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FDN and State Space Descriptions

When $ M_1=M_2=M_3=1$ in Eq.(2.10), the FDN (Fig.2.28) reduces to a normal state-space model1.3.7),

$\displaystyle \mathbf{x}(n+1)$ $\displaystyle =$ $\displaystyle \mathbf{A}\, \mathbf{x}(n) + \mathbf{u}_+(n)$  
$\displaystyle \mathbf{y}_+(n)$ $\displaystyle =$ $\displaystyle \mathbf{x}(n)
\protect$ (3.11)

The matrix $ \mathbf{A}={\bm \Gamma}\mathbf{Q}$ is the state transition matrix. The vector $ \mathbf{x}(n) = [x_1(n), x_2(n), x_3(n)]^T$ holds the state variables that determine the state of the system at time $ n$ . The order of a state-space system is equal to the number of state variables, i.e., the dimensionality of $ \mathbf{x}(n)$ . The input and output signals have been trivially redefined as

\mathbf{u}_+(n) &\isdef & \mathbf{u}(n+1)\\
\mathbf{y}_+(n) &\isdef & \mathbf{y}(n+1)

to follow normal convention for state-space form.

Thus, an FDN can be viewed as a generalized state-space model for a class of $ N$ th-order linear systems--``generalized'' in the sense that unit delays are replaced by arbitrary delays. This correspondence is valuable for analysis because tools for state-space analysis are well known and included in many software libraries such as with matlab.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University