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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}}{2j\omega - 1 + e^{-2j\omega}}
= e^{-2j\omega} \frac{e^{2j\omega} - 1 - 2j\omega }{e^{-2j\omega} - 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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