## Feedback Delay Networks (FDN)

The FDN can be seen as a *vector feedback comb filter*,^{3.10}obtained 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

(3.7) |

or, in frequency-domain vector notation,

(3.8) | |||

(3.9) |

where

### FDN and State Space Descriptions

When
in Eq.(2.10), the FDN (Fig.2.28)
reduces to a normal *state-space model* (§1.3.7),

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

When there is only one input signal , the input vector in Fig.2.28 can be defined as the scalar input times a vector of gains:

Note that when
, this system is capable of realizing
*any* transfer function of the form

*z*transform of the impulse response of the system.

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 [385].

### 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 , 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

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

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

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.

**Next Section:**

Allpass Filters

**Previous Section:**

Comb Filters