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Root-Power Waves

It is sometimes helpful to normalize the wave variables so that signal power is uniformly distributed numerically. This can be especially helpful in fixed-point implementations.

From (C.49), it is clear that power normalization is given by

\begin{displaymath}\begin{array}{rclrcl} \tilde{f}^{+}&\isdef & f^{{+}}/\sqrt{R}\qquad & \tilde{f}^{-}& \isdef & f^{{-}}/\sqrt{R}\\ \tilde{v}^{+}&\isdef & v^{+}\sqrt{R}\qquad & \tilde{v}^{-}& \isdef & v^{-}\sqrt{R} \end{array}\end{displaymath} (C.53)

where we have dropped the common time argument `$ (n)$ ' for simplicity. As a result, we obtain

\begin{displaymath}\begin{array}{rcccl} {\cal P}^{+}& = & f^{{+}}v^{+}&=& \tilde{f}^{+}\tilde{v}^{+} \\ &=&R(v^{+})^2 &=& (\tilde{v}^{+})^2 \\ &=&(f^{{+}})^2 / R&=& (\tilde{f}^{+})^2 \nonumber \end{array}\end{displaymath}    


\begin{displaymath}\begin{array}{rcccl} {\cal P}^{-}& = & -f^{{-}}v^{-}&=& -\tilde{f}^{+}\tilde{v}^{+} \\ &=&R(v^{-})^2 &=& (\tilde{v}^{-})^2 \\ &=&(f^{{-}})^2 / R&=& (\tilde{f}^{-})^2. \nonumber \end{array}\end{displaymath}    

The normalized wave variables $ \tilde{f}^\pm $ and $ \tilde{v}^\pm $ behave physically like force and velocity waves, respectively, but they are scaled such that either can be squared to obtain instantaneous signal power. Waveguide networks built using normalized waves have many desirable properties [175,173,436]. One is the obvious numerical advantage of uniformly distributing signal power across available dynamic range in fixed-point implementations. Another is that only in the normalized case can the wave impedances be made time varying without modulating signal power. In other words, use of normalized waves eliminates ``parametric amplification'' effects; signal power is decoupled from parameter changes.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University