Ideal band pass and low pass filters (DT)

Ideal filters have unit gain in the passband and zero gain in the stopband. A DT frequency response is periodic with period $2\pi$, so it is defined over $-\pi\le\omega\le\pi$. The impulse response is found by the inverse DTFT, $h[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}H(e^{j\omega})e^{j\omega n}d\omega$.

Ideal low pass filter

$$H_{LP}(e^{j\omega})=\begin{cases}1,&|\omega|\le\omega_c\\0,&\omega_c<|\omega|\le\pi\end{cases}$$ $$h_{LP}[n]=\frac{1}{2\pi}\int_{-\omega_c}^{\omega_c}e^{j\omega n}d\omega=\frac{\sin(\omega_c n)}{\pi n},\qquad h_{LP}[0]=\frac{\omega_c}{\pi}$$

Ideal band pass filter

Passband $\omega_1\le|\omega|\le\omega_2$:

           H(e^jw)
   +--+              +--+
   |  |              |  |  1
 --+--+------+-------+--+---> w
 -w2 -w1     0      w1  w2

It equals an LPF with cut-off $\omega_2$ minus an LPF with cut-off $\omega_1$:

$$h_{BP}[n]=\frac{\sin(\omega_2 n)-\sin(\omega_1 n)}{\pi n},\qquad h_{BP}[0]=\frac{\omega_2-\omega_1}{\pi}$$

Using $\sin A-\sin B=2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$, with centre $\omega_0=\frac{\omega_1+\omega_2}{2}$ and half-width $W=\frac{\omega_2-\omega_1}{2}$:

$$h_{BP}[n]=2\cos(\omega_0 n)\cdot\frac{\sin(Wn)}{\pi n}$$

So the band pass impulse response is a low pass sinc (cut-off $W$) modulated by $2\cos\omega_0 n$, which shifts the passband to $\pm\omega_0$.

Remarks

• Both responses are sinc-shaped, non-zero for $n<0$: the ideal filters are non-causal.
• They decay only as $1/n$, so they are not absolutely summable: not BIBO stable, hence not realizable exactly. Practical filters truncate (window) and delay $h[n]$.