A Single-Input, Single-Output (SISO) FDN

Define

$$\mathbf{u}(n) = \mathbf{B}u(n)$$

where $\mathbf{B}$ is $N\times 1$. Similarly, define

$$y(n) = c_1 x_1(n-M_1) + c_2 x_2(n-M_2) + c_3 x_3(n-M_3)$$

By state-space analysis, the transfer function is

$$H(z) = \mathbf{C}^T \mathbf{D}(z)\left[\mathbf{I}- \mathbf{A}\mathbf{D}(z)\right]^{-1}\mathbf{B}$$

where

$$\mathbf{D}(z) \mathrel{\stackrel{\Delta}{=}} \left[\begin{array}{... ... 0 & 0\\ [2pt] 0 & z^{-M_2} & 0\\ [2pt] 0 & 0 & z^{-M_3} \end{array}\right]$$

When $M_1=M_2=M_3=1$, this system can realize any transfer function of the form

$$H(z) = \frac{\beta_1z^{-1}+\beta_2z^{-2}+\beta_3z^{-3}}{1+a_1z^{-1}+a_2z^{-2}+a_3z^{-3}}.$$