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Von Neumann Analysis

The matrix

$\displaystyle A \mathrel{\stackrel{\mathrm{\Delta}}{=}}\left[\begin{array}{cc} 2\cos\omega\Delta & -1 \\ [2pt] 1 & 0 \end{array}\right]
$

can be called the state transition matrix corresponding to the state-space description determined by the choice of state vector

$\displaystyle {\boldmath {x}}(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\left[\begin{array}{c} U^{n}(\omega) \\ [2pt] U^{n-1}(\omega) \end{array}\right]
$

and the state update can be written more simply in vector form as $ {\boldmath {x}}(n+1) = A {\boldmath {x}}(n)$ . Note that the state-space description is indexed by frequency $ \omega$ , regarded as fixed.


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``Discrete-Time Lumped Models'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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