### 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

 (B.6)

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

Since is complex, the output is generally complex even when the input is real.

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

where, as always, denotes the unit step function:

Thus, the impulse response is simply a scale factor times the geometric sequence with the pole as its term ratio''. In general, is a sampled, exponentially decaying sinusoid at radian frequency . By setting somewhere on the unit circle to get

we obtain a 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,

we can arbitrarily set both the amplitude and phase of this phase-quadrature oscillator:

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-PolePartial Fraction Expansion

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

 (B.7)

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

which implies

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

Since is real, we must have , as we found above without assuming it. If , then is a real sinusoid created by the sum of two complex sinusoids spinning in opposite directions on the unit circle.
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