Stochastic process; WSS random signal through LTI

Stochastic process

A stochastic (random) process $X(t)$ is an ensemble (collection) of time functions, one sample function for each outcome of a random experiment. At any fixed time $t_1$, $X(t_1)$ is a random variable. Example: the noise voltage at the output of many identical receivers. It is described by its mean $m_X(t)=E[X(t)]$ and autocorrelation $R_X(t_1,t_2)=E[X(t_1)X(t_2)]$.

A process is wide-sense stationary (WSS) if (i) its mean is constant and (ii) its autocorrelation depends only on the time difference $\tau=t_2-t_1$: $R_X(t,t+\tau)=R_X(\tau)$.

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). Input $S_X(f)=\frac{N_0}{2}$, $\lvert H(f)\rvert^2=\frac{1}{1+(2\pi fRC)^2}$:

$$S_Y(f)=\frac{N_0/2}{1+(2\pi fRC)^2},\qquad R_Y(\tau)=\frac{N_0}{4RC}e^{-\lvert\tau\rvert/RC},\qquad P_Y=\frac{N_0}{4RC}$$

The filter "colours" the noise: the output is correlated over about one time constant $RC$, and its power becomes finite.

Cross-correlation between input and output (useful for system identification): $R_{XY}(\tau)=h(\tau)*R_X(\tau)$; with white-noise input $R_{XY}(\tau)=\frac{N_0}{2}h(\tau)$, so the impulse response can be measured by correlating output with input noise.