Impulse response; ideal LPF formula

Impulse response of an LTI system

The impulse response $h(t)$ of an LTI system is its output when the input is the unit impulse $\delta(t)$ and the system is initially at rest:

$$\delta(t) \;\to\; h(t)$$

Because any input can be written as $x(t) = \int x(\tau)\delta(t-\tau)d\tau$, linearity and time invariance give $y(t) = \int x(\tau)h(t-\tau)d\tau = x(t)*h(t)$. So $h(t)$ completely characterises an LTI system: causality ($h(t) = 0$ for $t<0$), stability ($\int|h|\,dt < \infty$) and frequency response ($H(j\omega) = \mathcal{F}\{h(t)\}$) can all be read from it.

Impulse response of the CT ideal low-pass filter

The ideal low-pass filter passes all frequencies below the cutoff $\omega_c$ with unit gain and rejects all others:

$$H(j\omega) = \begin{cases} 1, & |\omega| \le \omega_c \\ 0, & |\omega| > \omega_c \end{cases}$$

The impulse response is the inverse Fourier transform of $H(j\omega)$:

$$\begin{aligned} h(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} H(j\omega)e^{j\omega t}\,d\omega = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega t}\,d\omega \\ &= \frac{1}{2\pi}\cdot\frac{e^{j\omega_c t} - e^{-j\omega_c t}}{jt} = \frac{1}{2\pi}\cdot\frac{2j\sin\omega_c t}{jt} \\ &= \frac{\sin\omega_c t}{\pi t} = \frac{\omega_c}{\pi}\,\text{sinc}\!\left(\frac{\omega_c t}{\pi}\right) \end{aligned}$$

 H(jw)                     h(t)
   1 ______                    wc/pi
    |      |                    /\
    |      |           _  _    /  \    _  _
----+------+--> w    -' '-' '-/----\-' '-' '-> t
  -wc  0   wc               -pi/wc pi/wc

• $h(0) = \omega_c/\pi$; zero crossings at $t = k\pi/\omega_c$ ($k = \pm1, \pm2, \dots$).
• Width of the main lobe $2\pi/\omega_c$ shrinks as bandwidth $\omega_c$ grows.
• $h(t)$ exists for $t < 0$, so the ideal LPF is non-causal and not realizable; it is also of infinite duration.