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 [452, 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 .