--- title: "Autocorrelation properties; AC of white noise" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:autocorrelation-properties-white-noise" source: https://www.bctengineer.com/q/semester_6/communication_systems/autocorrelation-properties-white-noise site: BCT Engineering --- # Autocorrelation properties; AC of white noise Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2078 Chaitra, 3+5 marks; 2070 Bhadra (CS I), 3+3 marks (asked 2 times) - Video: [Autocorrelation Function | Communication System | Electronics and Communication (ECE) for GATE 2024](https://www.youtube.com/watch?v=c5dTgHIx89U) — BYJU'S Exam Prep GATE & ESE: EE,EC,IN,CS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/autocorrelation-properties-white-noise - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question List out any three properties of autocorrelation (AC) function. Mention the autocorrelation function of white noise. ## Short answer Autocorrelation is even: R(-τ)=R(τ); R(0) equals average power; |R(τ)| ≤ R(0). White noise has a constant power spectral density N0/2, and its autocorrelation function is (N0/2) δ(τ), indicating uncorrelated samples at any non-zero lag. ## Answer #### Properties of the autocorrelation function The **autocorrelation (AC) function** measures how similar a signal is to a time-shifted copy of itself. For a power signal or a wide-sense stationary (WSS) random process $X(t)$: $$ R_X(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}x(t)\,x(t+\tau)\,dt \quad\text{or}\quad R_X(\tau)=E[X(t)X(t+\tau)] $$ For an energy signal, $R_g(\tau)=\int_{-\infty}^{\infty}g(t)g(t+\tau)\,dt$. Important properties (any three answer the question): 1. **Even symmetry:** $R_X(-\tau)=R_X(\tau)$. 2. **Value at origin = mean-square value (power):** $R_X(0)=E[X^2(t)]=P$ (for an energy signal, $R_g(0)=E$). 3. **Maximum at origin:** $\lvert R_X(\tau)\rvert\le R_X(0)$ for all $\tau$. 4. **Fourier pair with PSD (Wiener–Khinchin):** $S_X(f)=\int_{-\infty}^{\infty}R_X(\tau)e^{-j2\pi f\tau}d\tau$ and $R_X(\tau)=\int_{-\infty}^{\infty}S_X(f)e^{j2\pi f\tau}df$. 5. **Periodicity:** if $x(t)$ is periodic with period $T_0$, $R_X(\tau)$ is also periodic with $T_0$. 6. **DC component:** if $X(t)$ has mean $m$ and no periodic part, $R_X(\tau)\to m^2$ as $\tau\to\infty$. #### Autocorrelation function of 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. Which property holds for an autocorrelation function? A. R(-tau) = R(tau) B. R(-tau) = -R(tau) C. R(tau) is always zero at tau = 0 D. R(tau) increases with |tau| Answer: A. The autocorrelation is an even function. (Key point: Even symmetry) 2. What does R(0) represent and where is its maximum? A. Mean value; minimum at the origin B. Energy only; no maximum C. Bandwidth; maximum at tau = T D. Mean-square value (power); maximum at the origin Answer: D. R(0) = E[X^2] = P and |R(tau)| <= R(0). (Key point: Value at origin) 3. What is the PSD of white noise? A. Zero outside a band B. Proportional to f C. Flat N0/2 for all frequencies D. Gaussian in f Answer: C. White noise has a constant two-sided PSD N0/2. (Key point: White noise PSD) 4. What is the autocorrelation of white noise? A. N0/2 B. (N0/2) delta(tau) C. N0 sinc(tau) D. (N0/2) cos(tau) Answer: B. The inverse transform of a constant is an impulse, so samples at different times are uncorrelated. (Key point: White noise autocorrelation) ## 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 - [Random process; white noise PSDF and autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-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 - [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× - [Short note: Autocorrelation function and properties](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-autocorrelation-properties) — Communication Systems, chapter 9, asked 2×