The signal flow graph for the general twozero filter is given in Fig.B.7, and the derivation of frequency response is as follows:
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 secondorder polynomial, we find that the zeros are located at
If the zeros are real [ ], then the twozero case reduces to two instances of our earlier analysis for the onezero. Assuming the zeros to be complex, we may express the zeros in polar form as and , where .
Forming a general twozero transfer function in factored form gives
from which we identify and , so that
is again the difference equation of the general twozero 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. (B.5) for the twopole resonator now applies to the notch width in the twozero filter.
Figure B.8 gives some twozero frequency responses obtained by setting to 1 and varying . The value of , is again . Note that the response is exactly analogous to the twopole resonator with notches replacing the resonant peaks. Since the plots are on a linear magnitude scale, the twozero amplitude response appears as the reciprocal of a twopole response. On a dB scale, the twozero response is an upsidedown twopole response.
