TV

    television
CPU

    central processing unit
CPRI

    common public radio interface
PLF

    predecessor leader following
WHO

    world health organization
PL

    platoon leader
LTE-V2V

    long term evolution V2V
PM

    platoon member
CAM

    cooperative awareness message
D2D

    device-to-device
TVWS

     TV white spaces
RSRQ

    reference signal received quality
RSSI

    received signal strength indicator
SWOT

    strengths, weaknesses, opportunities, and threats
AP

    access point
MS

    mobile station
ID

    identity
NCL

    neighbor cell list
WLAN

    wireless local area network
WiMAX

    worldwide interoperability for microwave access
LTE

    long term evolution
DMRS

    demodulation reference signal
UL

    uplink
DL

    downlink
DPC

    dirty paper coding
TTI

    transmission time interval
CSI

    channel state information
CoMP

    coordinated multi-point
ISM

    industrial, scientific and medical
3GPP

    3rd generation partnership project
M2M

    machine-to-machine
MTC

    machine type communication
eNB

    evolved node B
PRB

    physical resource block
PF

    proportional fair
RRA

    radio resource allocation
MCS

    modulation and coding scheme
SCM

    spatial channel model
SNR

    signal-to-noise ratio
PDF

    probability density function
GPS

    global positioning system
P2P

    peer to peer
MAC

    medium access control
OFDM

    orthogonal frequency division multiplexing
CCL

    common cell list
5G

    fifth generation
FDD

    frequency division duplex
TDD

    time division duplex
EPA

    equal power allocation
MR

    maximum ratio
PhD

    doctor of philosophy
3G

    third generation
4G

    fourth generation
ANACOM

    autoridade nacional de comunicações
AF

    array factor
ARPU

    average return per user
AHP

    analytic hierarchy process
B3G

    beyond 3G
BSc

    bachelor of science
BV

    beamforming vector
BWA

    broadband wireless access
CAPEX

    capital expenditure
COGEU

    cognitive radio systems for efficient sharing of TV white spaces in European context
CR

    cognitive radio
CV

    combining vector
DEnKF

    deterministic ensemble Kalman filter
DFT

    discrete Fourier transform
DSRC

    dedicated short-range communication
DVB-H

    handled digital video broadcast
DVB-T

    terrestrial digital video broadcast
DTT

    digital terrestrial TV
ETSI

    European telecommunications standards institute
FCC

    federal communications commission
FP7

    seventh framework programme
HM

    high mobility
HSPA

    high speed packet access
IA

    initial access
ICT

    information and communication technologies
IEEE

    institute of electrical and electronics engineers
DySPAN-SC

     IEEE DySPAN standards committee
ITU

    international telecommunications union
LAN

    local area network
LS

    least squares
LSFD

    large-scale fading decoding
MBMS

    multimedia broadcast multicast services
MVNO

    mobile virtual network operator
MT

    mobile terminal
MF

    matched filter
MSc

    master of science
NPV

    net present value
NRA

    national regulator authority
OFCOM

    office of communications
OMP

    orthogonal matching pursuit
OPEX

    operational expenditure
PHY

    physical
PMSE

    programme making and special events
PT

    Portugal
RRS

    reconfigurable radio systems
QoE

    quality of experience
SAE

    system architecture evolution
SDR

    software defined radio
SLA

    service level agreement
SVD

    singular-value decomposition
TETRA

    trans European trunked radio access
ULA

    uniform linear array
UHF

    ultra high frequency
UK

    United Kingdom
UMTS

    universal mobile telecommunications system
UPA

    uniform planar array
UTRA

     UMTS terrestrial radio access
USP

    unique selling point
USA

    United States of America
WG

    working group
Wi-Fi

    wireless fidelity
WRC

    world radiocommunication conference
WSD

    white space device
O&M

    operational & maintenance
IRR

    internal rate of return
EKF

    extended Kalman filter
EnKF

    ensemble Kalman filter
EPC

    evolved packet core
E-UTRAN

    evolved universal terrestrial radio access network
MME

    mobility management entity
PCRF

    policy and charging rules function
S-GW

    service gateway
P-GW

    packet gateway
EIRP

    equivalent isotropically radiated power
CBR

    constant bit rate
RRM

    radio resource management
HetNet

    heterogeneous network
CEPT

    European conference of postal and telecommunications administrations
ANATEL

    agência nacional de telecomunicações
RAT

    radio access technology
RAN

    radio access network
SC

    single connection
FS

    fast switching
DC

    dual connectivity
C-RAN

    cloud radio access network
NR

    new radio
mmW

    millimeter wave
KF

    Kalman filter
KPI

    key performance indicator
BB

    branch and bound
BER

    bit error rate
BS

    base station
CDF

    cumulative distribution function
CFN

    channel Frobenius norm
DCP

    difference of convex functions program
GTEL

    wireless telecommunications research group
GUB

    global upper bound
GLB

    global lower bound
HMSINR

    highest minimum signal-to-interference-plus-noise ratio (SINR)
HPBW

    half power beamwidth
HWCR

    highest weighted common rate (WCR)
HWSR

    highest weighted sum rate
IBC

    interference broadcast channel
IC

    interference channel
KKT

    Karush-Kuhn-Tucker
LB

    lower bound
LiDAR

    light detection and ranging
LOS

    line of sight
LP

    linear programming
MAP

    a maximum a posteriori probability
MILP

    mixed integer linear programming
mTRP

    multi transmission and reception point
MIMO

    multiple input multiple output
MINLP

    mixed integer nonlinear programming
MIP

    mixed integer programming
MISO

    multiple input single output
ML

    machine learning
MMSE

    minimum mean squared error
MRC

    maximum ratio combining
MRT

    maximum ratio transmission
MSE

    mean squared error
MU-MIMO

    multi-user multiple input multiple output
MU

    multi-user
NLP

    nonlinear programming
NLOS

    non-line of sight
NMSE

    normalized mean squared error
NP

    non-polynomial time
QoS

    quality of service
SCA

    successive convex approximation
SINR

    signal-to-interference-plus-noise ratio
SISO

    single input single output
SIMO

    single input multiple output
SV

    steering vector
TU

    typical urban
UAV

    unmanned aerial vehicles
UE

    user equipment
UKF

    unscented Kalman filter
JSFRA

    joint space-frequency resource allocation
UB

    upper bound
WINNER

    wireless world initiative new radio
WMMSE

    weighted minimum mean squared error
WCR

    weighted common rate
V2V

    vehicle-to-vehicle
V2I

    vehicle-to-infrastructure
V2P

    vehicle-to-pedestrian
V2X

    vehicle-to-everything
RF

    radio frequency
RL

    reinforcement learning
RRV

    rate-relaxation variable
DQN

    deep Q-network
ADMM

    alternating direction method of multipliers
SDP

    semidefinite programming
PARAFAC

    parallel factor analysis
V2N

    vehicle-to-network
BALS

    bilinear alternating least-squares
TALS

    trilinear alternating least-squares
RMSE

    root mean square error
ALS

    alternating least-squares
5G-PPP

    5G infrastructure public private partnership
SLNR-MAX

    maximum signal-to-leakage-plus-noise ratio
ReLU

    rectifier linear unit
ZF

    zero-forcing
RUE

    resource usage efficiency
5G-StoRM

    5G stochastic radio channel for dual mobility
UMi

    urban micro
DB

    database
KNN

    K-nearest neighbors
PCA

    principal component analysis
CIR

    channel impulse response
EE

    energy efficiency
SE

    spectral efficiency
LTE-A

    long term evolution advanced
NOMA

    non-orthogonal multiple access
FD

    full-duplex
OFDMA

    orthogonal frequency-division multiple access
PDPR

    pilot-to-data power ratio
AWGN

    additive white Gaussian noise
CFO

    carrier frequency offset
SER

    symbol error rate
M-QAM

    multilevel quadrature amplitude modulation
B5G

    beyond fifth generation
SSB

    synchronization signal block
gNB

    gNodeB
RSRP

    reference signal received power
PSS

    primary synchronization signal
SSS

    secondary synchronization signal
PBCH

    physical broadcast channel
PCI

    physical cell identity
RACH

    random access channel
RAR

    random access response
BPSK

    binary phase shift keying
TRP

    transmission reception point
LP-MMSE

    local partial MMSE
mMIMO

    massive multiple input multiple output
GP

    geometric programming
i.i.d.

    independent and identically distributed
SIC

    successive interference cancellation
LHS

    left hand side
SOC

    second-order cone
SOCP

    second-order cone program
FD

    full-duplex
SWIPT

    simultaneous wireless information and power transfer
WPT

    wireless power transfer
HD

    half-duplex
SI

    self-interference
IAI

    inter-AP interference
IUI

    inter-user interference
LMMSE

    linear minimum mean squared error
InH

    indoor hotspot
FPC

    fractional power control
TS

    time switching
6G

    sixth generation
B5G

    Beyond fifth generation
PS

    power splitting
i.i.d

    independent and identically distributed
CF

    cell-free

# Template for IEEE TVT LaTeX Submission

Iran M. Braga Jr., Roberto P. Antonioli, Gábor Fodor,
Yuri C. B. Silva and Walter C. Freitas Jr This work was supported in part by
Ericsson Research, Technical Cooperation Contract UFC.48, in part by the
Brazilian National Council for Scientific and Technological Development
(CNPq), in part by FUNCAP, and in part by CAPES/PRINT Grant
88887.311965/2018-00. Iran M. Braga Jr., Roberto P. Antonioli, Yuri C. B.
Silva and Walter C. Freitas Jr. are with the Wireless Telecom Research Group
(GTEL), Federal University of Ceará, Fortaleza 60455-760, Brazil. (e-mails:
{iran, antonioli, yuri, walter}@gtel.ufc.br.). Roberto P. Antonioli is also
with Instituto Atlântico, Fortaleza 60811-341, Brazil (e-mail:
roberto_antonioli@atlantico.com.br).Gábor Fodor is with Ericsson Research,
16480 Stockholm, Sweden, and also with the Division of Decision and Control,
KTH Royal Institute of Technology, 11428 Stockholm, Sweden (e-mail:
gabor.fodor@ericsson.com)Copyright (c) 2015 IEEE. Personal use of this
material is permitted. However, permission to use this material for any other
purposes must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>
# Efficient Battery Usage in Wireless-Powered Cell-Free Systems with Self-
Energy Recycling

Iran M. Braga Jr., Roberto P. Antonioli, Gábor Fodor,
Yuri C. B. Silva and Walter C. Freitas Jr This work was supported in part by
Ericsson Research, Technical Cooperation Contract UFC.48, in part by the
Brazilian National Council for Scientific and Technological Development
(CNPq), in part by FUNCAP, and in part by CAPES/PRINT Grant
88887.311965/2018-00. Iran M. Braga Jr., Roberto P. Antonioli, Yuri C. B.
Silva and Walter C. Freitas Jr. are with the Wireless Telecom Research Group
(GTEL), Federal University of Ceará, Fortaleza 60455-760, Brazil. (e-mails:
{iran, antonioli, yuri, walter}@gtel.ufc.br.). Roberto P. Antonioli is also
with Instituto Atlântico, Fortaleza 60811-341, Brazil (e-mail:
roberto_antonioli@atlantico.com.br).Gábor Fodor is with Ericsson Research,
16480 Stockholm, Sweden, and also with the Division of Decision and Control,
KTH Royal Institute of Technology, 11428 Stockholm, Sweden (e-mail:
gabor.fodor@ericsson.com)Copyright (c) 2015 IEEE. Personal use of this
material is permitted. However, permission to use this material for any other
purposes must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>
###### Abstract

This paper investigates wireless-powered cell-free systems, in which the users
send their uplink data signal while simultaneously harvesting energy from
network nodes and user terminals – including the transmitting user terminal
itself – by performing self-energy recycling. In this rather general setting,
a closed-form lower bound of the amount of harvested energy and the achieved
signal-to-interference-plus-noise ratio expressions are derived. Then, to
improve the energy efficiency, we formulate the problem of minimizing the
users’ battery energy usage while satisfying minimum data rate requirements.
Due to the non-convexity of the problem, a novel alternating optimization
algorithm is proposed, and its proof of convergence is provided. Finally,
numerical results show that the proposed method is more efficient than a
state-of-art algorithm in terms of battery energy usage and outage rate.

###### Index Terms:

Cell-free, EE, self-energy recycling, SWIPT.

##  I Introduction

Beyond fifth generation (B5G) and sixth generation (6G) networks are expected
to deal with a large number of devices, which are battery-limited. Thus,
energy management plays a crucial role in the current and next generations of
mobile networks. In this context, wireless power transfer (WPT) has garnered
several efforts in the industry and academia, and is considered a promising
technology for B5G and 6G networks. Indeed, WPT allows low-power devices,
including rechargeable batteries and energy-constrained wireless sensors, to
power up their battery by means of radio frequency signals. In terms of
enabling architectures, cell-free (CF) systems offer higher coverage
probabilities than cell-based networks, and provide surface-uniform signal
strengths, which reduces the impact of path losses experienced in wireless
scenarios. These advantageous features of CF systems improve the energy
harvesting opportunities of cell-edge users [1]. Specifically, the combination
of simultaneous wireless information and power transfer (SWIPT) and CF systems
is promising, since SWIPT is attractive in terms of enabling to carry
information and energy simultaneously [2].

SWIPT in CF systems has been investigated in some works from the literature
using the time switching (TS) protocol, such as in [1, 3, 2].Although the TS
protocol has low hardware complexity, it suffers from some performance loss in
terms of spectral efficiency (SE), because less resources are dedicated to
information transmission. In [4], the power splitting (PS) protocol was
studied, which splits the received signal into an energy-carrying signal and
information-detection signal. In general, the PS protocol achieves higher
performance than the TS protocol. However, the PS protocol receiver needs a
radio frequency signal splitter, which increases the hardware complexity.
Hence, the PS protocol is often not viable when dealing with constrained-size
devices [5].

Also, SWIPT and full-duplex (FD) can be combined to obtain advantages in terms
of spectral and energy efficiency. In contrast to conventional FD systems, in
which self-interference (SI) is harmful, SI can be beneficial in terms of
energy source for harvesting energy by exploiting self-energy recycling.
Indeed, self-energy recycling can provide an extra energy source for users in
addition to the energy sent from the access points. For instance, in [6] the
authors proposed relay selection algorithms with self-energy recycling to
improve outage probability in multi-relay FD networks with non-identically
distributed fading channels. The authors in [7] and [8] also studied self-
energy recycling in the context of multi-relay FD networks. However, as far as
we know, this is the first paper analyzing wireless-powered CF systems with
self-energy recycling.

Specifically, the main contributions of this work are:

  * •

We derive a closed-form lower bound of the amount of harvested energy and the
SINR expressions in wireless-powered CF systems with self-energy recycling.
Differently from the TS and PS protocols, in the proposed method more
resources are available for data transmission and more energy is harvested due
to self-energy recycling with low hardware complexity at the user-side;

  * •

We investigate the problem of minimizing the users’ battery energy usage with
minimum data rate demands;

  * •

Due to the non-convexity of the formulated optimization problem, we propose a
novel alternating optimization algorithm and provide its proof of convergence;
and

  * •

We analyze the performance using simulations and compare the proposed solution
with another state-of-art algorithm, showing that the proposed method is more
efficient in terms of battery energy usage and outage rate.

Notation: We use uppercase and lowercase boldface to denote matrices and
vectors, respectively. Plain letters are used for scalars.
${(\cdot)}^{\mathrm{T}}$, ${(\cdot)}^{\mathrm{H}}$ and ${(\cdot)}^{*}$ denote
the transpose, conjugate transpose and conjugate operations, respectively.
$\\{x_{i}\\}_{\forall i}$ denotes the set of elements $x_{i}$ for the values
of $i$ denoted by the subscript expression. $\mathbf{I}$ is the identity
matrix. $\operatorname{Re}\left\\{\cdot\right\\}$ takes the real value of a
complex number. $|\cdot|$ takes the absolute value of a complex number. The
expected value of a random variable is denoted by $\mathbb{E}[\cdot]$. Other
notational conventions are summarized in Table I:

TABLE I: Notation conventions. Notation | Description
---|---
$x_{m,k}$ | Variable related to the link between AP $m$ and user $k$
$\ddot{x}_{k,j}$ | Variable related to the link between user $k$ and user $j$
$\dddot{x}_{m,q}$ | Variable related to the link between AP $m$ and user $q$
$\bar{x}$ | Variable related to the line of sight (LOS) component
$x^{\text{p}}$ | Variable related to the pilot transmission
$x^{(\text{d})}$ | Variable related to the downlink transmission
$x^{(\text{u})}$ | Variable related to the uplink transmission

##  II System Model

We consider a CF system consisting of $M$ single-antenna APs and $K$ single-
antenna users. We define $\mathcal{M}$ and $\mathcal{K}$ as the sets of APs
and users, respectively. Furthermore, all APs are connected via fronthaul
links to a central processing unit (CPU). Each coherence interval is divided
into two phases. In the first phase, all users send their pilot sequences to
the APs, which estimate the channels to design precoding vectors for effective
energy transfer and data reception in the second phase. In the second phase,
all users send their data signal while harvesting energy from the APs and
interfering users. Let $\tau_{\text{c}}$ be the total number of samples per
coherence interval. Then, we have that
$\tau_{\text{c}}=\tau_{\text{p}}+\tau_{\text{u}}$, where $\tau_{\text{p}}$ and
$\tau_{\text{u}}$ are the length of the first and second phases, respectively.
Therefore, differently from the TS and PS protocols, in the proposed method
more resources are available for energy harvesting and data transmission as
they are both executed in the second phase, thus more energy is harvested due
to self-energy recycling. Also, no SI canceling is needed at the users as they
do not need to decode the data signal while transmitting data on the uplink,
which reduces the hardware complexity at the user-side.

We denote by $g_{m,k}\in\mathbb{C}$ the channel vector between user $k$ and AP
$m$, by $\ddot{g}_{k,j}\in\mathbb{C}$ the inter-user interference (IUI)
channel between user $k$ and user $j$, and by $\dddot{g}_{m,q}\in\mathbb{C}$
the channel between AP $m$ and AP $q$. Note that $\dddot{g}_{m,m}$ is the SI
channel at AP $m$ while $\dddot{g}_{m,q}$ for $m\neq q$ is referred to as the
inter-AP interference (IAI) channel, since uplink signals received at AP $m$
are corrupted by downlink signals sent from AP $q$. Similarly,
$\ddot{g}_{k,k}$ is the SI channel at user $k$. All channels are considered to
be constant in each time-frequency coherence interval and they are independent
and identically distributed (i.i.d) in different coherence intervals. In
addition, all channels are modeled as spatially uncorrelated Rician fading
channels with unknown phase shifts, i.e., each channel realization can be
expressed as

$\displaystyle g_{m,k}$
$\displaystyle=\bar{h}_{m,k}e^{j\theta_{m,k}}+h_{m,k}\,,$ (1)
$\displaystyle\ddot{g}_{k,j}$
$\displaystyle=\ddot{\bar{h}}_{m,k}e^{j\theta_{k,j}}+{\ddot{h}}_{k,j}\,,$ (2)
$\displaystyle\dddot{g}_{m,q}$
$\displaystyle=\dddot{\bar{h}}_{m,q}e^{j\theta_{m,q}}+{\dddot{h}}_{m,q}\,,$
(3)

where $\bar{h}_{m,k}e^{j\theta_{m,k}}$,
$\ddot{\bar{h}}_{k,j}e^{j\theta_{k,j}}$ and
$\dddot{\bar{h}}_{m,q}e^{j\theta_{m,q}}$ denote the LOS components and
$h_{m,k}$, ${\ddot{h}}_{k,j}$ and ${\dddot{h}}_{m,q}$ correspond to the non-
line of sight (NLOS) small-scale fading with
$h_{m,k}\sim\mathcal{C}\mathcal{N}({0},\beta_{m,k})$,
${\ddot{h}}_{k,j}\sim\mathcal{C}\mathcal{N}(0,\ddot{\beta}_{k,j})$ and
${\dddot{h}}_{m,q}\sim\mathcal{C}\mathcal{N}(0,\dddot{\beta}_{m,q})$, where
$\beta_{m,k}$, $\ddot{\beta}_{k,j}$ and $\dddot{\beta}_{m,q}$ are the large-
scale fading coefficients which account for path-loss and shadowing. We assume
that only
$\left\\{\bar{h}_{m,k},\ddot{\bar{h}}_{k,j},\dddot{\bar{h}}_{m,q},\beta_{m,k},\ddot{\beta}_{k,j},\dddot{\beta}_{m,q}\right\\}$
are known at the CPU, such as in [9, 1]. Indeed, we consider a more realistic
scenario where the phase shifts $\theta$ in the LOS components are unknown,
and assume that they are uniformly distributed in the interval
$\left[0,2\pi\right)$ [9, 1]. Finally, we have that

$\displaystyle w_{m,k}$
$\displaystyle\triangleq\mathbb{E}\left[g_{m,k}{g}^{*}_{m,k}\right]=\bar{h}_{m,k}{\bar{h}}^{*}_{m,k}+\beta_{m,k}\,,$
(4) $\displaystyle\ddot{w}_{k,j}$
$\displaystyle\triangleq\mathbb{E}\left[\ddot{g}_{k,j}{\ddot{g}}^{*}_{k,j}\right]=\ddot{\bar{h}}_{k,j}{\ddot{\bar{h}}}^{*}_{k,j}+\ddot{\beta}_{k,j}\,,$
(5) $\displaystyle\dddot{w}_{m,q}$
$\displaystyle\triangleq\mathbb{E}\left[\dddot{g}_{m,q}{\dddot{g}}^{*}_{m,q}\right]=\dddot{\bar{h}}_{m,q}{\dddot{\bar{h}}}^{*}_{m,q}+\dddot{\beta}_{m,q}\,.$
(6)

###  II-A Channel Estimation in the First Phase

We assume that $\tau_{\text{p}}$ mutually orthogonal pilot sequences
$\sqrt{\tau_{\text{p}}}\bm{\varphi}\in\mathbb{C}^{\tau_{\text{p}}\times 1}$
are used for channel estimation with $\lVert\bm{\varphi}\rVert^{2}~{}=~{}1$.
Thus, let $\sqrt{\tau_{\text{p}}\rho_{\text{p}}}\bm{\varphi}_{k}$ be the pilot
sequence assigned to user $k$, for $k=1,\ \ldots,\ K$, where $\rho_{\text{p}}$
is the transmit power of the pilot symbol, and $\mathcal{P}_{k}$ the subset of
users which are assigned the same pilot sequence as user $k$, including
itself. Then, the received pilot signal vector
$\mathbf{y}_{m}^{\text{p}}\in\mathbb{C}^{1\times\tau_{\text{p}}}$ at the AP
$m$, in the first phase, is given by

$\displaystyle\mathbf{y}_{m}^{\text{p}}$
$\displaystyle=\sum_{k=1}^{K}\sqrt{\tau_{\text{p}}\rho_{\text{p}}}g_{m,k}{\bm{\varphi}}^{\mathrm{H}}_{k}+\mathbf{n}_{m}^{\text{p}},$
(7)

where $\mathbf{n}_{m}^{\text{p}}\in\mathbb{C}^{1\times\tau_{\text{p}}}$ is the
receiver noise with independent $\mathcal{C}\mathcal{N}(0,\sigma^{2})$
entries, in which $\sigma^{2}$ is the noise power.

After obtaining the projection of $\mathbf{y}_{m}^{\text{p}}$ onto
$\bm{\varphi}_{k}$, given by
$\check{y}_{m,k}^{\text{p}}=\mathbf{y}_{m}^{\text{p}}\bm{\varphi}_{k}$, the
phase-unaware linear minimum mean squared error (LMMSE) estimate of
$\\{g_{m,k}\\}_{\forall(m,k)}$ is [9]

$\displaystyle\hat{g}_{m,k}$
$\displaystyle=\sqrt{\tau_{\text{p}}\rho_{\text{p}}}w_{m,k}\psi_{m,k}^{-1}\check{y}_{m,k}^{\text{p}},$
(8)

where
$\psi_{m,k}=\sum\limits_{j\in\mathcal{P}_{k}}\tau_{\text{p}}\rho_{\text{p}}w_{m,j}+\sigma^{2}.$
Note that the estimated channel $\hat{g}_{m,k}$ and the channel estimation
error $\tilde{g}_{m.k}=g_{m,k}-\hat{g}_{m,k}$ are independent variables
distributed as $\hat{g}_{m,k}\sim\mathcal{C}\mathcal{N}(0,\gamma_{m,k})$ and
$\tilde{g}_{m,k}\sim\mathcal{C}\mathcal{N}(0,c_{m,k})$, where
$\gamma_{m,k}=\tau_{\text{p}}\rho_{\text{p}}w_{m,k}^{2}\psi_{m,k}^{-1}$ and
$c_{m,k}=w_{m,k}-\gamma_{m,k}$.

###  II-B Wireless Power Transfer in the Second Phase

We consider a non-coherent energy harvesting, thus, the signal transmitted by
AP $m$ to user $k$ is given by

$\displaystyle
s_{m}^{\text{(d)}}=\sum_{k=1}^{K}\sqrt{p_{m,k}^{\text{(d)}}}\hat{g}_{m,k}s_{m,k},$
(9)

where $s_{q,k}\in\mathbb{C}^{\tau_{\text{p}}\times 1}$ is an energy signal
transmitted from the $q$-th AP to the $k$-th user, which are i.i.d with
$|s_{q,k}|^{2}=1$, and $p_{q,k}^{(\text{d})}$ is the power used by the $q$-th
AP to transmit to the $k$-th user. The transmission power for each AP should
satisfy the maximum power limit which is $P_{m}^{\max}$ in the long-term,
i.e.,

$\displaystyle\mathbb{E}\left[\left|s_{m}^{\text{(d)}}\right|^{2}\right]=\sum_{k=1}^{K}p_{m,k}^{\text{(d)}}\gamma_{m,k}\leq
P_{m}^{\max}.$ (10)

The received signal at user $k$ in the second phase is

$\displaystyle z^{\text{(d)}}_{k}$
$\displaystyle=\sum_{m=1}^{M}\sum_{j=1}^{K}\sqrt{p_{m,j}^{\text{(d)}}}{g}^{*}_{m,k}\hat{g}_{m,j}s_{q,j}+\sum_{j=1}^{K}\sqrt{\eta_{j}}\ddot{g}_{k,j}x_{j}+n_{k}^{\text{E}},$
(11)

where $n_{k}^{\text{E}}$ is the noise at the user $k$, which is neglected
because the noise floor is too low for energy harvesting [1]. The second term
is related to the signals coming from other users in the system. This occurs
because, differently from [1], we also transmit data while harvesting energy
at the user side. Observe also that this term includes the user $k$ itself,
which is the term related to the self-energy recycling that increases the
harvested energy. The $\eta_{j}$ and $x_{j}$ terms are related to the uplink
transmission and are defined in the next subsection.

In this work, we consider a linear energy harvesting model. While such a model
might be optimistic for practical applications, it allows us to obtain initial
insights about cell-free systems with self-energy recycling. In fact, as shown
[1], it is expected that the performance of cell-free systems when considering
a non-linear energy harvesting model is lower if compared to a linear model.
However, the study of cell-free systems with self-energy recycling and a non-
linear energy harvesting model is left for future works. Therefore, assuming
that $\mu\in[0,1]$ is the energy harvesting efficiency of the rectifier
circuit [1], the average harvested energy at user $k$ during the second phase
is

$\displaystyle E_{k}^{\text{(d)}}$
$\displaystyle=\tau_{\text{d}}\mu\mathbb{E}\left[\left|\sum_{m=1}^{M}\sum_{j=1}^{K}\sqrt{p_{m,j}^{\text{(d)}}}{g}^{*}_{m,k}\hat{g}_{m,j}s_{m,j}\right.\right.$
$\displaystyle\hskip
86.11084pt\left.\left.+\sum_{j=1}^{K}\sqrt{\eta_{j}}\ddot{g}_{k,j}x_{j}\right|^{2}\right]\,,$
(12)

where the second term is related to the signals coming from other users in the
system.

######  Proposition 1.

Assuming the phase-unaware LMMSE channel estimator in (8), the average energy
harvested for non-coherent downlink transmission is given in (13) at the top
of the next page.

###### Proof.

The proof follows similar steps as in [1], and the detailed proof is omitted
here due to the limitation of space. ∎

$\displaystyle E_{k}^{\text{(d)}}$
$\displaystyle=\mu\tau_{\text{d}}\left(\sum_{m=1}^{M}\sum_{j=1}^{K}{p_{m,j}^{\text{(d)}}}\gamma_{m,j}w_{m,k}+\sum_{j=1}^{K}\eta_{j}\ddot{w}_{k,j}\right.$
$\displaystyle\hskip
64.58313pt\left.+\left(\tau_{\text{p}}\rho_{\text{p}}\right)^{2}\sum_{m=1}^{M}\sum_{j\in\mathcal{P}_{k}}p_{m,j}^{\text{(d)}}\left(2\beta_{m,k}\operatorname{Re}\left\\{{\bar{h}}^{*}_{m,k}\psi_{m,j}^{-1}w_{m,j}\bar{h}_{m,k}w_{m,j}\psi_{m,j}^{-1}\right\\}+\beta_{m,k}^{2}\left|\psi_{m,j}^{-1}w_{m,j}\right|^{2}\right)\right)$
(13)

###  II-C Uplink Data Transmission in the Second Phase

In this phase, we also assume that all $K$ users simultaneously send their
data and receive energy from the APs. Thus, the received signal at the $m$-th
AP $y_{m}^{(\text{u})}\in\mathbb{C}$ is modeled as

$y_{m}^{(\text{u})}=\sum_{k=1}^{K}\sqrt{\eta_{k}}g_{m,k}x_{k}+\sum_{q=1}^{M}\sum_{k=1}^{K}\sqrt{p_{q,k}^{(\text{u})}\sigma_{m}^{\text{RSI}}}\dddot{g}_{m,q}\hat{g}_{q,k}s_{q,k}+n_{m}^{(\text{u})}\,,$
(14)

where $x_{k}\in\mathbb{C}$, with $\mathbb{E}\\{|x_{k}|^{2}\\}=1$, is the
transmitted data symbol by user $k$,
$\sigma_{m}^{\text{RSI}}\in\left[0,1\right)$ is the residual SI and inter-AP
interference111Similar to [10], we assume that the residual SI inter-AP
interference can be modeled in the same manner as for the SI. Indeed, the IAI
can be seen as the virtual residual SI of a virtual “large” base station where
all transmit and receive antennas are collocated due to the centralized
processing in the CPU. Moreover, there are practical mechanisms that allow us
to have residual SI in the order of -110 dB[11, 12, 13, 14], which practically
enables FD implementations. suppression level after all real-time
cancellations in analog-digital domains [15, 16],
$n_{m}^{(\text{u})}\sim\mathcal{C}\mathcal{N}(0,\sigma^{2})$ is the noise on
the received data signal, $\eta_{k}$ is the transmit power of the data symbol
and $p_{q,k}^{\text{(u)}}$ is the power allocated by AP $q$ to transmit to
user $k$ in the second phase. The second term is related to the residual SI at
AP $m$.

Also, each AP is able to perform local data detection that are passed to the
CPU for final decoding. By employing the phase-unaware LMMSE channel estimator
in (8) for maximum ratio combining (MRC) coherent detection, the local
estimate of $x_{k}$ at AP $m$ is given by:

$\displaystyle\tilde{x}_{m,k}$
$\displaystyle=\sum_{j=1}^{K}\sqrt{\eta_{j}}{\hat{g}}^{*}_{m,k}g_{m,j}x_{j}$
$\displaystyle\hskip
8.61108pt+\sum_{q=1}^{M}\sum_{j=1}^{K}\sqrt{p_{q,j}^{(\text{p})}\sigma_{m}^{\text{RSI}}}{\hat{g}}^{*}_{m,k}{\dddot{g}}_{m,q}\hat{g}_{q,j}s_{q,j}+n_{m}^{(\text{u})}.$
(15)

Once the local estimates are sent to the CPU, they are multiplied by a receive
filter coefficient $\alpha_{m,k}$, i.e.,

$\hat{x}_{k}=\sum_{m\in\mathcal{M}_{k}}\alpha_{m,k}\tilde{x}_{m,k}.$ (16)

Note that the receive filter coefficients can be optimized to maximize the SE
using only channel statistics since the CPU does not know the channel
estimates.

In addition, let us define the following vectors and matrices for ease of
notation:

$\displaystyle\bm{\alpha}_{k}\triangleq{\left[\alpha_{1,k},\ldots,\alpha_{M,k}\right]}^{\mathrm{T}}\in\mathbb{C}^{M\times
1},$
$\displaystyle\mathbf{b}_{k}\triangleq{\left[b_{1,k},\ldots,b_{M,k}\right]}^{\mathrm{T}}\in\mathbb{C}^{M\times
1},\quad b_{m,k}\triangleq\mathbb{E}\left[{\hat{g}}^{*}_{m,k}g_{m,k}\right],$
$\displaystyle\mathbf{C}_{k,j}\in\mathbb{C}^{M\times M},\quad
c_{k,j}^{m,m^{\prime}}\triangleq\mathbb{E}\left[{\hat{g}}^{*}_{m,k}g_{m,j}{g}^{*}_{m^{\prime},j}\hat{g}_{m^{\prime},k}\right],$
$\displaystyle\mathbf{D}_{k}\in\mathbb{C}^{M\times M},\quad
d_{m,k}\triangleq\mathbb{E}\left[{\hat{g}}^{*}_{m,k}n_{m}^{\text{(u)}}{\left(n_{m}^{\text{(u)}}\right)}^{*}\hat{g}_{m,k}\right],$
$\displaystyle\mathbf{F}_{k,q,j}\in\mathbb{C}^{M\times M},$ $\displaystyle
f_{k,j}^{m,m^{\prime}}\triangleq\mathbb{E}\left[{\hat{g}}^{*}_{m,k}\dddot{g}_{m,q}\hat{g}_{q,j}\sqrt{\sigma_{m}^{\text{RSI}}}\sqrt{\sigma_{m^{\prime}}^{\text{RSI}}}{\hat{g}}^{*}_{q,j}{\dddot{g}}^{*}_{m^{\prime},q}\hat{g}_{m^{\prime},k}\right],$

where $c_{k,j}^{m,m^{\prime}}$ and $f_{k,j}^{m,m^{\prime}}$ are the elements
of the line $m$ and column $m^{\prime}$ of the matrices $\mathbf{C}_{k,j}$ and
$\mathbf{F}_{k,q,j}$, respectively, and $\mathbf{D}_{k}$ is a diagonal matrix
with the $m$-th diagonal element being $d_{m,k}$. Thus, the SE can be computed
using proposition 2.

######  Proposition 2.

Assuming the phase-unaware LMMSE channel estimator in (8), the closed-form
expression for the achievable SE of user $k$ using MRC at the APs when
simultaneous data transmission and WPT is performed in CF systems is

$\displaystyle R_{k}=\tau_{\text{u}}/\tau_{\text{c}}\log_{2}(1+\Gamma_{k}),$
(17)

where $\Gamma_{k}$ is the effective SINR shown in (18) at the top of the next
page. The term $f_{k,q,j}^{m,m^{\prime}}$ is not present in [1] and arises in
our analyses due to the residual SI at the APs.

###### Proof.

Based on [9, 17, 18] where all additive interference is treated as a worst-
case Gaussian noise we can arrive at the result. The detailed proof is omitted
due to lack of space. ∎

$\displaystyle\Gamma_{k}=\frac{\eta_{k}\left|{\bm{\alpha}}^{\mathrm{H}}_{k}\mathbf{b}_{k}\right|^{2}}{{\bm{\alpha}}^{\mathrm{H}}_{k}\left(\sum\limits_{j=1}^{K}\eta_{j}\mathbf{C}_{k,j}\right)\bm{\alpha}_{k}-\eta_{k}\left|{\bm{\alpha}}^{\mathrm{H}}_{k}\mathbf{b}_{k}\right|^{2}+{\bm{\alpha}}^{\mathrm{H}}_{k}\left(\sum\limits_{q=1}^{M}\sum\limits_{j=1}^{K}p_{q,j}^{\text{d}}\mathbf{F}_{k,q,j}\right)\bm{\alpha}_{k}+{\bm{\alpha}}^{\mathrm{H}}_{k}\mathbf{D}_{k}\bm{\alpha}},$
(18) where $\displaystyle
b_{k}^{m}=\tau_{\text{d}}\rho_{\text{p}}{\bar{h}}^{*}_{m,k}\psi_{m,k}^{-1}w_{m,k}\bar{h}_{m,k}+\tau_{\text{d}}\rho_{\text{p}}\beta_{m,k}\psi_{m,k}^{-1}w_{m,k},\qquad
d_{k}^{m,m^{\prime}}=\left\\{\begin{matrix}\sigma^{2}\gamma_{m,k},&m=m^{\prime}\\\
0,&m\neq m^{\prime}\end{matrix}\right.$ $\displaystyle
c_{k,j}^{m,m^{\prime}}=\left\\{\begin{matrix}\gamma_{m,k}w_{m,j}+\tau_{\text{d}}^{2}\rho_{\text{p}}^{2}\left(2\beta_{m,j}\operatorname{Re}\left\\{{\bar{h}}^{*}_{m,j}\psi_{m,k}^{-1}w_{m,k}\bar{h}_{m,j}w_{m,k}\psi_{m,k}^{-1}\right\\}+\beta_{m,j}^{2}\left|w_{m,k}\psi_{m,k}^{-1}\right|^{2}\right)\left|{\bm{\phi}}^{\mathrm{H}}_{j}\bm{\phi}_{k}\right|^{2},&m=m^{\prime}\\\
\tau_{\text{d}}^{2}\rho_{\text{p}}^{2}\left[\left({\bar{h}}^{*}_{m,j}\psi_{m,k}^{-1}w_{m,k}\bar{h}_{m,j}+\beta_{m,j}\psi_{m,k}^{-1}w_{m,k}\right)\left({\bar{h}}^{*}_{m^{\prime},j}\psi_{m^{\prime},k}^{-1}w_{m^{\prime},k}\bar{h}_{m^{\prime},j}+\beta_{m^{\prime},j}\psi_{m^{\prime},k}^{-1}w_{m^{\prime},k}\right)\right]\left|{\bm{\phi}}^{\mathrm{H}}_{j}\bm{\phi}_{k}\right|^{2},&m\neq
m^{\prime}\end{matrix}\right.$ $\displaystyle
f_{k,q,j}^{m,m^{\prime}}=\left\\{\begin{matrix}\dddot{w}_{m,q}\sigma_{m}^{\text{RSI}}\left[\tau_{\text{d}}^{2}\rho_{\text{p}}^{2}\left(2\beta_{q,j}\operatorname{Re}\left\\{{\bar{h}_{q,j}}^{*}w_{m,k}\psi_{m,k}^{-1}\bar{h}_{q,j}\psi_{m,k}^{-1}w_{m,k}\right\\}+\beta_{q,j}^{2}\left|w_{m,k}\psi_{m,k}^{-1}\right|^{2}\right)+\gamma_{m,k}c_{q,j}\right]\left|{\bm{\phi}}^{\mathrm{H}}_{j}\bm{\phi}_{k}\right|^{2}\\\
+\dddot{w}_{m,q}\sigma_{m}^{\text{RSI}}\gamma_{m,k}\gamma_{q,j},&q=m\mbox{ and
}m=m^{\prime}\\\
\dddot{w}_{m,q}\gamma_{m,k}\sigma_{m}^{\text{RSI}}\gamma_{q,j},&q\neq m\mbox{
and }m=m^{\prime}\\\ 0,&m\neq m^{\prime}\end{matrix}\right.$

##  III Problem Formulation

In this paper we consider the problem of minimizing the users’ battery energy
usage while satisfying minimum data rate requirements. We assume that the
users are powered over the air by the APs and have a battery. The battery
energy is required to compensate for the randomness and intermittence of the
harvested energy, increasing the feasibility of the problem (i.e., serving all
users with their required minimum rate). Then, considering
$\eta_{k}=\eta_{k}^{\text{(e)}}+\eta_{k}^{\text{(b)}}$, where
$\eta_{k}^{\text{(e)}}$ and $\eta_{k}^{\text{(b)}}$ denote the power drawn
from the harvesting and the battery, respectively, the problem can be
formulated as

$\displaystyle\underset{p_{m,k}^{\text{(d)}},\eta_{k}^{\text{(e)}},\eta_{k}^{\text{(b)}},\bm{\alpha_{k}}}{\text{minimize}}$
$\displaystyle\sum_{k}^{K}\tau_{\text{u}}\eta_{k}^{\text{(b)}}$ (19a) subject
to $\displaystyle R_{k}\geq R_{k}^{\text{th}},$ $\displaystyle\forall k,$
(19b) $\displaystyle\tau_{\text{u}}\eta_{k}^{\text{(e)}}\leq
E_{k}^{\text{(d)}},\;\tau_{\text{u}}\eta_{k}^{\text{(b)}}\leq E_{k}^{\max},$
$\displaystyle\forall k,$ (19c)
$\displaystyle\sum_{k=1}^{K}p_{m,k}^{\text{(d)}}\gamma_{m,k}\leq
P_{m}^{\max},$ $\displaystyle\forall m,$ (19d)

where $E_{k}^{\max}$ is the maximum available energy at the battery of user
$k$ and $R_{k}^{\text{th}}$ is the minimum data rate demand of user $k$.

##  IV Proposed Solution

Note that problem (19) is non-convex, which makes the global optimal solution
computationally difficult to be obtained. Indeed, the non-convexity arises due
to constraints (19b) and the highly coupled variables. To deal with this
issue, note that the data rate constraints (19b) can be replaced by SINR
constraints as follows

$\Gamma_{k}\geq\Gamma_{k}^{\text{th}},\quad\forall k,$ (20)

where
$\Gamma_{k}^{\text{th}}=2^{\frac{\tau_{\text{c}}R_{k}^{\text{th}}}{\tau_{\text{u}}}}-1$.
Moreover, to deal with the highly coupled variables, we can resort to the
alternating optimization method, such as in [1]. Note that
$E_{k}^{\text{(d)}}$ as well as the numerator and denominator of SINR
expressions are linear in $\\{p_{m,k}^{\text{(d)}}\\}_{\forall m,k}$ and
$\\{\eta_{k}\\}_{\forall k}$, given the receiver filter coefficient vector
$\\{\bm{\alpha}_{k}\\}_{\forall k}$. Hence, for some given $\bm{\alpha}_{k}$,
the optimization problem can be shown to be quasilinear and its global optimum
solution can be found using standard solvers such as CVX [19]. In addition,
the receiver filter coefficients $\bm{\alpha}_{k}$ only affect the SINR of the
user $k$ and can be found in closed form for the given uplink data power
$\\{\eta_{k}\\}_{\forall k}$ by maximizing a generalized Rayleigh quotient
[1]. The complete alternating optimization algorithm can be seen in Algorithm
1.

Algorithm 1 Proposed solution.

1: Initialize $\\{\bm{\alpha}_{k}\\}_{\forall k}$ as all ones vector and set
$l=0$;

2: repeat

3: Replace constraints (19b) by (20);

4: Solve problem (19) by taking $\\{\bm{\alpha}_{k}\\}_{\forall k}$ as
constants;

5: Obtain the optimal $\\{\bm{\alpha}_{k}\\}_{\forall k}$ by maximizing a
generalized Rayleigh quotient;

6: Set $l\leftarrow l+1$;

7: until Convergence has been reached or $l>L_{\max}$.

######  Proposition 3.

The objective function of the original problem (19) converges by performing
algorithm 1.

###### Proof.

Note that the power values remain constant at each receiver filter
coefficients update in line 5. Moreover, for fixed power values, the receiver
filter coefficients optimization achieves an uplink data rate greater than or
equal to that of the previous iteration. Then, less (or equal) transmission
power will be needed at the next iteration. Hence, the objective function is
monotonically decreasing with each iteration. Since the objective is bounded
by the power and rate constraints, we can claim the convergence of (19a) with
algorithm 1. ∎

Proposition 3 shows the convergence of the objective function to a limit point
of a monotonically decreasing sequence. Since the original problem is non-
convex, global optimality of the provided solution cannot be guaranteed. Thus,
the proof of optimality of the proposed method is an open research topic and
left for future works. Nevertheless, the simulation results in Section V
demonstrate that Algorithm 1 provides significant performance gains compared
to the state-of-the-art solutions.

Furthermore, note that this method consists in a centralized approach in which
the CPU is responsible for all computations. Then, the CPU informs the values
of $\\{\eta_{k}^{(\text{(b)})},\eta_{k}^{(\text{(e)})}\\}_{\forall k}$ to the
users. Finally, the per-iteration computational complexity of Algorithm 1 is
dictated by solving the linear problem in line 4, which has a complexity
equivalent to $\mathcal{O}\left((MK+2K)^{3.5}\right)$.

##  V Numerical Results

We consider the uplink of a CF system with parameters
$\\{M,K,\tau_{\text{p}},\tau_{\text{c}}\\}=\\{64,4,2,200\\}$. We adopt the 3rd
generation partnership project (3GPP) indoor hotspot (InH) model in [1] with a
3.4 GHz carrier frequency and 20 MHz of bandwidth. The large-scale fading
coefficients, shadowing parameters, probability of LOS, and the Rician factors
are simulated based on [1]. The APs and users are uniformly distributed within
a square of 100 m $\times$ 100 m and a 4 m height difference between APs and
users is taken into account when calculating distances. A wrap-around
technique is applied to imitate a network with an infinite area. Moreover,
each user randomly selects a pilot from a predefined set of orthogonal pilots.
We set $E_{k}^{\max}=0.2$ J and $P_{m}^{\max}$ = 1 W. The pilot power is fixed
and set to 0.1 W. The noise power is given by $\sigma^{2}=-96$ dBm. The energy
harvesting efficiency of the rectifier circuit, $\mu$, is 0.5. Finally, for
comparison, we consider the solution proposed in [1] adapted to our problem in
which the TS protocol is adopted, i.e., it dedicates $\tau_{\text{d}}$ samples
of the coherent interval for energy harvesting (thus,
$\tau_{\text{c}}=\tau_{\text{p}}+\tau_{\text{d}}+\tau_{\text{u}}$). We denote
this solution as TS in the plots. Also, numerical results are obtained by 500
random realizations of APs and users locations.

In Fig. 1 we present the convergence of proposed algorithm with different
values of $\mu$. We can see that the proposed solutions converges to the final
solution with a low number of iterations. The key insight taken from Fig. 1 is
that the objective function improves as the energy harvesting efficiency of
the rectifier circuit increases since more energy is being harvested, which
decreases the energy usage from users’ batteries. Also, we observe that the
number of iterations needed for convergence increases as $\mu$ increases.

Figure 1: Convergence of proposed algorithm with $\sigma_{m}^{\text{RSI}}=-90$
dB and $R_{k}^{\text{th}}=2.5$ bit/s/Hz.

Since the proposed solution allows users to transmit data while simultaneously
receiving energy from the APs, it is affected by the residual SI at the APs.
Thus, Fig. 2 presents the fraction of energy from the battery versus the
residual SI. The fraction of energy from the battery is given by
$\tau_{\text{u}}\eta_{k}^{\text{(b)}}/(\tau_{\text{u}}\eta_{k}^{\text{(b)}}+\tau_{\text{u}}\eta_{k}^{\text{(e)}})$.
As we can see, the TS solution presents the same performance for all values of
$\sigma_{m}^{\text{RSI}}$ since it is not affected by the SI. The proposed
solution, in its turn, increases the fraction of energy used from the battery
as the residual SI increases. The reason behind this is that to avoid SI, the
APs allocate less power to the users, which affects the energy harvesting.
Thus, to fulfill the minimum data rates more energy is obtained from the
battery. Even so, it is worth highlighting that the proposed solution is able
to outperform the TS solution when the residual SI is less than -95 dB, which
is a practical assumption considering existing SI canceling mechanisms [11,
12, 13, 14], which allow us to have residual SI in the order of -110 dB.

Figure 2: Fraction of energy from the battery versus residual SI
($\sigma_{m}^{\text{RSI}}$) with $R_{k}^{\text{th}}=2.5$ bit/s/Hz.

Another aspect that should be highlighted is that while SI energy-harvesting
increases the harvested energy, the residual SI affects the data decoding. To
analyze this effect, Fig. 3 presents per-user SE versus the residual SI at the
APs. As it can be seen, the SE increases as the residual SI decreases. That is
expected because low residual SIs yield higher data decoding rate at the APs,
leading to higher SE. Moreover, the proposed method performs much better than
the solution proposed in [1] in terms of SE for values of residual SI lower
than -90 dB. This occurs because the baseline method harvests less energy, and
- to minimize the battery consumption - it satisfies, but does not exceed, the
data rate requirement. On the other hand, for lower and practical values of
residual SI, the proposed method is able to harvest sufficient energy for data
transmission without affecting the battery consumption. Since all harvested
energy is used for data transmission, the user achieves data rates higher than
the data requirements.

Figure 3: SE per user versus the residual SI at the APs
($\sigma_{m}^{\text{RSI}}$) with $R_{k}^{\text{th}}$ = 2.0 bits/s/Hz .

Last but not least, in Fig. 4 we present the outage rate versus the minimum
data rate requirements. We say that an outage event occurs when an algorithm
is not able to find a feasible solution that fulfills all constrains of
problem (19) for a given realization of APs and users’ locations. As expected,
the outage rate increases as the minimum data rate requirements increases for
all solutions. Indeed, the higher the minimum data rate requirements, the
harder it is to solve the problem. The TS solution degrades its performance as
the number of dedicated samples for energy harvesting increases. This occurs
because less samples are dedicated to data transmission, which affects the SE
of the users. Thus, the TS solution performs a trade-off between minimizing
the battery usage and fulfilling the rate demands. Meanwhile, we highlight
that the proposed solution achieves the best performance in terms of outage
with no trade-off required.

Figure 4: Outage rate versus minimum data rate requirements with
$\sigma_{m}^{\text{RSI}}=-100$ dB and $R_{k}^{\text{th}}=2.5$ bit/s/Hz.

##  VI Conclusions

In this paper, we studied wireless-powered CF systems in which users transmit
data signals to the APs while simultaneously harvesting energy from the
network nodes, including itself, by performing self-energy recycling. Due to
this new feature of self-energy recycling, new terms arise in the derived
equations, which have not been previously presented in the literature. The
problem of minimizing the users’ battery energy usage with minimum data rate
constraints was formulated, for which an alternating optimization algorithm
was proposed along with its proof of convergence. Simulation results showed
that the proposed method outperforms a state-of-art algorithm in terms of
battery energy usage and outage rate. Future works could consider coherent
energy transmission and non-linear energy harvesting.

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  *[TV]: television
  *[3G]: third generation
  *[IEEE]: institute of electrical and electronics engineers
  *[UMTS]: universal mobile telecommunications system
  *[SINR]: signal-to-interference-plus-noise ratio
  *[WCR]: weighted common rate
  *[B5G]: Beyond fifth generation
  *[6G]: sixth generation
  *[WPT]: wireless power transfer
  *[CF]: cell-free
  *[SWIPT]: simultaneous wireless information and power transfer
  *[TS]: time switching
  *[SE]: spectral efficiency
  *[PS]: power splitting
  *[FD]: full-duplex
  *[SI]: self-interference
  *[AP]: access point
  *[LOS]: line of sight
  *[APs]: access point
  *[CPU]: central processing unit
  *[IUI]: inter-user interference
  *[IAI]: inter-AP interference
  *[i.i.d]: independent and identically distributed
  *[NLOS]: non-line of sight
  *[LMMSE]: linear minimum mean squared error
  *[MRC]: maximum ratio combining
  *[3GPP]: 3rd generation partnership project
  *[InH]: indoor hotspot
  *[SIs]: self-interference


# PSFormer: Point Transformer for 3D Salient Object Detection

###### Abstract

We propose PSFormer, an effective point transformer model for 3D salient
object detection. PSFormer is an encoder-decoder network that takes full
advantage of transformers to model the contextual information in both multi-
scale point- and scene-wise manners. In the encoder, we develop a Point
Context Transformer (PCT) module to capture region contextual features at the
point level; PCT contains two different transformers to excavate the
relationship among points. In the decoder, we develop a Scene Context
Transformer (SCT) module to learn context representations at the scene level;
SCT contains both Upsampling-and-Transformer blocks and Multi-context
Aggregation units to integrate the global semantic and multi-level features
from the encoder into the global scene context. Experiments show clear
improvements of PSFormer over its competitors and validate that PSFormer is
more robust to challenging cases such as small objects, multiple objects, and
objects with complex structures.

Index Terms—  PSFormer, 3D salient object detection, Transformer, Point cloud

##  1 Introduction

While salient object detection (SOD) from 2D images has been extensively
studied [1, 2], there are very few efforts on SOD from 3D point clouds. This
is despite the fact that the rapid development of 3D acquisition technologies
has significantly simplified geometric modeling, and 3D point clouds become
more and more popular with wide applications of autonomous driving, and
Metaverse.

Unlike images in which salient objects are unchanged, point clouds can be
readily rotated. That means, one object may change from the salient to non-
salient object during rotation. PointSal [3] is the pioneering work of point
cloud salient object detection (PCSOD). PointSal takes full advantage of
multi-scale features and global semantics to locate salient objects. However,
all feature extraction modules of PointSal are implemented by multi-layer
perceptrons (MLPs), which seriously limits the capability of learning long-
range feature representations due to fixed receptive fields. When dealing with
an object with complex structures (see Figure 1 (A)), PointSal [3] fails to
capture the complete structure information.

Fig. 1: 3D heatmap visualization of feature maps (FM). Points that belong to
the same semantic part share more similar features by (B) PSFormer than (A)
PointSal [3].  Fig. 2: Pipeline of PSFormer. PSFormer is an encoder-decoder
structure, mainly consisting of PCT and SCT.

To deal with the challenges of small/multiple objects and objects with complex
structures, we propose a point transformer model for PCSOD, dubbed PSFormer.
PSFormer leverages multiple transformer modules to capture multi-scale point-
wise and scene-wise contextual information simultaneously. In detail, PSFormer
follows an encoder-decoder structure, including two types of transformer
modules. The first type is Point Context Transformer (PCT) and the second is
Scene Context Transformer (SCT). PCT is employed to model the local and global
point-wise relationships by two transformers; the local point-wise context can
describe the boundary structure and the global point-wise context assists in
distinguishing the seed belonging to different classes. SCT integrates the
multi-scale features from decoders to learn the scene-wise context
representation; the global scene context can locate the salient object from
the interfering background. Benefiting from PCT and SCT, PSFormer can
effectively model the long-range dependencies and learn the contextual
information. Therefore, it can be clearly observed in Figure 1 (B) that the
structural features of the 3D flower are completely distinguished from the
complex backgrounds via PSFormer.

To verify the effectiveness of PSFormer, we compare it with PointSal [3] and
five representative segmentation models on the PCSOD benchmark dataset and
achieve the best performance. Our contributions are summarised as follows:
(1) We propose a point transformer model for 3D salient object detection
(PSFormer). PSFormer aims to deal with the challenges of small objects,
multiple objects, and objects with complex structures.
(2) We formulate two types of transformers, i.e., Point Context Transformer
and Scene Context Transformer, capturing contextual information at the point
and scene levels.
(3) Experiments verify that PSFormer outperforms PointSal and five
segmentation models on the PCSOD dataset.

##  2 Methodology

###  2.1 Overview of PSFormer

We propose a novel transformer-based salient object detector for point clouds.
PSFormer models the context-dependent feature representations in both point
and scene levels. It leverages an encoder-decoder structure (see Figure 2).
The input point cloud is first fed into the encoder to progressively capture
high-level semantic features by continuous Point Context Transformer (PCT)
blocks. In PCT, two transformers ($\psi_{pre}$ and $\psi_{post}$) learn the
point-wise context information in the local and global region. The decoder
employs Scene Context Transformer (SCT) module to recover the resolution of
the input point cloud by the Upsampling-and-Transformer (UT) block and
integrate the scene contextual information via a Multi-Context Aggregation
(MCA) module.

###  2.2 Point Context Transformer

To construct the hierarchical feature representation for understanding the
semantic information of whole scenes, we follow PointNet++ [4] to build PCT in
a pyramid manner.

Given an input point cloud $\mathcal{P}$ = $\\{p_{1},p_{2},...,p_{N}\\}$, we
first generate a new subset $\\{p_{c_{1}},p_{c_{2}},...,p_{c_{M}}\\}$ by the
furthest point sampling (FPS) operation. In this new set, each point is
recognized as the centroid to choose $K$ closest points in the local region
within a given radius to form the point group. We denote
$\\{f_{i}|i\in{\mathcal{N}(p_{c_{t}})}\\}$ as the $t_{th}$ group with the
centroid $p_{c_{t}}$, where $f_{i}$ represents features of points in the
$t_{th}$ group. The transformer is formulated as:

$q_{i}=f_{i}W_{q},\quad{}k_{i}=f_{i}W_{k},\quad{}v_{i}=f_{i}W_{v},$ (1)
$y_{i}=Softmax(q_{i}\cdot{}k_{i}/\sqrt{d})v_{i},$ (2)
$Trans(f_{i})=o_{i}=f_{i}+FFN(y_{i}),$ (3)

where $W_{q}$, $W_{k}$ and $W_{v}$ are linear projections for query, key and
value terms, $d$ represents the feature dimension of key and value vectors,
$FFN(\cdot)$ is a feed-forward network.

These groups are first fed into the Feature Normalization (FN) module to
normalize the feature distributions in the local region. Then, the transformer
block $\psi_{pre}$ takes these groups as input to model the context
dependencies between points in the group, which describes the structural
features on the boundary. The max-pooling function aggregates the features of
neighbor points into the centroid to reduce the resolution and expand the
receptive field of the sampled seeds. Subsequently, the aggregated features
are fed into the transformer block $\psi_{post}$ to learn the correlations
among the sampled seeds, which assists in interfering the category of each
point. The whole operation is formulated as:

$y_{i}=\psi_{post}(\mathcal{A}(\psi_{pre}(f_{i,j}),|j\in{\mathcal{N}(i)})),$
(4) $\psi_{pre}(f_{i})=\psi_{post}(f_{i})=Trans(f_{i}).$ (5)

Fig. 3: Point Context Transformer (PCT).  Fig. 4: The qualitative of different
methods for PCSOD. Our PSFormer behaves better in handling challenging cases,
including small objects, multi-objects, and objects with complex structures.

###  2.3 Feature Normalization Module

The self-attention layer is the key component of Transformer, which has a
strong ability to learn the feature representations. But simply stacking self-
attention layers to learn the deeper features will decrease the accuracy and
robustness of performance. This is because points are sparse and irregular and
feature distributions among the different local groups are diverse, while
shared self-attention layers treat these groups equally. Inspired by [5], we
utilize a Feature Normalization (FN) module to assign different weights to
diverse groups. Let $\\{f_{i,j}\\}_{j=1,...,k}\in{\mathbb{R}^{k\times{b}}}$ be
the grouped set with the centroid ${f_{i}}\in{\mathbb{R}^{d}}$, we transform
the local grouped set by

$\sigma=\sqrt{\dfrac{1}{n\times{k\times{d}}}\sum\limits^{n}_{i=1}\sum\limits^{k}_{j=1}{(f_{i,j}-f_{i})^{2}}},$
(6)
$\\{f_{i,j}\\}=\alpha\odot\dfrac{\\{f_{i,j}\\}-f_{i}}{\sigma+\epsilon}+\beta,$
(7)

where the standard deviation $\sigma$ of all point cloud features are first
calculated to describe the offset across all groups and feature channels,
$\alpha$ and $\beta$ are learnable parameters to simulate the distribution in
different groups, $\odot$ is dot production and $\epsilon$ is a small number
to maintain the numerical stability [6, 7, 8].

Table 1: Comparison with the state-of-the-art methods on the PCSOD dataset.

Methods |  MAE $\downarrow$ |  F-measure $\uparrow$ |  E-measure $\uparrow$ |  IoU $\uparrow$
---|---|---|---|---
ASSANet [9] | 0.089 | 0.709 | 0.814 | 0.606
PointTransformer [10] | 0.075 | 0.762 | 0.848 | 0.670
PCT [11] | 0.069 | 0.770 | 0.846 | 0.652
PointMLP [5] | 0.065 | 0.792 | 0.875 | 0.702
PointNeXt [12] | 0.066 | 0.779 | 0.859 | 0.680
PointSal [3] | 0.069 | 0.769 | 0.851 | 0.656
Ours | 0.058 | 0.805 | 0.878 | 0.711

###  2.4 Scene Context Transformer

The high-level semantics and the multi-scale features are crucial for salient
detection tasks [13, 14, 15]. Thereby we integrate the global semantic and
multi-level features $\\{F_{l}\\}^{5}_{l=1}$ from encoders into the global
scene context via the Scene Context Transformer (SCT) module. Scene context
that describes the object distributions can locate the salient object. SCT
mainly includes two components, i.e., Upsampling-and-Transformer (UT) and
Multi-Context Aggregation (MCA).

UT upsamples the output $F^{i+1}$ from the previous UT block and concatenates
it with features $F^{i}$ of the corresponding PCT module using the short link.
Later, a transformer block excavates the inner relationships of the whole
features as

$F_{o}^{i}=Trans({C}({U}(F^{i+1}),F^{i})),$ (8)

where $U(\cdot)$ is an upsampling function and $C(\cdot,\cdot)$ is a
concatenation function. MCA directly takes outputs from all encoders as input.
We concatenate them together as the global scene context to assist in
predicting the salient object. Specifically, we first adopt a channel
compression operation for each output, which consists of the MLP layer and
Max-pooling. MLP compresses outputs to an identical feature dimension and Max-
pooling is employed to generate different vectors for succeeding concatenation
as

$F_{c}=C(Max-pooling(MLP(F^{i}))),i=1,...,5$ (9)

##  3 Experiments

###  3.1 Experimental Setup

Datasets. PCSOD [3] is a benchmark dataset for 3D salient object detection. It
includes 2,873 3D views from over one hundred scenes, where each view has
240,000 points. Following the widely used split ratio of 7:3, PCSOD is
randomly split into 2,000 training samples and 872 testing samples. Moreover,
PCSOD contains a certain amount of challenging samples, such as multiple
objects, small objects, complex structures and low illumination. We follow the
evaluation protocol proposed along with the dataset.

Implementation. We implement our model with Pytroch on an NVIDIA RTX 2080ti
GPU. The point clouds include 9-dimensional features that consist of spatial
coordinates, RGB colors and normalized coordinates. We randomly split the
complete 3D view into patches with 4,096 points, and treat these patches as
input. We train our model by the Adam optimizer in an end-to-end manner. The
total training epochs are 800 and the initial learning rate is 5e-4.

###  3.2 Comparison and Analysis

We compare our method with PointSal [3] and five segmentation methods [9, 11,
5, 12, 10].

Quantitative results. From Table 1, our method achieves the best performance
among all the methods on PCSOD. Our PSFormer outperforms the suboptimal model
PointMLP [5] in all metrics on the PCSOD testing set. Point Transformer [10]
and PCT [11] design various transformer modules on the point feature
extraction. Although these sophisticated local feature extractors already
learn the local context well, their performances are not effective enough due
to the lack of representation of the global scene context. PointMLP enhances
the ability to learn the point cloud feature representation by stacking more
residual feed-forward MLPs. But the fixed receptive field restricts the
representation power of MLPs.

Qualitative results. To further verify the effectiveness of PSFormer, we
visualize the predicted results on some challenging views, $e.g.$, structure-
complex objects (Figure 4 (b)), small objects (Figure 4 (c)) and multi-objects
(Figure 4 (d)). Our PSFormer can generate more accurate and complete
segmentation maps than the other methods.

Table 2: Ablation study on the PCSOD testing set.

Methods |  MAE $\downarrow$ |  F-measure $\uparrow$ |  E-measure $\uparrow$ |  IoU $\uparrow$
---|---|---|---|---
PSFormer (w/o FN) | 0.071 | 0.755 | 0.842 | 0.649
PSFormer (w/o $\psi_{pre}$) | 0.066 | 0.789 | 0.864 | 0.687
PSFormer (w/o $\psi_{post}$) | 0.063 | 0.799 | 0.873 | 0.700
PSFormer (w/o UT) | 0.069 | 0.775 | 0.858 | 0.678
PSFormer (w/o MCA) | 0.061 | 0.801 | 0.873 | 0.702
PSFormer (full) | 0.058 | 0.805 | 0.878 | 0.711

###  3.3 Ablation Study

We split out five main components from our model, and remove them one by one
to verify their effectiveness. They are FN, $\psi_{pre}$, $\psi_{post}$ self-
attention layers in PCT and UT, MCA modules in SCT. All results are reported
in Table 2. It can be observed that the contextual information captured by PCT
and SCT indeed improves the performance of our method.

##  4 Conclusion

3D salient object detection is a new topic, remaining many non-trivial
problems to solve. For the first time, we propose a transformer model for 3D
salient object detection from point clouds, namely PSFormer. Our PSFormer
enhances the ability to learn context-dependent feature representations at the
point and scene levels by introducing two different-type transformers. The
proposed Point Context Transformer (PCT) models hierarchical context-aware
features at the point level by two different transformer blocks. The proposed
Scene Context Transformer (SCT) captures the global scene context by
integrating the multi-scale contextual information from different-level PCT
modules. Thus, PSFormer is robust to the cases of small objects, multiple
objects, and objects with complex structures. Extensive experiments verify the
effectiveness of our method over its competitors.

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