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Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Modeling Examples
          Mass and Dashpot in Series
             Checking the WDF against the Analog Equivalent Circuit

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Checking the WDF against the Analog Equivalent Circuit

Let's check our result by comparing the transfer function from the input force to the force on the mass in both the discrete- and continuous-time cases.

For the discrete-time case, we have

$\displaystyle H_m(z) \isdef \frac{F_3(z)}{F(z)} = \frac{F^{+}_3(z) + F^{-}_3(z)}{F(z)} = (1-z^{-1}) \frac{F^{-}_3(z)}{F(z)} $

where the last simplification comes from the mass reflectance relation $ F^{+}_3(z) = -z^{-1}F^{-}_3(z)$. (Note that we are using the standard traveling-wave notation for the adaptor, so that the $ \pm$ signs are swapped relative to element-centric notation.)

We now need $ F^{-}_3(z)/F(z)$. To simplify notation, define the two coefficients as

\begin{eqnarray*} a &=& \frac{m}{m+\mu}\\ b &=& \frac{\mu}{m+\mu} \end{eqnarray*}

From Figure F.30, we can write

\begin{eqnarray*} F^{-}_3(z) &=& -a\left[F(z)-z^{-1}F^{-}_3(z)\right] + b\left[-... ...\,\,\quad F^{-}_3(z) &=& -a\frac{F(z)}{1-(a-b)z^{-1}F^{-}_3(z)} \end{eqnarray*}

Thus, the desired transfer function is

$\displaystyle H_m(z) = -a \frac{1-z^{-1}}{1-(a-b)z^{-1}} = -\frac{m}{m+\mu} \frac{1-z^{-1}}{1-\left(\frac{m-\mu}{m+\mu}\right)z^{-1}} $

We now wish to compare this result to the bilinear transform of the corresponding analog transfer function. From Figure F.27, we can recognize the mass and dashpot as voltage divider:

$\displaystyle H^a_m(s) = \frac{ms}{ms+\mu} $

Applying the bilinear transform yields

\begin{eqnarray*} H^a_m\left(\frac{1-z^{-1}}{1+z^{-1}}\right) &=& \frac{m\left(\... ...{-1}}{1 - \left(\frac{m-\mu}{m+\mu}\right)z^{-1}}\\ &=& H_m(z) \end{eqnarray*}

Thus, we have verified that the force transfer-function from the driving force to the mass is identical in the discrete- and continuous-time models, except for the bilinear transform frequency warping in the discrete-time case.

Previous: Mass and Dashpot in Series
Next: Wave Digital 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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