### 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].

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FDN Stability

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FDN and State Space Descriptions