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