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Proceedings of the 26th International Conference on Digital Audio Effects (DAFx23), Copenhagen, Denmark, 4 - 7 September 2023
HOW SMOOTH DO YOU THINK I AM: AN ANALYSIS ON THE
FREQUENCY-DEPENDENT TEMPORAL ROUGHNESS OF VELVET-NOISE
Jade Roberts1,2, Jon Fagerström1, Sebastian J. Schlecht1,2and Vesa Välimäki1
1Acoustics Lab, Department of Information and Communications Engineering
2Media Lab, Department of Art and Media
Aalto University
Espoo, Finland
jade.roberts@aalto.fi
ABSTRACT
Velvet noise is a sparse pseudo-random signal, with applications in
late reverberation modeling, decorrelation, speech generation, and
extending signals. The temporal roughness of broadband velvet
noise has been studied earlier. However, the frequency-dependency
of the temporal roughness has little previous research. This pa-
per explores which combinative qualities such as pulse density,
filter type, and filter shape contribute to frequency-dependent tem-
poral roughness. An adaptive perceptual test was conducted to
find minimal densities of smooth noise at octave bands as well as
corresponding lowpass bands. The results showed that the cut-
off frequency of a lowpass filter as well as the center frequency
of an octave filter is correlated with the perceived minimal den-
sity of smooth noise. When the lowpass filter with the lowest
cutoff frequency, 125 Hz, was applied, the filtered velvet noise
sounded smooth at an average of 725 pulses/s and an average of
401 pulses/s for octave filtered noise at a center frequency of 125
Hz. For the broadband velvet noise, the minimal density of smooth-
ness was found to be at an average of 1554 pulses/s. The results of
this paper are applicable in designing velvet-noise-based artificial
reverberation with minimal pulse density.
1. INTRODUCTION
Velvet noise is a sparse pseudo-random noise sequence, which
consists of ternary values (−1,0, and 1) [1] and has a constant
power spectrum [2]. Velvet noise was originally proposed by Kar-
jalainen and Järveläinen to model room reverberation [1]. It is
known that late reverberation resembles exponentially-decaying,
filtered white noise [3, 4, 5]. Broadband velvet noise has been
shown to retain its perceived smoothness with lower pulse densi-
ties in comparison to other types of sparse noise sequences [6]. At
2000 impulses per second, velvet noise has been shown to sound
smoother than Gaussian white noise (GWN) [6].
The perceived temporal smoothness of velvet noise has been
investigated mostly on broadband noise sequences [6]. Kar-
jalainen and Järveläinen [1] made an initial study on the frequency-
dependency of the temporal roughness, where lowpass filtered
velvet-noise, with a cutoff frequency of 1.5kHz, was shown to
sound smoother than GWN with a pulse density of 600 pulses/s.
This paper investigates further the frequency-dependent psychoa-
coustic temporal roughness of velvet-noise sequences. Having a
Copyright: © 2023 Jade Roberts et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution 4.0 International License, which
permits unrestricted use, distribution, adaptation, and reproduction in any medium,
provided the original author and source are credited.
clear understanding of the frequency-dependency of the temporal
roughness can help in the design and optimization of reverberation
models based on sparse noise sequences.
The earliest sparse-noise-based reverberation algorithm was
proposed by Rubak and Johansen [7, 8]. Their algorithm is based
on totally random noise (TRN), which is a type of sparse pseudo-
random noise with an equal probability of any sample having a
non-zero value. The proposed minimal pulse density for produc-
ing high-quality noise with the TRN was reported to be between
2000 −4200 pulses/s, however for a lowpass filtered noise with
cutoff at 8kHz.
Rubak and Johansen also proposed a recursive structure for
computational efficiency [7], which was further improved by Kar-
jalainen and Järveläinen [1] by replacing the TRN with velvet
noise and by introducing time-variation. The time variation was
introduced to reduce the periodicity of repeating the same short
velvet-noise sequence inside the recursive structure. A further
problem arises from the time-variability which creates warbling
especially on stationary input sounds [1, 9, 10]. An alternative so-
lution to mitigate the periodicity problem was proposed in [10],
where interleaved velvet-noise sequences hide the repetitiveness.
A different approach to reverberation modeling was taken in
[4, 5], where filtered velvet noise segments were concatenated to
model target late-reverberation. It was shown empirically that
lower pulse density could be used towards the end of the model
response, where the bandwidth was reduced. Recently, it was also
shown that colored velvet noise can be generated directly by con-
trolling the pulse location distribution [11, 2]. A practical algo-
rithm for generating velvet noise with a lowpass spectrum, called
dark velvet noise (DVN), was later proposed in [12]. The cut-
off of DVN can be varied in time to generate characteristic late-
reverberation, where the low frequencies decay slower than the
high frequencies.
Another application for velvet-noise is to implement an effi-
cient decorrelator [13, 14]. An optimization scheme for minimiz-
ing the spectral coloration introduced by the velvet-noise decorre-
lator was proposed in [14]. Velvet noise has been also used in hy-
brid reverb structures combining it with feedback delay networks
(FDN) [15, 16]. Short velvet-noise filters are applied either within
the feedback matrix [16] or at the inputs and outputs of the FDN
[15]. Additionally, velvet-noise has been used in vocoder-based
speech generation by serving as excitation signals [17].
In this paper, the perceptual temporal roughness of velvet
noise at octave bands as well as at lowpass bandwidths with the
octave band center frequencies as the cutoffs are studied in a per-
ceptual test. Additionally, the time-domain smearing of various
filter orders is investigated objectively to narrow down the filter
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0 0.002 0.004 0.006 0.008 0.01
Time (s)
Unfiltered 2 4 6 8
Figure 1: Velvet-noise sequence, filtered with an octave-band filter
at the center frequency Fc= 1.5kHz, using different filter orders
N∈ {2,4,6,8}from top to bottom. The consecutive plots are
offset for better visualization. The grid size Td= 0.5ms is shown
with a dotted line.
selection for the listening test. An adaptive perceptual test was
implemented in Matlab, where the subjects control the density of
test velvet noise signals. The test aims to find perceptual density
thresholds where each test signal still sounds as smooth as a refer-
ence velvet noise signal with the pulse density of 2000 pulses/s.
The rest of this paper is organized as follows. Section 2 gives
relevant background on velvet noise and temporal roughness. Sec-
tion 3 describes the design of the filters for the listening test. Sec-
tion 4 introduces the listening test setup. The results of the listen-
ing test are analyzed in Section 5. Section 6 concludes the paper
and comments the future work.
2. BACKGROUND
This section explains the concepts behind velvet noise and tempo-
ral roughness.
2.1. Velvet Noise
Velvet noise is a sparse random noise sequence comprised of only
sample values of −1, 0, and 1. Each frame contains a single ran-
domly placed impulse with randomized sign; the rest are zeros,
leading to a sparser noise than GWN. The number of nonzero im-
pulses per second is defined as the pulse density ρ, i.e.,
ρ=fs
Td
,(1)
where Tdrefers to the average distance between impulses mea-
sured in samples and fsrefers to the sampling rate. In this work, a
sampling rate of fs= 44100 Hz is used for all generated signals.
Karjalainen and Järveläinen [1] found that a pulse density of
ρ= 1500 pulses/s satisfied the aim of minimal pulse density and
maximal smoothness. This was the sweet spot for a perceived
smoother noise than GWN. To prevent gaps and clusters of sample
values which contribute to the perception of temporal roughness,
the impulse locations are determined as [6]
k(m) = ⌊mTd+r1(m)(Td−1)⌉,(2)
where mrefers to the pulse counter while r1(m)refers to the se-
quence of uniformly distributed random values between 0 and 1,
and ⌊·⌉ is the rounding operation. The velvet-noise sequence is
computed with
s(n) = (2⌊r2(m)⌉ − 1,when n=k(m)
0,otherwise,(3)
where nis the sample index and r2(m)is another uniform random
number sequence to decide when an impulse will be 1 or −1. In
Fig. 1, the black line at the top shows a broadband velvet noise
sequence. Here, the impulses of 1 or −1 appear only once in each
frame.
Additionally, velvet noise is featureless with a flat power spec-
trum [6]. The computing time of a convolution with velvet noise is
much shorter than that of GWN because it mainly consists of zeros
and where there are impulses in velvet noise, the ones and minus
ones are easy to multiply by [4, 5].
2.2. Temporal Roughness of Sparse Noise
Temporal roughness is a psychoacoustic quality of sparse noise se-
quences where the lower the pulse density the rougher the signal is
perceived [1, 6]. Temporal roughness has not been fully defined in
the literature but it might be directly related to the roughness which
is defined as the sensation caused by amplitude-modulated sine
waves with modulation frequencies between 15 −300 Hz range.
The sensation reaches its maximum around modulation frequency
of 70 Hz [18].
Rubak and Johansen [7] observed that when the delay of a
comb filter is increased above 25 ms one starts to perceive rough-
ness in the sound and the perception changes from the coloration to
the time-domain character. The value 25 ms corresponds to a fre-
quency of 40 Hz for the pulses of the comb filter, which falls close
to the modulation frequency causing maximal roughness sensa-
tion. Furthermore, it is reported that the modulation signal does
not have to be periodic to cause the perception of roughness [18]
The random assignment of pulses in sparse noise sequence can be
interpreted as pseudo-random amplitude modulation [19].
Velvet noise has the ability to sound smoother than GWN,
leading to the question of at which pulse densities is velvet-noise
perceived as smooth versus perceived as temporally rough. In pre-
vious studies on sparse noise, optimal pulse densities for still main-
taining perceived smoothness were researched in multiple ways
such as first-order lowpass filtering reverberation tails and using
totally random noise [7, 8] as well as using velvet-noise [1] which
gave results of 2000 −4200 pulses/s [7, 8] and 1500 pulses/s [1]
as optimal smoothness, respectively.
3. FILTER DESIGN
In order to investigate the frequency-dependent temporal rough-
ness of velvet noise, filtering is to be applied to the white velvet-
noise sequences. In this research, both lowpass and octave-band
filtering is applied. In this section, the properties of various filter
orders are investigated, to narrow down the filter parameters for
the perceptual test.
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0 0.002 0.004 0.006 0.008 0.01
Time (s)
Unfiltered 2 4 6 8
Figure 2: Velvet-noise sequence, filtered with a Butterworth low-
pass filter with the cutoff frequency Fc= 1.5kHz, using different
filter orders N∈ {2,4,6,8}, cf. Fig. 1.
125 250 500 1k 2k 4k 8k 16k
Frequency (Hz)
0
50
100
Effective length (ms)
2
4
6
8
Filter order
(a)
125 250 500 1k 2k 4k 8k 16k
Frequency (Hz)
0
50
100
Effective length (ms)
2
4
6
8
Filter order
(b)
Figure 3: Effective length of the (a) octave-band filters
and (b) lowpass filters at the center frequencies Fc∈
{125,250,500,1k, 2k, 4k, 8k, 16k}Hz, using different filter or-
ders N∈ {2,4,6,8}.
3.1. Time-domain Smearing
Time-domain smearing refers to the energy of a signal being
spread across a longer period of time during playback which also
means a loss of detail in the signal itself. Fig. 1 shows an ex-
ample velvet-noise sequence (black) and its filtered versions (col-
ored) with various filter orders of an octave band filter centered
at Fc= 1.5kHz. As the order of the filter increases the time
response gets more and more smeared in time. With the second-
order octave filter (blue) in Fig. 1, the pulse locations are still
visible. A similar trend is shown in Fig. 2, which shows the
velvet-noise sequence filtered with a lowpass Butterworth filter
with various filter orders. The second-order lowpass filter shows
more smearing than the second-order octave filter. This is why the
fourth-order octave filters were used to compare against the low-
pass filters.
Fig. 3a and Fig. 3b show the effective length of the octave-
band filter and the Butterworth lowpass filter, respectively. Again,
various filter orders are compared. The effective length is com-
puted with Matlab function impzlength with a tolerance of
−60 dB. Longer effective lengths of the filter will result in more
time-domain smearing. The overall trend in Fig. 3 is that higher
filter order and lower center frequency or cutoff frequency result
in longer effective length. Furthermore, the difference in effective
length between the lower and higher center frequencies grows with
the filter order. Thus, for the listening test design we opted to use
the second-order lowpass filters which introduce minimal smear-
ing. For the octave filters order four was used, since the steepness
of the second-order lowpass is most similar to that of a fourth-order
octave filter.
3.2. Listening Test Filters
Two types of filters were used in the final listening test: second-
order Butterworth lowpass filters and fourth-order octave-band fil-
ters. A lowpass filter attenuates frequencies above a specified cut-
off frequency and the frequencies below the cutoff are retained.
The Butterworth lowpass filter [20] is often used in audio be-
cause of its maximally flat magnitude response in the passband
and monotonic roll-off in the stopband. The magnitude responses
of the used octave filters and lowpass filters are shown in Fig. 4a
and Fig. 4b, respectively. Note: There is a slight shift of the low-
pass passband versus the passband of the octave filters.
As for octave filters, an octave means an interval where there
is a frequency ratio of 2:1, the upper frequency is twice the lower
frequency. This is important when using the filter which consists
of bandpass filters, the bandwidth of each filter will always be a 2:1
ratio. There are ten octave bands within the human hearing range.
Octave filters are used because they can be utilized for measuring
noise power at certain frequency ranges. They also give insight
into the human hearing of temporal roughness which was the aim
of this study. The magnitude response is shown in Fig. 4a. The
center and cutoff frequencies were calculated using base 10. The
nominal center frequencies of the octave filters used in the test are
shown in Table 1 and is also shown by the peaks/centers of the
octaves filter magnitude responses in Fig. 4a.
4. LISTENING TEST DESIGN
To test the perception of smoothness in velvet noise, a listening test
was employed using Matlab App. As shown in Fig. 5 of the test
user interface, there is a reference signal and a test signal. The ref-
erence signal was broadband velvet noise of 2000 pulses/s which
has been shown to be perceived as smooth [1]. The participant
was asked to find the lowest possible pulse density in which the
test signal sounded as smooth as the reference. There was not a
specific practice page, but testers were given a brief introduction
to the sliders and signals where they could ask questions about the
task. In this opportunity, they were able to familiarize themselves
with the loudness level of the signals, the buttons and the slider
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(a)
(b)
Figure 4: (a) Fourth-order octave and (b) second-order Butter-
worth lowpass filter magnitude responses with center/cutoff fre-
quencies one octave apart between 125 Hz and 16,000 Hz.
function. The test required the participant to move the slider ac-
cordingly to match their perception of where the velvet noise sig-
nal is still smooth, but on the edge of being rough. The movement
of the slider changed the pulse density of the velvet noise signal
Table 1: Nominal center frequencies used to test the effect of oc-
tave and lowpass filters. For the lowpass filters, the center fre-
quency was used as the cutoff frequency. Note: The lowest 2 oc-
taves of the audio range are omitted, only octaves 3 to 10 were
used in testing.
Octave band Nominal center frequency (Hz)
3 125
4 250
5 500
6 1000
7 2000
8 4000
9 8000
10 16000
Figure 5: User interface of the listening test. The participant
chooses the lowest density where the test sound is still as smooth
as the reference. The user can choose to play and stop the refer-
ence sound and the test sound. The participant can also go back
to the previous or go to the next page using the respective buttons.
and automatically played the test sound set to the chosen pulse
density. Each page’s velvet noise condition contained all possible
pulse densities between 50 and 2000 and the reference remained at
2000 pulses per second for each test page. A pulse density of 2000
was chosen for the reference as previous papers on sparse and vel-
vet noise respectively found already 1500 and 2000 as acceptably
smooth pulse densities [1, 6].
The test signals presented were either unfiltered, filtered
through an octave filter or through a lowpass Butterworth filter.
Sound examples of the test signals are available on the compan-
ion web page of this paper1The filters used center frequencies or
cutoff frequencies calculated from octave bands 3-10, shown in
Table 1. The listening test was composed of two linked MATLAB
apps, the first assessed octave filtering, while the second assessed
lowpass filtered and broadband velvet noise. The center frequency
change in the filters between pages was randomized. The lowest
two center frequency bands were omitted due to being too low for
users to perceive, and, for this reason, are not included in Table 1
either.
The loudness level for each signal was set based on the EBU R
128 standard at -23 LUFS (Loudness Unit Full Scale). LUFS is a
loudness measurement based on the human perception of loudness.
This was used over the counterpart of RMS (root-mean-square)
power which is the average power of a signal without any weight-
ing. Additionally, the use of LUFS was due to the frequency-
dependent nature of loudness LUFS can account for. Normalizing
the loudness between test signals was to ensure that rating was not
influenced by how loud the signal sounded such as the perception
that high frequencies are louder compared to low frequencies.
1http://research.spa.aalto.fi/publications/
papers/dafx23-vn- roughness
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Figure 6: Violin plot of minimal pulse density for smooth-sounding
lowpass filtered velvet noise.
The final listening test sound pressure level of the isolated
listening booths used for testing was calibrated to 60 dB [6] us-
ing a RA0045 G.R.A.S Ear Simulator. The simulator followed
IEC 60318-4 regulations. Participants listened to the signals using
Sennheiser HD-650 headphones.
There was a total of 34 listening test pages with 16 pages
testing octave-filtered velvet noise, 16 testing pages for lowpass-
filtered velvet noise, and two pages testing unfiltered velvet noise.
This was done so that each signal of same conditions: filter type
and center frequency or cutoff frequency, occurred twice during
the test to compare individual differences and assess the reliability
of the participants’ ratings.
In total, there were 12 participants between ages 19 and 42
who completed the octave filtered velvet noise tests and 18 partic-
ipants between ages 19 and 42 who completed the lowpass filtered
velvet noise test all with previous experience in a formal listen-
ing test. For each test signal, there were two pages, and a corre-
lation was calculated for each participant between the two pages
for each center frequency/cutoff frequency. This was done using
corrcoef in MATLAB. If the participant’s mean correlation co-
efficient was below 0.5, meaning their answers were too different
for the same stimuli to be considered, their data was discarded.
Fortunately, no participants had a correlation coefficient under 0.8
and therefore, all participant data was assessed.
5. RESULTS
The results in Fig. 6 and Fig. 7 show the rated pulse density in the
y-axis against the center frequencies of the 8 octave bands used
in this study on the x-axis. The rated pulse density values come
from the participant’s task of setting the slider to where they per-
ceive the lowest possible density where it still sounds as smooth
as the reference signal. In both violin plots [21], the central white
dot refers to the median of the data within the specific octave band
center frequency. The means are indicated by a horizontal line
in each violin. The median and mean values can be found in Ta-
bles 2 and 3. These values were calculated using an average of
Figure 7: Minimal pulse density for smooth-sounding octave fil-
tered velvet noise.
each participant’s answers for each filter band and filter type re-
spectively.The top and bottom of the thick grey line in the cen-
ter of each plot refers to the first and third quartiles. Both plots
seem to follow somewhat of a curve showing that there is some
frequency-dependence on the perception of smoothness. Espe-
cially the octave-filtered velvet noise was rated with a consistently
Table 2: Nominal cutoff frequencies used for lowpass cutoffs and
their corresponding median and mean results.
Cutoff frequency (Hz) Median (pulses/s) Mean (pulses/s)
125 668 725
250 912 896
500 1155 1120
1000 1242 1260
2000 1415 1350
4000 1487 1455
8000 1541 1509
16000 1571 1548
Broadband 1587 1554
Table 3: Nominal center frequencies used for octave filters and
their corresponding median and mean results.
Center frequency (Hz) Median (pulses/s) Mean (pulses/s)
125 290 401
250 368 433
500 447 557
1000 756 794
2000 1038 1035
4000 1107 1183
8000 1240 1212
16000 1315 1299
Broadband 1587 1554
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lower pulse density than for the low pass filtered velvet noise.
Fig. 6 demonstrates that lower cutoff frequencies allowed the
pulse density of the velvet noise to be lower than that of broadband
and octave-filtered signals. The signals were rated at nearly half
the pulse density of broadband in the lowest octave band center
frequency cutoff of 125 Hz, whereas in the highest human hearing
octave band frequency cutoff, there was little difference.
We found the broadband velvet noise had a median of 1587
pulses/s and a mean of 1554 pulses/s where the signal still sounded
as smooth as the reference signal, as shown in Fig. 6, which is
similar to results in the original velvet noise study [1], where 1500
pulses/s was found to be an optimal smoothness. The filter shapes
of the second-order Butterworth lowpass filters and the fourth-
order octave filters were similar, however, the octave-filtered vel-
vet noise had much lower ratings.
A repeated measures two-way analysis of variance (ANOVA)
test with factors of center or cutoff frequency and repetition was
used to assess the statistical significance of the octave-filtered vel-
vet noise and the lowpass-filtered velvet noise separately. the sec-
ond factor of repetition was tested to determine if there were sig-
nificant differences in how a participant rated the same stimuli.
For both filtered velvet noises, the dependent variable was pulse
density. ANOVA preconditions such that the data is normally dis-
tributed, the dependent variable of rated pulse density is on a con-
tinuous scale and the sphericity of the two trials per participant
showing equal amounts of variance were met. More specifically, a
Kolmogorov–Smirnov test was conducted on the rated pulse den-
sities for each center and cutoff frequency at a 1% significance
level which found the data to be normally distributed for both the
lowpass filtered data and the octave filtered data separately. A
Mauchly sphericity test was also conducted which found that for
each participant, the density ratings across the center or cutoff fre-
quencies were normally distributed.
For the octave-filtered noise, the repeated measures two-way
analysis was chosen to show the effect of center-frequencies and
repeating trials on participant pulse density rating. The two-
way ANOVA test showed that center-frequencies was significant
and repeated trials were not significant on pulse density with F-
statistics of F(9,99) = 79.801, p < .001 and F(1,11) =
.995, p > .001, respectively. The interaction between frequency
and repetition was also not significant with an F0-statistic of
F(9,99) = 1.609, p > .001.
For the lowpass-filtered noise, the two-way analysis was cho-
sen to show the effect of cutoff frequency and repetition of condi-
tions on pulse density ratings. There was no significant effect from
repetition, F(1,16) = .278, p > .001. or from the interaction be-
tween repetition and frequency, F(9,144) = 1.475,p > .001.
The two-way repeated measures ANOVA testing found only sig-
nificant effect from cutoff frequencies on pulse density rating such
that F(9, 144) = 354.15, p < .001.
Additionally, a post-hoc paired t-test was conducted with a
Bonferroni-Holm correction on both the lowpass and octave fil-
tered noise tests which showed that between neighboring center
or cutoff frequencies, the pulse density ratings were statistically
significant (p<.01) except between 4 kHz and 8 kHz as well as
between 8 kHz and 16 kHz. This means that pulse densities above
4 kHz show no important differences in pulse density ratings.
6. CONCLUSIONS
We studied the frequency-dependency of temporal roughness of
velvet noise in human noise perception. The results showed that
second-order lowpass filtering and fourth-order octave filtering,
the shape of the filter, allows for temporal smearing meaning that
the resulting signal can still seem smooth at lower densities than
that of broadband velvet noise. Octave-filtered noise at the lowest
frequencies showed even lower pulse density ratings than lowpass
filtered velvet noise. This study showed that artificial reverbera-
tion modeling which utilizes filtering and velvet noise can employ
lower pulse densities than what is currently used which allows for
more efficient computation.
7. ACKNOWLEDGMENTS
This work has been funded in part by the Nordic Sound and Mu-
sic Computing Network, NordForsk project number 86892. The
authors would like to thank Dr. Hanna Järveläinen for her helpful
comments and would also like to thank Nils Meyer-Kahlen in his
help with ANOVA testing and clarifications.
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