FDN Stability

Stability of the FDN is assured when the norm of the state vector $\mathbf{x}(n)$ decreases over time when the input signal is zero [204, ``Lyapunov stability theory'']. That is, a sufficient condition for FDN stability is

$$\left\Vert\,\mathbf{x}(n+1)\,\right\Vert < \left\Vert\,\mathbf{x}(n)\,\right\Vert, \protect$$ (2.12)

for all $n\geq 0$, where $\left\Vert\,\mathbf{x}(n)\,\right\Vert$ denotes some norm of $\mathbf{x}(n)$, and

$$\mathbf{x}(n+1) = \mathbf{A}\left[\begin{array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right].$$

Using the augmented state-space analysis mentioned above, inequality Eq. (1.12) holds under the $L2$ norm whenever the feedback matrix $\mathbf{A}$ in Eq. (1.6) satisfies [449]

$$\left\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert _2 < \left\Vert\,\mathbf{x}\,\right\Vert _2$$

where $\left\Vert\,\cdot\,\right\Vert _2$ denotes the $L2$ norm, defined by

$$\left\Vert\,\mathbf{x}\,\right\Vert _2 \isdef \sqrt{x_1^2+x_2^2+\dots+x_N^2}.$$

Thus, a wide variety of stable feedback matrices can be parametrized by

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

where $\mathbf{Q}$ is any orthogonal matrix, and ${\bm \Gamma}$ is a diagonal matrix having entries less than 1 in magnitude:

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

An alternative stability proof may be based on showing that an FDN is a special case of a passive digital waveguide network (derived in §G.13). 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.