FDN Stability

Stability of the FDN is assured when some *norm* [454] of
the state vector
decreases over time when the input signal is
zero [221, ``Lyapunov stability theory'']. That is, a
sufficient condition for FDN stability is

for all , where denotes the norm of , and

Using the augmented state-space analysis mentioned above, the inequality of Eq. (2.12) holds under the norm [454] whenever the feedback matrix in Eq. (2.6) satisfies [475]

for all , where denotes the

In other words, stability is guaranteed when the feedback matrix decreases the norm of its input vector.

The *matrix norm* corresponding to any vector norm
may be defined for the matrix
as

where denotes the norm of the vector . In other words, the matrix norm ``induced'' by a vector norm is given by the maximum of over all unit-length vectors in the space. When the vector norm is the norm, the induced matrix norm is often called the

where denotes the spectral norm of .

It can be shown [168] 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

where is any orthogonal matrix, and is a diagonal matrix having entries less than 1 in magnitude:

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University