Root-Power Waves

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Physical Audio Signal Processing

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The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

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A waveguide restricts wave propagation to a particular subspace, which is usually a line. Vibrating strings, woodwinds, and transmission lines are examples of one-dimensional waveguides. — Click for http://en.wikipedia.org/wiki/Waveguide

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

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In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

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Physical Audio Signal Processing

Alternative Wave Variables

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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 `

' for simplicity. As a result, we obtain

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

and

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

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 © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Root-Power Waves

``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

``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