# An optical elevator for precise delivery of cold atoms using an acousto-
optical deflector

Francesco Ferri1,2, Arthur La Rooij1,3, Claire Lebouteiller1, Pierre-Antoine
Bourdel1, Mohamed Baghdad1, Sylvain Schwartz1,4, Sébastien Garcia1,5, Jakob
Reichel1, Romain Long1 1 Laboratoire Kastler Brossel, ENS-Université PSL,
CNRS, Sorbonne Université, Collège de France, 24 rue Lhomond, 75005 Paris,
France  2 Present address: Institute for Quantum Electronics, ETH Zurich, 8093
Zurich, Switzerland  3 Present address: University of Strathclyde, Department
of Physics, SUPA, Glasgow G4 0NG, United Kingdom  4 Present address: DPHY,
ONERA, Université Paris-Saclay, 91123 Palaiseau, France  5 Present address:
Collège de France, 11 Place Marcelin Berthelot, 75321 Paris, France
<EMAIL_ADDRESS>
###### Abstract

We implement a simple method for fast and precise delivery of ultracold atoms
to a microscopic device, i.e. a Fabry-Perot microcavity. By moving a single
beam optical dipole trap in a direction perpendicular to the beam axis with an
acousto-optical deflector, we transport up to 1 million atoms within 100 ms
over $1\,$cm. Under these conditions, a transport efficiency above $95\%$ is
achieved with only minimal heating. The atomic cloud is accurately positioned
within the microcavity and transferred into an intra-cavity optical lattice.
With the addition of a secondary guiding beam, we show how residual sloshing
motion along the shallow axis of the trap can be minimized.

##  1 Introduction

The transport of cold neutral atoms has been realized in different
experimental contexts. One common motivation is to deliver cold atoms to a
“science chamber” benefiting from a lower vacuum pressure and/or better
optical access than a “loading chamber” where a magneto-optical trap (MOT) is
produced. Transport is also required to enable the interaction of cold atomic
samples with in-vacuum microscopic devices such as optical microcavities [1,
2, 3, 4, 5, 6] or micromechanical resonators [7, 8, 9], whose size precludes
the direct loading of a MOT in their vicinity.

Different transport schemes have been implemented, which are all based on
moving either magnetic of optical traps. Movable magnetic traps have been
created by successively activating different pairs of coils placed along a
path [10] or by mechanically moving magnetic coils on a rail [11]. Atom chip-
based magnetic conveyors have been implemented to transport small ensemble of
ultracold atoms with high positioning accuracy [12, 13, 14]. In all these
methods, the transport setup reduces the optical access to the atoms along one
or multiple directions.

The alternative method is to use optical dipole traps. The transport of a cold
atomic cloud has been achieved by moving the waist of a single-beam optical
trap along the optical axis. It has been realized by translating a focusing
lens [15], by using a pair of focus tunable lenses [16] or more recently by
using a Moiré lens [17]. These methods can achieve large transport distances
over tens of centimeters, but require either moving or deforming elements or
achieve limited transport efficiency. Alternatively, optical conveyor belts
have been developed specially in the context of Cavity Quantum Electrodynamics
(CQED) experiments [18, 1, 2, 3]. However, they are significantly more complex
to implement, requiring costly single frequency high power lasers as well as
interferometric phase stabilization, and are limited to few millimeters
transport distance except for specific demanding realizations [19, 20, 21]. To
conclude this short overview, we note that tight optical tweezers are used for
re-arrangement of single atom arrays but the typical displacements are in the
100$\,\mu$m range [22, 23, 24, 25, 26].

For experiments requiring the interactions of cold atoms with microscopic
devices, several requirements have to be fulfilled. First, the transporting
trap should be tight enough for an atomic ensemble to fit inside the target
device. Furthermore, the final position of the atomic cloud has to be
reproducible with micrometer accuracy. In addition, one would rather avoid
moving parts to prevent mechanical vibrations and consequent drifts. Finally,
the transport duration between the MOT and the device should be as short as
possible to limit atom losses and achieve fast experimental cycle times. For
our specific experiment, the endpoint of the transport is located at 12 mm
from the starting point where the MOT is obtained. This target endpoint is the
optical mode of a fiber Fabry-Perot microcavity (length $\simeq 140\,\mu$m,
waist $\simeq 8\,\mu$m), which is a key element for the generation of multi-
particle entangled states by CQED methods [27, 28, 29, 30].

Transporting an atomic cloud over a moderately large distance of $1\,$cm and,
at the same time, position it with micrometer accuracy is challenging. To
address this problem, we have developed a new and versatile solution by
designing an optical elevator based on a single-beam dipole trap and an
acousto-optical deflector (AOD), a device that has been used before but in
different contexts [31, 32, 33, 34]. This allows a robust transport with high
acceleration, low loss and minimal excitation as the atomic sample is
transported along a direction transverse to the beam axis, so a direction of
much stronger confinement than along the beam axis. This method is simple to
implement, requiring apart from the AOD only a lens and a high-power laser
(not necessarily monomode). It allows us to transport a cold atomic cloud over
one centimeter with acceleration up to $\sim 10\,$ m$\cdot$s-2 and efficiency
above 95$\%$, and to position it with sub-micrometer accuracy ($\sim
0.6\,\mu$m). It is also robust as no moving or deforming part is involved.

The paper is organized as follows. In section 2, we describe the qualitative
and quantitative features of our transport setup. In section 3, we
characterize the setup experimentally and discuss the results. We present in
section 4 an upgraded version of the setup with the addition of a guiding beam
to reduce the oscillations along the shallow axis of the trap, and conclude by
summarizing the results and potential further applications of our method.

##  2 Implementation of the optical elevator

The transport duration $T_{trans}$ is usually targeted to be as short a
possible in order to achieve fast cycle times but different effects limit the
minimal achievable duration. The finite trap depth puts an upper limit on the
maximal acceleration $a_{lim}$ of the atoms while keeping them trapped. For a
given transport distance $D$, this leads to a lower limit of the transport
duration $T_{lim}\propto\sqrt{\frac{D}{a_{lim}}}$. In addition, when the
transport duration gets closer to the oscillation period $T_{0}$ of the trap,
non-adiabatic processes induce oscillations of the cloud, heating and atom
loss. If a single laser beam is used for trapping, the transport can be
performed either perpendicularly or along the beam directions. We briefly
analyze the different limits of the transport duration for these two
geometries.

For transport along the longitudinal direction [35, 16], the maximal
acceleration scales as $a_{lim}^{\|}\propto\frac{U_{0}}{m_{Rb}z_{R}}$ where
$U_{0}$ is the trap depth, $m_{Rb}$ the mass of a Rb atom, and
$z_{R}=\frac{\pi w_{0}^{2}}{\lambda}$ is the Rayleigh range of the dipole trap
beam with $w_{0}$ the waist and $\lambda$ the wavelength of the trapping
light. We then find $\frac{T_{lim}^{\|}}{T_{0}}\propto\sqrt{\frac{D}{z_{R}}}$.
A typical example of longitudinal transport can be found in Ref. [35], where
$T_{lim}^{\|}<T_{0}$. Non-adiabatic effects are then dominant in the
longitudinal geometry and transport schemes based on shortcuts to adiabaticity
can be advantageous [36]. For transport along the transverse direction, the
maximal acceleration scales as
$a_{lim}^{\bot}\propto\frac{U_{0}}{m_{Rb}w_{0}}$. One then finds
$\frac{T_{lim}^{\bot}}{T_{0}}\propto\sqrt{\frac{D}{w_{0}}}$. For standard trap
parameters, $T_{lim}^{\bot}>T_{0}$ and so the transport is limited by the
finite trap depth, enabling a fast and simple transport of cold atoms, that
does not required involved shortcuts to adiabaticity nor phase-stabilized
standing-waves.

In the following, we implement a transport along the transverse direction by
designing an optical elevator based on an acousto-optical deflector (AOD) and
a lens. In figure 1, the transport method is presented schematically. The AOD
deflects a collimated red-detuned laser beam along the transverse direction:
the first diffracted order is used for the optical dipole trap, while all
other orders are blocked. An achromatic doublet with a focal distance $f$
focuses the collimated laser beam at the center of the science chamber along
the $x$-axis. The AOD is placed at the distance $f$ behind the lens, hence
sweeping the driving radio-frequency (RF) of the AOD produces a translation of
the focused beam perpendicular to the optical axis. Atoms are loaded in the
dipole trap at the lower position from the magneto-optical trap and
transported vertically along the $y$-axis by 12 mm up to the microcavity.

Figure 1:  (a) The setup for transporting cold atoms from the MOT to the fiber
mircrocavity. The method is based on a single-beam optical dipole trap at 1070
nm that move due to an acousto-optical deflector (AOD). (b) Graphics showing
the initial and final stage of the transport scheme. The optical trap (red) is
moving the atoms (yellow) from below the cavity where the MOT is produced, up
to the cavity where they can be transferred to the intra-cavity lattice
(green). (c) Absorption images along two orthogonal directions of the atoms
loaded in the dipole trap at the start of the transport after a time-of-flight
of 0.5 ms (average over 10 single-shot images).

Here we use cold 87Rubidium atoms which have a cooling transition at 780.24
nm. The red-detuned light for the optical dipole trap is provided by a 1070 nm
fiber laser with a maximal output power of 50 W [37], which is a linearly
polarized multimode laser [38, 39]. The laser output beam which is fiber
delivered has an initial beam diameter of 5.0 mm. The AOD has an aperture of
7.5 mm and an efficiency above $70$% over the whole scan angle of 85 mrad
[40]. The 2 inches diameter achromat has a focal length $f=250\,$mm, which in
combination with the scan angle of the AOD allows us to move the dipole trap
by up to 14 mm. The lens also sets the trap waist at $45\,\mu$m. This fulfills
two important requirements: it allows the transport beam to overlap with
several tens of sites of an intra-cavity optical lattice at a wavelength of
$1559\,$nm; on the other hand, it is small enough to fit into the $140\,\mu$m
long cavity without depositing excessive optical power on the fibers.

Due to the tight fit, any misalignment or extended transport duration can
cause the high-power laser to damage the micromirrors. We implement a safety
system to monitor both the intensity of the dipole trap beam and the frequency
of the AOD (representing the transverse position of the beam). It switches off
the laser beam if the dipole beam stays close the cavity for too long or when
the beam intensity is too large. The radio-frequency (RF) signals used to
drive the AOD are provided by a Direct Digital Synthesizer (DDS) [41]
controlled by an Arduino microcontroller [42] via serial communication. The
DDS is used in the linear sweep mode and the microcontroller performs a
dynamical update of the DDS sweeping slope at a rate of up to 20 kHz. This
allows the implementation of arbitrary velocity profiles for the transport,
where the effect of discretization on the atoms is fully negligible even for
transport duration as short as 10 ms [43]. Equivalently, discretization of the
profile is negligible if the number of steps is above 100.

##  3 Transport efficiency and heating

To load the dipole trap, the 1070 nm beam is overlapped with the MOT and a 5
ms molasses cooling is performed before switching off the cooling beams.
Approximately $10^{6}$ atoms are loaded at a power of $6\,$W, which
corresponds to an optical trap depth of $220\,\mu$K. The longitudinal and
transverse trap frequencies are found to be $1\,$kHz and $5\,$Hz respectively.
For a laser power of 6 W, the temperature of the cloud after loading is about
20$\,\mu$K. Residual heating of the atoms is observed on the timescale of 1 s
in the static trap. However this will not play a significant role during the
transport, as we achieve transport durations in the 100 ms range.

With the RF-control system described in the previous paragraph, an arbitrary
velocity profile can be implemented for the transport with a high level of
control. We typically use transport ramps which are discretized in 200 time
steps. Different velocity profiles have been investigated: the Blackman-Harris
function [44, 35], a polynomial of degree 2, a linear function and a constant
velocity profile. We observe a clear degradation of the transport if a
constant velocity profile is used, while only minor differences exist between
all the other profiles. In the following, we use the Blackman-Harris profile
(see Appendix), which for a given transport duration leads to a slightly
lowest heating of the atoms by about 2$\,\mu$K.

To characterize the transport with the optical elevator, the atoms are moved
by 10 mm and we measure the transport efficiency $\eta=\frac{N_{f}}{N_{i}}$
and the heating $\Delta\theta=\theta_{f}-\theta_{i}$ by absorption imaging.
Here $N_{i}$ and $N_{f}$ are respectively the number of atoms at the initial
position and at the final position of the ramp. The temperature increase
$\Delta\theta=\theta_{f}-\theta_{i}$ of the atoms is extracted from
$\theta_{f}$ and $\theta_{i}$ which are the temperature of the atoms at the
end and at the start of the transport. All temperatures are measured along the
$x$ axis by time-of-flight measurement. An hold time of $1\,$s was applied
after the transport to allow for thermalization. In figure 2, the efficiency
and the heating are shown as a function of the transport duration $T_{trans}$
for different values of the dipole laser power $P$. If $T_{trans}$ is kept
larger than $100$ ms, the transport efficiency is above $90\%$ and the heating
is kept small for all laser powers. To limit the amount of heat transferred to
the microcavity, a laser power of $P=6.0\,$W and a transport duration of
$T_{trans}=100\,$ms are considered optimal. With these parameters, $9.5\times
10^{5}$ atoms are transported with an efficiency of $(97.0\pm 0.5)\%$. The
cloud temperature is increased by $(2.5\pm 0.5)~{}\mu$K after transport. The
calculated maximal acceleration is $8.6\,$m$\cdot$s-2, which is an order of
magnitude larger than typical transport schemes along the longitudinal
direction. These results show that a fast, robust, and efficient transport can
be performed over a wide range of parameters without inducing excessive
heating.

Figure 2: (Left) Fraction of transported atoms as a function of the transport
duration $T_{trans}$. (Right) Heating of the atoms $\theta_{f}-\theta_{i}$
induced by the transport.

As $T_{trans}$ gets shorter than $50\,$ms, losses and heating become
significant. For the parameters of our transport setup, the minimal transport
time $T_{lim}$ due to the finite trap depth is 4.6 times larger than the
oscillation period $T_{0}$ along the transport direction. Therefore, we expect
that the minimal duration for an efficient transport is limited by the finite
trap depth. By computing the effective potential along the vertical direction
in the reference frame of the transport beam moving against gravity for
$P=6\,$W, we find that the atoms become untrapped when moving faster than 12
ms. This occurs at the time when the acceleration is maximal for a Blackman-
Harris velocity profile. For transport duration 12 ms $<T_{trans}<$ 40 ms, the
actual height of the potential barrier is strongly reduced by the acceleration
in the moving frame, inducing large atom loss.

Due to the large anisotropy of the trapping potential, we also observe
oscillations of the atomic cloud of $0.5\,$mm along the longitudinal weak axis
of the trap for $T_{trans}=100\,$ms (see Fig. 3). We attribute the excitation
of these oscillations to time-dependent thermal aberrations of the AOD and the
focusing lens which affect the trapping potential during the transport. These
oscillations lead to a less efficient and reproducible loading of the atoms in
the cavity mode. We address this issue in the following section.

##  4 Improved transport and cavity loading

To circumvent the aforementioned longitudinal oscillations, a second red-
detuned beam is introduced, orthogonal to the transport beam, creating a
crossed dipole trap. This second beam intersects both the center of the fiber
cavity and the center of the $1070\,$nm dipole trap at the MOT position, see
Figure 3 a. It acts as a guide for the sub-ensemble of atoms at the center of
the elongated $1070\,$nm trap. Since the guide beam passes through the
microcavity, its heating effect has to be minimized in order to protect the
mirror coatings as well as to maintain the cavity stabilization. For this
reason, we choose the wavelength of the guide beam to be $785.2\,$nm, which is
only $5\,$nm away from the 87Rb D2 line. As the beam waist is $50\,\mu$m, this
allows us to achieve a trap depth of 280$\,\mu$K with a reasonably low optical
power of 380 mW. This waist size guarantees at the same time low clipping from
the cavity mirrors and a small beam divergence along the path between the
cavity and the MOT.

Figure 3: (a) Scheme of the additional dipole trap (wavelength $785\,$nm) for
guiding the atoms in the $1070\,$nm transport trap to the fiber cavity. (b)
Sloshing amplitude of the atoms trapped in the 1070 nm trap (black, and in the
crossed dipole trap (red). The atomic clouds are detected after 0.5 ms free
expansion and fitted with a 2D Gaussian profile in two different regions of
interest; the error bars correspond to $1/4$ of the standard deviation. (c)
Absorption images after a time-of-flight of 0.5 ms of the atoms loaded in the
dimple trap at the MOT position (left) and at the cavity position (middle).
The right picture is an absorption image with a time-of-flight of 0.1 ms of
the atoms after being transferred into the 1559 nm lattice . The images result
from the average of 10 single-shot images. The color scales indicate the
optical depth. The grey rectangles represent the two optical fibers of the
microcavity.

To load the atoms in the crossed dipole trap, we proceed as previously
described with the additional ramping up of the guide beam power in 25 ms
before the molasse phase. With this configuration and a transport duration of
100 ms over 11 mm, the guide beam allows us to increase the atom number in the
cavity by $30\%$. A temperature increase of the atoms by 10$\,\mu$K is also
observed and deemed acceptable. It can be explained by the adiabatic
compression of the cloud as the size of the guide beam is reduced from
80$\,\mu$m at the MOT to 50$\,\mu$m inside the cavity.

The main advantage of the guide beam is visible in Fig.3b, where the
oscillations of the atomic ensemble along the weak axis of the 1070 nm trap
are shown for different transport durations. The sloshing amplitude of the
atoms in the bare 1070 nm trap is on the order of $0.5\,$mm for
$T_{trans}=100\,$ms. In the crossed dipole trap, the position of the sub-
ensemble is stable within a few tenths of microns, which is small compared to
the extension of the cloud itself, enabling the efficient and reproducible
loading of atoms inside the cavity.

In our optical microcavity, the atoms are trapped in an intra-cavity standing-
wave at 1559 nm, which ensures an equal and maximal coupling of all the atoms
with a resonant probing standing wave at 780 nm [45]. To avoid irreversible
damages to the cavity mirrors and to be less demanding for the cavity
stabilization, we reduce the optical power of both the transport and the guide
beams when moving the transport beam closer to the cavity. Starting at 2.5 mm
away from the cavity, we reduce exponentially their power to 15$\%$ of their
initial values, which leads to an approximately isotropic trap with calculated
trap frequencies of $(400,400,550)\,$Hz at the cavity position. We typically
end up with $10^{5}$ atoms in the crossed dipole trap inside the cavity at a
temperature of 25$\,\mu$K.

The atoms are then transferred adiabatically from the crossed dipole trap to
the intra-cavity lattice, by ramping down the transport and guide beam power
while increasing the 1559 nm intensity over 20 ms. As the 1559 nm is also used
for cavity stabilization, we also have to keep a minimal amount of power
inside the cavity corresponding to a trap depth of 10$\,\mu$K. In Fig. 3c, we
show an absorption image of a cold atoms cloud loaded in a 1 mK deep
intracavity lattice after a time-of-flight of 100$\,\mu$s. For such a short
time, the expansion is negligible compared to the initial size. We load
typically around 1600 atoms, and the cloud size is 60$\,\mu$m along the cavity
axis, corresponding to the loading of about 80 sites with more than 10 atoms
per site on average.

##  5 Conclusion

In this article, a novel scheme for transporting cold atoms into a Fabry-Perot
microcavity is presented. It is based on the realization of an optical
elevator with an acousto-optical deflector at the focus of a lens. We have
presented a new scheme for transporting ultracold atoms that enables a fast
transport at high acceleration by moving along the strong axis of the dipole
trap while avoiding any mechanical vibration due to the elimination of moving
parts. Clouds of $10^{6}$ atoms are transported over a distance of $10\,$mm in
$100\,$ms with efficiency above 95$\%$ and low residual heating. An additional
guide beam, orthogonal to the $1070\,$nm laser, is shown to reduce the effect
of atoms sloshing along the weak axis of the dipole trap. This moving crossed
dipole elevator has enabled us to deliver the atomic cloud with sub-micrometer
accuracy and can therefore be used to precisely position a cold atomic
ensemble in or nearby microscopic devices. In our lab, this fast and highly
reproducible scheme enables future studies of many-body QED entanglement
generation for quantum metrology [46] and quantum simulations [47]. This
scheme can be used advantageously in a wide range of applications where cold
atomic ensembles must be positioned with high accuracy, in particular near
nanodevices such as photonic waveguides and optomechanical resonators.

## Acknowledgements

We thank Constance Poulain for careful reading of the manuscript. This project
has received funding from: Agence Nationale de la Recherche (ANR) (SAROCEMA
project, ANR-14-CE32-0002); European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme Grant agreement No
671133 (EQUEMI project). This work has been supported by Region Ile-de-France
in the framework of DIM SIRTEQ. S. Schwartz acknowledges funding from the
European Union under the Marie Sklodowska Curie Individual Fellowship
Programme H2020-MSCA-IF-2014 (project No. 658253).

## Appendix

The Blackman-Harris profile of the speed used to transport the atoms over a
distance $D$ in a time $T$ is given by the following equation :

The Blackman-Harris profile describes the velocity of the atoms while they
move over a distance D in a time T with the following equation:

$v(t,T)=\frac{D}{T}\left(1-\frac{25}{21}\cos{\Big{(}2\pi\frac{t}{T}\Big{)}}+\frac{4}{21}\cos{\Big{(}4\pi\frac{t}{T}\Big{)}}\right).$
(1)

In figure 4, the acceleration, velocity and position are shown for $D=1$.

Figure 4: Position profile (black line), Velocity profile scaled by $21/50$
(blue dotted line), Acceleration profile scaled by $\times 0.116$ (red dashed
line) for a Blackman-Harris ramp.

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# Joint User Activity and Data Detection in Grant-Free NOMA using Generative
Neural Networks

Yixuan Zou, Zhijin Qin, Yuanwei Liu,
Queen Mary University of London, London, UK
{yixuan.zou, z.qin<EMAIL_ADDRESS>
###### Abstract

Grant-free non-orthogonal multiple access (NOMA) is considered as one of the
supporting technology for massive connectivity for future networks. In the
grant-free NOMA systems with a massive number of users, user activity
detection is of great importance. Existing multi-user detection (MUD)
techniques rely on complicated update steps which may cause latency in signal
detection. In this paper, we propose a generative neural network-based MUD
(GenMUD) framework to utilize low-complexity neural networks, which are
trained to reconstruct signals in a small fixed number of steps. By exploiting
the uncorrelated user behaviours, we design a network architecture to achieve
higher recovery accuracy with a low computational cost. Experimental results
show significant performance gains in detection accuracy compared to
conventional solutions under different channel conditions and user sparsity
levels. We also provide a sparsity estimator through extensive experiments.
Simulation results of the sparsity estimator showed high estimation accuracy,
strong robustness to channel variations and neglectable impact on support
detection accuracy.

##  I Introduction

The emerging Internet-of-Things (IoT) technologies accelerate the urge for
reliable high-capacity wireless communication systems in supporting the
massive connections among IoT devices and users. It brings challenges to the
traditional grant-based orthogonal multiple access (OMA) schemes due to the
limiting capacity of the conventional cellular systems and the high signalling
overhead caused by the complicated and time-consuming scheduling. To overcome
the aforementioned challenges, grant-free NOMA techniques have been proposed,
which allow multiple users to share common resources non-orthogonally using
controllable interference without complicated grant mechanism [1].

A typical grant-free NOMA system can support more users than the number of
orthogonal resources [1] and data packets can be transmitted in the next
available time slot without waiting for a grant. Since the BS has no prior
knowledge of user activity, MUD should be performed at the BS in addition to
signal detection. Exploiting the sporadic user activity, compressed sensing
(CS) has been considered as a standard framework for solving the MUD problems
[2]. In practice, users transmit information in a continuous manner and stay
in the communication system for several time slots. Hence, as a mean to
further improve the reconstruction performance, many recent works utilize the
temporal correlations in user activity and adopt the frame-wise joint sparsity
model in their MUD algorithms [3] [4]. In particular, [3] proposed a threshold
aided block sparsity adaptive subspace pursuit algorithm to improve recovery
accuracy and [4] proposed a sparsity-blind greedy algorithm to accommodate
practical scenarios. However, both approaches rely on complicated update steps
where the total number of iterations per recovery is unknown in either method.
It raises the concern of slow recovery speed, which is critical to the
latency-sensitive IoT applications.

Recently, a deep learning-based CS-MUD solution has been proposed, which
achieves faster recovery speed and higher detection accuracy compared to
conventional iterative algorithms [5]. This approach, however, requires
multiple trained networks to be stored at the BS to accommodate different
system overloading ratios, since the received signals are used as the network
inputs. [6] proposed a deep learning-based active user detection which detects
sparse supports without prior knowledge on sparsity. Similar to [5], this
approach also uses the received signals as network inputs, resulting in the
same requirements of different models for a different number of orthogonal
resources.

Compared to the typical neural network designs, generative networks, which
could deal with arbitrary input dimensions, serve as a more flexible solution
with fewer restrictions on applicable scenarios. To address the slow recovery
speed of generative models in CS problems, [7] applied model-agnostic meta-
learning (MAML) in the training algorithm so that the target signals can be
recovered in as few as three iterations instead of hundreds. The fast and
accurate signal recovery ability motivates the use of generative neural
networks in solving MUD problems. However, standard generative network
architectures [8] are not suitable for MUD. In typical communication system
setups, users are assumed to behave independently and transmit signals based
on random choices which make it difficult for generative networks to learn
compared to image datasets which exhibit strong spatial correlations.
Fortunately, the learning capability of generative networks can be improved by
larger input dimensions [9]. However, the sizes of standard generative
networks scale badly with input dimensions since fully-connected layers are
commonly used as the first layer. Moreover, large kernel sizes are generally
used in standard generative networks to capture the spatial features, whereas,
in communication data, there are limited spatial correlations hence small
kernel sizes are preferred.

In practice, it is usually difficult to gain knowledge of user sparsity prior
to MUD. Various efforts have been made on sparsity-blind CS approaches and a
common solution is to estimate the signal sparsity adaptively in the recovery
algorithm [3] [4]. However, these sparsity approximation methods are each
restricted to a certain type of recovery algorithms such as subspace pursuit-
based approaches [3] and orthogonal matching pursuit (OMP)-based approaches
[4]. Therefore, it is difficult to adopt the existing sparsity estimation
techniques in new CS recovery algorithms, especially for the non-iterative
deep learning-based approaches.

In this paper, we propose a generative neural network-based MUD (GenMUD)
framework based on the training algorithm in [7]. The main contributions are
as follows:

  1. 1.

We propose to use generative networks in solving MUD problems to cope with
different overloading factors with a single trained model.

  2. 2.

We design a low-complexity generative network architecture using 1x1
convolutional layers which significantly reduce the computational cost as
input dimensions are increased to achieve higher recovery accuracy.

  3. 3.

We obtain a closed-form low-complexity sparsity estimator which can be applied
to any MUD algorithm to realize sparsity-blind MUD without complicated and
iterative sparsity approximations.

  4. 4.

We show that the proposed approach outperforms conventional methods in terms
of detection accuracy, and the proposed sparsity estimator achieves high
accuracy under various channel conditions while making little impact on
support detection accuracy.

##  II System Model

###  II-A System Setup

We consider the uplink NOMA system with K users and one BS. Without loss of
generality, all users and the BS are assumed to be equipped with one single
antenna.

By performing channel coding and modulation, the transmitted symbol $x_{k}$ by
user $k$ is spread onto $M$ orthogonal subcarriers by a unique spreading
sequence $\boldsymbol{s_{k}}=(s_{1k},s_{2k},...,s_{Mk})^{T}\in\mathbb{C}^{M}$.
Particularly, we consider the overloaded NOMA system, i.e., $M<K$. For
inactive users, their transmitted symbol is equal to zero. The $M$ signals
received at the BS can be expressed individually as

$y_{m}=\sum_{k=1}^{K}g_{mk}s_{mk}x_{k}+n_{m},\ \ m=1,2,...,M\ ,$ (1)

where $g_{mk}$ denotes the channel gain of user $k$ transmitted over the $m$th
subcarrier and $n_{m}\sim\mathcal{CN}(0,\sigma^{2})$ denotes the Gaussian
noise with noise power $\sigma^{2}$. We adopt the Rayleigh fading channel
model whose channel gains are independent and identically distributed (i.i.d.)
complex Gaussian random variables, i.e.,
$g_{mk}\overset{i.i.d.}{\sim}\mathcal{CN}(0,1)\ \forall{m,k}$. To simplify the
expression, we express the received signal vector
$\boldsymbol{y}=(y_{1},...,y_{M})^{T}$ as

$\boldsymbol{y}=\boldsymbol{H}\boldsymbol{x}+\boldsymbol{n},$ (2)

where $\boldsymbol{x}=(x_{1},...,x_{K})^{T}$ is the transmitted signal vector,
$\boldsymbol{H}$ denotes the $M\times K$ channel matrix whose entries are
$h_{mk}=g_{mk}s_{mk}$, and $\boldsymbol{n}=(n_{1},...,n_{M})^{T}$ is the
Gaussian noise vector.

###  II-B Frame-wise Joint Sparsity Model

Generally, users transmit data in consecutive time slots and remain active or
inactive throughout the time frame [3] [4]. Motivated by the temporal
correlations in user activity, we extend the system from a single transmission
to a time frame model, known as the frame-wise joint sparsity model.

Given a time frame of length $J$, the common sparsity support $\mathcal{S}$ is
defined as

$supp(\boldsymbol{x}^{(1)})=supp(\boldsymbol{x}^{(2)})=\dots=supp(\boldsymbol{x}^{(J)})\triangleq\mathcal{S},$
(3)

where $supp(\boldsymbol{x}^{(j)})=\\{k\ |\ x^{(j)}_{k}\neq
0,k\in\\{1,...,K\\}\\}$. The number of active users during each transmission
is defined as $S=|\mathcal{S}|$. We thus achieve the frame-wise joint sparsity
model:

$\boldsymbol{Y}=\boldsymbol{HX+N},$ (4)

where
$\boldsymbol{Y}=[\boldsymbol{y}^{(1)},...,\boldsymbol{y}^{(J)}]\in\mathbb{R}^{M\times
J}$,
$\boldsymbol{X}=[\boldsymbol{x}^{(1)},...,\boldsymbol{x}^{(J)}]\in\mathbb{R}^{K\times
J}$ and
$\boldsymbol{N}=[\boldsymbol{n}^{(1)},...,\boldsymbol{n}^{(J)}]\in\mathbb{R}^{M\times
J}$. We restrict the length of the time frame to be shorter than the channel
coherence time, so that the channel matrix
$\boldsymbol{H}\in\mathbb{R}^{M\times K}$ remains constant throughout the
entire time frame.

The MUD problem now becomes a 2-dimensional CS problem, where the estimation
target is the signal matrix $\boldsymbol{X}$ with known channel matrix
$\boldsymbol{H}$ and the received signal matrix $\boldsymbol{y}$, which could
be formulated as solving the optimization problem:

$\underset{\boldsymbol{X}}{\text{min}}||\boldsymbol{Y}-\boldsymbol{H}\boldsymbol{X}||_{2}^{2}.$
(5)

Existing recovery methods consist of two major approaches,
$\ell_{1}$-minimization and greedy algorithms. $\ell_{1}$-minimization methods
[10] provide high recovery accuracy and theoretical performance guarantees but
suffer from heavy computational complexity and are sensitive to noise. Greedy
algorithms [11] have relatively lower complexity but usually require a large
number of measurements for exact recovery.

###  II-C Performance Metrics

Symbol error rate (SER) has been used as the main evaluation method for MUD
algorithms. It describes the general detection ability of the algorithm and
fails to reflect some specific targets of MUD. To achieve a more thorough
performance evaluation, we adopt three separate performance metrics: SER,
detection probability ($P_{d}$) and false alarm probability ($P_{fa}$). Given
the recovered symbol $\hat{X}^{(j)}_{k}$ for user $k$ in the $j$th time slot,
the performance metrics are introduced as follows:

  * •

SER: SER is defined as the ratio of incorrectly recovered symbols transmitted
by the active users to all symbols transmitted by the active users, given by

$\text{SER}=\frac{1}{S}\sum_{k\in\mathcal{S}}\mathds{1}_{\\{X^{(j)}_{k}\neq\hat{X}^{(j)}_{k}\\}}.$
(6)

  * •

$P_{d}$: $P_{d}$ is defined as the ratio of correctly detected active users to
all active users, given by

$P_{d}=\frac{1}{S}\sum_{k\in\mathcal{S}}\mathds{1}_{\\{\hat{X}^{(j)}_{k}\neq
0\\}}.$ (7)

  * •

$P_{fa}$: $P_{fa}$ is defined as the ratio of incorrectly detected inactive
users to all inactive users, given by

$P_{fa}=\frac{1}{K-S}\sum_{k\notin\mathcal{S}}\mathds{1}_{\\{\hat{X}^{(j)}_{k}\neq
0\\}}.$ (8)

.

##  III Generative Networks for Multi-user Detection

In this section, we first describe the MAML training procedure of generative
networks based on the algorithm developed in [7], then introduce the proposed
network architecture, followed by the proposed GenMUD framework.

###  III-A Generative Network Training Algorithm

MUD aims to obtain the user activity and decode the transmitted signal if the
user is active, which could be achieved by solving (5). To achieve higher
accuracy and lower complexity, we propose to use generative networks for
solving (5), which is described in the following.

Instead of directly solving (5), a map is built from a latent space to the
space of all possible transmitted signals as

${}\boldsymbol{X}=G_{\theta}(\boldsymbol{z}),$ (9)

where $\boldsymbol{z}$ represents a point in the latent space of arbitrary
dimensions and $G_{\theta}$ is a generative neural network. By searching for a
particular $\boldsymbol{\hat{z}}$ in the latent space that maps to an accurate
estimation $\boldsymbol{\hat{X}}$, the MUD problem is reformulated as:

${}\boldsymbol{\hat{z}}=\underset{\boldsymbol{z}}{\text{argmin}}||\boldsymbol{Y}-\boldsymbol{H}G_{\theta}(\boldsymbol{z})||_{2}^{2}\
.$ (10)

One common solution to the above non-convex optimization problem is to apply
gradient descent starting from a randomly sampled point
$\boldsymbol{z}\sim\mathcal{N}(0,1)$. Therefore, we can update
$\boldsymbol{\hat{z}}$ by

$\boldsymbol{\hat{z}}\leftarrow\boldsymbol{\hat{z}}-\alpha\frac{\partial||\boldsymbol{Y}-\boldsymbol{H}G_{\theta}(\boldsymbol{z})||_{2}^{2}}{\partial\boldsymbol{z}},$
(11)

where $\alpha$ indicates the learning rate. However, it usually requires
hundreds or even thousands of gradient descent steps until (11) converges.
Fortunately, a separate loss function for $G_{\theta}$ has been proposed and
minimized in conjunction with (9) [7]. This technique of optimizing an
optimization procedure is known as Model-Agnostic Meta-Learning (MAML) [12].

To formulate MAML in the context of MUD, we denote
$p_{\text{task}}(\mathcal{T})$ as the distribution of tasks where each task
$\mathcal{T}_{i}$ is to find an optimal $\boldsymbol{\hat{z}}_{i}$ that
approximates the target signal matrix $\boldsymbol{X}_{i}$. MAML is employed
by training the network weights $\boldsymbol{\theta}$ against the measurement
error over all tasks:

${}\underset{\theta}{\text{min}}\ \mathcal{L}_{G},\ \ \text{for}\
\mathcal{L}_{G}=\mathbb{E}_{\mathcal{T}_{i}\sim
p_{\text{task}}(\mathcal{T})}\left[||\boldsymbol{Y}_{i}-\boldsymbol{H}_{i}G_{\theta}(\boldsymbol{\hat{z}}_{i})||_{2}^{2}\right],$
(12)

where $\boldsymbol{\hat{z}}_{i}$ is the output of task $\mathcal{T}_{i}$,
i.e., the gradient descent steps in (11). To further increase the convergence
speed, the same optimization procedure is applied to the learning rate
$\alpha$.

An additional constraint is needed for $G_{\theta}$ since it can exploit (12)
and easily reach small loss by mapping all $G_{\theta}(\boldsymbol{\hat{z}})$
into the null space of $\boldsymbol{H}$. To address this problem, the
generator is trained against Restricted Isometry Property (RIP) through an
approximated RIP loss:

$\mathcal{L}_{H}=\mathbb{E}_{\boldsymbol{X}_{1},\boldsymbol{X}_{2}}\left[\Big{(}\big{|}\big{|}\boldsymbol{H}\boldsymbol{X}_{1}-\boldsymbol{H}\boldsymbol{X}_{2}\big{|}\big{|}_{2}-\big{|}\big{|}\boldsymbol{X}_{1}-\boldsymbol{X}_{2}\big{|}\big{|}_{2}\Big{)}^{2}\right],$
(13)

where $\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$ are sampled from different
stages in the training process. Specifically, we sample one true signal, two
generated signals each before and after the operation in (11), then compute an
average loss over the three pairs of signals. Algorithm 1 illustrates the
training procedure for $G_{\theta}$.

Algorithm 1 Generative Network Training Algorithm

0: Training data $\\{\boldsymbol{X}_{i}\\}_{i=1}^{N_{d}}$, channel matrix
$\\{\boldsymbol{H}_{i}\\}_{i=1}^{N_{d}}$, generator $G_{\theta}$, number of
latent update steps $T$

0: Trained generator $G_{\hat{\theta}}$, optimized learning rate
$\hat{\alpha}$ Initialize $\theta$, $\alpha$

1: repeat

2: for $i=1$ to $N_{d}$ do

3: Measure the signal
$\boldsymbol{y}_{i}\leftarrow\boldsymbol{H}_{i}\boldsymbol{x}_{i}+\boldsymbol{n}_{i}$

4: Sample $\boldsymbol{\hat{z}}_{i}\sim\mathcal{N}(0,\boldsymbol{I})$

5: for $t=1$ to $T$ do

6:
$\boldsymbol{\hat{z}}_{i}\leftarrow\boldsymbol{\hat{z}}_{i}-\alpha\frac{\partial}{\partial\boldsymbol{\hat{z}}}_{i}||\boldsymbol{y}_{i}-\boldsymbol{H}G_{\theta}(\boldsymbol{\hat{z}}_{i})||_{2}^{2}$

7: end for

8:
$\mathcal{L}_{G}=\frac{1}{N_{d}}\sum_{i=1}^{N_{d}}||\boldsymbol{y}_{i}-\boldsymbol{H}G_{\theta}(\boldsymbol{\hat{z}}_{i})||_{2}^{2}$

9: Compute $\mathcal{L}_{H}$ using Eq. (13)

10: Update
$\theta\leftarrow\theta-\frac{\partial}{\partial\theta}(\mathcal{L}_{G}+\mathcal{L}_{H})$

11: Update
$\alpha\leftarrow\alpha-\frac{\partial}{\partial\alpha}(\mathcal{L}_{G}+\mathcal{L}_{H})$

12: end for

13: until reaches the maximum training steps

14: Return $G_{\hat{\theta}}$, $\hat{\alpha}$

###  III-B Proposed Neural Network Architecture

Standard generative neural networks [8] are mainly designed for image data to
extract spatial correlations and hidden structures. To be applied to MUD data
which consists of random data points, larger input dimensions are required for
greater learning capability. However, the use of fully-connected layers causes
network sizes to increase dramatically with input dimensions, leading to high
computational complexity. Moreover, the large kernel sizes used in typical
networks introduce redundant network parameters when applied to MUD data, and
have the risk of overfitting. Hence, we propose a low-complexity generative
network designed specifically for MUD problems.

The proposed network has an input dimension of $4KJ\times 1$ and an output
dimension of $2K\times J$, i.e.,
$[\Re(\boldsymbol{X}),\Im(\boldsymbol{X})]^{T}$. The hidden layers consist of
three 1D convolutional layers all using a kernel size of 1. LeakyReLU
activation and batch-normalization are employed after each convolutional layer
except the last. The last layer employs Tanh activation to scale the outputs
to [-1, 1], as displayed in Fig 1.

The proposed architecture has three main differences compared to standard
generative networks:

  * •

Large input dimensions: For generative neural networks, larger input
dimensions are proved to increase the learning capability of the network at
the cost of computational complexity [9]. Given the considered model, we
choose a latent dimension of twice the size of the output, since a further
increase in the dimension improves accuracy by a negligible amount.

  * •

No fully-connected layers: Fully-connected layers scale badly with input
sizes, hence are too expensive for solving MUD problems in term of complexity.
In the proposed network, the input layer is followed directly with a
convolutional layer to increase the model scalability as the input dimensions
grow.

  * •

1x1 convolutional layers: Considering the independent user behaviours, we
propose to eliminate any spatial convolutions among the users. In particular,
we replace all the 2D convolutional layers with 1x1 1D convolutional layers of
input size $K$, i.e., the total number of users. The resulting network is
capable of generating the framewise joint sparsity feature without any inter-
user correlation.

Figure 1: Structure of the adopted neural network.

###  III-C GenMUD Framework

Given the received signals $\boldsymbol{Y}$, channel response matrix
$\boldsymbol{H}$ and sparsity level $S$, the MUD problem can be solved by the
trained model, including the trained generator $G_{\hat{\theta}}$ and the
optimized learning rate $\hat{\alpha}$.

A set of initial reconstructions are obtained by optimizing the latent
representation $\boldsymbol{z}$ from a random start using (11). Both the
dimension of $\boldsymbol{z}$ and the number of gradient descent steps $T$
must be the same values used in the training algorithm. The generator
$G_{\hat{\theta}}$ is treated as fixed during the latent updates and
$\hat{\alpha}$ is used as the learning rate.

Since neural networks output continuous numbers, we further map the outputs
into valid modulation symbols. We calculate the magnitude of each
reconstructed signal and, in each time slot, map the $S$ signals with greater
magnitudes to the nearest constellation symbol, and the rest to zero
indicating inactive users. Algorithm 2 illustrates the proposed GenMUD
algorithm.

Algorithm 2 GenMUD Algorithm

0: Received signals $\boldsymbol{Y}$, channel matrix $\boldsymbol{H}$, pre-
trained generator $G_{\hat{\theta}}$, number of latent update steps $T$,
optimized learning rate $\hat{\alpha}$

0: Reconstructed signals $\boldsymbol{\hat{X}}$

1: Sample $\boldsymbol{\hat{z}}\sim\mathcal{N}(0,\boldsymbol{I})$

2: for $t=1$ to $T$ do

3:
$\boldsymbol{\hat{z}}\leftarrow\boldsymbol{\hat{z}}-\hat{\alpha}\frac{\partial}{\partial\boldsymbol{\hat{z}}}||\boldsymbol{Y}-\boldsymbol{H}G_{\hat{\theta}}(\boldsymbol{\hat{z}})||_{2}^{2}$

4: end for

5: Initial reconstructions
$\boldsymbol{\tilde{X}}=G_{\hat{\theta}}(\boldsymbol{\hat{z}})$

6: Initialize output $\boldsymbol{\hat{X}}=\boldsymbol{0}_{K\times J}$

7: for $j=1$ to $J$ do

8: for $s=1$ to $S$ do

9: $p=\text{argmax}||\boldsymbol{\tilde{X}}^{(j)}||^{2}_{2}$

10: $\hat{X}^{(j)}_{p}=\\{\tilde{X}^{(j)}_{p}\text{mapped to the nearest
symbol}\\}$

11: $\tilde{X}^{(j)}_{p}=0$

12: end for

13: end for

14: Return $\boldsymbol{\hat{X}}$

##  IV Simulations

In this section, we evaluate the performance of the proposed GenMUD algorithm
in solving MUD problems compared to several existing solutions including
orthogonal matching pursuit (OMP) [11], a dynamic CS-based MUD method (DyCS)
[13] and basis pursuit de-noising (BPDN) algorithm [10]. The oracle least
squares (LS) algorithm, which assumes full knowledge of the sparse support,
including the sparsity level and the location of sparse support, is considered
as a benchmark method. For all considered methods, user sparsity is assumed to
be known at the BS unless otherwise stated. In the simulations, the total
number of users is $K=200$ and the transmitted signals are modulated by
Quadrature Phase Shift Keying (QPSK). The number of consecutive time slots is
fixed at $J=7$. Toeplitz random matrix is used to design the spreading matrix
so that RIP is satisfied with high probability [14].

###  IV-A Model Training

The proposed neural network is trained by the simulated dataset where the
number of active users is $S=40$ among the $K=200$ users, the length of the
time frame is $J=7$, SNR is set to 20 dB and the number of subcarriers is
$M=100$. Latent parameters are normalized before the first update and the
total number of latent update steps is $T=20$. The initial value for the
latent learning rate $\alpha$ is 0.01. Adam is used as the optimizer with
learning rate 0.0001. The batch size is 32.

###  IV-B Simulation Results for the MUD

Fig. 2 depicts the SER performance comparison of the considered algorithms
under different SNRs. In the simulations, the number of active users is $S=40$
and the number of subcarriers is $M=100$. In Fig. 2, the proposed GenMUD
algorithm outperforms OMP and DyCS significantly in terms of SER over the
whole range of SNR. In comparison to the BPDN approach, GenMUD shows
increasing performance gain as SNR increases from 0 dB to 14 dB. It is
observed that the proposed GenMUD demonstrates near oracle performance when
SNR is lower than 8 dB and is the only method other than LS that achieves an
SER lower than $1\times 10^{-3}$. Given that SNR is 20 dB during training, the
results indicate that the neural network has learned to minimize the influence
of noise during detection and provide high symbol detection accuracy through
iterative latent updates.

Figure 2: SER performance comparison against SNR. Figure 3: Detection
probability $P_{d}$ performance comparison against SNR. Figure 4: False alarm
probability $P_{fa}$ performance comparison against SNR.

Fig. 3 illustrates the detection performance, $P_{d}$, against SNRs for
different methods. The number of active users is $S=40$ and the number of
subcarriers is $M=100$. From the figure, the proposed GenMUD method achieves
the highest $P_{d}$ among all methods, except for oracle LS which assumes full
knowledge of the sparse support. As SNR becomes above 10 dB, the proposed
GenMUD achieves nearly 100% detection probability.

Fig. 4 compares the performance of the false alarm probability, $P_{fa}$,
against SNRs among the considered methods. It is noted that the proposed
GenMUD achieves the lowest $P_{fa}$ compared to OMP, BPDN and DyCS, and
approaches zero $P_{fa}$ as SNR increases beyond 10 dB. The results in Fig. 3
and Fig. 4 validate that the proposed GenMUD is capable of accurately
identifying active users from inactive users for all SNRs.

Figure 5: SER performance comparison against user sparsity.

Fig, 5 illustrates the effects of user sparsity on SER performance of the
considered methods under $\text{SNR}=6$ dB and $M=100$. For all the methods,
the SER performance degrades as the number of active users increases. However,
the proposed GenMUD exhibits consistently lower SER than OMP, DyCS and BPDN
throughout the range of SNR. Given that the proposed neural network is trained
under $S=40$ active users, the consistent performance gains of the proposed
method imply that the network has precisely captured the underlying
relationships between user activity and the received signals.

###  IV-C Sparsity Estimation

Sparsity estimation has been a challenging topic in MUD due to the difficulty
in measuring user sparsity prior to MUD. Most MUD algorithms, such as DyCS and
OMP, are designed based on known user sparsity which limits their applications
in practice. Recently, some sparsity-blind MUD algorithms are proposed with
different sparsity-blind strategies including approximation algorithms for
sparsity built inside the MUD algorithms [3] or stopping criteria for greedy
algorithms with no prior knowledge of user sparsity [4]. These sparsity
approximation methods are computationally expensive and do not apply to other
MUD frameworks.

Based on extensive experiment results and data analysis, we propose a sparsity
estimator given by

$\hat{S}=\mathbb{E}\left[\frac{\tau}{2(\tau+1)}||\boldsymbol{y}||_{2}^{2}\right],$
(14)

where $\tau$ is SNR in its linear scale and $\boldsymbol{y}$ is the $M\times
1$ signal vector received in a single time slot. In terms of a framewise model
of length $J$, the estimator is formulated as

$\hat{S}=\frac{1}{J}\sum_{j=1}^{J}\frac{\tau}{2(\tau+1)}||\boldsymbol{y}^{(j)}||_{2}^{2}.$
(15)

The estimation accuracy is evaluated by the normalized error ($E_{n}$), given
by

$E_{n}=\frac{|S-\hat{S}|}{S}.$ (16)

Figure 6: Normalized error ($E_{n}$) of the sparsity estimator against frame
length. Figure 7: Detection probability ($P_{d}$) and false alarm probability
($P_{fa}$) performance against SNR of the proposed GenMUD algorithm with known
sparsity or estimated sparsity.

Fig. 6 depicts the $E_{n}$ performance against frame length $J$ of the
proposed sparsity estimator in the framewise model. Results are plotted for a
different number of subcarriers $M$ and SNRs. As the frame length increases
from 1 to 50, $E_{n}$ of the proposed estimator decreases gradually from
around 0.05 to below 0.01 indicating a generally low estimation error. The
decrease in $E_{n}$ also indicates an inverse relationship between estimation
error and the number of time slots $J$. In practical scenarios, the BS can
perform a constant update of the estimated sparsity as new signals are
received to reduce error overtime. It is also observed that there is little
increase in $E_{n}$ as the communication environment becomes severe, i.e., as
$M$ decreases from 100 to 50 and SNR decreases from 10 dB to 0 dB, which
demonstrates the robustness of the proposed estimator to the varying
environment.

Fig. 7 illustrates the $P_{d}$ and $P_{fa}$ performance of the proposed GenMUD
with known sparsity and estimated sparsity. In the simulations, the number of
active users is $S=40$ and the number of subcarriers is $M=100$. It is
observed that, the proposed GenMUD with estimated sparsity achieves the same
$P_{fa}$ performance as when sparsity is known. For SNR smaller than 6 dB, the
sparsity estimator is used with no impact on $P_{d}$ performance, and for SNR
greater than 6 dB, the performance difference is unnoticeable. Fig. 7 shows
that the proposed sparsity estimator provides accurate approximation and has a
neglectable influence on support detection performance.

##  V Conclusions

In this paper, we investigated the problem of multi-user detection (MUD) for
grant-free non-orthogonal multiple access (NOMA) using compressed sensing (CS)
techniques. We proposed a generative neural network-based MUD (GenMUD)
framework and designed a generative neural network to achieve higher learning
capability with lower complexity by exploiting the independent user
behaviours. Simulation results showed that the proposed GenMUD method provided
better detection performance compared to conventional CS and MUD approaches in
terms of SER, detection probability and false alarm probability. We also
provided a sparsity estimator supported with simulation results demonstrating
low estimation error and negligible impact on support detection accuracy under
various communication settings.

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$l_{*}(\mathcal{O}_{C})$ and $i_{*}(\mathcal{O}_{C})$ are $\sigma$-stable. As
a consequence, all the already constructed pairs in Lemma 3.47 are Bridgeland
stability conditions.

######  Theorem 6.13.

The space of stability conditions
$\operatorname{Stab}(\mathcal{T}_{C})=S_{12}\cup S_{23}\cup S_{31}$ is a
connected, four dimensional complex manifold.

######  Proof 54.

Since $\mathcal{V}_{12}$ is connected, this implies that $S_{12}$ is also
connected.

Moreover, $S_{12}\cap S_{23}=S_{23}\cap S_{31}=S_{12}\cap S_{31}$ is not
empty. Therefore $\operatorname{Stab}(\mathcal{T}_{C})$ is connected.

##  Appendix A Recollement

In this appendix, we would like to stress the importance of the gluing
condition (3.5) of the hearts in the definition of the CP-glued stability
conditions. In [35, Proposition 2.8.12], we showed that CP-gluing of hearts
correspond to hearts constructed by classical recollement in the sense of [6]
where the gluing condition (3.5) is satisfied. We recall the definitions in
[6] below.

######  Definition A.1.

Let $\mathcal{X},\mathcal{Y},\mathcal{D}$ be triangulated categories.
$\mathcal{D}$ is said to be a recollement of $\mathcal{X}$ and $\mathcal{Y}$
if there are six triangulated functors as in the diagram

$\mathcal{X}$$\mathcal{D}$$\mathcal{Y}$$j_{*}=j_{!}$$j^{*}$$j^{!}$$i^{*}=i^{!}$$i_{!}$$i_{*}$

such that

  1. (1)

$(i^{*},i_{*})$, $(i_{!},i^{!})$, $(j^{*},j_{*})$, $(j_{!},j^{!})$ are adjoint
pairs;

  2. (2)

$j_{*}$, $i_{*}$, $i_{!}$ are full embeddings;

  3. (3)

$j^{!}\circ i_{*}=0$ (and thus also $i^{*}\circ j_{*}=0$ and $j^{*}\circ
i_{!}=0$);

  4. (4)

for every $T\in\mathcal{D}$ there are triangles

$\textstyle{i_{!}i^{!}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}j^{*}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{!}i^{!}T[1]}$$\textstyle{j_{!}j^{!}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{*}i^{*}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{!}j^{!}T[1].}$

Note that the functors of the definition of recollement satisfy the following
properties as a consequence of the vanishing condition (3.).

  * •

$\prescript{\perp}{}{\mathcal{X}}=\ker(j^{*})$ and $j^{*}\circ i_{!}=0$
implies that $i_{!}$ embeds the category $\mathcal{Y}$ as
$\prescript{\perp}{}{\mathcal{X}}$.

  * •

$\mathcal{X}^{\perp}=\ker(j^{!})$ and $j^{!}\circ i_{*}=0$ implies that
$i_{*}$ embeds the category $\mathcal{Y}$ as $\mathcal{X}^{\perp}$.

Hence, if $i_{*}$ denotes the natural embedding of $\mathcal{X}$ into
$\mathcal{D}$, we have $\mathcal{Y}=\mathcal{X}^{\perp}$. In fact, the
definition of recollement gives the two semiorthogonal decompositions
$\mathcal{D}=\langle\mathcal{X},\prescript{\perp}{}{\mathcal{X}}\rangle$ and
$\mathcal{D}=\langle\mathcal{X}^{\perp},\mathcal{X}\rangle$ associated to an
admissible full subcategory $\mathcal{X}\subset\mathcal{D}$.

######  Proposition A.2 ([35, Proposition 2.8.2]).

Let $\mathcal{D}$ be a triangulated category and let
$\mathcal{X}\subset\mathcal{D}$ be a full triangulated subcategory. Then,
$\mathcal{D}$ is a recollement of $\mathcal{X}$ and $\mathcal{X}^{\perp}$ if
and only if $\mathcal{X}$ is (left and right) admissible.

The following theorem shows how to construct t-structures from t-structures in
the smaller subcategories.

######  Theorem A.3 ([6, Theorem 1.4.10]).

Let $\mathcal{X},\mathcal{Y},\mathcal{D}$ be triangulated categories such that
$\mathcal{D}$ is a recollement of $\mathcal{X}$ and $\mathcal{Y}$ and assume
the notation of Definition A.1. Let $(\mathcal{X}^{\leq 0},\mathcal{X}^{\geq
0})$ and $(\mathcal{Y}^{\leq 0},\mathcal{Y}^{\geq 0})$ be t-structures on
$\mathcal{X}$ and $\mathcal{Y}$ respectively. Then there is a t-structure
$(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ in $\mathcal{D}$ defined by:

$\begin{array}[]{c}\mathcal{D}^{\leq 0}\coloneqq\\{T\in\mathcal{D}\mid
i^{*}T\in\mathcal{Y}^{\leq 0},j^{*}T\in\mathcal{X}^{\leq 0}\\}\\\
\mathcal{D}^{\geq 0}\coloneqq\\{T\in\mathcal{D}\mid i^{*}T\in\mathcal{Y}^{\geq
0},j^{!}T\in\mathcal{X}^{\geq 0}\\}.\end{array}$

If we write $\mathcal{A}_{\mathcal{X}}$ and $\mathcal{A}_{\mathcal{Y}}$ for
the corresponding hearts in $\mathcal{X}$ and $\mathcal{Y}$ respectively, we
denote by
$\operatorname{rec}(\mathcal{A}_{\mathcal{Y}},\mathcal{A}_{\mathcal{X}})\coloneqq\mathcal{D}^{\leq
0}\cap\mathcal{D}^{\geq 0}$.

######  Remark A.4.

From Remark 3.3, we know that all three subcategories $\mathcal{C}_{i}$ for
$i=1,2,3$ are admissible in $\mathcal{T}_{C}$. Therefore, by Proposition A.2
they define 3 recollements of $\mathcal{T}_{C}$. Given hearts
$\mathcal{A}_{i}$ (resp. $\mathcal{A}_{j}$) in $\mathcal{C}_{i}$ (resp.
$\mathcal{C}_{j}$), we will denote by
$\operatorname{rec}_{ij}(\mathcal{A}_{i},\mathcal{A}_{j})$ the heart in
$\mathcal{T}_{C}$ as recollement of $\mathcal{C}_{j}$ and $\mathcal{C}_{i}$
constructed as in Theorem A.2 for $ij\in\\{12,23,31\\}$.

In general, when a heart is constructed by CP-gluing we know it agrees with a
heart constructed from recollement. See [35, Proposition 2.8.12]. On the other
hand, at least in the context of holomorphic triples, a heart constructed by
recollement without satisfying CP-gluing condition does not accept a stability
function as shown in the example below.

######  Lemma A.5 (Jealousy Lemma).

Let $\mathcal{A}\subset\mathcal{T}_{C}$ be a heart constructed by recollement
of hearts $\mathcal{A}_{i}\subset\mathcal{C}_{i}$,
$\mathcal{A}_{j}\subset\mathcal{C}_{j}$ which do not satisfy CP-gluing
conditions. Then, $\mathcal{A}$ does not accept a stability function defined
on $K(\mathcal{A})$, i.e. $Z(\mathcal{A})\not\subset\overline{\mathbb{H}}$ for
every $Z\colon K(\mathcal{A})\rightarrow\mathbb{C}.$

###### Proof.

We give the proof for the case of $\mathcal{T}_{C}$ as recollement of
$\mathcal{C}_{2}$ and $\mathcal{C}_{1}$ and the other cases will follow by
acting with the Serre functor $S_{\mathcal{T}_{C}}$ (or its inverse) on
$\sigma$.

Let $\sigma=(Z,\mathcal{A}_{12})$ be a stability condition on
$\mathcal{T}_{C}$ such that
$\mathcal{A}_{12}\coloneqq\operatorname{rec}_{12}(\mathcal{A}_{1},\mathcal{A}_{2})$
is a heart in $\mathcal{T}_{C}$ defined by recollement from given hearts
$\mathcal{A}_{i}\coloneqq\operatorname{Coh}^{r_{i}}(C)$ in $\mathcal{C}_{i}$,
for $i=1,2$ that do not satisfy the CP-gluing condition (3.5), i.e. such that

(A.1) $r_{2}-1<r_{1}<r_{2}.$

First of all, we claim that the hearts $i_{*}\mathcal{A}_{1}$,
$j_{*}\mathcal{A}_{2}$ and $l_{*}\mathcal{A}_{1}$ are in $\mathcal{A}_{12}$.
Indeed, it follows from the definitions. Recall from the isomorphism of
Theorem 2.49 that we can identify $\mathcal{A}_{i}=\mathcal{P}(r_{i},r_{i}+1]$
for $i=1,2$, where $\mathcal{P}(0,1]=\operatorname{Coh}(C)$. Therefore,
$j_{*}\mathcal{P}(r_{2},r_{2}+1]\subset\mathcal{A}_{12}$ since $i^{*}j_{*}=0$
and by adjunction $j^{*}j_{*}=\operatorname{id}$ and
$j^{!}j_{*}=\operatorname{id}$. Also
$i_{*}\mathcal{P}(r_{1},r_{1}+1]\subset\mathcal{A}_{12}$, since
$j^{!}i_{*}=0$, $i^{*}i_{*}=\operatorname{id}$ and using
$j^{*}i_{*}=j^{*}j_{!}[1]$ and by (A.1),
$\mathcal{P}(r_{1}+1,r_{1}+2]\subset\mathcal{P}(r_{2},\infty)$. Similarly,
$l_{*}\mathcal{P}(r_{1},r_{1}+1]\subset\mathcal{A}_{12}$, since $l_{*}=i_{!}$,
$j^{*}i_{!}=0$, $i^{*}i_{!}=\operatorname{id}$ and using
$j^{!}i_{!}=l^{*}l_{*}$ and by (A.1),
$\mathcal{P}(r_{1},r_{1}+1]\subset\mathcal{P}(-\infty,r_{2}+1]$.

As in Remark 2.52, we write each $r_{i}=n_{i}+\theta_{i}$ for unique
$n_{i}\in\mathbb{Z}$ and $\theta_{i}\in[0,1)$. We assume $n_{2}$ to be (up to
shift) equal to $0$. Note that equation (A.1) implies that either $n_{1}=0$
and $\theta_{1}<\theta_{2}<\theta_{1}+1$ or $n_{1}=-1$ and
$\theta_{2}<\theta_{1}<\theta_{2}+1$.

If we look closely at the imaginary part of $Z$, it has the form $\Im
Z(r_{1},d_{1},r_{2},d_{2})=D_{1}d_{1}+D_{2}d_{2}+C_{1}r_{1}+C_{2}r_{2}$ with
$C_{i},D_{i}\in\mathbb{R}$, for $i=1,2$. The restrictions to the previous
hearts are

$\begin{array}[]{l}\Im Z\mid_{i_{*}\mathcal{A}_{1}}=D_{1}d+C_{1}r\\\ \Im
Z\mid_{j_{*}\mathcal{A}_{2}}=D_{2}d+C_{2}r\\\ \Im
Z\mid_{l_{*}\mathcal{A}_{1}}=(D_{1}+D_{2})d+(C_{1}+C_{2})r\end{array}$

for $d,r\in\mathbb{Z}$ with $r\geq 0$. We recall that if $Cr+Dd$ is the
imaginary part of a stability function on $\operatorname{Coh}^{\theta}$, then
the value $\theta$ is determined by the quotient $D/C$ which implies that
$\theta_{1}$ is determined by the quotients $D_{1}/C_{1}$ and
$(D_{2}+D_{1})/(C_{1}+C_{2})$. But these two quotients cannot determine the
same $\theta_{1}$ unless $\theta_{1}=\theta_{2}$, which contradicts the
assumption (A.1). ∎

######  Remark A.6.

The contradiction was a consequence of including 3 smaller hearts in the big
one. The only case when this situation does not create a contradiction is for
the extreme cases of CP-gluing conditions in Proposition 3.21.

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