### 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 *resonance*^{B.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 is the pole radius, is the bandwidth in Hertz (cycles per second), and is the sampling interval in seconds.

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

where and .

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Two-Zero

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One-Pole