--- title: "Impulse response of matched filter" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:matched-filter-awgn" source: https://www.bctengineer.com/q/semester_6/communication_systems/matched-filter-awgn site: BCT Engineering --- # Impulse response of matched filter Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2068 Jestha (old course), 10 marks; 2080 Bhadra (CS II), 8 marks; 2074 Asoj (CS II), 8 marks (asked 3 times) - Video: [What is a Matched Filter?](https://www.youtube.com/watch?v=Ci-EjiMJo3I) — Iain Explains Signals, Systems, and Digital Comms - 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-awgn - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Derive the expression for the impulse response of a matched filter for a AWGN channel. ## Short answer Maximizing the output SNR at sampling instant T via the Schwarz inequality yields H_opt(f)=k G*(f) e^{-j2πfT}. The impulse response is h_opt(t)=k g(T-t), the time-reversed and delayed version of the input pulse, called the matched filter. ## Answer A **matched filter** is the linear filter that maximises the peak output signal-to-noise ratio at the sampling instant when a known pulse is received in additive white Gaussian noise (AWGN). It is used as the optimum detector in digital receivers. #### Derivation Consider a known pulse $g(t)$, $0\le t\le T$, plus white Gaussian noise $w(t)$ of power spectral density $N_0/2$, applied to a linear time-invariant filter with impulse response $h(t)$ and transfer function $H(f)$. ```text x(t) = g(t) + w(t) --> [ h(t) ] --> y(t) = go(t) + n(t) | sample at t = T ``` **Signal output** at the sampling instant: $$ g_o(T)=\int_{-\infty}^{\infty}H(f)G(f)e^{j2\pi fT}df $$ **Noise output** power (white noise through $H(f)$): $$ E[n^2(t)]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df $$ **Peak pulse SNR** to be maximised: $$ \eta=\frac{\lvert g_o(T)\rvert^2}{E[n^2(t)]}=\frac{\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df} $$ **Schwarz inequality:** for any $\phi_1(f),\phi_2(f)$, $$ \left|\int\phi_1\phi_2\,df\right|^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df $$ with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=G(f)e^{j2\pi fT}$: $$ \left|\int H(f)G(f)e^{j2\pi fT}df\right|^2\le\int\lvert H(f)\rvert^2df\int\lvert G(f)\rvert^2df $$ Substituting into $\eta$: $$ \eta\le\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df=\frac{2E}{N_0} $$ using Rayleigh's energy theorem $E=\int\lvert G(f)\rvert^2df=\int g^2(t)dt$. **Optimum filter:** the maximum $\eta_{max}=2E/N_0$ is reached when equality holds: $$ H_{opt}(f)=k\,G^*(f)\,e^{-j2\pi fT} $$ **Impulse response:** take the inverse Fourier transform: $$ \begin{aligned} h_{opt}(t)&=k\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi f(T-t)}df\\ &=k\left[\int_{-\infty}^{\infty}G(f)e^{j2\pi f(T-t)}df\right]^*\quad(\text{real }g)\\ &=k\,g(T-t) \end{aligned} $$ So the impulse response of the optimum filter is the **time-reversed and delayed (by $T$) version of the input signal**, scaled by $k$. Because it is "matched" to the signal, it is called the **matched filter**. ```text g(t) h(t) = g(T - t) |\ /| | \ / | | \ / | +---+-----> t +---+---+----> t 0 T 0 T (flip about t = 0, then shift right by T) ``` #### Properties - Maximum output SNR $=2E/N_0$ depends only on the signal **energy**, not on its shape. - $\lvert H_{opt}(f)\rvert=k\lvert G(f)\rvert$: the filter passes strongly where the signal spectrum is strong. - The output $g_o(t)$ is proportional to the autocorrelation of $g(t)$ shifted by $T$, peaking at $t=T$; hence a matched filter is equivalent to a **correlator** (multiply by $g(t)$ and integrate over $0$ to $T$). - The delay $T$ makes $h(t)$ causal (zero for $t<0$). #### Example For a rectangular pulse $g(t)=A$, $0\le t\le T$, the matched filter is $h(t)=kA$ for $0\le t\le T$ (an integrate-and-dump filter). The output is a triangle peaking at $t=T$ with value $kA^2T=kE$, and $\eta_{max}=2A^2T/N_0$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a matched filter maximise? A. Output noise power B. Bandwidth of the pulse C. Peak output SNR at the sampling instant for a known pulse in AWGN D. Average signal amplitude Answer: C. It is the linear filter giving maximum peak pulse SNR at t = T. (Key point: Matched filter purpose) 2. What role does the Schwarz inequality play? A. It gives the noise PSD B. It bounds the numerator so SNR <= 2E/N0, with equality when H(f) = k G*(f) e^(-j2 pi fT) C. It converts time to frequency D. It shows h(t) is not causal Answer: B. Equality in Schwarz fixes the optimum H(f). (Key point: Schwarz inequality) 3. What is the impulse response of the matched filter? A. h(t) = k g(T - t), the time-reversed pulse delayed by T B. h(t) = k g(t) C. h(t) = k g(t + T) D. h(t) = k G(f) Answer: A. The inverse transform of k G*(f) e^(-j2 pi fT) is k g(T-t). (Key point: Impulse response) 4. On what does the maximum output SNR depend? A. Pulse shape and energy B. Peak amplitude only C. Pulse width only D. Only signal energy: 2E/N0 Answer: D. eta_max = 2E/N0 regardless of shape. (Key point: Maximum SNR) 5. What is the matched filter for a rectangular pulse of amplitude A and width T? A. A sinc response B. An impulse at t = 0 C. A rectangular response kA over 0 to T (integrate-and-dump), output peaking at t = T D. A ramp that ends at 0 Answer: C. Reversing and delaying a rectangle gives the same rectangle; the output is a triangle peaking at T. (Key point: Rectangular pulse) ## Related in Communication Systems - [Ergodic process; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aergodic-matched-filter) — Communication Systems, chapter 9 - [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× - [Matched filter impulse response is reverse delayed input](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-reverse-delayed) — Communication Systems, chapter 9, asked 2× - [Matched filter; impulse response of optimum detector](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-optimum-detector) — 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× - [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 ## 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 RC filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Arc-filter-impulse-response) — Signals and Systems, chapter 5, asked 2× - [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 LPF impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-step) — Signals and Systems, chapter 5, asked 5×