Householder Feedback Matrix
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 . 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: