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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
FLEXIBLE REAL-TIME REVERBERATION SYNTHESIS WITH ACCURATE PARAMETER
CONTROL
Karolina Prawda ∗
Acoustics Lab
Dept. of Signal Processing and Acoustics
Aalto University, Espoo, Finland
karolina.prawda@aalto.fi
Silvin Willemsen, Stefania Serafin
Multisensory Experience Lab
Dept. of Architecture, Design & Media Tech.
Aalborg University, Copenhagen, Denmark
{sil,sts}@create.aau.dk
Vesa Välimäki
Acoustics Lab
Dept. of Signal Processing and Acoustics
Aalto University, Espoo, Finland
vesa.valimaki@aalto.fi
ABSTRACT
Reverberation is one of the most important effects used in audio
production. Although nowadays numerous real-time implementa-
tions of artificial reverberation algorithms are available, many of
them depend on a database of recorded or pre-synthesized room
impulse responses, which are convolved with the input signal. Im-
plementations that use an algorithmic approach are more flexible
but do not let the users have full control over the produced sound,
allowing only a few selected parameters to be altered. The real-
time implementation of an artificial reverberation synthesizer pre-
sented in this study introduces an audio plugin based on a feed-
back delay network (FDN), which lets the user have full and de-
tailed insight into the produced reverb. It allows for control of
reverberation time in ten octave bands, simultaneously allowing
adjusting the feedback matrix type and delay-line lengths. The
proposed plugin explores various FDN setups, showing that the
lowest useful order for high-quality sound is 16, and that in the
case of a Householder matrix the implementation strongly affects
the resulting reverberation. Experimenting with delay lengths and
distribution demonstrates that choosing too wide or too narrow a
length range is disadvantageous to the synthesized sound quality.
The study also discusses CPU usage for different FDN orders and
plugin states.
1. INTRODUCTION
Artificial reverberation is one of the most popular audio effects. It
is used in music production, sound design, game audio, and movie
production to enhance dry recordings with the impression of space.
The development of digital artificial reverberation started nearly 60
years ago [1], and since then various improvements as well as dif-
ferent techniques have been developed [2]. The designs available
nowadays can be roughly divided into three groups: convolution
algorithms, delay networks, and physical room models [2, 3, 4].
The methods involving physical modeling simulate sound prop-
agation in a specific geometry. Due to their high computational
cost, though, they are used mostly in off-line computer simulations
of room acoustics [3]. Recent developments in hardware and soft-
ware technologies have also allowed computationally expensive
simulations, such as those based on 3-D finite-difference schemes,
to run in real time [5].
∗This work was supported by the “Nordic Sound and Music Computing
Network—NordicSMC”, NordForsk project number 86892.
Copyright: © 2020 Karolina Prawda et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution 3.0 Unported License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
The techniques convolving the input signal with a measured
room impulse response (RIR) produce rich, high-fidelity reverber-
ation. However, since the RIR samples serve as the coefficients of
a finite impulse-response (FIR) filter, with which the dry signal is
filtered, the computational cost is high, especially for long RIRs.
Another group of artificial reverberation algorithms is based
on networks of delay lines and digital filters. The first example
of such reverberators was introduced by Schroeder and Logan [1],
who used feedback-comb-filter structures to create a sequence of
decaying echoes. A similar architecture using allpass filters was
also proposed to ensure high echo density without spectral col-
oration. The development of such structures led to the invention of
feedback delay network (FDN) algorithms, which can be regarded
as a “vectorized” comb filter [2]. The FDN, as used in its current
form, was presented in the work of Jot and Chaigne [6, 7].
Over the years, many real-time implementations of artificial
reverberation algorithms have been developed. The designs that
use a convolution-based approach, however, depend on measured
or pre-synthesized RIRs convolved with the signal, which are col-
lected in groups of presets [3, 8, 9, 10]. Such Virtual Studio Tech-
nology (VST) plugins allow modifying the reverberation by modu-
lating, damping or equalizing the available RIRs. The possibilities
are, however, limited by the size of the RIR databases and there-
fore prove to be relatively inflexible.
Algorithmic reverb plugins that are based on delay network
designs are both computationally efficient and easily modulated,
thus providing more flexibility and freedom in producing reverber-
ated sounds [4, 11]. The available designs vary between simple so-
lutions allowing the user to change only a few parameters [12] and
complex architectures with an elaborate interface enabling control
over a wide range of variables [13]. Many of those plugins, how-
ever, still remain ambiguous about the reverberation they synthe-
size, allowing the user to set only the broadband decay parameter,
and rely on presets based on the types of rooms they are supposed
to imitate (e.g., Bright Room or Dark Chamber [14]). Usually,
they also lack the information about the reverberation algorithm
they use and its elements.
The present work proposes a real-time implementation of an
FDN algorithm with accurate control over the reverberation time
(RT) in ten octave frequency bands in the form of an audio plu-
gin. The graphical user interface (GUI) gives a thorough insight
into the attenuation filter’s magnitude response, corresponding RT
curve, and resulting impulse response (IR). The plugin also pro-
vides several possibilities to control the elements of the FDN struc-
ture, such as the feedback matrix and delay lines. It gives the user
a full view of the decay characteristics and quality of the synthe-
sized reverberation. The study also presents the effect that the type
and size of the feedback matrix and the lengths and distribution of
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
the delay lines have on the produced sound and the algorithm’s
performance.
This paper is organized as follows: Section 2 presents the
theory behind the FDN, and Section 3 shows the GUI of the im-
plemented plugin, describes the functionalities and user-controlled
parameters of the reverberator, presents the code structure and dis-
cusses the real-time computation issues. Section 4 shows and dis-
cusses results regarding the echo density produced by the imple-
mentation and the CPU usage of the plugin. Finally, Section 5
summarizes and concludes the work.
2. FEEDBACK DELAY NETWORK
Figure 1 presents a flow diagram of a conventional FDN, which is
expressed by the relation:
y(n) =
N
X
i
cisi(n) + dx(n),(1a)
si(n+Li) =
N
X
j
Ai,je
hi(n)sj(n) + bix(n),(1b)
where y(n)and x(n)are the output and input signal, respectively,
at time sample n,si(n)is the output of the ith delay line, and
Ai,j is the element of an N-by-Nfeedback matrix (or scattering
matrix) A, through which all the delay lines are interconnected.
Parameters biand cisymbolize input and output coefficients, re-
spectively, dis the direct-path gain, and e
hi(n)is the attenuation
filter of the ith delay line.
When designing an FDN, a common practice is to first ensure
that the energy of the system will not decay for any possible type
of delay. Therefore, the matrix Ashould be unilossless [15]. To
obtain a specific frequency-dependent RT, each of the delay lines
must be cascaded with an attenuation filter, which approximates
the target gain-per-sample expressed by
γdB(ω) =
−60
fsT60(ω),(2)
where T60(ω)is the target RT in seconds, ω= 2πf /fsis the
normalized angular frequency, fis the frequency in Hz, and fs
is the sampling rate in Hz. In order to ensure that all delay lines
approximate the same RT, the gain-per-sample for each of them
must be scaled by a respective delay in samples L. This implies
that the target magnitude response of the attenuation filter in dB is
defined as follows:
AdB(ω) = LγdB (ω).(3)
In order to provide an accurate approximation of the target
RT, and therefore to closely follow the AdB, the attenuation filter
used in the FDN implementation in the present study is a graphic
equalizer (GEQ), which controls the energy decay of the system in
ten octave bands, with center frequencies from 31.25 Hz to 16kHz.
The equalizer is composed of biquad filters [16] and designed with
the method proposed by Välimäki and Liski [17] with later modifi-
cations, such as the scaling by a median of gains and the adding of
a first-order high-shelf filter as proposed in [18]. The GEQ mag-
nitude response for the ith delay line is expressed in dB as
e
HdB,i(ej ω ) = g0+
M
X
m=1
HdB,i,m(ej ω )−
g0
M,(4)
Filter
EQDelayLine
Figure 1: Flow diagram of an FDN with Nequalized delay lines
and their Moctave-band biquad filters shown in detail. See Sec-
tions 2 and 3.5 for more details.
where g0is the broadband gain factor, HdB,i,m are the magni-
tude responses of the band filters, and m= 1,2, ..., M is the
frequency-band index with Mcontrolled frequency bands. The
time-domain representation e
hi(n)of e
HdB,i(ej ω )is used in Eq. (1b).
3. IMPLEMENTATION
This section describes the real-time implementation of late rever-
beration synthesis using an FDN and a modified GEQ as the atten-
uation filter. The algorithm has been implemented in the form of an
audio plugin in C++ using JUCE, an open-source cross-platform
application framework [19]. The plugin can be downloaded from
[20], and an explanatory demo video can be found in [21].
3.1. Control over RT Values
In the present implementation, the modified GEQ attenuation filter
allows controlling the RT values in ten frequency bands. In order
to utilize the whole potential of the filter, the GUI of the plugin is
equipped with ten vertical sliders, one for each frequency band, as
depicted in Fig. 2. By changing the value of each of the sliders, the
user is able to change the RT value for the corresponding frequency
band from 0.03 s to 15s with a 0.01-s step.
Since too large a difference between two consecutive RT val-
ues can cause instability [18, 22], two extra modes are imple-
mented for better control: the All Sliders and the Smooth modes.
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
(a) The attenuation filter’s response (red line) and the corresponding RT
curve (black line). No preset is selected, and the Smooth button is pressed.
(b) Reverberator IR. The Fix coeffs button has been pressed, and the preset’s
drop-down menu and sliders have been disabled (see Sec. 3.6).
Figure 2: GUI of the implemented FDN plugin.
The modes are activated by pressing the respective buttons, as in-
dicated in Fig. 2a, with the corresponding buttons on the GUI be-
ing highlighted in green. If a mode is activated when the other is
active, the latter will deactivate. The All Sliders mode allows the
user to set all the RT values to be the same by changing the slider
position in one of the frequency bands.
When the Smooth mode is activated, changing the value of one
RT will also adjust the RT in the other frequency bands. RT values
of bands closer to the band that is changed are more affected than
other RT values via the formula
T60[m] = T60,init [m] + T60[mc]−T60,init[mc]ϵ|m−mc|,(5)
where mcis the index of the currently adjusted slider, m= 1,2, ...,
Mis the slider number, T60 and T60,init are the final and initial RT
values, respectively, and ϵ= 0.6is a heuristically chosen scaling
factor.
Five typical reverberation presets were created: Small Room,
Medium Room,Large Room,Concert Hall, and Church. The first
three presets are based on the measurement results presented in
[23], whereas RT values for the last two are taken from [24]. All
examples are available in a drop-down list in the top part of the
GUI. If one of the sliders is changed, “– no preset –” is displayed
in the drop-down list, as shown in Fig. 2a.
The Impulse button at the bottom right of the GUI empties
the delay lines and feeds a Dirac delta into the system so that the
impulse response of the reverberator is produced as an output.
3.2. Response Plotting
The window in the upper-half part of the GUI displays plots that
inform the user about the state of the plugin. As seen in Fig. 2a, the
GUI can display the RT curve (black) and the corresponding mag-
nitude response of the attenuation filter (red), which are plotted in
real time based on the values set by the sliders. This provides the
user with an insight into the actual decay characteristics of the syn-
thesized reverberation, which may differ from the user-defined RT
values. This happens due to the limited ability of the attenuation
filter in following the target RT curve, especially when the differ-
ences between values set for the neighboring frequency bands are
big [18, 22]. Very extreme differences may lead to the filter’s mag-
nitude response reaching or exceeding 0 dB, which results in the
system’s instability. This state is signaled by the background color
of the window changing to light red. For the response, only one
delay line is used to retain real-time plotting. Due to the fact that
the attenuation filter adopts smaller values for shorter delay-line
lengths, the shortest delay line is chosen as it exhibits instability
sooner than the others.
The Show IR button located in the top right of the window al-
lows the user to toggle between the RT curve and filter’s response
plots and the reverberator’s IR plot, which is shown in Fig. 2b. As
opposed to the response, the longest delay line is used to calcu-
late the IR. Even though the effect of the scattering matrix, and
with that the effect of other delay lines, are not included, using the
longest delay line has been proven empirically to give a good in-
dication of the audible IR. The values displayed on the x-axis are
determined by the average slider value, i.e., a shorter reverb time,
results in a more detailed plot of the earlier seconds of the IR. Fur-
thermore, not every sample is drawn, but 1,000 data points spread
over the plot-range.
3.3. Choice of Delay Lengths and Distribution
Although FDN-based reverbs are nowadays among the most pop-
ular algorithmic reverbs, there is no clear rule on how to choose
the lengths of the delay lines [25, 26]. The common practice is
to choose the number of samples that are mutually prime and uni-
formly distributed between the maximum and minimum lengths to
avoid clustering of echoes [26].
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
Figure 3: The advanced settings window.
Through the Advanced Settings window shown in Fig. 3, the
distribution of delay-line lengths can be chosen through a drop-
down menu from four options: Random,Gaussian,Primes, and
Uniform. Whenever an option is selected, the delay-line lengths
are randomly generated based on the distribution selected and rounded
to the nearest integers. The generation can be repeated by clicking
on the Randomize button. Furthermore, the minimum (500 sam-
ples) and maximum (10,000 samples) delay-line lengths can be
controlled; the minimum difference between the two has been set
to 100 samples. Moreover, there is an option to have the lengths
pre-defined for each distribution so that the plugin will have the
same behavior every time it is used. The minimum and maximum
delay-line lengths have been empirically set to 1,500 and 4,500
samples, respectively (~30–100 ms at fs=44.1 kHz).
3.4. Choice of Feedback Matrix
The choice of the feedback matrix is crucial for the FDN algo-
rithm to work correctly. The popular matrix types used in FDN
implementations that fulfill the requirement of being unilossless
are Hadamard [27], Householder [27], random orthogonal, and
identity matrices [28]. Where the first three are chosen to en-
hance specific properties of the algorithm, e.g., density of the im-
pulse response, the identity matrix, however, reduces the FDN to
a Schroeder reverberator, or a parallel set of comb filters [6, 28].
The plugin presented in this study allows the user to choose be-
tween these four matrices through a drop-down menu and to learn
about the differences in the sound obtained by changing this part
of the FDN reverberator. Additionally, the order of the FDN, and
thus the size of the feedback matrix, can be changed. The avail-
able options are 2, 4, 8, 16, 32, and 64, which can be chosen from
a menu.
In the case of the Householder matrix type, the implementa-
tion of matrices of different sizes vary. For all orders except for
16, the matrix is constructed using following the formula:
AN=IN−
2
NuNuT
N,(6)
where uT
N= [1,...,1], and INis the identity matrix [27]. The
matrix of order 16, on the other hand, following [29], is con-
structed using the recursive embedding of matrix of order 4:
A16 =1
2
A4−A4−A4−A4
−A4A4−A4−A4
−A4−A4A4−A4
−A4−A4−A4A4
.(7)
As a result, the matrix of order 16 consists of the same values,
differing only in their sign.
3.5. Code Structure
The plugin is divided into two main components that run on dif-
ferent threads at different rates. Firstly, the DSP component run-
ning at 44,100 Hz (audio rate), is structured in the same fashion
as shown in Fig. 1. An FDN class contains the scattering matrix
A, vectors band cthat scale the inputs and outputs of each delay
line (marked as biin Eq. (1b) and ciin Eq. (1a), respectively, and
in the current implementation all set to 1), and Ninstances of the
EQDelayLine class. This class, in turn, contains a delay line of
length Li(implemented as a circular buffer) and Minstances of
the Filter class. This class does all the low level computation and
contains the filter states and coefficients bi,m and ai,m of the ith
delay line and the mth octave band.
Secondly, the GUI component running at 5 Hz is responsible
for the graphics and control of the FDN. Apart from the controls,
this component contains the Response class that is used to draw the
RT and gain curves and the IR shown in Figs. 2a and 2b. The fil-
ter coefficients necessary for drawing the curves are updated at the
aforementioned rate. This calculation also provides information
about the stability of the FDN and is used to trigger the light-red
background denoting instability. The Response class also contains
a single instance of the EQDelayLines class that is used to calcu-
late the IR.
Communication from the GUI to the DSP component happens
at a 5-Hz control rate, which has been found to be a great trade-off
between speed and quality of control. When changing any of the
non-RT controls, the GUI triggers flags that are outside of the pro-
cess buffer (512 samples) to avoid the manipulation of parameters
when sample-by-sample calculations are being made.
3.6. Real-time Considerations
The components of the plugin requiring most computations are the
(re-)calculation of the filter coefficients and the plotting of the re-
sponses. Even though the filter coefficients only need to be recal-
culated when the sliders’ values are changed, it is good practice for
a plugin to have the same CPU usage when its values are changed
as when its values are static to prevent unexpected spikes in the
CPU usage. Instead, a Fix coeffs (coefficients) button has been
implemented that, when clicked, will deactivate the preset’s drop-
down menu and the sliders (as shown in Fig. 2b). Furthermore,
the plugin will stop recalculating the plots and filter coefficients,
greatly decreasing CPU usage (see Sec. 4.2). The CPU usages of
both threads are shown at the top of the plugin.
When any change is made to the FDN, be it the order, delay-
line distribution or length, the delay lines and filter states are set to
zero to prevent any unwanted artifacts. Only the RT control works
in real time without emptying the delay lines and filter states.
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
Time (s)
0
16
32
DL number
(a) Distribution of delay-line outputs for the option Primes.
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
Time (s)
0
16
32
DL number
(b) Distribution of delay-line outputs for the option Uniform.
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
Time (s)
0
16
32
DL number
(c) Distribution of delay-line outputs for the option Random.
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
0
16
32
DL number
0 0.2 0.4 0.6 0.8 1
Time (s)
0
16
32
DL number
(d) Distribution of delay-line outputs for the option Gaussian.
Figure 4: Distribution of the outputs of 32 delay lines (without scattering) for the options (a) primes, (b) random, (c) uniform, and (d)
Gaussian, and the length range (top) pre-defined lengths (1,500–4,500 samples), (middle) lengths randomized over the entire range (500–
10,000 samples), and (bottom) lengths randomized over a narrow range (5,000–6,500 samples). Each dot marks the time when the given
delay line outputs a sample.
4. RESULTS AND DISCUSSION
This section presents results regarding the echo density produced
by and CPU usage of the plugin.
4.1. Echo Density
To achieve smooth reverberation, a sufficient echo density, i.e., the
number of echoes per time unit produced by the algorithm and
their distribution [26], should be obtained. Echo density is affected
by a few factors, such as the lengths and the distribution of the
delay lines, the type of the feedback matrix [30] and its size, all of
which are discussed below.
4.1.1. Delay Lengths
The choice of delay-line length-distribution can help avoid more
than one sample appearing at the system’s output at the same time
and a clustering of the echoes, since both of these phenomena
lower the echo density. Additionally, the range over which the
delay-line lengths are chosen also affects the quality of the synthe-
sized sound. The distribution of delay-line outputs over time, with-
out a scattering matrix (i.e., an identity feedback matrix is used),
is shown in Fig. 4 for all the options available in the plugin. In the
case of the randomized selection of the delay-line lengths (middle
and bottom panes of Figs. 4a–4d), the results show one of the pos-
sible configurations. The delay-line lengths used in the examples
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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
were sorted in ascending order.
The top panes of Figures 4a–4d show the outputs of the pre-
defined delay lines, which depict the typical behavior of the FDN
algorithm. The outputs become more diffused over time, mak-
ing the reverb smoother. It should be noticed, however, that when
using Uniform distribution, the chosen range is divided into por-
tions proportional to the FDN order, and the delay-line lengths are
chosen from such “bands”. This makes the consecutive delay lines
differ by a similar number of samples, and the possibility of output
samples overlapping or clustering is higher than with other distri-
butions. Choosing the Gaussian option, on the other hand, draws
the delay-line lengths from the normal distribution with the mean
being the midpoint between the range’s boundaries. This results in
choosing the lengths closer to the mean more often than those fur-
ther from it, as depicted in Fig. 4d, potentially causing clustering
of echoes and slowing down the increase of the echo density.
The distribution of outputs presented in the middle panes show
that when the delay-length range is very wide, the output is dif-
fused from the beginning. Since such decay is rarely met in reality,
it is useful when recreating only specific spaces [31]. Additionally,
very short delay lines create clusters of echoes and a huge portion
of the output samples overlap. They do not contribute to the in-
crease of echo density, but nevertheless add to the computation.
Such clusters are well visible in Figs. 4a and 4c. Moreover, the
attenuation applied to the short delay lines is usually small, and
therefore closer to 0 dB, which makes them more prone to causing
the system’s instability.
On the other hand, very long delay lines (10,000 samples trans-
lates to about 0.23 s for the 44.1-kHz sample rate) may not produce
a meaningful contribution to the synthesized reverb for low RT val-
ues. However, such long delay lines still add to the computation,
since the order of the FDN, and at the same time, the size of the
feedback matrix needs to be equal to the number of delay lines.
Using a very narrow range over which the delay-line lengths
are distributed results in clusters of samples arriving at the output
within a very short time, as seen in the bottom panes of Figs. 4a–
4d. Between the consecutive clusters, however, relatively long
silences occur. The synthesized reverberation tail diffuses very
slowly. Regardless of whether the delay-line lengths are chosen to
be prime, random, distributed normally or uniformly, choosing too
narrow a range results in low sound quality with clearly audible
segmentation and in the effect’s behavior resembling more that of
a single delay line than a reverb.
4.1.2. Feedback Matrix
The normalized echo densities for all types of matrices available
in the plugin were calculated, following the method presented in
[32, 33, 34], for orders 2–64 and the delay lines selected randomly
from the range between 1,500 and 4,500 samples (the same set of
delay lines was used for all calculations). To avoid bias caused by
the smearing of echoes due to the filtering, the attenuation filters
were not used in the calculations. The results are presented in
Fig. 5 which generally show that the echo density increases faster
with a higher FDN order than with a lower one.
When matrices of size 2 and 4 are used, the number of echoes
in the output of the reverberator increases slowly and may never
reach saturation, i.e., the moment when there is an echo at ev-
ery successive time unit [26]. Therefore, these low orders do not
produce smooth sound. In the case of an FDN of order 8, the
echo density build-up is slow, which results in audible artifacts in
Table 1: CPU usage for all FDN orders in the cases of unfixed
(plotting IR and EQ) and fixed coefficients.
FDN CPU usage (%)
Order Unfixed (IR) Unfixed (EQ) Fixed
2 18.4 11.0 3.1
4 19.8 12.0 5.4
8 22.7 15.2 7.9
16 28.6 22.2 13.3
32 46.1 40.2 30.4
64 110.5 100.1 92.5
synthesized reverbs for as long as one second. Thus, a matrix of
size 16 is the smallest that increases the number of echoes quickly
enough so that the resulting sound is perceived as smooth for all
matrix types (except for the identity matrix). For the Hadamard
and random matrices, a further rise in the size accelerates the echo
density build-up, as evident in Fig. 5c and 5d.
Interestingly, the Householder matrix excels with the order of
16 using the recursive embedding of Eq. (7). This can be explained
by the fact that for all other orders, the implementation follows
Eq. (6), which produces matrices in which the difference between
the diagonal and the rest of the elements grows proportionally to
the order. Effectively, this makes the FDN approach a bank of
decoupled comb filters, which results in high variability of echo
density for orders 32 and 64, as seen in Fig. 5b, leading to audible
artifacts in the reverberation. For the matrix of order 16, however,
the echo density increases fast and remains high once saturation is
reached.
Because the identity matrices produce a very low echo den-
sity that does not increase with time, as seen is Fig. 5a, they are
not well fitted for the FDN. Reverberation synthesized using such
matrices is always low-quality. Being also an identity matrix, the
Householder matrix of order 2 should be avoided as well.
4.2. CPU Usage
Table 1 shows the CPU usage for all implemented FDN orders for
three different plugin-states: unfixed coefficients plotting the IR,
unfixed coefficients plotting the EQ, and fixed coefficients (plot-
ting and recalculation of filter coefficients disabled). The perfor-
mance has been measured on a MacBook Pro with a 2.2 GHz Intel
i7 processor using Xcode’s time profiler [35].
For all plugin states, the CPU usage increases exponentially
with the FDN order. Furthermore, fixing the coefficients, and thus
disabling the plotting and filter-coefficient calculation, greatly de-
creases the plugin’s CPU usage. Comparing this to the unfixed EQ
case, an additional ~8.0% is added to the CPU usage, and when
plotting the IR versus the EQ, an additional ~7.5% is added to the
usage. This value, however, depends on the average reverb time
used. For testing, the Concert hall preset was used, which requires
calculating 2.5 s of sound for the IR plot. With a higher average
slider value, and thus a longer IR to be calculated and plotted, the
CPU usage also increases.
The smallest useful FDN order is 16, as stated in Sec. 4.1.2.
Table 1 shows that this order, or even 32, is unlikely to cause audi-
tory drop-outs, especially when the coefficients are fixed.
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0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
0
0.2
0.4
0.6
0.8
1
Echo density
Order 2
Order 4
Order 8
Order 16
Order 32
Order 64
(a) Identity matrix.
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
0
0.2
0.4
0.6
0.8
1
Echo density
(b) Householder matrix.
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
0
0.2
0.4
0.6
0.8
1
Echo density
(c) Hadamard matrix.
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
0
0.2
0.4
0.6
0.8
1
Echo density
(d) Random orthogonal matrix.
Figure 5: Normalized echo densities for four types of feedback matrices and different FDN orders.
5. CONCLUSIONS
The present study introduces the FDN-based artificial reverber-
ation synthesis plugin. The implementation allows control over
the decay characteristics of the sound in ten octave bands in real
timeand plots the corresponding RT curve, the attenuation filter’s
magnitude response, and the IR. Additionally, users can explore
different setups of the FDN by changing the type and size of the
feedback matrix, and the lengths and distribution of the delay lines.
Experiments with the delay-line lengths and their distributions
suggest that these parameters should always be used in a balanced
manner, that suit the target reverberation. A wrong choice may
result in the creation of clusters of output samples and a low echo
density, which is undesirable in a reverberator. Choosing the lengths
over a narrow range results in low-quality, segmented sound, which
diffuses slowly. Picking the right distribution of delay-line lengths
is also important.
The ability to choose from among different FDN orders shows
that the lowest useful order for high-quality sound processing is
16, as it a sufficiently provides fast echo density build-up to ob-
tain smooth reverberation without audible artifacts. Shifting be-
tween feedback matrix types proves that the identity matrix, even
though it is lossless, should not be used in such applications, since
the produced sound is fluttery. It also shows that, in the case of
the Householder matrix, implementation affects the reverberation.
Results show that using recursive embedding when constructing
the Householder matrix increases the echo density in the produced
reverberation.
6. ACKNOWLEDGMENTS
This work was initialized, when Karolina Prawda made a Short-
Term Scientific Mission to the Aalborg University Copenhagen
from October 28 to November 15, 2019.
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