### Choice of Lossless Feedback Matrix

As mentioned in §3.4, an ``ideal'' late reverberation
impulse response should resemble exponentially decaying noise
[314]. It is therefore useful when designing a
reverberator to start with an infinite reverberation time (the
``lossless case'') and work on making the reverberator a good ``noise
generator''. Such a starting point is ofen referred to as a
*lossless prototype*
[153,430]. Once
smooth noise is heard in the impulse response of the lossless
prototype, one can then work on obtaining the desired reverberation
time in each frequency band (as will be discussed in §3.7.4
below).

In reverberators based on feedback delay networks (FDNs), the smoothness of the ``perceptually white noise'' generated by the impulse response of the lossless prototype is strongly affected by the choice of FDN feedback matrix as well as the (ideally mutually prime) delay-line lengths in the FDN (discussed further in §3.7.3 below). Following are some of the better known feedback-matrix choices.

#### Hadamard Matrix

A second-order *Hadamard matrix* may be defined by

*e.g.*,

*mixing and scattering*property of the matrix.

As of version 0.9.30, Faust's `math.lib`^{4.12}contains a function called `hadamard(n)` for generating an
Hadamard matrix, where must be a power of . A
Hadamard feedback matrix is used in the programming example
`reverb_designer.dsp` (a configurable FDN reverberator)
distributed with Faust.

A Hadamard feedback matrix is said to be used in the IRCAM Spatialisateur [218].

#### 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:

#### Householder Reflections

For completeness, this section derives the Householder reflection
matrix from geometric considerations [451]. Let
denote
the *projection matrix* which orthogonally projects vectors onto
, *i.e.*,

*difference vector*between and , its orthogonal projection onto , since

*minus*this difference vector gives a

*reflection*of the vector about :

*reflecting*about --a so-called

*Householder reflection*.

#### Most General Lossless Feedback Matrices

As shown in §C.15.3, an FDN feedback matrix is lossless if and only if its eigenvalues have modulus 1 and its eigenvectors are linearly independent.

A *unitary matrix* is any (complex) matrix that is inverted
by its own (conjugate) transpose:

*Hermitian conjugate*(

*i.e.*, the complex-conjugate transpose) of . When is real (as opposed to complex), we may simply call it an

*orthogonal matrix*, and we write , where denotes matrix transposition.

All unitary (and orthogonal) matrices have unit-modulus eigenvalues and linearly independent eigenvectors. As a result, when used as a feedback matrix in an FDN, the resulting FDN will be lossless (until the delay-line damping filters are inserted, as discussed in §3.7.4 below).

#### Triangular Feedback Matrices

An interesting class of feedback matrices, also explored by Jot
[216], is that of *triangular
matrices*. A basic fact from linear algebra
is that triangular matrices (either lower or upper triangular) have
all of their eigenvalues along the diagonal.^{4.13} For example, the
matrix

It is important to note that not all triangular matrices are lossless. For example, consider

One way to avoid ``coupled repeated poles'' of this nature is to use non-repeating eigenvalues. Another is to convert to Jordan canonical form by means of a similarity transformation, zero any off-diagonal elements, and transform back [329].

**Next Section:**

Choice of Delay Lengths

**Previous Section:**

History of FDNs for Artificial Reverberation