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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
DELAY NETWORK ARCHITECTURES FOR ROOM AND COUPLED SPACE MODELING
Orchisama Das , Jonathan S. Abel , Elliot K. Canfield-Dafilou
Center for Computer Research in Music and Acoustics
Stanford University
Stanford, CA, USA
[orchi|abel|kermit]@ccrma.stanford.edu
ABSTRACT
Feedback delay network reverberators have decay filters associ-
ated with each delay line to model the frequency dependent re-
verberation time (T60) of a space. The decay filters are typically
designed such that all delay lines independently produce the same
T60 frequency response. However, in real rooms, there are mul-
tiple, concurrent T60 responses that depend on the geometry and
physical properties of the materials present in the rooms. In this
paper, we propose the Grouped Feedback Delay Network (GFDN),
where groups of delay lines share different target T60s. We use the
GFDN to simulate coupled rooms, where one room is significantly
larger than the other. We also simulate rooms with different ma-
terials, with unique decay filters associated with each delay line
group, designed to represent the T60 characteristics of a particular
material. The T60 filters are designed to emulate the materials’ ab-
sorption characteristics with minimal computation. We discuss the
design of the mixing matrix to control inter- and intra-group mix-
ing, and show how the amount of mixing affects behavior of the
room modes. Finally, we discuss the inclusion of air absorption
filters on each delay line and physically motivated room resizing
techniques with the GFDN.
1. INTRODUCTION
Feedback delay networks (FDNs) are efficient structures for syn-
thesizing room impulse responses (RIRs). RIRs consist of a set of
sparse early reflections which increase in density over time, build-
ing toward late reverberation where the impulse density is high and
statistically Gaussian. Feedback delay networks are composed of
delay lines in parallel, which are connected through a feedback
matrix (or mixing matrix), which is unitary to conserve system en-
ergy [1]. Jot proposed adding shelf filters to the delay lines to yield
a desired frequency dependent T60 [2, 3]. Since then, FDNs have
become one of the most popular structures for synthesizing rever-
beration due to the relative efficiency of the approach. Recent re-
search on FDNs has focused on mixing matrix design to increase
echo density [4], modal analysis [5, 6], time-varying FDNs [7],
directional FDNs [8], and reverberation time control by accurate
design of the decay filters [9, 10].
In this paper, we propose a new delay network architecture for
physically informed room modeling. We also provide an alternate
design technique to the one proposed in [10], where a 10-band
graphic equalizer consisting of cascaded second order peak-notch
Copyright: © 2020 Orchisama Das, Jonathan S. Abel, and Elliot K. Canfield-
Dafilou . This is an open-access article distributed under the terms of the Creative
Commons Attribution 3.0 Unported License, which permits unrestricted use, distribu-
tion, and reproduction in any medium, provided the original author and source are
credited.
M11 M12
M21 M22
d
b1
b2
z−τ1
z−τ2
g1(z)
g2(z)
c1
c2
+
u(n)
y(n)
|
1
|
1
|
1
|
N1
|
N1
|
N1
|
N2
|
N2
|
N2
|
N1
|
N2
|
N1
|
N2
Figure 1: GFDN block diagram.
IIR filters was fit to a desired T60 response. Our argument is that
the T60 response of a room depends on the physical configuration
of the room, and room modes at the same frequency need not share
the same T60. The T60 response of each material depends on its
frequency dependent absorption coefficients and its volume to sur-
face area ratio, according to the Sabine theory of late-field rever-
beration [11]. We propose a new architecture, called the grouped
feedback delay network (GFDN), where groups of delay lines have
the same target T60 response associated with them. These filters
are designed to be low-order filters consisting of cascaded shelf
and resonant biquad filters. Low order filters significantly reduce
the computation required in [10], where each delay line has an IIR
filter of order 20. The interaction among the different delay line
groups is controlled by a block mixing matrix. As applications
of the proposed GFDN, we model coupled rooms, a single room
composed of different materials, and propose an efficient means
of incorporating air absorption and a physically accurate method
to resize the modeled rooms.
In §2, we introduce the structure of the GFDN, and discuss
block mixing matrix design. In §3, we use the GFDN to simu-
late the impulse response of a large room coupled with a smaller
room, such as box seating in an opera hall. Two different sets
of decay filters, associated with two delay line groups are used,
and the mixing matrix is parameterized by a coupling coefficient,
which effectively controls the amount of coupling between the two
rooms. In §4, a single room made of different materials is mod-
eled with the GFDN. Delay line groups have different T60 filters
associated with each material. T60 filter design according to mate-
rial absorption characteristics is discussed in §4.1. The amount of
mixing controls the behavior of the GFDN modes. Modal analysis
as a function of the mixing matrix shows how the T60 characteris-
tics of different materials interact as occupancy of a room changes
DAFx.1
Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
[6]. In §5, we build upon the method proposed in [12] and discuss
efficient room resizing with the GFDN by taking into considera-
tion air absorption, delay line lengths and T60 filters. The paper is
concluded in §6.
2. GROUPED FEEDBACK DELAY NETWORKS
A standard feedback delay network consists of Ndelay lines of
length τiseconds i= 1,2,...,N, each with its associated decay
filter, gi(z), connected through an N×Nfeedback matrix, M.
For a frequency dependent T60(z), the decay filter gains are related
to the delay line length as
gi(z) = 0.001 exp τi
T60(z).(1)
The same T60(z)is used to design the decay filters in all N
delay lines. In the proposed grouped feedback delay network ar-
chitecture, we use different T60(z)for each set of delay lines.
In Fig. 1, a GFDN with two sets of delay lines are shown. For
a total of Ndelay lines, N1delay lines have a decay response,
T601(z), and N2delay lines have a decay response, T602(z), such
that N1+N2=N. The two groups of decay filter gains, g1(z)
and g2(z)are calculated according to the different T60(z)s. The
mixing matrix Mis now an N×Nblock matrix made of the sub-
matrices, Mij ∈RNi×Nj, i, j = 1,2. With ci,bi,gi∈CNi×1
and τi∈RNi×1, the transfer function of Fig. 1, H(z), can be
written as
H(z) = Y(z)
U(z)
=d+c1c2g1(z)z−τ10
0g2(z)z−τ2
I−g1(z)z−τ10
0g2(z)z−τ2M11 M12
M21 M22−1b1
b2!.
(2)
The mixing matrix determines the amount of coupling be-
tween various delay lines. This property controls the rate at which
the echo density increases. A room with many objects and com-
plex geometry will mix faster than an empty room with simple
geometry. The mixing matrix can be designed to have a desired
mixing time according to the method in [6], where the Kronecker
product of a 2×2rotation/reflection matrix (parameterized by an
angle θ) with itself is taken log2(N)times to give an N×Nor-
thonormal matrix, M(θ)
R(θ) = cos θsin θ
−sin θcos θ
MN×N(θ) = R(θ)⊗R(θ)⊗. . . R(θ).
(3)
A well-diffused room with fast mixing time can be achieved by
a scaled Hadamard mixing matrix (θ=π
4). Similarly, a “room”
with no mixing and no increase in echo density can be synthesized
by an Identity mixing matrix (θ= 0). The parameter θcan be
chosen to give a desired mixing time, where θ=π
4yields the
maximum amount of mixing and smaller positive values give less
mixing.
In the GFDN, we can choose different, independent θval-
ues for each delay line group (the diagonal submatrices M11 and
V1V2
R2
R1
S
Figure 2: Coupled rooms
M22). The off-diagonal submatrices (M12 and M21) then control
how strongly coupled the groups are to each other. This gives us
independent control over the intra- and inter-group mixing charac-
teristics. The design of these submatrices will be described thor-
oughly in the following section.
3. COUPLED ROOMS
Two or more rooms can be coupled through an acoustically trans-
parent aperture. If the acoustic source is present in the smaller
room with a shorter decay time, the sound will travel to the larger
room and spill back into the first room. Such a configuration is
shown in Fig. 2. The resulting impulse response will have a non-
exponential decay. The first part of the decay has a steeper slope
due to the short decay rate of the first room, whereas the latter
part has a gentler slope representing the longer decay rate of the
second room. This is known as the Double-slope effect (DSE).
The physics of sound propagation in coupled rooms was studied
in [13]. The effect of the volume ratio, absorption ratio and aper-
ture size on the double slope profile was studied in [14]. Coupled
spaces are ubiquitous in the real world. They are found in con-
cert halls, opera halls, and churches [15] where columns, arches,
domes, etc., divide the space into two or more subspaces with dif-
ferent absorption properties.
3.1. Coupled mixing matrix design
The mixing matrix is crucial in simulating coupled rooms since
it controls diffusion within each room and among the rooms. A
method for mixing matrix design in coupled rooms has been sug-
gested in [16], where a sign-agnostic Procrustes method is used to
convert an arbitrary matrix to its nearest orthonormal form. Here,
we take a different approach. The diagonal submatrices that rep-
resent mixing in rooms 1and 2respectively can be characterized
by two mixing angles, θ1and θ2depending on the occupancy of
the rooms. The off-diagonal matrices represent the coupling be-
tween two rooms, and can be represented by matrices R12,R21 ,
multiplied by a scalar, α, which represents the amount of coupling.
M=M(θ1)αR12
αR21 M(θ2).(4)
This coupled mixing matrix is required to be orthonormal by
design. Using this criteria, i.e., MTM=I, we come up with the
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
following constraints:
1. R12 and R21 need to be orthonormal.
2. M(θ1)TR12 +RT
21M(θ2) = 0 ⇒
R21 =−M(θ2)RT
12M(θ1).
3. Mneeds to be scaled by 1
√1+α2.
Let R12 =M(θ1)1
2M(θ2)1
2=M(θ1
2)M(θ2
2). Therefore,
R21 =−M(θ2
2)M(θ1
2). Now, the orthonormal mixing matrix
is
M=1
√1 + α2M(θ1)αM(θ1
2)M(θ2
2)
−αM(θ2
2)M(θ1
2)M(θ2).
(5)
Let 1
√1+α2= cos ϕand α
√1+α2= sin ϕ, then our mixing matrix
is characterized by a coupling angle,ϕ∈[0,π
4]radians. When
ϕ= 0, we get minimum coupling (diagonal M), and when ϕ=
π
4, we get maximum coupling between the two rooms. The final
parameterized coupled mixing matrix is
M(θ1, θ2, ϕ) = cos ϕM(θ1) sin ϕM(θ1
2)M(θ2
2)
−sin ϕM(θ2
2)M(θ1
2) cos ϕM(θ2).
(6)
3.2. Evaluation
To simulate coupled rooms, R1and R2, we design an 8delay line
GFDN, with 4delay lines each representing the smaller and larger
room, with the source placed in R2and listener placed in R1.
The source and listener locations are determined by the b1, b2
and c1, c2coefficients respectively. The T60 filters of the two
rooms are first order low shelf filters parameterized by the DC and
Nyquist gains and transition frequency. The smaller room, R1, (in
blue) has a shorter decay time, so its T60(0) = 1 s, T60(∞) =
0.2s and fT= 1 kHz. The larger room, R2(in red) has T60(0) =
3s, T60(∞) = 1 s and fT= 4 kHz. The decay filters, g1(z),g2(z)
calculated according to (1) are shown in Fig. 3a.
The impulse responses of the coupled GFDN1as a function
of linearly spaced coupling angles (normalized by π
4) is shown in
Fig. 3b. As expected, when ϕ= 0, the rooms are decoupled and
the GFDN gives zero output. Increasing ϕincreases diffusion be-
tween the two rooms, giving denser reverb. The normalized echo
density (NED) [17], which is a perceptual measure of reverbera-
tion that helps quantify when early decay switches to late reverb,
is plotted in black. The NED plots show that denser reverberation
is achieved more quickly as ϕincreases. The effect of the smaller
room dominates in the coupled room RIRs as ϕincreases. This ap-
pears to go against the finding in [14], where subjects of a listening
test perceived more reverberance as coupling coefficient increased.
However, in our case the source is in the smaller room whereas the
listener is in the bigger room. So, the perceived reverberance will
decrease as coupling coefficient increases.
Additionally, we calculate the slopes of the two-stage decay
of the synthesized RIR. We do this by fitting the sum of two de-
caying exponentials and a constant to the energy envelope of the
1All sound examples are available at https://ccrma.stanford.
edu/~orchi/FDN/GFDN/GFDN.html
102103104
Frequency (Hz)
0.5
1
2
T60 (s)
102103104
Frequency (Hz)
-8
-6
-4
-2
0
T60 Filter gain (dB)
Room 1 (small)
Room 2 (big)
(a) Desired T60 response (left). Delay line T60 filters (right).
102103
Time (ms)
0
1
0
1
0
1
0
1
0
1
Amplitude
coupling =0
coupling =0.25
coupling =0.5
coupling =0.75
coupling =1
Larger room
Coupled rooms
Smaller room
(b) Impulse responses for different coupling coefficients. Nor-
malized Echo Density (NED) in black.
Figure 3: Coupled Rooms modeled with GFDN
synthesized RIR
henv(t) = γ0+γ1exp −t
T1+γ2exp −t
T2.(7)
We use MATLAB’s fmincon to find the decay rates T1, T2,
and update γusing weighted least squares (with more weight on
the tail). Two-stage decay of the RIR, with the fitted curve can be
seen in Fig. 4a. The ratio of T60 s calculated from T1and T2, as
a function of the coupling coefficient, is shown in Fig. 4b. This is
known as the decay ratio [14]. A larger decay ratio indicates more
coupling.
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
0 0.5 1 1.5 2
Time (s)
-100
-80
-60
-40
-20
0
Magnitude (dB)
(a)
0 0.2 0.4 0.6 0.8 1
/( /4)
1.5
2
2.5
3
3.5
4
Decay Ratio
(b)
Figure 4: Top - Two-stage decay in coupled GFDN impulse re-
sponse for ϕ=π
4. Red line indicates energy envelope, yellow line
is the curve fit and the black dotted lines are the 2-stage decay fits.
Bottom - Decay Ratio (ratio of T60s of two rooms) v/s normalized
coupling angle.
4. SINGLE ROOM WITH DIFFERENT MATERIALS
As described in [11], the acoustic energy density of a room, w(t),
with volume V, and absorbing surface area A, decays exponen-
tially as a function of time
w(t) = w0e−t
τ
τ=V
gcA
T60 =−2 log (0.001)τ
,(8)
where cis the speed of sound, and gis a geometric constant. A
room is typically constructed of several absorbing materials, each
with its unique frequency dependent absorption S(ω), and surface
area, a. The T60 of the room, and of the individual materials is
given by
Material Frequency (Hz)
125 250 500 1000 2000 4000
Plywood 0.28 0.22 0.17 0.22 0.10 0.11
Glass 0.35 0.25 0.18 0.12 0.07 0.04
Carpet 0.02 0.06 0.14 0.37 0.66 0.65
Air 0.10 0.30 0.60 1.00 1.90 5.80
Table 1: Absorption coefficients of different materials as function
of frequency. Absorption coefficients of air is the ANSI standard
at 20◦C, 30 −50% humidity.
T60room (ω) = −2 log (0.001) V
gc PiaiSi(ω)
T60mati(ω) = −2 log (0.001) V
gcaiSi(ω)
.(9)
To model such a room with the GFDN, we associate different
T60 filters with each group of delay lines, corresponding to differ-
ent materials present in the room. Groups of delay lines that share
the same T60 filter represent surfaces in the room that are made
of the same material. T60 corresponding to a material depends on
the volume to surface area ratio V
a, and the materials’ absorption
characteristics S(ω). This is unlike standard FDNs, where a single
T60 filter representing the room’s reverberation time is associated
with all delay lines. This yields significant computational savings,
because T60 filters associated with most materials can be repre-
sented by simple low order IIR filters. However, a single T60 filter
based on room geometry (9), would require a very high-order fil-
ter. A physical motivation behind this design choice is that in real
rooms, multiple modes at the same frequency can have different
decay rates, that depend on the properties and distances between
surfaces from which the acoustic waves get reflected.
4.1. T60 filter design
By specifying the absorption coefficients and the volume to sur-
face area ratio in a room, T60 filters for several materials can be
designed. Table 1 shows the absorption coefficients of three com-
mon construction materials and air in octave bands. In Fig. 5, filter
fits to T60 responses of plywood, glass, carpet and air is shown.
Often, (Fig. 5b, 5d), a simple first order shelf filter is enough
to model the desired T60 response. The first order shelf filter, pa-
rameterized by its DC gain, γ0, Nyquist gain, γπand transition
frequency ωT, is given by
H(s) = √γ0γπs
ωT+γ0
γπ1
2
s
ωTγ0
γπ1
2+ 1
.(10)
For materials like plywood that have resonant shelf-like T60
characteristics (Figs. 5a), we can cascade a peak/notch biquad filter
with a first order shelf filter. Thus, a third order filter is needed.
The transfer function of the peak (or notch) biquad, parameterized
by its peak frequency ωc, gain at peak frequency γωcand quality
factor Q, is given by
H(s) = s
ωc2+γωc
Qs
ωc+ 1
s
ωc2+1
Qs
ωc+ 1
.(11)
DAFx.4
Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
50 100 200 500 1000 2000 5000 10000 20000
Frequency (Hz)
-1.2
-1
-0.8
-0.6
-0.4
Material T60 (dB)
(a) Plywood
50 100 200 500 1000 2000 5000 10000 20000
Frequency (Hz)
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
Material T60 (dB)
(b) Glass
50 100 200 500 1000 2000 5000 10000 20000
Frequency (Hz)
-0.8
-0.6
-0.4
-0.2
Material T60 (dB)
(c) Carpet
50 100 200 500 1000 2000 5000 10000 20000
Frequency (Hz)
-0.25
-0.2
-0.15
-0.1
-0.05
Material T60 (dB)
(d) Air
Figure 5: T60 filter fits to different materials for a delay line length of 10 ms. Circles represent theoretical T60 values calculated according
to Sabine’s equation.
The details of converting these analog filter coefficients to digital
filter coefficients is given in [18].
Some materials have a steeper T60 slope, and a first or second
order shelf filter is not enough to model their responses (Fig. 5c).
For such materials, we use the method in [19] to cascade multiple
second order shelf filters (NObiquads per octave) to achieve a
desired transition bandwidth in number of octaves β, upper cutoff
frequency ωu, DC gain γ0, and Nyquist gain γπ. The total number
of biquads needed is N=⌈βNO⌉. The center frequencies of the
Nfilters are ωcµ= 2−βωu, and the DC and Nyquist gains of
each of the filters is γ0,πµ=N
√γ0,π. For Q=1
√2, the transfer
function of the cascaded shelf filters is given by
H(s) =
N−1
Y
µ=0
Hµ(s)
Hµ(s) = s
ωcµ2γ
1
2
0µ+s
ωcµ(γ0µγπµ)1
4
Q+γ
1
2
πµ
s
ωcµ2γ−1
2
0µ+s
ωcµ(γ0µγπµ)−1
4
Q+γ−1
2
πµ
.
(12)
For designing the T60 filter of carpet (Fig. 5c), we chose β= 3
and NO= 1, giving a total of 3biquads, and a filter order of 6.
4.2. Evaluation
We synthesize the RIR of a 5×5×5m3cubical room with a
carpeted floor (25 m2), a glass window on a wall (8m2), and ply-
wood on the ceiling and rest of the walls (77 m2) with a 16 delay
line GFDN, with 8delay lines dedicated to modeling plywood, and
4delay lines for carpet and glass each. We vary the mixing matrix
from minimum to maximum occupancy (identity to Hadamard).
T60s for smaller mixing angles are longer. The modal decompo-
sition of the GFDN, calculated according to [5], for four different
mixing angles is shown in Fig. 6. As per our previous work [6],
mode dampings approach each other and mode frequencies repel
as mixing increases. This effect is clearly visible, as mode T60
responses start by resembling the individual filter characteristics
of the three materials for no mixing, but as mixing increases, they
scatter towards each other. For a fully mixed GFDN, the T60 s are
well mixed and converge within a narrow band, thus giving a more
diffused RIR.
However, the unusually high T60 of the modes produce very
long RIRs. This is because we have not taken into account the
effect of air absorption in the room, which is discussed in the fol-
lowing section.
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
20 50 100 200 500 1000 2000 5000 20000
Frequency (Hz)
100
101
Pole T60 (s)
(a) Fraction mixing = 0
20 50 100 200 500 1000 2000 5000 20000
Frequency (Hz)
100
101
Pole T60 (s)
(b) Fraction mixing = 0.25
20 50 100 200 500 1000 2000 5000 20000
Frequency (Hz)
100
101
Pole T60 (s)
(c) Fraction mixing = 0.5
20 50 100 200 500 1000 2000 5000 20000
Frequency (Hz)
100
101
Pole T60 (s)
(d) Fraction mixing = 1
Figure 6: GFDN modes (T60 v/s frequency) as function of mixing
matrix angle. Increase in mixing causes modes to approach each
other in damping and scatter.
5. AIR ABSORPTION AND ROOM RESIZING
As described by Sabine’s equation (8), reverberation time is re-
lated to volume and surface area of the room. In small rooms,
the reverberation time and characteristic are predominantly a re-
sult of the materials properties, but in larger rooms, the effect of
air absorption becomes significant as the volume increases. The
architecture of the GFDN described above can be useful for real-
istically modeling rooms of different sizes. In addition to having
each delay line group represent a single material, we can cascade
an air absorption filter (first-order shelf) with each delay line. Nat-
urally, this increases the filter order for each delay line, however it
improves our ability to model the reverberation characteristics of
realistic rooms.
Fig. 7 shows the frequency responses of the T60 filters, im-
pulse responses, and spectrograms of the room described in §4.2
with and without the effect of air absorption. Since this is a medium
sized room, the effect of inclusion of air has a noticeable effect in
the damping the high frequencies. The inclusion of air absorp-
tion also has a significant effect in making the reverberation sound
natural and less metallic.
Now, say we have a GFDN that models a room that we like
but we want to increase or decrease the size of the room. Since
each delay line encodes the size, materials properties, and ratio
of absorbing surface area to volume, we can resize the room by
scaling the delay line lengths, τand recalculating the g(z)filter
coefficients to account for the changes in surface area and volume.
Let L0by the nominal length of the original room. To scale
the size of the room by a factor of L, we proportionally scale the
delay line lengths
τscaled =L
L0
τoriginal .(13)
Then, based on the new absorbing surface area, volume, and delay
line lengths, we recalculate the target T60s and the g(z)filters for
each delay line as described in §4. For small rooms, the material
properties will be more significant and in large rooms, air absorp-
tion will more significant.
Updating the delay line lengths scales the room mode frequen-
cies and fixes the temporal spacing of the early reflections and
mixing time while modifying the filters correctly updates the tar-
get frequency dependent T60s for a room of the new volume and
surface area. This means that we do not need to modify the mix-
ing matrix. Note that the method for room size control here is
a refinement to the methods described in [12] since we explicitly
have filters for materials and air absorption in each delay line. If
real-time room size control is desired, we can forgo modifying the
delay line lengths to avoid pitch shift artifacts. Moreover, if we
start with the geometry of the room we are modeling, it is easy
to scale the dimensions independently. For example, we can raise
the roof by modifying the delay line lengths associated with the
room’s height only. We have to modify the g(z)filters associated
with air absorption since the room’s volume is changed, however
we only need to change the materials filters associated with the
changes to the surface area of the walls as the floor and ceiling do
not change. Alternatively, we do not need to explicitly know the
geometry of the room. Since we group some delay lines together,
we can modify individual groups to modify different components
of the room abstractly.
Fig. 8 shows the impulse response, modes, and spectrogram
for the GFDN designed for the medium sized room described in
§4.2. We additionally show impulse responses and spectrograms
DAFx.6
Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
102103104
Frequency in Hz
-5
-4
-3
-2
-1
0
Magnitude in dB
without air
with air
(a)
0 50 100 150 200 250 300
Time in milliseconds
-1
-0.5
0
0.5
1
1.5
Amplitude of IRs
without air
with air
(b)
(c)
Figure 7: Top - T60 filter responses of a GFDN designed to model
a medium sized room without and with air absorption. Eight de-
lay lines are used to model plywood, four for carpet, and four for
glass. Middle - impulse responses of the GFDN without and with
air absorption. Bottom - spectrograms of the GFDN without and
with air absorption.
0 50 100 150 200 250 300
Time in milliseconds
-1
0
1
2
3
Amplitude of IRs
room scale: 0.5
room scale: 1
room scale: 2
(a)
20 50 100 200 500 1000 2000 5000 20000
Frequency (Hz)
100
101
Pole T60 (s)
room scale: 0.5
room scale: 1
room scale: 2
(b)
(c)
Figure 8: Top - impulse responses of a GFDN designed to model
a medium sized room and scaled to have its nominal length halved
and doubled. Middle - modes of the same rooms. Bottom - spec-
trograms of the same rooms.
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
of the same room, scaled to have the nominal length halved and
doubled. One can clearly see how the room sizing operation effects
the mode frequencies, early reflection spacing, mixing time, and
frequency dependent T60s.
6. CONCLUSION
In this paper, we have proposed the Grouped Feedback Delay Net-
work, which has different decay filters in different groups of delay
lines, motivated by the fact that in real rooms neighboring modes
do not have a single T60. Instead, they are distributed in a band.
We have used the GFDN to synthesize RIRs of coupled rooms,
where one room is significantly larger than the other. We have also
discussed the design of a parameterized orthonormal coupled mix-
ing matrix that controls the occupancy of the individual rooms and
amount of coupling between them. Single rooms composed of dif-
ferent materials (or different absorbing surfaces) have been mod-
eled with the GFDN. Delay line attenuation filters have been de-
signed to represent T60 characteristics of different absorbing mate-
rials in the room, instead of the space as a whole. Unlike [10], our
filters are of a lower order than a 10-band GEQ; hence the GFDN is
computationally more efficient. Modal analysis has shown that the
mode T60s of the synthesized RIRs follow the individual groups’
decay response when there is no mixing, and approach each other
as mixing increases, as previously investigated in [6]. Finally, we
have discussed the effect of air absorption in attenuating the over-
all T60 response of the modeled room. Methods for room resizing
by altering the delay line lengths and decay filter gains have been
proposed. We have provided relevant sound examples wherever
applicable.
The GFDN cannot be used to exactly match the measured T60
response of a particular space. However, it is a cheap way to gener-
ate approximate, physically informed RIRs when the configuration
of the room is known. Therefore, we think it will find applications
in VR audio, where cheap, approximate and dynamic artificial re-
verberation is a requirement. Perceptual evaluation of GFDN RIRs
with those synthesized by ray-tracing/image-source/FDTD meth-
ods is a topic we leave open for future work.
7. ACKNOWLEDGEMENT
The authors would like to thank Sebastian J. Schlecht for sharing
his FDN Toolbox [20] written in MATLAB, which was used to im-
plement FDN modal analysis with the Ehrlich-Aberth algorithm.
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