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Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables
          State Conversions

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State Conversions

In §H.3.6, an arbitrary string state was converted to traveling displacement-wave components to show that the traveling-wave representation is complete, i.e., that any physical string state can be expressed as a pair of traveling-wave components. In this section, we revisit this topic using force and velocity waves.

By definition of the traveling-wave decomposition, we have

\begin{eqnarray*} f&=&f^{{+}}+f^{{-}}\\ v&=&v^{+}+v^{-}. \end{eqnarray*}

Using Eq.$ \,$(H.46), we can eliminate $ v^{+}=f^{{+}}/R$ and $ v^{-}=-f^{{+}}/R$, giving, in matrix form,

$\displaystyle \left[\begin{array}{c} f \\ [2pt] v \end{array}\right] = \left[\b... ...ay}\right] \left[\begin{array}{c} f^{{+}} \\ [2pt] f^{{-}} \end{array}\right]. $

Thus, the string state (in terms of force and velocity) is expressed as a linear transformation of the traveling force-wave components. Using the Ohm's law relations to eliminate instead $ f^{{+}}= Rv^{+}$ and $ f^{{-}}=-Rv^{-}$, we obtain

$\displaystyle \left[\begin{array}{c} f \\ [2pt] v \end{array}\right] = \left[\b... ...d{array}\right]\left[\begin{array}{c} v^{+} \\ [2pt] v^{-} \end{array}\right]. $

To convert an arbitrary initial string state $ (f,v)$ to either a traveling force-wave or velocity-wave simulation, we simply must be able to invert the appropriate two-by-two matrix above. That is, the matrix must be nonsingular. Requiring both determinants to be nonzero yields the condition

$\displaystyle 0 < R < \infty. $

That is, the