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Physical Audio Signal Processing
    Wave Digital Filters
       Adaptors for Wave Digital Elements
          Two-Port Series Adaptor for Force Waves

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Two-Port Series Adaptor for Force Waves

Figure F.6a illustrates a generic two-port description of the series adaptor.

Figure F.6: a) Two-port description of the adaptor implementing a series connection between reference impedances $ R_1$ and $ R_2$. b) Corresponding series force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form.
\includegraphics[width=\twidth]{eps/lAdaptorSeries}

As discussed in §7.2, a series connection is characterized by a common velocity and forces which sum to zero at the junction:

\begin{eqnarray*} && f_1(n) + f_2(n) = 0\\ && v_1(n) = v_2(n) \isdef v_J(n) \end{eqnarray*}

The derivation can proceed exactly as for the parallel junction in §F.2.1, but with force and velocity interchanged, i.e., $ f\leftrightarrow v$, and with impedance and admittance interchanged, i.e., $ R\leftrightarrow \Gamma $. In this way, we may take the dual of Eq.$ \,$(F.14) to get

\begin{eqnarray*} v^{-}_1 &=& -\rho v^{+}_1 + (1+\rho) v^{+}_2\\ v^{-}_2 &=& (1-\rho)v^{+}_1 + \rho v^{+}_2 \end{eqnarray*}

diagrammed in Fig.F.7. Converting back to force wave variables via $ f^{{+}}_i=R_iv^{+}_i$ and $ f^{{-}}_i=-R_iv^{-}_i$, and noting that $ (1+\rho)R_1/R_2 = 1-\rho$, we obtain, finally,

\begin{eqnarray*} f^{{-}}_1 &=& \rho f^{{+}}_1 - (1-\rho) f^{{+}}_2\\ f^{{-}}_2 &=& -(1+\rho)f^{{+}}_1 - \rho f^{{+}}_2 \end{eqnarray*}

as diagrammed in Fig.F.6b. The one-multiply form is now

\begin{eqnarray*} f^{{-}}_1 &=& -f^{{+}}_2 + \rho(f^{{+}}_1 + f^{{+}}_2)\\ f^{{-}}_2 &=& -f^{{+}}_1 - \rho(f^{{+}}_1 + f^{{+}}_2). \end{eqnarray*}

Figure F.7: Series velocity scattering junction in Kelly-Lochbaum form.
\includegraphics[scale=0.9]{eps/lscat_vel_series_renum}

Previous: Reflection Free Port
Next: General Series Adaptor for Force Waves

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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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