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Chapters

Physical Audio Signal Processing
    The Digital Waveguide Oscillator
      
             Eigenvalues in the Undamped Case

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Eigenvalues in the Undamped Case

When $ g=1$, the eigenvalues reduce to

$\displaystyle {\lambda_i}= c\pm j\sqrt{1-c^2} $

Assuming $ \left\vert c\right\vert<1$, the eigenvalues can be expressed as

$\displaystyle {\lambda_i}= c\pm j\sqrt{1-c^2} = \cos(\theta) \pm j\sin(\theta) = e^{\pm j\theta} \protect$ (J.17)

where $ \theta=\omega T$ denotes the angular advance per sample of the oscillator. Since $ c\in(-1,1)$ corresponds to the range $ \theta\in(-\pi,\pi)$, we see that $ c$ in this range can produce oscillation at any digital frequency.

For $ \left\vert c\right\vert>1$, the eigenvalues are real, corresponding to exponential growth and/or decay. (The values $ c=\pm 1$ were excluded above in deriving Eq.$ \,$(J.17).)

In summary, the coefficient $ c$ in the digital waveguide oscillator ($ g=1$) and the frequency of sinusoidal oscillation $ \omega $ is simply

$\displaystyle \fbox{$\displaystyle c= \cos(\omega T)$}. $

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written by 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