# SWIPT-Enabled Multiple Access Channel: Effects of Decoding Cost and Non-
linear EH Model

Pouria Nezhadmohammad, Mohsen Abedi, Mohammad Javad Emadi, and Risto Wichman
This research work was partly supported by Academy of Finland under the grant
334000.P. Nezhadmohammad and M. J. Emadi are with the Department of Electrical
Engineering, Amirkabir University of Technology, Tehran 159163-4311, Iran
(E-mails:{pourianzh, mj.emadi}@aut.ac.ir).M. Abedi and R. Wichman are with the
Department of Electrical Engineering, Aalto University, Espoo 02150, Finland
(E-mails:{mohsen.abedi, risto.wichman}@aalto.fi).

###### Abstract

We studied power splitting-based simultaneous wireless information and power
transfer (PS-SWIPT) in multiple access channels (MAC), considering the
decoding cost and non-linear energy harvesting (EH) constraints at the
receiving nodes to study practical limitations of an EH communication system.
Under these restrictions, we formulated and analyzed the achievable rate and
maximum departure regions in two well-studied scenarios, i.e., a classical PS-
SWIPT MAC and a PS-SWIPT MAC with user cooperation. In the classical PS-SWIPT
MAC setting, closed-form expressions for the optimal values of the PS factors
are derived for two fundamental decoding schemes: simultaneous decoding and
successive interference cancellation. In the PS-SWIPT MAC with user
cooperation, the joint optimal power allocation for users as well as the
optimal PS factor are derived. This reveals that one decoding scheme
outperforms the other in the classical PS-SWIPT MAC, depending on the function
type of the decoding cost. Finally, it is shown that the cooperation between
users can potentially boost the performance of a PS-SWIPT MAC under decoding
cost and non-linear EH constraints. Moreover, effects of the decoding cost
functions, non-linear EH model and channel quality between the users are
studied, and performance characteristics of the system are discussed.

###### Index Terms:

Energy harvesting(EH), power splitting-based simultaneous wireless information
and power transfer (PS-SWIPT), multiple access channel (MAC), decoding cost,
maximum departure region, non-linear EH model.

##  I Introduction

ENERGY harvesting (EH) is a promising technique in the next generation of
wireless communication. The benefit it offers by energizing low-power devices
using environmental energy resources makes it an emerging alternative
technology in green wireless networking. Energy resources are classified into
two types; ambient and dedicated [1]. The use of ambient resources, e.g.,
wind, solar, heat, etc. in the conventional EH systems has introduced a vast
uncertainty regarding the harvested energy level, since most ambient resources
are usable only in specific environments under special conditions. Therefore,
to guarantee the quality of service (QoS) and EH stability, dedicated
resources such as radio frequency (RF) signals have emerged [2].

Recent studies in wireless networks demonstrate the ever-increasing demand for
high energy efficiency in modern communication systems, especially in fifth-
generation (5G) networks and beyond. Resultantly, a growing number of studies
are being undertaken in search of efficient ways to remotely charge wireless
devices. To this end, RF signals have exhibited a strong potential to not only
transmit information but also to simultaneously transfer energy, underpinning
the theory of simultaneous wireless information and power transfer (SWIPT)
[3]. Specifically, since high data rates exponentially increase the power
consumption of wireless devices and degrades the battery lifetime, SWIPT has
become an effective technology to tackle the contradiction between the high
data rate and energy conservation [4]. In this direction, the performance of
SWIPT is investigated in the new 5G frequencies [5] and multi-carrier non-
orthogonal multiple access (MC-NOMA) networks [6]. Furthermore, authors in [7]
characterize information-energy capacity region in SWIPT binary symmetric
channels. However, although the SWIPT systems offer unprecedented benefits to
modern communication networks, they suffer from fundamental performance
limitations as well. To address these limitations, authors in [8] analyzed the
channel output feedback in a SWIPT-based two-user Gaussian interference
channel. Unfortunately, the feedback at most doubles the energy transmission
rate at high signal-to-interference-plus-noise ratio (SINR) in a SWIPT-based
two-user Gaussian MAC [9].

To leverage the SWIPT systems in practice, four major techniques are proposed
in the literature: time switching (TS), power splitting (PS), antenna
switching (AS), and spatial switching (SS) [10]. By employing TS, the receiver
switches in time between information decoding (ID) and EH mode. In PS, the
receiver splits the received signal into two different streams with different
power levels, one for EH and the other for ID (see [10, 11]), whereas the
other two schemes are applied to multi-antenna configurations [11]. In
addition to these techniques, for the case of multi-user scenarios, authors in
[12, 9, 8] propose a signal design method performed at the transmitter sides
to not only transfer information to the receiver, but also to construct a
specific signal for energy harvester, that means a signal splitting is applied
at the transmitter. However, in the conventional PS-SWIPT concept, especially
the single user communications or the broadcast scenarios, a specific signal
design for energy transfer is not considered [10, 13, 14, 11], i.e., the
transmitted signal only conveys the information. Therefore, the destination
performs the PS scheme to decode the message while harvesting energy as well.

To compare PS with other practical techniques especially TS, several scenarios
are investigated in the literature. Authors in [15] and [16] studied both PS
and TS techniques for SWIPT-based multiple-input-single-output (MISO) and
massive multiple-input-multiple-output (MIMO) channels, respectively.
Furthermore, in a SWIPT-based decode-and-forward (DF) relay channel, [17]
investigates PS and AS techniques and [18] takes into account the temperature
constraints. Moreover, the problem of power allocation in SWIPT systems is
studied with a cooperative NOMA [19], PS NOMA [20], and a two-user MAC [21],
whereas [22] and [23] analyze power allocation in a PS-SWIPT device-to-device
underlaid cellular network carried out from practical and theoretical points
of view, respectively. Interestingly, although several studies indicate that
PS entails a higher complexity of implementation due to the PS factor
optimization, it outperforms TS owing to the logarithmic form of the rate-
energy function, especially in high signal-to-noise ratio (SNR) communications
and delay constrained cases as the both energy and information are partly
wasted in TS [10, 6]. As a result, we concentrated on PS and disregarded the
TS scheme in this paper, since our goal is to enlarge the achievable
rate/departure region.

Although a linear EH model has been conventionally considered in most
literature, recent investigations exposed a resource allocation mismatch in
the practical implementation approaches [24]. To be realistic, EH hardware
reveals a non-linear behavior due to physical limitations, such as storage
imperfections, non-linear circuits, processing costs, non-ideal energy
conversion efficiency, etc. [25, 26]. Accordingly, the realistic efficiency
models of EH circuits must be considered which is investigated in [27]. Here,
the practical SWIPT antennas with non-linear EH model are equipped with
electronic devices to convert the received RF signal power to a direct current
(DC) through impedance matching and passive filtering [28, 29]. To show that
PS still has superiority over TS, authors in [30] proved that in a SWIPT-based
MIMO broadcasting channel with a non-linear EH model, PS achieves a larger
average rate-region than TS. Similarly, authors in [31] experimentally showed
that at a high SNR level, energy-throughput regions of the PS receiver are
considerably larger than the TS receiver. Additionally, authors in [32] remark
that the PS receiver architecture in hybrid access point (H-AP) network
requires less power to meet the information and energy requirements at the
user’s end, and resultantly, more PS users can be served than the ones with
TS. However, owing to a higher input RF power in TS, the EH efficiency of TS
is larger than that of PS users. In practice, the EH operation is observed to
be non-linear due to the saturation effect[33]. However, this effect may be
rectified by placing several EH circuits in parallel, yielding a sufficiently
large linear conversion region in practice; thus, the EH-conversion function
can also be modeled with a linear function [34, 35]. Authors in [36] claim
that the RF-EH circuits can only harvest energy when the received power is
higher than a specific sensitivity level. As a result, determining an optimal
PS factor in PS-SWIPT systems is instrumental in order to maximize the
throughput and minimize the energy consumption.

On the other hand, the required power to decode the incoming data rate has
practically a significant impact on the performance of the low power EH
systems. Generally, _decoding cost_ at the destination depends on processing
algorithms, circuit designs, and architectures which tends to dominate the
other sinks of power (e.g. ADC, DAC, encoding, modulation/demodulation,
amplification, etc.) for a high data rate communication [37]. Authors in [38]
indicated that in VLSI devices, the decoding cost power is lower-bounded by
the number of message-passing iterations among processing elements $n$, amount
of energy consumed at each iteration $E_{node}$, the number of computational
nodes $m$, the channel dispersion, and the desired upper-bound on the decoding
error probability $P_{e}$. The impact of decoding cost on the performance of
the conventional EH systems is extensively studied in the literature [39, 40,
41, 42, 43, 44]. Decoding cost as a new constraint was recently investigated
in point-to-point, MAC, and broadcast channels with offline EH policies [39].
Furthermore, the decoding cost effect is considered in EH point-to-point
channel with an energy helper [40], EH MAC [39], EH two-way channels [42],
data cooperation between users [41], and SWIPT-based two-hop half-duplex DF
relay channel [43]. Authors in [38] proved that by applying VLSI and LDPC
codes, the decoding cost appears as an increasing quadratic function of the
decoding rate. However, unlike the EH conversion function, many different
codes with multiple decoding algorithms for each code yields a variety of
decoding cost function. In all recent publications [39, 40, 41, 42, 43], the
decoding cost is assumed as an increasing convex function of data rate to make
the optimization problems convex and tractable [44]. However, a general non-
decreasing decoding cost function in terms of the data rate has not yet been
studied in recent works.

A review of the existing literature did not yield any work that investigates
the impact of the decoding cost in the PS-SWIPT MAC scenarios under non-linear
EH receivers. To capture the practical restrictions of the physical system in
this paper, we focus on the problem of PS-SWIPT by taking into account the
non-decreasing decoding costs incurred at the non-linear EH nodes. The
predominant potential contributions of this paper are twofold. First, we
maximize the rate departure region for a two-user classical PS-SWIPT MAC with
two different decoding strategies, i.e., simultaneous decoding and successive
interference cancellation (SIC). Subsequently, we formulated the maximum
departure region boundaries (MDRBs) with power allocation in a PS-SWIPT MAC
with user cooperation. Second, we derived closed-form expressions for optimal
PS factors and sum rates in the two PS-SWIPT MAC scenarios. Unlike the
previous works wherein an increasing convex form of decoding cost is assumed
to ease the optimization problems, we assume a general non-decreasing form of
decoding cost function. Resultantly, it is proven that although our problems
are non-convex, global solutions can be derived in closed-form via solving
nonlinear system of equations. Finally, the simulation results are presented,
which provide invaluable insights into the PS-SWIPT systems with regard to the
decoding costs and non-linear EH model.

The rest of the paper is structured as follows. Section II describes Classical
PS-SWIPT MAC. Section III presents the PS-SWIPT MAC with user cooperation.
Finally, the numerical results and conclusions are included in Sections IV and
V, respectively.

##  II Classical PS-SWIPT MAC

###  II-A System Model

We study a two-user fading MAC wherein the destination employs PS technique to
harvest energy from the received signals and utilize the harvested energy for
decoding the users’ information. As depicted in Fig. 1, users 1 and 2 encode
the messages $m_{1}$ and $m_{2}$, respectively, then the users’ signals are
transmitted to the destination. To enlarge the achievable rate regions for
both of the classical PS-SWIPT MAC and the cooperative one, it is assumed that
both users simultaneously transmit their signals. Thus, in what follows, all
the achievable rate regions are derived based on the NOMA transmissions.
Besides, perfect channel state information (CSI) is assumed at the nodes to
derive performance upper bound of the considered systems. Here, the
destination aims to efficiently harvest energy from the received signal in the
EH block for the decoding process to maximize the achievable rate region.

The received signal at the destination is

$\displaystyle Y=h_{1}X_{1}+h_{2}X_{2}+Z,$ (1)

where $h_{1}$ and $h_{2}$ indicate the channel gains, $X_{1}$ and $X_{2}$ are
the transmitted symbols 111Since the focus of our work is to consider the
decoding costs effects on the performance of the SWIPT MAC with and without
cooperating users in presence of non-linear EH destination, we have only
considered the PS scheme at the destination without considering cooperative
signal design for the energy transfer [8, 9]. Our approach could be extended
to consider coherent energy-signal transmission by the two users, as well. by
the first and the second user with $\mathbb{E}\\{{|{X_{1}}|}^{2}\\}=P_{1}$ and
$\mathbb{E}\\{{|{X_{2}}|}^{2}\\}=P_{2}$, and $Z\sim\mathcal{CN}(0,N)$ is
complex additive white gaussian noise (AWGN) at the destination. The
destination splits the received signal into EH stream $Y_{EH}$ and ID one
$Y_{ID}$ as

$\displaystyle Y_{EH}=\sqrt{\rho}Y,\hskip
2.84544ptY_{ID}=\sqrt{1-\rho}Y+Z_{p},$ (2)

where $\rho\in[0,1]$ refers to the unit-less PS factor and
$Z_{p}\sim\mathcal{CN}(0,N_{p})$ represents the signal processing noise at the
power splitter. So, the harvested power at the destination is given by

$\displaystyle
P_{EH}=\psi\big{(}\rho(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+N)\big{)},$ (3)

wherein $\psi(.)$ models the output DC power of the rectifier in terms of the
input RF power $P_{in}$ defined by

$\displaystyle\psi(P_{in})=\psi^{DC}(P_{in})=\dfrac{\Psi(P_{in})-P_{max}^{DC}\theta}{1-\theta},$
(4)

which is a zero-crossing, non-decreasing [13], and upper-bounded EH conversion
function with the logistic function

$\displaystyle\Psi(P_{in})=\dfrac{P_{max}^{DC}}{1+e^{-q_{1}(P_{in}-q_{2})}},$
(5)

where $P_{max}^{DC}$ refers to the maximum output DC power and
$\theta=(1+e^{q_{1}q_{2}})^{-1}$ is a parameter to ensure zero input/zero
output of the EH block. Practically, the constant values $q_{1}$ and $q_{2}$
depend on the hardware elements of the rectifier and are determined via curve
fitting methods, [30, 28].

######  Remark 1.

In case of placing several EH circuits in parallel, the EH conversion function
is modeled with a linear zero-crossing function $\psi(P_{in})=\eta P_{in}$,
wherein $\eta\in[0,1]$ refers to the fixed energy conversion efficiency, [34,
35].

On the other hand, the required decoding power in the ID block is modeled as
$\varphi(R)$ which is a non-decreasing function of the achievable rate $R$.

Figure 1:  Classical PS-SWIPT MAC.

###  II-B Simultaneous Decoding Scheme

Assume that the destination applies the well-known simultaneous decoding
strategy to recover the users’ messages [45]. Therefore, the achievable rate
region is constrained by

$\displaystyle
R_{1}\leq\frac{1}{2}\log\left(1+\dfrac{(1-\rho)|h_{1}|^{2}P_{1}}{(1-\rho)N+N_{p}}\right),$
(6) $\displaystyle
R_{2}\leq\frac{1}{2}\log\left(1+\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{(1-\rho)N+N_{p}}\right),$
(7) $\displaystyle
R_{1}+R_{2}\leq\frac{1}{2}\log\left(1+\dfrac{(1-\rho)(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2})}{(1-\rho)N+N_{p}}\right),$
(8) $\displaystyle R_{1}+R_{2}\leq\varphi^{-1}\Big{(}P_{EH}\Big{)}=$
$\displaystyle\varphi^{-1}\Big{(}\psi\big{(}\rho(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}$
(9) $\displaystyle+N)\big{)}\Big{)},$

where (6) and (7) refer to the achievable rate constraints for the first and
the second user, respectively, while (8) denotes the sum rate constraint.
Also, the decoding cost constraint (9) utilizes the fact that $\varphi(.)$ is
a non-decreasing function.

######  Theorem 1.

The MDRB is formed by

$\displaystyle\begin{cases}R_{1}^{*}=\varphi^{-1}\big{(}\psi(\rho
a)\big{)}-R_{2}^{*},\\\ \hskip
156.49014pt\rho_{1}^{*}\leq\rho\leq\rho_{c}^{*}\\\
R_{2}^{*}=\frac{1}{2}\log\left(1+\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{(1-\rho)N+N_{p}}\right),\par\end{cases}$
(10) $\displaystyle\begin{cases}R_{2}^{*}=\varphi^{-1}\big{(}\psi(\rho
a)\big{)}-R_{1}^{*},\\\ \hskip
156.49014pt\rho_{2}^{*}\leq\rho\leq\rho_{c}^{*}\\\
R_{1}^{*}=\frac{1}{2}\log\left(1+\dfrac{(1-\rho)|h_{1}|^{2}P_{1}}{(1-\rho)N+N_{p}}\right),\end{cases}$
(11)

and

$R_{1}^{*}+R_{2}^{*}=\frac{1}{2}\log\left(1+\dfrac{(1-\rho_{c}^{*})(a-N)}{(1-\rho_{c}^{*})N+N_{p}}\right),$
(12)

where

$\displaystyle\rho_{c}^{*}=\Gamma_{c}^{-1}(N_{p}),\rho_{1}^{*}=\Gamma_{1}^{-1}(N_{p}),\rho_{2}^{*}=\Gamma_{2}^{-1}(N_{p}),$

and

$\displaystyle\Gamma_{c}(x)=2^{2\varphi^{-1}\big{(}\psi(xa)\big{)}}\Big{(}(1-x)N+N_{p}\Big{)}-(1-x)a,$
$\displaystyle\Gamma_{1}(x)=\Gamma_{c}(x)+(1-x)|h_{1}|^{2}P_{1},$
$\displaystyle\Gamma_{2}(x)=\Gamma_{c}(x)+(1-x)|h_{2}|^{2}P_{2},$
$\displaystyle\hskip 18.49411pta=|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+N.$

###### Proof.

To form the MDRB constrained by (6)-(9), we find the pairs of rates ($R_{1}$,
$R_{2}$) who are infeasible to be added at the same time. Since $\varphi(.)$
and $\psi(.)$ are non-decreasing functions, the right terms of (6), (7), and
(8) are decreasing, while the right term of (9) is increasing in $\rho$. The
pair of rates satisfying both (8) and (9) with strict inequality does not fall
on the MDRB, as we can decrease $\rho$ with small enough $\epsilon>0$ so that
all the constraints (6)-(9) are satisfied with strict inequality. Then, we can
feasibly increase both $R_{1}$ and $R_{2}$ with $\epsilon/M$ for a large
enough $M>0$, which contradicts the definition of the MDRB. As a result, we
consider three assumptions: I) Assume (8) dominates (9), i.e., (9) is
satisfied with strict inequality, while (8) is satisfied with equality. To
show the contradiction here, we can feasibly decrease $\rho$ with $\epsilon$
and increase both $R_{1}$ and $R_{2}$ with $\epsilon/M$; II) Assume (9)
dominates (8). With this assumption, we have four cases. First, (6) and (7)
are satisfied with equality. This case is already infeasible as it violates
(8). Second, (6) and (7) are satisfied with strict inequality. Here, we can
feasibly increase $\rho$, $R_{1}$, and $R_{2}$ with $\epsilon$, $\epsilon/M$,
and $\epsilon/M$, respectively, indicating that this case also does not form
the MDRB. Third, (7) dominates (6). This case, formulated in (10), partly
generates the MDRB as any small change in $\rho$ results in a violation of (7)
or (9). Forth, (6) dominates (7). This case formulated in (11), similarly
generates another part of the MDRB; III) Assume both (8) and (9) are satisfied
with equality. This assumption formulated in (12), also partly creates the
MDRB as any change in $\rho$ violates (8) or (9). ∎

Now, we characterize the convexity of the maximum departure region, bounded by
the MDRB in Theorem 1.

######  Remark 2.

Since $\varphi(.)$ and $\psi(.)$ are both increasing, we derive
$dR_{1}^{*}/dR_{2}^{*}\leq-1$ on the MDRB part, implying that the maximum sum
rate subject to (10) occurs in $\rho_{c}^{*}$. Using similar arguments, the
maximum sum rate subject to (11) occurs also in $\rho_{c}^{*}$. Moreover, if
$\varphi^{-1}\big{(}\psi(x)\big{)}$ is a convex function within the interval
$a\rho_{1}^{*}\leq x\leq a\rho_{c}^{*}$, then
$d^{2}R_{1}^{*}/dR_{2}^{*^{2}}\leq 0$ and the MDRB part calculated by (10) is
convex one. Otherwise, using time sharing technique, the convex hull of (10)
yields the maximum departure region. The same argument is verified for (11).

Depart from the maximum departure region, an important goal in PS-SWIPT MAC is
to optimize the PS factor for achievable sum rate maximization. To this end,
the optimization problem is formulated as

$\displaystyle\mathbf{P}\mathbf{1}:\max_{{\rho,R_{1},R_{2}}}$ $\displaystyle
R_{1}+R_{2}$ (13) s.t. $\displaystyle(\ref{simulr1})-(\ref{simulphi}),\hskip
2.84544pt\text{and}\hskip 2.84544pt0\leq\rho\leq 1,$ (14)

which is a non-convex optimization problem whose solution is a point on the
MDRB calculated in Theorem 1.

######  Lemma 1.

Optimally in $\mathbf{P}\mathbf{1}$, the whole harvested energy is consumed at
the ID block, i.e., (9) is satisfied with equality. Moreover, (8) is also
satisfied with equality.

###### Proof.

Suppose (9) is satisfied with strict inequality. The right terms of (6)-(8)
and the right term of (9) are decreasing and increasing in $\rho$,
respectively. So, we can reduce $\rho$ with a small enough $\epsilon>0$ such
that all the constraints (6)-(9) are satisfied with strict inequality. Thus,
both $R_{1}$ and $R_{2}$ can be added with no constraints violation, which
enhances the sum rate and contradicts the optimality. On the other hand,
suppose (8) is satisfied with strict inequality. Similarly, we can feasibly
increase $\rho$ in a small enough value, letting $R_{1}$ and $R_{2}$ increase,
and therefore the sum rate. Thus, Lemma 1 follows. ∎

######  Theorem 2.

The optimal PS factor $\tilde{\rho}$ in $\mathbf{P}\mathbf{1}$ is derived by
solving the equation

$\varphi\Big{(}\frac{1}{2}\log\big{(}1+\dfrac{(1-\rho)(a-N)}{(1-\rho)N+N_{p}}\big{)}\Big{)}=\psi\big{(}\rho
a\big{)}.$ (15)

Then, the optimal sum-rate is given by

$\tilde{R_{1}}+\tilde{R_{2}}=\frac{1}{2}\log\Big{(}1+\dfrac{(1-\tilde{\rho})(a-N)}{(1-\tilde{\rho})N+N_{p}}\Big{)},$
(16)

wherein $0\leq\tilde{\rho}\leq 1$ and $\tilde{R_{1}}+\tilde{R_{2}}$ exist and
are unique.

###### Proof.

From Lemma 1, the right terms of (8) and (9) are equal in the optimal solution
of $\mathbf{P}\mathbf{1}$, and thus, $\tilde{\rho}$ satisfies (15). The left
side of (15) is decreasing with $\rho$ with strictly positive and zero values
by setting $\rho=0$ and $\rho=1$, respectively. On the other hand, the right
side of (15) is increasing with $\rho$ with zero and strictly positive values
by setting $\rho=0$ and $\rho=1$, respectively. Hence, the optimal PS factor
$0\leq\tilde{\rho}\leq 1$ exists and is unique. Then, we derive the the
optimal sum rate in (16) by substituting $\tilde{\rho}$ in the right term of
(8). ∎

######  Remark 3.

It is revealed that in simultaneous decoding scheme,
$\tilde{\rho}=\rho_{c}^{*}$. Furthermore, the optimal sum rate
$\tilde{R_{1}}+\tilde{R_{2}}$ is upper-bounded by $\varphi^{-1}(\psi(a))$.

######  Remark 4.

Mostly in literature, the decoding cost is assumed a convex function of the
form $\varphi(R)=\beta(2^{2R}-1)$. Moreover, as a worst case of a practical
PS-SWIPT systems, $N$ is neglected as $N\ll N_{p}$ since the splitted signal
for the information recovery passes through several standard operation to be
converted from the RF band to baseband. During this process, additional noise
called signal processing noise is present due to the phase-offsets, non-
linearities, etc. [29].

As a special case as described in Remark 1 and 4, we assume $\psi(P_{in})=\eta
P_{in}$, $\varphi(R)=\beta(2^{2R}-1)$, and $N\ll N_{p}$. Then, the optimal PS
factor in $\mathbf{P}\mathbf{1}$ is derived as

$\tilde{\rho}=\frac{\beta(a-N)}{\beta(a-N)+\eta aN_{p}},$

which is an unit-less scalar ranged between $0$ and $1$.

###  II-C Successive Interference Cancellation Decoding Scheme

Assume that the destination applies SIC technique to decode the messages
received by the users, [45]. To decode the first user’s message, the
destination treats the second user’s signal as a noise. So, the receiver
cancels out the recovered signal of the first user and decodes the second
user’s signal. Accordingly, the achievable rate region is confined by

$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{2}\log\Big{(}1+\dfrac{(1-\rho)|h_{1}|^{2}P_{1}}{(1-\rho)N_{u}+N_{p}}\Big{)},$
(17) $\displaystyle R_{2}$
$\displaystyle\leq\frac{1}{2}\log\Big{(}1+\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{(1-\rho)N+N_{p}}\Big{)},$
(18) $\displaystyle\varphi(R_{1})+\varphi(R_{2})$
$\displaystyle\leq\psi\big{(}\rho(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+N)\big{)}.$
(19)

where $N_{u}=|h_{2}|^{2}P_{2}+N$, and (19) indicates the decoding cost
restriction resulting from the sequential decoding in SIC.

######  Theorem 3.

The MDRB is calculated as

$\displaystyle\begin{cases}R_{1}^{*}=\varphi^{-1}\Big{(}\psi\big{(}\rho
a\big{)}-\varphi(R_{2}^{*})\Big{)},\\\ \hskip
156.49014pt\rho_{1}^{*}\leq\rho\leq\rho_{c}^{*}\\\
R_{2}^{*}=\frac{1}{2}\log\Big{(}1+\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{(1-\rho)N+N_{p}}\Big{)},\par\end{cases}$
(20)

and

$\displaystyle\begin{cases}R_{2}^{*}=\varphi^{-1}\Big{(}\psi\big{(}\rho
a\big{)}-\varphi(R_{1}^{*})\Big{)},\\\ \hskip
156.49014pt\rho_{2}^{*}\leq\rho\leq\rho_{c}^{*}\\\
R_{1}^{*}=\frac{1}{2}\log\Big{(}1+\dfrac{(1-\rho)|h_{1}|^{2}P_{1}}{(1-\rho)N_{u}+N_{p}}\Big{)},\end{cases}$
(21)

where

$\displaystyle\rho_{c}^{*}=\Gamma_{c}^{-1}(a),\rho_{1}^{*}=\Gamma_{1}^{-1}(a),\rho_{2}^{*}=\Gamma_{2}^{-1}(a),$

and

$\displaystyle\Gamma_{c}(x)=$
$\displaystyle\dfrac{1}{x}\psi^{-1}\Bigg{(}\varphi\left(\frac{1}{2}\log\left(1+\dfrac{(1-x)|h_{1}|^{2}P_{1}}{(1-x)N_{u}+N_{p}}\right)\right)$
$\displaystyle+\varphi\left(\frac{1}{2}\log\left(1+\dfrac{(1-x)|h_{2}|^{2}P_{2}}{(1-x)N+N_{p}}\right)\right)\Bigg{)},$
$\displaystyle\Gamma_{1}(x)=$
$\displaystyle\dfrac{1}{x}\psi^{-1}\Bigg{(}\varphi\left(\frac{1}{2}\log\left(1+\dfrac{(1-x)|h_{2}|^{2}P_{2}}{(1-x)N+N_{p}}\right)\right)\Bigg{)},$
$\displaystyle\Gamma_{2}(x)=$
$\displaystyle\dfrac{1}{x}\psi^{-1}\Bigg{(}\varphi\left(\frac{1}{2}\log\left(1+\dfrac{(1-x)|h_{1}|^{2}P_{1}}{(1-x)N_{u}+N_{p}}\right)\right)\Bigg{)},$
$\displaystyle a=$ $\displaystyle|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+N.$

###### Proof.

Following the similar steps as in the proof of Theorem 1, the MDRB in SIC is
created by the pairs of rates ($R_{1}^{*},R_{2}^{*}$) who satisfy (19) and one
of (17) or (18) with equality. ∎

Now, by changing the decoding order, we firstly decode the second user’s
message. Then, we cancel this data out from the received signal, and finally
decode the first user’s message. So, by swapping the indices 1 and 2 in the
achievable rate region (17)-(19), we derive the new MDRB
$(\hat{R}_{1},\hat{R}_{2})$. Then, applying the time-sharing technique over
the two MDRBs resulting from each decoding order, the convex hull of
$(R_{1}^{*},R_{2}^{*})\cup(\hat{R}_{1},\hat{R}_{2})$ forms the maximum
departure region.

It is worth pointing out that the destination in SIC scheme prefers to
primarily decode the data from the user with better channel gain, and
thereafter, to decode the other data. So without loss of generality, we assume
that the channel between the first user and the destination is stronger than
the second one. Thus, the sum rate optimization problem is written as

$\displaystyle\mathbf{P}\mathbf{2}:\max_{{\rho,R_{1},R_{2}}}$ $\displaystyle
R_{1}+R_{2}$ s.t.
$\displaystyle(\ref{successr1}),(\ref{successr2}),(\ref{successphi}),\hskip
2.84544pt\text{and}\hskip 2.84544pt0\leq\rho\leq 1,$

which is a non-convex problem whose solution is not as straightforward as
$\mathbf{P}\mathbf{1}$, due to the unfavorable left term in (19). However,
using the fact that the both functions $\varphi(.)$ and $\psi(.)$ are non-
decreasing, we characterize the following lemma.

######  Lemma 2.

Optimally in $\mathbf{P}\mathbf{2}$, the harvested energy at the destination
is fully consumed, i.e., (19) is satisfied with equality. Moreover, at least
one of the constraints (17) or (18) are satisfied with equality.

###### Proof.

Assume that (19) is satisfied with strict inequality in the optimal solution
of $\mathbf{P}\mathbf{2}$. The contradiction is proven by decreasing $\rho$
with small enough $\epsilon>0$ and increasing $R_{1}$ and $R_{2}$ with
$\epsilon/M$ for a large enough $M>0$. Then, noting that (19) is satisfied
with equality, we assume that both (17) and (18) are satisfied with strict
inequality in the optimal solution. The contradiction here is shown by
feasibly adding $\rho$, $R_{1}$, and $R_{2}$ with $\epsilon$ and $\epsilon/M$,
respectively. So, Lemma 2 follows. ∎

Lemma 2 implies that in the optimal solution of $\mathbf{P}\mathbf{2}$, the
both constraints (17) and (18) are not necessarily satisfied with equality at
the same time. As a result, the optimal solution to $\mathbf{P}\mathbf{2}$ is
derived through Theorem 4.

######  Theorem 4.

In $\mathbf{P}\mathbf{2}$, the optimal PS factor $\tilde{\rho}$ is calculated
as

$\displaystyle\tilde{\rho}=$ $\displaystyle
ARGMAX\big{(}f_{1}(\rho_{1}^{*}),f_{2}(\rho_{2}^{*}),f_{1}(\rho_{11}),...,f_{1}(\rho_{1n}),$
(22) $\displaystyle
f_{2}(\rho_{21}),...,f_{2}(\rho_{2m}),f_{1}(\rho_{c}^{*}),f_{2}(\rho_{c}^{*})\big{)},$

and the optimal sum-rate is derived by

$\displaystyle\tilde{R}_{1}+\tilde{R}_{2}=$ $\displaystyle
MAX\big{(}f_{1}(\rho_{1}^{*}),f_{2}(\rho_{2}^{*}),f_{1}(\rho_{c}^{*}),f_{2}(\rho_{c}^{*}),$
(23) $\displaystyle
f_{1}(\rho_{11}),...,f_{1}(\rho_{1n}),f_{2}(\rho_{21}),...,f_{2}(\rho_{2m})\big{)},$

where $\rho_{1}^{*}<\rho_{1i}<\rho_{c}^{*}$ and
$\rho_{2}^{*}<\rho_{2j}<\rho_{c}^{*}$, for $1\leq i\leq n$ and $1\leq j\leq
m$, denote the $i^{\text{th}}$ and $j^{\text{th}}$ answers to the differential
equations $f^{\prime}_{1}(x)=0$ and $f^{\prime}_{2}(x)=0$, respectively,
wherein

$\displaystyle
f_{1}(x)=\varphi^{-1}\Big{(}\psi\big{(}xa\big{)}-\varphi\big{(}g_{1}(1-x)\big{)}\Big{)}+g_{1}(1-x),$
$\displaystyle
f_{2}(x)=\varphi^{-1}\Big{(}\psi\big{(}xa\big{)}-\varphi\big{(}g_{1}(1-x)\big{)}\Big{)}+g_{2}(1-x)$
$\displaystyle
g_{1}(x)=\frac{1}{2}\log\big{(}1+\dfrac{x|h_{2}|^{2}P_{2}}{xN+N_{p}}\big{)},$
$\displaystyle
g_{2}(x)=\frac{1}{2}\log\big{(}1+\dfrac{x|h_{1}|^{2}P_{1}}{xN_{u}+N_{p}}\big{)}.$

###### Proof.

See Appendix A. ∎

######  Lemma 3.

As a special case as described in Remark 1 and 4, we assume $\psi(P_{in})=\eta
P_{in}$, $\varphi(R)=\beta(2^{2R}-1)$, and $N\ll N_{p}$. Then, the optimal PS
factor in $\mathbf{P}\mathbf{2}$ is calculated as

$\tilde{\rho}=0.5\left(1+\dfrac{\beta B^{2}+\beta CN_{p}+\eta
aN_{p}^{2}-\Delta}{\beta B^{2}+\eta aN_{p}B}\right),$ (24)

and the optimal sum rate is derived as

$\displaystyle\tilde{R}_{1}+\tilde{R}_{2}=$
$\displaystyle\dfrac{1}{2}\log\Big{(}1+\dfrac{(1-\tilde{\rho})A}{(1-\tilde{\rho})N_{u}+N_{p}}\Big{)}$
(25)
$\displaystyle+\frac{1}{2}\log\Big{(}1+\dfrac{(1-\tilde{\rho})B}{N_{p}}\Big{)}$

where

$\displaystyle A=$
$\displaystyle|h_{1}|^{2}P_{1},~{}~{}~{}B=|h_{2}|^{2}P_{2},~{}~{}~{}C=A+B,$
(26) $\displaystyle\Delta=$ $\displaystyle N_{p}\big{[}(\eta aB-\beta
C)^{2}+\eta a(\eta aN_{p}^{2}+2\eta aN_{p}B$ $\displaystyle+2\beta
CN_{p}+4\beta B^{2})\big{]}^{0.5}.$

###### Proof.

From Lemma 2 and Appendix B. ∎

##  III PS-SWIPT MAC with user cooperation

###  III-A System Model

Here, we consider a two-user PS-SWIPT MAC with user cooperation wherein the
destination acquires energy for decoding data by applying PS scheme over the
received signals transmitted from both users. As depicted in Fig. 2, users
potentially enhance the departure region by creating a common information (see
[41], [46]) subject to the decoding cost at all receiving nodes, i.e., both
users and the destination. Note that in this work, the terms user and encoder
are used interchangeably. Also, perfect CSI is considered.

By utilizing the well-known Block Markov Encoding (BME) for MAC with
cooperative encoders at each transmission time block, both encoders transmit
their codewords not only based on their own message, but also based on the
decoded message of the other encoder which is decoded in the previous block,
see [45], [47] and [48]. So, at each transmission block, both encoders
transmit common and fresh messages, while the destination constructively
receives the common message transmitted from both encoders. Then, since the
common message is known at the encoders, each encoder is able to decode the
fresh part of the other encoders’ message. This procedure is possible thanks
to BME and backward decoding at the destination which is developed for the
classical relay channel, and for the MAC with noisy feedback. So, the
transmitted codeword by each user is given by

$\displaystyle X_{i}=\sqrt{P_{ij}}X_{ij}+\sqrt{P_{ui}}X_{u},\hskip
14.22636pti=1,2,\hskip 5.69046ptj=1,2,\hskip 5.69046pti\neq j,$

Figure 2: PS-SWIPT MAC with user cooperation.

where $X_{ij}$ with $\mathbb{E}\\{{|{X_{ij}}|}^{2}\\}=P_{ij}$ denotes the
transmitted symbol of the fresh information at user $i$ to be decoded at user
$j$ and to create the symbol $X_{u}$, the common information with power
$P_{ui}$. Therefore, $P_{i}=P_{ij}+P_{ui}$ refers to the total transmission
power at user $i$. Now, the received signals at users and the destination are
given by

$\displaystyle Y_{j}=h_{ij}X_{i}+Z_{j},\hskip 28.45274pti=1,2,\hskip
5.69046ptj=1,2,\hskip 5.69046pti\neq j,$ $\displaystyle
Y=h_{1}X_{1}+h_{2}X_{2}+Z,$

where $h_{j}$ indicates the channel gain between user $j$ and the destination
and $h_{ij}$ shows the channel gain from user $i$ to $j$. Besides,
$Z_{j}\sim\mathcal{CN}(0,N_{j})$ and $Z\sim\mathcal{CN}(0,N)$ represent AWGN
terms at user $j$ and the destination, respectively. However, it is assumed
that the user-user links are stronger than user-destination one, i.e., $N\gg
N_{j}$. Similarly, by applying the PS method, the received signal at the
destination is splitted into two different signals through the PS block. Thus,
the signals $Y_{EH}$ for EH and $Y_{ID}$ for ID are formulated as

$\displaystyle Y_{EH}=\sqrt{\rho}Y,$ (27) $\displaystyle
Y_{ID}=\sqrt{1-\rho}Y+Z_{P},$ (28) $\displaystyle
P_{EH}=\psi\Big{(}\rho\big{(}S+N\big{)}\Big{)},$ (29)

where $S=|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+2h_{1}h_{2}\sqrt{P_{u1}P_{u2}}$
refers to the received RF signal power, $P_{EH}$ is the total harvested power,
and $Z_{p}\sim\mathcal{CN}(0,N_{p})$ denotes the PS signal processing noise.

###  III-B Maximum Departure Region and Optimal Power Allocation

To obtain the maximum departure region with the set of powers
$\boldsymbol{P}=\\{P_{12},P_{21},P_{u1},P_{u2}\\}$, a pair of achievable rates
$(R_{1},R_{2})$ satisfies

$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{2}\log\left(1+\dfrac{|h_{12}|^{2}P_{12}}{N_{2}}\right),$
(30) $\displaystyle R_{2}$
$\displaystyle\leq\frac{1}{2}\log\left(1+\dfrac{|h_{21}|^{2}P_{21}}{N_{1}}\right),$
(31) $\displaystyle R_{1}+R_{2}$
$\displaystyle\leq\frac{1}{2}\log\left(1+\dfrac{(1-\rho)S}{(1-\rho)N+N_{p}}\right),$
(32) $\displaystyle R_{1}+R_{2}$
$\displaystyle\leq\varphi^{-1}\left(P_{EH}\right),$ (33)
$\displaystyle\varphi_{1}(R_{2})+P_{1}$ $\displaystyle\leq P_{U1},$ (34)
$\displaystyle\varphi_{2}(R_{1})+P_{2}$ $\displaystyle\leq P_{U2}.$ (35)

The inequalities (30)-(32) denote the achievable rate constraints at PS-SWIPT
MAC with user cooperation and (33)-(35) represent the decoding cost
constraints at the receiver nodes. The decoding cost function $\varphi_{1}(.)$
and $\varphi_{2}(.)$ at users are defined similar to $\varphi(.)$ at the
destination and the pair of $P_{U1}$ and $P_{U2}$ refers to the amount of
power available at user 1 and user 2, respectively.

In this section, we firstly characterize the MDRB by forming the weighted sum
rate as

$R_{\mu}=\mu_{1}R_{1}+\mu_{2}R_{2},$

subject to all the constraints (30)-(35), [46]. As proven in[41, Lemma 1],
there exists an optimal policy to achieve the MDRB, where (30) and (31) hold
with equality. Therefore, for any $\mu_{1},\mu_{2}\geq 0$, the optimization
problem is formulated as

$\displaystyle\mathbf{P}\mathbf{3}:\max_{\rho,\boldsymbol{P}}\hskip 5.69046pt$
$\displaystyle\mu_{1}g\left(bP_{12}\right)+\mu_{2}g\left(cP_{21}\right)$ (36)
s.t. $\displaystyle\varphi_{1}\big{(}g(cP_{21})\big{)}+P_{12}+P_{u1}\leq
P_{U1},$ (37)
$\displaystyle\varphi_{2}\big{(}g(bP_{12})\big{)}+P_{21}+P_{u2}\leq P_{U2},$
(38) $\displaystyle
g\left(bP_{12}\right)+g(cP_{21})\leq\varphi^{-1}\Big{(}\psi\big{(}\rho(S+N)\big{)}\Big{)},$
(39) $\displaystyle g\left(bP_{12}\right)+g\left(cP_{21}\right)\leq\
g\Big{(}\dfrac{(1-\rho)S}{(1-\rho)N+N_{p}}\Big{)},$ (40) $\displaystyle
0\leq\rho\leq 1,$ (41) $\displaystyle P_{12},P_{21},P_{u1},P_{u2}\geq 0.$ (42)

where $g(x)\triangleq\frac{1}{2}\log(1+x)$, $b=\frac{|h_{12}|^{2}}{N_{2}}$,
and $c=\frac{|h_{21}|^{2}}{N_{1}}$.

######  Lemma 4.

In the optimal solution of $\mathbf{P}\mathbf{3}$, all the constraints
(37)-(40) hold with equality.

###### Proof.

On the user side, suppose some energy is left at the first user in the optimal
solution and thus, (37) holds with strict inequality. So, we can increase
$P_{u1}$ with a small enough value $\epsilon>0$ such that (37) is still
satisfied with strict inequality. Noting that the functions $\varphi(.)$,
$\varphi_{1}(.)$, $\varphi_{2}(.)$, $\psi(.)$, and $g(.)$ are all increasing
and $S$ includes an addition term as a function of the multiplication of
$P_{u1}$ and $P_{u2}$, (39) and (40) are also satisfied with strict
inequality. Then, there exist a large enough value $M>0$ to add and subtract
$P_{21}$ and $P_{u2}$ with $\epsilon/M$, respectively, such that $S$ increases
and the left term of (38) is kept unchanged, while (37), (39), and (40) are
not violated. This act increases $P_{21}$, and thus, the objective function
which contradicts the optimality. Therefore, (37) is satisfied with equality.
Similarly, we can prove that (38) is also satisfied with equality. On the
destination side, we assume that (39) is satisfied with strict inequality.
Then, for a small enough $\epsilon$ and large enough $M$, we can decrease
$\rho$, $P_{u1}$, and $P_{u2}$ with $\epsilon$, $\epsilon/M$, and
$\epsilon/M$, respectively, such that the right term of (40) increases and
(39) is still satisfied with strict inequality. Now, all the constraints
(37)-(40) are satisfied with strict inequality. Then, there is large enough
$N>0$ such that adding the both $P_{21}$ and $P_{12}$ with $\epsilon/MN$ does
not violate non of the constraints (37)-(40), showing the contradiction.
Therefore, (39) is also satisfied with equality in the optimal solution.
Finally, assuming that (40) is satisfied with strict inequality, we increase
$\rho$ and decrease $P_{u1}$ and $P_{u2}$ with $\epsilon$, $\epsilon/M$, and
$\epsilon/M$, respectively to show the contradiction. As a result, (40) is
also satisfied with equality in the optimal solution of
$\mathbf{P}\mathbf{3}$. ∎

It is worth noting that decreasing $P_{u1}$ and $P_{u2}$ in the proof of Lemma
4 is feasible as long as $P_{u1},P_{u2}>0$. Otherwise,
$P_{u1}^{*}=P_{u2}^{*}=0$ in the optimal solution of $\mathbf{P}\mathbf{3}$,
or in other words, it is not optimal to employ cooperation anymore. As a
result of Lemma 4, we solve $\mathbf{P}\mathbf{3}$ using the non-linear system
of four equations in (37)-(40) and four unknowns in $\boldsymbol{P}$ as a
function of $\rho$. Then, we equate the derivative of the objective function
in terms of $\rho$ with zero, yielding the the optimal PS factor $\rho^{*}$.
Accordingly, the optimal set of powers $P_{12}^{*}$,
$P_{21}^{*}$,$P_{u1}^{*}$, and $P_{u2}^{*}$ are derived. However, to find a
closed form solution, we reduce $\mathbf{P}\mathbf{3}$ into

$\displaystyle\mathbf{P}\mathbf{4}:\max_{\rho,\boldsymbol{P}}\hskip 5.69046pt$
$\displaystyle\mu_{1}g\left(bP_{12}\right)+\mu_{2}g\left(cP_{21}\right)$ (43)
s.t. $\displaystyle\beta cP_{21}+P_{12}+P_{u1}\leq P_{U1},$ (44)
$\displaystyle\beta bP_{12}+P_{21}+P_{u2}\leq P_{U2},$ (45)
$\displaystyle\beta\Big{(}bP_{12}+cP_{21}+bcP_{12}P_{21}\Big{)}\leq\eta\rho(S+N),$
(46) $\displaystyle g\left(bP_{12}\right)+g\left(cP_{21}\right)\leq\
g\Big{(}\dfrac{(1-\rho)S}{(1-\rho)N+N_{p}}\Big{)},$ (47) $\displaystyle
0\leq\rho\leq 1,$ (48) $\displaystyle P_{12},P_{21},P_{u1},P_{u2}\geq 0.$ (49)

wherein the exponential functions
$\varphi(.)=\varphi_{1}(.)=\varphi_{2}(.)=\beta(2^{2R}-1)$ and a linear EH
model $\psi(P_{in})=\eta P_{in}$ as in described in Remark 1 and 4 are
assumed. Using Lemma (4), we solve $\mathbf{P}\mathbf{4}$ by maximizing the
objective function subject to (44)-(47) satisfied with equality. So,

$\displaystyle P_{12}=P_{U1}-P_{u1}-\beta cP_{21},$ (50) $\displaystyle
P_{21}=P_{U2}-P_{u2}-\beta bP_{12}.$ (51)

By substituting $P_{21}$ and $P_{12}$ in the right terms of (50) and (51), we
have

$\displaystyle P_{12}=\frac{P_{U1}-\beta cP_{U2}-P_{u1}+\beta
cP_{u2}}{1-\beta^{2}bc},$ (52) $\displaystyle P_{21}=\frac{P_{U2}-\beta
bP_{U1}-P_{u2}+\beta bP_{u1}}{1-\beta^{2}bc}.$ (53)

Then, the objective function of $\mathbf{P}\mathbf{3}$ can be rewritten as

$\small\mu_{1}g\left(b\frac{A-P_{u1}+\beta
cP_{u2}}{1-\beta^{2}bc}\right)+\mu_{2}g\left(c\frac{B-P_{u2}+\beta
bP_{u1}}{1-\beta^{2}bc}\right),$ (54)

where $A=P_{U1}-\beta cP_{U2}$ and $B=P_{U2}-\beta bP_{U1}$. Now, by taking
derivative of (54) with respect to $P_{u1}$ and $P_{u2}$, we have

$\displaystyle bc\Big{(}\mu_{1}+\mu_{2}\beta^{2}bc\Big{)}\tilde{P}_{u2}-\beta
b^{2}c\Big{(}\mu_{1}+\mu_{2}\Big{)}\tilde{P}_{u1}=C_{1},$ (55) $\displaystyle
bc\Big{(}\mu_{2}+\mu_{1}\beta^{2}bc\Big{)}\tilde{P}_{u1}-\beta
bc^{2}\Big{(}\mu_{1}+\mu_{2}\Big{)}\tilde{P}_{u2}=C_{2},$ (56)

where

$\displaystyle C_{1}=\mu_{1}b\Big{(}1-\beta^{2}bc+Bc\Big{)}-\mu_{2}\beta
bc\Big{(}1-\beta^{2}bc+Ab\Big{)},$ $\displaystyle
C_{2}=\mu_{2}c\Big{(}1-\beta^{2}bc+Ab\Big{)}-\mu_{1}\beta
bc\Big{(}1-\beta^{2}bc+Bc\Big{)}.$

######  Theorem 5.

In $\mathbf{P}\mathbf{4}$, the optimal points
($\tilde{P}_{u1}$,$\tilde{P}_{u2}$) and the optimal PS factor $\tilde{\rho}$
can be written as

$\displaystyle\begin{pmatrix}\tilde{P}_{u1}\\\
\tilde{P}_{u2}\end{pmatrix}=\frac{1}{ED-FC}\begin{pmatrix}F&D\\\
E&C\end{pmatrix}\begin{pmatrix}C_{1}\\\ C_{2}\end{pmatrix},$ (57)
$\displaystyle\tilde{\rho}=\frac{\beta\left(b\tilde{P}_{12}+c\tilde{P}_{21}+bc\tilde{P}_{12}\tilde{P}_{21}\right)}{\eta(\tilde{S}+N)},$
(58)

where

$\displaystyle C=\beta b^{2}c\Big{(}\mu_{1}+\mu_{2}\Big{)},\hskip 2.84544pt$
$\displaystyle D=bc\Big{(}\mu_{1}+\mu_{2}\beta^{2}bc\Big{)},$ $\displaystyle
E=bc\Big{(}\mu_{2}+\mu_{1}\beta^{2}bc\Big{)},\hskip 2.84544pt$ $\displaystyle
F=\beta bc^{2}\Big{(}\mu_{1}+\mu_{2}\Big{)},$
$\displaystyle\hat{S}=|h_{1}|^{2}(\tilde{P}_{12}+\tilde{P}_{u1})+|h_{2}|^{2}(\tilde{P}_{21}+\tilde{P}_{u2})+2h_{1}h_{2}\sqrt{\tilde{P}_{u1}\tilde{P}_{u2}}.$

Then, the optimal values ($\tilde{P}_{12}$,$\tilde{P}_{21}$) are derived by
substituting $\tilde{P}_{u1}$ and $\tilde{P}_{u2}$ into (52) and (53).

###### Proof.

The optimal powers ($\tilde{P}_{u1}$,$\tilde{P}_{u2}$) are derived by solving
the system of linear equations (55) and (56). Then, the optimal PS factor
$\tilde{\rho}$ is obtained by setting (46) with equality. ∎

######  Remark 5.

The optimal solution in Theorem 5 is valid only for
$\tilde{P}_{12},\tilde{P}_{21},\tilde{P}_{u1},\tilde{P}_{u2}\geq 0$.
Otherwise, it is not optimal to employ cooperation anymore. In that case, the
classical PS-SWIPT is the recommended framework to enhance the sum rate.
Furthermore, if $\beta^{2}bc=1$, then $ED=FC$ and the optimal powers
($\tilde{P}_{u1}$, $\tilde{P}_{u2}$, $\tilde{P}_{12}$, $\tilde{P}_{21}$) are
not unique.

(a) $\varphi(R)=\beta(2^{2R}-1)$

(b)  $\varphi(R)=\beta\log(2R+1)$

(c) $\varphi(R)=\beta(2R)$

(d) $\varphi(R)=\varphi_{0}$

Figure 3: Maximum departure region of the classical PS-SWIPT MAC for
simultaneous and SIC decoding schemes and four decoding cost forms.

##  IV Numerical Results and Discussions

In this section, we evaluate the performance of PS-SWIPT MAC systems under a
non-linear EH model and decoding cost constraints. Specifically, to gather a
practical overview of the systems, we consider the non-linear function of the
EH rectifier. To this end, we set the circuity parameters of the EH rectifier
in (5) as $q_{1}=1500$ and $q_{2}=0.0022$ with $P_{max}^{DC}=24$mW (see [49,
35, 50, 30, 28]). Furthermore, we assume that the users, with power sources
$P_{1}=P_{2}=0.5$W, are located at $d_{1}=d_{2}=3$m from the destination and
the channel gains are defined by $h_{1}=d_{1}^{-\alpha}$ and
$h_{2}=d_{2}^{-\alpha}$, with the propagation exponent $\alpha=2$. Finally, as
described in Remark 4, we apply the worst case in practical PS-SWIPT by
setting $N_{p}=-30$dB and $N=-60$dB to satisfy $N_{p}\gg N$.

Fig. 3 compares the maximum departure region of the classical PS-SWIPT MAC for
simultaneous decoding and SIC schemes with representative decoding cost
functions: convex, concave, linear, and constant ones. Note that the constant
decoding cost function might be interpreted as the conventional _signal
processing cost_ in the literature irrespective of the incoming rate [42]
and[51]. As observed in Fig. 3a, the SIC scheme outperforms the simultaneous
one with the convex decoding cost, as a smaller amount of power is consumed
for decoding the same pair of rates. While setting a concave decoding cost
function, Fig. 3b indicates that the simultaneous decoding outperforms the SIC
scheme. However, Fig. 3c shows that both the decoding schemes display the same
performance with linear decoding cost function. Finally, setting constant
values for the decoding cost as shown in Fig. 3d, the cost for the SIC scheme
is doubled compared to the simultaneous one, due to the sequential decoding
manner in SIC. Here, if $\varphi_{0}>\psi(a)/2\simeq P_{max}^{DC}/2=12$mW, the
rate region significantly shrinks in SIC as the destination carries enough
power to decode only one of the user’s messages. Moreover, setting
$\varphi_{0}>\psi(a)\simeq P_{max}^{DC}=24$mW results in an empty rate region
for both the decoding schemes, as the EH block lacks enough power for the
decoding process.

(a) Simultaneous Decoding

(b) SIC

Figure 4:  Maximum achievable sum rate in terms of the PS factor in Classical
PS-SWIPT MAC with the decoding cost function $\varphi(R)=\beta(2^{2R}-1)$.

Fig. 4 illustrates the maximum achievable sum rate in terms of the PS factor
in the Classical PS-SWIPT MAC system with the convex decoding cost function
for both the decoding schemes. It is observed that increasing $\rho$ saturates
the EH rectifier for a large $\beta$, which results in an upper-bound maximum
achievable sum rate at which the bound is decreasing by increasing $\beta$ in
both the decoding schemes. Moreover, in case the decoding factor is as small
as $\beta=-30dB$, the necessary power for decoding is much less than
$\psi(a)\simeq P_{max}^{DC}$. As a result, the EH rectifier approximately
performs within the linear functionality.

Finally, Fig. 5 investigates the effect of the user-user channel gain values
and $\beta$ in the performance of the classical PS-SWIPT MAC and PS-SWIPT MAC
with cooperation. In the cooperation case, we set the user-user channel
parameters as $N_{1}=N_{2}=N$ and $h_{12}=h_{21}=h_{u}$, and the channel
between the users and destination similar to the classical case. Further, we
assume $\varphi_{1}(R)=\varphi_{2}(R)=\varphi(R)$ as the convex decoding cost
function. It is observed that a large $\beta$ suppresses the maximum departure
region in both the classical and cooperation systems irrespective of the
channel gain. Therefore, although the cooperation outperforms the classical
system for $h_{u}\geq 0.004$, this superiority fades as $\beta$ increases such
that by setting $\beta=-21$dB, the system does not benefit from the
cooperation among the users anymore. On the other hand, if the user-user
channel gain drops to a low value as $h_{u}<0.004$, the classical PS-SWIPT MAC
outperforms the cooperation system, regardless of $\beta$.

(a) $\beta=-30dB$

(b) $\beta=-27dB$

(c) $\beta=-24dB$

(d) $\beta=-21dB$

Figure 5: Maximum departure region comparison between Classical PS-SWIPT and
PS-SWIPT MAC with cooperation for different values of user-user channel gain
and decoding cost function $\varphi(R)=\beta(2^{2R}-1)$.

##  V CONCLUSIONS

We studied two scenarios in the PS-SWIPT MAC wherein the PS scheme was
employed at the destination while all the receiving nodes were subjected to
decoding cost under non-linear EH model constraints. In the classical PS-SWIPT
MAC scenario, we considered two different decoding schemes, i.e., simultaneous
decoding and SIC. In each scheme, we initially calculated the maximum
departure region; then, the optimal PS factors and the maximum sum rates were
derived. The numerical results indicated that by changing the forms of
decoding cost function, the performance of both SIC scheme and simultaneous
decoding significantly vary especially in the case of constant decoding cost
function. Moreover, we derived an analytical optimal solution to jointly
optimizing PS factors and power allocation in order to maximize the sum rate
in PS-SWIPT MAC with user cooperation. Numerically, the results showed that it
is not always efficient to employ cooperation in the PS-SWIPT MAC which is a
significant impact of decoding cost and rectifier saturation on the
performance of the system.

## APPENDIX A

Sketch of proof for Theorem 4 : Firstly, the solution to
$\mathbf{P}\mathbf{2}$ falls on the MDRB characterized by (20) and (21) or on
it’s Convex Hull Boundary (CHB). To further shrink the feasible set, we assume
a pair of points $p^{*}=(R_{1}^{*},R_{2}^{*})$ and
$\hat{p}=(\hat{R}_{1},\hat{R}_{2})$ on the MDRB characterized by (20) and/or
(21). Then for any $0\leq\lambda\leq 1$, the sum-rate of any point on the line
segment between $p^{*}$ and $\hat{p}$ is expressed as

$\displaystyle\|\lambda p^{*}+(1-\lambda)\hat{p}\|_{1}$ $\displaystyle=\lambda
R_{1}^{*}+(1-\lambda)\hat{R}_{1}+\lambda R_{2}^{*}+(1-\lambda)\hat{R}_{2}$
(59)
$\displaystyle=\lambda(R_{1}^{*}+R_{2}^{*})+(1-\lambda)(\hat{R}_{1}+\hat{R}_{2})$
$\displaystyle\leq
max\big{(}R_{1}^{*}+R_{2}^{*},\hat{R}_{1}+\hat{R}_{2}\big{)},$

where $\|.\|_{1}$ stands for $L_{1}$ norm, indicating that a convex hull does
not increase the maximum sum-rate. As a result, regardless of the system’s
parameters and the form of the functions $\varphi(.)$ and $\psi(.)$, the
optimal point in $\mathbf{P}\mathbf{2}$ falls on the MDRB expressed by (20) or
(21). Thus, confining the feasibility set of $\mathbf{P}\mathbf{2}$ to the
MDRB, we reformulate $\mathbf{P}\mathbf{2}$ as

$\displaystyle\mathbf{P}\mathbf{2-1}:\max_{{\rho}}~{}~{}$ $\displaystyle
R_{1}^{*}+R_{2}^{*}$ s.t.
$\displaystyle[\rho,R_{1}^{*},R_{2}^{*}]\subseteq(\ref{simulregion1})\cup(\ref{simulregion3}).$

Since the feasible set of $\mathbf{P}\mathbf{2-1}$ consists of the union of
the sets satisfying (20) or (21), we can separately maximize
$R_{1}^{*}+R_{2}^{*}$ with the feasible sets (20) and (21), and then to find
the maximum of their solutions. So, we rewrite $\mathbf{P}\mathbf{2-1}$ as

$\displaystyle\mathbf{P}\mathbf{2-2}:\max_{{\rho,\hat{\rho}}}$
$\displaystyle~{}~{}~{}\text{MAX}(R_{1}^{*}+R_{2}^{*},\hat{R}_{1}+\hat{R}_{2})$
s.t.
$\displaystyle[\rho,R_{1}^{*},R_{2}^{*}]\subseteq~{}(\ref{simulregion1}),$
$\displaystyle[\hat{\rho},\hat{R}_{1},\hat{R}_{2}]\subseteq~{}(\ref{simulregion3}),$

where MAX(.) stands for the maximum value function. Then using the equations
(20) and (21), $\mathbf{P}\mathbf{2-2}$ is rewritten as

$\displaystyle\mathbf{P}\mathbf{2-3}:\max_{{\rho,\hat{\rho}}}$
$\displaystyle~{}~{}~{}\text{MAX}(f_{1}(\rho),f_{2}(\hat{\rho}))$ s.t.
$\displaystyle\rho_{1}^{*}\leq\rho\leq\rho_{c}^{*},$
$\displaystyle\rho_{2}^{*}\leq\hat{\rho}\leq\rho_{c}^{*},$

wherein the first and the second argument of MAX(.) are independent and non-
convex. As a result, solving the derivative equations $f^{\prime}_{1}(x)=0$
and $f^{\prime}_{2}(x)=0$, Theorem 4 follows.

## APPENDIX B

Sketch of proof for Lemma 3: From Lemma 2, (19) is satisfied with equality.
Now, we can add $R_{1}$ and $R_{2}$ by increasing $\rho$ with a small enough
value with no constraints violation. So, at least one of (17) or (18) holds
with equality. First, suppose (18) is satisfied with equality and (17) holds
with strict inequality. Then, we have

$R_{1}=\frac{1}{2}\log\Big{(}1+\dfrac{\eta\rho
a}{\beta}-\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{(1-\rho)N+N_{p}}\Big{)}.$ (60)

So, we have three conditions. First, we show that (60) satisfies (17), by a
substituting and removing the logarithms as

$\dfrac{\eta\rho
a}{\beta}-\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{N_{p}}\leq\dfrac{(1-\rho)|h_{1}|^{2}P_{1}}{(1-\rho)|h_{2}|^{2}P_{2}+N_{p}},$
(61)

wherein $N_{p}\gg N$. Simplifying (61), we derive an equation as

$\displaystyle-B(\eta aN_{p}+\beta B)\rho^{2}+\big{(}\eta
aN_{p}(B+N_{p})+\beta(2B^{2}$ (62)
$\displaystyle+CN_{p})\big{)}\rho-\beta(B^{2}+AN_{p}+BN_{p})\leq 0,$

wherein $A$, $B$, and $C$ are defined in Lemma 3. The left term of (62) is a
quadratic function of $\rho$. Since both addition and multiplication of the
roots in the quadratic function are positive, the both roots are positive.
Denoting the smaller root by $\rho_{1}$, the feasible $\rho$ in (62) falls
within the interval $0<\rho\leq\rho_{1}<1$, where

$\rho_{1}=0.5\left(1+\dfrac{\beta B^{2}+\beta CN_{p}+\eta
aN_{p}^{2}-\Delta}{\beta B^{2}+\eta aN_{p}B}\right),$

wherein $\Delta$ is a real positive value defined in Lemma 3. The second
condition is to have $\varphi(R_{2})\leq P_{EH}$ by satisfying

$\dfrac{\beta(1-\rho)|h_{2}|^{2}P_{2}}{N_{p}}\leq\eta\rho a,$

which gives the second feasibility interval $\rho\in[\rho_{2},1]$, where

$\rho_{2}=\dfrac{\beta B}{\beta B+\eta aN_{p}}.$

As a result, noting that $\rho_{2}<\rho_{1}$, the feasible PS factor is
bounded as $0<\rho_{2}\leq\rho\leq\rho_{1}<1$.

Finally, we prove that the sum rate is increasing in $\rho$ in the feasible
interval. From (60), we have

$\displaystyle\begin{aligned}
R_{1}+R_{2}=&\frac{1}{2}\log\Big{(}1+\dfrac{\eta\rho
a}{\beta}-\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{N_{p}}\Big{)}\\\
&+\frac{1}{2}\log\Big{(}1+\dfrac{(1-\rho)|h_{2}|^{2}P_{2}}{N_{p}}\Big{)}\\\
&=\frac{1}{2}\log\Big{(}1+\dfrac{\eta\rho a}{\beta}+\dfrac{\eta
a|h_{2}|^{2}P_{2}(1-\rho)}{\beta N_{p}}\\\
&-\dfrac{(1-\rho)^{2}|h_{2}|^{4}P_{2}^{2}}{N_{p}^{2}}\Big{)}.\end{aligned}$
(63)

Note that (63) is increasing in $\rho$ iff the argument of the logarithm
function is increasing. Thus from (63),
$\dfrac{d\left(R_{1}+R_{2}\right)}{d\rho}\geq 0$ is satisfied in the interval
$0<\rho\leq\rho_{c}<1$, where

$\rho_{c}=0.5\left(1+\dfrac{\beta B^{2}+\eta aN_{p}^{2}}{(\beta B+\eta
aN_{p})B}\right).$

Now, to prove that the sum rate is increasing in the interval
$[\rho_{2},\rho_{1}]$, we show $\rho_{1}\leq\rho_{c}$ as

$\dfrac{\beta(B^{2}+CN_{p})+\eta aN_{p}^{2}-\Delta}{B(\beta B+\eta
aN_{p})}\leq\dfrac{\beta B^{2}+\eta aN_{p}^{2}}{B(\beta B+\eta aN_{p})},$ (64)

which is derived from $\beta CN_{p}<\Delta$. Hence, setting $\rho=$ $\rho_{1}$
is feasible and results in the maximum sum rate. In other words, (17) is also
satisfied with equality.

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