### 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

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

from which we identify and , so that

*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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Complex Resonator

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Two-Pole