--- title: "Short note: Autocorrelation function and properties" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:short-note-autocorrelation-properties" source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-autocorrelation-properties site: BCT Engineering --- # Short note: Autocorrelation function and properties Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2079 Chaitra (CS I), 5 marks; 2067 Shrawan (CS I), 5 marks (asked 2 times) - Video: [What is Autocorrelation?](https://www.youtube.com/watch?v=hOvE8puBZK4) — 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/short-note-autocorrelation-properties - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Write a short note on autocorrelation function and its properties. ## Short answer The autocorrelation function measures a signal's similarity to a delayed version of itself. For a power signal or WSS process, R(τ)=E[X(t)X(t+τ)]. Its key properties: even symmetry, maximum at origin equal to average power, Fourier pair with power spectral density (Wiener–Khinchin), periodic if the signal is periodic. ## Answer 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, $R_g(\tau)=\int_{-\infty}^{\infty}g(t)g(t+\tau)\,dt$. It tells how fast a signal changes: a slowly varying signal stays correlated for large $\tau$, a rapidly varying one loses correlation quickly. It is used to find the PSD, to detect periodic signals buried in noise, and in correlation receivers and radar. **Properties:** 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 $\tau\to\infty$. **Example:** white noise has $R(\tau)=\frac{N_0}{2}\delta(\tau)$; samples at different times are uncorrelated. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the autocorrelation function indicate? A. Similarity of a signal with a time-shifted copy of itself B. Spectrum width only C. Signal amplitude only D. Phase of the carrier Answer: A. It measures how correlated a signal remains as the shift grows. (Key point: What AC measures) 2. What is the relation between R(tau) and PSD? A. PSD is the derivative of R B. PSD equals R squared C. They are unrelated D. They are a Fourier transform pair Answer: D. S(f) is the Fourier transform of R(tau). (Key point: Wiener-Khinchin) 3. What happens to R(tau) if x(t) is periodic with period T0? A. R(tau) decays to zero B. R(tau) becomes impulsive C. R(tau) is periodic with T0 D. R(tau) is constant Answer: C. Periodicity carries over, which helps detect periodic signals buried in noise. (Key point: Periodic signals) 4. Which statement is correct? A. R(0) is the minimum value B. R(0) is the power and |R(tau)| <= R(0) C. R(0) is always zero D. |R(tau)| >= R(0) Answer: B. The maximum occurs at the origin. 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