Householder Feedback Matrix

One choice of lossless feedback matrix $\mathbf{A}_N$ for FDNs, especially nice in the $4\times4$ case, is a specific Householder reflection proposed by Jot [201]:

$$\mathbf{A}_N = \mathbf{I}_N - \frac{2}{N}\underline{u}_N\underline{u}_N^T \protect$$ (3.2)

where $\underline{u}_N^T = [1, 1, \dots, 1]$ can be interpreted as the specific vector about which the input vector is reflected in $N$-dimensional space (followed by a sign inversion). More generally, the identity matrix $\mathbf{I}_N$ can be replaced by any $N\times N$ permutation matrix [145, p. 126].

It is interesting to note that when $N$ is a power of 2, no multiplies are required [406]. For other $N$, only one multiply is required (by $2/N$).

Another interesting property of the Householder reflection $\mathbf{A}_N$ given by Eq. (2.2) (and its permuted forms) is that an $N\times N$ matrix-times-vector operation may be carried out with only $2N-1$ additions (by first forming $\underline{u}_N^T$ times the input vector, applying the scale factor $2/N$, and subtracting the result from the input vector).

An example implementation of a Householder FDN for $N=3$ is shown in Fig. 2.7. As observed by Jot [145, p. 216], this computation is equivalent to $N$ parallel feedback comb filters with one new feedback path from the output to the input through a gain of $-2/N$.

Figure 2.7: FDN using a Householder-reflection feedback matrix.

An unusual feature of the Householder feedback matrix $A_N$ is that for $N\neq 2$, 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 $N=4$, the matrix entries all have the same magnitude:

$$\mathbf{A}_4 = \frac{1}{2} \left[\begin{array}{rrrr} 1 & -1 & -1 ... ... -1 & 1 & -1 & -1\\ -1 & -1 & 1 & -1\\ -1 & -1 & -1 & 1 \end{array}\right].$$

Only the $N=4$ case is ``balanced'' in this way. For larger $N$, the diagonal becomes larger than the off-diagonal elements, and as $N$ becomes very large, the FDN approaches a bank of decoupled parallel comb filters.

Due to the elegant balance of the $N=4$ Householder feedback matrix, Jot [200] proposes an $N=16$ FDN based on an embedding of $N=4$ feedback matrices:

$$\mathbf{A}_{16} = \frac{1}{2} \left[\begin{array}{rrrr} \mathbf{A... ...\mathbf{A}_4 & -\mathbf{A}_4 & -\mathbf{A}_4 & \mathbf{A}_4 \end{array}\right]$$

Another method is to replace each of the four delay lines in an FDN(4) by a Gerzon vector allpass (see §1.8.5) which is $4\times4$ and contains four delay lines.

Householder Reflections.

This subsection derives the Householder reflection matrix from geometric considerations [422]. Let $\mathbf{P}_{\underline{u}}$ denote the projection matrix which orthogonally projects vectors onto ${\underline{u}}$, i.e.,

$$\mathbf{P}_{\underline{u}}= \frac{\underline{u}\,\underline{u}^T}... ...frac{\underline{u}\,\underline{u}^T}{\left\Vert\,\underline{u}\,\right\Vert^2}$$

and

$$\mathbf{P}_{\underline{u}}\, \underline{x}= \underline{u}\,\frac{... ...<\underline{u},\underline{x}\right>}{\left\Vert\,\underline{u}\,\right\Vert^2}$$

specifically projects $\underline{x}$ onto $\underline{u}$. Since the projection is orthogonal, we have

$$\left<\underline{x}-\mathbf{P}_{\underline{u}}\underline{x},\unde... ...}-\mathbf{P}_{\underline{u}})\underline{x},\underline{u}\right>=\underline{0}.$$

We may interpret $(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}$ as the difference vector between $\underline{x}$ and $\mathbf{P}_{\underline{u}}\underline{x}$, its orthogonal projection onto $\underline{u}$, since

$$(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}+ \mathbf{P}_{\underline{u}}\underline{x}= \underline{x}$$

and we have $(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}\perp \mathbf{P}_{\underline{u}}\underline{x}$ by definition of the orthogonal projection. Consequently, the projection onto $\underline{u}$ minus this difference vector gives a reflection of the vector $\underline{x}$ about $\underline{u}$:

$$\underline{y}= \mathbf{P}_{\underline{u}}\underline{x}- (\mathbf{... ...line{u}})\underline{x}= (2\mathbf{P}_{\underline{u}}- \mathbf{I})\underline{x}$$

Thus, $\underline{y}$ is obtained by reflecting $\underline{x}$ about $\underline{u}$--a so-called Householder reflection.