Peaking and band-stop filter

In the same way, a peaking filter is obtained from a shelving filter by rotating the pole and the zero, and by providing a conjugate pole and zero, as shown in Figure 8.16. If the desired center frequency is $\omega$, and the radii of the pole and zero (as for the shelving filter) are $p$ and $q$, then we place the the upper pole and zero at

$${P_1} = p \cdot (\cos \omega + i \sin \omega)$$

$${Q_1} = q \cdot (\cos \omega + i \sin \omega)$$

As a special case, placing the zero on the unit circle gives a band-stop filter; in this case the gain at the center frequency is zero. This is analogous to the one-pole, one-zero high-pass filter above.

Figure 8.16: A peaking filter: (a) pole-zero diagram; (b) frequency response. Here the filter is set to attenuate by 6 decibels at the center frequency.