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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 [158] 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,

Let $ \underline{x}(n)$ denote the $ N\times 1$ input vector with components $ x_i(n), i=1,\dots,N$ , and let $ \underline{X}(z)=[X_1(z),\dots,X_N(z)]$ denote the corresponding vector of z transforms. Denote the $ N\times 1$ output vector by $ \underline{y}(n)$ . 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)
$

which leads to the matrix transfer function

$\displaystyle \mathbf{H}(z) = [\mathbf{I}+ g \mathbf{U}(z)]^{-1}[\overline{g}\mathbf{I}+ \mathbf{U}(z)]
$

where $ \mathbf{I}$ denotes the $ N\times N$ identity matrix, and $ \mathbf{U}(z)$ denotes any paraunitary matrix transfer function [502], [452, Appendix C].

Note that to avoid implementing $ \mathbf{U}(z)$ twice, $ \mathbf{H}(z)$ should be realized in vector direct-form II, viz.,

\begin{eqnarray*}
\underline{v}_d(n) &=& \mathbf{U}(d)\underline{v}(n) = {\cal Z}_n^{-1}\left\{\mathbf{U}(z)\underline{v}(z)\right\}\\
\underline{v}(n) &=& \underline{x}(n) - g\underline{v}_d(n)\\
\underline{y}(n) &=& \underline{v}(n) + \overline{g}\underline{v}_d(n)
\end{eqnarray*}

where $ d$ denotes the unit-delay operator ( $ d^k x(n)\isdef x(n-k)$ ).

To avoid a delay-free loop, the paraunitary matrix must include at least one pure delay in every row, i.e., $ \mathbf{U}(z) = z^{-1}
\mathbf{U}^\prime(z)$ where $ \mathbf{U}^\prime(z)$ is paraunitary and causal.

In [158], Gerzon suggested using $ \mathbf{U}(z)$ of the form

$\displaystyle \mathbf{U}(z) = \mathbf{D}(z) \mathbf{Q}
$

where $ \mathbf{Q}$ is a simple $ N\times N$ orthogonal matrix, and

$\displaystyle \mathbf{D}(z) = \left[ \begin{array}{ccccc} z^{-m_1} & 0 & 0 & \dots & 0\\ 0 & z^{-m_2} & 0 & \dots & 0\\ 0 & 0 & z^{-m_3} & \dots & 0\\ \vdots & \vdots & \vdots & \ddots& \vdots\\ 0 & 0 & 0 & \dots & z^{-m_N} \end{array} \right] \protect$ (3.17)

is a diagonal matrix of pure delays, with the lengths $ m_i$ chosen to be mutually prime (as suggested by Schroeder [420,421] for a series combination of Schroeder allpass sections). This structure is very close to the that of typical feedback delay networks (FDN), but unlike FDNs, which are ``vector feedback comb filters,'' the vectorized Schroeder allpass is a true multi-input, multi-output (MIMO) allpass filter.

Gerzon further suggested replacing the feedback and feedforward gains $ \pm g$ by digital filters $ \pm G(z)$ 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 [219].


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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