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See Also

Embedded SystemsFPGAElectronics

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

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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

$\displaystyle \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] x_3(n)\end... ...gin{array}{c} u_1(n) \\ [2pt] u_2(n) \\ [2pt] u_3(n)\end{array}\right] \protect$ (3.6)

with the outputs given by

$\displaystyle \left[\begin{array}{c} y_1(n) \\ [2pt] y_2(n) \\ [2pt] y_3(n)\end... ...array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right],$ (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
Previous: Time Varying Comb Filters
Next: FDN and State Space Descriptions

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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