---
title: "Stochastic process; WSS random signal through LTI"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:stochastic-wss-through-lti"
source: https://www.bctengineer.com/q/semester_6/communication_systems/stochastic-wss-through-lti
site: BCT Engineering
---

# Stochastic process; WSS random signal through LTI

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2076 Asoj (CS II), 2+10 marks
- Video: [12.12 LTI Systems with WSS Random Input](https://www.youtube.com/watch?v=neBow2aUE18) — M. Akif YAZICI
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/stochastic-wss-through-lti
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

What do you mean by Stochastic Process? Explain with necessary derivation passage of wide-sense random signals through a LTI.

## Short answer

A stochastic process is an ensemble of time functions where each outcome yields a sample function; it is wide-sense stationary (WSS) if mean is constant and autocorrelation depends only on time difference τ. When a WSS process passes through an LTI system, the output is WSS with mean m_Y = m_X H(0), autocorrelation R_Y(τ) = h(τ)*h(-τ)*R_X(τ), and PSD S_Y(f) = |H(f)|² S_X(f).

## Answer

#### 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 $H(f)$:

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

```text
  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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a stochastic process?
   A. A single deterministic time function
   B. An ensemble of time functions, one per outcome, where X(t1) is a random variable
   C. A process with zero mean
   D. A signal with constant power only
   Answer: B. A stochastic process is the collection of sample functions; fixing t gives a random variable. (Key point: Stochastic process)

2. Which pair of conditions defines a wide-sense stationary process?
   A. Constant mean and autocorrelation depending only on tau
   B. All joint pdfs time invariant
   C. Zero mean and unit variance
   D. Constant PSD and constant mean
   Answer: A. WSS needs only a constant mean and R_X(t, t+tau) = R_X(tau). (Key point: WSS conditions)

3. What does the autocorrelation of the LTI output equal?
   A. h(tau) * R_X(tau)
   B. h(tau) * h(-tau) * R_X(tau)
   C. H(f) R_X(tau)
   D. R_X(tau) h(tau)^2
   Answer: B. Double integration of h(a)h(b)R_X(tau+a-b) is convolution with h(tau) and h(-tau). (Key point: Output autocorrelation)

4. If the input is white noise with PSD N0/2, what is R_XY(tau)?
   A. (N0/2) h(tau)
   B. (N0/2) H(f)
   C. N0 h(-tau)
   D. (N0/2) delta(tau)
   Answer: A. R_XY = h * R_X = h * (N0/2)delta = (N0/2) h(tau), so h(t) can be measured by correlation. (Key point: Cross-correlation for system identification)

## Close variants set in other exams

- [Ergodic process; WSS random signals through LTI](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aergodic-wss-through-lti) — Communication Systems, chapter 9, asked 2×

## Related in Communication Systems

- [Random process; LTI output is WSS if input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-lti-wssp) — Communication Systems, chapter 9
- [NEB; mean and AC of WSS signal through LTI](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-wssp-lti-mean-ac) — Communication Systems, chapter 9