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Stable Feedback Matrices

The matrix

$\displaystyle \mathbf{A}= {\bm \Gamma}\mathbf{Q}

always gives a stable FDN when $ \mathbf{Q}$ is an orthogonal matrix, and $ {\bm \Gamma}$ is a diagonal gain matrix having entries less than 1 in magnitude:

$\displaystyle {\bm \Gamma}= \left[ \begin{array}{cccc}
g_1 & 0 & \dots & 0\\
0 & 0 & \dots & g_N
\end{array}\right], \quad \left\vert g_i\right\vert<1.

It is also possible to express FDNs as special cases of digital waveguide networks, in which case stability depends on the network being passive. Smith and Rocchesso 1994 This analysis reveals that the FDN is lossless if and only if the feedback matrix $ \mathbf{A}$ has unit-modulus eigenvalues and linearly independent eigenvectors (see the Rocchesso and Smith 1996 for details).

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``Computational Acoustic Modeling with Digital Delay'', by Julius O. Smith III and Nelson Lee,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2008-06-05 under the Creative Commons License (Attribution 2.5), by Julius O. Smith III and Nelson Lee
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University