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Modified Coupled Form Resonator

Robust behavior in fixed-point implementations using short word lengths, including perfect stability in the undamped case, can be obtained by replacing the complex multiply in Equations (12-13) with a damped version of the so-called ``modified coupled form'' (MCF) sine oscillator [2]:

$\displaystyle x(n+1)$ $\displaystyle =$ $\displaystyle r_1 [x(n) - \epsilon y(n)] + b_1 u(n)$ (27)
$\displaystyle y(n+1)$ $\displaystyle =$ $\displaystyle r_1 [\epsilon x(n+1) + y(n)] + b_2 u(n)$ (28)

where $ \epsilon = 2\sin(\theta_1/2)$, and nominally $ b_1=1$ and $ b_2=0$. This recursion is highly insensitive to round-off error. When excited by an impulse ( $ u(n)=\delta(n)$) with no damping ($ r_1=1$), the signals $ [x(n),y(n)]$, viewed as coordinates in the complex plane, follow a fixed elliptical orbit over time. As $ \epsilon\to0$, the ellipse approaches a perfect circle. The complex multiply algorithm of Equations (12-13), on the other hand, generates an exact circle for all frequencies in the absence of quantization errors.3The component sinusoids $ x(n)$ and $ y(n)$ remain samples of pure sinusoids at the same frequency, as in the complex multiply case. The elliptical orbit is due only to the components having a relative phase difference other than 90 degrees. The relative phase approaches 90 degrees as the oscillation frequency (pole angle) approaches zero.

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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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