Normalizing Two-Pole Filter Gain at Resonance

The question we now pose is how to best compensate the *tunable*
two-pole resonator of §B.1.3 so that its peak gain is the same for
all tunings. Looking at Fig.B.17, and remembering the graphical
method for determining the amplitude response,^{B.6} it is intuitively
clear that we can help matters by adding two *zeros* to the
filter, one near dc and the other near
. A zero exactly at dc
is provided by the term
in the transfer function numerator.
Similarly, a zero at half the sampling rate is provided by the term
in the numerator. The series combination of both zeros
gives the numerator
. The complete
second-order transfer function then becomes

corresponding to the difference equation

Checking the gain for the case , we have

which is better behaved, but now the response falls to zero at dc and rather than being heavily boosted, as we found in Eq. (B.14).

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