Relation of FDNs to State-Space Models

When the delay lines are only 1 sample long, a standard state-space model results:

$$\begin{eqnarray*} \mathbf{x}(n+1) &=& \mathbf{A}\mathbf{x}(n) + \mathbf{u}(n)\\ \mathbf{y}(n) &=& \mathbf{x}(n)\nonumber \protect \end{eqnarray*}$$

• The matrix $\mathbf{A}={\bm \Gamma}\mathbf{Q}$ is the state transition matrix
• The vector $\mathbf{x}(n) = [x_1(n), x_2(n), x_3(n)]^T$ contains the state variables
• The state vector $\mathbf{x}(n)$ completely determines the state of the system at time $n$.
• The length of the state vector $\mathbf{x}(n)$ is the order of the linear system.

When the delay-lines are not unit length, the state-space description of an FDN expands in a simple way.

Matlab includes many tools for state-space analysis and simulation, especially in the Control Systems Tool Box.