Two-Zero

The signal flow graph for the general two-zero filter is given in Fig.10.8, and the derivation of frequency response is as follows:

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

The parameters $b_1$ and $b_2$ 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

$$z = \frac{-b_1 \pm \sqrt{b_1^2 - 4 b_0 b_2}}{2b_0}$$

If the zeros are real [ $(b_1/2)^2 \geq b_2$], 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 $Re^{j\theta_c}$ and $Re^{-j\theta_c}$, where $\theta_c = \omega_c T = 2\pi f_c T$.

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

$$\begin{eqnarray*} H(z) &=& b_0 (1 - Re^{j\theta_c} z^{-1}) (1 - Re^{-j\theta_c} z^{-1})\\ &=& b_0 [1 - 2R\cos(\theta_c) z^{-1}+ R^2 z^{-2}] \end{eqnarray*}$$

from which we identify $b_1/b_0 = - 2R \cos(\theta_c)$ and $b_2/b_0 = R^2$, so that

$$y(n) = b_0\{ x(n) - [2R \cos(\theta_c)]x(n - 1) + R^2 x(n - 2)\}$$

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

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

Figure 10.9 gives some two-zero frequency responses obtained by setting $b_0$ to 1 and varying $R$. The value of $\theta _c$, is again $\pi /4$. 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