Starting with the three 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, stop-band, and peaking filters. First we develop the band-pass
filter. Suppose we want a center frequency at radians and a
bandwidth of . We take the low-pass filter with cutoff frequency
; its pole is located, for small values of , roughly at
.
Now rotate this value by radians in the complex plane, i.e.,
multiply by the complex number
. The new pole
is at:
The peak is approximately (not exactly) at the desired center frequency
, and the frequency response drops by 3 decibels approximately
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):