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
Note that when
, this system is capable of realizing
any transfer function of the form
By elementary state-space analysis [452, Appendix E], the transfer function can be written in terms of the FDN system parameters as
where
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
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 [388].