Properties of Reflection Coefficients

$$\zbox{k_i\mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{ R_i-R_{i-1}}{R_i+R_{i-1}}}$$

• $k_i\in[-1,1]$
• $\left\vert k_i\right\vert>1$ implies either $R_i$ or $R_{i-1}$ is negative
• A negative impedance is active
• A positive impedance is passive
• Digital waveguide networks are typically stable if and only if they are passive
• Digital waveguide networks (including lattice and ladder digital filters) are stable if and only if all reflection coefficients are bounded by $1$ in magnitude
• Can define $\theta_i \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sin^{-1}(k_i)$ for passive junctions