Band-pass filter

Taking the filter types shown above, which all have real-valued poles and zeros, we now transform them to operate on bands located off the real axis. The low-pass, high-pass, and shelving filters will then become band-pass, band-stop, and peaking filters. First we will develop the band-pass filter. Suppose we want a center frequency at $\omega$ radians, and a bandwidth of $\beta$. We take the low-pass filter with cutoff frequency $\beta$; its pole is located, for small calues of $\beta$, roughly at $p - 1 - \beta$. Now rotate this value by $\omega$ radians in the complex plane, by multiplying by the complex number $\cos \omega + i \sin \omega$. The new pole is at:

$${p_1} = (1 - \beta) (\cos \omega + i \sin \omega)$$

To get a real-valued output, this must be paired with another pole:

$${p_2} = \overline{p_1} = (1 - \beta) (\cos \omega - i \sin \omega)$$

The resulting pole-zero plot is as shown in Figure 8.15.

Figure 8.15: Two-pole band-pass filter: (a) pole-zero diagram; (b) frequency response.

The peak is approximately (not exactly) at the desired center frequency $\omega$ and the 3-dB points approximately $\beta$ radians above and below it. It is often desirable to normalize the filter to have a peak gain near unity; this is done by multiplying the input or output by the product of the distances of the two poles to the peak on the circle, or (very approximately):

$$\beta * (\beta + 2 \omega)$$

For some applications it is desirable to add a zero at the points $1$ and $-1$, so that the gain drops to zero at angular frequencies $0$ and $\pi$.