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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{\rh... ...ine{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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Reflectance and Transmittance of a Yielding String Termination