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FDN and State Space Descriptions

When $ M_1=M_2=M_3=1$, the FDN in Fig. 1.22 reduces to a standard state-space description [426, Appendix E] for a three-input, three-output, order 3 system:

$\displaystyle \mathbf{x}(n+1)$ $\displaystyle =$ $\displaystyle \mathbf{A}\mathbf{x}(n) + \mathbf{u}_+(n)$  
$\displaystyle \mathbf{y}_+(n)$ $\displaystyle =$ $\displaystyle \mathbf{x}(n)
\protect$ (2.11)

The matrix $ \mathbf{A}={\bm \Gamma}\mathbf{Q}$ is called the state transition matrix. The vector $ \mathbf{x}(n) = [x_1(n), x_2(n), x_3(n)]^T$ holds the state variables which completely determine the state of the system at time $ n$.2.7 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

\begin{eqnarray*}
\mathbf{u}_+(n) &\isdef & \mathbf{u}(n+1)\\
\mathbf{y}_+(n) &\isdef & \mathbf{y}(n+1)
\end{eqnarray*}

in order to obtain standard 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 since tools for state-space analysis are well known (and included in software such as Matlab, among others).

An introduction to state-space models is given in [426, available online]. A more detailed treatment appears in [204].


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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