--- title: "Matched filter; impulse response of optimum detector" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:matched-filter-optimum-detector" source: https://www.bctengineer.com/q/semester_6/communication_systems/matched-filter-optimum-detector site: BCT Engineering --- # Matched filter; impulse response of optimum detector Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2079 Bhadra (CS II), 5+5 marks - Video: [L 8 | Introduction to Matched Filter | SNR of Matched Filter | Transfer Function of Matched Filter |](https://www.youtube.com/watch?v=UGvAPqwiwkc) — Dopamine - 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-optimum-detector - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Define matched filter. Find the impulse response of optimum detector in the presence of additive white noise. ## 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. Key features: maximum output SNR $=2E/N_0$; $H(f)=kS^*(f)e^{-j2\pi fT}$; it maximises SNR but does not preserve pulse shape (output is the autocorrelation of the pulse). #### Impulse response of optimum detector in additive white noise An **optimum detector** is a receiver (filter followed by sampler and threshold device) that makes the decision with the **minimum probability of error**. For a known pulse in additive white Gaussian noise, this is achieved by the filter that **maximises the output signal-to-noise ratio at the sampling instant**; this optimum filter is the **matched filter**. **Set-up.** The received signal is $x(t)=s(t)+w(t)$, $0\le t\le T$, where $s(t)$ is a known pulse and $w(t)$ is white noise of two-sided PSD $N_0/2$. It passes through an LTI filter $h(t)\leftrightarrow H(f)$ and is sampled at $t=T$: ```text s(t)+w(t) +--------+ y(t) sample y(T) ----------->| h(t) |---------->--/ ----> decision +--------+ at t=T ``` Output $y(t)=s_o(t)+n_o(t)$ with $$ s_o(T)=\int_{-\infty}^{\infty}H(f)S(f)e^{j2\pi fT}df,\qquad E[n_o^2]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df $$ We want $H(f)$ that maximises the output peak signal-to-noise ratio $$ \eta=\frac{\lvert s_o(T)\rvert^2}{E[n_o^2]}=\frac{\left\lvert\int H(f)S(f)e^{j2\pi fT}df\right\rvert^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df} $$ **Schwarz inequality:** $\left\lvert\int\phi_1\phi_2\,df\right\rvert^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$, with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=S(f)e^{j2\pi fT}$: $$ \eta\le\frac{\int\lvert H\rvert^2df\int\lvert S\rvert^2df}{\frac{N_0}{2}\int\lvert H\rvert^2df}=\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert S(f)\rvert^2df=\frac{2E}{N_0} $$ (using Rayleigh's theorem, $E$ = pulse energy). The maximum is reached when $$ H_{opt}(f)=k\,S^*(f)\,e^{-j2\pi fT} $$ **Impulse response.** Take the inverse transform. For real $s(t)$, $S^*(f)=S(-f)$, which is the transform of $s(-t)$; the factor $e^{-j2\pi fT}$ is a delay of $T$: $$ \begin{aligned} h_{opt}(t) &= k\int_{-\infty}^{\infty}S^*(f)e^{-j2\pi fT}e^{j2\pi ft}df \\ &= k\int_{-\infty}^{\infty}S(-f)e^{-j2\pi f(T-t)}df \\ &= k\,s(T-t) \end{aligned} $$ So the optimum (matched) filter impulse response is the input pulse **reversed in time and delayed by $T$**. Choosing the sampling instant $T$ equal to the pulse duration makes $h(t)$ causal. ```text s(t) h(t) = s(T - t) |\ /| | \ / | | \ / | ---+---+---> t ---+---+---> t 0 T 0 T ``` **Results:** maximum SNR $=2E/N_0$ depends only on pulse energy, not on pulse shape; the filter's frequency response is the conjugate of the pulse spectrum, so it gives the most weight to the frequencies where the signal is strongest. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the impulse response of the matched filter for pulse s(t) and sampling time T? A. h(t) = k s(t) B. h(t) = k s(T - t) C. h(t) = k s(t - T) D. h(t) = k s(-t - T) Answer: B. The pulse is reversed in time and delayed by T. (Key point: Matched filter impulse response) 2. Which result is used to find the filter that maximises output SNR? A. Nyquist criterion B. Parseval only C. Central limit theorem D. Schwarz inequality, with equality when H(f) = k S*(f) e^{-j2 pi f T} Answer: D. Equality in the Schwarz inequality requires phi1 proportional to conjugate of phi2, giving the matched filter. (Key point: Mathematical tool) 3. What is the maximum output peak SNR of the matched filter? A. E / N0 B. 2E / N0 C. E / 2N0 D. 4E / N0 Answer: B. eta <= (2/N0) integral |S(f)|^2 df = 2E/N0; it depends only on pulse energy. (Key point: Maximum SNR) 4. Which statement about the matched filter is correct? A. It preserves the pulse shape B. Its output SNR depends on the pulse shape C. It maximises SNR at the sampling instant but the output is the pulse autocorrelation D. It is optimal only for coloured noise Answer: C. Matched filter output is the autocorrelation of the pulse; shape is not preserved, SNR is maximised for white noise. (Key point: Definition features) ## Related in Communication Systems - [Optimum detector impulse response in additive white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response) — Communication Systems, chapter 9, asked 2× - [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× - [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 - [Optimum detector impulse response; Hilbert transform](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response-hilbert) — Communication Systems, chapter 9 - [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× - [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×