#### Householder Feedback Matrix

One choice of lossless feedback matrix
for FDNs, especially
nice in the case, is a specific *Householder
reflection* proposed by Jot [217]:

where can be interpreted as the specific vector about which the input vector is reflected in -dimensional space (followed by a sign inversion). More generally, the identity matrix can be replaced by any permutation matrix [153, p. 126].

It is interesting to note that when is a power of 2, no multiplies are required [430]. 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 [153, 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*:

*decoupled*parallel comb filters.

Due to the elegant balance of the Householder feedback matrix, Jot [216] proposes an FDN based on an embedding of feedback matrices:

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Householder Reflections

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Hadamard Matrix