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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Modeling Examples
          Wave Digital Mass-Spring Oscillator
             WD Mass-Spring Oscillator at Half the Sampling Rate

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WD Mass-Spring Oscillator at Half the Sampling Rate

Under the bilinear transform, the $ s=\infty$ maps to $ z=-1$ (half the sampling rate). It is therefore no surprise that given $ \omega_0=\infty$ ($ \rho=-1$), inspection of Fig.F.35 reveals that any alternating sequence (sinusoid sampled at half the sampling rate) will circulate unchanged in the loop on the right, which is now isolated. Let $ x_2(n) = (-1)^n x_0$ denote this alternating sequence. The loop on the left receives $ - 2 x_2(n)$ and adds $ - x_1(n-1)$ to it, i.e., $ x_1(n+1)= - x_1(n) - 2 x_2(n) = -x_1(n) - 2x_2(0)(-1)^n$. If we start out with $ x_1(0)=0$ and $ x_2(0)=x_0$, we obtain $ x_1 = [0,-2x_0, 4x_0, -6x_0, \ldots]$, or

$\displaystyle x_1(n) = (-1)^n 2 n x_0, \quad n=0,1,2,3,\ldots\,. $

However, the physical spring force is well behaved, since

$\displaystyle f_k(n) = f^{{+}}_k(n) + f^{{-}}_k(n) = x_1(n+1) + x_1(n) = (-1)^{n+1}2 x_0 $

As a check, the mass force is found to be

\begin{eqnarray*} f_m(n) &=& x_2(n+1) - x_2(n)\\ &=& (-1)^{n+1}x_0 - (-1)^n x_0\\ &=& (-1)^{n+1}x_0 + (-1)^{n+1}x_0\\ &=& (-1)^{n+1}2 x_0, \end{eqnarray*}

which agrees with the spring, as it must.

Previous: DC Analysis of the WD Mass-Spring Oscillator
Next: Linearly Growing State Variables in WD Mass-Spring Oscillator

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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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