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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:

Difference equation: & $y(n) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2)$\\ [5pt]
{\it z} transform: & $Y(z) = b_0 X(z) + b_1 z^{-1}X(z) + b_2 z^{-2}X(z)$\\ [5pt]
Transfer function: & $H(z) = b_0+b_1z^{-1}+b_2z^{-2}$\\ [5pt]
Frequency response: & $H(e^{j\omega T}) = b_0+b_1e^{-j\omega T}+b_2e^{-j2\omega T}$\\ [5pt]
Amplitude response: & $G^2(\omega) = [b_0 + b_1 \cos(\omega T) + b_2 \cos(2\omega T)]^2$\\
&$\qquad\qquad + [-b_1 \sin(\omega T) - b_2 \sin(2\omega T)]^2$\\ [8pt]
Phase response: & $\Theta(\omega) = \tan^{-1}\left[\displaystyle\frac{-b_1 \sin(\omega T)
- b_2 \sin(2\omega T)}{b_0 + b_1 \cos(\omega T) + b_2 \cos(2\omega T)}\right]$

Figure B.7: Signal flow graph for the general two-zero filter
$ y(n) = b_0x(n) + b_1x(n - 1) + b_2x(n - 2)$ .

As discussed in §5.1, 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

$\displaystyle 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

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}]

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

$\displaystyle 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.(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 $ 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 B.8: Frequency response of the two-zero filter
$ y(n) = x(n) - 2R\cos (\theta _c) x(n - 1) + R^2 x(n - 2)$
with $ \theta _c$ fixed at $ \pi /4$ and for various values of $ R$ . (a) Amplitude response. (b) Phase response.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University