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 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

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

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).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University