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Eigenvalues in the Undamped Case
When
, the eigenvalues reduce to
Assuming

, the eigenvalues can be expressed as
 |
(J.17) |
where

denotes the angular advance per sample of the
oscillator. Since

corresponds to the range

, we see that

in this range can produce
oscillation at any digital frequency.
For
, the eigenvalues are real, corresponding to
exponential growth and/or decay. (The values
were
excluded above in deriving Eq.
(J.17).)
In summary, the coefficient
in the digital waveguide oscillator
(
) and the frequency of sinusoidal oscillation
is simply
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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