Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Classic WDF Wave Variables

We have been using our usual traveling-wave decomposition of force and velocity waves:

\begin{eqnarray*}
f(t) &=& f^{{+}}(t)+f^{{-}}(t) \eqsp R_0v^{+}(t) - R_0v^{-}(t)\\ [5pt]
v(t) &=& v^{+}(t)+v^{-}(t) \eqsp \frac{f^{{+}}(t)}{R_0} - \frac{f^{{-}}(t)}{R_0}
\end{eqnarray*}

where $ R_0$ is the wave impedance of the medium, or

$\displaystyle \left[\begin{array}{c} f(t) \\ [2pt] v(t) \end{array}\right]
= \left[\begin{array}{cc} R_0 & -R_0 \\ [2pt] 1 & 1 \end{array}\right] \left[\begin{array}{c} v^{+}(t) \\ [2pt] v^{-}(t) \end{array}\right]
= \left[\begin{array}{cc} 1 & 1 \\ [2pt] \frac{1}{R_0} & -\frac{1}{R_0} \end{array}\right] \left[\begin{array}{c} f^{{+}}(t) \\ [2pt] f^{{-}}(t) \end{array}\right]
$

Inverting these gives

\begin{eqnarray*}
\left[\begin{array}{c} v^{+}(t) \\ [2pt] v^{-}(t) \end{array}\right] &=& \frac{1}{2}\left[\begin{array}{cc} 1/R_0 & 1 \\ [2pt] -1/R_0 & 1 \end{array}\right]\left[\begin{array}{c} f(t) \\ [2pt] v(t) \end{array}\right]\\ [5pt]
\left[\begin{array}{c} f^{{+}}(t) \\ [2pt] f^{{-}}(t) \end{array}\right] &=& \frac{1}{2}\left[\begin{array}{cc} 1 & R_0 \\ [2pt] 1 & -R_0 \end{array}\right]\left[\begin{array}{c} f(t) \\ [2pt] v(t) \end{array}\right]
\end{eqnarray*}

In the WDF literature, the second case is typically used, multiplied by 2, and replacing force and velocity by voltage and current:

\begin{eqnarray*}
a(t) &=& v(t)+R_0\,i(t)\\
b(t) &=& v(t)-R_0\,i(t)
\end{eqnarray*}

where $ v(t)$ is now voltage and $ i(t)$ denotes current. Thus, $ a(t)=2v^+(t)$ and $ b(t)=2v^-(t)$ (doubled voltage traveling-wave components)


Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download WaveDigitalFilters.pdf
Download WaveDigitalFilters_2up.pdf
Download WaveDigitalFilters_4up.pdf

``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2017-06-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]