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See Also

Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Digital Waveguide Theory
       Non-Cylindrical Acoustic Tubes
          Cylinder with Conical Cap
             Reflectance Magnitude

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Reflectance Magnitude

We have shown that the conical cap reflectance has no poles in the strict right-half plane. For passivity, we also need to show that its magnitude is bounded by unity for all $ s$ on the $ j\omega $ axis.

We have

$\displaystyle R_J(j\omega) = \frac{1 - e^{-2j\omega} - 2j\omega e^{-2j\omega}}{... ...mega} - 1 + 2j\omega} = e^{-2j\omega} \frac{\overline{D(j\omega)}}{D(j\omega)} $

so that $ \left\vert R_J(j\omega)\right\vert = 1$, which is exactly lossless, as expected.

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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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