One choice of lossless feedback matrix
for FDNs, especially
nice in the
case, is a specific Householder
reflection proposed by Jot [218]:
It is interesting to note that when
is a power of 2, no multiplies
are required [434]. For other
, only one multiply is
required (by
).
Another interesting property of the Householder reflection
given by Eq.(3.4) (and its permuted forms) is that an
matrix-times-vector operation may be carried out with only
additions (by first forming
times the input vector, applying
the scale factor
, and subtracting the result from the input
vector). This is the same computation as physical wave
scattering at a junction of identical waveguides (§C.8).
An example implementation of a Householder FDN for
is shown in
Fig.3.11. As observed by Jot [154, p.
216], this computation is equivalent to
parallel feedback comb filters with one new feedback path from the
output to the input through a gain of
.
A nice feature of the Householder feedback matrix
is that
for
, all entries in the matrix are nonzero. This
means every delay line feeds back to every other delay line, thereby
helping to maximize echo density as soon as possible.
Furthermore, for
, all matrix entries have the same
magnitude:
Only the
Due to the elegant balance of the
Householder feedback matrix,
Jot [217] proposes an
FDN based on an embedding of
feedback matrices:
Another method is to replace each of the four delay lines in an FDN(4) by a Gerzon vector allpass (see §2.8.5) which is