As mentioned in §3.4, an ``ideal'' late reverberation impulse response should resemble exponentially decaying noise . 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.
A second-order Hadamard matrix may be defined by
As of version 0.9.30, Faust's math.lib4.12contains 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 .
Householder Feedback Matrix
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 . 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:
A unitary matrix is any (complex) matrix that is inverted by its own (conjugate) transpose:
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).
An interesting class of feedback matrices, also explored by Jot , 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 .
Choice of Delay Lengths
History of FDNs for Artificial Reverberation