DSPRelated.com
Free Books
   Free Books    Physical Audio Signal Processing  

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 model (§1.3.7),

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

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

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.

Next Section:
Single-Input, Single-Output (SISO) FDN
Previous Section:
Time Varying Comb Filters