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Power-Complementary Reflection and Transmission

We can show that the reflectance and transmittance of the yielding termination are power complementary. That is, the reflected and transmitted signal-power sum to yield the incident signal-power.

The average power incident at the bridge at frequency $ \omega $ can be expressed in the frequency domain as $ F^{+}(e^{j\omega T})\overline{V^{+}(e^{j\omega T})}$ . The reflected power is then $ F^{-}\overline{V^{-}} =
-\left\vert\hat{\rho}_f\right\vert^2F^{+}\overline{V^{+}}$ . Removing the minus sign, which can be associated with reversed direction of travel, we obtain that the power reflection frequency response is $ \left\vert\hat{\rho}_f\right\vert^2$ , which generalizes by analytic continuation to $ \hat{\rho}_f(s)\hat{\rho}_f(-s)$ . The power transmittance is given by

$\displaystyle F_b\overline{V_b}
\eqsp (\hat{\tau}_fF^{+})\overline{(1-\hat{\rho}_f)V^{+}}
\eqsp [(\hat{\rho}_f+1)F^{+}](1-\overline{\hat{\rho}_f}\overline{V^{+}})
\eqsp (1-\left\vert\hat{\rho}_f\right\vert^2)F^{+}\overline{V^{+}}

which generalizes to the $ s$ plane as

$\displaystyle F_b(s)V_b(-s) = \left[1-\hat{\rho}_f(s)\hat{\rho}_f(-s)\right]F^{+}(s)V^{+}(-s)

Finally, we see that adding up the reflected and transmitted power yields the incident power:

$\displaystyle -F^{-}(s)V^{-}(-s) + F_b(s)V_b(-s) \eqsp F^{+}(s)V^{+}(-s)

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University