The FDN can be seen as a vector feedback comb filter,3.10obtained by replacing the delay line with a diagonal delay matrix (defined in Eq.(2.10) below), and replacing the feedback gain by the product of a diagonal matrix times an orthogonal matrix , as shown in Fig.2.28 for . The time-update for this FDN can be written as
with the outputs given by
or, in frequency-domain vector notation,
The matrix is the state transition matrix. The vector holds the state variables that determine the state of the system at time . The order of a state-space system is equal to the number of state variables, i.e., the dimensionality of . The input and output signals have been trivially redefined as
to follow normal convention for state-space form.
Thus, an FDN can be viewed as a generalized state-space model for a class of th-order linear systems--``generalized'' in the sense that unit delays are replaced by arbitrary delays. This correspondence is valuable for analysis because tools for state-space analysis are well known and included in many software libraries such as with matlab.
Single-Input, Single-Output (SISO) FDN
Note that when , this system is capable of realizing any transfer function of the form
The more general case shown in Fig.2.29 can be handled in one of two ways: (1) the matrices can be augmented to order such that the three delay lines are replaced by unit-sample delays, or (2) ordinary state-space analysis may be generalized to non-unit delays, yielding
In FDN reverberation applications, , where is an orthogonal matrix, for reasons addressed below, and is a diagonal matrix of lowpass filters, each having gain bounded by 1. In certain applications, the subset of orthogonal matrices known as circulant matrices have advantages .
Stability of the FDN is assured when some norm  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 , where denotes the norm of , and
for all , where denotes the norm, defined by
The matrix norm corresponding to any vector norm may be defined for the matrix as
where denotes the spectral norm of .
It can be shown  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
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.