## Feedback Delay Networks (FDN)

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

 (3.6)

with the outputs given by

 (3.7)

or, in frequency-domain vector notation,
 (3.8) (3.9)

where

 (3.10)

### FDN and State Space Descriptions

When in Eq.(2.10), the FDN (Fig.2.28) reduces to a normal state-space model1.3.7),

 (3.11)

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:

where is an matrix. Similarly, a single output can be created by taking an arbitrary linear combination of the components of . An example single-input, single-output (SISO) FDN for is shown in Fig.2.29.

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

By elementary state-space analysis [449, Appendix E], the transfer function can be written in terms of the FDN system parameters as

where denotes the identity matrix. This is easy to show by taking the 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

where denotes the matrix transpose of , and

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].

### 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.

Next Section:
Allpass Filters
Previous Section:
Comb Filters