# Neural network for multi-exponential sound energy decay analysis

Georg Götz<EMAIL_ADDRESS>Ricardo Falcón Pérez  Sebastian J. Schlecht
Also at: Media Lab, Department of Art and Media, Aalto University, Otakaari 5,
00250 Espoo, Finland.  Ville Pulkki  Aalto Acoustics Lab, Department of Signal
Processing and Acoustics, Aalto University, P.O. Box 13100, 00076 Aalto,
Finland

###### Abstract

An established model for sound energy decay functions (EDFs) is the
superposition of multiple exponentials and a noise term. This work proposes a
neural-network-based approach for estimating the model parameters from EDFs.
The network is trained on synthetic EDFs and evaluated on two large datasets
of over $20\,000$ EDF measurements conducted in various acoustic environments.
The evaluation shows that the proposed neural network architecture robustly
estimates the model parameters from large datasets of measured EDFs, while
being lightweight and computationally efficient. It achieves an almost 30-fold
speed-up compared to state-of-the-art methods. An implementation of the
proposed neural network is publicly available.

††preprint: Götz et al.

##  I Introduction

Room impulse responses (RIRs) are signals that describe the sound recorded by
a receiver in a room in response to an impulsive excitation. They characterize
how room, source, and receiver affect sound on the investigated transmission
path, implicitly assuming a linear time-invariant system. The RIR decays
gradually, thus causing the well-known acoustic phenomenon called
reverberation [1].

Due to pronounced fluctuations in the RIR, the energy-time curve may be hard
to interpret and difficult to use in further analyses. Schroeder proposed a
backwards integration procedure for obtaining smooth steady-state energy decay
functions (EDFs) [2], which are also called Schroeder decay curvesfunctions.
Schroeder decay curvesfunctions are frequently used in architectural acoustics
to calculate the reverberation time, which is the time until the sound energy
in an enclosure has decreased by $60\text{\,}\mathrm{dB}$. The reverberation
time is commonly determined for multiple frequency bands by fitting a straight
line to the band-limited logarithmic EDF [3, 4].

Measured RIRs are usually contaminated with noise, for instance, due to the
measurement equipment, ambient sound, or quantization. The noise inevitably
affects the EDF [5] and causes errors when estimating the reverberation time
based on a straight-line fit [6]. Several approaches were proposed to counter
the effect of noise on the reverberation time estimation [5, 7, 8, 3].
Alternatively, Xiang [9] and Karjalainen et al. [10] proposed to include an
additional noise term in the model and perform a non-linear regression.

In coupled rooms or rooms with a considerably non-uniform absorption material
distribution, sound energy can decay with multiple decay rates [11, 12, 13,
1]. For this reason, Xiang and Goggans model EDFs with multiple exponential
decays and a noise term [14].

This paper presents a neural-network-based approach for fitting multiple
exponential decays and a noise term to an EDF. Despite being trained on a
fully synthetic dataset, we show that such a neural network structure can
robustly analyze real-world measurements. Fully synthetic training datasets
can be easily generated and subsequently extended to different scenarios.
Previous methods for multi-exponential sound energy decay analysis rely on
iterative algorithms. Our approach has the advantage of being fully
deterministic at runtimeinference time and requiring no user-tuning, while
being robust and computationally efficient. Therefore, the neural network
structure is especially appealing for room acoustic analysis and modeling with
machine-learning-based approaches, for which it is essential to achieve robust
performance on large datasets without manual intervention. The proposed
network is lightweight, allowing it to be implemented on mobile devices.
Furthermore, the neural-network-based structure allows efficient up-scaling
and parallelization on modern hardware with dedicated graphical processing
units (GPUs).

Our contribution is threefold. Firstly, we present a lightweight and
computationally efficient neural-network-based structure for sound energy
decay analysis that achieves an almost 30-fold speed-up compared to state-of-
the-art methods, while maintaining a comparable fitting performance.that
achieves a comparable fitting performance as state-of-the-art methods.
Secondly, we evaluate the proposed network and previous decay analysis
approaches on two large datasets of more than $20\,000$ EDFs. Thirdly, we
provide an open-source decay analysis toolbox for MATLAB and Python,
comprising the neural network structure and our implementations of the other
evaluated multi-slope decay analysis methods.

The remainder of this paper is organized as follows. Section II states the
problem formulation, and Section III provides an overview of prior work.
Section IV describes the proposed neural network and its training in detail.
Section V presents an evaluation of the proposed network on two large datasets
of real-world measurements and compares its performance to other state-of-the-
art approaches. Section VI discusses the results. Section VII details the
publicly available decay analysis toolbox, and Section VIII concludes the
paper.

##  II Problem formulation

Smooth energy decay functions (EDFs) can be obtained from RIRs via the
backwards integration procedure proposed by Schroeder [2]. The EDF $d(t)$ of
an RIR $h(t)$ is calculated by

$d(t)=\frac{1}{E}\ \sum_{\tau=t}^{L}h^{2}(\tau)\,,\quad\textrm{with}\quad
E=\sum_{\tau=1}^{L}h^{2}(\tau)\,,$ (1)

where $t$ is the sample index and $L$ is the number of samples in the EDF,
which is also called the upper limit of integration (ULI).

EDFs can be modeled as a sum of multiple exponential decays and a noise term
[14]. The model $d_{K}(t)$ of the EDF $d(t)$ is then given as [14, 15] $-13.8$
instead of formulation with natural logarithm for readability

$d_{K}(t)=N_{0}(L-t)+\sum_{i=1}^{K}A_{i}\left[e^{\frac{-13.8\cdot
t}{f_{\mathrm{s}}T_{i}}}-e^{\frac{-13.8\cdot
L}{f_{\mathrm{s}}T_{i}}}\right]\,,$ (2)

where $T_{i}$ and $A_{i}$ are the decay [time should be a better term, because
a rate would be the entire fraction without the $t$]ratetime and the amplitude
of the $i$th exponential decay, respectively, $\textrm{ln}(\cdot)$ denotes the
natural logarithm, $N_{0}$ is the amplitude of the noise term, [to make clear
where the constant value in the exponents of Eq. (2) comes from]the constant
$-13.8=\textrm{ln}(10^{-6})$ ensures that the sound energy has decayed to
$-60\text{\,}\mathrm{dB}$ after $T_{i}$ seconds, $f_{\mathrm{s}}$ is the
sampling frequency of the RIR, and $K$ is the model order, i.e., the number of
exponential decays in the model. The constant second term in the square
brackets accounts for the finite upper limit of integration and can be
neglected for large $L$ [15].

Estimating the parameters $T_{i}$ and $A_{i}$ is a crucial task for various
problems in room acoustics. For example, the reverberation time can be
determined by estimating the parameter $T_{1}$ for an EDF model with $K=1$.
Decay models of higher order have successfully been used to measure the
absorption coefficients of materials [16] or characterize the sound decay of
coupled spaces [17, 18].

##  III Prior work

###  III.1 Sound energy decay analysis

Previous approaches for estimating the parameters $T_{i}$, $A_{i}$, and
$N_{0}$ differ mainly regarding the underlying model order. For model order
$K=1$, linear regression is commonly used to determine the reverberation time
as a straight line fit to the band-limited logarithmic EDF [3, 4]. In this
case, the noise term of the model is neglected, i.e., $N_{0}=0$. To get
accurate estimates for $T_{1}$, the effect of the noise has to be countered by
noise subtraction [5] or truncation of the RIR before backwards integration
[7, 8, 3]. The noise term $N_{0}$ can be included in the model by using non-
linear regression [9, 10].

The sound decay of coupled rooms or rooms with considerably non-uniform
absorption material distribution can usually not be modeled with a single
decay rate [11, 12, 13, 1]. In such cases, model orders $K>1$ need to be
considered. Xiang and Goggans proposed a Bayesian framework to determine the
model parameters $T_{i}$, $A_{i}$, and $N_{0}$ for $K\geq 1$ [14]. The
Bayesian formulation can also determine the most probable model order $K$
given the measured EDF [19]. Numerous works have advanced the approach by
investigating more accurate and efficient algorithms for estimating the
parameters or determining the model order [20, 19, 16].

Previous studies have evaluated the performance of single-slope EDF fitting
software [21, 22, 23]. Katz [23] used a single measured lecture theater RIR,
which was also used in a later study by Cabrera et al. [21] together with
artificial single-slope responses. Álvarez-Morales et al. [22] studied the
software behavior on a slightly larger set of RIRs ($15$ receiver positions
$\times$ $2$ source positions, measured in a single auditorium). While Katz
[23] still reported considerable differences among reverberation time
estimation software in 2004, the later studies [21, 22] found that
reverberation time is consistently estimated within perceptual limits by the
more recent software implementations in most cases. In all studies, the most
inconsistent reverberation time estimates were obtained for low-frequency
bands [21, 22, 23].

###  III.2 Convolutional neural networks in acoustics

Figure 1: An example of the proposed neural network structure for the
parameter estimation of the sound decay model in Eq. (2). Throughout this
paper, we refer to this particular architecture as DecayFitNet. The network
outputs are preliminary values for the decay times $\widetilde{T}_{i}$, decay
amplitudes $\widetilde{A}_{i}$, and noise term $\widetilde{N}_{0}$.
Additionally, the network returns the values $P_{K=i}$, which quantify the EDF
model order prediction. The input length and maximum model order for the
current implementation are $M=100$ samples and $\widehat{K}_{\textrm{max}}=3$,
respectively.

The recent advances in machine learning and artificial intelligence have
pushed the state of the art in many different fields. Machine learning has
traditionally been of interest in speech recognition [24], natural language
processing [25], and music information retrieval [26]. More recent
applications in acoustics include source localization and tracking [27], blind
room acoustic parameter estimation [28], sound field scattering from 3D
objects [29, 30], and the inverse problem of object geometry regression from
the scattered sound field [31]. For more details, the reader is referred to
the thorough review by Bianco et al. [32]. Although the applications, models,
and task definitions vary considerably, most of the previously mentioned works
share a common approach: a machine learning model, typically a neural network,
extracts features and predicts the parameters of a parametric model.

Most of these applications can be divided into either classification or
regression tasks, where the main difference is the output domain [33]. For
classification, the goal is to select one or several categories, given a
specific input. For regression, the goal is to predict one or several
continuous values. The study by Fernández-Delgado et al. [34] includes a
detailed analysis of many regression methods and applications. Depending on
the network architecture and training procedure, some tasks can be formulated
as regression or classification tasks. In this paper, the decay parameter
prediction is treated as a regression task, where we estimate the decay
parameters as continuous values. On the other hand, the model order prediction
is treated as a classification task with the categories “1 active decay
slope”, “2 active decay slopes”, and so forth.

##  IV Proposed method

In this paper, we propose a neural network structure for estimating the
parameters $T_{i}$, $A_{i}$, and $N_{0}$ of the model in Eq. (2). We show that
such a network can be trained on a fully-synthetic training dataset to make
predictions on real-world measurements. In the following section, we will
first describe an example of such a network architecture. Subsequently, we
provide details on the synthetic training dataset, the utilized loss
functions, and the hyperparameters that can be used to train the network.

###  IV.1 Network architecture

Erroneous ReLU activations on final layer have been deleted from Fig. 1 An
example of the proposed neural network structure for estimating the sound
decay parameters of the model in Eq. (2) is depicted in Figure 1. We refer to
this particular architecture as DecayFitNet in the remainder of this paper.
The network takes EDFs as its input and returns preliminary estimates
$\widetilde{T}_{i}$, $\widetilde{A}_{i}$, and $\widetilde{N}_{0}$ for the
decay [time should be a better term, because a rate would be the entire
fraction in Eq. (2) without the $t$]ratestimes, decay amplitudes, and noise
term, respectively. Furthermore, it outputs the values $P_{K=i}$, which
quantify the model order prediction. For example, by applying the logistic
function

$f(x)=\frac{1}{1+e^{-x}}\,,$ (3)

we could get the probabilities $f(P_{K=i}{})=[0.75,0.15,0.10]$, which are
bound between $0$ and $1$, and indicate that the network predicts the model
order $\widehat{K}$ to be $1,2$, or $3$ with the probabilities $0.75$, $0.15$,
and $0.10$, respectively. In the current implementation, we restrict the
maximum model order to $\widehat{K}_{\textrm{max}}=3$, although higher model
orders could also be supported in the future.

Erroneous ReLU activations of final layers have been deleted from Fig. 1.

The network consists of a common base and individual output branches for the
different estimated parameters. The base contains a sequence of three one-
dimensional convolutional layers with intermediate max-pooling layers [35]
along the time-axis and three fully-connected layers. The output branches
consist of two fully-connected layers each. Rectified Linear Units (ReLU) [36]
are used as activation functions after each network layer, excluding the
output layers.

This type of network architecture is ubiquitous for many different
applications and tasks. In theory, a network that only consists of fully-
connected layers and nonlinearities can be seen as a universal approximator
[37]. Such networks are also called multi-layer perceptron (MLP). In practice,
the flexibility of MLPs is limited by the amount of training data, size of the
network, and training procedure. Most recent deep learning approaches favor
convolutional layers to reduce the number of weights needed [33]. Furthermore,
models with subsequent blocks of convolutional layers, nonlinearities, and
pooling layers are biologically inspired by the visual processing system found
in many living beings [38]. The convolutional layers act as feature extractors
by learning filters that process the input signal in such a way as to maximize
the information required for the task. The max-pooling layers behave like a
downsampling operation that focuses on the most active features, thus reducing
dimensionality and redundancies. Consecutive blocks represent progressively
higher-level features of the input signal. The shared fully-connected layers
recombine the extracted features in a non-linear way. Finally, the task-
specific fully-connected layers act as independent regressors.

Using a shared common core for all tasks has several advantages over using
independent networks for each task. First, this reduces the system complexity
for both training and inference, as the number of computations is reduced.
More importantly, the shared core will tend to learn useful features across
all tasks and not overfit any of them [39]. This property can improve
generalization and robustness to noisy data.

###  IV.2 Pre-processing

A neural-network-based structure for decay analysis requires some pre-
processing steps. In the following, we detail the pre-processing steps for our
example implementation DecayFitNet.

As the network requires a fixed-length input, we propose three pre-processing
steps on the EDF $d(t)$ to be analyzed:

  1. (i)

The last $5\text{\,}\mathrm{\char 37\relax}$ of the EDF are excluded, i.e.,
the samples $d(t>0.95L)$ are discarded. This step is motivated by the
statistical uncertainty of the last EDF samples. This uncertainty is inherent
to the Schroeder backwards integration method because, at the end of the EDF,
only few RIR samples are integrated.

  2. (ii)

The result is resampled to a fixed length $M$. In our example implementation
DecayFitNet, we set $M=100$ samples. We apply fractional resampling.

  3. (iii)

The EDF is converted into logarithmic scale (in $\mathrm{dB}$) and normalized
with the biggest absolute EDF sample value of the entire training dataset. The
normalization ensures that every EDF sample of the training dataset lies in
the interval $\left[-1,\,1\right]$. The required normalization factor must be
saved to a normalization file after the training procedure is completed. This
way, the normalization can also be applied during inference.

###  IV.3 Post-processing of estimates

The preliminary parameter estimates are processed with the following output
transformations to yield the final parameter estimates $\widehat{T}_{i}$,
$\widehat{A}_{i}$, and $\widehat{N}_{0}$:

$\displaystyle\widehat{K}=$ $\displaystyle\ \textrm{argmax}_{i}(P_{K=i}{})$
(4a) $\displaystyle\widehat{T}_{i}=$ $\displaystyle\
\frac{\widetilde{T}_{i}^{2}+1}{M}\ \frac{L}{f_{\textrm{s}}}$ (4b)
$\displaystyle\widehat{A}_{i}=$ $\displaystyle\
\begin{cases}\widetilde{A}_{i}^{2}&\textrm{ for}\ i\leq\widehat{K}\\\
0&\textrm{ for}\ i>\widehat{K}\end{cases}$ (4c)
$\displaystyle\widehat{N}_{0}=$ $\displaystyle\ \frac{M}{L}\
10^{-\widetilde{N}_{0}}\,.$ (4d)

We use these transformations to ensure that the final predictions
$\widehat{T}_{i}>$0\text{\,}\mathrm{s}$$, $\widehat{A}_{i}\geq 0$, and to
achieve better numerical stability during the training while covering a large
dynamic range of background noise levels $\widehat{N}_{0}$. While
$\widehat{T}_{i}$ is the estimated decay time in seconds, the preliminary
estimate $\widetilde{T}_{i}$ is the decay time in samples. Therefore, the
value $\frac{\widetilde{T}_{i}^{2}+1}{M}$ is the decay time relative to the
neural network input length $M$, where squaring and adding $1$ results in
$\widehat{T}_{i}{}>0$, thus avoiding division by zero in the exponential terms
[c.f. Eq. (2)]. Due to the resampling in the pre-processing stage, the decay
[time should be a better term, because a rate would be the entire fraction in
Eq. (2) without the $t$]ratetime estimates must be readjusted to the original
time scale by multiplying with $\frac{L}{f_{\textrm{s}}}$. For the same
reason, the noise value predictions are scaled by $\frac{M}{L}$. The
amplitudes of all exponential terms that have a higher order than the
predicted model order $\widehat{K}$ are set to zero, thus effectively removing
their contribution to the predicted EDF fit. Although the amplitude values
$A_{i}$ can cover a large dynamic range, our preliminary experiments found
that the training converges to better results when we predict
$\widehat{A}_{i}$ on a linear scale, as opposed to the logarithmic scale that
we use to predict the noise values $\widehat{N}_{0}$.

The estimated fit $\widehat{d}_{\widehat{K}}(t)$ is obtained by inserting the
network predictions $\widehat{T}_{i}$, $\widehat{A}_{i}$, and
$\widehat{N}_{0}$ into Eq. (2).

###  IV.4 Synthetic training dataset

A large quantity of EDFs with various combinations of the decay parameters
$T_{i}$, $A_{i}$, and $N_{0}$ [c.f. Eq. (2)] is required to train the proposed
neural network structure. Synthetic EDFs are an efficient way of collecting
many different EDFs that cover the variety of real-world measurements. In the
following, we want to detail how such a training dataset can be synthesized.

A large fully-synthetic dataset of $150\,000$ EDFs was generated to train the
DecayFitNet. It can be split into three equally-sized subsets, consisting of
EDFs with model orders of $1$, $2$, and $3$, respectively. The data generation
included three steps. Firstly, we randomly assigned values for the decay
parameters $T_{i}$, $A_{i}$, and $N_{0}$. Secondly, for model orders bigger
than $1$, we checked if the drawn values were different enough to produce a
proper multi-slope EDF and redraw values if necessary. Lastly, Gaussian noise
was octave-band filtered, shaped, and backwards-integrated to obtain a
synthetic EDF with the previously drawn $T_{i}$, $A_{i}$, and $N_{0}$ values.
Details about these steps are elaborated in the following.

Values for the decay [time should be a better term, because a rate would be
the entire fraction in Eq. (2) without the $t$]ratestimes $T_{i}$ were drawn
from the uniform distribution

$T_{i}\in\mathcal{U}(0.1\ T_{\textrm{EDF}},\,1.5\ T_{\textrm{EDF}})\,,$ (5)

where $T_{\textrm{EDF}}=\frac{L}{f_{\textrm{s}}}$. In other words, the drawn
$T_{i}$ values are between $10\text{\,}\mathrm{\char
37\relax}150\text{\,}\mathrm{\char 37\relax}$ of the input EDF length
$T_{\textrm{EDF}}$. For our training dataset, we set
$T_{\textrm{EDF}}=$10\text{\,}\mathrm{s}$$ and
$f_{\textrm{s}}=$48\text{\,}\mathrm{kHz}$$. Due to the resampling of arbitrary
length EDFs during inference, our dataset allows predicting different $T_{i}$
value ranges for different input EDF lengths. For instance, for a
$2.5\text{\,}\mathrm{s}$ long input EDF, the network can predict $T_{i}$
values with lower and upper limits corresponding to
$0.1\times$2.5\text{\,}\mathrm{s}$=$0.25\text{\,}\mathrm{s}$$ and
$1.5\times$2.5\text{\,}\mathrm{s}$=$3.75\text{\,}\mathrm{s}$$, respectively.
In other words, the resampling allows our network to operate on various
reverberation time ranges, depending on the length of the input EDF. After
drawing the values, they were ordered, such that $T_{i+1}\geq T_{i}$.

In our training dataset we wanted to cover an extensive dynamic range.
Therefore, we assigned

$N_{0}=10^{a_{\textrm{noise}}},\quad\textrm{with}\quad
a_{\textrm{noise}}\in\mathcal{U}(-14,-3)\,,$ (6)

corresponding to noise values between $-140\text{\,}\mathrm{dB}$ and
$-30\text{\,}\mathrm{dB}$.

As the decay amplitudes $A_{i}$ typically cover a large dynamic range as well,
we assigned the values as

$A_{i}{}=10^{a},\quad\textrm{with}\quad a\in\mathcal{U}(-3,\,0)\,,$ (7)

corresponding to amplitude values between $-30\text{\,}\mathrm{dB}$ and
$0\text{\,}\mathrm{dB}$. The $A_{i}$ values were normalized, such that they
sum up to unity, i.e., $\sum\limits_{i=1}^{K}A_{i}=1$. Finally, they were
ordered, such that $A_{i}\geq A_{i+1}$.

If the desired model order $K$ was bigger than $1$, the initial $T_{i}$ and
$A_{i}$ values were checked for a sufficient multi-slope characteristic. This
step was crucial because the random assignment could result in almost
identical $T_{i}$ and $A_{i}$ values for the different slopes, thus generating
a single-slope EDF, although a multi-slope EDF was desired. Therefore, we
introduced the constraints

$\displaystyle T_{i+1}$ $\displaystyle\geq 1.5\ T_{i}$ (8a)
$\displaystyle\frac{A_{i}}{A_{i+1}}$ $\displaystyle\geq 10^{\frac{3}{K}}\,.$
(8b)

The constraint of Eq. (8a) ensured that the different slopes have considerably
different decay rates. By applying the constraint of Eq. (8b), we aimed to
distribute the amplitude values $A_{i}$ over the entire range from
$-30\text{\,}\mathrm{dB}0\text{\,}\mathrm{dB}$, thus preventing very similar
values. New values for $T_{i}$ and $A_{i}$ were randomly drawn from the above
distributions until both constraints were fulfilled. A visual inspection of
some resulting EDFs showed that both constraints together ensured a distinct
multi-slope character.

Instead of directly inserting the resulting $T_{i}$, $A_{i}$, and $N_{0}$
values into Eq. (2), the final synthetic EDF was generated by applying the
backwards integration [c.f. Eq. (1)] on decaying Gaussian noise. This
additional step introduced small random fluctuations into the synthetic EDFs
to improve generalization after training the neural network. Four steps were
necessary to obtain the final synthetic EDF. Firstly, the $A_{i}$ and $N_{0}$
values had to be scaled to account for the backwards integration

$\displaystyle A_{i,\textrm{synth}}$ $\displaystyle=13.8\cdot
T_{\textrm{EDF}}\ \frac{A_{i}}{T_{i}}$ (9a) $\displaystyle
N_{0,\textrm{synth}}$ $\displaystyle=M\ N_{0}\,.$ (9b)

Secondly, a synthetic energy response was generated as

$\displaystyle h_{\textrm{synth}}^{2}(t)=$ $\displaystyle
N_{0,\textrm{synth}}\cdot(f_{\textrm{s}}\cdot T_{\textrm{EDF}}-t)\cdot
g_{0}^{2}(t)$ (10) $\displaystyle+\sum_{i=1}^{K}A_{i,\textrm{synth}}\cdot
g_{i}^{2}(t)\cdot e^{\frac{-13.8\cdot t}{f_{\mathrm{s}}T_{i}}}\,,$

where $g_{0}(t),g_{1}(t),...,g_{K}(t)$ is Gaussian noise that is filtered with
a random octave-band filter, and re-normalized to zero mean and unit variance.
For the data generation we assumed $f_{\textrm{s}}=$48\text{\,}\mathrm{kHz}$$
and $T_{\textrm{EDF}}=$10\text{\,}\mathrm{s}$$. Thirdly, the response
$h_{\textrm{synth}}(t)$ was backwards-integrated according to Eq. (1). Lastly,
just as described in Section IV.2, the last $5\text{\,}\mathrm{\char
37\relax}$ of the EDF samples were discarded, and the result was resampled to
a length of $M=100$ samples.

###  IV.5 Loss function

We propose to use a loss function consisting of three parts for training the
proposed neural network structure.
The first part is the EDF loss, defined as the mean absolute error (MAE)
between the analyzed EDF $d_{\textrm{dB}}(t)$ and the estimated fit
$\widehat{d}_{\widehat{K},\textrm{dB}}(t)$

$\mathcal{L}_{\textrm{EDF}}=\frac{1}{M}\sum_{t=0}^{M-1}\lvert
d_{\textrm{dB}}(t)-\widehat{d}_{\widehat{K},\textrm{dB}}(t)\rvert\,,$ (11)

where the last $5\text{\,}\mathrm{\char 37\relax}$ of EDC samples are excluded
for both EDFs (c.f. Section IV.2), both EDFs are converted to a logarithmic
scale (in $\mathrm{dB}$), and $\lvert.\rvert$ denotes the absolute value.
The second part of the loss is the noise loss, defined as the absolute error
between the ground truth and the estimated noise exponent

$\mathcal{L}_{\textrm{noise}}=\lvert\textrm{log}_{10}(N_{0})-\textrm{log}_{10}(\widehat{N}_{0})\rvert\,.$
(12)

The third part of the loss is the model order loss, defined as the cross-
entropy loss

$\mathcal{L}_{\textrm{order}}=-P_{K=\widehat{K}}+\textrm{ln}\left(\sum_{i=1}^{\widehat{K}_{\textrm{max}}}e^{P_{K=i}}\right)\,.$
(13)

In Eq. (13), $P_{K=i}$ quantifies which probability the network assigns to the
model order $i$, where $K$ is the true model order. We include the model order
loss to teach the network to predict the correct number of slopes in an EDF.
This measure should prevent the network from outputting multiple similar
slopes in cases where a single slope would fit the EDF sufficiently well.
Finally, the total loss $\mathcal{L}$ for training the proposed neural network
is

$\mathcal{L}=\mathcal{L}_{\textrm{EDF}}+\mathcal{L}_{\textrm{noise}}+\mathcal{L}_{\textrm{order}}\,.$
(14)

###  IV.6 Training

We train the DecayFitNet for $200$ epochs using the Adam optimizer [40] with
an initial learning rate of $1\times 10^{-3}$ and a weight decay [41] of
$3\times 10^{-4}$. Additionally, we apply cosine annealing with warm restarts
[42] using a schedule of $40$ epochs between the restarts.

##  V Evaluation

In our evaluation, we use the DecayFitNet on two large datasets and compare
its EDF fitting performance with a publicly available toolbox and our own
implementation of the Bayesian decay analysis framework. The following section
describes the details of our evaluation.

###  V.1 Baseline methods

We compare the DecayFitNet with two other decay analysis approaches. The first
baseline method is based on a non-linear regression model of a single
exponential and a noise term [10]. The second baseline method is based on
slice sampling for Bayesian decay analysis [20]. Details about the
implementation of both methods are presented in the following.

####  V.1.1 Non-linear regression

This baseline method uses the publicly available toolbox implemented by
Karjalainen et al. [10]. In initial experiments, we found that the performance
of the toolbox depends considerably on the choice of the fitting scale. This
issue was already observed by Karjalainen et al., which is why they proposed
to fit the EDF $d(t)$ on a power scale [10]. This means that the non-linear
regression is carried out on the scaled EDF $d_{\textrm{scale}}(t)=d^{s}(t)$,
thus adding the adjustable hyperparameter $s$ to the method. In our
evaluation, we use $s=0.5$ as suggested by the developers of the toolbox [10].
Additionally, we use an improved variant, where a grid search is carried out
over the interval $s\in\left[0.2,\ 0.8\right]$ to find the best fit regarding
the mean squared error (MSE). In both variants, we only use the EDF below
$-5\text{\,}\mathrm{dB}$, which is common practice for reverberation time
estimation.

####  V.1.2 Bayesian decay analysis

For the second baseline method of our evaluation, we implemented a slice-
sampling-based Bayesian decay analysis [20] in MATLAB and Python. The code is
contained in our Decay Analysis Toolbox (c.f. Section VII). Our implementation
is based on the fully parameterized Bayesian formulation using the likelihood
$\ell$ defined as [20]

$\ell=\Gamma\left(\frac{L}{2}\right)\frac{(2\pi\mathcal{E})^{-L/2}}{2}\,,$
(15)

where $\Gamma(\cdot)$ is the gamma function, $L$ is the number of EDF samples,
and $\mathcal{E}$ quantifies the error between the measured EDF $d(t)$ and the
model $d_{K}(t)$ as defined in Eq. (2):

$\mathcal{E}=\frac{1}{2}\sum_{t=1}^{L}\left[d(t)-d_{K}(t)\right]^{2}\,.$ (16)

No prior information about the parameter values $T_{i}$, $A_{i}$, and $N_{0}$
is available before the decay analysis. Therefore, we assign a uniform prior
and estimate the model parameters by maximizing the likelihood $\ell$ over the
parameter space.

For this purpose, we apply the slice sampling algorithm [20], because a grid
search over all parameter combinations would be computationally infeasible. In
our analysis, we let the slice sampling algorithm run for $50$ iterations and
restrict the search space as follows

$\displaystyle T_{i}$
$\displaystyle\in\,\mathcal{U}($0.1\text{\,}\mathrm{s}$,\,$3.5\text{\,}\mathrm{s}$)$
(17a) $\displaystyle A_{i}$ $\displaystyle=10^{a},$
$\displaystyle\textrm{with}\quad a$ $\displaystyle\in\,\mathcal{U}(-3,\,0)$
(17b) $\displaystyle N_{0}$ $\displaystyle=10^{a_{\textrm{noise}}},$
$\displaystyle\textrm{with}\quad a_{\textrm{noise}}$
$\displaystyle\in\,\mathcal{U}(-10,\,-2)\,,$ (17c)

where each dimension is discretized into $100$ points (equally-spaced in
$T_{i}$, logarithmically-spaced in $A_{i}$, and $N_{0}$). In the first
iteration, the decay parameters are initialized with random values from the
search space. The algorithm proceeds by repeatedly sampling each decay
parameter in turn.

In our evaluation, we use this framework to fit models of orders $K=1,2,3$ to
the measured EDFs. As suggested in previous work [19], we apply the Bayesian
information criterion (BIC)inverse Bayesian information criterion (IBIC) to
determine the lowest possible model order that fits the data well. Although it
differs from the regular BIC by a negative sign, the definitions are sometimes
used interchangeably. By choosing the model with the maximum BICIBIC, the
Bayesian framework balances the degree of fit and a potential over-
parameterization. The BICIBIC for the model of order $K$ is given as [19]

$\textrm{IBIC}_{K}=2\,\textrm{ln}(\hat{\ell}_{K})-(2K+1)\,\textrm{ln}(L)\,,$
(18)

where $\hat{\ell}_{K}$ is the maximum-likelihood determined with the slice-
sampling algorithm, and $L$ is the number of EDF samples.

###  V.2 Datasets

We use two large RIR datasets for our evaluation. The first dataset is the
_Motus dataset_ [43, 44], which contains 3320 RIRs measured in various
acoustic conditions. More precisely, the RIRs were measured in a single room
(approx. volume of $60\text{\,}\mathrm{m}^{3}$), where the furniture amount
and placement were varied between measurements to generate 830 unique
furniture combinations. The dataset features reverberation times between
$0.5\text{\,}\mathrm{s}$ and $2\text{\,}\mathrm{s}$ at
$1\text{\,}\mathrm{kHz}$. Due to the complex geometries, the dataset also
features acoustic wave phenomena like scattering. While most of its RIRs have
a single-slope characteristic, only approximately $40$ RIRs have pronounced
multi-slope EDFs due to non-uniform absorption material distributions.

The second dataset is the _Room Transition dataset_ [45, 46]. It contains
measurements of four room transitions with a spatial resolution of
$5\text{\,}\mathrm{cm}$, including positions with occluded line-of-sight
between source and receiver. Due to the room coupling, a large number of EDFs
in the dataset exhibit a multi-slope characteristic, where the amplitudes of
the slopes vary considerably with the receiver position between the rooms.
Furthermore, it was shown that the Room Transition dataset features complex
acoustic phenomena of coupled room transitions, such as the portaling effect
and distinctive direct-to-reverberant ratio transitions [45]. Our evaluation
excluded the transition “Office to anechoic chamber”, thus using only 1212
RIRs of the dataset. The remaining room transitions are “Meeting room to
hallway”, “Office to kitchen”, and “Office to stairwell”. We cut away the last
$0.1\text{\,}\mathrm{s}$ of all RIRs because they include a Hanning fade-out
window that disturbs the fitting process.

Both datasets contain higher-order Ambisonic RIR. The following analyses are
based on the omnidirectional channel and the six octave bands from
$125\text{\,}\mathrm{Hz}4000\text{\,}\mathrm{Hz}$. A preliminary analysis of
the estimated $\widehat{N}_{0}$ values showed that both datasets have similar
average signal-to-noise ratios of approximately $75\text{\,}\mathrm{dB}$.

###  V.3 Results

We evaluated the DecayFitNet with respect to its fitting performance and
computational complexity. The results are presented in the following section.

####  V.3.1 Example fits

Figure 2 shows example fits obtained with the proposed DecayFitNet for two
measured EDFs. The figure also includes the corresponding fits obtained with
the previously described grid search variant of the Karjalainen toolbox and
our implementation of the slice-sampling-based Bayesian decay analysis. All
approaches fit the single-slope EDF of Figure 2 equally well. Figure 2 depicts
the resulting fits for a multi-slope EDF. The fits obtained with the proposed
DecayFitNet architecture and the Bayesian decay analysis show good agreement
with the measured EDF. In contrast, the Karjalainen toolbox is based on a
single exponential plus noise model, thus being unable to fit EDFs with more
than one slope. In the depicted scenario, the Karjalainen toolbox returns a
slope between the two distinct slopes. Additionally, it overestimates the
noise floor.

Figure2a.pdf0.45(a) Figure2b.pdf0.45(b)

Figure 2: Examples of fitting (a) a single-slope (from Motus dataset:
measurement #$500$, loudspeaker $1$, $1\text{\,}\mathrm{kHz}$ octave-band) and
(b) a multi-slope EDF (from Room Transition dataset: Meeting room to hallway,
source in room, no line of sight, $25\text{\,}\mathrm{cm}$ position,
$1\text{\,}\mathrm{kHz}$ octave-band). The measured EDFs are fitted with the
proposed DecayFitNet, the grid-search variant of the Karjalainen toolbox [10],
and our implementation of the slice-sampling-based Bayesian decay analysis
[20]. Analogously to the rest of our evaluation, the last
$5\text{\,}\mathrm{\char 37\relax}$ of EDF samples are excluded from the plot.

####  V.3.2 Error analysis

Table 1: Median and $99\text{\,}\mathrm{\char 37\relax}$ quantiles of the mean squared error between analyzed EDF and estimated fits. We compare the proposed DecayFitNet with the publicly available toolbox by Karjalainen et al. [10] and our own implementation of the slice-sampling-based Bayesian decay analysis [20]. The different fitting approaches are evaluated on two large, publicly available datasets. Dataset |  | Karjalainen
---
(standard)
| Karjalainen
---
(grid)
| Bayesian
---
(50 iterations)
DecayFitNet
| median |  $99\text{\,}\mathrm{\char 37\relax}$ q. | median |  $99\text{\,}\mathrm{\char 37\relax}$ q. | median |  $99\text{\,}\mathrm{\char 37\relax}$ q. | median |  $99\text{\,}\mathrm{\char 37\relax}$ q.
Motus | $0.68\text{\,}\mathrm{dB}$ | $6.69\text{\,}\mathrm{dB}$ | $0.56\text{\,}\mathrm{dB}$ | $5.32\text{\,}\mathrm{dB}$ | $0.23\text{\,}\mathrm{dB}$ | $1.11\text{\,}\mathrm{dB}$ | $0.27\text{\,}\mathrm{dB}$ | $1.74\text{\,}\mathrm{dB}$
Room Transition | $0.89\text{\,}\mathrm{dB}$ | $45.56\text{\,}\mathrm{dB}$ | $0.73\text{\,}\mathrm{dB}$ | $22.53\text{\,}\mathrm{dB}$ | $0.35\text{\,}\mathrm{dB}$ | $1.89\text{\,}\mathrm{dB}$ | $0.47\text{\,}\mathrm{dB}$ | $2.75\text{\,}\mathrm{dB}$

Slice-sampling-based decay analysis was added to caption of Table I.Table 1
summarizes the fitting results on the entire datasets. The table shows medians
and $99\text{\,}\mathrm{\char 37\relax}$ quantile values of the mean squared
error (MSE) between measured octave-band filtered EDFs and obtained fits. The
last $5\text{\,}\mathrm{\char 37\relax}$ of EDF samples are excluded from the
MSE calculation, following the reasoning described in Section IV.2.
Furthermore, $11$ RIRs of the Motus dataset were excluded from the analysis
because they featured transient noise artifacts. Such artifacts introduce
large discontinuities into the corresponding EDFs, which violate the model of
Eq. (2). Consequently, all tested fitting approaches had large MSEs for EDFs
with such artefacts, allowing us to detect and exclude these measurements.
The table indicates that the Karjalainen toolbox, its grid-search variant, the
slice-sampling-based Bayesian analysis, and the proposed DecayFitNet
accurately fit the Motus dataset EDFs, with median MSEs of
$0.68\text{\,}\mathrm{dB}$ or lower. However, the $99\text{\,}\mathrm{\char
37\relax}$ quantile values indicate that the spread of achieved MSE values is
higher for the Karjalainen toolbox than for the proposed DecayFitNet and the
Bayesian analysis. This suggests that the proposed DecayFitNet achieves
slightly more robust fitting on a dataset with mostly single-slope EDFs than a
well-established single-slope fitting toolbox. Furthermore, its performance in
terms of median errors is comparable to an existing multi-slope fitting
approach. However, the DecayFitNet is slightly less robust than the Bayesian
decay analysis as indicated by the increased $99\text{\,}\mathrm{\char
37\relax}$ quantile value.

The results for the Room Transition dataset show a similar trend. While median
values do not differ considerably between the four approaches, the standard
Karjalainen toolbox and its grid-search variant exhibit increased
$99\text{\,}\mathrm{\char 37\relax}$ quantile values of
$45.56\text{\,}\mathrm{dB}$ and $22.53\text{\,}\mathrm{dB}$, respectively.
This considerable variability in fitting performance can be attributed to the
insufficient model order for fitting multi-slope EDFs. In contrast, the
$99\text{\,}\mathrm{\char 37\relax}$ quantile value for fitting the Room
Transition dataset with the proposed DecayFitNet is $2.75\text{\,}\mathrm{dB}$
and therefore only slightly higher than for the Motus dataset. The Bayesian
analysis achieves a similar performance, although being a bit more robust as
indicated by the lower $99\text{\,}\mathrm{\char 37\relax}$ quantile value.
Nevertheless, this result suggests that the DecayFitNet can robustly fit large
quantities of multi-slope EDFs.

Figure 3 supports these findings. It shows violin plots of the MSE values that
were the basis for the calculations of Table 1. Figure 3 shows that all
approaches achieve low MSE values for the Motus dataset across all frequency
bands. Most MSE values lie below $10\text{\,}\mathrm{dB}$, and the spread of
values below $10\text{\,}\mathrm{dB}$ is slightly bigger for the Karjalainen
toolbox than for the DecayFitNet and the Bayesian analysis. Figure 3 shows the
results for the Room Transition dataset. The plots exhibit larger MSE values
for the two Karjalainen toolbox approaches, with many data points well above
$10\text{\,}\mathrm{dB}$. A more thorough analysis of the high MSE values
revealed that they can be attributed to multi-slope EDFs, for which the model
order of the Karjalainen toolbox is too low. In contrast, the proposed
DecayFitNet and the Bayesian analysis achieve low fitting errors for the Room
Transition dataset, with all MSE values below $10\text{\,}\mathrm{dB}$. The
Bayesian decay analysis exhibits slightly reduced MSE value spreads compared
to the DecayFitNet.

Figure3a.pdf(a) Figure3b.pdf(b)

Figure 3: Violin plots of the mean squared error between measured octave-band-
filtered EDFs and the corresponding fits obtained with different approaches.
The evaluation is based on the (a) Motus dataset and the (b) Room Transition
dataset. Light gray circles indicate individual data points, and white circles
indicate the median. MSE values greater than $50\text{\,}\mathrm{dB}$ have
been excluded from the plot for clarity.

####  V.3.3 Computational efficiency analysis

The execution time of the DecayFitNet depends largely on the number of model
parameters, i.e., the learned weights and biases of the different layers. With
approximately $677\,000$ parameters, the DecayFitNet is rather lightweight and
could run on mobile devices.

Table 2 compares the computational efficiency of the evaluated decay analysis
approaches. It shows how much time the different approaches require to analyze
the entire Motus dataset of $19\,920$ EDFs ($3320$ RIRs $\times\ 6$ octave
bands). The evaluation was carried out on a modern laptop computer (model year
2019) without a dedicated graphics processing unit (GPU). Whereas the standard
version of the Karjalainen toolbox required almost two hours for the analysis,
the Bayesian analysis and the proposed DecayFitNet were considerably faster.
For the latter two approaches, we averaged the elapsed time of $100$ runs over
the entire dataset to account for temporary drops of processing power that
could bias the result. The table shows that the DecayFitNet achieves an almost
30-fold speed-up compared to the Bayesian analysis. This speed-up can be
attributed to the lightweight structure of the DecayFitNet and the iterative
nature of the slice-sampling-based Bayesian analysis.The table shows that the
neural-network-based analysis can process a large dataset with $19\,920$ EDFs
within seconds, which can be attributed to the lightweight structure of the
DecayFitNet. However, it is important to note that the presented numbers
should only be understood as an orientation for the reader. It is obvious that
optimized implementations are an interesting engineering task for future work,
which could reduce the computation times of all approaches even further. For
example, due to the CNN-based architecture, our approach can easily be scaled
up and parallelized to efficiently process large amounts of data with modern
GPUs.

Apart from the execution times, Table 2 also features two measures that
provide additional insights into the computational efficiency of the
approaches on other machines. The execution time of the slice-sampling-based
Bayesian analysis depends largely on how fast the algorithm converges to the
parameter space region surrounding the maximum likelihood. Therefore, we
determined the average number of likelihood evaluations for the analysis of a
single EDF. This measure indicates how many steps the algorithm takes through
the parameter space until a satisfactory solution is found.

Table 2: Runtime efficiency comparison of the evaluated decay analysis algorithms. For the Bayesian decay analysis and the DecayFitNet, the elapsed time for analyzing the entire Motus dataset ($19\,920$ EDFs in total) is calculated as the average over $100$ runs. Karjalainen (standard) |  |
---|---|---
|  $6281\text{\,}\mathrm{s}$ ($1.75\text{\,}\mathrm{h}$) |
Bayesian |  |
|  $733\text{\,}\mathrm{s}$ ($12.2\text{\,}\mathrm{min}$) |
DecayFitNet |  |
| $26.6\text{\,}\mathrm{s}$ |

##  VI Discussion

The results presented in Section V.3 indicate that the proposed neural network
structure can robustly fit single-slope and multi-slope EDFs on large datasets
without prior parameter tuning or supervision.

Our evaluation was based on two large datasets of more than $1000$ RIRs each,
corresponding to more than $20\,000$ EDFs across $6$ octave bands in total.
They feature various acoustic conditions, such as varying amounts and
placements of absorptive materials, diffraction and scattering from the room
geometry, and room coupling. On both evaluated datasets, the DecayFitNet and
the Bayesian decay analysis outperform the toolbox by Karjalainen et al. [10],
which is a well-established toolbox for fitting EDFs with a single slope and a
noise term. It is important to note that the Karjalainen toolbox cannot fit
multi-exponential decays, thus explaining the degraded performance on the Room
Transition dataset, which contains many multi-slope EDFs. In contrast, the
Karjalainen toolbox performed well on the Motus dataset, which primarily
contains single-slope decays. Nevertheless, it exhibited a considerably larger
spread of errors than the DecayFitNet and the Bayesian decay analysis. The
latter two approaches performed similarly well on both datasets, although the
Bayesian analysis is slightly more robust as indicated by the lower error
spreads.

Furthermore, it is interesting to note that the DecayFitNet is completely
deterministic at runtimeinference time, because it always applies the same set
of parameters to its inputs. This property limits the number of executed
operations to a fixed value and ensures that the same results are obtained for
repeated runs. In contrast, iterative algorithms like slice-sampling exhibit
some degree of uncertainty, depending on the random steps taken throughout the
execution. This uncertainty affects the consistency of results and the
execution time alike.

To the best of the authors’ knowledge, at this point, there is no study
examining the performance of state-of-the-art decay analysis algorithms on
large amounts of data. We found that the slice-sampling-based Bayesian
analysis performs well on large-scale datasets, despite the iterative nature
of the approach. As a fully-deterministic alternative approach, we have
proposed a neural network structure. With the presented DecayFitNet, potential
numerical difficulties have moved from the application (and the user) to the
network training stage. The network training has to be performed only once,
and the pre-trained network can subsequently be used by users without any
additional effort. This shift decreases the required user tuning and also
reduces the computational complexity at run-timeinference time. The efficiency
and low user tuning of the neural network structure are balanced by the
decreased possibilities of adjusting an incorrectly fitted EDF. While other
approaches have adjustable parameters to fix a wrong EDF fit, the output of
the neural network cannot be changed unless it is trained with new data.

Lastly, we found that the presented DecayFitNet achieves an almost 30-fold
speed-up compared to the slice-sampling-based Bayesian decay analysis.can
analyze a large dataset of almost $20\,000$ EDFs in less than
$30\text{\,}\mathrm{s}$. ThisThe high computational efficiency improvement is
an important step toward bringing decay analysis algorithms to mobile devices.
Furthermore, it benefits the processing of large datasets, where GPU-
accelerated machines can leverage the full potential of our CNN-based
architecture. In principle, the Bayesian formulation allows the user to find a
satisfactory fit for every EDF, as long as the parameter space is searched
long enough. In some cases, this may require increasing the number of
iterations in the slice-sampling algorithm, thus also increasing the
computational costs. In our experiments, we found that $50$ iterations
sufficiently explore the parameter space to yield good EDF fits in most cases,
whereas the fitting performance degraded quickly with a decreasing number of
iterations. While the fitting performance degraded quickly with a decreasing
number of iterations, the execution time remained high compared to the
proposed neural network approach. This can be explained by the iterative
exploration of the parameter space, where the first iterations evaluate the
likelihood function more often than latter iterations.

##  VII Decay Analysis Toolbox

As part of this work, we provide an open source toolbox for both Python and
Matlab. The toolbox includes a pretrained neural network model, a code
interface to interact with the model, our implementation of the slice-
sampling-based Bayesian decay analysis, as well as some utility functions. The
main goal of the toolbox is to present a simple package to estimate multi-
slope EDFs that can be used with minimal code. Although a pretrained model is
provided, the toolbox also includes the ability to train and export a model
using a custom dataset. This could be useful in scenarios where the
generalization of the pretrained model is not accurate enough. The source code
and documentation for the toolbox are available at https://github.com/georg-
goetz/DecayFitNet/.

Figure 4:  Workflow and structure of the Decay Analysis Toolbox to estimate
the EDF parameters from a RIR.

The structure and workflow of the toolbox are presented in Figure 4. The main
functionality includes:

  * •

Impulse responses pre-processing for the neural-network-based decay analysis
(as described in Section IV.2) .

  * •

Estimation of parameters $T_{i}$, $A_{i}$, and $N_{0}$ from Eq. (2) by doing a
forward pass of the pre-trained model, or by applying the slice-sampling-based
Bayesian decay analysis.

  * •

Generation of the fitted EDF using the estimated parameters.

  * •

Training and export of the DecayFitNet neural network model.

The toolbox exports the neural network model and operates using the Open
Neural Network Exchange format (ONNX) [47]. ONNX is an open and common format
that allows for fully-trained machine learning models to be distributed and
utilised by a variety of frameworks and platforms. While the toolbox only
includes interfaces for Python and Matlab, a trained model exported to the
ONNX format can be supported by many applications, requiring only the porting
of the pre-processing and EDF generation code. The provided model was trained
using the synthetic dataset described in Section IV.4.

##  VIII Conclusions

This paper proposed a neural network structure for fitting multi-exponential
sound energy decay functions (EDFs). It was shown that such a network can be
trained on a dataset of synthetically generated EDFs. We presented the
DecayFitNet as an example architecture of the proposed approach. A large-scale
evaluation applied the DecayFitNet and two comparable state-of-the-art methods
on two large datasets of real-world measurements with more than $1000$ RIRs
each, corresponding to over $20\,000$ EDFs across $6$ octave bands. The
analyzed datasets featured various acoustic conditions, such as varying
amounts of absorptive material, diffraction and scattering from the room
geometry, and room coupling.

The results of our evaluation indicate that the proposed neural network
structure can robustly fit single-slope and multi-slope EDFs without prior
parameter tuning or supervision by the user. Additionally, the presented
DecayFitNet is fully-deterministic during inference time and computationally
efficient, being capable of analyzing almost $20\,000$ EDFs in less than $30$
seconds on a modern laptop computer (2019). Our evaluation indicates that the
DecayFitNet achieves an almost 30-fold speed-up compared to state-of-the-art
multi-slope decay analysis algorithmsrobustly estimates the model parameters
from large datasets of measured EDFs, while exhibiting a comparablywhile
achieving a comparable but slightly less robust fitting performance compared
to state-of-the-art multi-slope decay analysis algorithms. A decay analysis
toolbox has been made publicly available for the audio community.

The DecayFitNet and its corresponding toolbox may benefit future research on
room acoustic analysis and modeling. Data-heavy machine-learning-based
approaches may leverage its full potential regarding computational efficiency
and robustness.

## Acknowledgments

The project has received funding from from the Academy of Finland, project no.
317341. We acknowledge the computational resources provided by the Aalto
Science-IT project.

## References

  * [1] H. Kuttruff, _Room Acoustics_ , 4th ed. (Spon Press, London, UK, 2000).
  * [2] M. R. Schroeder, “New Method of Measuring Reverberation Time,” J. Acoust. Soc. Am. 37(3), 409–412 (1965) 10.1121/1.1909343.
  * [3] ISO 3382-1, “Acoustics - Measurement of room acoustic parameters - Part 1: Performance Spaces,” Standard, International Organization for Standardization (ISO), Geneva, Switzerland (2009).
  * [4] ISO 3382-2, “Acoustics - Measurement of room acoustic parameters - Part 2: Reverberation time in ordinary rooms,” Standard, International Organization for Standardization (ISO), Geneva, Switzerland (2008).
  * [5] W. T. Chu, “Comparison of reverberation measurements using Schroeder’s impulse method and decay‐curve averaging method,” J. Acoust. Soc. Am. 63(5), 1444–1450 (1978) 10.1121/1.381889.
  * [6] D. R. Morgan, “A parametric error analysis of the backward integration method for reverberation time estimation,” J. Acoust. Soc. Am. 101(5), 2686–2693 (1997) 10.1121/1.418557.
  * [7] A. Lundeby, T. E. Vigran, H. Bietz, and M. Vorländer, “Uncertainties of Measurements in Room Acoustics,” Acta Acustica united with Acustica 81(4), 344–355 (1995).
  * [8] M. Guski and M. Vorländer, “Measurement Uncertainties of Reverberation Time caused by Noise,” in _Proceedings of the International Conference on Acoustics (ICA), the 39th annual congress of DEGA, and the 40th annual congress of AIA_ , DEGA, Merano, Italy (2013), pp. 2067–2070.
  * [9] N. Xiang, “Evaluation of reverberation times using a nonlinear regression approach,” J. Acoust. Soc. Am. 98(4), 2112–2121 (1995) 10.1121/1.414460.
  * [10] M. Karjalainen, P. Antsalo, A. Mäkivirta, T. Peltonen, and V. Välimäki, “Estimation of Modal Decay Parameters from Noisy Response Measurements,” J. Audio Eng. Soc. 50(11), 867–878 (2002).
  * [11] C. F. Eyring, “Reverberation time measurements in coupled rooms,” J. Acoust. Soc. Am. 3(2), 181–206 (1931) 10.1121/1.1915555.
  * [12] F. V. Hunt, L. L. Beranek, and D. Y. Maa, “Analysis of Sound Decay in Rectangular Rooms,” J. Acoust. Soc. Am. 11(1), 80–94 (1939) 10.1121/1.1916010.
  * [13] E. Nilsson, “Decay Processes in Rooms with Non-Diffuse Sound Fields Part I: Ceiling Treatment with Absorbing Material,” Building Acoustics 11(1), 39–60 (2004) 10.1260/1351010041217220.
  * [14] N. Xiang and P. M. Goggans, “Evaluation of decay times in coupled spaces: Bayesian parameter estimation,” J. Acoust. Soc. Am. 110(3), 1415–1424 (2001) 10.1121/1.1390334.
  * [15] N. Xiang, P. M. Goggans, T. Jasa, and M. Kleiner, “Evaluation of decay times in coupled spaces: Reliability analysis of Bayeisan decay time estimation,” J. Acoust. Soc. Am. 117(6), 3707–3715 (2005) 10.1121/1.1903845.
  * [16] J. Balint, F. Muralter, M. Nolan, and C.-H. Jeong, “Bayesian decay time estimation in a reverberation chamber for absorption measurements,” J. Acoust. Soc. Am. 146(3), 1641–1649 (2019) 10.1121/1.5125132.
  * [17] H. Pu, X. Qiu, and J. Wang, “Different sound decay patterns and energy feedback in coupled volumes,” J. Acoust. Soc. Am. 129(4), 1972–1980 (2011) 10.1121/1.3553223.
  * [18] Z. Sü Gül, E. Odabaş, N. Xiang, and M. Çalışkan, “Diffusion equation modeling for sound energy flow analysis in multi domain structures,” J. Acoust. Soc. Am. 145(4), 2703–2717 (2019) 10.1121/1.5095877.
  * [19] N. Xiang, P. Goggans, T. Jasa, and P. Robinson, “Bayesian characterization of multiple-slope sound energy decays in coupled-volume systems,” J. Acoust. Soc. Am. 129(2), 741–752 (2011) 10.1121/1.3518773.
  * [20] T. Jasa and N. Xiang, “Efficient estimation of decay parameters in acoustically coupled-spaces using slice sampling,” J. Acoust. Soc. Am. 126(3), 1269–1279 (2009) 10.1121/1.3158934.
  * [21] D. Cabrera, J. Xun, and M. Guski, “Calculating Reverberation Time from Impulse Responses: A Comparison of Software Implementations,” Acoustics Australia 44(2), 369–378 (2016) 10.1007/s40857-016-0055-6.
  * [22] L. Álvarez-Morales, M. Galindo, S. Girón, T. Zamarreño, and R. M. Cibrián, “Acoustic Characterisation by Using Different Room Acoustics Software Tools: A Comparative Study,” Acta Acustica united with Acustica 102(3), 578–591 (2016) 10.3813/aaa.918975.
  * [23] B. F. G. Katz, “International Round Robin on Room Acoustical Impulse Response Analysis Software 2004,” Acoustics Research Letters Online 5(4), 158–164 (2004) 10.1121/1.1758239.
  * [24] M. Alam, M. Samad, L. Vidyaratne, A. Glandon, and K. Iftekharuddin, “Survey on deep neural networks in speech and vision systems,” Neurocomputing 417, 302–321 (2020) 10.1016/j.neucom.2020.07.053.
  * [25] T. Young, D. Hazarika, S. Poria, and E. Cambria, “Recent trends in deep learning based natural language processing” (2017) Available online: https://arxiv.org/abs/1708.02709.
  * [26] H. Purwins, B. Li, T. Virtanen, J. Schluter, S.-Y. Chang, and T. Sainath, “Deep learning for audio signal processing,” IEEE Journal of Selected Topics in Signal Processing 13(2), 206–219 (2019) 10.1109/jstsp.2019.2908700.
  * [27] P.-A. Grumiaux, S. Kitić, L. Girin, and A. Guérin, “A Survey of Sound Source Localization with Deep Learning Methods” (2021) Available online: https://arxiv.org/abs/2109.03465.
  * [28] J. Eaton, N. D. Gaubitch, A. H. Moore, and P. A. Naylor, “Estimation of room acoustic parameters: The ACE challenge,” IEEE/ACM Trans. Audio, Speech, Language Process. 24(10), 1681–1693 (2016) 10.1109/TASLP.2016.2577502.
  * [29] Z. Fan, V. Vineet, H. Gamper, and N. Raghuvanshi, “Fast Acoustic Scattering Using Convolutional Neural Networks,” in _Int. Conf. Acoust., Speech, Sig. Proc. (ICASSP)_ , Online conference (2020), pp. 171–175, 10.1109/icassp40776.2020.9054091.
  * [30] S. Wirler, S. J. Schlecht, and V. Pulkki, “Machine Learning Based Auralization of Rigid Sphere Scattering,” in _International Conference on Immersive and 3D Audio (I3DA)_ , Online conference (2021), 10.1109/i3da48870.2021.9610951.
  * [31] Z. Fan, V. Vineet, C. Lu, T. W. Wu, and K. McMullen, “Prediction of object geometry from acoustic scattering using convolutional neural networks,” in _Int. Conf. Acoust., Speech, Sig. Proc. (ICASSP)_ , Online conference (2021), pp. 471–475, 10.1109/ICASSP39728.2021.9414743.
  * [32] M. J. Bianco, P. Gerstoft, J. Traer, E. Ozanich, M. A. Roch, S. Gannot, and C.-A. Deledalle, “Machine learning in acoustics: Theory and applications,” The Journal of the Acoustical Society of America 146(5), 3590–3628 (2019) 10.1121/1.5133944.
  * [33] I. J. Goodfellow, Y. Bengio, and A. Courville, _Deep Learning_ (MIT Press, Cambridge, MA, USA, 2016) Available online: http://www.deeplearningbook.org.
  * [34] M. Fernández-Delgado, M. Sirsat, E. Cernadas, S. Alawadi, S. Barro, and M. Febrero-Bande, “An extensive experimental survey of regression methods,” Neural Networks 111, 11–34 (2019) 10.1016/j.neunet.2018.12.010.
  * [35] D. Scherer, A. Müller, and S. Behnke, “Evaluation of Pooling Operations in Convolutional Architectures for Object Recognition,” in _Proceedings of the International Conference on Artificial Neural Networks (ICANN), Part III_ , Lecture Notes in Computer Science, Springer, Thessaloniki, Greece (2010), pp. 92–101, 10.1007/978-3-642-15825-4_10.
  * [36] V. Nair and G. E. Hinton, “Rectified Linear Units Improve Restricted Boltzmann Machines,” in _Proceedings of the 27th International Conference on Machine Learning (ICML-10)_ , Haifa, Israel (2010), pp. 807–814.
  * [37] A. Pinkus, “Approximation theory of the MLP model in neural networks,” Acta Numerica 8, 143–195 (1999).
  * [38] Y. Lecun and Y. Bengio, “Convolutional networks for images, speech and time series,” in _The Handbook of Brain Theory and Neural Networks_ , edited by M. A. Arbib (The MIT Press, 1995), pp. 255–258.
  * [39] H. Bilen and A. Vedaldi, “Integrated perception with recurrent multi-task neural networks,” in _Proceedings of the 30th International Conference on Neural Information Processing Systems_ , NIPS’16, Curran Associates Inc., Red Hook, NY, USA (2016), p. 235–243.
  * [40] D. P. Kingma and J. L. Ba, “Adam: A Method for Stochastic Optimization,” in _International Conference on Learning Representations (ICLR)_ , San Diego, CA, USA (2015).
  * [41] I. Loshchilov and F. Hutter, “Decoupled Weight Decay Regularization,” in _International Conference on Learning Representations (ICLR)_ , New Orleans, LA, USA (2019).
  * [42] I. Loshchilov and F. Hutter, “SGDR: Stochastic Gradient Descent with Warm Restarts,” in _International Conference on Learning Representations (ICLR)_ , Toulon, France (2017).
  * [43] G. Götz, S. J. Schlecht, and V. Pulkki, “A dataset of higher-order Ambisonic room impulse responses and 3D models measured in a room with varying furniture,” in _International Conference on Immersive and 3D Audio (I3DA)_ , Online conference (2021), pp. 1–8, 10.1109/i3da48870.2021.9610933.
  * [44] G. Götz, S. J. Schlecht, and V. Pulkki, “Motus: A dataset of higher-order Ambisonic room impulse responses and 3D models measured in a room with varying furniture (Version 1.0)” (2021), 10.5281/zenodo.4923187 Zenodo.
  * [45] T. McKenzie, S. J. Schlecht, and V. Pulkki, “Acoustic Analysis and Dataset of Transitions Between Coupled Rooms,” in _Int. Conf. Acoust., Speech, Sig. Proc. (ICASSP)_ , Online conference (2021), pp. 481–485, 10.1109/icassp39728.2021.9415122.
  * [46] T. McKenzie, S. J. Schlecht, and V. Pulkki, “A dataset of measured spatial room impulse responses for the transition between coupled rooms (Version 1.2)” (2021), 10.5281/zenodo.4636068 Zenodo.
  * [47] J. Bai, F. Lu, K. Zhang, _et al._ , “ONNX: Open neural network exchange” , https://github.com/onnx/onnx (2019).