Figure B.5: Signal flow graph for the general two-pole filter
$ y(n) = b_0 x(n) - a_1 y(n - 1) - a_2 y(n - 2)$.
\begin{figure}\input fig/twopole.pstex_t

The signal flow graph for the general two-pole filter is given in Fig.B.5. We proceed as usual with the general analysis steps to obtain the following:

Difference equation: & $y(n) = b_0 x(n) - a_1 y(n-1)...
... \displaystyle\frac{b_0}{1+a_1e^{-j\omega T}+a_2e^{-j2\omega T}}$

The numerator of $ H(z)$ is a constant, so there are no zeros other than two at the origin of the $ z$ plane.

The coefficients $ a_1$ and $ a_2$ are called the denominator coefficients, and they determine the two poles of $ H(z)$. Using the quadratic formula, the poles are found to be located at

$\displaystyle z = -\frac{a_1}{2} \pm \sqrt{\left(\frac{a_1}{2}\right)^2 -a_2}.

When the coefficients $ a_1$ and $ a_2$ are real (as we typically assume), the poles must be either real (when $ (a_1/2)^2\geq a_2$) or form a complex conjugate pair (when $ (a_1/2)^2<a_2$).

When both poles are real, the two-pole can be analyzed simply as a cascade of two one-pole sections, as in the previous section. That is, one can multiply pointwise two magnitude plots such as Fig.B.4a, and add pointwise two phase plots such as Fig.B.4b.

When the poles are complex, they can be written as

p_1 &=& x_p + j y_p\\
p_2 &=& x_p - j y_p = \overline{p}_1

since they must form a complex-conjugate pair when $ a_1$ and $ a_2$ are real. We may express them in polar form as



R&=&\sqrt{x_p^2 + y_p^2}\,>\,0\\

$ R$ is the pole radius, or distance from the origin in the $ z$-plane. As discussed in Chapter 8, we must have $ R<1$ for stability of the two-pole filter. The angles $ \pm\theta_c$ are the poles' respective angles in the $ z$ plane. The pole angle $ \theta _c$ corresponds to the pole frequency $ \omega_c$ via the relation

$\displaystyle \theta_c = \omega_c T = 2\pi f_c T

where $ T$ denotes the sampling interval. See Chapter 8 for a discussion and examples of pole-zero plots in the complex $ z$-plane.

If $ R$ is sufficiently large (but less than 1 for stability), the filter exhibits a resonanceB.2 at radian frequency $ \omega_c = 2\pi f_c = \theta_c/T$. We may call $ \omega_c$ or $ f_c$ the center frequency of the resonator. Note, however, that the resonance frequency is not usually the precise frequency of peak-gain in a two-pole resonator (see Fig.B.9 on page [*]). The peak of the amplitude response is usually a little different because each pole sits on the other's ``skirt,'' which is slanted. (See §B.1.5 and §B.6 for an elaboration of this point.)

Using polar form for the (complex) poles, the two-pole transfer function can be expressed as

$\displaystyle H(z)$ $\displaystyle =$ $\displaystyle \frac{b_0}{(1-Re^{j\theta_c}z^{-1})(1-Re^{-j\theta_c}z^{-1})}$  
  $\displaystyle =$ $\displaystyle \frac{b_0}{1 - 2 R \cos(\theta_c)z^{-1}+ R^2 z^{-2}}
\protect$ (B.1)

Comparing this to the transfer function derived from the difference equation, we may identify

a_1 &=& -2R \cos(\theta_c)\\
a_2 &=& R^2.

The difference equation can thus be rewritten as

$\displaystyle y(n) = b_0 x(n) + [2 R \cos(\theta_c)] y(n - 1) - R^2 y(n - 2). \protect$ (B.2)

Note that coefficient $ a_2$ depends only on the pole radius R (which determines damping) and is independent of the resonance frequency, while $ a_1$ is a function of both. As a result, we may retune the resonance frequency of the two-pole filter section by modifying $ a_1$ only.

The gain at the resonant frequency $ \omega=\omega_c$, is found by substituting $ z = e^{j\theta_c}=e^{j\omega_c T}$ into Eq.$ \,$(B.1) to get

$\displaystyle G(\omega_c) \isdef \left\vert H(e^{j\theta_c})\right\vert$ $\displaystyle =$ $\displaystyle \left\vert\frac{b_0}{(1-R)(1-Re^{-j2\theta_c})}\right\vert$  
  $\displaystyle =$ $\displaystyle \frac{\vert b_0\vert}{(1-R)\sqrt{1-2R\cos(2\theta_c)+R^2}}
\protect$ (B.3)

See §B.6 for details on how the resonance gain (and peak gain) can be normalized as the tuning of $ \omega_c$ is varied in real time.

Since the radius of both poles is $ R$, we must have $ R<1$ for filter stability8.4). The closer $ R$ is to 1, the higher the gain at the resonant frequency $ \omega_c = 2\pi f_c$. If $ R=0$, the filter degenerates to the form $ H(z) = b_0$, which is a nothing but a scale factor. We can say that when the two poles move to the origin of the $ z$ plane, they are canceled by the two zeros there.

Resonator Bandwidth in Terms of Pole Radius

The magnitude $ R$ of a complex pole determines the damping or bandwidth of the resonator. (Damping may be defined as the reciprocal of the bandwidth.)

As derived in §8.5, when $ R$ is close to 1, a reasonable definition of 3dB-bandwidth $ B$ is provided by

$\displaystyle B$ $\displaystyle \isdef$ $\displaystyle - \frac{\ln(R)}{\pi T}$ (B.4)
$\displaystyle R$ $\displaystyle =$ $\displaystyle e^{- \pi B T}
\protect$ (B.5)

where $ R$ is the pole radius, $ B$ is the bandwidth in Hertz (cycles per second), and $ T$ is the sampling interval in seconds.

Figure B.6 shows a family of frequency responses for the two-pole resonator obtained by setting $ b_0 = 1$ and varying $ R$. The value of $ \theta _c$ in all cases is $ \pi /4$, corresponding to $ f_c =
f_s/8$. The analytic expressions for amplitude and phase response are

G(\omega)\! &=&
\!\frac{b_0}{\sqrt{[1 + a_1 \cos(\omega T) + a...
... + a_1 \cos(\omega T) + a_2 \cos(2\omega T)}\right]\qquad(b_0>0)

where $ a_1 = - 2R \cos(\theta_c)$ and $ a_2 = R^2$.

Figure B.6: Frequency response of the two-pole filter
$ y(n) = x(n) + 2R \cos (\theta _c) y(n - 1) - R^2 y(n - 2)$
with $ \theta _c$ fixed at $ \pi /4$ and for various values of $ R$. (a) Amplitude response. (b) Phase response.
\begin{figure}\input fig/kfig2p23.pstex_t

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