FDN Time Update

$$\left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] x_3(n)\end... ...eft[\begin{array}{c} u_1(n) \\ [2pt] u_2(n) \\ [2pt] u_3(n)\end{array}\right],$$

Outputs given by

$$\left[\begin{array}{c} y_1(n) \\ [2pt] y_2(n) \\ [2pt] y_3(n)\end... ...rray}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right]$$

In frequency-domain vector notation,

$$\begin{eqnarray*} \mathbf{X}(z) &=& {\bm \Gamma}\mathbf{Q}\mathbf{D}(z) \mathbf{... ...+ \mathbf{U}(z)\\ \mathbf{Y}(z) &=& \mathbf{D}(z) \mathbf{X}(z) \end{eqnarray*}$$

where

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