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ADAPTING ARTIFICIAL REVERBERATION ARCHITECTURES FOR
B-FORMAT SIGNAL PROCESSING
Joseph Anderson1, Sean Costello2
1 School of Arts & New Media, University of Hull, UK (j.anderson@hull.ac.uk)
2 Valhalla DSP, USA (sean@valhalladsp.com)
Abstract: Auralisation ray-traced volume modeling reverberation methods presently appear to be a preferred approach
for generating surround sound reverberation. However, a wide literature describing various architectures for artificial
reverberation filters is extant. Many of these alternative delay-line methods give distinct performance advantages, in
that they are not reliant upon convolution or volume modeling, and allow a variety of tonal and spatial parameters to
be modified independently. The authors describe a method to expand or adapt these known architectures to an
Ambisonic B-format context, and provide an illustrative example native B-format reverberator. Central to this
approach is the adaptation of scattering junctions to a B-format context and the use of B-format spatial image
transformation techniques.
Key words: Ambisonic, Reverberation
1 INTRODUCTION
Our present discussion aims to describe the methods the
authors have used to develop native B-format reverberator
networks. One of the authors, Anderson, is a composer
who works with B-format sound recordings. With Kyai
Pranaja [1] Anderson began to experiment with artificial
reverberation of B-format soundfields, which was initially
approached by applying separate mono reverberators to
each channel (W, X, W, Z) of the B-format signal. While
in some cases this may generate a ‘compositionally
suitable’ soundfield, the results are not necessarily wholly
satisfactory.
It is with work on Pacific Slope and Mpingo [1] that the
task of developing a satisfactory reverberation for B-
format was revisited, this time involving collaboration
with Costello. As B-format acquisition is not a necessarily
a simple task, requiring a Soundfield microphone and
associated 4-channel recorder, maintaining appropriate
‘Ambisonic-ness’ of the acquired sound recording was an
important goal in this effort.
Firstly, this implies a suitable network should both input
and output B-format. That is, the network should not
simply be a mono in reverberator acting on the mono part
(W) of the input soundfield. Secondly, the network should
somehow ‘make sense’ in terms of how reverberant signal
is distributed across the B-format soundfield. That is,
energy should spread throughout the soundfield and be
diffused wholly through the space. Thirdly, advantages of
B-format soundfield representation should be observed.
Being ‘B-format native’ should imply certain
opportunities—from the development of the network
itself (scattering matrices) to the ergonomics B-format
imaging matrices offer (control of early reflections and
late reverberation).
This paper is primarily concerned with networks suitable
for 1st order Ambisonic soundfields, but it is expected 2nd
order networks may be developed in a similar way.
2 ARCHITECTURE
Our principle approach to adapting reverberation
architectures involves a translation from B-format to the
A-format domain, the ‘directional representation’ of the
soundfield. In A-format, the outputs for each channel of a
reverberation network should be mutually decorrelated.
2.1. Decorrelation by differing delay lengths
Ideally, the decorrelation of the A-format reverb outputs
is created by different delay times, not just phase
differences. In many published reverb examples [2], [3],
[4] the outputs of the reverberation network are created by
summing together the delay network outputs using an
orthogonal matrix, such that the outputs have equal
energy from all delay lines, and differ in phase. Using
such an output structure in the A-format reverb may often
result in unintended cancellations in the A-B format
matrix.
2.2. Delay output options
The outputs can be taken from the ends of different delay
lines, such as the parallel delay lines in a feedback delay
network (FDN), or parallel combs. Alternately, the
outputs can be obtained by weighted delay taps from the
recursive reverberation network as shown by Dattorro [5].
AMBISONICS SYMPOSIUM 2009
June 25-27, Graz!
Page 2 of 5
2.3. Early Reflections via tapped delay lines
Early reflections can be generated by tapped and weighted
delay lines, as long as each output channel has mutually
decorrelated delay lengths. In practice, and for the
Ambisonic periphonic case (full, with-height 3D), it has
proven effective to stagger the delay lengths of each
channel, such that the output tap delay lengths are
positioned in the soundfield vertices of the A-format
tetrahedron, located in space as follows: front-left-up,
front-right-down, back-left-down, back-right-up. This is
repeated for a convincing set of reflections.
2.4. Early Reflections via cascaded unitary blocks
It is also possible to generate early reflections from
cascaded unitary stages [6], [7]. In such a system, parallel
unitary operators, such as delay lines, allpass filters, or a
passthrough signal, are combined via unitary matrices. By
cascading several stages of the parallel operators and
matrices, a high echo density can be obtained. The
outputs can be taken from the outputs of the final unitary
stage, or can be taken from weighted taps within the
network (such as after each block's delay lines, or from
the individual unitary matrices). Such a network can also
be used to feed the inputs of a recursive late reverb
network, as the outputs of the cascaded unitary stages
preserve the total input energy.
2.5. Adapting existing structures to A-format
As an example of adapting an existing reverberation
algorithm to A-format, we can start with the 4-channel
FDN reverb published by Puckette [7], as similar
algorithms can be found in the Pd distribution, as well as
in algorithms dating back to the 1980's that were in use at
IRCAM, and later distributed under the Jimmies title for
Max/MSP. The Puckette algorithm takes a stereo input,
and uses cascaded unitary stages as described above to
quickly build the early echo density. The outputs of the
cascaded unitary stages are fed into the inputs of four
parallel delay lines, which are scaled and fed through
lowpass filters,. The outputs of the four scaled lowpassed
delay lines are combined via a unitary matrix, the outputs
of which are fed back into the four delay line inputs.
2.6. Expanding 2-channel reverb to 4-channels
The cascaded unitary stages found in Puckette's algorithm
are made up of a delay line, in parallel with a passthrough
path, that are combined via a
!
2"2
rotation matrix with
its rotation angle set to
!
"
/ 4
(the gain normalization is
left out of Puckette's algorithm for simplicity). To work
with A-format input signals, the cascaded unitary stages
can be expanded to 4 parallel branches. The parallel
branches can be 4 parallel delay lines, 3 delay lines and a
passthrough path, series combinations of delay lines and
allpass delays, etc. The 3 parallel delays, 1 pass-through
path structure has the advantage of being able to cascade
multiple stages while still having a zero-delay path
through the structure, which can be advantageous when
feeding the late reverb (to avoid an excessive delay of the
onset of late reverberation).
3 ADAPTING TO A-FORMAT
Internally, the adapted network operates in A-format.
We’ll need to change domain from B-format to A-format
to enter the network, and then from A-format to B-format
on exiting, as early reflections and late reverberation.
In viewing the problem as how best to adapt Puckette’s
structure to an A-format space, we’ll repeat the
observation that for an ideally diffuse reverberation, all
channels of A-format should be mutually decorrelated.
Additionally, we’ll need to observe that the canonical
scaling on the W channel of B-format is
!
1 2
, which is
not ideal in this instance. This canonical scaling is usually
referred to as an engineering consideration. For horizontal
only soundfields (pantophonic), the scaling is a suitable
choice, and when on considers B-format was initially
intended as a successor of the ‘quad’ formats of the 1970s
[8], this choice is credible. However, for a full periphonic
soundfield (3D), a normalised scaling on W of
!
1 3
is
more appropriate. We’ll refer to signals using this scaling
as ‘W-normalised’.
3.1. Normalised A-format
For our purposes, we’ll define two matrices for changing
the signal domain from B-format to W-normalised A-
format (1) and back again (2).
!
1
6
1
2
1
2
1
2
1
6
1
2"1
2"1
2
1
6"1
2
1
2"1
2
1
6"1
2"1
2
1
2
#
$
%
%
%
%
%
%
&
'
(
(
(
(
(
(
(1)
!
3
8
3
8
3
8
3
8
1
2
1
2"1
2"1
2
1
2"1
2
1
2"1
2
1
2"1
2"1
2
1
2
#
$
%
%
%
%
%
&
'
(
(
(
(
(
(2)
A useful feature to note of the W-normalised A-B matrix
is that each point of the A-format tetrahedron encodes on
the surface of the Ambisonic sphere, a characteristic
appropriate for the encoding of early reflections.
3.2. Scattering
In making a choice for a suitable scattering matrix, the
question remains as to what matrix may be regarded as
equivalent in 4-channel A-format to a 2-channel rotation
by
!
"
/ 4
. Puckette [9] has given a hint when he discusses
power conservation in complex delay networks in
showing a cascade of 2-channel rotations to scatter more
than two channels. In reviewing the 2-channel early
reflection unitary cascade of Puckett’s algorithm, each
rotation of
!
"
/ 4
in the early reflection stages may be
regarded as a ‘maximally diffusive’ [10] scattering
network of the form shown below (3).
Page 3 of 5
!
1
2"1
2
1
2
1
2
#
$
%
%
&
'
(
(
(3)
This matrix may be regarded as a
!
2"2
Householder
reflection matrix and is of the kind suggested by Jot and
others [4] , [10], [11]. As 1st order A-format consists of
four channels, a
!
4"4
Householder matrix (4) is an
appropriate choice for scattering in our adapted network.
!
1
2"1
2"1
2"1
2
"1
2
1
2"1
2"1
2
"1
2"1
2
1
2"1
2
"1
2"1
2"1
2
1
2
#
$
%
%
%
%
%
&
'
(
(
(
(
(
(4)
3.3. Geometric interpretation of scattering
If we consider the effect of (4) on the signal in the B-
format domain, we’ll see that it is equivalent to inverting
the sign of W. Interpreting this action spatially, on each
pass through the scattering matrix, the signal is reflected
through the origin to the opposite side of the sphere.
Visualising the A-format tetrahedron, one sees a pass
through the scattering matrix reorients the tetrahedron so
that each vertex is reflected, to be distributed between the
three opposite vertices—resulting in a maximally
diffusive scattering. That is, after a pass through, each
vertex is re-oriented so that it is then strongly split across
the delay lines positioned at the opposite three vertices.
3.4. Scattering in higher orders
The authors suspect that a similar procedure may be
followed for adapting networks for high order Ambisonic
(HOA) reverberation. (However, at this point an
investigation has not been made.) In particular, it is
suspected that the geometric interpretation of reflecting
the soundfiled about the origin will similarly result in
maximally diffusive scattering.
4 CONTROLLING THE DECAY CURVE AND
SPATIALISATION OF EARLY REFLECTIONS
One drawback of the cascaded unitary matrix approach to
early reflections is that, as the number of cascaded blocks
is increased, the amplitude response of the output grows
closer to a Gaussian bell curve, with its characteristic fade
in and fade out. This can be partially alleviated by taking
the outputs as taps from within the cascaded blocks, either
as taps from the individual delay lines, from the outputs
of the branches pre-scattering matrices, or from the
outputs of the scattering matrices themselves. The outputs
can be weighted so as to produce a variety of amplitude
responses, within limits, as the blocks further down the
cascade will have a more Gaussian amplitude response. If
a specific early reflections amplitude response is required,
the cascaded unitary blocks approach should be replaced
with tapped delay lines, with one delay line for each input
channel, and with each delay line having weighted taps
that are sent to each of the 4 output channels as required.
4.1. Bringing early reflections into B-format
Early reflections may be tapped off from each early
reflection stage and panned into the resulting B-format
output at azimuth
!
"
and elevation
!
"
using the familiar
Ambisonic encoding matrix. (5)
!
1
2
cos
"
cos
#
cos
"
sin
#
sin
"
$
%
&
&
&
&
&
'
(
)
)
)
)
)
(5)
While such a procedure allows detailed control of the
position of each early reflection in the soundfield, the
authors regard this to be ‘over specified’ in many cases.
A more convenient approach is to use the W-normalised
A-B matrix (2), and variants of this matrix, to position
early reflections in the soundfield. This approach places
reflections at the vertices of a tetrahedron and may be
regarded as maximally diffuse in a geometric sense. The
resulting reflections may then be steered using rotation (6)
(shown across the Z-axis) and the focus transform (7), or
a combination of both. The early reflection cascade is
illustrated in figure 1, as described, with the ‘early
reflection positioning network’ including gain scaling and
imaging.
!
1 0 0 0
0 cos
"
#sin
"
0
0 sin
"
cos
"
0
0 0 0 1
$
%
&
&
&
&
'
(
)
)
)
)
(6)
!
1
1+sin
"
1
2
sin
"
1+sin
"
#
$
%
&
'
( 0 0
2sin
"
1+sin
"
#
$
%
&
'
( 1
1+sin
"
0 0
0 0 1)sin
"
1+sin
"
0
0 0 0 1)sin
"
1+sin
"
*
+
,
,
,
,
,
,
,
,
,
,
,
-
.
/
/
/
/
/
/
/
/
/
/
/
(7)
The focus transform is a dominance [12] variant
developed by one of the authors (Anderson). For this
current application it brings particular advantages, in that
reflections can be directed or ‘focused’ along the X-axis.
For focus,
!
"
refers to the image distortion half-angle. A
value of
!
0
leaves the image unchanged, where
!
"
/ 4
focuses (or compresses) the image to front-centre. If one
is interested in modeling concert hall reverberation,
choosing a value of
!
"
towards
!
"
/ 4
for the first
reflection stage will give a compressed image. Then, with
each successive early reflection tap-off, decreasing
!
"
towards
!
0
successively opens up each reflection group.
Page 4 of 5
z-m1a
z-m1b
z-m1c
Scattering
Matrix
B<->A
Format
Matrix
W
X
Y
Z
Early
Reflection
Positioning
Matrix
z-m2a
z-m2b
z-m2c
Scattering
Matrix
Early
Reflection
Positioning
Matrix
To B-
Format
Sum
To B-
Format
Sum
Early Reflections Stage 1 Early Reflections Stage 2
To Additional
Early Reflections
Stages and Late
Reverb Inputs
Figure 1: Early reflection cascade
5 LATE REVERBERATION
The late reverberation network [7] used in this example
suffers from a low initial echo density, as well as a fairly
low modal density, due to the length of the delay lines. By
replacing each of the delay lines with several allpass
delays in series, with a straight delay line pre or post
allpass, the echo density of the network can build at a
much higher rate, and the total delay of each branch can
be made high enough to obtain the required modal
density. The perceived modal density can be improved by
modulated one or more of the delays in each branch [3],
[5]. The outputs of the late reverb can be taken from the
end of each branch, pre-scattering matrix, and pre-scaling
and lowpass filter if desired. Alternately, the outputs can
be taken from taps within each branch, or after each
allpass in the branch. The outputs from each branch can
be sent to a single A-format channel, or scattered between
the channels as desired.
z-(lateDelay 1a)
lateDiffusion
Coef
z-(lateDelay 1b) z-(lateDelay 1c)
z-1
-a1
b0
b1
RT60 Filter
z-(lateDelay 1d +mod1)
-lateDiffusion
Coef
lateDiffusion
Coef
lateDiffusion
Coef
-lateDiffusion
Coef
-lateDiffusion
Coef
z-(lateDelay 2a)
lateDiffusion
Coef
z-(lateDelay 2b) z-(lateDelay 2c)
z-1
-a1
b0
b1
RT60 Filter
z-(lateDelay 2 d+mod2)
-lateDiffusion
Coef
lateDiffusion
Coef
lateDiffusion
Coef
-lateDiffusion
Coef
-lateDiffusion
Coef
z-(lateDelay 3a)
lateDiffusion
Coef
z-(lateDelay 3b) z-(lateDelay 3c)
z-1
-a1
b0
b1
RT60 Filter
z-(lateDelay 3d +mod3)
-lateDiffusion
Coef
lateDiffusion
Coef
lateDiffusion
Coef
-lateDiffusion
Coef
-lateDiffusion
Coef
z-(lateDelay 4a)
lateDiffusion
Coef
z-(lateDelay 4 b) z-(lateDelay 4c)
z-1
-a1
b0
b1
RT60 Filter
z-(lateDelay 4 d+mod4)
-lateDiffusion
Coef
lateDiffusion
Coef
lateDiffusion
Coef
-lateDiffusion
Coef
-lateDiffusion
Coef
Householder
Scattering
Matrix
From Early
Reflections
Stages
Figure 2: Late reverberation network
5.1. Bringing late reverb into B-format
Tapping out late reverb as described, the resulting A-
format should be as decorrelated as the network adapted
is able to provide. We can choose the A-B matrix (2)
previously discussed, or other tetrahedral orientations are
possible. The authors prefer the variant below (8) which
places the vertices of the tetrahedron at front-left, front-
right, back-up and back-down.
!
6
4
6
4
6
4
6
4
1
2
1
2"1
2"1
2
1
2"1
2
0 0
0 0 1
2"1
2
#
$
%
%
%
%
%
%
&
'
(
(
(
(
(
(
(8)
As one would expect, it is possible to further steer the late
reverberation as one chooses with the dominance
transform (9) [12]. The forward scaling is
!
"
, which can
be represented as the forward gain of the soundfield,
!
gforward
, in dB. (10)
!
1
2
"
+1
"
#
$
% &
'
( 1
8
"
)1
"
#
$
% &
'
( 0 0
1
2
"
)1
"
#
$
% &
'
( 1
2
"
+1
"
#
$
% &
'
( 0 0
0 0 1 0
0 0 0 1
*
+
,
,
,
,
,
,
-
.
/
/
/
/
/
/
(9)
!
"
=10
gforward
20
(10)
In application, if a concert hall reverb is desired, it may be
suitable to steer the late reverberation the rear of the
soundfield. To do so, one would choose a negative value
for
!
gforward
; -3 dB reduces the gain at the front of the late
reverberant soundfield by 3 dB and increases the gain at
the back by 3 dB, giving a 6 dB difference between front
and back.
Combining this approach with the late reverberation and
the steering of the early reflections discussed previously,
with care, can result in a convincing hall.
5.2. Higher order approaches for late reverberation
An alternate approach for the late reverberation is to use a
higher order feedback delay network, where each A-
format channel output is taken as a sum of the outputs of
some of the delays in the network. Puckette includes a 16-
channel FDN in the Pd distribution. This network can be
adapted to A-format by dividing the delays into groups of
four, summing the outputs from those delays, and using
the summed outputs as the late reverberation outputs for
the respective A-format channels. The feedback paths can
remain the same, and the outputs taken pre-scattering
matrix. An example of this approach can be found in [13].
6 CONCLUSION
This paper presented an approach to adapting known
artificial reverberation networks to a B-format context,
creating native Ambisonic reverberators. Resulting
reverberators both input and output B-format, distribute
signal energy appropriately through the resulting
reverberant soundfield, and take advantage of the
ergonomics Ambisonic imaging techniques offer for
shaping the spatial impression of a soundfield.
Page 5 of 5
A stereo network was adapted as an example. Similarly a
variety of known stereo and multi-channel reverberators
may be adapted to create a number of native B-format
reverberators with varying characteristics.
7 ACKNOWLEDGEMENTS
The authors would like to thank Dave Malham for initial
introductions, discussions and inspirations.
REFERENCES
[1] J. Anderson, Epiphanie Sequence, Sargasso
SCD28056, 2008. Audio CD.
[2] M.R. Schroeder, “Natural-sounding artificial
reverberation,” J. Audio Eng.. Soc., 10(3), 1962, 219-
233.
[3] J. Stautner and M. Puckette, “Designing multichannel
reverberators,” Comput. Music J., 6(2), 1982, 52–65.
[4] J.-M. Jot and A. Chaigne, “Digital delay networks for
designing artificial reverberators,” Proc. 90th Conv.
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[5] J. Dattorro, “Effect Design, Part 1: Reverberator and
Other Filters”, J. Audio Eng.. Soc., 45(9), 1997, 660-
684.
[6] M. Gerzon, “Synthetic Stereo Reverberation, Parts I
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[8] R. Elen, “Whatever happened to Ambisonics?”, Audio
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http://crca.ucsd.edu/~msp/techniques/latest/book-
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[10] D. Rocchesso, “Maximally Diffusive Yet
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[11] D. Rocchesso and J.O. Smith, “Circulant and
Elliptic Feedback Delay Networks for Artificial
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[12] P.S. Cottrell, “On the Theory of the Second-
Order Soundfield Microphone”, Ph.D. Thesis,
University of Reading, 2002, 105-107, 170-174.
[13] S. Costello, “B-Format Reverb”, DXARTS 567 /
Sound in Space, Online. Available:
http://www.dxarts.washington.edu/courses/567/08WI
N/BFRev.rtf. [Accessed: Jun. 2009]
