--- title: "Ideal and RC filtering of white noise" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:rc-filtering-white-noise" source: https://www.bctengineer.com/q/semester_6/communication_systems/rc-filtering-white-noise site: BCT Engineering --- # Ideal and RC filtering of white noise Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2075 Chaitra (CS II), 8 marks - Video: [Practical RC Filtered White Noise](https://www.youtube.com/watch?v=jBmiHS7a8zE) — Electronic Waves - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/rc-filtering-white-noise - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Explain and compare ideal and practical RC filtering of white noise with respect to change in autocorrelation function. ## Short answer Ideal low-pass filtering of white noise yields a sinc autocorrelation R(τ)=N₀B sinc(2Bτ) with zeros at τ=k/(2B), so samples at Nyquist rate are uncorrelated. Practical RC filtering produces an always-positive exponential autocorrelation R(τ)=(N₀/(4RC))e^{-|τ|/RC} with no zero crossings, indicating correlated samples for all τ. ## Answer White noise ($S_W(f)=N_0/2$, $R_W(\tau)=\frac{N_0}{2}\delta(\tau)$) has infinite power and zero correlation time. Passing it through a filter $H(f)$ gives $S_Y(f)=\lvert H(f)\rvert^2\frac{N_0}{2}$ and $R_Y(\tau)=\mathcal{F}^{-1}\{S_Y(f)\}$: the noise becomes **band-limited (coloured)** and its samples become correlated. #### Ideal low-pass filtering Ideal LPF, gain 1 for $\lvert f\rvert\le B$, zero elsewhere: $$ S_Y(f)=\begin{cases}\frac{N_0}{2}, & \lvert f\rvert\le B\\ 0, & \text{otherwise}\end{cases} $$ $$ \begin{aligned} R_Y(\tau) &= \int_{-B}^{B}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\cdot\frac{\sin2\pi B\tau}{\pi\tau} \\ &= N_0B\,\frac{\sin2\pi B\tau}{2\pi B\tau} \end{aligned} $$ - Power $R_Y(0)=N_0B$. - $R_Y(\tau)$ is a sinc with zeros at $\tau=\pm\frac{k}{2B}$, $k=1,2,\dots$: samples taken at the Nyquist rate $2B$ are **uncorrelated** (independent if Gaussian). - It oscillates (takes negative values) and decays slowly as $1/\tau$. #### Practical RC low-pass filtering $H(f)=\frac{1}{1+j2\pi fRC}$, so $$ S_Y(f)=\frac{N_0/2}{1+(2\pi fRC)^2} $$ Using the pair $e^{-a\lvert\tau\rvert}\leftrightarrow\frac{2a}{a^2+(2\pi f)^2}$ with $a=1/RC$: $$ R_Y(\tau)=\frac{N_0}{4RC}\,e^{-\lvert\tau\rvert/RC} $$ - Power $R_Y(0)=\frac{N_0}{4RC}$, i.e. NEB $=\frac{1}{4RC}=\frac\pi2f_{3dB}$. - $R_Y(\tau)$ is a decaying exponential: always positive, **never exactly zero**, falls to $1/e$ at $\tau=RC$. Correlation time $\approx RC$. ```text R(tau) ideal LPF: sinc RC LPF: exponential N0B . N0/4RC . / \ /|\ . / \ . / | \ -.-/-----\-.-- tau ----'--+--'---- tau -1/2B 1/2B -RC 0 RC ``` #### Comparison | Point | Ideal LPF (bandwidth $B$) | RC LPF ($f_{3dB}=1/2\pi RC$) | |---|---|---| | Output PSD | Flat up to $B$, sharp cut-off | Lorentzian, gradual roll-off | | Autocorrelation | $N_0B\,\text{sinc}(2B\tau)$ | $\frac{N_0}{4RC}e^{-\lvert\tau\rvert/RC}$ | | Output noise power | $N_0B$ | $N_0/4RC=N_0\cdot\frac\pi2f_{3dB}$ | | Zero crossings of $R(\tau)$ | At $\tau=k/2B$ | None | | Sign of $R(\tau)$ | Positive and negative | Always positive | | Decay of $R(\tau)$ | Slow, $\propto1/\tau$ | Fast, exponential | | Realisable | No (non-causal) | Yes | In both cases a narrower bandwidth gives a **wider** autocorrelation (longer correlation time) and less noise power. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the output autocorrelation of white noise through an ideal LPF of bandwidth B? A. (N0/4RC) exp(-|tau|/RC) B. N0 B sinc(2B tau) C. (N0/2) delta(tau) D. N0 B exp(-2B|tau|) Answer: B. Inverse transform of a flat PSD over |f| < B is N0 B sin(2 pi B tau)/(2 pi B tau). (Key point: Ideal LPF autocorrelation) 2. Which describes R_Y(tau) for white noise through an RC low-pass filter? A. Sinc with zero crossings B. Delta function C. (N0/4RC) e^{-|tau|/RC}: always positive, never exactly zero, 1/e at tau = RC D. Constant for all tau Answer: C. S_Y is Lorentzian and its inverse transform is a two-sided decaying exponential. (Key point: RC filter autocorrelation) 3. What happens when the filter bandwidth is reduced? A. Correlation time shortens and power rises B. Correlation time and power are unchanged C. Autocorrelation becomes a delta function D. Autocorrelation becomes wider (longer correlation) and noise power is less Answer: D. Narrower bandwidth means a wider R(tau) and lower power (N0 B or N0/4RC). 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