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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Digital Waveguide Theory
       Non-Cylindrical Acoustic Tubes
          Generalized Scattering Coefficients


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Generalized Scattering Coefficients

Generalizing the scattering coefficients at a multi-tube intersection (§C.12) by replacing the usual real tube wave impedance $ R_i=\rho c/A_i$ by the complex generalized wave impedance

$\displaystyle R(x)=-\frac{\rho c(x)}{1 + \frac{\mbox{ln}'A(x)}{\mbox{ln}'P(x)}} $

from Eq.$ \,$(C.152), or, as a special case, the conical-section wave impedance $ R_A^\pm (s)=[\rho c/A(x)]/[s/(s \pm 1/t_x)]$ from Eq.$ \,$(C.151), we obtain the junction-pressure phasor [436]

$\displaystyle P_J = \left(G_J + \sum_{i=1}^N G_i^-\right)^{-1} \sum_{i=1}^N \left(G_i^+ + G_i^- \right)P_i^+ $

where $ G_i^+ $ is the complex, frequency-dependent, incoming, acoustic admittance of the $ i$th branch at the junction, $ G_i^-$ is the corresponding outgoing acoustic admittance, $ P_i^+$ is the incoming traveling pressure-wave phasor in branch $ i$, $ P_i^- = P_J - P_i^+$ is the outgoing wave, and $ G_J$ is the admittance of a load at the junction, such as a coupling to another simulation. For generality, the formula is given as it appears in the multivariable case.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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