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Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          Kelly-Lochbaum Scattering Junctions

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Kelly-Lochbaum Scattering Junctions

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

$\displaystyle f_{i-1}(t,cT)$ $\displaystyle =$ $\displaystyle f_i(t,0)$ (H.61)
$\displaystyle v_{i-1}(t,cT)$ $\displaystyle =$ $\displaystyle v_i(t,0)$  

where velocity is defined as positive to the right on both sides of the junction. Force (or stress or pressure) is a scalar while velocity is a vector with both a magnitude and direction (in this case only left or right). Equations (H.59), (H.60), and (H.61) 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):
$\displaystyle f^{{+}}_i(t)$ $\displaystyle =$ $\displaystyle \left[1+k_i(t) \right]f^{{+}}_{i-1}(t-T) - k_i(t) f^{{-}}_i(t)$  
$\displaystyle f^{{-}}_{i-1}(t+T)$ $\displaystyle =$ $\displaystyle k_i(t)f^{{+}}_{i-1}(t-T) + \left[1-k_i(t)\right]f^{{-}}_i(t)$ (H.62)

where

$\displaystyle k_i(t) \isdef \frac{ R_i(t)-R_{i-1}(t) }{R_i(t)+R_{i-1}(t) }$ (H.63)

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 [301].

Figure H.21: The Kelly-Lochbaum scattering junction.
\includegraphics[scale=0.9]{eps/Fkl}

The scattering equations are illustrated in Figs. H.20b and H.21. In