--- title: "Impulse response of ideal low pass filter" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:ideal-lpf-impulse-response" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ideal-lpf-impulse-response site: BCT Engineering --- # Impulse response of ideal low pass filter Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2080 Chaitra, 5 marks; 2079 Chaitra, 5 marks; 2071 Bhadra, 5 marks; 2070 Magh, 5 marks (asked 4 times) - Video: [Ideal LPF Impulse Response](https://www.youtube.com/watch?v=fCNLMPL2XI8) — John Buck - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ideal-lpf-impulse-response - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question Derive the expression for impulse response of ideal low pass filter (with cutoff frequency ω_c) and discuss. ## Short answer The impulse response of an ideal low-pass filter with cutoff ω_c is h(t) = sin(ω_c t)/(π t). It is non-causal and infinite in extent, thus physically unrealizable; its peak at t=0 is ω_c/π and zero crossings occur at t = kπ/ω_c. ## Answer The ideal low-pass filter passes all frequencies below the cutoff $\omega_c$ with unit gain and blocks all others: $$ H(j\omega) = \begin{cases} 1, & |\omega| \le \omega_c \\ 0, & |\omega| > \omega_c \end{cases} $$ (zero phase taken for simplicity). #### Derivation Taking the inverse Fourier transform: $$ \begin{aligned} h(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} H(j\omega)e^{j\omega t}\,d\omega = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega t}\,d\omega \\ &= \frac{1}{2\pi}\left[\frac{e^{j\omega t}}{jt}\right]_{-\omega_c}^{\omega_c} = \frac{1}{2\pi}\cdot\frac{e^{j\omega_c t} - e^{-j\omega_c t}}{jt} \\ &= \frac{1}{2\pi}\cdot\frac{2j\sin\omega_c t}{jt} = \frac{\sin\omega_c t}{\pi t} \end{aligned} $$ In sinc form: $$ h(t) = \frac{\omega_c}{\pi}\cdot\frac{\sin\omega_c t}{\omega_c t} = \frac{\omega_c}{\pi}\,\text{sinc}\!\left(\frac{\omega_c t}{\pi}\right) $$ ```text H(jw) h(t) 1 ______ wc/pi | | /\ | | _ _ / \ _ _ ----+------+--> w -' '-' '-/----\-' '-' '-> t -wc 0 wc -pi/wc pi/wc zeros at t = k.pi/wc ``` #### Discussion - **Peak:** at $t = 0$, $h(0) = \omega_c/\pi$ (limit of $\sin\omega_c t/\pi t$). - **Zero crossings:** at $t = k\pi/\omega_c$, $k = \pm1, \pm2, \dots$. A larger $\omega_c$ gives a narrower main lobe (time–bandwidth inverse relation). - **Non-causal:** $h(t) \ne 0$ for $t < 0$; the sinc extends to $-\infty$. So the filter would have to respond before the impulse is applied, and it is **not physically realizable**. - **Infinite duration** and oscillating tails (ringing). Its step response shows overshoot (Gibbs phenomenon, about 9%). - With linear phase $H(j\omega) = e^{-j\omega t_0}$ in the passband, $h(t) = \dfrac{\sin\omega_c(t-t_0)}{\pi(t-t_0)}$: a delayed sinc that is still non-zero for $t<0$. - Practical filters (Butterworth, Chebyshev, RC) approximate it with a finite transition band. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is h(t) of the ideal LPF obtained? A. Fourier transform of H(jω) over all ω B. Differentiating H(jω) C. Inverse Fourier transform of H(jω) over |ω| <= ωc D. Laplace transform of u(t) Answer: C. h(t) = (1/2π)∫_{-ωc}^{ωc} e^{jωt} dω. (Key point: Method) 2. What is h(t)? A. sin(ωc t)/(π t) B. sin(ωc t)/t C. cos(ωc t)/(π t) D. e^{-ωc t}u(t) Answer: A. Evaluating the integral gives sin(ωc t)/(π t). (Key point: Result) 3. What is h(0)? A. 0 B. ωc/π C. 1 D. π/ωc Answer: B. The limit of sin(ωc t)/(πt) as t -> 0 is ωc/π. (Key point: Peak value) 4. Why is the ideal LPF not physically realizable? A. h(t) is zero everywhere B. It has too small a bandwidth C. h(t) is a pure delay D. h(t) is non-zero for t < 0 (non-causal) Answer: D. The sinc extends to negative time, so output precedes input. (Key point: Realizability) ## Close variants set in other exams - [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× ## Related in Signals and Systems - [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 - [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× - [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 - [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× - [Ideal LPF impulse response; realizable?](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-realizable) — 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 ## Related in other subjects - [RC low pass filter operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Arc-low-pass-filter) — Basic Electronics Engineering, chapter 1, asked 5× - [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× - [Ergodic process; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aergodic-matched-filter) — Communication Systems, chapter 9 - [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×