The signal flow graph for the general two-zero filter is given in
Fig.B.7, and the derivation of frequency response is as
Signal flow graph for the general two-zero filter
As discussed in §5.1,
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
Forming a general two-zero transfer function in factored form gives
from which we identify
, so that
is again the difference equation
of the general two-zero filter with
complex zeros. The frequency
, is now viewed as a notch
, or antiresonance frequency
. The closer R is to 1,
the narrower the notch centered at
The approximate relation between bandwidth and given in
Eq.(B.5) for the two-pole resonator now applies to the notch
width in the two-zero filter.
Figure B.8 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.
Frequency response of the two-zero filter
fixed at and for various values of .
(a) Amplitude response.
(b) Phase response.
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