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

$\displaystyle H(z) = \frac{g}{1-pz^{-1}} \protect$

(B.6)

where $p=Re^{j\theta_c}$ is the single complex pole, and

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

$\displaystyle y(n) = g x(n) + p y(n-1).$

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

$\displaystyle h(n) = u(n) g p^n,$

where, as always,

denotes the unit step function:

$\displaystyle u(n) \isdef \left\{\begin{array}{ll} 1, & n\geq 0 \\ [5pt] 0, & n<0 \\ \end{array}\right.$

Thus, the impulse response is simply a scale factor

times the geometric sequence

with the pole

as its ``term ratio''. In general, $p^n = R^n e^{j\omega_c nT}$ is a sampled, exponentially decaying sinusoid at radian frequency $\omega_c=\theta_c/T$ . By setting

somewhere on the unit circle to get

$\displaystyle p \isdef e^{j\omega_c T},$

we obtain a complex sinusoidal oscillator at radian frequency $\omega_c$ 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:

$\begin{eqnarray*} \mbox{re}\left\{h(n)\right\} &=& u(n) g \cos(\omega_c n T)\\ \mbox{im}\left\{h(n)\right\} &=& u(n) g \sin(\omega_c n T) \end{eqnarray*}$

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,

$\displaystyle g \isdef A e^{j\phi}$

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

$\begin{eqnarray*} \mbox{re}\left\{h(n)\right\} &=& u(n) A \cos(\omega_c n T + \p... ...mbox{im}\left\{h(n)\right\} &=& u(n) A \sin(\omega_c n T + \phi) \end{eqnarray*}$

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 $\omega_c$ , but not both. As a result, the resonance frequency $\omega_c$ 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 $\theta_c \isdef \omega_c T = 0$ , $\pi/2$ , or $\pi$ . See §B.6 for more on peak-gain versus resonance-gain (and how to normalize them in practice).

Subsections

Two-Pole Partial Fraction Expansion

Previous: Two-Zero
Next: Two-Pole Partial Fraction Expansion

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Elementary Audio Digital Filters
       Elementary Filter Sections
          Complex Resonator