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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          Plane-Wave Scattering
             Reflection Coefficient

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Reflection Coefficient

Define the reflection coefficient of the scattering junction as

$\displaystyle \zbox {\rho = \frac{R_2-R_1}{R_1+R_2} = \frac{\mbox{Impedance Step}}{\mbox{Impedance Sum}}.} $

Then we get the following scattering relations in terms of $ \rho$ for pressure waves:

\begin{eqnarray*} p^+_2 &=& (1+\rho)p^+_1\\ [3pt] p^-_1 &=& \rho\,p^+_1 \end{eqnarray*}

Signal flow graphs for pressure and velocity are given in Fig.C.16.

Figure C.16: Signal flow graph for the pressure and velocity components of a plane wave scattering at an impedance discontinuity $ R_1$:$ R_2$.
\includegraphics{eps/planewavescatdm2}

It is a simple exercise to verify that signal power is conserved by checking that $ p^+_1v^{+}_1 = p^+_2v^{+}_2 + ( - p^-_1v^{-}_1)$. (Left-going power is negated to account for its opposite direction-of-travel.)

So far we have only considered a plane wave incident on the left of the junction. Consider now a plane wave incident from the right. For that wave, the impedance steps from $ R_2$ to $ R_1$, so the reflection coefficient it ``sees'' is $ -\rho$. By superposition, the signal flow graph for plane waves incident from either side is given by Fig.C.17. Note that the transmission coefficient is one plus the reflection coefficient in either direction. This signal flow graph is often called the ``Kelly-Lochbaum'' scattering junction [297].

Figure C.17: Signal flow graph for plane waves incident on either the left or right of an impedance discontinuity. Also shown are delay lines corresponding to sampled traveling plane-wave components propagating on either side of the scattering junction.
\includegraphics{eps/planewavescatdmfull}

There are some simple special cases:

  • $ R_2=\infty\,\,\Rightarrow\,\,\rho = 1\quad$ (e.g., rigid wall reflection)
  • $ R_2=0\,\,\Rightarrow\,\,\rho = -1\quad$ (e.g., open-ended tube)
  • $ R_2=R_1\,\,\Rightarrow\,\,\rho = 0\quad$ (no reflection)

Previous: Scattering Solution
Next: Plane-Wave Scattering at an Angle

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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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