--- title: "Matched filter approximation using RC low pass filter" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:matched-filter-rc-approximation" source: https://www.bctengineer.com/q/semester_6/communication_systems/matched-filter-rc-approximation site: BCT Engineering --- # Matched filter approximation using RC low pass filter Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2072 Kartik (CS II), 2+6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/matched-filter-rc-approximation - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Define Matched filter. Explain the approximation of the matched filter for a rectangular pulse using a single pole RC low pass filter with variable bandwidth. ## Short answer A matched filter maximizes peak SNR for a known signal in white noise, with impulse response h(t)=k s(T-t). For a rectangular pulse, an RC low-pass filter approximates the integrate-and-dump; optimum 3 dB bandwidth f_c ≈ 0.2/T gives SNR within 0.89 dB of the matched filter. ## Answer #### Matched filter A **matched filter** is a linear filter whose impulse response is matched to (a time-reversed, delayed copy of) a known input pulse $s(t)$, $h(t)=k\,s(T-t)$, so that it gives the **maximum possible peak signal-to-noise ratio** at the sampling instant $t=T$ when the pulse is received in additive white noise. It is the optimum detector for digital signals in AWGN; in practice it is built as a correlator. #### RC low-pass filter as approximate matched filter The exact matched filter for a rectangular pulse is an integrate-and-dump circuit, which is not always convenient. A simple **single-pole RC low-pass filter** can approximate it; we find the best bandwidth and the loss. **Input pulse:** $s(t)=A$ for $0\le t\le T$ (energy $E=A^2T$), white noise PSD $N_0/2$. **Filter:** $H(f)=\dfrac{1}{1+jf/f_c}$, $f_c=\dfrac{1}{2\pi RC}$ (3 dB bandwidth, variable). **Signal output.** The step response of the RC filter rises as $A(1-e^{-t/RC})$; it is maximum at the end of the pulse, $t=T$: $$ s_o(T)=A\left(1-e^{-T/RC}\right)=A\left(1-e^{-2\pi f_cT}\right) $$ ```text s(t) s_o(t) A +-----+ A | ..-. peak at t=T | | | .' '. | | | .' '.. --+-----+--> t --+-'-----+-------'--> t 0 T 0 T ``` **Noise output power:** $$ N_o=\frac{N_0}{2}\int_{-\infty}^{\infty}\frac{df}{1+(f/f_c)^2}=\frac{N_0}{2}\cdot\pi f_c $$ **Output SNR:** $$ \text{SNR}_o=\frac{A^2\left(1-e^{-2\pi f_cT}\right)^2}{\pi N_0f_c/2}=\frac{2A^2T}{N_0}\cdot\frac{2\left(1-e^{-x}\right)^2}{x},\qquad x=2\pi f_cT $$ **Compared with the matched filter** ($\text{SNR}_{max}=2E/N_0=2A^2T/N_0$): $$ \rho(x)=\frac{\text{SNR}_o}{\text{SNR}_{max}}=\frac{2(1-e^{-x})^2}{x} $$ **Effect of variable bandwidth:** - $f_c$ too small: the filter cannot follow the pulse, $s_o(T)$ is small, SNR low. - $f_c$ too large: signal fully passes but much extra noise enters, SNR low. - An optimum lies between. Setting $d\rho/dx=0$ gives $(1+2x)e^{-x}=1$, solved numerically: $x\approx1.256$. $$ f_cT=\frac{1.256}{2\pi}\approx0.2,\qquad \rho_{max}=\frac{2(1-e^{-1.256})^2}{1.256}\approx0.815 $$ **Result:** the best RC filter has $f_c\approx\dfrac{0.2}{T}$ (i.e. $RC\approx0.8T$) and gives an SNR only $10\log_{10}0.815\approx-0.89$ dB, i.e. **less than 1 dB below** the ideal matched filter. Hence a simple RC filter is a good practical substitute for the matched filter of a rectangular pulse. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A matched filter for pulse s(t) has impulse response: A. h(t) = k s(t) and gives minimum noise B. h(t) = k s(T - t), giving maximum peak SNR at t = T in white noise C. h(t) = k ds/dt D. h(t) = k, a constant over the pulse and nothing outside Answer: B. The matched filter is a scaled time-reversed, delayed copy of the pulse and maximises peak SNR at the sampling instant. (Key point: Definition of matched filter) 2. The exact matched filter for a rectangular pulse is: A. A high-pass filter B. An integrate-and-dump (correlator) circuit C. An ideal low-pass filter D. A single-pole RC filter Answer: B. A rectangular pulse reversed is rectangular, so the matched filter integrates over the pulse duration. (Key point: Exact filter for rectangular pulse) 3. What happens if the RC filter's bandwidth f_c is too large? A. The signal output at T becomes zero B. The signal passes fully but extra noise enters, lowering SNR C. The noise is reduced below the matched filter value D. The output SNR exceeds 2E/N0 Answer: B. Too wide a bandwidth lets in extra noise (N_o proportional to f_c) without more signal; too narrow reduces s_o(T). The optimum lies between. (Key point: Effect of RC bandwidth) 4. The best single-pole RC approximation uses x = 2πf_cT ≈ 1.256. What is the result? A. f_cT ≈ 0.2 and SNR about 0.815 of the matched filter value (loss under 1 dB) B. f_cT ≈ 1.256 and SNR equal to the matched filter C. f_cT ≈ 0.2 and SNR half of the matched filter value D. f_cT ≈ 2 and a loss of 3 dB Answer: A. f_cT = 1.256/(2π) ≈ 0.2 and rho_max ≈ 0.815, about -0.89 dB, so RC is a good substitute. (Key point: Optimum RC result) ## Close variants set in other exams - [Matched filter approximation using ideal LPF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-ideal-lpf) — Communication Systems, chapter 9, asked 2× ## Related in Communication Systems - [Noise equivalent bandwidth of RC low pass filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-rc-lowpass) — Communication Systems, chapter 9 - [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 - [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 - [Impulse response of matched filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-awgn) — Communication Systems, chapter 9, asked 3× - [Optimum detector; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-impulse-response) — Communication Systems, chapter 9, asked 3× - [Optimum detector; matched filter; autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-matched-autocorrelation) — Communication Systems, chapter 9, asked 3× ## 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×