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One-Multiply Scattering Junctions

By factoring out $ k_i(t)$ in each equation of (C.60), we can write

$\displaystyle f^{{+}}_i(t)$ $\displaystyle =$ $\displaystyle f^{{+}}_{i-1}(t-T) + f_{{\Delta}}(t)$  
$\displaystyle f^{{-}}_{i-1}(t+T)$ $\displaystyle =$ $\displaystyle f^{{-}}_i(t) + f_{{\Delta}}(t)$ (C.62)


$\displaystyle f_{{\Delta}}(t) \isdef k_i(t)\left[f^{{+}}_{i-1}(t-T) - f^{{-}}_i(t) \right]$ (C.63)

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.C.21, and it is known as the one-multiply scattering junction [299].

Figure C.21: The one-multiply scattering junction.

Another one-multiply form is obtained by organizing (C.60) as

$\displaystyle f^{{+}}_i(t)$ $\displaystyle =$ $\displaystyle f^{{-}}_i(t) + \alpha_i(t)\tilde{f_d}(t)$  
$\displaystyle f^{{-}}_{i-1}(t+T)$ $\displaystyle =$ $\displaystyle f^{{+}}_i(t) - \tilde{f_d}(t)$ (C.64)

$\displaystyle \alpha_i(t)$ $\displaystyle \isdef$ $\displaystyle 1+k_i(t)$  
$\displaystyle \tilde{f_d}(t)$ $\displaystyle \isdef$ $\displaystyle f^{{+}}_{i-1}(t-T) - f^{{-}}_i(t).$ (C.65)

As in the previous case, only one multiplication and three additions are required per junction. This one-multiply form generalizes more readily to junctions of more than two waveguides, as we'll see in a later section.

A scattering junction well known in the LPC speech literature but not described here is the so-called two-multiply junction [299] (requiring also two additions). This omission is because the two-multiply junction is not valid as a general, local, physical modeling building block. Its derivation is tied to the reflectively terminated, cascade waveguide chain. In cases where it applies, however, it can be the implementation of choice; for example, in DSP chips having a fast multiply-add instruction, it may be possible to implement the inner loop of the two-multiply, two-add scattering junction using only two instructions.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University