One choice of lossless feedback matrix for FDNs, especially nice in the case, is a specific Householder reflection proposed by Jot [201]:
It is interesting to note that when is a power of 2, no multiplies are required [406]. For other , only one multiply is required (by ).
Another interesting property of the Householder reflection given by Eq. (2.2) (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).
An example implementation of a Householder FDN for is shown in Fig. 2.7. As observed by Jot [145, 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 .
An unusual 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 , the matrix entries all have the same magnitude:
Due to the elegant balance of the Householder feedback matrix, Jot [200] proposes an FDN based on an embedding of feedback matrices:
Householder Reflections.
This subsection derives the Householder reflection matrix from geometric considerations [422]. Let denote the projection matrix which orthogonally projects vectors onto , i.e.,