---
title: "Ideal LPF impulse response; realizable?"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:ideal-lpf-impulse-realizable"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ideal-lpf-impulse-realizable
site: BCT Engineering
---

# Ideal LPF impulse response; realizable?

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2073 Bhadra, 4+2 marks
- Video: [#5.1 Calculate the impulse response of an ideal low-pass filter.](https://www.youtube.com/watch?v=-nYBIvMkHI8) — Quick Solvers
- 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-realizable
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Derive Formula to calculate the impulse response of continuous time ideal low pass filter. Is this system practically realizable or not.

## Short answer

Ideal LPF has H(ω)=Ke^{-jωt0} for |ω|<ωc, else 0. Inverse FT gives h(t)=Kωc/π sinc[ωc(t−t0)]. It is not practically realizable because sinc is nonzero for t<0, so non-causal.

## Answer

#### Impulse response of an ideal low pass filter

An **ideal low pass filter (LPF)** passes all frequencies below the cutoff $\omega_c$ without distortion and completely blocks higher frequencies:

$$
H(\omega) = \begin{cases} K\,e^{-j\omega t_0}, & |\omega| < \omega_c \\ 0, & |\omega| > \omega_c \end{cases}
$$

(constant gain $K$ and linear phase, i.e. distortionless in the pass band).

```text
 |H(w)|
   K  +-----------+
      |           |
 -----+-----+-----+-------> w
    -wc     0     wc
```

The impulse response is the inverse Fourier transform:

$$
\begin{aligned}
h(t) &= \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} K\,e^{-j\omega t_0}\,e^{j\omega t}\,d\omega \\
&= \frac{K}{2\pi}\left[\frac{e^{j\omega(t-t_0)}}{j(t-t_0)}\right]_{-\omega_c}^{\omega_c} \\
&= \frac{K}{2\pi}\cdot\frac{2j\sin\omega_c(t-t_0)}{j(t-t_0)} \\
&= \frac{K\sin\omega_c(t-t_0)}{\pi(t-t_0)} \\
&= \frac{K\omega_c}{\pi}\,\mathrm{sinc}\left[\omega_c(t-t_0)\right]
\end{aligned}
$$

where $\mathrm{sinc}(x) = \sin x / x$. The peak value $K\omega_c/\pi$ occurs at $t = t_0$, and zeros occur at $t = t_0 \pm n\pi/\omega_c$.

```text
            h(t)
             /\    peak K*wc/pi at t0
   .  ..   /  \   ..  .
 --'--''--/----\--''--'--> t
          |  t0 |
   tails extend to t -> -inf
```

#### Is it practically realizable?

**No.** A physically realizable system must be causal, i.e. $h(t) = 0$ for $t < 0$. The sinc response extends from $t = -\infty$ to $+\infty$, so it is non-zero for $t < 0$ for any finite delay $t_0$: the filter would respond before the impulse is applied. Hence the ideal LPF is **non-causal and not realizable**. (The Paley–Wiener criterion also fails, because $|H(\omega)|$ is exactly zero over a band.) Practical filters only approximate it: Butterworth and Chebyshev filters accept a finite transition band and non-linear phase, while FIR filters use a large delay and truncate (window) the sinc tails.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Ideal LPF H(ω)=K e^(-jωt0), |ω|<ωc. The h(t)?
   A. K sin(ωc t)/(πt)
   B. K sin(ωc t0)/π
   C. K sin(ωc(t-t0))/(π(t-t0))
   D. K e^(-t0)u(t)
   Answer: C. Time shift by t0. (Key point: Ideal LPF with delay)

2. Peak is at:
   A. t=0 with K
   B. t=t0 with K
   C. t=π/ωc with ωc
   D. t=t0 with Kωc/π
   Answer: D. sinc max. (Key point: Peak)

3. Realizable?
   A. No, tails extend to t→-∞ so non-causal
   B. Yes with t0>0
   C. Yes with large K
   D. Yes if ωc finite
   Answer: A. Non-zero for t<0 for any finite t0. (Key point: Realizable?)

4. Practical filters like Butterworth:
   A. Are exact
   B. Approximate the ideal with a transition band
   C. Have infinite cutoff
   D. Have zero phase
   Answer: B. Approximations. (Key point: Practical)

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