Net Signal Power at a Two-Port Scattering Junction

A junction is passive if the power flowing away from it does not exceed the power flowing into it

$$\begin{eqnarray*} \underbrace{\frac{[f^{{+}}_i(t)]^2}{R_i(t)} + \frac{[f^{{-}}_{i-1}(t+T)]^2}{R_{i-1}(t)}}_{\hbox{outgoing power}} \leq \underbrace{\frac{[f^{{+}}_{i-1}(t-T)]^2}{R_{i-1}(t)} + \frac{[f^{{-}}_i(t)]^2}{R_i(t)}}_{\hbox{incoming power}} \end{eqnarray*}$$

Let ${\hat f}$ denote the finite-precision version of $f$ . Then a sufficient condition for junction passivity is

$$\begin{eqnarray*} \left\vert{\hat f}^{+}_i(t)\right\vert&\leq&\left\vert f^{{+}}_i(t)\right\vert \\ \left\vert{\hat f}^{-}_{i-1}(t+T)\right\vert&\leq&\left\vert f^{{-}}_{i-1}(t+T)\right\vert \end{eqnarray*}$$