### 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 .^{3.15}In 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))

*paraunitary matrix transfer function*[500], [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 [157], Gerzon suggested using of the form

*orthogonal*matrix, and

is a diagonal matrix of pure delays, with the lengths chosen to be mutually prime (as suggested by Schroeder [417] 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 [218].

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Example Allpass Filters