--- title: "Matched filter approximation using ideal LPF" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:matched-filter-ideal-lpf" source: https://www.bctengineer.com/q/semester_6/communication_systems/matched-filter-ideal-lpf site: BCT Engineering --- # Matched filter approximation using ideal LPF Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2070 Chaitra (CS II), 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-ideal-lpf - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Explain the approximation of the matched filter for a rectangular pulse using an Ideal low pass filter with variable bandwidth. ## Short answer An ideal LPF with variable bandwidth approximates the matched filter for a rectangular pulse. Optimum bandwidth B ≈ 0.685/T yields output SNR ratio ρ ≈0.825 compared to the matched filter, only about0.84 dB degradation. ## Answer The true matched filter for a rectangular pulse is an integrate-and-dump filter, giving $(S/N)_{max} = 2E/N_0$. A simpler practical receiver uses an **ideal LPF** whose bandwidth $B$ can be chosen; the question is how close it can come. **Set-up.** Pulse $p(t) = A$ for $|t| \le T/2$ (energy $E = A^2T$), white noise PSD $N_0/2$, ideal LPF of unit gain and bandwidth $B$. Pulse spectrum: $$ P(f) = AT\,\frac{\sin \pi fT}{\pi fT} $$ ```text |H(f)| 1 +-----------+ | | ----+-----+-----+-----> f -B 0 B ``` **Peak output signal** (at the pulse centre $t=0$): $$ \begin{aligned} s_o(0) &= \int_{-B}^{B} AT\frac{\sin\pi fT}{\pi fT}\,df = \frac{2A}{\pi}\int_0^{\pi BT}\frac{\sin x}{x}\,dx \\ &= \frac{2A}{\pi}\,\text{Si}(\pi BT) \end{aligned} $$ **Output noise power:** $N_o = \dfrac{N_0}{2}\cdot 2B = N_0B$ **Output SNR** $$ \left(\frac{S}{N}\right)_{LPF} = \frac{4A^2\,\text{Si}^2(\pi BT)}{\pi^2N_0B} $$ **Compare with the matched filter**, $(S/N)_{MF} = 2A^2T/N_0$: $$ \rho = \frac{(S/N)_{LPF}}{(S/N)_{MF}} = \frac{2\,\text{Si}^2(\pi BT)}{\pi^2\,BT} $$ **Effect of bandwidth** - $B$ too small: pulse energy is cut off, peak signal is small. - $B$ too large: little extra signal but noise grows in proportion to $B$. - An optimum exists in between. Evaluating $\rho$ numerically: | $BT$ | 0.5 | 0.685 | 1.0 | 1.5 | |---|---|---|---|---| | $\rho$ | 0.762 | 0.825 | 0.695 | 0.349 | **Result:** the best choice is $B \approx 0.685/T$ (about $0.7/T$), giving $\rho \approx 0.825$, i.e. only about **0.84 dB** worse than the true matched filter. Hence an ideal LPF with bandwidth about $0.7 \times$ the bit rate is a good, simple approximation to the matched filter for rectangular pulses. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is an ideal LPF used to approximate the matched filter for a rectangular pulse? A. It gives higher SNR than the matched filter B. It is a simpler practical receiver whose bandwidth B can be chosen C. It removes all noise D. It needs no knowledge of the pulse Answer: B. The true matched filter is integrate-and-dump; the LPF is simpler and the question is how close it gets by choosing B. (Key point: Why an LPF approximation) 2. What happens if the LPF bandwidth B is made too large? A. Peak signal drops sharply B. Little extra signal is gained but noise grows in proportion to B C. Noise power falls D. The pulse energy is cut off Answer: B. Noise power is N_0 B, so it keeps increasing while the peak signal saturates. (Key point: Bandwidth trade-off) 3. The best LPF bandwidth for a rectangular pulse of width T is about A. 0.2/T B. 0.7/T C. 1.5/T D. 3/T Answer: B. The ratio rho is maximum (about 0.825) at BT of about 0.685. (Key point: Optimum bandwidth) 4. At the optimum bandwidth, the ideal LPF output SNR is about what fraction of the matched filter SNR? A. 0.825, about 0.84 dB worse B. 0.5, about 3 dB worse C. 0.95, about 0.2 dB worse D. 1.0, no loss Answer: A. rho is about 0.825 at BT = 0.685, which is roughly a 0.84 dB loss. (Key point: Loss relative to matched filter) ## Close variants set in other exams - [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× ## 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 ## Related in other subjects - [Impulse response of ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-response) — Signals and Systems, chapter 5, asked 5× - [Impulse response of DT ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-discrete-time) — Signals and Systems, chapter 6 - [Ideal low pass and high pass filters](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afrequency-response-lp-hp-filters) — Signals and Systems, chapter 5 - [Ideal band pass and low pass filters (DT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-bandpass-lowpass-dt) — Signals and Systems, chapter 6