### 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:

*denominator coefficients*, and they determine the two

*poles*of . Using the quadratic formula, the poles are found to be located at

*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

*polar form*as

*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

*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

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