Modal Representation

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Modal Representation

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

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

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

$\begin{eqnarray*} x_1(n+1) &=& \lambda _1 x_1(n) + b_1 u(n)\\ x_2(n+1) &=& \lam... ...y(n) & = & c_1 x_1(n) + c_2 x_2(n) + \dots + c_N x_N(n) + d u(n) \end{eqnarray*}$

Thus, the diagonalized state-space system consists of 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.

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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 http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
State Space Filters
Modal Representation