Two-Pole
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:

The numerator of is a constant, so there are no zeros other
than two at the origin of the
plane.
The coefficients and
are called the denominator
coefficients, and they determine the two poles of
.
Using the quadratic formula, the poles are found to be located at





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

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

where

is the pole radius, or distance from the origin in the
-plane. As discussed in Chapter 8, we must have
for
stability of the two-pole filter. The angles
are the
poles' respective angles in the
plane. The pole angle
corresponds to the pole frequency
via the
relation



If is sufficiently large (but less than 1 for stability), the
filter exhibits a resonanceB.2 at
radian frequency
. We may call
or
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
Comparing this to the transfer function derived from the difference equation, we may identify
The difference equation can thus be rewritten as
Note that coefficient depends only on the pole radius R (which
determines damping) and is independent of the resonance frequency,
while
is a function of both. As a result, we may retune
the resonance frequency of the two-pole filter section by modifying
only.
The gain at the resonant frequency
, is found by
substituting
into
Eq.
(B.1) to get
See §B.6 for details on how the resonance
gain (and peak gain) can be normalized as the tuning of is
varied in real time.
Since the radius of both poles is , we must have
for filter
stability (§8.4). The
closer
is to 1, the higher the gain at the resonant frequency
. If
, the filter degenerates to the form
, which is a nothing but a scale factor. We can say that
when the two poles move to the origin of the
plane, they are
canceled by the two zeros there.
Resonator Bandwidth in Terms of Pole Radius
The magnitude 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 is close to 1, a reasonable
definition of 3dB-bandwidth
is provided by
where



Figure B.6 shows a family of frequency responses for the
two-pole resonator obtained by setting and varying
. The
value of
in all cases is
, corresponding to
. The analytic expressions for amplitude and phase response are
![\begin{eqnarray*}
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)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1385.png)
where
and
.
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Two-Zero
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One-Pole