--- title: "Random process; white noise PSDF and autocorrelation" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:random-process-white-noise" source: https://www.bctengineer.com/q/semester_6/communication_systems/random-process-white-noise site: BCT Engineering --- # Random process; white noise PSDF and autocorrelation Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2072 Kartik (CS II), 2+4 marks; 2069 Chaitra (CS II), 4 marks (asked 2 times) - Video: [What is White Gaussian Noise (WGN)?](https://www.youtube.com/watch?v=QfUQMzHfbxs) — Iain Explains Signals, Systems, and Digital Comms - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/random-process-white-noise - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question What do you mean by Random process? Explain white noise with its PSDF and autocorrelation function. ## Answer #### Random process A **random (stochastic) process** $X(t,s)$ is a collection (ensemble) of time functions, one for each outcome $s$ of a random experiment. At a fixed time $t_1$, $X(t_1)$ is a random variable; for a fixed outcome it is an ordinary waveform (a sample function). Example: thermal noise voltages across many identical resistors; the received signal in a communication channel. It is described by its mean $m_X(t)=E[X(t)]$, autocorrelation $R_X(t_1,t_2)$ and PSD. #### White noise **White noise** is an idealised random noise whose power spectral density is **flat (constant) over all frequencies**, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed **Gaussian, zero-mean** (AWGN). **PSD (two-sided):** $$ S_W(f)=\frac{N_0}{2}\quad\text{for all } f $$ where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature. **Autocorrelation** (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$): $$ R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau) $$ ```text S_W(f) R_W(tau) | ^ (N0/2) delta N0/2+----------------- | | | ----+-----------------> f ------+------> tau 0 0 ``` Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so **any two samples of white noise, however close, are uncorrelated** (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a random process? A. A single deterministic waveform B. An ensemble of time functions, one per outcome of a random experiment C. A signal with a constant mean only D. A noise with flat PSD Answer: B. At a fixed time it is a random variable; for a fixed outcome it is a sample function. (Key point: Random process meaning) 2. What is the two-sided PSD of white noise? A. N0 only for |f| < W B. Proportional to f^2 C. N0/2 for all frequencies D. Zero outside one frequency Answer: C. White noise has constant PSD N0/2 over all f. (Key point: White noise PSD) 3. What is the autocorrelation of white noise? A. (N0/2) delta(tau) B. N0/2 for all tau C. N0 sinc(tau) D. N0 exp(-|tau|) Answer: A. The inverse Fourier transform of a constant is an impulse. (Key point: White noise autocorrelation) 4. What does R(tau) = 0 for tau not 0 imply? A. The noise has zero power B. The noise is periodic C. The noise has a nonzero mean D. Samples at different times are uncorrelated, however close Answer: D. An impulse ACF means no correlation between distinct samples; total power is infinite, so it is only a model. (Key point: Meaning of delta ACF) ## Related in Communication Systems - [White noise PSDF and AC; properties of autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Awhite-noise-psdf-ac-properties) — Communication Systems, chapter 9 - [White noise: relation between PSDF and AC function](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Awhite-noise-psdf-ac-relation) — Communication Systems, chapter 9 - [Autocorrelation properties; AC of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-properties-white-noise) — Communication Systems, chapter 9, asked 2× - [Parseval's theorem; PSD–autocorrelation relation for white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-psd-autocorrelation-white-noise) — Communication Systems, chapter 9 - [Short note: White noise and its PSDF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-white-noise-psdf) — Communication Systems, chapter 9 - [ESD, PSD and autocorrelation of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-psd-white-noise-autocorrelation) — Communication Systems, chapter 9, asked 2×