Stable Feedback Matrices

The matrix

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

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:

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

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).