Velvet-Noise Feedback Delay Network

Abstract

Artificial reverberation is an audio effect used to simulate the acoustics of a space while controlling its aesthetics, particularly on sounds recorded in a dry studio environment. Delay-based methods are a family of artificial reverberators using recirculating delay lines to create this effect. The feedback delay network is a popular delay-based reverberator providing a comprehensive framework for parametric reverberation by formalizing the recirculation of a set of interconnected delay lines. However, one known limitation of this algorithm is the initial slow build-up of echoes, which can sound unrealistic, and overcoming this problem often requires adding more delay lines to the network. In this paper, we study the effect of adding velvet-noise filters, which have random sparse coefficients, at the input and output branches of the reverberator. The goal is to increase the echo density while minimizing the spectral coloration. We compare different variations of velvet-noise filtering and show their benefits. We demonstrate that with velvet noise, the echo density of a conventional feedback delay network can be exceeded using half the number of delay lines and saving over 50% of computing operations in a practical configuration using low-order attenuation filters.

Full text

Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
VELVET-NOISE FEEDBACK DELAY NETWORK
Jon Fagerström1, Benoit Alary1, Sebastian J. Schlecht1,2and Vesa Välimäki1∗
1Acoustics Lab, Dept. of Signal Processing and Acoustics
2Media Lab, Dept. of Media
Aalto University
Espoo, Finland
jon.fagerstrom@aalto.fi
ABSTRACT
Artificial reverberation is an audio effect used to simulate the acous-
tics of a space while controlling its aesthetics, particularly on sounds
recorded in a dry studio environment. Delay-based methods are
a family of artificial reverberators using recirculating delay lines
to create this effect. The feedback delay network is a popular
delay-based reverberator providing a comprehensive framework
for parametric reverberation by formalizing the recirculation of
a set of interconnected delay lines. However, one known limita-
tion of this algorithm is the initial slow build-up of echoes, which
can sound unrealistic, and overcoming this problem often requires
adding more delay lines to the network. In this paper, we study the
effect of adding velvet-noise filters, which have random sparse co-
efficients, at the input and output branches of the reverberator. The
goal is to increase the echo density while minimizing the spec-
tral coloration. We compare different variations of velvet-noise
filtering and show their benefits. We demonstrate that with velvet
noise, the echo density of a conventional feedback delay network
can be exceeded using half the number of delay lines and saving
over 50% of computing operations in a practical configuration us-
ing low-order attenuation filters.
1. INTRODUCTION
Artificial reverberation algorithms have been developed for almost
60 years, starting from the first algorithm by Schroeder [1]. For
many years, the feedback delay network (FDN) has been one of
the most popular methods to create artificial reverberation [2, 3].
This paper proposes a novel FDN structure for increasing its echo
density.
The idea of interconnecting multiple allpass filters through a
matrix, which is the underlying principle of the FDN, was first
introduced by Gerzon [4]. Stautner and Puckette further devel-
oped the idea of a recirculating network of delays [5], and Jot and
Chaigne later extended it to the formal design that we now know
as the FDN [2]. The analysis and improvement of the FDN remain
active areas of research today [6, 7, 8, 9, 10], and very recently, the
FDN was extended into the spatial domain [11, 12, 13].
The number and lengths of the delay lines are among the main
questions when designing an FDN reverberator. The modal den-
sity of the synthetic response is positively correlated with the total
∗This research has been funded by the Nordic Sound and Music Com-
puting Network—NordicSMC (NordForsk project no. 86892) and by the
Academy of Finland (ICHO project, grant no. 296390).
Copyright: © 2020 Jon Fagerström 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.
length of all the delay lines [2]. On the other hand, shorter de-
lays help build up the echo density faster, but can lead to metallic
timbre in the impulse response due to a lower modal density [14].
Increasing the number of delay lines improves the echo density [2]
but also increases the number of arithmetic operations required per
sample.
In practical applications, low-order attenuation filters are usu-
ally used to ensure a faster decay at high frequencies, which mim-
ics the acoustics of a room. For precise control of the reverbera-
tion time of different frequency bands, high-order attenuation fil-
ters within the FDN are necessary [15, 16, 17, 18]. However, in-
creasing the complexity of the attenuation filter greatly impacts the
computational cost per delay line. Thus, the number of delay lines
in the system must be minimized while retaining a sufficiently high
echo and modal density.
Traditionally allpass filters have been used to increase the echo
density of a reverberator [1, 19]. However, smearing problems
have been reported for transient sounds [19]. Other ways to im-
prove the FDN include introducing time-varying elements in the
structure, such as modulated delay lines [20], allpass filters [3],
or a time-varying feedback matrix [21, 22, 23]. Time-varying de-
lay lines lead to imprecise control of the decay time, whereas an
FDN with time-varying allpass filters is not guaranteed to be sta-
ble [23]. Since a time-varying feedback matrix is less likely to
cause artifacts in the reverberation sound, this method has been
found to improve the sound quality of the reverberation tail [23].
Another approach is to introduce short delays in the feedback ma-
trix, so that each matrix element consists of a gain and a delay
[24]. Also, separate early reflection modules using finite impulse
response (FIR) filters for FDNs have been suggested [25]. How-
ever, the magnitude spectrum of these filters should be designed to
minimize undesirable spectral coloration.
In this paper, we propose a novel reverberator structure with
improved echo density, consisting of a conventional FDN structure
with sparse velvet-noise filters, called the Velvet-noise Feedback
Delay Network (VFDN). Velvet noise has been previously applied
in audio processing to model the reverberation [26, 27, 28, 29] and
to design computationally efficient decorrelation filters [30, 31].
The proposed method allows for a delay network using fewer but
longer delay lines, while retaining a suitable echo density buildup.
Using fewer delays reduces the computational cost and allows for
more accurate attenuation filters, whereas the longer delays ensure
a suitable modal density is retained.
The rest of this paper is organized as follows. Sec. 2 presents
the previous ideas used in this study, including the basics of FDN,
echo-density estimation, and velvet noise. Sec. 3 introduces the
novel reverberation structure and discusses different ways of ap-
plying velvet-noise filtering. Sec. 4 analyzes the results of this
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z-m1
z-m2
z-m3
+
+
+
X(z) Y(z)+
+
g1
g2
g3
b1
b2
b3
c1
c2
c3
Figure 1: Block diagram of a conventional FDN of size N= 3.
work, and Sec. 5 conclude the paper. Spectral coloration caused
by velvet-noise filters is studied analytically in the Appendix.
2. BACKGROUND
This section presents the basic FDN structure, the echo-density
measure used in this study, and velvet-noise decorrelators.
2.1. Feedback Delay Network
The FDN consists of a set of recirculating delay lines intercon-
nected through a feedback matrix A(Fig. 1), that defines the re-
circulating gains for each connection [2]. By ensuring that this
matrix is orthogonal, a lossless prototype is obtained, which redis-
tributes the output energy of one delay line to the input of all delay
lines. The lossless nature of this system permits the parametric
control of the decay rate by using a target reverberation time T60,
corresponding to the time it takes to reach 60 dB of attenuation in
the decay.
The output sample y(n)of the recursive system, for an input
x(n), is formulated as
y(n) =
N
X
i=1
cigisi(n),(1)
si(n+mi) =
N
X
j=1
Aij gjsj(n) + bix(n),(2)
where biand ciare the input and output coefficients, respectively,
Aij is the feedback matrix element, giis the attenuation gain, and
siare the output states of each delay line. The T60 specification
is used to compute the appropriate givalues used to attenuate the
output of each delay line based on its length mi.
The transfer function of the FDN is
H(z) = Y(z)
X(z)=c⊤Dm(z)−1−A−1b,(3)
where band care vectors containing the input and output gains,
Dm(z) = diagG1(z)z−m1, G2(z)z−m2, ..., GN(z)z−mN, and
Ais the feedback matrix. In practice, T60 is often specified at var-
ious frequencies, such as at octave bands, and then each gain gi
must be replaced with an attenuation filter Gi(z), which can be a
graphic equalizer [16, 17, 18].
2.2. Estimating the Echo Density
The echo density of an FDN impulse response is a measure of
the number of echoes over time. For a lossless prototype FDN,
a straightforward way to estimate it is the empirical echo density
[32], which is computed by counting the amount of impulses per
time frame at the output of the FDN.
Recently, Tukuljac et al. proposed another method for estimat-
ing the echo density of an impulse response [33]. This method,
called the sorted density (SD) measure, was developed for com-
plex acoustic scenes where the empirical echo density did not work
reliably. The steps for computing the SD estimate are briefly de-
scribed below.
First, the direct sound in the impulse response is removed.
Then, the impulse response is converted to an echogram, i.e., e(n) =
h(n)2and normalized to factor out the energy decay. This normal-
ized echogram is then analyzed with a sliding window by comput-
ing the SD within each window. Finally, this density is normalized
with the expected value of Gaussian noise so that an SD of 1 indi-
cates the expected density of Gaussian noise. Thus, the SD mea-
sure yields an echo density measure that increases until it reaches
a plateau close to 1.
2.3. Velvet-Noise Decorrelators
A velvet-noise sequence (VNS) is a pseudo-random signal, com-
parable to white noise, using as few non-zero values as possible
[26, 34]. By taking advantage of the sparsity of the signal, com-
puting its time-domain convolution with another signal becomes
very efficient [28, 29]. Conceptually, the first step in generating
velvet noise is to create a sequence of evenly spaced impulses at
a selected density [26]. The sign and location of each impulse are
then randomized, but impulses still remain within a given interval,
having a range dictated by the desired impulse density. Figs 2(a)
and 3(a) show an example of a VNS and its magnitude spectrum,
respectively.
For a given density ρand sampling rate fs, the average spacing
between two neighboring impulses in a VNS is
Td=fs/ρ, (4)
which is called the grid size [34]. The total number of impulses in
a VNS of length Ls(in samples) is
M=LsTd.(5)
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0
1
(a)
0 5 10 15 20 25 30
Time [ms]
-1
0
1
(b)
0 5 10 15 20 25 30
Time [ms]
Figure 2: Examples of (a) a non-decaying (VN15) and (b) an opti-
mally decaying velvet-noise (OVN15) sequence (M= 15).
The sign of each impulse is
s(m) = 2 round(r1[m]) −1,(6)
where m= 0,1,2, ..., M −1is the impulse index, the round
function is the rounding operation, and r1(m)is a random number
between 0 and 1. The location of the mth impulse of the VNS is
calculated as
k(m) = round[mTd+r2[m](Td−1)],(7)
where r2(m)is also a random number between 0 and 1 [34].
To convolve a VNS with another signal, we exploit the sparsity
of the sequence, representing about 98% of the sequence for a den-
sity of ρ= 1000 at fs= 44100 Hz, which allows a very efficient
time-domain convolution computation [30]. By storing the VNS
as a series of indices of the non-zero elements, all mathematical
operations involving zeros can be skipped. The convolution with a
basic VNS does not require multiplications either, only additions
and subtractions [28, 29].
Furthermore, VNS sequences have been found to be suitable
to decorrelate audio signals [30] by applying an exponentially de-
caying gain to each impulse to prevent the smearing of transients.
For a given decay constant α, the gains are expressed as
se(m) = e−αms(m)r3(m),(8)
where r3(m)is a random gain between 0.5 and 2.0 [30, 31]. The
sparse convolution operation with a signal x(n)can be written as
x∗se=
M−1
X
m=0
x[n−k(m)]se(m),(9)
where the asterisk (∗) denotes the discrete convolution.
Since VNS filters do not have an exactly flat magnitude re-
sponse, as seen in Fig. 3(a), they introduce a minor coloration to
audio signals. For this reason, optimizing the random values is
recommended to minimize the spectral deviation [31]. Instead of
simply choosing random values, the impulse sign r1(m), the im-
pulse location r2(m), and the impulse gain r3(m)are specified
by a nonlinear optimization scheme. For the resulting optimized
velvet-noise decorrelators, a peak magnitude-response deviation
50 100 250 500 1k 2k 3k 5k 10k
Frequency[Hz]
-10
-5
0
5
10
Magnitude [dB]
(a)
50 100 250 500 1k 2k 3k 5k 10k
Frequency [Hz]
-10
-5
0
5
10
Magnitude [dB]
(b)
Figure 3: Smoothed magnitude spectra of the signals in Fig. 2: (a)
a non-decaying and (b) an optimally decaying VNS.
of less than 1 dB can be achieved when third-octave smoothing is
applied [31]. Fig. 2(b) shows an example optimized VNS, that
decays approximately exponentially. Fig. 3(b) presents its magni-
tude spectrum, that has only a few dB of ripple, much less than in
the spectrum of the non-optimized VNS in Fig. 3(a).
In this work, VNS filters are used to increase the echo den-
sity in artificial reverberation, a goal different from decorrelation.
Some of the same requirements as above still apply, such as the
desire for computational efficiency, minimizing the smearing of
transients, and the need for a flat magnitude response.
3. PROPOSED STRUCTURE
The novel VFDN structure is an extension of the conventional
FDN in Fig. 1. As shown in Fig. 4, the input and output gains
biand ci, respectively, are replaced by sparse VNS filters bi(z)
and ci(z). The VNS filters used here are relatively short, and their
main purpose is to increase the echo density in the final output,
which is otherwise sparse in the beginning of the impulse response
due to the exponential nature of the recirculation. The transfer
function of the proposed VFDN structure is
H(z) = c(z)⊤Dm(z)−1−A−1b(z),(10)
where c(z)and b(z)are vectors containing the VNSs for the out-
put and input delay lines, respectively. The form of the transfer
function stays similar to the conventional FDN, as seen by com-
paring (3) and (10). Since the VNS filters are placed outside the
feedback loops in Fig. 4, they cannot affect the decay rate of the
system.
3.1. Optional Configurations
The VFDN offers three distinct configurations. One configuration
is to use a single set of VNS filters connected at either the input,
using the bi(z), or the output, using the ci(z)of each delay line.
Alternatively, the VNS filters can be connected to both the input
and output, using both bi(z)and ci(z).
When considering the absolute echo density of the VFDN out-
put signal, each convolution operation with the VNS multiplies the
number of echoes. If a single set of VNS filters is used at either
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z-m1
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+
+
+
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+
g1
g2
g3
c1(z)
c2(z)
c3(z)
b1(z)
b2(z)
b3(z)
Figure 4: Block diagram of the proposed VFDN having a VNS filter in each input and output branch.
the input or output, and each VNS filter has 2Mimpulses, the
multiplier is approximately
Esingle = 2M. (11)
Here, the superposition of impulses that inevitably occurs is not
accounted for, but we assume for simplicity that each impulse ap-
pears separately in the impulse response.
If each VNS contains Mimpulses when using sets of VNSs
at both the inputs and outputs, the multiplier of the absolute echo
density is approximately
Eboth =M2.(12)
Both configurations retain the same computational cost since they
both use a total of 2Mimpulses. Since Eboth > Esingle when
M > 1, dividing the total number of impulses equally to both the
input and output increases the echo density more when compared
to using the same total number of impulses at either just the inputs
or outputs.
Fig. 5 further demonstrates the benefit of using VNSs at both
the input and output of each delay line. There, VN1and VN2are
two VNSs with 15 impulses each, where the former can be thought
to be connected at the input and the latter connected at the output
VNS filters. The bottom pane in Fig. 5 shows the resulting con-
volved sequence, which is much denser than the two individual
VNSs with almost M2= 152= 225 pulses. This convolution
result is equivalent to the filtering observed when a signal goes
through both filters in the structure. Superimposed impulses result
in samples with a value of 2 in Fig. 5 (bottom).
We could also consider having two sets of VNS filters con-
nected in series either at the input or output. However, the result-
ing spectral coloration is flatter when having the filters at both the
input and output. This difference in the coloration between these
two configurations is derived in Appendix 7.2.
Fig. 6 shows the echo density plot for the three optional con-
figurations of the VFDN. Each of the configurations has the same
total number of impulses in the velvet sequences (M= 30), which
also means they have the same computational cost. All other pa-
rameters, such as the feedback matrix and the delay-line lengths,
remain unchanged in this comparison. The normalized echo den-
sities of the single input or output VFDN configurations are equiv-
alent, as expected. However, when using a configuration of the
VFDN that combines both input and output VNSs, the echo den-
sity grows faster. We observe that the measured echo density agrees
with (11) and (12), demonstrating the benefits of using this config-
uration.
3.2. Velvet-Noise Sequence Types
There are two types of VNSs to be used in a VFDN, the basic
non-decaying sequences containing +1’s and −1’s only and opti-
mized decaying VNSs. The advantage of the non-decaying VNS
is that it does not require multiplications in the convolution, and
that all its impulses are equally tall. Therefore, no computation
is necessary for perceptually negligibly small impulses that do not
contribute to the overall echo density. The advantage of the de-
caying VNS is that it can be optimized to have a practically flat
magnitude response. Unfortunately, towards its tail, the sample
values get smaller, so some of them may not contribute to the in-
creased echo density, as they may become inaudible at the output
signal. It should be noted that, on modern hardware, some of these
operations can be performed in parallel, which will improve the
performance.
The optimized sequences used in this paper are the optimized
VN sequences from [31]. They are 30-ms long with 15 or 30 im-
pulses each and are denoted OVN15 and OVN30, respectively.
The non-decaying sequences are 10-ms long with 15 impulses
each and are denoted VN15. Initially, 30-ms-long non-decaying
sequences were also considered, but they were found to cause tran-
sient smearing. Also, the improvement in the echo-density growth
is superior with the 10-ms sequences.
4. RESULTS
The main motivation behind this research was to improve the echo
density build-up of a conventional FDN. To quantify the resulting
improvements, we compared the normalized echo density of two
conventional FDNs, the proposed VFDN structures, and another
recent extended FDN structure. The spectral coloration and imple-
mentation costs are also discussed. Audio examples are available
at http://research.spa.aalto.fi/publications/papers/dafx20-vfdn/ us-
ing the web-audio player from [35].
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0
1
VN1
0 100 200 300 400 500
Time [samples]
-1
0
1
VN2
0 100 200 300 400 500
Time [samples]
-2
-1
0
1
2
VN1 VN2
0 100 200 300 400 500 600 700 800 900
Time [samples]
Figure 5: Two velvet-noise sequences (VN1, VN2) and their convolution.
0 100 200 300 400 500 600 700
Time [ms]
0
0.2
0.4
0.6
0.8
1
Normalized Echo Density
FDN16
VFDN single (in)
VFDN single (out)
VFDN both
Figure 6: Growth of the normalized echo density over time with a
conventional FDN and three optional VFDN configurations.
4.1. Improvements in Echo Density
FDN structures with 32 and 16 delay lines, denoted as FDN32 and
FDN16, respectively, were used as the target and baseline methods
in this study. Fig. 7 shows the normalized echo density of the
FDN32 and FDN16. FDN32 is considered to produce a sufficiently
dense impulse response, whereas FDN16 has an impulse response
that is slightly too sparse and would benefit from improvement.
First, the target FDN32 was created with prime-number delay-line
lengths (839, 881, 929, 971, 1013, 1049, 1091, 1123, 1181, 1223,
1277, 1301, 1361, 1423, 1451, 1487, 1531, 1571, 1609, 1657,
1699, 1747, 1789, 1861, 1889, 1949, 1997, 2029, 2083, 2129,
2161, and 2237).
The delay-line lengths of FDN16 were then computed by sum-
ming each of the two consecutive delay-line lengths of FDN32 and
rounding them to the closest prime number length (1721, 1901,
2063, 2213, 2399, 2579, 2789, 2939, 3109, 3271, 3449, 3643,
3833, 4027, 4211, and 4397). This allowed keeping all the delay-
line lengths prime, while having the total delay length in both con-
figurations close to equal. Retaining the same total delay length
0 100 200 300 400 500 600 700
Time [ms]
0
0.2
0.4
0.6
0.8
1
Normalized Echo Density
FDN32
FDN16
DFM
Proposed, OVN15
Proposed, OVN30
Proposed, VN15
Figure 7: Normalized echo density of two conventional FDNs, a
DFM-FDN [24], and the three proposed VFDN structures. The
OVN30 configuration has VNSs only at the output, and the OVN15
and VN15 configurations have VNSs both at the input and the out-
put.
between different configurations ensures the modal density is not
lowered, which can result in poor sound quality. The feedback
matrices are random orthogonal matrices.
All the proposed structures in Fig. 7 have the same delay lines
and feedback matrix as the FDN16. The delay feedback matrix
(DFM) corresponds to a recently proposed FDN structure having
delay lines in its feedback matrix [24]. Fig. 7 shows the growth
of the echo density of the VFDN16 with the short VN15 sequences
is even faster than that of the target FDN32. When using the opti-
mized OVN15 instead, the resulting echo-density growth still sur-
passes the DFM structure and gets close to the target FDN32. As
expected, the OVN30 configuration with VNS filters only on the
output, introduces fewer echoes.
As mentioned earlier, since the decaying attenuation in an OVN
sequence can preserve the transients better, the length used for the
VN and OVN filters are 10 and 30 ms, respectively. Additionally,
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Frequency [Hz]
0
0.5
1
1.5
STD Power [dB]
(c)
50 100 250 500 1k 2k 3k 5k 10k
Frequency [Hz]
-1.5
-1
-0.5
0
0.5
1
1.5
Power[dB]
(b)
50 100 250 500 1k 2k 3k 5k 10k
Frequency [Hz]
-4.5
-3
-1.5
0
1.5
3
4.5
Power [dB]
(a)
OVN15
VN15
OVN30
Figure 8: (a) Example power spectra, (b) mean, and (c) standard
deviation of the smoothed power spectra for 500 instances.
we have experimented with a Schroeder allpass (SAP) filter struc-
ture with seven filters in series at each delay output [28]. The de-
lay lengths of the SAP filter are 630, 555, 442, 209, 140, 64, and 1
samples, and the coefficient is 0.7 for all allpass filters. The benifit
in the echo density was comparable to a VFDN structure with the
same computational cost. However, for transient sounds the se-
ries SAP smears the signal, as is evident from the provided online
sound examples. These results demonstrate that the proposed idea
of inserting VNS filters at the input and output branches of the
baseline method FDN16 substantially improves its echo density.
4.2. Spectral Coloration
Although the VNS filters improve the echo density, they also in-
troduce some coloration [30, 31], which appears in the response of
the reverberator. The amount of coloration depends on the partic-
ular sequences used, but some types of VNS filters introduce more
coloration than others [30, 31].
The spectral coloration introduced by the VNS filters placed
at the inputs and outputs of the FDN can be analyzed using the
following method. The transfer function in (10) can be rewritten
as
H(z) = 1⊤diag(c(z))P(z) diag(b(z))1,(13)
where P(z) = Dm(z)−1−A−1denotes the loop transfer
Table 1: Computational costs of different FDN and VFDN config-
urations.
Configuration ADD MUL Total Saving
FDN32 1280 1440 2720 Reference
FDN16 384 464 848 69%
DFM [24] 384 4464 848 69%
OVN15 864 912 1776 35%
OVN30 at outputs 864 912 1776 35%
VN15 864 432 1296 52%
function, or more succinctly
H(z) = 1⊤(P(z)◦Γ(z))1,(14)
where Γ(z) = c(z)b(z)⊤is a frequency-dependent gain matrix
and ◦denotes the Hadamard product, i.e., the element-wise mul-
tiplication of two matrices. The spectral coloration E(ω)is the
frequency-dependent energy ratio of the FDN with and without
the gain matrix Γ(z). If the matrix entries in P(z)are uncorre-
lated and of the same energy, then only the input and output filters
determine the spectral coloration, i.e.,
E(ω) = ∥Γ(eıω )◦P(eıω )∥2
F
∥P(eıω)∥2
F
=∥c(eıω)∥2∥b(eıω )∥2
N2.(15)
More details on this derivation are given in Appendix 7.1. Al-
though the loop transfer function P(z)is not entirely uncorrelated,
for the FDN configuration tested, the equation above yields an ac-
curate estimate of the spectral coloration with the mean broadband
error being 0.02 dB.
To compare the impact of the different variants of the pro-
posed method on the spectrum, we computed the spectral col-
oration E(ω)of 500 configurations for each of the proposed VNS
type. One configuration includes 32 sequences for VN15 and OVN15,
having 16-input and 16-output filters, and 16 sequences for OVN30,
which consist only of the output filters. Each of the VN15 se-
quences are random, whereas the optimized OVN15 and OVN30
sequences were picked randomly out of a set of 500 pregenerated
sequences. Fig. 8(a) shows example spectral coloration plots of
a single instance of each proposed VNS configuration. Fig. 8(b)
and (c) show the mean, and standard deviation (STD) of the spec-
tral coloration of the 500 instances, respectively.
Of the tested options, OVN30 sequences placed only at the
delay-line outputs introduce the least amount of coloration to the
system, followed by the OVN15 sequences placed at both input
and output branches. The random VN15 sequences are the most
coloring, but still their STD value remains less than 1.5 dB, as seen
in Fig. 8(c). However, a formal perceptual study is necessary to
determine whether this deviation is problematic in a practical set-
ting, in which frequency-dependent attenuation is applied. Fur-
thermore, since the spectral deviation is constant, it is possible to
introduce a set of equalization filters at the final outputs to com-
pensate for the coloration.
4.3. Computational Cost
Table 1 shows the number of operations per output sample for the
configurations presented in Fig. 7. These numbers are computed
for a practical setup using fourth-order attenuation filters consist-
ing of a second-order low-shelf, a second-order high-shelf, and
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gain for middle frequencies. The FDN32 structure is used as the
reference method to calculate the amount of savings, since it is
the target we used for the echo density measure. A comparison
between the VFDN16 structures implemented using OVN15 and
VN15 highlights the added cost of using decaying VNSs, which
shows in the number of multiplications in Table 1. The proposed
configuration using OVN15 and VN15 save 35% and 52% of op-
erations, respectively, so they both still use considerably less oper-
ations than the reference method FDN32.
Although counting additions and multiplications separately may
not be relevant on modern hardware, Table 1 follows the typical
method of presenting the computational cost and from which also
the Multiply and Accumulate (MAC) operations can be derived.
Here, we used a dense random orthogonal matrix in the FDN re-
circulation path. However, computational cost can be saved by
using special matrices, such as the Hadamard matrix [36], which
would reduce the computational benefit of using VN.
5. CONCLUSION
This paper proposes inserting velvet-noise filters at the input and
output branches of an FDN to increase its echo density during the
beginning of the impulse response. The sparseness of the impulse
response is a known limitation of the FDN. This work shows that
with the proposed VFDN an even faster growth in the echo density
can be obtain than with the doubling of the number of delay lines
in a conventional FDN. The short velvet-noise filters lead to com-
putational savings, as they can be convolved very efficiently with
a digital signal.
Various configurations for the VFDN are proposed in this work.
A configuration with VNS filters both at the input and at the output
of the FDN was shown to be a particularly effective solution, since
the echo density increases more than when having VNS sequences
with the same number of impulses only at the input or output of
the FDN.
Non-decaying and decaying VNS filters were compared in the
VFDN. The non-decaying sequences were found to help the echo
density of the VFDN grow more rapidly than the decaying ones.
However, the non-decaying VNSs cause more coloration than non-
decaying sequences optimized to have a flat spectrum. Equaliza-
tion of the resulting response of the VFDN to compensate for the
coloration is left for future work.
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7. APPENDIX
7.1. Spectral Deviation
Let w1(n)and w2(n)be two Gaussian noise sequences of length
L. Then, the energy of wj(n)is given by
∥wj(n)∥2
2=
L
X
n=0
|wj(n)|2.(16)
If w1(n)and w2(n)are uncorrelated and have each a normalized
energy of 1, then the energy of the scaled sum is
∥γ1w1(n) + γ2w2(n)∥2
2=|γ1|2+|γ2|2,(17)
where γ1and γ2are two scalar gains. This relation can be readily
extended to a summation of an N×Nmatrix of uncorrelated and
normalized noise sequences W(n)and a matrix of scalar gains Γ,
i.e.,





N
X
i,j=0
Γij Wij (n)




2
2
=
N
X
i,j=0
|Γij |2,(18)
or more succinctly
∥Γ◦W(n)∥2
F=∥Γ∥2
F,(19)
where ∥·∥Fdenotes the Frobenius norm. Now, for Γ=cb⊤,
which is a rank-1 matrix, the Frobenius norm is expressed as
∥Γ∥2
F=∥c∥2∥b∥2.(20)
The spectral coloration is the energy ratio between summation
with and without gain matrix Γ, i.e.,
E=∥Γ◦W(n)∥2
F
∥W(n)∥2
F
=∥c∥2∥b∥2
N2.(21)
Analogously, the same considerations can be made for individual
frequency bands, and a frequency-dependent spectral coloration
from a frequency-dependent gain matrix can be derived.
7.2. Serialized Structure
If the gains cand bare replaced with VNS filters v1(z)and v2(z),
respectively, we note in Sec. 4.2 that the gain matrix in (14) be-
comes
Γ(z) = v1(z)v2(z)⊤,(22)
and the resulting spectral coloration is given in (15). However, if
we connect the filters v1(z)and v2(z)in series at the input side of
the FDN, the resulting gain matrix can be written as
Γ(z) = 1[v1(z)◦v2(z)]⊤,(23)
and, similarly, if the series connection of filters v1(z)and v2(z)
are placed at the output side, the gain matrix becomes
Γ(z) = [v1(z)◦v2(z)]1⊤.(24)
The resulting spectral coloration for the gain matrices in (23) and
(24) is then
E(ω) = ∥1∥2∥v1(eıω )◦v2(eıω )∥2
N2.(25)
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