### Two-Zero

The signal flow graph for the general two-zero 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 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 .

Forming a general two-zero transfer function in factored form gives

from which we identify and , so that

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.(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.

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Two-Pole