Gerzon Nested MIMO Allpass
An interesting generalization of the single-input, single-output Schroeder allpass filter (defined in §2.8.1) was proposed by Gerzon [157] for use in artificial reverberation systems.
The starting point can be the first-order allpass of Fig.2.31a on
page , or the allpass made from two comb-filters depicted
in Fig.2.30 on
page
.3.15In either case,
- all signal paths are converted from scalars to vectors of dimension
,
- the delay element (or delay line) is replaced by an arbitrary
unitary matrix frequency response.3.16
Let
denote the
input vector with components
, and let
denote
the corresponding vector of z transforms. Denote the
output
vector by
. The resulting vector difference equation becomes,
in the frequency domain (cf. Eq.
(2.15))
![$\displaystyle \underline{Y}(z) = \overline{g} \underline{X}(z) + \mathbf{U}(z)\underline{X}(z) - g \mathbf{U}(z)\underline{Y}(z)
$](http://www.dsprelated.com/josimages_new/pasp/img643.png)
![$\displaystyle \mathbf{H}(z) = [\mathbf{I}+ g \mathbf{U}(z)]^{-1}[\overline{g}\mathbf{I}+ \mathbf{U}(z)]
$](http://www.dsprelated.com/josimages_new/pasp/img644.png)
![$ \mathbf{I}$](http://www.dsprelated.com/josimages_new/pasp/img558.png)
![$ N\times N$](http://www.dsprelated.com/josimages_new/pasp/img252.png)
![$ \mathbf{U}(z)$](http://www.dsprelated.com/josimages_new/pasp/img645.png)
Note that to avoid implementing
twice,
should
be realized in vector direct-form II, viz.,
![\begin{eqnarray*}
\underline{v}_d(n) &=& \mathbf{U}(d)\underline{v}(n) = {\cal Z...
...line{y}(n) &=& \underline{v}(n) + \overline{g}\underline{v}_d(n)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img647.png)
where denotes the unit-delay operator (
).
To avoid a delay-free loop, the paraunitary matrix must include at
least one pure delay in every row, i.e.,
where
is paraunitary and causal.
In [157], Gerzon suggested using
of the form
![$\displaystyle \mathbf{U}(z) = \mathbf{D}(z) \mathbf{Q}
$](http://www.dsprelated.com/josimages_new/pasp/img651.png)
![$ \mathbf{Q}$](http://www.dsprelated.com/josimages_new/pasp/img531.png)
![$ N\times N$](http://www.dsprelated.com/josimages_new/pasp/img252.png)
is a diagonal matrix of pure delays, with the lengths
![$ m_i$](http://www.dsprelated.com/josimages_new/pasp/img653.png)
Gerzon further suggested replacing the feedback and feedforward gains
by digital filters
having an amplitude response
bounded by 1. In principle, this allows the network to be arbitrarily
different at each frequency.
Gerzon's vector Schroeder allpass is used in the IRCAM Spatialisateur [218].
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Example Allpass Filters