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Modal Representation

When the state transition matrix $ A$ is diagonal, we have the so-called modal representation. In the single-input, single-output (SISO) case, the general diagonal system looks like

$\displaystyle \left[\!\!\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \\ [2pt] \vdots \\ [2pt] x_{N-1}(n+1)\\ [2pt] x_N(n+1)\end{array}\!\!\right]$ $\displaystyle =\!\!\!\!$ \begin{displaymath}\left[\!
\begin{array}{ccccc}
\lambda _1 & 0 & 0 & \cdots & 0 \\ [2pt]
0 & \lambda _2 & 0 & \cdots & 0 \nonumber \\ [2pt]
\vdots & \vdots & \ddots & \vdots & \vdots \nonumber \\ [2pt]
0 & 0 & 0 & \lambda _{N-1} & 0 \nonumber \\ [2pt]
0 & 0 & 0 & 0 & \lambda _N \nonumber \\ [2pt]
\end{array}\!\right]
\left[\!\!\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] \vdots \\ [2pt] x_{N-1}(n)\\ [2pt] x_N(n)\end{array}\!\!\right]
+
\left[\!\!\begin{array}{c} b_1 \\ [2pt] b_2 \\ [2pt] \vdots \\ [2pt] b_{N-1}\\ [2pt] b_N\end{array}\!\!\right] u(n)\nonumber\end{displaymath}  
$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle C {\underline{x}}(n) + du(n)$  
  $\displaystyle =$ $\displaystyle [c_1, c_2, \dots, c_N]{\underline{x}}(n) + d u(n).
\protect$ (G.21)

Since the state transition matrix is diagonal, the modes are decoupled, and we can write each mode's time-update independently:

\begin{eqnarray*}
x_1(n+1) &=& \lambda _1 x_1(n) + b_1 u(n)\\
x_2(n+1) &=& \lambda _2 x_2(n) + b_2 u(n)\\
&\vdots& \\
x_N(n+1) &=& \lambda _N x_N(n) + b_N u(n)\\ [5pt]
y(n) & = & c_1 x_1(n) + c_2 x_2(n) + \dots + c_N x_N(n) + d u(n)
\end{eqnarray*}

Thus, the diagonalized state-space system consists of $ N$ parallel one-pole systems. See §9.2.2 and §6.8.7 regarding the conversion of direct-form filter transfer functions to parallel (complex) one-pole form.



Subsections
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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2015-03-04 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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