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Freely available online PAPERS
J. Fagerstr¨
om, S. J. Schlecht, And V. V¨
alim¨
aki,
“Non-Exponential Reverberation Modeling Using Dark Velvet Noise,”
J. Audio Eng. Soc., vol. 72, no. 6, pp. 370–382 (2024 Jun.).
https://doi.org/10.17743/jaes.2022.0138.
Non-Exponential Reverberation Modeling Using
Dark Velvet Noise
JON FAGERSTR ¨
OM,1,∗
(jon.fagerstrom@aalto.fi)
SEBASTIAN J. SCHLECHT,1,2AES Member,
(sebastian.schlecht@aalto.fi)
AND VESA V ¨
ALIM ¨
AKI,1AES Fellow
(vesa.valimaki@aalto.fi)
1Acoustics Lab, Department of Information and Communications Engineering, Aalto University, Espoo, Finland
2Media Lab, Department of Art and Media, Aalto University, Espoo, Finland
Previous research on late-reverberation modeling has mainly focused on exponentially
decaying room impulse responses, whereas methods for accurately modeling non-exponential
reverberation remain challenging. This paper extends the previously proposed basic dark-
velvet-noise reverberation algorithm and proposes a parametrization scheme for modeling late
reverberation with arbitrary temporal energy decay. Each pulse in the velvet-noise sequence
is routed to a single dictionary filter that is selected from a set of filters based on weighted
probabilities. The probabilities control the spectral evolution of the late-reverberation model
and are optimized to fit a target impulse response via non-negative least-squares optimization.
In this way, the frequency-dependent energy decay of a target late-reverberation impulse
response can be fitted with mean and maximum reverberation-time errors of 4% and 8%,
respectively, requiring about 50% less coloration filters than a previously proposed filtered-
velvet-noise algorithm. Furthermore, the extended dark-velvet-noise reverberation algorithm
allows the modeled impulse response to be gated, the frequency-dependent reverberation time
to be modified, and the model’s spectral evolution and broadband decay to be decoupled. The
proposed method is suitable for the parametric late-reverberation synthesis of various acoustic
environments, especially spaces that exhibit a non-exponential energy decay, motivating its
use in musical audio and virtual reality.
0 INTRODUCTION
Artificial reverberation algorithms have been developed
since the 1960s, starting with Schroeder’s original algo-
rithm [1, 2]. Schroeder’s algorithm, as well as many that
followed, are based on the assumption that the late reverber-
ation part of a room impulse response (IR) can be modeled
with exponentially decaying filtered white noise [2−5].
However, non-exponentially decaying reverberation can be
observed in forests [6, 7], coupled rooms [8−10], and the
famous gated reverb sound from the 1980s [11].
A room IR can be divided into three perceptually mo-
tivated parts, the direct sound, early reflections, and late
reverberation [12]. The study presented in this paper fo-
cuses on the modeling of the late-reverberation part. In this
paper, the authors propose a novel artificial reverberator
capable of modeling target late reverberation with arbitrary
energy decay and spectral evolution.
A notable branch of artificial reverberation algorithms is
based on pseudo-random noise. Rubak and Johansen pro-
*To whom correspondence should be addressed, email:
jon.fagerstrom@aalto.fi, Last updated: April 11, 2024
posed using sparse random noise to model exponentially de-
caying Gaussian noise [4, 13]. However, the resulting sparse
finite-impulse-response (FIR) filters placed inside a feed-
back loop would still need over 10,000 filter coefficients to
produce a smooth reverberation IR, i.e. reverberation IR that
does not sound rough. Later, Karjalainen and J¨
arvel¨
ainen
introduced velvet noise [5], which becomes smooth broad-
band noise with a pulse density of 1,500−2,000 pulses/s
[5, 14]. To reduce computational costs, several velvet-
noise−based algorithms employ a feedback structure, lim-
iting them to generating only exponential decay [5, 15, 16].
It is important to make a distinction between the psy-
choacoustic quantity roughness, measured in asper [17],
and the term “temporal roughness” used in this work to
describe the perceived quality of sparse noise. The former
is defined as the auditory sensation caused by amplitude-
modulated pure tones, with modulation frequencies within
the range of 15 to −300 Hz [17]. The latter is only loosely
defined in previous literature as the sensation when a sparse
noise sequence is not perceived as sounding smooth [5, 14].
As Meyer-Kahlen et al. [18] pointed out, the random as-
signment of the pulses of a sparse noise sequence can be
interpreted as pseudo-random amplitude modulation.
370 J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun.
PAPERS NON-EXPONENTIAL REVERBERATION MODELING
The feedback delay network (FDN) [19] is a generaliza-
tion of the comb-filter−based reverberator, which is still
actively studied today [20−23]. Hybrid reverberators com-
bining an FDN and velvet noise have also been proposed,
which place the velvet-noise filters at the inputs and out-
puts [24] or within the feedback matrix of an FDN [22] to
increase the echo density. However, in its basic form, the
FDN can produce only an exponential decay. Combining
two FDNs with different parameters allows for generating
various non-exponential attenuation patterns, such as fade-
in control or two-stage decay [25−27, 10]. An extended
method based on the FDN has been proposed to synthesize
double-slope decays of coupled rooms [23, 9]. However,
no FDN-based method is capable of synthesizing reverber-
ation, which has an arbitrary and non-exponential energy
decay.
Karjalainen and J¨
arvel¨
ainen proposed a modal reverber-
ator structure for modeling late reverberation [28], and the
idea was later refined by Abel et al. [29]. The modal re-
verberator is implemented with a parallel combination of
mode filters, whose resonant frequencies and damping co-
efficients are tuned to match those of the target space. The
number of modes to produce high-quality reverberation is
suggested to be between 1,000 and 2,000 modes, based on
informal listening. Wells recommends the use of a much
larger number of modes [30]. The modal reverberators are
best suited for exponentially decaying reverberation. How-
ever, implementing two-stage decays and fade-ins is also
possible by tuning the damping of a portion of the mode fil-
ters [26]. Recently, modal synthesis has been proposed for
the resynthesis of denoised anisotropic late reverberation
with multi-slope decays by Hold et al. [31]. The multi-
slope decays, however, are still a linear combination of
exponential decays.
Holm et al. introduced an FIR-filter–based algorithm
called the filtered-velvet-noise (FVN) reverberator [32] that
V¨
alim¨
aki et al. refined later [33]. The FVN models a target
late-reverberation IR with concatenated filtered-noise seg-
ments of different lengths. The filters are designed based
on the time-frequency analysis of a target IR. The variable-
length windowing mitigates audible transitions between the
consecutive filters. The FIR-based FVN structure allows
manipulating the IR by changing the lengths of the VN
block, e.g., lengthening or shortening the decay time [33].
In this paper, the authors’ previous work on the dark-
velvet-noise (DVN) reverberator [34] is extended to fit its
IR to a measured target late-reverberation IR. An extension
to the original DVN algorithm is introduced, which replaces
its recursive running-sum filters with arbitrary dictionary
filters. Furthermore, the uniform probability for a pulse to
be connected to a certain filter is set as a free parameter. The
proposed method fits the extended DVN model to a target
response via non-negative least-squares (NNLS) optimiza-
tion [35], which is applied for the first time to model rever-
beration. The resulting model is parametric and facilitates
various perceptually relevant modifications. Additionally,
the authors investigate the temporal roughness properties
of the proposed extended DVN and propose a scheme for
mitigating it.
The proposed extended DVN model IR is subjected to
an objective evaluation: First, the original target IRs are
compared in terms of spectro-temporal fit and reverbera-
tion time (T60) estimation in the case where the target IR
has an exponential energy decay, with IRs synthesized with
optimized extended DVN model instances. It is demon-
strated that the new method provides a good objective fit in
parametrizing two distinctly different target reverberation
IRs, those of a concert hall and outdoor space coupled to
a cave opening. Next, the proposed method is compared to
the previously proposed FVN method [33], and the FVN
method is formulated as a special case of the proposed
method.
The rest of this paper is organized as follows. SEC.1sum-
marizes the previously proposed DVN algorithm [34]. SEC.
2 proposes the novel extension of the DVN structure. SEC.
3 discusses the proposed reverberation modeling scheme,
step by step, including the NNLS optimization scheme.
SEC. 4 presents an objective performance evaluation when
using the proposed method for modeling a target IR and
discusses the flexibility of the extended DVN in generat-
ing parametric modifications of the modeled IR. SEC.5
concludes the paper.
1 DARK VELVET NOISE
This section provides the relevant background on the pre-
viously proposed DVN algorithm [34]. The basics of velvet
noise and dark velvet noise serve as the basis for developing
the proposed method for late-reverberation synthesis.
Original velvet noise [5] is a sparse pseudo-random
noise, which consists of sparsely placed unit impulses with
uniformly distributed signs. The main design parameter of
a velvet-noise sequence is its pulse density ρin pulses per
second. Based on the desired pulse density, the grid size Td
can be computed as
Td=fs
ρ,(1)
where fsis the sample rate in hertz. Within each grid seg-
ment, a single unit impulse occurs.
Whereas the original velvet noise has a white magni-
tude spectrum [36], DVN is an extension of velvet noise
that has a low-pass spectrum [34]. The low-pass spectrum
is achieved by a random modulation of the pulse width w
along the DVN sequence, i.e., the unit impulses of the orig-
inal velvet noise are replaced by square pulses of varying
width. In practice, the pulse-width modulation is imple-
mented using a discrete set of recursive running-sum filters,
one for each required pulse width [34]. The DVN sequence
is given as
h(n)=s(m)fork(m)≤n<k(m)+w(m),
0 otherwise,(2)
where nis the sample index, mis the pulse index, s(m)is
the sign of the mth pulse, and k(m) is the location of the
mth pulse. When w(m)≡1, Eq. (2) gives the original velvet
noise sequence where each pulse is a unit impulse.
J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun. 371
FAGERSTR ¨
OM ET AL. PAPERS
Fig. 1. (a) Example of a basic DVN sequence, and (b) its PSD
normalized to 0 dB. The dashed lines show the grid spacing mTd
at the sample rate fs=48 kHz.
The pulse locations of DVN are computed as
k(m)=mT
d+r1(m)(Td−w(m)),(3)
where ·is the rounding operator and r1(m) is a uniform
random number between 0 and 1. This formulation assures
that the pulses do not overlap each other, given that 1 ≤
w(m)≤Td.
The pulse widths of DVN (in samples) are computed as
w(m)=r2(m)(wmax −wmin )+wmin,(4)
where r2(m) is an uniform random number between 0 and
1, and wmax and wmin are the maximum and minimum pulse
widths, respectively. The sign of each pulse is computed by
[5]:
s(m)=2r3(m)−1,(5)
where r3(m) is a uniform random number between 0 and 1.
The first 8 ms (fs=48 kHz1) of an example basic DVN
sequence and the power spectral density (PSD) of the cor-
responding infinitely long DVN sequence are shown in
Figs. 1(a) and Fig. 1(b), respectively.
2 EXTENDED DARK VELVET NOISE
In this section, an extension to the previously proposed
DVN [34] is presented. The proposed extension replaces
the previously used recursive running-sum filters with ar-
bitrary dictionary filters, whose probabilities are set as a
free parameter. Additionally, the temporal roughness of the
proposed extension is discussed.
1The authors used the sample rate of fs=48 kHz throughout
this work.
Fig. 2. Structure of the proposed extended DVN convolution with
Qdictionary filters and Mpulses. Each pulse is multiplied by
againg(m) and routed to the input of one dictionary filter, as
indicated by the matrix. Each dot in the routing matrix represents
a connection path.
2.1 Generalization of Dark Velvet Noise
As the first step of the generalization, the recursive
running-sum filters are replaced with arbitrary dictionary
filters F(z), allowing the generation of colored noise of de-
sired spectral shape. Fig. 2 shows the block diagram of the
proposed extended DVN convolution structure with Qdic-
tionary filters F(z) and MQpulses. The decay-envelope
gains g(m) parametrize the broadband energy decay. Fi-
nally, in contrast to the basic DVN, the uniform probabili-
ties of each dictionary filter were set as a free parameter.
The filter probabilities for a single pulse are denoted by
the vector
p=p1,p2,..., pQT≥0,with
Q
q=1
pq=1,(6)
where [ ·]Tis the transpose operation.
A list of filter indices for each pulse is determined based
on the pulse-filter probabilities pas
φ(m)=f(p(m)) ∈{1,2, ..., Q},(7)
where f(·) is any function that selects the pulse filter based
on the probabilities p. The resolved list of filter indices φ
is visualized in matrix form in Fig. 2, where the one-hot
column vectors show the filter selection for each pulse.
Eq. (2) can be reformulated so that the velvet-noise se-
quence is split into Qsub-sequences. Each sub-sequence
includes the pulses routed to one of the dictionary filters,
corresponding to a single row of the connection matrix of
Fig. 2. The qth sub-sequence is then given as
vq(n)=s(m)g(m)forn=k(m)∧φ(m)=q,
0 otherwise.(8)
The transfer function of the extended DVN can now be
written as
H(z)=
Q
q=1
Vq(z)Fq(z),(9)
372 J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun.
PAPERS NON-EXPONENTIAL REVERBERATION MODELING
Fig. 3. (a) Extended DVN IR with the underlying pulse locations
and signs indicated by asterisks. (b) The corresponding PSD, com-
puted using Eq. (10) and normalized to 0 dB. The used dictionary
filters correspond to the ones presented in Fig. 4(a).
where Vq(z) is the velvet-noise subsequence routed to the
qth dictionary filter Fq(z). The time-dependent PSD of the
extended DVN sequence is described by the weighted mean
magnitude response of the dictionary filters:
|H(m,ω)|=
Q
q=1
Fq(ejω)pq(m),(10)
where Fq(ejω) is the magnitude response of the qth dictio-
nary filter, and the probabilities pqare defined in Eq. (6).
The PSD of the extended DVN sequence, Eq. (10), is inde-
pendent of the pulse density ρ, and the equation holds true
as the occurrences of dictionary filters are uncorrelated due
to the randomized selection of the filter and placement of
pulses.
Figs. 3(a) and 3(b) show an example of an extended DVN
sequence and its PSD, respectively. The pulse locations
are the same as in the basic DVN sequence in Fig. 1(a).
Note that the pulses are no longer located within the grid
segments of the basic DVN and are smeared in time.
2.2 Mitigating Temporal Roughness
In this work, and in previous studies on velvet noise [5,
14], temporal roughness describes the perceived quality of
sparse noise. For the original velvet noise, which consists
of randomly placed unit impulses, the perceived temporal
roughness is simply inversely proportional to its pulse den-
sity ρ. Based on informal listening experiments the basic
DVN method, which uses the recursive running-sum filters,
shows similar behavior in terms of temporal roughness [34].
In this work, however, the authors used high-order fil-
ters with a potentially steep spectral decay. In combination
with the naive implementation of the pulse-filter selection
[see Eq. (7)], this could lead to perceivable temporal rough-
Fig. 4. (a) Magnitude responses of the example second-order
dictionary filters Fq(z) with q=1, 2, ..., 10, and the spectrograms
of the resulting extended DVN sequence with (b) a totally random
and (c) greedy filter assignment.
ness. In particular, roughness issues may originate from the
random switching between filters that have large energy
differences within certain frequency bands, as seen in the
spectrogram of Fig. 4(b).
To mitigate issues related to temporal roughness, a two-
step solution is proposed. First, the filter energies are nor-
malized to minimize the temporal roughness, by ensuring
the broadband filter energy does not fluctuate when switch-
ing between different filters. Next, a greedy filter assign-
ment is proposed. The naive pulse filter selection is based
on a weighted uniform random number and is given as
φ(m)=arg max
q
{rq(m)pq(m)},(11)
where rq(m) is a uniform random number in the range [0,
1].
The following greedy filter assignment is proposed in-
stead:
φ(m)=arg max
q
{(τq(m)+rq(m)) pq(m)},(12)
where is a free parameter controlling the amount of ran-
domization and τqis a sample index (i.e., time) when the
qth dictionary filter was last selected. The value τqis up-
J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun. 373
FAGERSTR ¨
OM ET AL. PAPERS
dated sequentially based on the previously selected pulse
filter with
τq(m+1) =0forφ(m)=q,
τq(m)+1 otherwise.(13)
When the qth dictionary filter stays inactive, its weight τq
grows, thus making it more likely for the greedy assignment
to pick that filter.
The roughness problem is best visualized by synthesiz-
ing stationary noise with uniform probabilities pq≡1/Q,
i.e., a pulse routing to any of the defined dictionary filters
with uniform probability. Fig. 4(a) shows the magnitude
responses of an example set of ten second-order dictionary
filters. However, the spectrogram in Fig. 4(b) highlights the
frequency-dependent roughness problem that remains after
the normalization when the naive pulse filter selection of
Eq. (11) is used. The spectrograms were computed using
Fast Fourier Transform of length 2,048, with a 256-sample
Hann window with 50% overlap. The frequency axis in
Fig. 4(b) was limited from 1 to 20 kHz for better visualiza-
tion of the sparsity, which creates audible temporal rough-
ness in the noise sequence. The naive filter assignment
is based on uniform probabilities, and thus, the resulting
sub-sequences defined by Eq. (8) resemble totally random
noise, which was shown to sound rougher than velvet noise
[5, 14].
Fig. 4(c) shows the spectrogram of the extended DVN se-
quence generated based on the same uniform probabilities,
but now applying the greedy filter assignment of Eq. (12)
instead of the random assignment. The resulting extended
DVN sequence is visually and audibly much smoother.
However, there is a trade-off since the greedy assignment
creates some periodicity in the sequence. Note that the PSD
of the sequences in Figs. 4(b) and 4(c) is the same. The noise
sequences in Figs. 4(b) and 4(c) can be listened to on the
companion web page of this paper.2
3 REVERBERATION MODELING
The proposed framework for parametrizing a target late-
reverberation IR using the novel extended DVN structure
is presented in this section. The previously proposed FVN
method is shown to be a special case of the extended DVN
algorithm.
3.1 Preprocessing and Analysis
To parameterize a late-reverberation IR with the extended
DVN algorithm, the short-time Fourier transform (STFT)
was used to obtain a time-frequency representation of the
target IR. The late-reverberation IR of the Promenadi Hall
[37], Pori, Finland, serves as an example target IR through-
out this section. The T60 of the target IR, evaluated at octave
bands between 125 and 8,000 Hz, ranges from 2.7 s at the
lowest band to 1.2 s at the highest band [37].
In the following, the number of STFT time frames is de-
noted with T. A Fast Fourier Transform length of 2,048 was
2https://github.com/Ion3rik/dark-velvet-noise-reverb.
Fig. 5. Example magnitude responses of the pre-filter and com-
plementary post-filter of an extended DVN model instance tuned
to the Promenadi Hall IR.
used. The design choice in the analysis step is to choose a
suitable frame size and overlap amount to capture the tem-
poral and spectral changes in sufficient detail. The choice
may vary depending on the characteristics of the target IR.
Typically, larger rooms with longer decay times will tend
to have smooth exponentially decaying time envelopes for
which a shorter window size yields good results. Small
rooms or outdoor spaces with short reverb time tend to have
a more detailed envelope that heavily affects the timbre of
the IR, thus requiring a denser framing for the analysis to
yield a more detailed decay envelope for the extended DVN
model.
As a preprocessing step, linear prediction (LP) was ap-
plied on the first time frame of the STFT representation
of the target IR. The solution of the LP gives the allpole
post-filter 1/R(z) coefficients. The coefficients of the LP
filters were obtained using MATLAB’s lpc function. A
similar LP modeling approach was applied previously by
Holm et al. [32]. A filter order of 10 was used in this case
for LP modeling since this was found to be sufficient for
capturing the low-pass characteristic of late reverberation
in a previous study [33].
However, the low-order LP filter cannot model any pos-
sible low-frequency rolloff present in the target IR. Thus,
in this work, an additional first-order DC-blocker filter [38]
was fitted to the target IR to improve the model’s spec-
tral accuracy at low frequencies. The inverse filter of the
allpole filter, i.e., the pre-filter, was applied to the whole tar-
get IR to whiten it. The whitening ensures that the target IR
starts with a flat magnitude spectrum. Thus, the post-filter
takes care of matching the initial coloration of the target
IR, whereas the dictionary filters concentrate on modeling
the time-dependent relative spectral change.
The magnitude response of the post-filter is shown in
Fig. 5 (solid line). The response is the combined response
of the tenth-order allpole filter and first-order DC-blocker
filter. In this example, the effect of the DC blocker is visible
as a slight cut below 100 Hz. The corresponding pre-filter
magnitude response of the target IR is shown as a dashed
line in Fig. 5.
3.2 Dictionary-Filter Design
The main design question was to define the dictionary
filters to be used in the extended DVN structure shown in
Fig. 2. In this work, the dictionary filters were obtained di-
rectly from the analysis stage, reminiscent of the FVN algo-
374 J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun.
PAPERS NON-EXPONENTIAL REVERBERATION MODELING
Fig. 6. Normalized magnitude responses of the dictionary filters
used for modeling the late-reverberation IR of the Promenadi
Hall. Here, the dictionary consists of ten second-order allpole
filters extracted from the analyzed target reverberation. The filter
energies are normalized to one.
rithm [33], by approximating each analysis frame response
with a second-order allpole filter. The coefficients of the
second-order allpole filter were estimated using an LP. The
single tenth-order pre-filter allows applying second-order
filters in the dictionary instead of the tenth-order filters used
in the previously proposed FVN method [33].
Applying a subset of the analyzed filters that were loga-
rithmically spaced in time was found to yield good results.
Due to the logarithmic spacing, closely spaced frame fil-
ters are picked from the beginning of the analyzed IR, and
more sparsely spaced filters toward the end. The justifica-
tion for the logarithmic spacing was inspired by the results
of V¨
alim¨
aki et al. [33]. They observed that the magnitude
spectrum of late reverberation typically varies rapidly at the
beginning of the IR and slower toward the end of the IR
[33].
Fig. 6 shows example magnitude responses of the set
of dictionary filters used to model the Promenadi Hall late-
reverberation IR. In this example, the target IR was analyzed
in 50 frames, and the logarithmically spaced subset of ten
dictionary filters included the filters 1, 2, 3, 5, 7, 10, 15, 23,
34, and 50. The filter energies were normalized to one. Note
that the first dictionary filter in Fig. 6, which is estimated
from the first frame of the pre-whitened target IR, has a
practically flat magnitude response and can be omitted in
the implementation without causing an audible effect.
3.3 Solving the Filter Probabilities
After designing the dictionary filters, their magnitude
responses need to be fit to the target magnitude response
by solving an NNLS problem. NNLS is a constrained ver-
sion of the least-squares optimization problem, which is a
convex problem with linear constraints [35].
The dictionary filter magnitude responses are denoted by
the matrix |F(ejω)|, where the columns contain the magni-
tude response of each dictionary filter at discrete frequen-
cies. The NNLS solution yields the activation vector z.The
values of coefficients zare constrained to be positive or
zero. The optimization problem has the form
min
z
F(ejω)z−|h(ωk)|
2
2,subject to z≥0,(14)
where ωkare the discrete frequencies and |h(ωk)|is
the target magnitude response. In this work, MATLAB’s
lsqnonneg function was used to solve the optimization
Fig. 7. (a) Probability matrix Pand (b) the corresponding resolved
pulse filters for modeling the Promenadi Hall late-reverberation
IR.
problem. To compose the activation matrix for the whole
time-frequency representation of the target IR, the opti-
mization problem, cf. Eq. (14), was solved for each time
step separately. The activation matrix has the form
Z=z(1),z(2),...,z(T),(15)
where z(t) are the column vectors containing the solved
activations for each analysis frame t.
Due to the formalization of the proposed extended DVN
structure, the solved activation matrix Zwas normalized to
obtain the probability of each dictionary filter for each time
step. The normalization gains are computed as
γ(t)=||z(t)||1=
Q
i=1
|z(t)|.(16)
The probability matrix is then given as
P=Z1
γ
,(17)
where is the Hadamard product and γ=
[γ(1),γ(2),...,γ(T)]T. After the normalization, each
column vector p(t) of the probability matrix fulfills Eq.
(6).
Fig. 7(a) shows the probability matrix Pobtained via
normalization of the activation matrix solution for the target
IR. Dark areas indicate the most probable filter routings for
each analysis time frame. The general trend shows that
the higher probabilities shift from the first dictionary filter
toward the last dictionary filter.
3.4 Sparse Synthesis with Dark Velvet Noise
The first step of the synthesis is to generate a velvet-noise
sequence that has the same length as the target IR with the
desired pulse density ρ. In this work, the authors used a
time-dependent density starting from ρ=2,000 pulses/s
J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun. 375
FAGERSTR ¨
OM ET AL. PAPERS
Fig. 8. Normalization gains γ(t) per frame.
and linearly decreasing toward ρ=500. The decreasing
density allows maintaining a low computational load of the
algorithm without introducing temporal roughness to the
reverberation [33]. The pulse locations and signs of the se-
quence were computed using Eqs. (3) and (5), respectively.
Conveniently, the normalization gains γ(t) defined in
Eq. (16) describe the broadband energy decay envelope of
the target IR. Fig. 8 shows the normalization gains γ(t)
corresponding to the probability matrix Pof Fig. 7. On
a logarithmic scale, the normalization gains γ(t) decrease
approximately linearly over time, except for their beginning
and end—a trend that corresponds to the exponential decay
of the target IR.
Because the synthesis requires a discrete number of
pulses, the probability matrix Pand the normalization gains
γwere interpolated to match the number of pulses M.The
pre-computed pulse locations kwere used as query points,
and linear interpolation was applied. The interpolated nor-
malization gains are denoted by
γ. The per-pulse decay
gains of the extended DVN convolution structure shown in
Fig. 2 are obtained as
g(m)=γ(m)Td(m),(18)
where the multiplication with the square root of the grid
size Td(m) compensates for the change in energy due to the
time-dependent density.
Finally, the pulse filters are resolved using Eq. (12), based
on the interpolated probability matrix. Fig. 7(b) shows the
resolved pulse filters computed for the probability matrix of
Fig. 7(a). High concentrations of pulses [see Fig. 7(b)] are
likely routed to filters with high probabilities [see Fig. 7(a)],
whereas areas of low probability, i.e., the areas of lighter
color, show sparser patterns of filter selection.
3.5 Comparison to Filtered Velvet Noise
The reverberation modeling framework presented above
resembles the previously proposed FVN algorithm [32, 33].
In this section, a comparison between the two methods is
provided.
In the FVN algorithm, the target IR is segmented into
non-overlapping frames of different lengths, and an allpole
LP filter is extracted from each segment [33]. In the ex-
tended DVN method, the extracted LP filters can be used
as both the target and dictionary responses. In this special
case, the solution of the NNLS problem returns a diagonal
probability matrix, since the best fit is obtained by simply
activating the extracted segment filters in order.
Fig. 9(a) shows the first ten segments of the probability
Fig. 9. First ten frames of the (a) probability matrix Pand corre-
sponding resolved pulse filters (b) with filter interpolation, resem-
bling uniformly segmented FVN, and (c) with probability inter-
polation, unique to the extended DVN. The presented dictionary
filters are the LP allpole target filters of each frame.
matrix Pobtained using the matched dictionary filters ex-
tracted from the target IR. Fig. 9(b) shows the correspond-
ing filter assignment for each pulse when the filters are
assigned using the per-frame probabilities and the resulting
filter list is interpolated to yield a filter routing for each
pulse. In this configuration, the extended DVN essentially
implements the FVN algorithm, where the filter is switched
at the segment boundaries. However, the constant segment
length results in periodic switching of the dictionary filter,
which can cause audible disturbance in the synthesized IR
[33]. The switching periodicity was mitigated in the FVN
algorithm by using a non-uniform segment length.
The extended DVN presents another option for the filter
assignment, where the probability matrix is first interpo-
lated, and the filters are then assigned per pulse based on
the interpolated probability matrix. Fig. 9(c) shows the cor-
responding filter assignment for each pulse. In this case, the
switching is not discrete at each segment boundary. Instead,
a smoother mix is obtained between the consecutive filters.
Informal listening experiments suggested that the mixing
mitigates the problem of periodic disturbances even when
using uniform segmentation. Sound examples of the noise
sequences shown in Fig. 9(b) and 9(c) are available online.2
376 J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun.
PAPERS NON-EXPONENTIAL REVERBERATION MODELING
Fig. 10. The left-column sub-figures show the spectrograms of the Promenadi Hall and their modeled counterparts: (a) target IR and
estimated IRs based on (c) the FVN model and (e) extended DVN model. The T60 estimates of the target IR (dashed) and modeled IRs
(solid) are overlayed on the spectrograms, indicating a better fit using the proposed method. The right-column sub-figures present (b)
the target IR, (d) FVN model instance IR, and (f) extended DVN model instance IR. The direct part and early reflections of the IR, both
of which are not modeled, are shown in gray.
4EVALUATION
In this section, an objective evaluation of the extended
DVN reverberation algorithm is presented. The method was
applied to model the late-reverberation part of two target
IRs from different spaces. The early IR parts containing
the direct sound and early reflections were not modeled
but directly adopted from the original data. The two spaces
have distinctive acoustical characteristics and were selected
to provide challenging conditions to test the generalizability
of the proposed reverberation algorithm.
4.1 Target Impulse Responses
The first of the two target IRs is the high-quality mea-
surement of the IR of the Promenadi concert hall in Pori,
Finland, conducted by Merimaa et al. [37]. The late part
of the IR was modeled after 110 ms as a test case to
allow a direct comparison with the previously proposed
advanced FVN model [33]. V¨
alim¨
aki et al. [33] deter-
mined the late reverberation of the Pori Hall IR to start
after 110 ms based on preliminary testing. The second tar-
get IR has been recorded in Creswell Crags [39], where
the IR has a strong second echo. The specific IRs the au-
thors chose from the two datasets are “s1 r3 o.wav” and
“8 rgrundymouth sgrundypath.wav.”
4.2 Modeling Concert-Hall Reverberation
In this section, the authors compare IRs synthesized with
their extended DVN and the previously proposed FVN [33]
objectively to the target IR of the Promenadi Hall. The
objective accuracy was analyzed in terms of spectrogram
comparison and T60 estimation. The extended DVN model
of the Promenadi Hall was parametrized using the config-
uration described in SEC.3.
Figs. 10(b) and 10(f) show the target IR of the Promenadi
Hall and the IR synthesized with an optimized instance of
the extended DVN model, respectively. In Fig. 10, the early
parts of the IR are shown in gray and the late reverberation
in black. The target IR envelope is visually highly similar
to the envelope of the extended DVN model instance IR.
The temporal resolution of the extended DVN model enve-
lope is determined by the STFT window length, which was
selected to be 85 ms with 50% overlap. Lengthening the
STFT window would result in a smoother envelope. The
previously proposed advanced FVN [33] model instance
IR, shown in Fig. 10(d), has a slightly longer decay com-
pared to the target IR envelope.
The estimated T60 curves are overlayed on the spectro-
grams in Fig. 10, where the solid line shows the T60 of
the models and the dashed line shows the target T60. The
mean and maximum T60 errors measured in the frequency
J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun. 377
FAGERSTR ¨
OM ET AL. PAPERS
Fig. 11. Spectrogram of the Creswell Crags (a) target IR and (c) the extended DVN model. Energy (in decibels) of the Cresswell Crags
(b) target IR and (d) the corresponding extended DVN model. The late and early reflections, after the direct sound (1 ms), are modeled.
range of 20–16,000 Hz of the FVN model instance are 14%
and 28%, respectively. The extended DVN model instance
in Fig. 10(e) follows the frequency-dependent decay of the
target IR well across all frequencies with mean and maxi-
mum T60 errors of 4% and 8%, respectively.
Overall, the extended DVN model instance achieved a
better spectro-temporal fit than the FVN model instance us-
ing only ten dictionary filters, whereas the advanced FVN
model uses 20 coloration filters. This implies that the pro-
posed method is both more accurate and more efficient in
modeling a late-reverberation IR than the previously pro-
posed FVN method.
4.3 Modeling Non-Exponential Reverberation
The authors evaluate objectively the accuracy of the syn-
thesized IR of the extended DVN model instance in model-
ing the Cresswell Crags target IR. The target IR and its spec-
trogram are shown in Figs. 11(b) and 11(a), respectively.
There is no other parametric method that can successfully
model target IRs containing a single echo that results in two
exponential energy decays. The analysis presented in this
section is intended to demonstrate the variable capabilities
of the proposed method.
The extended DVN model uses ten second-order allpole
dictionary filters, a single 12th-order post-filter, and two
cascaded first-order DC blockers. The order of the post-
filter components was increased to obtain a better spectral
fit to the target IR. The analysis methods are identical to the
ones used in SEC. 4.2, except for the T60 analysis, which is
not a meaningful metric for the target IR in question. For
this example, the extended DVN was used to synthesize the
entire target IR except for the direct sound (first 1 ms), since
the target IR does not show any prominent early reflections.
Fig. 4(c) shows the spectrogram of the extended DVN
model instance IR. The DVN model provides an accurate
approximation of the overall shape of the target spectro-
gram. However, a slightly longer decay is seen in Fig. 11(c)
around 1 kHz. The largest difference between the target and
model instance spectrograms is seen at low frequencies, at
100 Hz and just above it, where the DC-blocker filters out
the low-frequency noise visible in the target spectrogram
of Fig. 11(a).
The IR of the extended DVN model instance in Fig. 11(d)
now features a more detailed energy-decay envelope com-
pared to that of the concert hall reverberation DVN model
instance in Fig. 10(f). This is due to the smaller frame
length (5.3 ms) with 50% overlap for the STFT analy-
sis. The smaller frame length and hop size are beneficial
in modeling the more complex double-stage decay of the
Crasswell Crags target reverberation. The envelope shape
of the extended DVN model instance in Fig. 10(d) follows
well that of the target IR in Fig. 11(b). In summary, this
design example demonstrates the ability of the proposed
method to model a non-exponential IR.
4.4 Modification of the Proposed Model
One of the benefits of the parametric extended DVN is its
flexibility concerning modification possibilities of the syn-
thesized IR. The direct modification of IRs used in convo-
lution reverberation has been proposed by Canfield-Dafilou
and Abel [40] to change the perceived room size of the orig-
inal IR. Modifications such as time stretching and temporal
envelope modification have been demonstrated already us-
ing the previously proposed FVN algorithm [32, 33].
The extended DVN lends itself just as well to time
stretching and envelope modifications. Furthermore, the
achievable level of detail in envelope modifications with
the extended DVN algorithm is vaster than with the previ-
ously proposed FVN algorithm, since the former method
relies on pulses scaled by individual gains. The choice of
378 J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun.
PAPERS NON-EXPONENTIAL REVERBERATION MODELING
Fig. 12. Spectrograms of the modified Promenadi Hall extended
DVN model instance with (a) gated reverb, (b) modified T60
[compared to the original in Fig. 10(e)], (c) reversed spectral
evolution, and (d) reversed decay. The modifications are applied
only to the late part, which is modeled, except for the reversed
decay, where the early parts are also reversed and placed after
the modeled part. The T60 estimates of the original target IR
(dashed) and modified model IR (solid) are overlayed on top of
the spectrogram in (b).
envelope resolution is a trade-off between computational
cost and envelope detail. Time stretching with the extended
DVN is implemented by changing the length of the gener-
ated velvet-noise sequence and interpolating the probability
matrix Pto match the modified pulse locations. In the fol-
lowing examples, three more modifications achievable with
the extended DVN algorithm are highlighted.
Fig. 12 shows the spectrograms of three modifications
of the Promenadi Hall extended DVN model instance.
Fig. 12(a) illustrates the effect of the gated reverb modifi-
cation made to the extended DVN model instance IR. Since
the extended DVN is implemented as an FIR structure, gen-
erating a gated reverb is achieved simply by truncating part
of the delay-line coefficients from the end of the model
instance IR.
In Fig. 12(b), the authors slowed down the spectral
change relative to the original model instance of Fig. 10(e).
With the proposed method, the relative spectral change of
the late reverberation can be adjusted by manipulating the
probability matrix P, cf. Eq. (17). The slowing down of the
decay was achieved by taking a subset of columns from the
beginning of the original probability matrix, i.e., the matrix
Pα=p(1) p(2) ... p(ˆ
T),(19)
where ˆ
T=Tαand 0 ≤α≤1. The new matrix Pα
was then interpolated to fit the original pulse locations.
In Fig. 12(b), the resulting T60 estimate is overlayed on
the spectrogram, where the dashed line shows the original
target T60 and the solid line shows the modified T60, where
the high frequencies ring considerably longer than in the
target.
The final modification example is the time-reversed re-
verb effect, which has two versions. Since the spectral
change is parametrized by the probability matrix Pand
the energy decay with g(m) of Eq. (18), they are decoupled.
In Fig. 12(c), the authors have time reversed the filter list
φ, cf. Eq. (7), of the extended DVN model instance. The
resulting IR retains its overall energy decay, but the spec-
tral change is now from dark to bright. Implementing the
opposite version of the two flip operations is also a possi-
bility. Flipping the decay envelope while using the original
filter list results in an IR whose energy rises while retain-
ing the original spectral change, as shown in Fig. 12(d).
Sound examples of the modifications presented in Fig. 12
are provided together with the MATLAB application that
can recreate them.2
The brightening effect and gated effect can be imple-
mented also with recursive algorithms given that an ac-
curate model is first obtained. However, the gated reverb
requires either a second parallel recursive algorithm to be
run to cancel the remaining part of the IR [25] or, alterna-
tively, an additional noise gate to mute the output of the
reverberator. On the other hand, the reversed energy decay
is impossible to implement in a recursive manner, since it
requires growth of energy over time, which would result in
instability in a recursive implementation.
5 CONCLUSION
In this paper, an extension of the previously proposed
DVN algorithm was presented. A parametric reverberator
was developed based on the extended DVN structure utiliz-
ing sparse velvet-noise convolution. By replacing the square
pulses of the previously proposed DVN method with arbi-
trary dictionary filters and setting the probability of each
dictionary filter as a free parameter, the extended DVN
model can be fitted to a target IR via NNLS optimization.
The authors additionally showed that a previously proposed
J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun. 379
FAGERSTR ¨
OM ET AL. PAPERS
FVN reverberation algorithm [33] is a special case of the
proposed extended DVN reverberator.
The authors evaluated the proposed method objectively
and demonstrated its capability to accurately model the late
reverberation of two distinctly different spaces, a large con-
cert hall and a coupled space in which sound decays in a
non-exponential manner. No previously proposed paramet-
ric reverberation algorithm exists for modeling the latter
target IR. The authors assessed the spectro-temporal fit of
optimized instances of the proposed extended DVN model
to the two target IRs objectively, and the proposed method
produced an accurate model in both cases. For the concert
hall target IR, the extended DVN yielded a better spectro-
temporal fit than the previously proposed FVN algorithm,
when using half the number of filters. The method’s capa-
bility was demonstrated by providing practically relevant
examples of model IR modification, such as slowing down
the decay in a frequency-dependent way and gating the
model IR.
All in all, the proposed extended DVN reverberator is ap-
plicable to synthesizing late reverberation of various spaces
while providing perceptually meaningful parametric con-
trol. Future work could investigate a more general approach
to specify a broad set of dictionary filters and find a sparse
solution, instead of using the empirically designed filters.
6 ACKNOWLEDGMENT
This research is part of the activities of the Nordic Sound
and Music Computing Network (NordicSMC; NordForsk
project no. 86892).
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OM ET AL. PAPERS
THE AUTHORS
Jon Fagerstr¨
om Sebastian J. Schlecht Vesa V¨
alim¨
aki
Jon Fagerstr¨
om received his M.Sc. degree in electrical en-
gineering, majoring in acoustics and audio technology,
from Aalto University, Espoo, Finland, in 2020. He is
currently working toward a doctoral degree at the Acous-
tics Lab, Aalto University. His research interests include
sparse-noise modeling, decorrelation filters, artificial re-
verberation, and reverberation perception.
•
Sebastian J. Schlecht is a Professor of Practice for Sound
in Virtual Reality at the Acoustics Lab, Department of In-
formation and Communications Engineering and Media
Labs, Department of Art and Media, of Aalto University,
Finland. He received the Diploma in Applied Mathemat-
ics from the University of Trier, Germany, in 2010 and an
M.Sc. degree in Digital Music Processing from the School
of Electronic Engineering and Computer Science at Queen
Mary University of London, UK, in 2011. In 2017, he
received a doctoral degree at the International Audio Lab-
oratories Erlangen, Germany, on artificial spatial reverber-
ation and reverberation enhancement systems. From 2012
to 2019, Dr. Schlecht was also external research and de-
velopment consultant and lead developer of the 3D Reverb
algorithm at the Fraunhofer IIS, Erlangen, Germany.
•
Ve sa V ¨
alim¨
aki received his D.Sc. degree in electrical
engineering from the Helsinki University of Technology,
Espoo, Finland, in 1995. In 1996, he was a Postdoctoral
Researcher at the University of Westminster, London, UK.
In 2001–2002, he was a Professor of signal processing at
the Pori unit of the Tampere University of Technology, Fin-
land. In 2008–2009, he was a Visiting Scholar at Stanford
University. He is a Full Professor of audio signal processing
and the Vice Dean for Research in electrical engineering
at Aalto University. He is a Fellow of the Institute of Elec-
trical and Electronics Engineers (IEEE). In 2015–2020, he
was a Senior Area Editor of the IEEE/ACM Transactions
on Audio, Speech, and Language Processing. Since 2020,
Prof. V¨
alim¨
aki has been the Editor-in-Chief of the Journal
of the Audio Engineering Society.
382 J. Audio Eng. Soc., Vol. 72, No. 6, 2024 Jun.
