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

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

which generalizes to the $s$ plane as

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

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

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