FDN Stability
Stability of the FDN is assured when some norm [451] of
the state vector
decreases over time when the input signal is
zero [220, ``Lyapunov stability theory'']. That is, a
sufficient condition for FDN stability is
for all
![$ n\geq0$](http://www.dsprelated.com/josimages_new/pasp/img270.png)
![$ \left\Vert\,\mathbf{x}(n)\,\right\Vert$](http://www.dsprelated.com/josimages_new/pasp/img566.png)
![$ \mathbf{x}(n)$](http://www.dsprelated.com/josimages_new/pasp/img547.png)
![$\displaystyle \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].
$](http://www.dsprelated.com/josimages_new/pasp/img567.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ L2$](http://www.dsprelated.com/josimages_new/pasp/img568.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
for all
![$ \mathbf{x}$](http://www.dsprelated.com/josimages_new/pasp/img571.png)
![$ \left\Vert\,\cdot\,\right\Vert _2$](http://www.dsprelated.com/josimages_new/pasp/img572.png)
![$ L2$](http://www.dsprelated.com/josimages_new/pasp/img568.png)
![$\displaystyle \left\Vert\,\mathbf{x}\,\right\Vert _2 \isdef \sqrt{x_1^2+x_2^2+\dots+x_N^2}.
$](http://www.dsprelated.com/josimages_new/pasp/img573.png)
![$ L2$](http://www.dsprelated.com/josimages_new/pasp/img568.png)
The matrix norm corresponding to any vector norm
may be defined for the matrix
as
![$\displaystyle \left\Vert\,\mathbf{A}\,\right\Vert \isdef \max_{\mathbf{x}\neq \...
...\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert}{\left\Vert\,\mathbf{x}\,\right\Vert}
$](http://www.dsprelated.com/josimages_new/pasp/img575.png)
![$ \left\Vert\,\mathbf{x}\,\right\Vert$](http://www.dsprelated.com/josimages_new/pasp/img576.png)
![$ \mathbf{x}$](http://www.dsprelated.com/josimages_new/pasp/img571.png)
![$ \vert\vert\,\mathbf{A}\mathbf{x}\,\vert\vert $](http://www.dsprelated.com/josimages_new/pasp/img577.png)
![$ \mathbf{x}$](http://www.dsprelated.com/josimages_new/pasp/img571.png)
![$ L2$](http://www.dsprelated.com/josimages_new/pasp/img568.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
where
![$ \left\Vert\,\mathbf{A}\,\right\Vert _2$](http://www.dsprelated.com/josimages_new/pasp/img579.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
It can be shown [167] that the spectral norm of a matrix
is given by the largest singular value of
(``
''), and that this is equal to the
square-root of the largest eigenvalue of
, where
denotes the matrix transpose of the real matrix
.3.11
Since every orthogonal matrix
has spectral norm
1,3.12 a wide variety of stable
feedback matrices can be parametrized as
![$\displaystyle \mathbf{A}= {\bm \Gamma}\mathbf{Q}
$](http://www.dsprelated.com/josimages_new/pasp/img586.png)
![$ \mathbf{Q}$](http://www.dsprelated.com/josimages_new/pasp/img531.png)
![$ {\bm \Gamma}$](http://www.dsprelated.com/josimages_new/pasp/img530.png)
![$\displaystyle {\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.
$](http://www.dsprelated.com/josimages_new/pasp/img587.png)
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
has unit-modulus eigenvalues
and linearly independent eigenvectors.
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Allpass from Two Combs
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Single-Input, Single-Output (SISO) FDN