### FDNStability

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

 (3.12)

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 [451] whenever the feedback matrix in Eq.(2.6) satisfies [473]

 (3.13)

for all , where denotes the norm, defined by

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 spectral norm. Thus, Eq.(2.13) can be restated as

 (3.14)

where denotes the spectral norm of .

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

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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