Feedback Delay Networks (FDN)

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Feedback Delay Networks (FDN)

**Figure 1.22:** Order 3 MIMO Feedback Delay Network (FDN).
$\begin{figure}\input fig/FDNMIMO.pstex_t \end{figure}$

The FDN can be seen as a vector feedback comb filter,^2.6obtained by replacing the delay line with a diagonal delay matrix (defined in Eq. $\,$ (1.10) below), and replacing the feedback gain by the product of a diagonal matrix ${\bm \Gamma}$ times an orthogonal matrix $\mathbf{Q}$ , as shown in Fig.1.22 for . The time-update for this FDN can be written as

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

(2.6)

with the outputs given by

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

(2.7)

or, in frequency-domain vector notation,

$\displaystyle \mathbf{X}(z)$	$\displaystyle =$	$\displaystyle {\bm \Gamma}\mathbf{Q}\mathbf{D}(z)\mathbf{X}(z) + \mathbf{U}(z)$	(2.8)
$\displaystyle \mathbf{Y}(z)$	$\displaystyle =$	$\displaystyle \mathbf{D}(z) \mathbf{X}(z)$	(2.9)

where

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

(2.10)

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Physical Audio Signal Processing
Acoustic Modeling with Digital Delay
Feedback Delay Networks (FDN)