FDN Stability

Stability of the FDN is assured when some norm [454] of the state vector $\mathbf{x}(n)$ decreases over time when the input signal is zero [221, ``Lyapunov stability theory'']. That is, a sufficient condition for FDN stability is

$$\left\Vert\,\mathbf{x}(n+1)\,\right\Vert < \left\Vert\,\mathbf{x}(n)\,\right\Vert, \protect$$ (3.12)

for all $n\geq0$ , where $\left\Vert\,\mathbf{x}(n)\,\right\Vert$ denotes the 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, the inequality of Eq.(2.12) holds under the $\ensuremath{L_2}$ norm [454] whenever the feedback matrix $\mathbf{A}$ in Eq.(2.6) satisfies [476]

$$\left\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert _2 < \left\Vert\,\mathbf{x}\,\right\Vert _2 \protect$$ (3.13)

for all $\mathbf{x}$ , where $\left\Vert\,\cdot\,\right\Vert _2$ denotes the $\ensuremath{L_2}$ norm, defined by

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

In other words, stability is guaranteed when the feedback matrix decreases the $\ensuremath{L_2}$ norm of its input vector.

The matrix norm corresponding to any vector norm $\vert\vert\,\cdot\,\vert\vert$ may be defined for the matrix $\mathbf{A}$ as

$$\left\Vert\,\mathbf{A}\,\right\Vert \isdef \max_{\mathbf{x}\neq \mathbf{0}} \frac{\left\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert}{\left\Vert\,\mathbf{x}\,\right\Vert}$$

where $\left\Vert\,\mathbf{x}\,\right\Vert$ denotes the norm of the vector $\mathbf{x}$ . In other words, the matrix norm ``induced'' by a vector norm is given by the maximum of $\vert\vert\,\mathbf{A}\mathbf{x}\,\vert\vert$ over all unit-length vectors $\mathbf{x}$ in the space. When the vector norm is the $\ensuremath{L_2}$ norm, the induced matrix norm is often called the spectral norm. Thus, Eq.(2.13) can be restated as

$$\left\Vert\,\mathbf{A}\,\right\Vert _2 < 1 \protect$$ (3.14)

where $\left\Vert\,\mathbf{A}\,\right\Vert _2$ denotes the spectral norm of $\mathbf{A}$ .

It can be shown [168] that the spectral norm of a matrix $\mathbf{A}$ is given by the largest singular value of $\mathbf{A}$ (`` $\left\Vert\,\mathbf{A}\,\right\Vert _2=\sigma_1(\mathbf{A})$ ''), and that this is equal to the square-root of the largest eigenvalue of $\mathbf{A}\mathbf{A}^T$ , where $\mathbf{A}^T$ denotes the matrix transpose of the real matrix $\mathbf{A}$ .^3.11

Since every orthogonal matrix $\mathbf{Q}$ has spectral norm 1,^3.12 a wide variety of stable feedback matrices can be parametrized as

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

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

$${\bm \Gamma}= \left[ \begin{array}{cccc} g_1 & 0 & \dots & 0\\ 0 & g_2 & \dots & 0\\ \vdots & \vdots & \ddots& \vdots\\ 0 & 0 & \dots & g_N \end{array}\right], \quad \left\vert g_i\right\vert<1.$$

An alternative stability proof may be based on showing that an FDN is a special case of a passive digital waveguide network (derived in §C.15). 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.