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Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          Normalized Scattering Junctions

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Normalized Scattering Junctions

Figure H.23: The normalized scattering junction.
\includegraphics[scale=0.9]{eps/scatnlf}

Using (H.55) to convert to normalized waves $ \tilde{f}^\pm $, the Kelly-Lochbaum junction (H.62) becomes

$\displaystyle \tilde{f}^{+}_i(t)$ $\displaystyle =$ $\displaystyle \sqrt{1-k_i^2(t)} \tilde{f}^{+}_{i-1}(t-T) - k_i(t) \tilde{f}^{-}_i(t)$  
$\displaystyle \tilde{f}^{-}_{i-1}(t+T)$ $\displaystyle =$ $\displaystyle k_i(t)\tilde{f}^{+}_{i-1}(t-T) + \sqrt{1-k_i^2(t)}\tilde{f}^{-}_i(t)$ (H.68)

as diagrammed in Fig.H.23. This is called the normalized scattering junction [301], although a more precise term would be the ``normalized-wave scattering junction.''

It is interesting to define $ \theta_i \isdef \sin^{-1}(k_i)$, always possible for passive junctions since $ -1\leq k_i\leq 1$, and note that the normalized scattering junction is equivalent to a 2D rotation:

$\displaystyle \tilde{f}^{+}_i(t)$ $\displaystyle =$ $\displaystyle \cos(\theta_i) \tilde{f}^{+}_{i-1}(t-T) - \sin(\theta_i) \tilde{f}^{-}_i(t)$  
$\displaystyle \tilde{f}^{-}_{i-1}(t+T)$ $\displaystyle =$ $\displaystyle \sin(\theta_i)\tilde{f}^{+}_{i-1}(t-T) + \cos(\theta_i)\tilde{f}^{-}_i(t)$ (H.69)

where, for conciseness of notation, the time-invariant case is written.

While it appears that scattering of normalized waves at a two-port junction requires four multiplies and two additions, it is possible to convert this to three multiplies and three additions using a two-multiply ``transformer'' to power-normalize an ordinary one-multiply junction [441].

The transformer is a lossless two-port defined by [136]

$\displaystyle f^{{+}}_i(t)$ $\displaystyle =$ $\displaystyle g_i f^{{+}}_{i-1}(t-T)$  
$\displaystyle f^{{-}}_{i-1}(t+T)$ $\displaystyle =$ $\displaystyle \frac{1}{g_i}f^{{-}}_i(t)$ (H.70)

The transformer can be thought of as a device which steps the wave impedance to a new value without scattering; instead, the traveling signal power is redistributed among the force and