--- title: "White noise PSDF and AC; properties of autocorrelation" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:white-noise-psdf-ac-properties" source: https://www.bctengineer.com/q/semester_6/communication_systems/white-noise-psdf-ac-properties site: BCT Engineering --- # White noise PSDF and AC; properties of autocorrelation Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2068 Bhadra (CS I), 4+4 marks - Video: [WHITE NOISE (OR) WHITE GAUSSIAN NOISE](https://www.youtube.com/watch?v=1KkbpsV6WLk) — PRAKASAM TUTORIALS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/white-noise-psdf-ac-properties - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Define white noise with PSDF and auto correlation function. State the properties of auto correlation function. ## Short answer White noise has flat PSD S(f) = N0/2 for all f, leading to autocorrelation function R(τ) = (N0/2)δ(τ). Key properties of autocorrelation include even symmetry, maximum at zero lag equal to average power, a Fourier pair with PSD via Wiener–Khinchin, periodicity for periodic signals, and convergence to the squared mean for large lags. ## Answer #### White noise, its PSDF and AC function **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. #### Properties of the autocorrelation function 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 $\tau\to\infty$. White noise satisfies these: $\frac{N_0}{2}\delta(\tau)$ is even, maximum at $\tau=0$, and its Fourier transform is the flat PSD. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What defines white noise? A. Constant PSD over all frequencies B. Constant amplitude in time C. Non-zero mean D. PSD confined to one band Answer: A. Like white light, all frequencies contribute equally. (Key point: White noise definition) 2. How is R(tau) of white noise obtained? A. Square of the PSD B. Integral of the PSD over time C. Inverse Fourier transform of the PSD, giving (N0/2) delta(tau) D. Derivative of the PSD Answer: C. Wiener-Khinchin with F^-1{1} = delta. (Key point: ACF derivation) 3. Where does the ACF have its maximum magnitude? A. At tau = T0/2 B. At tau = 0 C. At tau tending to infinity D. At the first zero crossing Answer: B. |R(tau)| <= R(0). (Key point: Max of ACF) 4. Which pair of ACF properties is correct? A. R(-tau) = R(tau) and R(0) = mean-square power B. R(-tau) = -R(tau) and R(0) = 0 C. R(tau) is odd and R(0) is the mean D. R(tau) is periodic for all processes Answer: A. The ACF is even and R(0) equals E[X^2]. (Key point: ACF symmetry and power) ## Related in Communication Systems - [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× - [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× - [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× - [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