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
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).