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KellyLochbaum Scattering Junctions
Conservation of energy and mass dictate that, at the impedance
discontinuity, force and velocity variables must be continuous
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
(
C.57),
(
C.58), and (
C.59) imply the following
scattering
equations (a derivation is given in the next section for the more
general case of
waveguides meeting at a junction):
where

(C.61) 
is called the
th
reflection coefficient. Since
, we have
. It can be shown that if
, then either
or
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
in magnitude
[
297].
Figure C.20:
The KellyLochbaum scattering
junction.

The scattering equations are illustrated in Figs. C.19b and
C.20. In linear predictive coding of speech [482], this
structure is called the KellyLochbaum scattering junction, and
it is one of several types of scattering junction used to implement
lattice and ladder digital filter structures
(§C.9.4,[297]).
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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.