###
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].

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