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Feedback Delay Networks (FDN)

Figure 2.28: Order 3 MIMO Feedback Delay Network (FDN).
\includegraphics[width=\twidth]{eps/FDNMIMO}

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

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

with the outputs given by

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

or, in frequency-domain vector notation,
$\displaystyle \mathbf{X}(z)$ $\displaystyle =$ $\displaystyle {\bm \Gamma}\mathbf{Q}\mathbf{D}(z)\mathbf{X}(z) + \mathbf{U}(z)$ (3.8)
$\displaystyle \mathbf{Y}(z)$ $\displaystyle =$ $\displaystyle \mathbf{D}(z) \mathbf{X}(z)$ (3.9)

where

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



Subsections
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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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