--- title: "Optimum detector; matched filter impulse response" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:matched-filter-impulse-response" source: https://www.bctengineer.com/q/semester_6/communication_systems/matched-filter-impulse-response site: BCT Engineering --- # Optimum detector; matched filter impulse response Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2079 Chaitra, 2+6 marks; 2072 Chaitra (CS II), 2+5 marks (asked 2 times) - Video: [Matched Filter- Communication Systems](https://www.youtube.com/watch?v=DKHOfYFwK6Q) — sam bytes - 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-impulse-response - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question What do you mean by optimum detector? Show that the impulse response of the matched filter is reverse delayed version of the input signal. ## Short answer An optimum detector minimizes error probability by maximizing output SNR with a matched filter. For a known pulse g(t), the filter impulse response is h(t)=k g(T-t), i.e., the time-reversed and delayed input signal, derived from the Schwarz inequality. ## Answer #### Optimum detector An **optimum detector** is a receiver that makes decisions with the **minimum probability of error** for given signals and noise. In AWGN this is achieved by first passing the received signal through a filter that maximises the output SNR at the sampling instant (the matched filter or an equivalent correlator), then comparing the sample with a threshold (maximum-likelihood decision). #### Matched filter impulse response 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 $t<0$). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is an optimum detector? A. A receiver with maximum bandwidth B. A receiver that decides with minimum probability of error C. A receiver with no filter D. One that amplifies the signal most Answer: B. In AWGN, it is a matched filter or correlator followed by a threshold decision. (Key point: Optimum detector) 2. What quantity is maximised in the derivation? A. Peak pulse SNR |go(T)|^2 / E[n^2] B. Output noise power C. Pulse bandwidth D. Signal phase Answer: A. The filter is chosen to maximise eta at the sampling instant. (Key point: SNR to maximise) 3. Equality in the Schwarz inequality gives which H(f)? A. H(f) = k G(f) B. H(f) = k / G(f) C. H(f) = k e^(j2 pi fT) D. H(f) = k G*(f) e^(-j2 pi fT) Answer: D. Equality needs H proportional to the conjugate of G(f)e^(j2 pi fT). (Key point: Optimum transfer function) 4. What does the matched filter impulse response show? A. h(t) = k g(t - T) B. h(t) = k g(-t) with no delay C. h(t) = k g(T - t): time-reversed input delayed by T D. h(t) = k dg/dt Answer: C. The delay T makes the reversed pulse causal. (Key point: Reverse delayed result) 5. Why is a matched filter equivalent to a correlator? A. It multiplies by noise B. Its output is the autocorrelation of g(t), peaking at t = T C. It differentiates the signal D. It has no peak Answer: B. Sampling at T gives the integral of x(t)g(t) over 0 to T. (Key point: Correlator equivalence) ## Close variants set in other exams - [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 Communication Systems - [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 - [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 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 impulse response; Hilbert transform](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response-hilbert) — 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×