Exponentially-Weighted Energy Dispersion Index for the Nonlinear Interference Analysis of Finite-Blocklength Shaping

Kaiquan Wu, Student Member, IEEE
Gabriele Liga, Member, IEEE
Yunus Can Gültekin Member, IEEE, and
Alex Alvarado Member, IEEE

Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
Corresponding author: K. Wu (email: k.wu@tue.nl).

A metric called exponentially-weighted energy dispersion index (EEDI) is proposed to explain the blocklength-dependent effective signal-to-noise ratio (SNR) in probabilistically shaped fiber-optic systems. EEDI is better than energy dispersion index (EDI) at capturing the dependency of the effective SNR on the blocklength for long-distance transmission.

§ INTRODUCTION

Probabilistic amplitude shaping (PAS) [1] can realize near capacity-achieving transmission for the additive white Gaussian noise (AWGN) channel. In fiber optical communications, significant shaping gains over uniform quadrature amplitude modulation (QAM) are achieved by the AWGN-optimal PAS [2, 3]. However, these gains are undermined by the nonlinear interference (NLI) penalty, since shaping can enhance NLI effects with respect to uniform signaling [4]. This penalty is enhanced by i) using a Maxwell-Boltzmann distribution, and ii) the temporal structure of the transmitted shaped symbols. The penalty from the former can be reduced by optimizing the distribution of the constellation symbols to be more NLI-tolerant [4, 5, 6]. The penalty from the latter can be instead mitigated by manipulating the temporal structure of the symbol sequences [7, 8]. The amplitude shaper in PAS imposes a hidden temporal structure on the symbols, and thus, the symbols can no longer be treated as independent identically distributed (i.i.d.). It was found that the temporal structure caused by short shaping blocklengths can provide effective signal-to-noise ratio (SNR) gains, due to a weaker presence of nonlinearities [9]. Therefore, a straightforward approach would be to simply employ short blocklengths [10, 11]. The NLI mitigation is intuitively explained by the fact that using short blocklengths avoids multiple consecutive occurrences of high-energy symbols, and thus, induces less NLI [12, 13].

Recently, in [14] we analyzed the statistical properties of symbols generated by constant composition distribution matching (CCDM) [15] with finite blocklengths. Inspired by the behavior of time-domain first-order perturbation NLI models [16] and the finite memory Gaussian noise (GN) model [17], we found that the variations of the windowed symbol energy are crucial for the NLI generation. We also proposed in [14] a precise metric to quantify the effect of energy variations on the NLI magnitude, the so-called energy dispersion index (EDI). One drawback of EDI is that all symbol energies within a time window are assumed to be equally important. This assumption does not reflect the reality as interfering symbols far away from the symbol of interest are expected to have smaller impact on the NLI impinging on that symbol than those nearby.

In this paper, we propose a refined version of the EDI, which we call exponentially-weighted EDI or EEDI. EEDI takes into account the fact that the NLI contribution from different symbol energies varies depending on their relative delay with respect to the symbol of interest. Our contribution in this paper is to verify that by weighting the interfering symbol energies properly, EEDI is a better effective SNR estimator than EDI at long transmission distances.

§ WEIGHTED ENERGY IN FIBER CHANNEL MODEL

In our previous study, we assumed that the NLI generation is dominated by symbol energies within a finite time window [14]. In this paper, we extend this time window to infinity and we introduce a decay factor to weigh the effect of temporal separation between symbols on the NLI generation. Using the first-order perturbation model and assuming single channel transmission, the NLI term $Z_{\text{NLI},0}$, which is modeled as additive noise on the transmitted symbol $X_0$, can be expressed as <cit.>

\begin{equation}\label{KnMd0}
Z_{\text{NLI},0}= \jmath \gamma \sum_{h=-\infty}^{\infty} \sum_{j=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} \! S_{h, j, l} X_{h} X_{j} X_{l}^{*}.
\end{equation}

In (<ref>), $\gamma$ is the nonlinear coefficient, and the complex kernels $S_{h, j, l}$ determine the self-phase modulation contribution of the symbol triple product $X_{h} X_{j} X_{l}^{*}$ based on the temporal separation of its factors. Note that the NLI term for the cross-channel interference can be expressed in the same form as in (<ref>) <cit.>. Since the perturbation terms $S_{h, j, l}$ satisfying $j=l$ have large magnitude <cit.>, we can approximate (<ref>) as

\begin{equation}\label{pmd}
Z_{\text{NLI},0}\approx \jmath \gamma \, \sum_{h=-\infty}^{\infty} X_{h} \underbrace{\sum_{l=-\infty}^{\infty} S_{h, l, l} |X_{l}|^2.}_\text{Weighted Sum of Energies}
\end{equation}

It can be seen in (<ref>) that the NLI experienced by $X_{0}$ is determined by the weighted sum of symbol energies. The variance of this term determines the variance of the induced NLI. This term indicates that all transmitted symbols generate NLI proportional to their energies.
However, these energies are weighted by $S_{h, l, l}$, whose magnitude change as a function of the index $l$ [18]. In general, $S_{h, l, l}$ slowly decays as the offset $|l|$ increases [7].

§ EXPONENTIALLY-WEIGHTED EDI

An illustration of the weighted sum of energy $G_i^\lambda$

To reflect the effect of the $S_{h, l, l}$ in (<ref>), we heuristically assume that its magnitude decays exponentially with increasing $|l|$. Let $\lambda$ be a forgetting factor, where $0\leq\lambda\leq 1$, and let $X_{l}$ be the symbol $|l|$ symbol periods away from $X_{0}$. We assume that the NLI contributions associated with $|X_{l}|^2$ can be expressed as $|S_{h, l, l}|=\lambda^{|l|} |S_{h, 0, 0}|$.
Then, we design a variable $G_i^\lambda$ to capture the weighted sum of symbol energies around symbol $X_i$, which is defined as

\begin{equation}\label{weng}
G_i^\lambda\define\sum_{l=-\infty}^{\infty} \lambda^{|l|} |X_{i+l}|^2.
\end{equation}

We can then express the NLI $Z_{\text{NLI},i}$ in the product between $G_i^{\lambda}$ and $\sum_{h=-\infty}^{\infty} X_{h+i}|S_{h+i, 0, 0}|$ which is still expected to be somehow correlated to $G_i^{\lambda}$.

Fig. <ref> illustrates how $G_i^\lambda$ is obtained. The central symbol is the most important in the NLI generation (with red color), and thus, is weighted by $1$. The contribution of the adjacent symbols that are further away by $|l|$ symbol periods, which is determined by $\lambda^{|l|}$, decays exponentially (shown with a faded color). As $i$ changes, the weighted infinite-window will slide through the symbols to obtain a number of weighted energies $G_i^\lambda$. Then, the EEDI $\hat{\Psi}_\text{Exp}$ is defined as the sample variance over the sample mean of the weighted sum of energy, i.e.,

\begin{equation}\label{expEDI}
\hat{\Psi}_\text{Exp} \define \frac{\TVar{\WE}}{\TExp{\WE}}.
\end{equation}

For $\lambda=1$, since all the symbol energies are weighted by $1$, EEDI is equivalent to EDI with infinite widow, and thus $\hat{\Psi}_\text{Exp}=0$. For $\lambda=0$, $\hat{\Psi}_\text{Exp}$ will converge to $\Exp{|X|^2}(\Phi-1)$, where $\Phi$ represents the standardized fourth moment (a.k.a. kurtosis) of the input symbols. Kurtosis is the NLI predictor proposed by the enhanced GN model, which assumes only i.i.d. transmitted symbols [19]. EEDI can be viewed as a refined version of kurtosis that can account for the interaction between non-i.i.d. symbols and the channel memory.

Effective SNR (left axis), EEDI, and EDI in [14] (right axis) vs. blocklength. The transmission distances are 80, 320 and 1600 km with the launch powers $-1.5$, $-2.0$ and $-3.0$ dBm, respectively. The EEDI is shown in dB and inverted for convenience of comparison. Error bars for effective SNRs represent 95% confidence interval. The circled areas in (b)–(c) show the nonlinear behavior of EEDI and effective SNR.

§ NUMERICAL RESULTS

Similar to our previous work [14], we study how well EEDI can predict the blocklength-dependent effective SNR. A single-polarization multi-span wavelength division multiplexing (WDM) system is simulated by using the split-step Fourier method. The system has a span length of 80 km, a fiber loss of 0.19 dB/km, a dispersion parameter of 17 ps/nm/km, and a nonlinear parameter of 1.37 1/W/km. Moreover, a $5\times32$ Gbaud 64-QAM transmission with root-raised-cosine pulse with 10% roll-off and 50 GHz spacing is considered. The central channel is the one of interest for our analysis. The attenuation after each span is compensated by an erbium-doped fiber amplifier (EDFA) with a noise figure of 6 dB. At the receiver, the channel of interest is processed using ideal chromatic dispersion compensation, matched filtering and sampling.

For the PAS with 64-QAM, we employ CCDM ranging from ultra short ($n=10$) to long ($n=10,000$) blocklengths. The amplitude distribution $[0.4,0.3,0.2,0.1]$ is used over the amplitudes $\{1, 3, 5, 7\}$. The amplitudes on in-phase and quadrature dimensions are independently generated. For each transmission, the same blocklength $n$ is used for all WDM channels. At the transmitter, we measure the EEDI of the symbols. At the receiver, we evaluate the effective SNR. Pairs of EEDI and effective SNR at different blocklengths are used to obtain their Pearson's correlation coefficient <cit.> $r_p$. The absolute value of coefficient $|r_p|=1$ indicates perfect correlation, while $|r_p|=0$ indicates no correlation.

EEDI in Fig. <ref> is computed using the optimal forgetting factor, which will be discussed in Fig. <ref>. Fig. <ref> shows that both EEDI and EDI predict effective SNR well with absolute correlation coefficient $|r_p|>0.99$ for three distances. To the left of the vertical dotted line, the effective SNR decreases significantly as blocklength $n$ increases. We call this the blocklength-dependent region [14]. Then, the effective SNR decreases slowly until it reaches a floor. For the sake of comparison with the effective SNR, the EEDI is shown in dB and its $y$-axis is inverted, and the EDI from our previous work [14] is shifted vertically by a constant such that it is aligned with EEDI at $n=10,000$. EEDI and EDI have very similar performance in terms of $|r_p|$. However, compared to EDI, one noticeable improvement of EEDI is that the nonlinear decrease of effective SNR at short blocklengths is much better predicted. This nonlinear decrease can be seen by the circled areas in Fig. <ref> (b)–(c). By contrast, EDI only predicts a linear decay of the effective SNR in these regions.

The optimal forgetting factor $\lambda^{*}$ used in Fig. <ref> was chosen such that the $|r_p|$ between EEDI and the effective SNR is maximized. To this end, as shown in Fig. <ref>, $\lambda^{*}$ is obtained by exhaustive search from $0.6\leq \lambda < 1$ at a step size of $10^{-4}$. Note that the $x$-axis in Fig. <ref> represents $1-\lambda$, and $|r_p|$ reaches its peak at $1-\lambda^{*}$. As distance increases from 80 to 1600 km, $\lambda^{*}$ increase from $0.9014$ to $0.9921$, which means that the decaying becomes slower, and more symbols are heavily involved in the nonlinear interaction. For all investigated cases, $|r_p|$ peaks at values very close to $1$, indicating almost perfect correlation between EEDI and effective SNR for $\lambda=\lambda^{*}$.

Finally, Fig. <ref> shows $1-\lambda^{*}$ at various distances. The $\lambda^{*}$ at each distance is obtained with $|r_p|$ at least $0.994$. Fig. <ref> shows that as the transmission distance increases from 80 km to 400 km, $1-\lambda^{*}$ decreases significantly and begins to decrease at a slower rate. The inset figures of Fig. <ref> show that at 80 and 1600 km, around $30$ and $404$ symbol energies are weighted more than $20\%$, respectively.

Absolute value of Pearson's linear correlation coefficient $|r_p|$ between EEDI and effective SNR vs. $1-\lambda$. The optimal value of $\lambda$ is denoted by $\lambda^{*}$.

$1-\lambda^{*}$ (all $\lambda^{*}$ obtained with $|r_p|> 0.994$) at distances from 80 km to 1600 km.

§ CONCLUSIONS

We conclude that by using exponential weighting method, EEDI evaluated with the optimal forgetting factor is capable of reflecting the impact of blocklength and distance on the NLI. In addition, EEDI shows superior performance over EDI in terms of predicting the effective SNR for long-distance transmission. Future work will focus on the robustness of the EEDI at longer transmission distances and larger WDM bandwidth.

§ ACKNOWLEDGMENTS

The work of K. Wu, Y.C. Gültekin and A. Alvarado has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 757791). The work of G. Liga has received funding from the EuroTechPostdoc programme under the European Union's Horizon 2020 research and innovation programme (Marie Skłodowska grant agreement No 754462).

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# DATA FUSION FOR AUDIOVISUAL SPEAKER LOCALIZATION:
EXTENDING DYNAMIC STREAM WEIGHTS TO THE SPATIAL DOMAIN

###### Abstract

Estimating the positions of multiple speakers can be helpful for tasks like
automatic speech recognition or speaker diarization. Both applications benefit
from a known speaker position when, for instance, applying beamforming or
assigning unique speaker identities. Recently, several approaches utilizing
acoustic signals augmented with visual data have been proposed for this task.
However, both the acoustic and the visual modality may be corrupted in
specific spatial regions, for instance due to poor lighting conditions or to
the presence of background noise. This paper proposes a novel audiovisual data
fusion framework for speaker localization by assigning individual dynamic
stream weights to specific regions in the localization space. This fusion is
achieved via a neural network, which combines the predictions of individual
audio and video trackers based on their time- and location-dependent
reliability. A performance evaluation using audiovisual recordings yields
promising results, with the proposed fusion approach outperforming all
baseline models.

Index Terms—  audiovisual speaker localization, data fusion, dynamic stream
weights

##  1 Introduction

Data fusion is a process aimed at combining several data sources to accomplish
a task more reliably than it would be possible with just one individual
sensor. This has several useful applications, ranging from medical diagnostics
[1] to robotics [2] and smart buildings [3]. Tracking one or more speakers by
means of acoustic and visual sensors also relies on an appropriate fusion of
sensory inputs. This process is performed unconsciously in human listening,
since humans naturally combine visual and acoustic stimuli in estimating the
position of a person speaking or a sound source emitting noise [4].

Recent works in the domain of audiovisual speaker localization showed
promising results utilizing different data fusion strategies. For example, [5,
6] introduce a variational Bayesian approximation to optimally merge acoustic
and visual data for combined localization and tracking. A related approach
introduced in [7] uses an expectation maximization (EM) algorithm for weighted
clustering in the audiovisual observation space. Other probabilistic
techniques for audiovisual speaker tracking based on particle filters were
proposed in [8, 9]. Additionally, the work in [10] combined audiovisual
speaker localization with speech separation on a robotic platform in a real-
time processing system. A similarly designed audiovisual fusion system for
smart rooms [11] allows for joint localization and identification of persons
present in a room.

Fig. 1: Frame from the AVDIAR dataset [12], overlayed with speaker positions
predicted by the proposed fusion model. Note that the model utilizes a
discrete grid to represent the localization space, where each grid-cell is
associated with a speaker presence probability, described in Sec. 2.3.

A classical approach to combine different sensors is the Kalman filter (KF)
[13], where inference in a linear dynamical system (LDS) is carried out to
estimate all latent variables of interest; such as positions, velocities or
accelerations, based on noisy measurement data. Extending this concept to
dynamic stream weights (DSWs) allows us to weight the contributions of each
input based on their instantaneous reliability. This concept was originally
proposed in the context of audiovisual automatic speech recognition (ASR)
[14], and has proven valuable for speaker identification [15], but was also
recently adopted for speaker localization and tracking [16, 17].

All of these approaches specify DSWs as time-dependent weighting factors for
the acoustic and visual modalities. In this study, we introduce a fundamental
update: Since the reliability of the acoustic and visual streams may also vary
over time and _location_ , we extend the idea of DSWs to the spatial
dimension, as depicted in Fig. 1. Therefore, we introduce a spatial weighting
matrix to replace the scalar DSWs used in previous works. Compared to
previously proposed approaches, spatial DSWs can effectively capture specific
position-dependent sensor reliability properties, e.g., bright lighting
conditions coming from a certain direction or directional acoustic noise
sources. Whereas scalar DSWs can only provide a global estimate of sensor
reliability, the model proposed here provides a more fine-grained location-
dependent framework for data fusion.

##  2 System description

The proposed data fusion network consists of two building-blocks: the
individual unimodal tracking networks and the DSW-based data fusion network,
as depicted in Fig 2.

###  2.1 Audio tracker

We adopt the unimodal audio tracker from the DOAnet architecture [18]. Here, a
multi-channel short-time Fourier transform (STFT) is performed on the
individual microphone signals. Time-frequency analysis is conducted on the
6-channel acoustic inputs using a window length of 1920 samples with no
overlap and a fast Fourier transform (FFT) size of 2048. This results in a
frame length of $40\,\text{ms}$, matching the length of one video frame in the
utilized dataset. A context of one frame in both temporal directions around
the $t$-th time step is used, which results in a 3-dimensional acoustic input
tensor $\boldsymbol{\mathrm{X}}_{\text{A},t}$ of dimensions $3\times
1024\times 12$, representing the time, frequency and channel dimensions. The
third dimension is spanned by the stacked amplitude and phase of all 6 audio
channels.

Similar to the DOAnet architecture, the acoustic input tensor is fed through 4
convolutional stages first. Herein, each stage is composed of a 2-dimensional
convolution with $64$ filters and a kernel size of $3\times 3$, batch
normalization, rectified linear unit (ReLU) nonlinearity, max-pooling of size
$1\times 8$, and a dropout layer with a rate of $0.5$. Subsequently, the
tensor is flattened, yielding a dimensionality of $3\times 128$, and input to
a bidirectional gated recurrent unit (GRU) with two layers. The output of
these recurrent layers is then fed to a classification network, composed of
three fully connected layers with $2048$, $1024$ and $512$ neurons and ReLUs,
respectively. Lastly, the output layer is constructed using another fully
connected layer with $480$ neurons and sigmoid activations. The outputs are
reshaped to a grid with $20\times 24$ cells, as depicted in Fig. 1. The grid
size was chosen to roughly represent the size of a face in the utilized
dataset. Using a binary indicator variable $\zeta_{i,j}$ corresponding to the
$i$-th row and $j$-th column of the grid, the probability that a speaker is
located at this particular grid point given only the current acoustic
observation can be expressed as
$z_{\text{A},t}^{(i,j)}=p(\zeta_{i,j}|\boldsymbol{\mathrm{X}}_{\text{A},t})$.

###  2.2 Video tracker

Video tracking is conducted using the _you only look once_ (YOLO) framework
proposed in [19], which achieves a high detection rate in combination with
real-time tracking capabilities. The implementation of YOLOFace is used, which
is based on YOLOv3 [20]. Before an input video frame
$\boldsymbol{\mathrm{X}}_{\text{V},t}$ is processed by the video tracker, it
is first resized to $832\times 832$ pixels to meet the requirements of YOLOv3.
We adapt the bounding box output to the $20\times 24$ grid size. The grid size
can be set to arbitrary values if one is willing to retrain the network. We
evaluated the localization performance using several grid sizes. However, we
did not find any significant differences in localization performance. We
decided to set the size of the individual grid-cells to roughly match the size
of a human face in the dataset. The video tracker’s output probability is
denoted as
$z_{\text{V},t}^{(i,j)}=p(\zeta_{i,j}|\boldsymbol{\mathrm{X}}_{\text{V},t})$.

Fig. 2: High-level overview of the proposed data fusion framework. The
acoustic and visual inputs $\boldsymbol{\mathrm{X}}_{\text{A},t}$ and
$\boldsymbol{\mathrm{X}}_{\text{V},t}$ are first processed via unimodal
trackers and then combined in a dedicated fusion network, producing a grid of
speaker presence probabilities $\boldsymbol{\mathrm{Y}}_{t}$ as output. Fig.
3: Architecture of the acoustic and visual DSW estimation networks, taking as
input the observations $\boldsymbol{\mathrm{Z}}_{\text{A},t}$ and
$\boldsymbol{\mathrm{Z}}_{\text{V},t}$ obtained from the acoustic and visual
trackers and producing as output the corresponding DSWs
$\boldsymbol{\mathrm{\Lambda}}_{\text{A},t}$ and
$\boldsymbol{\mathrm{\Lambda}}_{\text{V},t}$ covering the spatial grid.

###  2.3 Data fusion

At each time $t$, a conventional data fusion strategy would handle all
acoustic and visual tracker outputs equally. In the log-domain, this _flat
fusion_ strategy can be expressed as

$y_{t}^{(i,j)}=\log z_{\text{A},t}^{(i,j)}+\log z_{\text{V},t}^{(i,j)},$ (1)

where $y_{t}^{(i,j)}$ represents the log-likelihood of a speaker being present
in grid-cell $(i,j)$ at time-step $t$, taking into account both acoustic and
visual observations. Note that the derivation of Eq. (1) is based on assuming
statistical independence between acoustic and visual observations, as well as
imposing an uninformative prior on $\zeta_{i,j}~{}\forall~{}i,j$, cf. [16].
However, when working with audiovisual input data, modalities may not be
equally informative in all spatial regions due to, e.g., challenging lighting
conditions or a directional noise source. In these cases, it should be
possible to weight each source individually to lessen or increase the impact
of an input. This process must be dynamic, in the sense that the network
should be able to adapt over space and time. A _weighted fusion_ approach

$y_{t}^{(i,j)}=\lambda_{\text{A},t}^{(i,j)}\log
z_{\text{A},t}^{(i,j)}+\lambda_{\text{V},t}^{(i,j)}\log
z_{\text{V},t}^{(i,j)}$ (2)

is proposed here, where $\lambda_{\text{A},t}^{(i,j)}$,
$\lambda_{\text{V},t}^{(i,j)}\in[0,\,1]$ denote the location- and time-
dependent DSWs for the acoustic and visual modalities, respectively. In this
manner, we are proposing an extension of the DSW idea, originally introduced
to handle only time-variant sensor reliability, to the spatial domain,
allowing us to learn how different locations are covered more or less reliably
by the available modalities. All relevant variables can be expressed in matrix
notation as

$\boldsymbol{\mathrm{Y}}_{t}=\begin{bmatrix}y_{t}^{(i,j)}\end{bmatrix}_{i,j},\,\boldsymbol{\mathrm{Z}}_{m,t}=\begin{bmatrix}z_{m,t}^{(i,j)}\end{bmatrix}_{i,j},\,\boldsymbol{\mathrm{\Lambda}}_{m,t}=\begin{bmatrix}\lambda_{m,t}^{(i,j)}\end{bmatrix}_{i,j},$

with $m\in\\{\text{A},\text{V}\\}$, each covering the entire $20\times 24$
grid. The actual data fusion is performed conjointly with estimating the DSWs
in Eq. (2) and the subsequent estimation of the speaker positions.

###  2.4 Weight estimation

The individual acoustic and visual tracking networks discussed in Secs. 2.1
and 2.2, serve as inputs to the fusion neural network. The architecture of the
corresponding DSW estimation networks is depicted in Fig. 3. Its output is
subsequently combined with the outputs from both unimodal trackers according
to Eq. (2). To further refine the resulting speaker presence probability grid,
it is fed into a dedicated refinement network based on the image restoration
architecture proposed in [21]. The parameterization of the model [21] is
adapted to the specific requirements of this work. In particular, the input
and output size is set to $20\times 24$, matching the size of the probability
grid. Each convolutional layer consists of 128 filters with a kernel size of
$3\times 3$ and padding. The refinement operation is valuable to remove
outliers and noisy estimates from the output grid, which should only contain
peaks in grid cells close to the true speaker positions.

##  3 Experimental setup

In this section, we define the experimental settings and metrics111A Python
implementation of all algorithms described in this paper is publicly available
at https://github.com/rub-ksv/spatial-stream-weights.

(a) Audio Baseline

(b) Spatially invariant DSWs

(c) Weighted Fusion

Fig. 4: Benchmark results plotted with respect to the threshold. The dashed
lines represent the best unimodal baseline results.

###  3.1 Dataset

The AVDIAR [12] dataset is employed to evaluate the tracking capabilities of
the proposed data fusion strategy. It contains 27 audiovisual sequences, which
were recorded in three different rooms with a duration between 10 seconds and
3 minutes. The overall duration of the dataset is about 27 minutes. The
recordings were conducted using 6 microphones attached to a dummy head and two
cameras located right below the head. The microphone signals were acquired
using a sampling rate of 48 kHz. The cameras cover a field of view of
$97\degree\times 80\degree$ at 25 frames per second. The video signals have a
resolution of $720\times 450$ pixels. In this work, only one video signal from
the left camera is used. The dataset was partitioned according to a training,
validation and test split utilizing approximately $75\%$, $10\%$ and $15\%$ of
the available data, respectively. The splitting procedure ensured that all
subsets included audiovisual sequences with one, two, three and four
simultaneously visible speakers.

###  3.2 Implementation details

All trainable weights were initialized using the Glorot initialization scheme
[22]. In our experiments, the Adam optimizer [23] with a batch size of 64, a
learning rate of 0.001 and early stopping yielded the best results. The fusion
networks were trained for 20 epochs, with early stopping triggering in the
first five epochs with a patience of 4. The audio tracking networks were
trained for 100 epochs, with early stopping triggering after 30 epochs with a
patience of 20.

###  3.3 Evaluation metrics

Localization error (LE) and frame recall (FR), as proposed in [18, 24], were
utilized as evaluation metrics in this work. The LE metric was slightly
modified to cope with the discrete grid structure used here. Therefore, the
Euclidean distance between a ground-truth speaker position and the center
point of a grid cell where a speaker was detected (both represented as pixel
coordinates), was utilized. This distance was computed for all possible
combinations of detected speakers and ground-truth speaker positions at each
time-frame. If fewer speakers were predicted than are actually present, the
prediction vector was filled with the next highest activation, to have at
least one prediction per target. To calculate the localization error for an
estimated speaker position given the corresponding ground-truth, a cost matrix
was optimized using the Hungarian algorithm [25]. The resulting distances were
summed up and divided by the total number of speakers present at the
particular time-frame, yielding the average distance per speaker. This process
is repeated and averaged for the complete test set to obtain the final LE
metric.

The FR represents the ability of the network to predict the correct number of
speakers in a specific frame. If the estimated number of speakers matches the
ground-truth in that frame, the metric is set to one, otherwise it is zero. In
the proposed framework, the number of speakers is set to the number of
posterior grid probabilities surpassing the specified detection threshold,
which has to be set empirically or via, e.g., a grid-search approach.
Potentially, it might be included into an end-to-end optimization pipeline.
Therefore, in Sec. 4.1, different threshold values are evaluated for this
purpose.

The FR cannot differentiate between missed and excessive numbers of speakers.
Therefore, we additionally count the average number of false positives (FPs)
for each frame. This metric can be used to better understand and analyze
causes of any detection issues. A low FR in combination with a high FP
suggests that the network is not performing well due to the threshold being
too low. On the other hand, in combination with a low FP it seems reasonable
that either the threshold is too high, pruning too many activations, or that
the detector is not working properly.

##  4 Results and discussion

In the following, we present and analyze the results of our baseline systems
and the proposed fusion network architecture.

###  4.1 Baseline methods

The results of our two single-modality baselines, i.e. audio- and video-only
tracking systems, are provided in the first two rows of Tab. 1. As expected,
the video-only baseline is generally superior in good visual conditions. The
audio-only system outperforms video localization when faces are occluded or
outside the field-of-view.

With the single-modality audio tracker and a threshold of 0.75, an LE of 11.93
pixels was achieved, while the FR was quite low at 23%. On average, a frame
contained 1.07 false positives. Fig. 4 (a) shows the performance of the audio
tracker as a function of the threshold. The LE shows that the audio tracker
predicts speakers with an error of roughly half a grid point. However, the low
FR indicates that many false positives are included in the tracking result.
Even with a threshold of 0.75, on average, one false positive is included in
every frame.

| LEpx | FR | FP
---|---|---|---
Audio-only (Sec. 2.1) | 11.93 | 0.24 | 0.072
Video-only (Sec. 2.2) | 19.95 | 0.83 | 0.006
Flat fusion (Eq. 1) | 7.65 | 0.82 | 0.003
Table 1: Results of the baseline trackers. For audio only the results for a
threshold of 0.75 are given. For audio-visual tracking, we show the results of
spatially flat fusion at a threshold of 0.75.

In comparison to the audio-only system, the video tracker in the second row in
Tab. 1 unsurprisingly achieves a significant higher FR of 83.45%, due to its
access to a highly informative modality, in line with its good benchmark
results in other works [20, 26]. Note that no threshold is needed for the
video tracker, as its output is binary. Additionally, the number of false
positives per frame is lower, at 0.006. The LE is still below the size of one
grid point with 19.95 pixels distance on average.

The face tracking capabilities lead to a good result with YOLO detecting the
right number of faces in most cases. This is also reflected by the number of
false positives, which are close to zero.

Finally, the third row shows results of our naive audio-visual baseline system
with the flat fusion, implemented according to Eq. (1). As can be seen, the
performance improves already, giving a lower localization error and false
positive rate than the best uni-modal system at a virtually identical frame
recall to the better-performing video setup.

###  4.2 Spatially invariant fusion

Fig. 4 (b) shows the results for the spatially invariant dynamic stream
weighting approach. In this case, the DSWs in Eq. (2) are simplified by using
a time-variant but spatially invariant scalar, i.e.:
$y_{t}^{(i,j)}=\lambda_{t}~{}\log
z_{\text{A},t}^{(i,j)}+(1-\lambda_{t})~{}\log z_{\text{V},t}^{(i,j)}$.

The dashed lines in Fig. 4 (b) represent the best reported single-modality
baseline results for the LE and FR, respectively. The spatially invariant
fusion is able to achieve a similar, but slightly worse FR of 0.82. Taking the
threshold into consideration, the FR rises from 0.1 to above 0.7 for
thresholds of 0.5 to 0.6. The reason is that the low activations around 0.5 of
the audio-only tracker are carried over to the spatially invariant fusion and
the flat fusion as well. Beyond a threshold of 0.6, the FR increases slowly
but steadily.

Comparing the LEs in Fig. 4 (b) with the reported values in Tab. 1, we can
observe that incorporating additional information to the data fusion layer
leads to further improvements in the model’s ability to extrapolate the
ground-truth location of the speakers: With temporal information, we see an
improvement from 11.9 to 10.1 pixels. With the flat fusion strategy reported
in Tab. 1, we would see a lower LE, reduced from 10.1 to 7.7 pixels.

Through multi-modal fusion, the number of false positives also improves w.r.t.
the respective best-performing unimodal system, which is again similarly
observable for the flat fusion strategy. Note that lower thresholds generally
return worse results and choosing a higher threshold helps to mask out
outliers. Increasing the threshold beyond 0.55 does not lead to any further
changes in the LEs and FPs, indicating that outliers are sufficiently
suppressed at this threshold.

###  4.3 Spatially weighted fusion

Fig. 4 (c) shows the performance of the weighted fusion system. At a threshold
of 0.75, the network achieves a FR of 0.92, clearly outperforming all the
other strategies. Again, the low performance at a threshold of 0.5 mentioned
in Sec. 4.2 is still visible.

In terms of the LE, the best result was achieved with 4.10 pixels distance at
a threshold of 0.75. For all thresholds above 0.5, nearly constant results
were measured. The FPs per frame were relatively low, at an average of 0.028
over-predictions per frame.

Overall, the spatially weighted fusion strategy with a location-dependent
matrix achieved the best performance. The target speakers were located with a
high precision, showing the lowest LE measured across all experiments. Even
though the number of false positives was increased slightly, the overall
capabilities of the network are significantly better compared to the other
systems.

Finally, comparing Fig. 4 (b) and (c), the questions arises, whether the
improved performance may be caused by the ability to focus on different
modalities in different parts of the room and whether this ability may be
helpful in dealing with multiple speakers. To investigate these questions, we
compared the performance metrics for audio frames containing only a single
speaker with audio frames containing $4$ different speakers. As hypothesized,
we achieved the same results (LE = 1.31, FR = 1 and FP = 0) for both fusion
strategies, when only a single speaker is involved. In contrast, when 4
speakers are present, the performances differ significantly. While the
performance of the spatially weighted fusion strategy remains stable (LE =
0,53, FR = 0,99, FP = 0), the LE of the spatially invariant fusion strategy
suffers (LE = 10,4, FR = 0,81, FP = 0). While based on comparatively little
data, this analysis appears to indicate that spatial weighting is better
equipped for dealing with multi-speaker scenarios.

##  5 Conclusion

In this paper, we introduced a novel deep-learning-based data-fusion strategy
by extending the idea of dynamic temporal stream weights to the spatial
domain. Experiments show that the unimodal baseline video tracker based on the
YOLO-framework achieved accurate results as a stand-alone system, while,
unsurprisingly, using only the unimodal audio tracker led to a significantly
higher false positive rate. However, by fusing acoustic and visual
information, it was possible to achieve an improved tracking result, with a
10% increase of the frame recall and a far more accurate localization, showing
an error of 3.83 pixels on average. The results show that the audiovisual
tracking system benefits from the proposed data fusion strategy, and that
adding spatial dependence of stream weights is highly beneficial compared to a
purely temporal dependence, as in standard dynamic stream weighting.

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#  Hindrance in heavy-ion fusion for lighter systems of astrophysical interest

Vinay Singh1, Joydev Lahiri1, Partha Roy Chowdhury2 and D. N. Basu1 1Variable
Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, INDIA  2
Chandrakona Vidyasagar Mahavidyalaya, Chandrakona Town, Paschim Medinipur,
West Bengal 721201, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The hindrance in fusion of heavy-ion reactions crops up in the region of
extreme sub-barrier energies. This phenomenon can be effectively analyzed
using a simple diffused barrier formula derived assuming a Gaussian
distribution of fusion barrier heights. Folding the Gaussian barrier
distribution with the classical expression for the fusion cross section for a
fixed barrier, the fusion cross section is obtained. The energy dependence of
the fusion cross section provides good description to the existing data on
sub-barrier heavy-ion fusion for lighter systems of astrophysical interest.
Using this simple formula, an analysis has been presented from 16O + 18O to
12C + 198Pt, all of which were measured down to $<$ 10 $\mu$b. The agreement
of the present analysis with the measured values is better than those
calculated even from the sophisticated coupled channels calculations. The
relatively smooth variation of the three parameters of this formula implies
that it may be exploited to estimate the excitation function or to extrapolate
cross sections for pairs of interacting nuclei which are yet to be measured.
Possible extensions of the present methodology and its limitations have also
been discussed.

Keywords: Fusion reactions; Barrier distribution; Excitation function;
Nucleosynthesis.

###### pacs:

25.70.-z; 25.60.Pj; 26.20.Np; 97.10.Cv

##  I Introduction

The phenomenon of fusion hindrance in heavy-ion fusion reactions for lighter
systems may have important consequences on the nuclear processes taking place
in astrophysical scenarios. The hindrance can seriously affect the energy
generation by heavy-ion fusion reactions occurring in the region of extreme
sub-barrier energies, encompassing reactions involving lighter systems, such
as the reactions occurring in the stage of carbon and oxygen burning in heavy
stars Ro91 ; Da98 ; Wa97  and their evolution and elemental abundances. It is
well known that fusion excitation functions cannot be satisfactorily explained
assuming penetration through a single, well-defined barrier in the total
potential energy of a colliding nucleus-nucleus system. In order to reproduce
shapes of the fusion excitation functions, especially at low near-threshold
energies, it is necessary to assume coexistence of different barriers, a
situation that is naturally accounted for in the description of fusion
reactions in terms of coupled channel calculations involving coupling to
various collective states We91 ; Ste95 ; Ti98 ; Tr01 ; St06 ; St07 .

The aim of the present work is to obtain the nuclear fusion cross sections for
heavy-ion fusion for lighter systems of astrophysical interest. The
phenomenological description of the fusion excitation functions is achieved by
assuming a Gaussian shape of the barrier distribution treating the mean
barrier and its variance as free parameters and folding it with the classical
expression for the fusion cross section for a fixed barrier with the distance
corresponding to the location of the interaction barrier as another free
parameter. The free parameters are then determined individually for each of
the reactions by comparing the predicted fusion excitation function with
experimental data. The energy dependence of the fusion cross section, thus
obtained, provides good description to the existing data on sub-barrier fusion
and capture excitation functions for lighter heavy-ion systems of
astrophysical interest.

##  II The fusion barrier distribution

It is well known that the energy dependence of the fusion cross sections can
not be well estimated assuming simply the penetration through a well-defined
barrier in one-dimensional potential of a colliding nucleus-nucleus system.
The heavy-ion fusion cross sections require interpretation Ro91  in terms of a
distribution of potential barriers. The smoothening due to the quantum
mechanical barrier penetration replaces set of discrete barriers by an
effective continuous distribution. In order to reproduce shapes of
experimentally observed fusion excitation functions, particularly at low,
near-threshold energies, it is necessary to assume a distribution of the
fusion barrier heights, the effect that results from the coupling to other
than relative distance degrees of freedom. This is naturally achieved in
coupled-channel calculations, involving the coupling to the lowest collective
states in both colliding nuclei. The structure effects in the barrier
distributions are neglected in the present work and for the distribution of
the fusion barrier heights, a Gaussian shape for the barrier distribution
$D(B)$ is assumed Wi04 . The barrier distribution is, therefore, given by

$D(B)=\frac{1}{\sqrt{2\pi}\sigma_{B}}\exp\Big{[}-\frac{(B-B_{0})^{2}}{2\sigma_{B}^{2}}\Big{]}$
(1)

where the two parameters, the mean barrier $B_{0}$ and the distribution width
$\sigma_{B}$, to be determined individually for each reaction.

Figure 1: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 12C+24Mg.

Figure 2: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 12C+30Si.

##  III Fusion cross section calculation

In order to provide a systematic analysis of the data on the fusion excitation
functions, a simple formula for the cross section for overcoming the potential
energy barrier is derived. The energy dependence of the fusion cross section
is obtained by folding the barrier distribution Wi04 ; Ca11  provided by
Eq.(1), with the classical expression for the fusion cross section given by

$\displaystyle\sigma_{f}(B)=$ $\displaystyle\pi
R_{B}^{2}\Big{[}1-\frac{B}{E}\Big{]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}B\leq E$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}B\geq E$ (2)

where $R_{B}$ denotes the relative distance corresponding to the position of
the barrier approximately, which yields

$\displaystyle\sigma_{c}(E)=\int_{E_{0}}^{\infty}\sigma_{f}(B)D(B)dB$
$\displaystyle=\int_{E_{0}}^{B_{0}}\sigma_{f}(B)D(B)dB+\int_{B_{0}}^{E}\sigma_{f}(B)D(B)dB$
(3) $\displaystyle=\pi
R_{B}^{2}\frac{\sigma_{B}}{E\sqrt{2\pi}}\Big{[}\xi\sqrt{\pi}\Big{\\{}{\rm
erf}(\xi)+{\rm erf}(\xi_{0})\Big{\\}}+e^{-\xi^{2}}-e^{-\xi_{0}^{2}}\Big{]}$

where $E_{0}=0$ for positive $Q$ value reactions and $E_{0}=Q$ for negative
$Q$ value reactions, $Q$ value being the sum of the rest masses of fusing
nuclei minus rest mass of the resultant fused nucleus,

$\displaystyle\xi$ $\displaystyle=\frac{E-B_{0}}{\sigma_{B}\sqrt{2}}$
$\displaystyle\xi_{0}$ $\displaystyle=\frac{B_{0}-E_{0}}{\sigma_{B}\sqrt{2}}$
(4)

and erf($\xi$) is the Gaussian error integral of argument $\xi$. The
parameters $B_{0}$ and $\sigma_{B}$ along with $R_{B}$ is to be determined by
fitting Eq.(3) along with Eq.(4) to a given fusion excitation function. In the
derivation of formula Eq.(3), the quantum effect of sub-barrier tunneling is
not accounted for explicitly. However, the influence of the tunneling on shape
of a given fusion excitation function is effectively included in the width
parameter $\sigma_{B}$.

The fusion cross section formula of the Eq.(3) obtained by using the diffused-
barrier, is a very elegant parametrization of the cross section for a process
of overcoming the potential-energy barrier. Hence, it can be successfully used
for analysis and predictions of the fusion excitation functions of light,
medium and moderately heavy systems, especially in the range of sub-barrier
energies.

In case of light and medium systems, surmounting the barrier automatically
guarantees fusion of the colliding nuclei leading to formation of the compound
nucleus. The term capture is used to refer the process of overcoming the
interaction barrier in a nucleus-nucleus collision, followed by formation of a
composite system. In general, the composite system undergoes fusion only in a
fraction $f$ of the capture events. For light and medium systems, $f\approx
1$, and almost all the capture events lead to fusion resulting fusion cross
sections to be practically identical with the capture cross sections. However,
for very heavy systems, only a small fraction ($f<1$) of capture events
ultimately lead to fusion while for the remaining part of the events, the
system re-separates prior to equilibration and clear distinction between
fusion and capture cross sections then becomes necessary. Therefore, for very
heavy systems, when the overcoming the barrier does not guarantee fusion,
predictions based on Eq.(3) provide the capture cross section.

Figure 3: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 12C+198Pt.

Figure 4: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 16O+18O.

##  IV  Results and discussion

The sub-barrier fusion excitation functions of heavy-ion fusion for lighter
systems of astrophysical interest have been analyzed using a simple diffused
barrier formula (given by Eq.(1)) derived by folding the Gaussian barrier
distribution with the classical expression for the fusion cross section for a
fixed barrier. The same set of target-projectile combinations have been
selected for which heavy ion sub-barrier fusion has been recently Mo97 ; Mo20
; Ra20 ; We21  studied. The values of mean barrier height $B_{0}$, width
$\sigma_{B}$ and the effective radius $R_{B}$ have been obtained using the
least-square fit method. These values are listed in Table-I arranging it in
order of increasing projectile mass.

Figure 5: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 28Si+64Ni.

Figure 6: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 58Ni+58Ni.

The task of estimating $\sigma_{c}(E)$ rests upon predicting the values of the
mean barrier height $B_{0}$, the width of the barrier height distribution
$\sigma_{B}$ and the effective radius $R_{B}$ for a particular reaction. Since
$B_{0}$ is essentially the mean height of the Coulomb barrier, it should be a
function of the Coulomb parameter $z=Z_{1}Z_{2}/(A_{1}^{1/3}+A_{2}^{1/3})$ in
the vicinity of the barrier. The second quantity, the effective barrier radius
$R_{B}$, quite obviously should depend upon
$r_{0}(A_{1}^{1/3}+A_{2}^{1/3})=r_{0}A_{12}$ where $r_{0}$ is the nuclear
radius parameter. The extrapolation of the trend of $\sigma_{B}$ is more
difficult, which primarily arises out of nuclear deformation, nuclear
vibrations and quantum mechanical barrier penetrability. It may be observed
from Table-I that the mean barrier height $B_{0}$ increases with the Coulomb
parameter $z$ for all cases while the effective radius $R_{B}$ also increases
with $A_{12}$ except for 16O+18O and 28Si+64Ni and interestingly, as may be
seen from Figs.-1-7, for 16O+18O case, the theoretical calculations are
slightly off at lower energies as well.

Table 1:  The extracted values of the mean barrier height $B_{0}$, the width of the barrier height distribution $\sigma_{B}$ and the effective radius $R_{B}$, deduced from the analysis of the measured fusion excitation functions. The table is arranged in order of increasing projectile mass. Reaction | Refs. | $z$ | $A_{12}$ | $\sigma_{B}$ | $B_{0}$ | $R_{B}$
---|---|---|---|---|---|---
|  |  |  | [MeV] | [MeV] | [fm]
12C+24Mg | Mo20  | 13.916 | 5.174 | 0.815 | 11.483 | 6.646
12C+30Si | Mo97  | 15.565 | 5.397 | 1.090 | 13.540 | 8.300
12C+198Pt | Ra20  | 57.650 | 8.118 | 1.749 | 55.140 | 11.179
16O+18O | Mo20  | 12.450 | 5.141 | 0.859 | 9.797 | 7.743
28Si+64Ni | We21  | 55.709 | 7.037 | 1.402 | 50.403 | 7.182
58Ni+58Ni | Ra20  | 101.269 | 7.742 | 2.275 | 98.278 | 8.550
64Ni+64Ni | Ra20  | 98.000 | 8.000 | 1.466 | 92.646 | 8.862

In Figs.-1-7, the measured fusion excitation functions represented by full
circles are compared with the predictions of the diffused barrier formula
depicted by the solid lines. The systems of 12C+24Mg, 12C+30Si, 12C+198Pt,
16O+18O, 28Si+64Ni, 58Ni+58Ni and 64Ni+64Ni are illustrated in Figs.-1-7. It
may be easily perceived from these figures that precisely measured fusion
excitation functions provide systematic information on the essential
characteristics of the interaction potential, viz. the mean barrier width
$\sigma_{B}$ and height $B_{0}$ of its distribution, for nucleus-nucleus
collisions. The fusion or capture cross sections for planning experiments can
also be predicted by using Eq.(3) and theoretically obtained values of the
parameters $B_{0}$ and $\sigma_{B}$.

As seen in the Figs.-1-7, the present theoretical description provides
excellent fits to the experimental data. This implies that for the chosen set
of nuclei almost all the capture events lead to fusion resulting fusion cross
sections to be practically identical with the capture cross sections.
Moreover, the Gaussian form for the barrier distribution describes sub-barrier
fusion cross sections quite well justifying the beyond single barrier model
arising out of tunneling, deformation and vibration of nuclei. Although
theoretically the concept of a barrier distribution is valid under certain
approximations, the good fits to the experimental data shown certainly imply
that in reactions involving heavy-ion fusion for lighter systems of
astrophysical interest, barrier distribution remains a meaningful concept.

Figure 7: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 64Ni+64Ni.

It is pertinent to mention here that although the single-Gaussian
parameterization [Eq.(1)] for barrier distribution is reasonably successful in
providing a good description of fusion process in general, neither the formula
derived for fusion cross-section nor the method using the barrier distribution
can be put to use for all fusing systems. One can visualize from Eq.(3) that
the excitation function is all the time a monotonically rising function of
energy. This puts a limitation on Eqs.(1,2) which can not be used for
describing fusion reactions at higher energies when incomplete fusion as well
as deep-inelastic scattering can cause a lowering of the fusion cross section.
Similar limitation arises for lighter systems (e.g. 12C+12C, 12C+16O, 16O+16O
etc.) as well when excitation functions possess oscillations and resonance
structures. Possibility of better agreement with data may be explored by
opting a more intricate formula for barrier distribution, which, however, will
bring in more additional adjustable parameters than just three used in the
present work. Such refinements for the barrier distribution of Eq.(1) include
distributions with different widths on the low or high energy sides, a
moderation of the exponent in the Gaussian distribution or multi-component
distributions.

##  V  Summary and conclusion

In the region of sub-barrier energies, the fusion reaction cross sections have
been estimated spanning a broad energy range. To comprehend the conditions of
overcoming the potential energy barrier in nuclear collisions and to obtain a
systematic knowledge on the essential characteristics of the interacting
potential, viz. the mean barrier height $B_{0}$ and its width $\sigma_{B}$ of
distribution, a set of precisely measured fusion excitation functions has been
studied for two colliding nuclei. A Gaussian distribution function for the
barrier heights is assumed to derive a simple diffused-barrier formula. The
values of the mean barrier height $B_{0}$, the width $\sigma_{B}$ and the
effective radius $R_{B}$ are determined using the method of least-square fit.

The present fusion cross section formula can be used to calculate the cross
sections for overcoming the barrier in collisions of very heavy systems in
order to calculate the overcoming-the-barrier cross section for a given
projectile-target combination. For calculating the production cross sections
of superheavy nuclei, the prediction of the capture excitation functions or
sticking can be used in the sticking-diffusion-survival model An93  as one of
three basic ingredients. The reasonably good fit of the present theoretical
description to the experimental data implies two important facts that for the
investigated set of nuclei almost all the capture events ultimately lead to
fusion and the idea of the Gaussian distribution of barrier provides good
description of the sub-barrier fusion cross sections. Although the single-
Gaussian parameterization for barrier distribution is reasonably successful in
providing a good description of fusion process in general, neither the formula
derived for fusion cross-section nor the method using the barrier distribution
can be put to use for all fusing systems. One can visualize from Eq.(3) that
the excitation function is all the time a monotonically rising function of
energy. This puts a limitation on Eqs.(1,2) which can not be used for
describing fusion reactions at higher energies when incomplete fusion as well
as deep-inelastic scattering can cause a lowering of the fusion cross section.
Similar limitation can arise for lighter systems also when excitation
functions possess oscillations and resonance structures. Possibility of better
agreement with data may be explored by opting a more intricate formula for
barrier distribution, which, however, will bring in more additional adjustable
parameters than just three used in the present work. Such improvements for the
barrier distribution can be realized through distributions with different
widths on the low or high energy sides, multi-component distributions or a
modification of the exponent in the Gaussian distribution.

## References

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# An ODMA-Based Unsourced Random Access Scheme with a Multiple Antenna
Receiver

Mert Ozates Mert Ozates’ work is supported by the Federal Ministry of
Education and Research of Germany in the programme of “Souverän. Digital.
Vernetzt” joint Project 6G-RIC, project identification number: 16KISK026. IHP
- Leibniz Institute for High Performance Microelectronics, 15236 Frankfurt
(Oder), Germany
Email<EMAIL_ADDRESS>Mohammad Kazemi  Department of
Electrical and Electronic Engineering, Imperial College London
Email<EMAIL_ADDRESS>Tolga M. Duman  Dept. of Electrical and
Electronics Engineering, Bilkent University, Bilkent 06800, Ankara, Turkey
Email<EMAIL_ADDRESS>
###### Abstract

We investigate the unsourced random access scheme assuming that the base
station is equipped with multiple antennas, and propose a high-performing
solution utilizing on-off-division multiple access. We assume that each user
spreads its pilot sequence and polar codeword to the pilot and data parts of
the transmission frame, respectively, based on a transmission pattern. The
iterative receiver operation consists of pilot and pattern detection followed
by channel vector and symbol estimation, polar decoding, and successive
interference cancellation. Numerical findings demonstrate that the proposed
scheme has superior performance compared to the state-of-the-art in various
antenna settings.

###### Index Terms:

Unsourced random access, on-off division multiple access, massive MIMO.

##  I Introduction

Next-generation wireless communication systems require high reliability, high
connection density, and low latency. This makes massive machine-type
communications (mMTC) [1] a key feature of 5G and beyond 5G (B5G) systems. In
mMTC, a massive number of devices (e.g., millions of devices per
$\text{km}^{2}$) with low transmit power and short payloads sporadically
communicate with a base station (BS) without any coordination, i.e., only a
small fraction of them are active at any given time.

Conventional grant-based multiple access approaches where the BS assigns fixed
resources (time, frequency, code, etc.) to each user become infeasible in mMTC
applications such as the Internet-of-things (IoT) due to the excessive delay
and signaling overhead of scheduling a massive number of users. To address
this problem, a new grant-free random access paradigm called unsourced random
access (URA) is introduced in [2]. In URA, the devices share the same
codebook; hence, the user identities are removed, which allows for an
arbitrarily large number of total users. The receiver aims to recover a list
of the transmitted messages regardless of the user identities, and the per-
user probability of error (PUPE) is adopted as the main error metric.

In [2], URA over a real-valued Gaussian multiple access channel (MAC) is
studied, and a random coding achievability bound quantifying the required
energy-per-bit for a given active user load is derived. Several practical
transmission schemes are proposed to approach this bound based on dividing the
transmission frame into slots [3, 4], coded compressed sensing [5], or
spreading the channel-coded bits by random signatures [6, 7]. Furthermore, the
on-off division multiple access (ODMA) [8] where a small fraction of the frame
is utilized by each user is applied to the URA set-up in [9, 10], offering a
promising performance with low complexity.

The fading MAC, which is a more practical model compared to the Gaussian MAC,
is widely studied in URA, where a single-antenna receiver is considered either
over flat [11, 12] or frequency-selective fading channels [13]. Besides, the
spatial multiplexing gains of massive multiple-input multiple-output (MIMO)
technology are utilized to cope with the high multiuser interference in URA.
As a result, many recent works on URA examine the scenario with a massive MIMO
receiver. Some typical methods in this framework are coded compressed sensing
combined with non-Bayesian activity detection [14], tensor-based modulation
[15], and the employment of pilots in a part of slot or frame for user
separation [18, 17, 16].

In this paper, we tackle the problem of URA with multiple antennas at the BS
and develop an energy-efficient solution based on ODMA. In our proposed
scheme, we assume that the channel frame is divided into the pilot and data
parts, and the transmission in both parts is performed in an ODMA manner.
Namely, each user randomly and independently selects a pilot sequence from a
common codebook to transmit in the active transmission indices of the pilot
part determined by a pilot transmission pattern. In the data part, a polar
encoded and quadrature phase shift keying (QPSK) modulated sequence is sent
based on a data transmission pattern determining the locations of the codeword
elements in the data part of the frame. The transmitted messages are recovered
by first detecting the active pilot sequences and transmission patterns using
the generalized orthogonal matching (gOMP) algorithm and then employing linear
minimum mean square (LMMSE) channel estimation, maximal ratio combining (MRC),
single-user decoding, and successive interference cancellation (SIC)
iteratively. We demonstrate the effectiveness of the proposed solution via
numerical examples, showing that it outperforms the state-of-the-art for
multiple antenna configurations, including massive MIMO. For instance, it can
decrease the required energy-per-bit up to 2.5 dB for massive MIMO and 1.5 dB
for the multiple antenna case compared to the state-of-the-art, and it can
support up to 1400 active users in the massive MIMO scenario. Note that our
proposed scheme is similar to that of [10]. However, we generalize the scheme
in [10] by equipping the receiver side with multiple antennas (while [10]
assumes a single antenna receiver) and employing ODMA in the pilot part of the
frame as well, and modify the related algorithmic steps accordingly.

The rest of the paper is organized as follows. We introduce the system model
in Section II, and give the details of the proposed solution in Section III.
We present a numerical evaluation of our scheme in Section IV, and give some
conclusions in Section V.

##  II System Model

We consider a massive random access scenario where $K_{T}$ users sporadically
communicate with a common BS, i.e., only $K_{a}$ of them are active at a given
time where $K_{a}\ll K_{T}$. The active users transmit $B$ bits of information
over a quasi-static fading channel to a BS with $M$ antennas through a channel
frame of length $n$. The received signal at the BS is

$\mathbf{Y}=\sum\limits_{i=1}^{K_{a}}\mathbf{x}_{i}\mathbf{h}_{i}+\mathbf{Z},$
(1)

where $\mathbf{x}_{i}\in\mathbb{C}^{n\times 1}$ is the transmitted signal of
the $i$-th user, $\mathbf{h}_{i}\in\mathbb{C}^{1\times M}$ is the channel
vector of the $i$-th user consisting of independent and identically
distributed (i.i.d.) elements that are constant throughout the transmission
frame, $\mathbf{Y}\in\mathbb{C}^{n\times M}$ is the received signal, and
$\mathbf{Z}\in\mathbb{C}^{n\times M}$ is the circularly symmetric complex
additive white Gaussian noise (AWGN) with i.i.d. zero mean elements.

In our proposed model, each user transmits a pilot signal of length $n_{p}$ in
the first $n_{p}^{\prime}$ time instances, and utilizes $n_{d}$ time instances
out of the remaining $n-n_{p}^{\prime}$ ones for the data transmission as
depicted in Fig. 1, which illustrates the ODMA idea. The pilot and data
signals have an average power constraint of $P_{p}$ and $P_{d}$, respectively,
hence, the required energy-per-bit is

$\frac{E_{b}}{N_{0}}=\frac{n_{p}P_{p}+n_{d}P_{d}}{BN_{0}},$ (2)

where $N_{0}$ is the noise variance, i.e., the variance of each element of
$\mathbf{Z}$. The purpose of the receiver is to generate a list of the decoded
messages
$\mathcal{L}=\\{\mathbf{\hat{m}}_{1},\dots,\mathbf{\hat{m}}_{K_{a}}\\}$. The
system PUPE $P_{e}$ is evaluated in terms of the misdetection and false alarm
probabilities $P_{\text{md}}$ and $P_{\text{fa}}$, respectively, which can be
calculated as

$P_{\text{md}}=\frac{1}{K_{a}}{\mathbb{E}\bigg{[}\sum\limits_{i\in\mathcal{K}_{a}}\mathbbm{1}_{\\{\mathbf{m}_{i}\notin\mathcal{L}\\}}\bigg{]}},$
(3)

$P_{\text{fa}}=\mathbb{E}\left[\frac{\absolutevalue{\mathcal{L}\setminus\\{\mathbf{m}_{i}:i\in\mathcal{K}_{a}\\}}}{\absolutevalue{\mathcal{L}}}\right],$
(4)

where $\mathbf{m}_{i}$ is the message of the $i$-th user, $\mathcal{K}_{a}$ is
the set of active users, $\mathbbm{1}_{\\{\ \cdot\\}}$ is the indicator
function, $\absolutevalue{\cdot}$ denotes the cardinality of a set, and the
expectation is taken over the randomness of the fading and noise processes.
The PUPE can be obtained by summation of these two probabilities, i.e.,
$P_{e}=P_{\text{md}}+P_{\text{fa}}$.

Figure 1: The proposed transmit signal structure of multiple users. The
utilized time instances in the pilot and data parts are shown by green and red
boxes, respectively.

##  III Proposed Scheme

###  III-A Encoding

We assume that each user divides its message bits into two parts,
$\mathbf{m}_{p}$ and $\mathbf{m}_{d}$, respectively. The first part
$\mathbf{m}_{p}$ determines the pilot and pattern indices of the users. For
the transmission in the pilot part, a user picks the
$ind(\mathbf{m}_{p})=\text{dec}(\mathbf{m}_{p})+1$-th column of the common
non-orthogonal pilot matrix $\mathbf{A}\in\mathbb{C}^{n_{p}\times N}$ as its
pilot sequence, where $N=2^{B_{p}}$, $B_{p}$ is the length of
$\mathbf{m}_{p}$, and $\text{dec}(.)$ denotes binary-to-decimal conversion.
The columns of $\mathbf{A}$ are normalized to have a Euclidean norm of
$\sqrt{n_{p}P_{p}}$ to satisfy the power constraint. The selected pilot
sequence is then transmitted in the first $n_{p}^{\prime}$ time instances of
the transmission frame based on a pilot transmission pattern, that is the
$ind(\mathbf{m}_{p})$-th column of the pilot transmission pattern matrix
$\mathbf{P}_{\text{pilot}}\in\mathbb{R}^{n_{p}^{\prime}\times N}$ with $n_{p}$
non-zero elements (activity indices) in each column. The indices of non-zero
elements show the time instances where the pilot symbols will be located,
while the rest are the idle indices. Note that we have dropped the user index
for ease of presentation. The pilot part of the received signal can be written
as

$\mathbf{Y}_{p}=\sum\limits_{i=1}^{K_{a}}{s}_{p}(\mathbf{a}_{i})\mathbf{h}_{i}+\mathbf{Z}_{p},$
(5)

where $\mathbf{Y}_{p}\in\mathbb{C}^{n_{p}^{\prime}\times M}$,
$\mathbf{a}_{i}\in\mathbb{C}^{n_{p}\times 1}$ is the pilot signal of the
$i$-th user, ${s}_{p}(\mathbf{a}_{i})\in\mathbb{C}^{n_{p}^{\prime}\times 1}$
is the pilot part of the transmitted signal of the $i$-th user, $s_{p}(.)$ is
a mapper distributing the pilot signal to the pilot part of the frame, and
$\mathbf{Z}_{p}\in\mathbb{C}^{n_{p}^{\prime}\times M}$ is the sub-matrix
consisting of the first $n_{p}^{\prime}$ rows of $\mathbf{Z}$.

The second part $\mathbf{m}_{d}$ is encoded by a $(n_{c},B-B_{p}+r)$ polar
code and modulated by QPSK to be transmitted in the last $n-n_{p}$ instances,
where $n_{c}$ is the code length and $r$ is the number of cyclic redundancy
check (CRC) bits. CRC is used to ensure that the decoded sequence is a valid
codeword to avoid error propagation in an iterative decoder. The encoded and
modulated sequence is then distributed to the data part of the frame based on
a data transmission pattern, that is the $ind(\mathbf{m}_{p})$-th column of
the data pattern matrix
$\mathbf{P}_{\text{data}}\in\mathbb{R}^{(n-n_{p}^{\prime})\times N}$ with
$n_{d}$ non-zero elements in each column, denoting the locations of the
codeword elements. The received signal in the data part can be written as

$\mathbf{Y}_{d}=\sum\limits_{i=1}^{K_{a}}{s}_{d}(\mathbf{c}_{i})\mathbf{h}_{i}+\mathbf{Z}_{d},$
(6)

where $\mathbf{Y}_{d}\in\mathbb{C}^{(n-n_{p}^{\prime})\times M}$,
${s}_{d}(\mathbf{c}_{i})\in\mathbb{C}^{(n-n_{p}^{\prime})\times 1}$ is the
transmitted data signal of the $i$-th user, $s_{d}(.)$ is a mapper that
distributes the modulated polar codeword of the $i$-th user $\mathbf{c}_{i}$
with $n_{d}$ non-zero elements in $\\{\sqrt{P_{d}/2}(\pm 1\pm j)\\}$ to the
data part, and $\mathbf{Z}_{d}$ is the matrix of the last $n-n_{p}^{\prime}$
rows of $\mathbf{Z}$. The encoding procedure is illustrated in Fig. 2.

Figure 2: Encoding process for a user in the proposed scheme.

###  III-B Decoding Procedure

####  III-B1 Joint Active Pilot and Pattern Detection

In order to decode the transmitted message sequences, the active pilot
sequences should be detected first. Since $K_{a}\ll N$, this is essentially a
compressed sensing problem, and we employ the gOMP algorithm [19] to detect
the active pilots.

Since the pilot sequences are sent in the active indices of the pilot part,
the pilot signal of any specific user occupies a fraction of the pilot part.
Hence, the pilot codebook $\mathbf{A}$ cannot be directly used for activity
detection. To solve this issue, we extend the pilot codebook $\mathbf{A}$
using the pilot transmission patterns. Namely, we define an extended pilot
codebook $\mathbf{A^{\prime}}\in\mathbb{C}^{n_{p}^{\prime}\times N}$ and
construct the $i$-th column of it by putting the elements of the $i$-th column
of $\mathbf{A}$ to the activity indices dictated by the $i$-th column of
$\mathbf{P}_{\text{pilot}}$, while the rest of the column elements are zero.
We then utilize $\mathbf{A^{\prime}}$ as an effective pilot codebook to detect
the active pilots and pilot patterns jointly, as $\mathbf{A}^{\prime}$ has the
information of pilot transmission patterns as well.

In the first step of each gOMP iteration, the correlation of the effective
pilots and the received signal is obtained as

$\mathbf{C}=\mathbf{A^{\prime}}^{H}\mathbf{Y}_{p}^{(k)},$ (7)

where $\mathbf{C}\in\mathbb{C}^{N\times M}$ and $\mathbf{Y}_{p}^{(k)}$ is the
residual pilot signal at the $k$-th gOMP iteration initialized as
$\mathbf{Y}_{p}^{(1)}=\mathbf{Y}_{p}$. The Euclidean norm of each row of
$\mathbf{C}$ gives the decision metric for the corresponding pilot. We then
add the indices of the $\lceil\frac{K_{a}+\Delta}{n_{\text{OMP}}}\rceil$
elements having the maximum decision metric to the output list of gOMP, where
$n_{\text{OMP}}$ is the number of gOMP iterations and $\Delta$ is a small
positive integer. In the second step, the effect of the detected pilots is
subtracted as

$\mathbf{Y}_{p}^{(k+1)}=\mathbf{Y}_{p}-\mathbf{A^{\prime}}_{\hat{{I}}^{(k)}}(\mathbf{A^{\prime}}_{\hat{{I}}^{(k)}}^{H}\mathbf{A^{\prime}}_{\hat{{I}}^{(k)}}+N_{0}\mathbf{I}_{\absolutevalue{{\hat{I}}^{(k)}}})^{-1}\mathbf{A^{\prime}}_{\hat{{I}}^{(k)}}^{H}\mathbf{Y}_{p},$
(8)

where $\mathbf{A^{\prime}}_{\hat{{I}^{(k)}}}$ is the column set of
$\mathbf{A^{\prime}}$ determined by $\hat{{I}^{(k)}}$, $\hat{I}^{(k)}$ is the
set of detected pilot indices up to the $k$-th iteration, and $\mathbf{I}_{B}$
is the $B\times B$ identity matrix. The iterations continue until
$n_{\text{OMP}}$ iterations are performed, and the output list of the detected
pilots as well as pattern indices $\hat{I}$ is obtained as they are the same
for a specific user since both of them are determined by the first $B_{p}$
bits, which are also decoded in this step.

####  III-B2 Channel Vector Estimation

Given the set of effective active pilot sequence estimates
${\mathbf{\hat{A^{\prime}}}}$, we estimate the channel vectors of the active
users by employing an LMMSE solution as

$\mathbf{\hat{H}}=(\mathbf{\hat{A^{\prime}}}^{H}\mathbf{\hat{A^{\prime}}}+N_{0}\mathbf{I}_{\hat{K_{a}}})^{-1}\mathbf{\hat{A^{\prime}}}^{H}\mathbf{Y}_{p}^{(j)},$
(9)

where $\hat{K_{a}}$ is the cardinality of ${\hat{I}}$ and
$\mathbf{Y}_{p}^{(j)}$ is the residual pilot signal in the $j$-th decoding
iteration.

####  III-B3 Symbol Estimation and Channel Decoding

Given the channel vector and data pattern estimates $\mathbf{\hat{H}}$ and
$\mathbf{\hat{P}}_{\text{data}}$, the other data bits are decoded by first
estimating user symbols by MRC and then passing them to a single-user decoder.
For this purpose, using MRC, we obtain the symbol estimates of the $i$-th user
by

$\hat{\mathbf{s}}_{i}=\mathbf{Y}_{d}^{(j)}\left(\mathbf{\hat{p}}_{\text{data}}(i)\right)\mathbf{\hat{h}}_{i}^{H},$
(10)

where $\mathbf{Y}_{d}^{(j)}(\mathbf{\hat{p}}_{\text{data}}(i))$ is the set of
rows of $\mathbf{Y}_{d}^{(j)}$ specified by the active data indices determined
by $\mathbf{\hat{p}}_{\text{data}}(i)$, and
$\mathbf{\hat{p}}_{\text{data}}(i)$ and $\mathbf{\hat{h}}_{i}$ are the data
pattern and channel vector estimates of the $i$-th user, respectively. We then
pass these symbol estimates to a channel decoder employing successive
cancellation list decoding (SCLD), where the log-likelihood ratio (LLR) values
are calculated by treating the symbol estimates as the output of a single-user
channel using the fact that the users are separated by MRC. The decoded
sequence is assumed to be valid and added to the output list if the CRC check
is satisfied.

####  III-B4 Successive interference cancellation

Due to the high multiuser interference, SIC is an essential step in URA
schemes to achieve low error rates. In the case that the initially estimated
channel vectors are used directly for SIC, it is performed as

$\mathbf{Y}^{(j+1)}=\mathbf{Y}^{(j)}-\mathbf{\hat{X}}_{\hat{\mathcal{D}}}\mathbf{\hat{H}}_{\hat{\mathcal{D}}},$
(11)

Figure 3: Decoding process of the proposed scheme.

where $\mathbf{Y}^{(j)}$ is the residual received signal at the $j$-th
iteration, $\mathcal{\hat{D}}$ is the set of the successfully decoded
sequences, $\mathbf{\hat{X}}_{\hat{\mathcal{D}}}$ is the matrix including the
re-constructed transmitted symbols, and $\mathbf{\hat{H}}_{\hat{\mathcal{D}}}$
is the set of channel vector estimates of the successfully decoded users.
Another approach is to utilize the re-encoded and modulated decoded message
sequences to re-estimate the channel vectors and perform SIC using them as

$\mathbf{\hat{H}}_{\text{re}}=(\mathbf{\hat{X}}_{\hat{\mathcal{D}}}^{H}\mathbf{\hat{X}}_{\hat{\mathcal{D}}}+N_{0}\mathbf{I}_{\absolutevalue{\mathcal{\hat{D}}}})^{-1}\mathbf{\hat{X}}_{\hat{\mathcal{D}}}^{H}\mathbf{Y}^{(j)},$
(12)

$\mathbf{Y}^{(j+1)}=\mathbf{Y}^{(j)}-\mathbf{\hat{X}}_{\hat{\mathcal{D}}}\mathbf{\hat{H}}_{\text{re}}.$
(13)

Algorithm 1 Decoding algorithm of the proposed scheme

1:Input: $\mathbf{Y}_{p}$, $\mathbf{Y}_{d}$, $\mathbf{A}$,
$\mathbf{P}_{\text{pilot}}$, $\mathbf{P}_{\text{data}}$, $\Delta$,
$n_{\text{max}}$, $n_{\text{OMP}}$

2:Construct the extended pilot codebook $\mathbf{A^{\prime}}$ by $\mathbf{A}$
and $\mathbf{P}_{\text{pilot}}$.

3:Activity detection:

4:for $k=1,2,\ldots n_{\text{OMP}}$ do

5: $\mathbf{C}=\mathbf{A^{\prime}}^{H}\mathbf{Y}_{p}^{(k)}$.

6: Get the decision metrics by the Euclidean norm of rows of $\mathbf{C}$,
take the
$i_{\text{OMP}}=\left\lceil{({{K_{a}}}+\Delta)/n_{\text{OMP}}}\right\rceil$
indices with the maximum decision metrics and add them to ${\hat{I}}$.

7: Subtract the effect of the detected indices by (8).

8:end for

9:Iterative decoding:

10:for $j=1,2,\ldots n_{\text{max}}$ do

11: Set $\mathcal{\hat{D}}=\emptyset$ , and
$\hat{K}_{a}=\absolutevalue{\mathcal{\hat{I}}}$.

12: Channel vector estimation by (9).

13: for $i=1,2,\ldots,\hat{K}_{a}$ do

14: MRC: Estimate $\mathbf{\hat{s}}_{i}$ by (10).

15: Pass $\mathbf{\hat{s}}_{i}$ to the polar decoder $\rightarrow$
$\mathbf{\hat{m}}_{i}$.

16: CRC check.

17: end for

18: if $\absolutevalue{\mathcal{\hat{D}}}=0$ then

19: Algorithm termination.

20: end if

21: for i in $\mathcal{\hat{D}}$ do

22: Re-encode and modulate $\mathbf{\hat{m}}_{i}$

23: end for

24: SIC:

25: Interference cancellation by (11) or re-estimation and interference
cancellation by (12) and (13).

26:end for

27:Output: List of the decoded messages

Our extensive simulations show that the SIC method combined with the data-
aided channel estimation is superior to one with the initially estimated
channel vectors in the lower multiuser interference regimes since the non-
decoded users act as interference. A comparison of these two methods is
provided in Section IV.

After SIC, the pilot and pattern indices of the successfully decoded messages
are removed from $\hat{I}$, which is given back to the channel estimation
stage along with the residual signal. The decoding iterations continue until
no valid codewords are detected in the current iteration or a predetermined
number of iterations $n_{\text{max}}$ are reached. The decoding process of the
proposed scheme is depicted in Fig. 3, where $\mathbf{\hat{S}}$ is the set of
user symbol estimates, and a pseudo-code of the decoding process is provided
in Algorithm 1.

###  III-C Complexity Analysis

We provide a computational complexity analysis in terms of the number of
multiplications per decoding iteration in this subsection. The complexity of
the correlation step of gOMP in (7) is $\mathcal{O}(MNn_{p})$, which is the
dominant complexity of activity detection. The complexities of channel
estimation and MRC are $\mathcal{O}(K_{a}^{2}n_{p})$ and
$\mathcal{O}(K_{a}Mn_{d})$, respectively, and re-estimation step for SIC has a
complexity $\mathcal{O}(K_{a}^{2}(n_{p}+n_{d}))$, dominating the complexity of
our proposed scheme. Note that one can implement SCLD by only additions and
subtractions [20]. The complexity of this operation is
$\mathcal{O}(K_{a}n_{L}n_{c}\hskip 1.42262pt\log n_{c})$, where $n_{L}$ is the
list size.

###  III-D Collisions

In the URA schemes, the active users pick a pilot, preamble, or pattern
sequence from a common codebook based on a part of their message bits as a
typical approach. Therefore, when the message bits that are allocated for this
purpose are the same for some different users, a collision event happens. If
the channel is a Gaussian MAC, the collision of two users is recovered with a
high probability [6]. For the effect of collisions on the system performance
in the fading MAC scenario and a detailed analysis of collision probability,
one can refer to [18].

##  IV Numerical Results

We evaluate the performance of the proposed scheme in this section. We take
$n=3200$, $B=100$, $\epsilon=0.05$ for $M=50$ and $\epsilon=0.1$ for $M=8$,
and assume Rayleigh fading channels with i.i.d. channel coefficients across
the users and the antennas for a more straightforward comparison with the
existing literature, where $\epsilon$ is the target PUPE. We utilize 5G polar
codes and set the code length to 1024, the CRC length to 16, the list size of
SCLD to 128, and $n_{\text{OMP}}=4$. We employ a sub-sampled discrete Fourier
transform (DFT) matrix as the pilot codebook which reduces the complexity of
the correlation step of gOMP to $\mathcal{O}(MN\text{log}N)$ by converting
matrix multiplications to DFT operations [21]. Furthermore, we assume that the
active indices for each transmission pattern are uniformly and randomly
generated, and choose $B_{p}$ such that the probability of collision of more
than two users is negligible.

We assess the energy efficiency by calculating the required $E_{b}/N_{0}$ by
performing a 2D search over $P_{p}$ and $P_{d}$ and comparing it with those of
some state-of-the-art solutions. In the massive MIMO regime where $M=50$, we
take $B_{p}=15$ for $K_{a}<800$ and $B_{p}=16$ otherwise, and $n_{p}=600$ and
$n_{p}^{\prime}=1000$ for $100\leq K_{a}\leq 600$, $n_{p}=800$ and
$n_{p}^{\prime}=1200$ for $K_{a}=800$ and $n_{p}=1000$ and
$n_{p}^{\prime}=1300$ for $K_{a}>800$ in our simulations. Also, we try both of
the SIC methods discussed in Sec. III-B and get the minimum of the resulting
required $E_{b}/N_{0}$ values for a given active user load. The results in
Fig. 4 demonstrate that our proposed scheme offers the best performance
compared to the state-of-the-art for $K_{a}\geq 800$ while it can support up
to 1400 users. Namely, it outperforms FASURA [16] and SNOP-URA [18] by a
margin of as high as 4.5 dB and 1.5 dB for $K_{a}\leq 1000$, respectively, and
sparse Kronecker product (SKP) coding scheme in [22] by up to 2.5 dB for
$K_{a}\leq 1400$. Note that in the SKP coding scheme, PUPE is measured by only
misdetections; hence, its performance can be slightly worsened if false alarms
are also considered.

In the multiple antenna regime where $M=8$, the proposed scheme has excellent
performance as illustrated in Fig. 5, where we take $n_{p}=800$ and
$n_{p}^{\prime}=1200$, and $B_{p}=15$. It performs better than FASURA and SKP
coding by up to 5 and 3 dB for $K_{a}\leq 250$, respectively, and it offers a
superior performance compared to SNOP-URA by up to 1.5 dB for $K_{a}\leq 300$.

Figure 4: Comparison of the required $E_{b}/N_{0}$ versus the number of active
users for $M=50$ and $P_{e}\leq 0.05.$ Figure 5: Comparison of the required
$E_{b}/N_{0}$ versus the number of active users for $M=8$ and $P_{e}\leq 0.1.$

We also compare the mean square error (MSE) performance of candidate SIC
solutions introduced in Sec. III-B normalized over the number of antennas
using the same parameters in Fig. 4. For simplicity, we only consider the
first iteration. The MSE results in Fig. 6 show that the data-aided channel
estimation is more beneficial for most of the active user regime, however, the
initial channel estimates can be used in the high multiuser interference
regime instead of them. The effect of this result on system performance can
also be seen in Fig. 6, showing that re-estimating the channel vectors using
the decoded symbols provides a performance advantage for $K_{a}\leq 1000$,
however, one should utilize the initially estimated channel vectors for
$K_{a}>1000$ as it provides a similar or better performance with a lower
complexity.

Figure 6: Comparison of the MSE and required $E_{b}/N_{o}$ versus number of
active users for different SIC methods.

##  V Conclusions

We study URA with multiple antennas at the receiver side, and propose an ODMA-
based scheme where the channel frame is divided into two parts, assuming that
the active users distribute their pilot and data symbols to the first and
second parts, respectively, based on a transmission pattern. We employ gOMP to
detect active pilots and patterns, LMMSE channel estimation, and single-user
polar decoding to recover the transmitted messages. Numerical examples
demonstrate that our proposed scheme offers a higher energy efficiency
compared to the state-of-the-art for different numbers of receive antennas,
and that it can be employed in environments with a very high connection
density.

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