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Chapters

Physical Audio Signal Processing
    Passive Impedances
       Passive String Terminations
          Reflection Transfer Function

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Reflection Transfer Function

Given the bridge driving-point impedance $ R_b(s)$, how do we incorporate it in a digital waveguide model? Let $ \hat R_b(z)$ denote the digital impedance transfer function. If the bilinear transform was used, we have

$\displaystyle \hat R_b(z) \isdef R_b\left(\frac{2}{T} \frac{1-z^{-1}}{1+z^{-1}}\right)= \frac{F_b(z)}{V_b(z)} $

where $ F_b(z)$ is the $ z$ transform of the force applied to the bridge, in discrete time, and $ V_b(z)$ is the $ z$ transform of the bridge vertical velocity. Since the bridge and string move together, $ V_b(z)=V(z)$. Also, the force applied to the bridge by the string is the one which acts to the left, or $ f_l(t,0)$. Since we chose the right-acting force $ f_r$ for our force wave variable $ f$, we have $ F_b(z) = -F(z,0)$. We have thus related the impedance of the termination to quantities wholly within the string wave state:

$\displaystyle \hat R_b(z) = -\frac{F(z)}{V(z)} = -\frac{F^{+}(z) + F^{-}(z)}{V^{+}(z)+V^{-}(z)} = -R\frac{F^{+}(z) + F^{-}(z)}{F^{+}(z)-F^{-}(z)} $

In a simulation context, we know the bridge reflectance $ \hat R_b(z)$ and string wave impedance $ R$, and we know the