### Further Extensions

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?

#### Spatialization of Reverberant Reflections

While we did not go into the subject here, the early reflections
should be *spatialized* by including a head-related transfer
function (HRTF) on each tap of the early-reflection delay line
[248].^{4.21}

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.

#### Distribution of Mode Frequencies

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].

#### The Digital Waveguide Mesh for Reverberation

A special case of digital waveguide networks known as the
*digital waveguide mesh*
has also been proposed for use in artificial reverberation systems
[396,518].

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.

#### Time Varying Reverberators

In real rooms, thermal convention currents cause the propagation path
delays to vary over time [58]. 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].

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