Impulse response of ideal low pass filter

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:

$$h(t) = \frac{\omega_c}{\pi}\cdot\frac{\sin\omega_c t}{\omega_c t} = \frac{\omega_c}{\pi}\,\text{sinc}\!\left(\frac{\omega_c t}{\pi}\right)$$

 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.