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Reflectance of an Impedance

Let $ R(s)$ denote the driving-point impedance of an arbitrary continuous-time LTI system. Then, by definition, $ F(s)=R(s)V(s)$ where $ F(s)$ and $ V(s)$ denote the Laplace transforms of the applied force and resulting velocity, respectively. The wave variable decomposition in Eq.$ \,$ (C.74) gives

$\displaystyle F(s)$ $\displaystyle =$ $\displaystyle R(s) V(s)$  
$\displaystyle \,\,\Rightarrow\,\,F^{+}(s) + F^{-}(s)$ $\displaystyle =$ $\displaystyle R(s) \left[V^{+}(s) + V^{-}(s)\right]$  
  $\displaystyle =$ $\displaystyle R(s) \left[\frac{F^{+}(s) - F^{-}(s)}{R_0}\right]$  
$\displaystyle \,\,\Rightarrow\,\,F^{-}(s) \left[\frac{R(s)}{R_0}+1\right]$ $\displaystyle =$ $\displaystyle F^{+}(s) \left[\frac{R(s)}{R_0}-1\right]$  
$\displaystyle \,\,\Rightarrow\,\,F^{-}(s)$ $\displaystyle =$ $\displaystyle F^{+}(s) \left[\frac{R(s)-R_0}{R(s)+R_0}\right]$  
  $\displaystyle \isdef$ $\displaystyle F^{+}(s)\, \hat{\rho}_f(s)
\protect$ (C.75)

We may call $ \hat{\rho}_f(s)$ the reflectance of impedance $ R(s)$ relative to $ R_0$ . For example, if a transmission line with characteristic impedance $ R_0$ were terminated in a lumped impedance $ R(s)$ , the reflection transfer function at the termination, as seen from the end of the transmission line, would be $ \hat{\rho}_f(s)$ .

We are working with reflectance for force waves. Using the elementary relations Eq.$ \,$ (C.73), i.e., $ F^{+}(s) = R_0V^{+}(s)$ and $ F^{-}(s) = -R_0V^{-}(s)$ , we immediately obtain the corresponding velocity-wave reflectance:

$\displaystyle \hat{\rho}_v(s) \isdefs \frac{V^{-}(s)}{V^{+}(s)} \eqsp \frac{-F^{-}(s)/R_0}{F^{+}(s)/R_0}
\eqsp - \frac{F^{-}(s)}{F^{+}(s)}
\eqsp - \hat{\rho}_f(s)
$


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA