### Complex Resonator

Normally when we need a resonator, we think immediately of the
two-pole resonator. However, there is also a
*complex one-pole resonator*
having the transfer function

where is the single complex pole, and is a scale factor. In the time domain, the complex one-pole resonator is implemented as

Since the impulse response is the inverse *z* transform of the
transfer function, we can write down the impulse response of the
complex one-pole resonator by recognizing Eq.(B.6) as the
closed-form sum of an infinite geometric series, yielding

*unit step function*:

*complex sinusoidal oscillator*at radian frequency rad/sec. If we like, we can extract the real and imaginary parts separately to create both a sine-wave and a cosine-wave output:

These may be called *phase-quadrature sinusoids*, since their
phases differ by 90 degrees. The phase quadrature relationship for
two sinusoids means that they can be regarded as the real and
imaginary parts of a complex sinusoid.

By allowing to be complex,

The frequency response of the complex one-pole resonator differs from
that of the two-pole *real* resonator in that the resonance
occurs only for one positive or negative frequency , but not
both. As a result, the resonance frequency is also the
frequency where the *peak-gain* occurs; this is only true in
general for the complex one-pole resonator. In particular, the peak
gain of a real two-pole filter does not occur exactly at resonance, except
when
, , or . See
§B.6 for more on peak-gain versus resonance-gain (and how to
normalize them in practice).

#### Two-Pole Partial Fraction Expansion

Note that every real two-pole resonator can be broken up into a sum of two complex one-pole resonators:

where and are constants (generally complex). In this ``parallel one-pole'' form, it can be seen that the peak gain is no longer equal to the resonance gain, since each one-pole frequency response is ``tilted'' near resonance by being summed with the ``skirt'' of the other one-pole resonator, as illustrated in Fig.B.9. This interaction between the positive- and negative-frequency poles is minimized by making the resonance sharper ( ), and by separating the pole frequencies . The greatest separation occurs when the resonance frequency is at one-fourth the sampling rate ( ). However, low-frequency resonances, which are by far the most common in audio work, suffer from significant overlapping of the positive- and negative-frequency poles.

To show Eq.(B.7) is always true, let's solve in general for and given and . Recombining the right-hand side over a common denominator and equating numerators gives

The solution is easily found to be

where we have assumed im, as necessary to have a resonator in the first place.

Breaking up the two-pole real resonator into a parallel sum of two
complex one-pole resonators is a simple example of a *partial
fraction expansion* (PFE) (discussed more fully in §6.8).

Note that the inverse *z* transform of a sum of one-pole transfer
functions can be easily written down by inspection. In particular,
the impulse response of the PFE of the two-pole resonator (see
Eq.(B.7)) is clearly

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The BiQuad Section

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Two-Zero