## 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

### 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:*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

*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

*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

**Next Section:**

Allpass Filters

**Previous Section:**

Comb Filters