Gerzon Nested MIMO Allpass

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Gerzon Nested MIMO Allpass

An interesting generalization of the single-input, single-output Schroeder allpass filter (defined in §1.8.1) was proposed by Gerzon [160] for use in artificial reverberation systems.

The starting point can be the first-order allpass of Fig.1.25a on page , or the allpass made from two comb-filters depicted in Fig.1.24 on page .^2.12In either case,

all signal paths are converted from scalars to vectors of dimension ,
the delay element (or delay line) is replaced by an arbitrary $N\times N$ unitary matrix frequency response^2.13

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. $\,$ (1.15))

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

which leads to the

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Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Allpass Filters
          Gerzon Nested MIMO Allpass