Kelly-Lochbaum Scattering Junctions

Conservation of energy and mass dictate that, at the impedance discontinuity, force and velocity variables must be continuous

$$f_{i-1}(t,cT) = f_i(t,0)$$ (G.59)

$$v_{i-1}(t,cT) = v_i(t,0)$$

where velocity is defined as positive to the right on both sides of the junction. (Force/stress/pressure is scalar.) Equations (G.57), (G.58), and (G.59) imply the following scattering equations (a derivation is given in the next section for the more general case of $N$ waveguides meeting at a junction):

$$f^{{+}}_i(t) = \left[1+k_i(t) \right]f^{{+}}_{i-1}(t-T) - k_i(t) f^{{-}}_i(t)$$

$$f^{{-}}_{i-1}(t+T) = k_i(t)f^{{+}}_{i-1}(t-T) + \left[1-k_i(t)\right]f^{{-}}_i(t)$$ (G.60)

where

$$k_i(t) \isdef \frac{ R_i(t)-R_{i-1}(t) }{R_i(t)+R_{i-1}(t) }$$ (G.61)

is called the $i$th reflection coefficient. Since $R_i(t)\geq 0$, we have $k_i(t)\in[-1,1]$. It can be shown that if $\vert k_i\vert>1$, then either $R_i$ or $R_{i-1}$ is negative, and this implies an active (as opposed to passive) medium. Correspondingly, lattice and ladder recursive digital filters are stable if and only if all reflection coefficients are bounded by $1$ in magnitude [275].

Figure G.17: The Kelly-Lochbaum scattering junction.

The scattering equations are illustrated in Figs. G.16b and G.17. In linear predictive coding of speech [460], this structure is called the Kelly-Lochbaum scattering junction, and it is one of several types of scattering junction used to implement lattice and ladder digital filter structures [275].