Householder Feedback Matrices

• One choice of feedback matrix $\mathbf{M}_N$ for FDNs is a specific Householder reflection proposed by Jot (1992):
  $$\mathbf{M}_N = \mathbf{I}_N - \frac{2}{N} \underbar{u}_{N} \underbar{u}^{T}_{N}$$
  where $\underbar{u}^{T}_{N} = [1, 1, \ldots, 1]$ is the specific vector about which the input vector is reflected in $N$-dimensional space.

• In addition to being lossless and not requiring any multiplies when $N$ is a power of 2 (for fixed-point implementations), the Householder matrix is attractive because the feedback matrix-times-channel-vector operation can be computed with only $2N-1$ additions (by first forming $\underbar{u}^{T}_{N}$ times the input vector, applying the scale factor $2/N$, and subtracting the result from the input vector).