Gerzon Nested MIMO Allpass
- 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))
Note that to avoid implementing twice, should be realized in vector direct-form II, viz.,
In , Gerzon suggested using of the form
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 .
Example Allpass Filters