#### 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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