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Homogeneous Solution

Equations (1-3) can be modified to compute the response of a finite $ Q$ filter simply by changing $ r_1$ to some value different from unity. From Eq.$ \,$(1), $ \vert z(n)\vert$ will be $ r_1$ raised to the $ n$th power, and $ y(n)$ will be

$\displaystyle y(n)=r_1^n\sin(n\theta_1) \protect$ (4)

If $ r_1$ is less than one, Eq.$ \,$(4) shows the homogeneous solution to Eq.$ \,$(1) is a sinusoid times a decaying exponential. This solution is the response of a two pole filter.

Equations (2-3) are unchanged except that the coefficients $ x_1$ and $ y_1$ become

$\displaystyle x_1$ $\displaystyle =$ $\displaystyle r_1 \cos(\theta_1) \protect$ (5)
$\displaystyle y_1$ $\displaystyle =$ $\displaystyle r_1 \sin(\theta_1) \protect$ (6)

If $ y(n)$ is the input to a digital-to-analog converter operating at a sampling rate of $ f_s$ samples per second, then $ x_1$ and $ y_1$ can be written
$\displaystyle x_1$ $\displaystyle =$ $\displaystyle e^{-\frac{1}{\tau f_s}}
\cos\left(2\pi \frac{f}{f_s}\right) \protect$ (7)
$\displaystyle y_1$ $\displaystyle =$ $\displaystyle e^{-\frac{1}{\tau f_s}}
\sin\left(2\pi \frac{f}{f_s}\right) \protect$ (8)

$\displaystyle \tau$ $\displaystyle \mathrel{\stackrel{\mathrm{\Delta}}{=}}$ $\displaystyle \hbox{time in seconds for filter to decay to $1/e$} \protect$ (9)
$\displaystyle f$ $\displaystyle \mathrel{\stackrel{\mathrm{\Delta}}{=}}$ $\displaystyle \hbox{resonant frequency of filter in Hz} \protect$ (10)
$\displaystyle f_s$ $\displaystyle \mathrel{\stackrel{\mathrm{\Delta}}{=}}$ $\displaystyle \hbox{sampling rate in samples per second} \protect$ (11)

Equations (7-10) are valid for $ f$ less than $ f_s/2$. For $ f$ between $ f_s/2$ and $ f_s$, the generated frequency will be reflected about $ f_s/2$.

Negative values of $ \tau$ will generate exponentially increasing sinusoids which can be musically useful.

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Download smac03maxjos.pdf

``Methods for Synthesizing Very High Q Parametrically Well Behaved Two Pole Filters'', by Max Mathews and Julius Smith, Proceedings of the Stockholm Musical Acoustics Conference (SMAC-03), pp. 405-408, August 6-9, 2003.
Copyright © 2008-03-12 by Max Mathews and Julius Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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