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Circulant Feedback Delay Networks

Consider the class of circulant feedback matrices having the form

\begin{displaymath}
{\bf A} = ~\left[ \begin{array}{llll}
a(0) & a(1) & \dots &...
...ots \\
a(1) & \dots & a(N-1) & a(0) \\
\end{array} \right]
\end{displaymath}

This class of matrices gives rise to a class of FDNs we call Circulant Feedback Delay Networks (CFDN). The following two facts can be proved [3]:

Fact 1: If a matrix is circulant, it is normal, i.e., ${\bf A}^\ast {\bf A}={\bf A}{\bf A}^\ast $.

Fact 2: If a matrix is circulant and lossless, it is unitary.

It is well known that every circulant matrix is diagonalized by the Discrete Fourier Transform (DFT) matrix [3]. This implies that the eigenvalues of ${\bf A}$ can be computed by means of the DFT of the first row:

\begin{displaymath}
\{\lambda ({\bf A})\} = \{{\bf A} (k)\} = DFT([a(0) \dots a(N-1) ]^T) \nonumber
\end{displaymath}

where $\{\lambda ({\bf A})\}$ denotes the set of all eigenvalues of ${\bf A}$, and $\{{\bf A}(k)\}$ denotes the set of complex DFT samples obtained from taking the DFT of $\{a(\cdot)\}$.



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``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in IEEE Transactions on Speech and Audio, vol. 5, no. 1, pp. 51-60, Jan. 1996.

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Copyright © 2005-03-10 by Davide Rocchesso and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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