Feedback Delay Networks (FDN)

Figure 2.28: Order 3 MIMO Feedback Delay Network (FDN).

The FDN can be seen as a vector feedback comb filter,^3.10obtained by replacing the delay line with a diagonal delay matrix (defined in Eq.(2.10) below), and replacing the feedback gain $g$ by the product of a diagonal matrix ${\bm \Gamma}$ times an orthogonal matrix $\mathbf{Q}$ , as shown in Fig.2.28 for $N=3$ . The time-update for this FDN can be written as

$$\left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] x_3(n)\end{array}\right] = \left[\begin{array}{ccc} g_1 & 0 & 0\\ [2pt] 0 & g_2 & 0\\ [2pt] 0 & 0 & g_3 \end{array} \right] \left[\begin{array}{ccc} q_{11} & q_{12} & q_{13}\\ [2pt] q_{21} & q_{22} & q_{23}\\ [2pt] q_{31} & q_{32} & q_{33} \end{array} \right] \left[\begin{array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right] + \left[\begin{array}{c} u_1(n) \\ [2pt] u_2(n) \\ [2pt] u_3(n)\end{array}\right] \protect$$ (3.6)

with the outputs given by

$$\left[\begin{array}{c} y_1(n) \\ [2pt] y_2(n) \\ [2pt] y_3(n)\end{array}\right] = \left[\begin{array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right],$$ (3.7)

or, in frequency-domain vector notation,

$$\mathbf{X}(z) = {\bm \Gamma}\mathbf{Q}\mathbf{D}(z)\mathbf{X}(z) + \mathbf{U}(z)$$ (3.8)

$$\mathbf{Y}(z) = \mathbf{D}(z) \mathbf{X}(z)$$ (3.9)

where

$$\mathbf{D}(z) \isdef \left[\begin{array}{ccc} z^{-M_1} & 0 & 0\\ [2pt] 0 & z^{-M_2} & 0\\ [2pt] 0 & 0 & z^{-M_3} \end{array} \right]. \protect$$ (3.10)