Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Chapters

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index


Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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