### 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  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)) which leads to the matrix transfer function where denotes the identity matrix, and denotes any paraunitary matrix transfer function , [449, Appendix C].

Note that to avoid implementing twice, should be realized in vector direct-form II, viz., 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 , Gerzon suggested using of the form where is a simple orthogonal matrix, and (3.17)

is a diagonal matrix of pure delays, with the lengths chosen to be mutually prime (as suggested by Schroeder  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 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 .

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