f)  |------------->
  R_X, S_X     +--------------+  R_Y, S_Y

1. Mean of output. Expectation and integration can be interchanged:

$$\begin{aligned} E[Y(t)] &= \int_{-\infty}^{\infty}h(\alpha)E[X(t-\alpha)]\,d\alpha = m_X\int_{-\infty}^{\infty}h(\alpha)\,d\alpha \\ &= m_X\,H(0) \quad(\text{a constant, independent of } t) \end{aligned}$$

2. Autocorrelation of output.

$$\begin{aligned} R_Y(t,t+\tau) &= E[Y(t)Y(t+\tau)] \\ &= E\left[\int h(\alpha)X(t-\alpha)d\alpha\int h(\beta)X(t+\tau-\beta)d\beta\right] \\ &= \int\!\!\int h(\alpha)h(\beta)\,E[X(t-\alpha)X(t+\tau-\beta)]\,d\alpha\,d\beta \\ &= \int\!\!\int h(\alpha)h(\beta)\,R_X(\tau+\alpha-\beta)\,d\alpha\,d\beta \end{aligned}$$

because $X$ is WSS, its autocorrelation depends only on the time difference $(t+\tau-\beta)-(t-\alpha)=\tau+\alpha-\beta$. The result depends only on $\tau$, not on $t$:

$$R_Y(\tau)=h(\tau)*h(-\tau)*R_X(\tau)$$

3. Mean-square value $E[Y^2]=R_Y(0)=\int\!\!\int h(\alpha)h(\beta)R_X(\alpha-\beta)d\alpha d\beta$ is finite and constant.

Since the mean is constant and the autocorrelation depends only on $\tau$, $Y(t)$ is also wide-sense stationary.

4. Output PSD. Taking the Fourier transform of $R_Y(\tau)$ ($h(\tau)\to H(f)$, $h(-\tau)\to H^*(f)$):

$$S_Y(f)=H(f)H^*(f)S_X(f)=\lvert H(f)\rvert^2S_X(f)$$

and the output power is $P_Y=R_Y(0)=\int_{-\infty}^{\infty}\lvert H(f)\rvert^2S_X(f)\,df$. The phase of $H(f)$ has no effect on the output PSD.

Example (white noise through RC low-pass filter). $S_Y(f)=\frac{N_0/2}{1+(2\pi fRC)^2}$, $R_Y(\tau)=\frac{N_0}{4RC}e^{-\lvert\tau\rvert/RC}$, output power $\frac{N_0}{4RC}$.

Cross-correlation: $R_{XY}(\tau)=h(\tau)*R_X(\tau)$; for white-noise input, $R_{XY}(\tau)=\frac{N_0}{2}h(\tau)$, so correlating output with input measures $h(t)$.