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:
![$\displaystyle \mathbf{u}(n) = \mathbf{B}u(n)
$](http://www.dsprelated.com/josimages_new/pasp/img551.png)
![$ \mathbf{B}$](http://www.dsprelated.com/josimages_new/pasp/img552.png)
![$ N\times 1$](http://www.dsprelated.com/josimages_new/pasp/img553.png)
![$ N$](http://www.dsprelated.com/josimages_new/pasp/img20.png)
![$ \mathbf{y}(n)$](http://www.dsprelated.com/josimages_new/pasp/img554.png)
![$ N=3$](http://www.dsprelated.com/josimages_new/pasp/img532.png)
Note that when
, this system is capable of realizing
any transfer function of the form
![$\displaystyle H(z) = \frac{\beta_1z^{-1}+\beta_2z^{-2}+\beta_3z^{-3}}{1+a_1z^{-1}+a_2z^{-2}+a_3z^{-3}}.
$](http://www.dsprelated.com/josimages_new/pasp/img556.png)
![$\displaystyle H(z) = \mathbf{C}^T(z\mathbf{I}- \mathbf{A})^{-1}\mathbf{B}
$](http://www.dsprelated.com/josimages_new/pasp/img557.png)
![$ \mathbf{I}$](http://www.dsprelated.com/josimages_new/pasp/img558.png)
![$ 3\times 3$](http://www.dsprelated.com/josimages_new/pasp/img559.png)
The more general case shown in Fig.2.29 can be handled in one of
two ways: (1) the matrices
can be augmented
to order
such that the three delay lines are replaced
by
unit-sample delays, or (2) ordinary state-space analysis
may be generalized to non-unit delays, yielding
![$\displaystyle H(z) = \mathbf{C}^T \mathbf{D}(z)\left[\mathbf{I}- \mathbf{A}\mathbf{D}(z)\right]^{-1}\mathbf{B}
$](http://www.dsprelated.com/josimages_new/pasp/img562.png)
![$ \mathbf{C}^T$](http://www.dsprelated.com/josimages_new/pasp/img563.png)
![$ \mathbf{C}$](http://www.dsprelated.com/josimages_new/pasp/img564.png)
![$\displaystyle \mathbf{D}(z) \isdef \left[\begin{array}{ccc} z^{-M_1} & 0 & 0\\ [2pt] 0 & z^{-M_2} & 0\\ [2pt] 0 & 0 & z^{-M_3} \end{array}\right]. \protect$](http://www.dsprelated.com/josimages_new/pasp/img539.png)
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