--- title: "Ideal LPF impulse and step response" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:ideal-lpf-impulse-step" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ideal-lpf-impulse-step site: BCT Engineering --- # Ideal LPF impulse and step response Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2071 Magh, 4+4 marks - 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-step - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question Derive the expression for impulse response and step response of ideal low pass filter. ## Short answer The impulse response of an ideal LPF (cutoff ω_c, delay t0) is h(t) = sin[ω_c(t-t0)] / [π(t-t0)], a sinc function. Its step response is s(t) = 1/2 + (1/π) Si[ω_c(t-t0)], showing ~9% overshoot and rise time inversely proportional to bandwidth. ## Answer #### Impulse response of the ideal LPF The ideal LPF has constant gain and linear phase in the pass band and zero gain outside: $$ H(\omega) = \begin{cases} e^{-j\omega t_0}, & |\omega| \le \omega_c \\ 0, & |\omega| > \omega_c \end{cases} $$ (unit gain taken; for gain $K$, multiply the results by $K$). Inverse Fourier transform: $$ \begin{aligned} h(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} H(\omega)e^{j\omega t}d\omega = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega(t-t_0)}d\omega \\ &= \frac{1}{2\pi}\cdot\frac{e^{j\omega_c(t-t_0)} - e^{-j\omega_c(t-t_0)}}{j(t-t_0)} \\ &= \frac{\sin\omega_c(t-t_0)}{\pi(t-t_0)} = \frac{\omega_c}{\pi}\,\mathrm{sinc}\left[\omega_c(t-t_0)\right] \end{aligned} $$ - Peak value $\omega_c/\pi$ at $t = t_0$. - Zero crossings at $t = t_0 + n\pi/\omega_c$, $n = \pm1, \pm2, \dots$ - Main lobe width $2\pi/\omega_c$: a wider bandwidth gives a narrower, taller pulse. - $h(t) \ne 0$ for $t < 0$, so the filter is non-causal (not physically realizable). ```text h(t) /\ wc/pi . .-. / \ .-. . -'--' '--/----\--' '--'--> t t0 ``` #### Step response of the ideal LPF The step response is the running integral of the impulse response: $$ s(t) = \int_{-\infty}^{t} h(\lambda)\,d\lambda = \int_{-\infty}^{t} \frac{\sin\omega_c(\lambda-t_0)}{\pi(\lambda-t_0)}\,d\lambda $$ Let $x = \omega_c(\lambda - t_0)$, so $d\lambda = dx/\omega_c$ and $\lambda - t_0 = x/\omega_c$: $$ \begin{aligned} s(t) &= \frac{1}{\pi}\int_{-\infty}^{\omega_c(t-t_0)} \frac{\sin x}{x}\,dx \\ &= \frac{1}{\pi}\int_{-\infty}^{0} \frac{\sin x}{x}\,dx + \frac{1}{\pi}\int_{0}^{\omega_c(t-t_0)} \frac{\sin x}{x}\,dx \end{aligned} $$ Using $\int_{-\infty}^{0} \frac{\sin x}{x}dx = \frac{\pi}{2}$ and the **sine integral** $\mathrm{Si}(y) = \int_0^{y}\frac{\sin x}{x}dx$: $$ s(t) = \frac{1}{2} + \frac{1}{\pi}\,\mathrm{Si}\left[\omega_c(t-t_0)\right] $$ **Features:** - $s(t_0) = 1/2$; $s(-\infty) = 0$; $s(\infty) = 1$ (since $\mathrm{Si}(\pm\infty) = \pm\pi/2$). - **Rise time:** the slope at $t_0$ is $h(t_0) = \omega_c/\pi$, so $t_r \approx \pi/\omega_c = 1/(2B)$ where $B = \omega_c/2\pi$ Hz. Rise time is inversely proportional to bandwidth. - **Overshoot and ringing:** the response overshoots by about 9% (peak about 1.09) and oscillates around 1. This is the Gibbs phenomenon; it does not shrink as $\omega_c$ increases. - The response starts before $t = 0$, again showing the filter is non-causal. ```text s(t) 1.09| _ 1.0 | - - - / \_/~~~~~~~ 0.5 | / |~~\_ / 0 +----------------------> t t0 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which H(w) describes the ideal low pass filter used in the derivation? A. Constant magnitude and linear phase e^(-jw t0) for |w| <= wc, zero outside B. Magnitude 1 for all w with zero phase C. Zero for |w| <= wc and e^(-jw t0) outside D. Magnitude 1/(1 + jw) for all w Answer: A. The ideal LPF has unit gain and linear phase in the pass band and zero gain elsewhere. (Key point: Ideal LPF definition) 2. What is the impulse response of the ideal LPF? A. wc/pi e^(-wc t) u(t) B. sin(wc(t - t0)) / (pi (t - t0)), a sinc pulse centred at t0 C. u(t - t0) - u(t - t0 - wc) D. delta(t - t0) Answer: B. The inverse Fourier transform of a rectangular band with linear phase is a sinc centred at t0 with peak wc/pi. (Key point: Impulse response) 3. Why is the ideal LPF not physically realizable? A. Its gain is greater than 1 B. Its h(t) is non-zero for t < 0, so it is non-causal C. Its h(t) is infinite at t = 0 D. Its phase is nonlinear Answer: B. The sinc tails extend to negative time, so the output starts before the input is applied. (Key point: Non-causality) 4. What is the step response of the ideal LPF? A. 1 - e^(-wc t) B. wc/pi sinc(wc t) C. 1/2 + (1/pi) Si[wc(t - t0)] D. 1/2 + (1/pi) sin[wc(t - t0)] Answer: C. s(t) is the running integral of h(t), which gives 1/2 plus the sine integral term over pi. (Key point: Step response) ## Close variants set in other exams - [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× ## 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 - [Ideal LPF realizability; step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-realizability-step) — Signals and Systems, chapter 5 - [Distortionless transmission; ideal LPF step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adistortionless-ideal-lpf-step) — Signals and Systems, chapter 5 - [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× - [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 ## Related in other subjects - [Impulse response usefulness; impulse response of LPF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-usefulness-lpf) — Communication Systems, chapter 1 - [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 - [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×