One-Multiply Scattering Junctions

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One-Multiply Scattering Junctions

By factoring out in each equation of (H.62), we can write

$\displaystyle f^{{+}}_i(t)$	$\displaystyle =$	$\displaystyle f^{{+}}_{i-1}(t-T) + f_{{\Delta}}(t)$
$\displaystyle f^{{-}}_{i-1}(t+T)$	$\displaystyle =$	$\displaystyle f^{{-}}_i(t) + f_{{\Delta}}(t)$	(H.64)

where

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

(H.65)

Thus, only one multiplication is actually necessary to compute the transmitted and reflected waves from the incoming waves in the Kelly-Lochbaum junction. This computation is shown in Fig.H.22, and it is known as the one-multiply scattering junction [301].

**Figure H.22:** The one-multiply scattering junction.
$\includegraphics[scale=0.9]{eps/Fom}$

Another one-multiply form is obtained by organizing (H.62) as

$\displaystyle f^{{+}}_i(t)$	$\displaystyle =$	$\displaystyle f^{{-}}_i(t) + \alpha_i(t)\tilde{f_d}(t)$
$\displaystyle f^{{-}}_{i-1}(t+T)$	$\displaystyle =$	$\displaystyle f^{{+}}_i(t) - \tilde{f_d}(t)$	(H.66)

where

$\displaystyle \alpha_i(t)$	$\displaystyle \isdef$	$\displaystyle 1+k_i(t)$
$\displaystyle \tilde{f_d}(t)$	$\displaystyle \isdef$

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Chapters

Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          One-Multiply Scattering Junctions