One-Filter Scattering Junction

The scattering relations above can be said to be in ``Kelly-Lochbaum form.'' The general relation $\hat{\tau}_f = 1+\hat{\rho}$ can be used to simplify to a one-filter dynamic scattering junction

$$\begin{eqnarray*} F^{-}_1 &=& \hat{\rho}F^{+}_1 + (1+\hat{\rho}) F^{-}_2 \;=\; F... ...at{\rho}F^{-}_2 \;=\; F^{+}_1 + \hat{\rho}\cdot(F^{+}_1+F^{-}_2) \end{eqnarray*}$$

The one-filter form follows from the observation that $\hat{\rho}\cdot(F^{+}_1+F^{-}_2)$ appears in both computations, and therefore need only be implemented once:

$$\begin{eqnarray*} F^{+}&\isdef & \hat{\rho}\cdot(F^{+}_1+F^{-}_2)\\ [5pt] F^{-}_... ...(1+\hat{\rho}) F^{+}_1 + \hat{\rho}F^{-}_2 \;=\; F^{+}_1 + F^{+} \end{eqnarray*}$$

Signal Flow Diagram: