Impulse response of DT ideal low pass filter

An ideal discrete-time low pass filter passes all frequencies below a cut-off $\omega_c$ with unit gain and blocks the rest. Its frequency response over one period ($-\pi\le\omega\le\pi$) is

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

and it repeats every $2\pi$.

        H(e^jw)
          1 +-----+
            |     |
 ----+------+-----+------+---> w
    -pi   -wc  0  wc     pi

Derivation

The impulse response is the inverse DTFT:

$$\begin{aligned} h[n] &= \frac{1}{2\pi}\int_{-\pi}^{\pi}H(e^{j\omega})e^{j\omega n}\,d\omega = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c}e^{j\omega n}\,d\omega \\ &= \frac{1}{2\pi}\left[\frac{e^{j\omega n}}{jn}\right]_{-\omega_c}^{\omega_c} = \frac{1}{2\pi}\cdot\frac{e^{j\omega_c n}-e^{-j\omega_c n}}{jn} \\ &= \frac{1}{2\pi}\cdot\frac{2j\sin(\omega_c n)}{jn} = \frac{\sin(\omega_c n)}{\pi n}, \qquad n\ne 0 \end{aligned}$$

At $n=0$:

$$h[0]=\frac{1}{2\pi}\int_{-\omega_c}^{\omega_c}d\omega=\frac{\omega_c}{\pi}$$

which is also the limit of $\frac{\sin\omega_c n}{\pi n}$ as $n\to0$. So

$$h[n]=\frac{\sin(\omega_c n)}{\pi n}=\frac{\omega_c}{\pi}\,\mathrm{sinc}\!\left(\frac{\omega_c n}{\pi}\right)$$

where $\mathrm{sinc}\,x=\frac{\sin \pi x}{\pi x}$ (the normalised sinc).

Remarks

• $h[n]$ is a sampled sinc: maximum $\omega_c/\pi$ at $n=0$, zero crossings where $\omega_c n$ is a multiple of $\pi$.
• $h[n]\ne0$ for $n<0$, so the ideal LPF is non-causal.
• $\sum|h[n]|$ diverges (decays only as $1/n$), so it is not BIBO stable. Hence it cannot be built exactly; practical filters truncate/window $h[n]$ and delay it.
• Example: for $\omega_c=\pi/2$, $h[n]=\frac{\sin(\pi n/2)}{\pi n}$: $h[0]=0.5$, $h[\pm1]=1/\pi\approx0.318$, $h[\pm2]=0$, $h[\pm3]=-1/(3\pi)\approx-0.106$.