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Physical Audio Signal Processing
    Introduction to Physical Signal Models
       Physical Models
          Modal Representation
             Typical State-Space Diagonalization Procedure

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Typical State-Space Diagonalization Procedure

As discussed in [449, p. 362] and exemplified in §C.17.6, to diagonalize a system, we must find the eigenvectors of $ A$ by solving

$\displaystyle A\underline{e}_i = \lambda_i \underline{e}_i $

for $ \underline{e}_i$, $ i=1,2$, where $ \lambda_i$ is simply the $ i$th pole (eigenvalue of $ A$). The $ N$ eigenvectors $ \underline{e}_i$ are collected into a similarity transformation matrix:

$\displaystyle E= \left[\begin{array}{cccc} \underline{e}_1 & \underline{e}_2 & \cdots & \underline{e}_N \end{array}\right] $

If there are coupled repeated poles, the corresponding missing eigenvectors can be replaced by generalized eigenvectors.2.12 The $ E$ matrix is then used to diagonalize the system by means of a simple change of coordinates:

$\displaystyle \underline{x}(n) \isdef E\, \tilde{\underline{x}}(n) $

The new diagonalized system is then
$\displaystyle \tilde{\underline{x}}(n+1)$ $\displaystyle =$ $\displaystyle \tilde{A}\, \tilde{\underline{x}}(n) + {\tilde B}\, \underline{u}(n)$  
$\displaystyle \underline{y}(n)$ $\displaystyle =$ $\displaystyle {\tilde C}\, \tilde{\underline{x}}(n) + {\tilde D}\,\underline{u}(n),$ (2.13)

where
$\displaystyle \tilde{A}$ $\displaystyle =$ $\displaystyle E^{-1}A E$  
$\displaystyle {\tilde B}$ $\displaystyle =$ $\displaystyle E^{-1}B$  
$\displaystyle {\tilde C}$ $\displaystyle =$ $\displaystyle C E$  
$\displaystyle {\tilde D}$ $\displaystyle =$ $\displaystyle D. \protect$ (2.14)

The transformed system describes the same system as in Eq.$ \,$(1.8) relative to new state-variable coordinates $ \tilde{\underline{x}}(n)$. For example, it can be checked that the transfer-function matrix is unchanged.

Previous: Force-Driven-Mass Diagonalization Example
Next: Efficiency of Diagonalized State-Space Models

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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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