Figure 10.8:
Signal flow graph for the general two-zero filter

.

The parameters and are called the numerator
coefficients, and they determine the two zeros. Using the
quadratic formula for finding the roots of a second-order polynomial,
we find that the zeros are located at

If the zeros are real [
], then the two-zero case
reduces to two instances of our earlier analysis for the
one-zero. Assuming the zeros to be complex, we may express the zeros
in polar form as
and
, where
.

is again the difference equation of the general two-zero filter with
complex zeros. The frequency , is now viewed as a notch
frequency, or antiresonance frequency. The closer R is to 1,
the narrower the notch centered at .

The approximate relation between bandwidth and given in
Eq. (10.5) for the two-poleresonator now applies to the notch
width in the two-zero filter.

Figure 10.9 gives some two-zero frequency responses obtained by
setting to 1 and varying . The value of , is again
. Note that the response is exactly analogous to the two-pole
resonator with notches replacing the resonant peaks. Since the plots
are on a linear magnitude scale, the two-zero amplitude response
appears as the reciprocal of a two-pole response. On a dB scale, the
two-zero response is an upside-down two-pole response.

Figure 10.9:
Frequency response of the two-zero filter

with
fixed at and for various values of .
(a) Amplitude response.
(b) Phase response.