We can apply the transformation to convert the Butterworth filter into a high-quality
band-pass filter with center frequency . A further transformation
can then be applied to shift the center frequency to any desired value
between 0 and . The transformation will be of the form,

where and are real numbers and not both are zero. This is a particular case of the general form given above for unit-circle-preserving rational functions. We have and , and the top and bottom halves of the unit circle are transformed symmetrically (if goes to then goes to ). The qualitative effect of the transformation is to squash points of the unit circle toward or .

In particular, given a desired center frequency ,
we wish to choose so that:

If we leave as before, and let be the transfer function for a low-pass Butterworth filter, then the combined filter with transfer function will be a band-pass filter with center frequency . Solving for and gives:

The new transfer function, , will have poles and zeros (if is the degree of the Butterworth filter ).

Knowing the transfer function is good, but even better is knowing the locations
of all the poles and zeros of the new filter, which we need to be able to
compute it using elementary filters. If is a pole of the transfer
function
, that is, if , then must
be a pole of . The same goes for zeros. To find a pole or zero of
we set , where is a pole or zero of , and solve for .
This gives:

(Here and are as given above and we have used the fact that ). A sample pole-zero plot and frequency response of are shown in Figure 8.20.