--- title: "Noise equivalent bandwidth of RC low pass filter" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:neb-rc-lowpass" source: https://www.bctengineer.com/q/semester_6/communication_systems/neb-rc-lowpass site: BCT Engineering --- # Noise equivalent bandwidth of RC low pass filter Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2080 Baisakh (CS II), 4+6 marks - Video: [Noise Equivalent Bandwidth](https://www.youtube.com/watch?v=NHSE908Rhbc) — 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/neb-rc-lowpass - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Define Noise Equivalent bandwidth. Find noise equivalent bandwidth of the first order RC low pass filter. ## Short answer Noise equivalent bandwidth B_N = ∫|H(f)|²df / |H(0)|² equals the bandwidth of an ideal rectangular filter with same peak gain passing equal noise power. For a first-order RC low-pass filter H(f) = 1/(1+j2πfRC), B_N = 1/(4RC) = (π/2)f_3dB ≈ 1.57 f_3dB. Output noise power is N_0/(4RC). ## Answer #### Noise equivalent bandwidth The **noise equivalent bandwidth (NEB)** $B_N$ of a filter is the bandwidth of an **ideal rectangular filter**, with the same peak (usually zero-frequency or centre) gain $\lvert H(0)\rvert$, that passes the **same total noise power** as the actual filter when both are fed with white noise. White noise $N_0/2$ into $H(f)$ gives output power $$ P_{actual}=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df=N_0\int_0^{\infty}\lvert H(f)\rvert^2df $$ The ideal filter of gain $\lvert H(0)\rvert$ and bandwidth $B_N$ gives $P_{ideal}=N_0B_N\lvert H(0)\rvert^2$. Equating: $$ B_N=\frac{\int_0^{\infty}\lvert H(f)\rvert^2df}{\lvert H(0)\rvert^2} $$ ```text |H(f)|^2 |H(0)|^2 +---------+ equal areas | ... | | .. | | .|.. | | ...... --+---------+----------> f 0 B_N ``` NEB lets us write the output noise power simply as $N=N_0B_N\lvert H(0)\rvert^2$ (or $kTB_N$ per unit gain), which is used in noise figure and SNR calculations. For a band-pass filter $\lvert H(0)\rvert$ is replaced by the centre-frequency gain $\lvert H(f_0)\rvert$. #### NEB of first-order RC low-pass filter ```text vin o---/\/\/\---+---o vout R | === C | gnd o------------+---o ``` For the RC low-pass filter (series $R$, shunt $C$): $$ H(f)=\frac{1}{1+j2\pi fRC},\qquad \lvert H(f)\rvert^2=\frac{1}{1+(2\pi fRC)^2},\qquad \lvert H(0)\rvert=1 $$ $$ \begin{aligned} B_N &= \int_0^{\infty}\frac{df}{1+(2\pi fRC)^2}\qquad (\text{let } u=2\pi fRC,\ df=\tfrac{du}{2\pi RC}) \\ &= \frac{1}{2\pi RC}\int_0^{\infty}\frac{du}{1+u^2}=\frac{1}{2\pi RC}\cdot\frac{\pi}{2} \\ &= \frac{1}{4RC} \end{aligned} $$ Since the 3 dB bandwidth is $f_{3dB}=\frac{1}{2\pi RC}$, $$ B_N=\frac{\pi}{2}f_{3dB}\approx1.57\,f_{3dB} $$ The NEB is larger than the 3 dB bandwidth because the RC response rolls off slowly (only $-20$ dB/decade) and passes noise well beyond $f_{3dB}$. Output noise power $=N_0B_N=\frac{N_0}{4RC}$. **Example:** $R=1\ \text{k}\Omega$, $C=1\ \mu\text{F}$: $f_{3dB}=159.2$ Hz, $B_N=\frac{1}{4\times10^{-3}}=250$ Hz. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the noise equivalent bandwidth of a filter represent? A. The frequency where gain falls by 3 dB B. The width of the main lobe of H(f) C. The bandwidth of an ideal rectangular filter with the same peak gain passing the same white-noise power D. The bandwidth where phase is linear Answer: C. NEB is defined by equal output noise power for white input: B_N = integral of |H|^2 df (0 to inf) / |H(0)|^2. (Key point: Definition of NEB) 2. What is the NEB of a first-order RC low-pass filter? A. 1/(2 pi RC) B. 1/(4RC) C. 1/(RC) D. pi/(RC) Answer: B. Integral of df/(1+(2 pi f RC)^2) from 0 to inf = (1/2piRC)(pi/2) = 1/(4RC). (Key point: NEB of RC filter) 3. How is B_N of the RC low-pass related to f_3dB? A. B_N = f_3dB B. B_N = (pi/2) f_3dB, about 1.57 f_3dB C. B_N = f_3dB / 2 D. B_N = 2 f_3dB Answer: B. With f_3dB = 1/(2 pi RC), 1/(4RC) = (pi/2) f_3dB. (Key point: NEB versus 3 dB bandwidth) 4. Why is B_N larger than f_3dB for the RC filter? A. Its gain exceeds 1 at low frequency B. The response rolls off slowly (-20 dB/decade) and passes noise beyond f_3dB C. The filter is non-linear D. The phase response adds noise Answer: B. Slow roll-off means noise beyond f_3dB still contributes, so an equivalent ideal filter must be wider. (Key point: Why NEB is larger) ## Related in Communication Systems - [Matched filter approximation using RC low pass filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-rc-approximation) — Communication Systems, chapter 9, asked 2× - [Noise equivalent bandwidth; matched filter is time reversed](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-matched-filter-reverse) — Communication Systems, chapter 9 - [Noise equivalent bandwidth; optimum detector impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-optimum-detector) — Communication Systems, chapter 9 - [Short note: Noise equivalent bandwidth](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-neb) — Communication Systems, chapter 9 - [Moments of random variable; NEB of RC-LPF; output noise power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amoments-neb-rc-lpf) — Communication Systems, chapter 9 - [Ideal and RC filtering of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arc-filtering-white-noise) — Communication Systems, chapter 9 ## Related in other subjects - [RC low pass filter operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Arc-low-pass-filter) — Basic Electronics Engineering, chapter 1, asked 5× - [Transfer function of RC low-pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Arc-lowpass-transfer-function) — Signals and Systems, chapter 5, asked 2× - [RC low and high pass filters](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Afilter-lpf-hpf-transfer) — Basic Electronics Engineering, chapter 1 - [RC high pass filter operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Arc-high-pass-filter) — Basic Electronics Engineering, chapter 1, asked 5×