Feedback Delay Networks (FDN) were introduced earlier in §2.7. An example is shown in Fig.2.29 on page . After a brief historical summary, this section will cover some practical considerations for the use of FDNs as reverberators.
Feedback delay networks were first suggested for artificial reverberation by Gerzon , who proposed an ``orthogonal matrix feedback reverberation unit''. He noted that individual feedback comb filters yielded poor quality, but that several such filters could sound good when cross-coupled. An ``orthogonal matrix feedback'' around a parallel bank of delay lines was suggested as a means of obtaining maximally rich cross-coupling. He was especially concerned with good stereo spreading of the reverberation at a time when most artificial reverberators sought merely to decorrelate the reverberation in each output channel.
Later, and apparently independently, Stautner and Puckette  suggested a specific four-channel FDN reverberator and gave general stability conditions for the FDN. They proposed the feedback matrix
More recently, Jot [217,216] developed a systematic FDN design methodology allowing largely independent setting of reverberation time in different frequency bands. Using Jot's methodology, FDN reverberators can be polished to a high degree of quality, and they are presently considered to be among the best choices for high-quality artificial reverberation.
Jot's early work was concerned only with single-input, single-output (SISO) reverberators. Later work  with Jullien and others at IRCAM was concerned also with spatializing the reverberation.
An example FDN reverberator using three delay lines is shown in Fig.3.10. It can be seen as an FDN (introduced in §2.7), plus an additional low-order filter applied to the non-direct signal. This filter is called a ``tonal correction'' filter by Jot, and it serves to equalize modal energy irrespective of the reverberation time in each band. In other words, if the decay time is made very short in some band, will have a large gain in that band so that the total energy in the band's impulse-response is unchanged. This is another example of orthogonalization of reverberation parameters: In this case, adjustments in reverberation time, in any frequency band, do not alter total signal energy in the impulse response in that band.
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 .
Following Schroeder's original insight, the delay line lengths in an FDN ( in Fig.3.10) are typically chosen to be mutually prime. That is, their prime factorizations contain no common factors. This rule maximizes the number of samples that the lossless reverberator prototype must be run before the impulse response repeats.
The delay lengths should be chosen to ensure a sufficiently high mode density in all frequency bands. An insufficient mode density can be heard as ``ringing tones'' or an uneven amplitude modulation in the late reverberation impulse response.
A rough guide to the average delay-line length is the ``mean free path'' in the desired reverberant environment. The mean free path is defined as the average distance a ray of sound travels before it encounters an obstacle and reflects. An approximate value for the mean free path, due to Sabine, an early pioneer of statistical room acoustics, is
Since the order of a system equals the number of poles, we have that is the number of poles on the unit circle in the lossless prototype. If the modes were uniformly distributed, the mode density would be modes per Hz. Schroeder  suggests that, for a reverberation time of 1 second, a mode density of 0.15 modes per Hz is adequate. Since the mode widths are inversely proportional to reverberation time, the mode density for a reverberation time of 2 seconds should be 0.3 modes per Hz, etc. In summary, for a sufficient mode density in the frequency domain, Schroeder's formula is
Prime Power Delay-Line Lengths
Suppose we are initially given desired delay-line lengths arranged in ascending order so that
Achieving Desired Reverberation Times
A lossless prototype reverberator, as in Fig.3.10 when , has all of its poles on the unit circle in the plane, and its reverberation time is infinity. To set the reverberation time to a desired value, we need to move the poles slightly inside the unit circle. Furthermore, due to air absorption (§2.3,§B.7.15), we want the high-frequency poles to be more damped than the low-frequency poles . As discussed in §2.3, this type of transformation can be obtained using the substitution
where denotes the filtering per sample in the propagation medium (a lowpass filter with gain not exceeding 1 at all frequencies).4.14Thus, to set the FDN reverberation time to at frequency , we want propagation through samples to result in attenuation by dB, i.e.,
Solving for , the propagation attenuation per-sample, gives
The last form comes from ln, where denotes the time constant of decay (time to decay by ) , i.e.,
Series expanding and assuming samples ( seconds) provides the practically useful approximation
Conformal Map Interpretation of Damping Substitution
The relation [Eq.(3.7)] can be written down directly from [Eq.(3.5)] by interpreting Eq.(3.5) as an approximate conformal map  which takes each pole , say, from the unit circle to the point . Thus, the new pole radius is approximately , where the approximation is valid when is approximately constant between the new pole location and the unit circle. To see this, consider the partial fraction expansion  of a proper th-order lossless transfer function mapped to :
Happily, while we may not know precisely where our poles have moved as a result of introducing the per-sample damping filter , the relation [Eq.(3.6)] remains exact at every frequency by construction, as it is based only on the physical interpretation of each unit delay as a propagation delay for a plane wave across one sampling interval , during which (zero-phase) filtering by is assumed (§2.3). More generally, we can design minimum-phase filters for which , and neglect the resulting phase dispersion.
In summary, we see that replacing by everywhere in the FDN lossless prototype (or any lossless LTI system for that matter) serves to move its poles away from the unit circle in the plane onto some contour inside the unit circle that provides the desired decay time at each frequency.
A general design guideline for artificial reverberation applications  is that all pole radii in the reverberator should vary smoothly with frequency. This translates to having a smooth frequency response. To see why this is desired, consider momentarily the frequency-independent case in which we desire the same reverberation time at all frequencies (Fig.3.10 with real , as drawn). In this case, it is ideal for all of the poles to have this decay time. Otherwise, the late decay of the impulse response will be dominated by the poles having the largest magnitude, and it will be ``thinner'' than it was at the beginning of the response when all poles were contributing to the output. Only when all poles have the same magnitude will the late response maintain the same modal density throughout the decay.
Damping Filters for Reverberation Delay Lines
In an FDN, such as the one shown in Fig.3.10, delays appear in long delay-line chains . Therefore, the filter needed at the output (or input) of the th delay line is (replace by in Fig.3.10).4.15 However, because is so close to in magnitude, and because it varies so weakly across the frequency axis, we can design a much lower-order filter that provides the desired attenuation versus frequency to within psychoacoustic resolution. In fact, perfectly nice reverberators can be designed in which is merely first order for each [314,217].
Let denote the desired reverberation time at radian frequency , and let denote the transfer function of the lowpass filter to be placed in series with the th delay line which is samples long. The problem we consider now is how to design these filters to yield the desired reverberation time. We will specify an ideal amplitude response for based on the desired reverberation time at each frequency, and then use conventional filter-design methods to obtain a low-order approximation to this ideal specification.
This is the same formula derived by Jot  using a somewhat different approach.
Now that we have specified the ideal delay-line filter in terms of its amplitude response in dB, any number of filter-design methods can be used to find a low-order which provides a good approximation to satisfying Eq.(3.9). Examples include the functions invfreqz and stmcb in Matlab. Since the variation in reverberation time is typically very smooth with respect to , the filters can be very low order.
First-Order Delay-Filter Design
The first-order case is very simple while enabling separate control of low-frequency and high-frequency reverberation times. For simplicity, let's specify and , denoting the desired decay-time at dc () and half the sampling rate ( ). Then we have determined the coefficients of a one-pole filter:
where denotes the th delay-line length in seconds. These two equations are readily solved to yield
The truncated series approximation
Orthogonalized First-Order Delay-Filter Design
denotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.4.16
Multiband Delay-Filter Design
In §3.7.5, we derived first-order FDN delay-line filters which can independently set the reverberation time at dc and at half the sampling rate. However, perceptual studies indicate that reverberation time should be independently adjustable in at least three frequency bands . To provide this degree of control (and more), one can implement a multiband delay-line filter using a general-purpose filter bank [370,500]. The output, say, of each delay line is split into bands, where is recommended, and then, from Eq.(3.6), the gain in the th band for a length delay-line can be set to
In the previous section, a ``graphical equalizer'' was used to set the reverberator decay time independently in each spectral band slice. While this gives much control over decay time, there is no control over the initial spectral gain in each band. Therefore, another good place for a graphical equalizer is at the reverberator input or output. Such an equalizer allows control of the initial spectral coloration of the reverberator. In the example of Fig.3.10, a spectral coloration equalizer is most efficiently applied to the input signal , before entering the FDN (but after splitting off the direct signal to be scaled by and added to the output), or the output of in Fig.3.10.
As discussed in §C.15, the FDN using a Householder-reflection feedback matrix is equivalent to a network of digital waveguides intersecting at a single scattering junction [463,464,385]. The wave impedance in the th waveguide is simply , the th element of the axis-of-reflection vector . The choice corresponds to all of the waveguides having the same impedance (the ``isotrophic junction'' case).
The Faust example reverb_designer.dsp brings up a FDN reverberator in which the signal out of each delay line is split into five bands so that can be controlled independently in each band. The 16 delay-line lengths are distributed exponentially between a minimum and maximum length set by two min/max-length sliders, but rounded to the nearest integer-power of a distinct prime, as introduced above in §3.7.3). The FDN reverberator is implemented in Faust's effect.lib. The band-splitting is carried out by the filterbank function in Faust's filter.lib.
The Faust function filterbank(order,freqs) implements a filter bank having the needed properties using Butterworth lowpass/highpass band-splitting arranged in a dyadic tree (normally a good choice for audio filter banks). That is, the whole spectrum is split at the highest crossover frequency, the lowpass region is then split into two bands at the next crossover frequency down, and so on, splitting the lowpass band at each stage in the dyadic tree [455,500]. The number of poles in each Butterworth lowpass/highpass filter is specified by order, and freqs contains a list of desired crossover frequencies separating the bands. A certain amount of dispersion is also introduced, since the filter bank is causal and delay-equalized (so that the bands may be summed without phase cancellation artifacts at the band edges). Also note that the lower bands are effectively produced by higher order filters than the upper bands. When the reverberation time is longer than the dispersion delay, the dispersion should not be audible as such, although it can affect the ``sound'' of the reverberation. In general, however, artificial reverberators normally benefit from additional allpass dispersion.
Figure 3.12 shows the block diagram of a FDN reverberator made from Faust's reverb_designer.dsp by changing 16 to 4. Figure 3.13 shows the Faust block diagram of the associated Hamard matrix multiplication. As it shows, multiplication by a Hadamard matrix can be implemented (ignoring the normalizing scale factor) as a series of block sums and differences (often called butterflies or shufflers) in which the block size decreases by a factor of 2 each stage. Figures for the remaining components of the reverberator may be perused via the shell command faust2firefox reverb_designer.dsp followed by clicking on the blocks in the browser.
A FOSS4.17 reverberator that combines elements of Schroeder (§3.5) and FDN reverberators (§3.7) is zita-rev1,4.18written in C++ for Linux systems by Fons Adriaensen. A Faust version of the zita-rev1 stereo-mode functionality is zita_rev1 in Faust's effect.lib. A high-level block diagram appears in Fig.3.14.
The main high-level addition relative to an 8th-order FDN reverberator is the block labeled allpass_combs in Fig.3.14. This block inserts a Schroeder allpass comb filter (Fig.2.30) in series with each delay line. In zita-rev1 (as of this writing), the allpass-comb feedforward/feedback coefficients are all set to . The delay-line lengths and other details are readily found in the freely available source code (or by browsing the Faust-generated block diagram).
In zita-rev1, the damping filter for each delay line consists of a low-shelf filter ,4.19in series with a unique first-order lowpass filter that sets the high-frequency to be half that of the middle-band at a particular frequency (specified as ``HF Damping'' in the GUI). Since the filter is constrained to be a lowpass, for , i.e., the decay time gets shorter at higher frequencies.
Viewing the resulting damping filter as a three-band filter bank (§3.7.5), let and denote the desired band gains at dc and ``middle frequencies'', respectively.4.20 Then the low shelf may be set for a desired dc-gain of , and its input (or output) signal multiplied by to obtain the resulting filter
The lowpass filter is also first order, and to provide half the middle-band at the beginning of the ``high'' band, the lowpass should ``break'' to a gain of at the ``HF Damping'' frequency specified in the GUI. A unity-dc-gain one-pole lowpass has the form 
Schroeder's original structures for artificial reverberation were comb filters and allpass filters made from two comb filters. Since then, they have been upgraded to include specific early reflections and per-sample air-absorption filtering (Moorer, Schroeder), precisely specified frequency dependent reverberation time (Jot), and a nearly independent factorization of ``coloration'' and ``duration'' aspects (Jot). The evolution from comb filters to feedback delay networks (Gerzon, Stautner, Puckette, Jot) can be seen as a means for obtaining greater richness of feedback, so that the diffuseness of the impulse response is greater than what is possible with parallel and/or series comb filters. In fact, an FDN can be seen as a richly cross-coupled bank of feedback comb filters whenever the diagonal of the feedback matrix is nonzero. The question then becomes what aspects of artificial reverberation have not yet been fully addressed?
Some kind of spatialization may be needed also for the late reverberation. A true diffuse field (§3.2.1) consists of a sum of plane waves traveling in all directions in 3D space. Since we do not know how to achieve this effect using current systems for reverberation, the typical goal is to simply extract uncorrelated outputs from the reverberation network and feed them to the various output channels, as discussed in §3.5. However, this is not ideal, since the resulting sound field consists of wavefronts arriving from each of the speakers, and it is possible for the reverberation to sound like it is emanating from discrete speaker locations. It may be that spatialization of some kind can better fool the ear into believing the late reverberation is coming from all directions.
Another way in which current reverberation systems are ``artificial'' is the unnaturally uniform distribution of resonant modes with respect to frequency. Because Schroeder, FDN, and waveguide reverbs are all essentially a collection of delay lines with feedback around them, the modes tend to be distributed as the superposition of the resonant modes of feedback comb filters. Since a feedback comb filter has a nearly harmonic set of modes (see §2.6.2), aggregates of comb filters tend to provide a uniform modal density in the frequency domain. In real reverberant spaces, the mode density increases as frequency squared, so it should be verified that the uniform modes used in a reverberator are perceptually equivalent to the increasingly dense modes in nature. Another aspect of perception to consider is that frequency-domain perception of resonances actually decreases with frequency. To summarize, in nature the modes get denser with frequency, while in perception they are less resolved, and in current reverberation systems they stay more or less uniform with frequency; perhaps a uniform distribution is a good compromise between nature and perception?
At low frequencies, however, resonant modes are accurately perceived in reverberation as boosts, resonances, and cuts. They are analogous to early reflections in the time domain, and we could call them the ``early resonances.'' It is interesting that no system for artificial reverberation except waveguide mesh reverberation (of which the author is aware) explicitly attempts precise shaping of the low-frequency amplitude response of a desired reverberant space, at least not directly. The low-frequency response is shaped indirectly by the choice of early reflections, and the use of parallel comb-filter banks in Schroeder reverberators serves also to shape the low-frequency response significantly. However, it would be possible to add filters for shaping more carefully the low-frequency response. Perhaps a reason for this omission is that hall designers work very hard to eliminate any explicit resonances or antiresonances in the response of a room. If uneven resonance at low frequencies is always considered a defect, then designing for a maximally uniform mode distribution, as has been discussed for the high-frequency modes, would be ideal also at low frequencies. Quite the opposite situation exists when designing ``small-box reverberators'' to simulate musical instrument resonators [428,203]; there, the low-frequency modes impart a characteristic timbre on the low-frequency resonance of the instrument (see Fig.3.2).
Digital Waveguide Reverberators
It was mentioned in §3.7.8 above that FDNs can be formulated as special cases of Digital Waveguide Networks (DWN) (see Appendix C for a fuller development of DWNs). Specifically, an FDN is obtained from a DWN consisting of a single scattering junction (§C.15). It follows that the DWN paradigm provides a more generalized framework in which to pursue further improvements of reverberation architecture. For example, when multiple FDNs are embedded within a single DWN, it becomes possible to richly cross-couple them in an energy-controlled manner in order to create richer recursive structures than either alone. General DWNs were proposed for artificial reverberation in [430,433].
As discussed in §2.4, a digital waveguide (bidirectional delay line) can be considered a computational acoustic model for traveling waves in opposite directions. A mesh of such waveguides in 2D or 3D can simulate waves traveling in any direction in the space. As an analogy, consider a tennis racket in which a rectilinear mesh of strings forms a pseudo-membrane.
A major advantage of the waveguide mesh for reverberation applications is that wavefronts are explicitly simulated in all directions, as in real reverberant spaces. Therefore, a true diffuse field can be developed in the late reverberation. Also, the echo density grows with time and the mode density grows with frequency in a natural manner for the 2D and 3D mesh. Finally, the low-frequency modes of the reverberant space can be simulated very precisely (for better or worse).
The computational cost of a waveguide mesh is made tractable relative to more conventional finite-difference simulations by (1) the use of multiply-free scattering junctions and (2) very coarse meshes. Use of a coarse mesh means that the ``physical modeling'' aspects of the mesh are only valid at low frequencies. As practical matter, this works out well because the ear cannot hear mode tuning errors at high frequencies. There is no error in the mode dampings in a lossless reverberator prototype, because the waveguide mesh is lossless by construction. Therefore, the only errors relative to an ideal simulation of a lossless membrane or space are (1) mode tuning error, and (2) finite band width (cut off at half the sampling rate). The tuning error can be understood as due to dispersion of the traveling waves in certain directions [518,399]. Much progress has been made on the problem of correcting this dispersion error in various mesh geometries (rectilinear, triangular, tetrahedral, etc.) [521,398,399].
See §C.14 for an introduction to the digital waveguide mesh and a few of its properties.
In real rooms, thermal convention currents cause the propagation path delays to vary over time . Therefore, for greater physical accuracy, the delay lines within a digital reverberator should vary over time. From a more practical perspective, time variation helps to break up and obscure unwanted repetition in the late reverberation impulse response [430,104].