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,

• all signal paths are converted from scalars to vectors of dimension $N$ ,

• the delay element (or delay line) is replaced by an arbitrary $N\times N$ unitary matrix frequency response.^3.16

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))

$$\underline{Y}(z) = \overline{g} \underline{X}(z) + \mathbf{U}(z)\underline{X}(z) - g \mathbf{U}(z)\underline{Y}(z)$$

which leads to the matrix transfer function

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

where $\mathbf{I}$ denotes the $N\times N$ identity matrix, and $\mathbf{U}(z)$ denotes any paraunitary matrix transfer function [504], [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

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

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

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