Frequency response; ideal LPF impulse response

Frequency response of a discrete-time LTI system

The frequency response $H(e^{j\Omega})$ of a discrete-time LTI system is the DTFT of its impulse response $h[n]$:

$$H(e^{j\Omega}) = \sum_{n=-\infty}^{\infty} h[n]\,e^{-j\Omega n}$$

If $x[n] = e^{j\Omega n}$, then $y[n] = H(e^{j\Omega})\,e^{j\Omega n}$, so $H(e^{j\Omega})$ gives the gain $|H|$ and phase shift $\angle H$ at each frequency. Also $Y(e^{j\Omega}) = H(e^{j\Omega})X(e^{j\Omega})$. It is always periodic in $\Omega$ with period $2\pi$.

Example: $h[n] = a^n u[n]$, $|a| < 1$ gives $H(e^{j\Omega}) = \dfrac{1}{1 - a e^{-j\Omega}}$.

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

• Peak value $h(0) = \omega_c/\pi$; zeros at $t = k\pi/\omega_c$, $k \ne 0$.
• $h(t) \ne 0$ for $t < 0$, so the ideal LPF is non-causal and cannot be built exactly.