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Chapters

Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Elements
          A Physical Derivation of Wave Digital Elements
             Reflectance of a General Lumped Waveguide Termination

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Reflectance of a General Lumped Waveguide Termination

Calculate the reflectance of the terminated waveguide. That is, find the Laplace transform of the return wave divided by the Laplace transform of the input wave going into the waveguide. In general, the reflectance of an impedance step for force waves (voltage waves in the electrical case) is

$\displaystyle \fbox{$\displaystyle S_R(s) \isdef \frac{F^{-}(s)}{F^{+}(s)} = \frac{R(s)-R_0}{R(s)+R_0}$} \protect$ (Q.1)

This is easily derived from continuity constraints across the junction. Specifically, referring to Fig.Q.1b, let $ F_R(s) = F^{+}_R(s) + F^{-}_R(s)$ denote the physical force and its traveling-wave components within the ``pseudo-infinitesimal-generalized-waveguide'' defined by the element impedance $ R(s)$, with the `$ +$' superscript denoting a right-going wave.Q.1 Similarly, let $ V(s) = V^{+}(s)+V^{-}(s)$ denote the velocity and its component wave variables on the side of the junction at impedance $ R_0$, and let $ V_R(s) = V^{+}_R(s)+V^{-}_R(s)$ denote the corresponding quantities on the element-side of the junction at impedance $ R(s)$. Again, the `$ +$' superscri