--- title: "Impulse response usefulness; impulse response of LPF" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:impulse-response-usefulness-lpf" source: https://www.bctengineer.com/q/semester_6/communication_systems/impulse-response-usefulness-lpf site: BCT Engineering --- # Impulse response usefulness; impulse response of LPF Communication Systems · Chapter 1: Introduction - Asked in: 2072 Asoj (CS I), 2+3+1 marks - Video: [What is the impulse response and pulse response?](https://www.youtube.com/watch?v=u6ceFafHTCA) — Discretised - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/impulse-response-usefulness-lpf - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question What do you mean by the impulse response of a system? Why is the impulse response of a system quite useful in the design of communication systems? What is the impulse response of low-pass filter? ## Short answer Impulse response is the output to a unit impulse input, fully describing an LTI system. It enables output calculation via convolution, yields frequency response, and aids filter/matched-filter design. An ideal low-pass filter (bandwidth B) has h(t)=2B sinc[2B(t-t_d)]; a practical RC filter has h(t)=(1/RC)e^{-t/RC}u(t). ## Answer #### Impulse response The **impulse response** $h(t)$ of a system is the output obtained when a unit impulse $\delta(t)$ is applied at the input, with the system initially at rest. For an LTI system it fully describes the system. #### Why it is useful in communication system design - **Output for any input:** $y(t) = x(t) * h(t)$, so one measurement or calculation of $h(t)$ predicts the response to any signal (speech, pulses, modulated waves). - **Frequency response:** its Fourier transform is the transfer function $H(f)$, which gives the bandwidth, amplitude and phase response; this tells whether a channel or filter will distort the signal and whether an equalizer is needed. - **Pulse spreading and ISI:** $h(t)$ of a channel shows how much a transmitted pulse spreads, which sets the maximum symbol rate; pulse-shaping (raised-cosine) filters are designed from it. - **Matched filter design:** the optimum receiver filter has $h(t) = s(T - t)$. - **Causality and stability checks:** realizable only if $h(t) = 0$ for $t < 0$; stable if $\int|h(t)|dt < \infty$. - **Noise analysis:** output PSD $S_y(f) = |H(f)|^2S_x(f)$. #### Impulse response of a low-pass filter For an **ideal LPF** of bandwidth $B$ (gain 1, delay $t_d$), $h(t) = 2B\,\text{sinc}[2B(t - t_d)]$, a sinc pulse centred at $t_d$ (non-causal, so not realizable). For a practical **RC low-pass filter**, $h(t) = \frac{1}{RC}e^{-t/RC}u(t)$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the impulse response h(t)? A. The output when the input is a unit step B. The output to a unit impulse with the system initially at rest C. The Fourier transform of the input D. The input needed to give a unit output Answer: B. h(t) is the response to delta(t). (Key point: Definition) 2. How does h(t) give the output for any input? A. y = x + h B. y = x * h (convolution) C. y = x / h D. y = x h in time domain Answer: B. For LTI systems y(t)=x(t)*h(t), and Y(f)=X(f)H(f). (Key point: Output for any input) 3. For realizability, what must h(t) satisfy? A. h(t)=0 for t<0 B. h(t)=0 for t>0 C. h(t) must be symmetric D. h(t) must be constant Answer: A. A causal (realizable) system has h(t)=0 for t<0. (Key point: Matched filter / causality) 4. The impulse response of an ideal low-pass filter of bandwidth B is: A. A decaying exponential B. A rectangular pulse C. A sinc pulse 2B sinc[2B(t-t_d)], non-causal D. An impulse at t=0 Answer: C. Ideal LPF has a sinc impulse response which extends to negative time, so is not realizable; the RC filter has the exponential. (Key point: Ideal LPF) ## Related in Communication Systems - [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 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× - [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 - [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× - [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× - [Ergodic process; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aergodic-matched-filter) — 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 - [Impulse response; ideal LPF formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-response-ideal-lpf) — Signals and Systems, chapter 5, asked 2× - [Frequency response; ideal LPF impulse response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-cutoff-impulse) — Signals and Systems, chapter 5, asked 2×