# Efficient Coded Cooperative Networks with Energy Harvesting and Wireless
Power Transfer

Nan Qi, Ming Xiao, Theodoros A. Tsiftsis, Mikael Skoglund, and Huisheng Zhang
Nan Qi and Huisheng Zhang are with the School of Electronics and Information,
Northwestern Polytechnical University, China (e-mail<EMAIL_ADDRESS>zhanghusheng@nwpu.edu.cn). Ming Xiao, Mikael Skoglund are with the School of
Electrical Engineering of KTH, Royal Institute of Technology, Stockholm,
Sweden (e-mail: {mingx, skoglund}@kth.se). Nan Qi is also with KTH. T. A.
Tsiftsis is with the School of Engineering, Nazarbayev University, 010000
Astana, Kazakhstan and with the Department of Electrical Engineering,
Technological Educational Institute of Central Greece, 35100 Lamia, Greece
(e-mails<EMAIL_ADDRESS>tsiftsis@teiste.gr).

###### Abstract

The optimum off-line energy management scheme for multi-user multi-relay
networks employing energy harvesting and wireless energy transfer is studied.
Specifically, the users are capable of harvesting and transferring energy to
each other over consecutive transmissions, though they have no fixed energy
supplies. Meanwhile, network coding for the users’ messages is conducted at
the relays to enable cooperative transmission with source nodes in independent
but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ fading
channels. Therefore, a simultaneous two level cooperation, i.e., information-
level and energy-level cooperation is conducted. The problem of energy
efficiency (EE) maximization under constraints of the energy causality and a
predefined outage probability threshold is formulated and shown to be non-
convex. By exploiting fractional and geometric programming, a convex form-
based iterative algorithm is developed to solve the problem efficiently.
Close-to-optimal power allocation and energy cooperation policies across
consecutive transmissions are found. Moreover, the effects of relay locations
and wireless energy transmission efficiency are investigated and the
performance comparison with the current state of solutions demonstrates that
the proposed policies can manage the harvested energy more efficiently.

###### Index Terms:

Convex optimization, energy harvesting and cooperation, energy efficiency,
outage probability, Nakagami-$m$ fading, and network coding.

##  I Introduction

###  I-A Motivation and Related Works

In wireless sensor networks (WSNs) or wireless body area networks (WBAN),
energy harvesting (EH) is a sustainable approach to prolong the network
lifetime. Specifically, the EH technique enables the nodes to harvest energy
from nature such as solar, wind, and vibration, and refill their energy-
constrained batteries. However, the energy arrival is highly dependent on the
environment such as the weather and location. On one hand, the variable
weather makes the harvested energy at an individual node intermittent
available. On the other hand, distinct locations may lead to some energy-
deprived nodes as well as energy-abundant nodes. Both facts cause the
inefficiency energy usage of the whole network. Fortunately, wireless energy
transfer [1]-[3] provides a possibility to share the harvested energy among
the nodes. That is, it offers extra energy supplement to enhance the data
transmission in case that energy harvested from nature is not adequate. All
these yield the integrated energy harvesting and transferring (IEHT)
techniques.

IEHT netowrks continues to attract considerable research interest recently
[1]-[17]. Simultaneous wireless information and power transfer (SWIPT) was
widely considered [4]-[9]. Two types of SWIPT protocols are 1) power splitting
(PA) [4]-[6], where the receiver splits the received signal into two parts for
decoding information and harvesting energy and 2) time-switching (TS) [8],
[9], where the receiver switches between decoding information and harvesting
energy. These two protocols, which allow receivers harvest energy and receive
messages from the same radio frequency (RF) signal have been widely applied in
Amplify-and-Forward (AF)/Decode-and-Forward(DF) based cooperative relaying
[4], orthogonal frequency division multiple access (OFDMA) [5], [6] and
single-input multiple-output (SIMO), multiple-input single-output (MISO) [8]
and multiple-input multiple-output (MIMO) [7] setups. With the Lagrange dual
method, the end-to-end rate, sum-rate or EE are maximized by optimizing the
power [4], [5], [6] or time fractions [8] for harvesting energy or decoding
information.

In the above two protocols, RF signals are dually exploited for delivering
energy as well as transmitting information. However, their practical
implementation may not be easily fulfilled currently given the following
limitations of the current state-of-the-art of electronic circuits: 1) the
operating power of the energy harvesting unit is much higher than that of the
information processing units ($-10$dBm for energy harvesters versus $-60$dBm
for information receivers [1], [17]); 2) the TS policy requires a strict
synchronization process and a non-continuous information transmission; 3) the
PS policy requires appropriate PS circuits that increase the complexity and
cost of the hardware, and hardware non-idealities can significant efficiency
loss of the PS strategy, as explained in [7]. This raises a demand for _non-
overlapping energy harvesting, transferring and information transmitting
(Non.o.-IEHT)_ techniques. That is, wireless energy transfer is maintained by
a separate unit and is independent of the energy harvesting and information
transmission, as in [10], [12]-[17]. The authors in [10], [12]-[17] focused on
maximizing the throughput for parallel fading subchannels [13], two-way
cahnnels without [10] or with relays [15] and MISO network settings [16]. The
battery storage can be finite [10] or infinite [12]-[14]. The directional
water-filling algorithm obtained via the Lagrange dual method is widely
adopted to obtain the closed-form solutions in [10], [12]-[16]. In [17], the
weighted sum-rate maximization problem in energy harvesting analog network
coding (ANC) based TWR has been investigated. By applying the semi-definite
relaxation and successive convex optimization, the optimum beamforming vector
and transmitting rates have been determined.

The above literatures assumed that one relay only assisted one transmitter-
destination transmission. However, in the multi-user multi-relay scenarios,
the RF signals broadcasted by each user can be received by multiple relays; on
the other hand, one relay may receive multiple signals from different users
and then coordinate their signals. Therefore, without extra power cost, the
system performance can be greatly improved by utilizing the potential of the
spatial diversity of distributed relays and the coordination of the source
signals at the relays. One particularly effective way to coordinate source
signals is to utilize physical layer network coding (NC) [9], [20]-[22], which
inherently poses an information-level cooperation. In the presence of network
coding, user messages are linearly combined over Galois field (GF) to enable
sources to cooperate and transmit messages simultaneously. The published works
[10], [12]-[17] considered the network coding that is operated in GF(2).
Recent works illustrated that, if the linear combination is performed over a
large finite field, benefits in terms of diversity order or even energy
efficiency can be obtained [18], [20]-[22]. Particularly, to achieve the full
diversity order for a group of cooperative users, the concept of maximum
diversity network coding (MDNC) was proposed in [21] and [22]. It was shown
that an $M$-user $N$-relay network based on MDNC can achieve the full
diversity order (i.e., $N+1$ and $N-M+1$ in the presence or absence of the
direct source node-destination channels, respectively). It was also proved in
[21] that MDNC can provide the network with a larger outage capacity than the
dynamic network coding and analog network coding in the high SNR region.

We consider Non.o.-IEHT in two-hop multi-user multi-relay systems, where
network coding over high Galois field is also performed, thereby creating a
simultaneous two level cooperation, i.e., information- and energy-level
cooperation. In this way, the potential of energy efficiency (EE) and wireless
resources are expected to be fully exploited. However, to the best of our
knowledge, very few works studied energy flow management for such network
settings. The authors of the published works [9], [21] and [22] were mainly
concerned about the diversity order (i.e., the exponent of SNR in the upper
bound); however, the above policies may result in a degraded energy efficiency
since they only considered the outage probability performance and energy cost
was ignored. In our previous work [20], we presented the energy efficient MDNC
networks with Rayleigh fading environment, where power allocation and relay
selection were jointly adopted. Nevertheless, the networks are consisted of
conventional nodes that cannot harvest or transfer energy. The energy was not
fully exploited in the sense that it can neither be shared among the users nor
optimized across consecutive transmissions. Moreover, the algorithm is not
applicable for the more general Nakagami-$m$ fading environment. An important
and pertinent work on energy harvesting coded networks is [9], where the time-
switching based energy transferring protocol has been applied and the time
fraction (for harvesting energy or decoding information) was optimized to
minimize the network outage probability over one single time slot. However, it
assumed that the outage probability was the same for all inter-user channels
and the energy-depletion policy was adopted that did not allow energy
accumulation at the nodes. The algorithm in [9] is not feasible in the
networks where the inter-channels are independent but not necessarily
identically distributed (i.n.i.d.). On the other hand, as we will show in
Section V-C, the energy-depletion policy is not optimal for the consecutive
transmission scenarios since the energy was not optimized along the time
dimension. It is thus being observed that, for network-coded systems employing
energy harvesting and transferring, the energy efficient energy flow
management which allows extra harvested energy to be accumulated and stored in
the batteries for its future usage is still an open problem.

###  I-B Contributions

In this paper, we study the high Galois field network-coded relaying systems
with Non.o.-IEHT techniques. For our considered network model, we focus on
careful management of the energy flow and answering the following questions:
1) To maximize the energy efficiency, how much harvested energy at one
specific user should be stored for future usage and how much energy should be
transferred to/obtained from other users in every individual transmission
period? and 2) How to allocate the data transmitting power among the
cooperative users and relays such that the EE can be maximized? Specifically,
our main contributions are listed as below:

(1)

We respectively derive the outage probability, energy consumption and EE for
the networks that are coded over Galois field. The EE maximizing problem
satisfying the energy causality and the pre-defined outage probability
threshold is exactly formulated. Different from the similar network coding
scenario published in [9], [20]-[22], energy accumulation is allowed at the
users rather than depleted over one transmission; meanwhile, PA and energy
cooperation policies are jointly optimized across consecutive transmissions.
That is, energy can flow in time from the past to the future, and in space
from one user to the other users. Thus, as we will show later, energy
causality constraints take a new form and the underline optimization problem
will be completely new and different from the state of arts.

(2)

We consider the Nakagami-$m$ channel, which well models various cellular
environments, including the non-line-of-sight (NLOS) and line-of-sight (LOS)
channels [23]. Moreover, for generality purposes, the channels are assumed to
be i.n.i.d. and the path-loss related to the transmission distance is also
incorporated.

(3)

The optimization problem is shown to be NP-hard. The Lagrange dual method
widely adopted in [5], [6], [8], [10], [12]-[17], however, is not feasible in
our network coding scheme. Instead, to efficiently obtain close-to-optimal
solutions, the relaxation and approximation methods are exploited. Finally, a
convex form-based iterative algorithm is developed by combining the geometric
programming and non-linear fractional programming.

(4)

The tradeoff between the EE and outage probability is derived. Moreover,
energy cooperation and power allocation policy results are illustrated. The EE
gains from NC and energy transferring are also analysed. Additionally, the
impacts of the relay locations and the wireless energy transmission efficiency
are also investigated.

The rest of the paper is organized as follows. In Section II, we present the
system model. EE maximization problem formulation is given in Section III.
Then, we reformulate the problem and propose an algorithm in Section IV. The
analytical and simulation results are presented in Section V. Section VI
concludes this paper.

##  II System Model

We consider a network with $N$ relays and one destination. There are $M$ users
in the network which intend to transmit their independent messages to the
destination with the assistance of $N$ full-duplex relays, as depicted in Fig.
1. All the nodes are all equipped with a single antenna. It is assumed that
there is no direct connection between users and the destination due to the
long communication distance or the presence of physical obstacles.

The destination and relays both have fixed power supplies, while the batteries
at the users have to be refilled externally or by the energy transferred from
other users. There are separate units for wireless energy transferring, energy
harvesting and information transmitting such that they are performed
independently and concurrently at one user [10], [14], [15].

Figure 1: Network coded cooperative transmission.

In what follows, we illustrate the channel model, information transmission,
energy harvesting and cooperation models, respectively.

###  II-A Channel Model

Slowly varying flat fading channels that follow Nakagami-$m$ distribution are
considered. We note that Nakagami-$m$ fading is a general channel model in the
sense that variations of the severity of any fading channel can be expressed
as Nakagami-$m$ distribution by changing the fading parameter, $m$, from $1/2$
to $+\infty$ [23]. Let $U_{i}$, $i\in\\{1,2,\cdots,M\\}$, and $R_{j}$,
$j\in\\{1,2,\cdots,N\\}$, represent the $i$th user and $j$th relay,
respectively. Then $U_{i}$-$R_{j}$ channel coefficient is represented as

$h_{ij}=|\Upsilon_{ij}|\sqrt{d_{ij}^{-\beta_{ij}}}\exp(j\varphi_{h_{ij}}),$
(1)

where $h_{{ij}}$ is the channel gain that combines the path-loss and
Nakagami-$m$ fading; $|\Upsilon_{ij}|$ is the Nakagami-$m$ fading envelope;
$\varphi_{h_{ij}}$ is the phase of the $U_{i}$-$R_{j}$ channel that is assumed
uniformly distributed over the range of $[0,2\pi)$; $d_{ij}^{-\beta_{ij}}$
denotes the path loss; ${d_{ij}}$ is the distance and ${\beta_{ij}}$ is the
channel path loss exponent.

The probability distribution function (pdf) of $|\Upsilon_{ij}|$ can be given
as [24]

$f_{|\Upsilon_{ij}|}(x)=\frac{{2m^{m}x^{2m-1}}}{{\Gamma(m)\Omega_{h_{ij}}^{m}}}\exp(-\frac{{x^{2}}}{{\Omega_{h_{ij}}}}),x>0,$
(2)

where $\Gamma(\cdot)$ is the gamma function; ${{\Omega_{h_{ij}}}}$ is the
average channel gain represented as
${{\Omega_{h_{ij}}}}=\mathbb{E}[{|\Upsilon_{ij}|^{2}}]$ and
$\mathbb{E}\\{\cdot\\}$ is the expectation operator.

Similar to $h_{{ij}}$ in (1), the $R_{j}$-destination channel coefficient,
denoted as ${g_{j}}$, also combines the path-loss and Nakagami-$m$ channel
fading, i.e.,

$g_{j}=|\Upsilon_{j}|\sqrt{d_{j}^{-\beta_{j}}}\exp(j\varphi_{g_{j}}),$ (3)

where $|\Upsilon_{j}|$, $\varphi_{g_{j}}$, $d_{j}$ and $\beta_{j}$ are
parameters for the $R_{j}$-destination channel and denote the channel fading
envelope, channel phase, distance and path loss exponent, respectively.
$|\Upsilon_{j}|$ also follows the Nakagami-$m$ distribution. The average
channel gain is represented as
${{\Omega_{g_{j}}}}=\mathbb{E}[{|\Upsilon_{j}|^{2}}]$.

We assume that perfect channel state information (CSI) is available at the
receivers, while the transmitters only have the knowlege of
${{\Omega_{h_{ij}}}}$ and ${{\Omega_{g_{j}}}}$ ($\forall i,j$). In the
i.n.i.d. fading environment, ${{\Omega_{h_{ij}}}}$ and ${{\Omega_{g_{j}}}}$
($\forall i,j$) may differ from each other.

###  II-B Information Transmission Scheme

All nodes operate in time division multiple access (TDMA), which is also
adpoted in [9] and [18]. Thus, there is no interference among information
transmissions.

As shown in Fig. 1, the whole transmission consists of two hops.

1) The First Hop: User-relay Transmission

The message of $U_{i}$ ($\forall i$) is denoted as $S_{i}$. Suppose that all
user messages are of the same length111We note that this assumption is made
for simplifying illustration. The system model can be extended to general
cases where different users may have different message lengths. More
specifically, if different users have different message lengths, we can divide
the messages into shorter ones such that the lengths of shorter messages are
the same and some users have more messages while some have fewer messages.
Then, the users with fewer messages may not participate in all transmission
rounds., denoted as $|S|$. Additionally, we assume that all users and relays
transmit information with a fixed rate $\alpha_{0}$ bits per second222Our
model and algorithm are also applicable for different fixed rates on different
channels. The rates affect the values of the data transmitting time and outage
probability. However, different rates have no impact in the convexity of
$\mathbf{P3}$ presented in Section IV. Hence, the analysis and proposed scheme
are still feasible.. Take $U_{i}$ as an example. $S_{i}$ is first protected by
channel coding and then modulated into a unit power-signal, denoted as
$X(S_{i})$. Then $X(S_{i})$ is broadcast to all relays, which takes
$T=|X(S_{i})|/\alpha_{0}$ seconds ($|\cdot|$ means the number of bits in
$X(S_{i})$).

$R_{j}$ receives the signal from $U_{i}$ as follows:

$F_{ij}=h_{{ij}}\sqrt{p_{i}}X(S_{i})+z_{{ij}},$ (4)

where ${p_{i}}$ is the transmitting power for the channel codeword at $U_{i}$;
$z_{ij}\sim{\mathcal{N}}(0,N_{0,ij}B)$ denotes the AWGN; $N_{0,ij}$ is the
one-sided power spectral density and $B$ is the bandwidth.

The achievable rate for the channel between $U_{i}$ and $R_{j}$ is

$C_{ij}=B{\log_{2}}(1+\frac{|{h_{ij}}{|^{2}}p_{i}}{{N_{0,ij}}B}),$ (5)

where $|{h_{ij}}|$ is the amplitude of ${h_{ij}}$. An outage event occurs in
the $U_{i}$-$R_{j}$ channel when the fixed data transmission rate is larger
than the Shannon capacity [21], i.e.,

$C_{ij}<\alpha_{0}.$ (6)

If no outage event happens in the $U_{i}$-$R_{j}$ channel, $R_{j}$ will decode
$F_{ij}$ into ${S_{i}}$. In this way, $R_{j}$ tries to obtain all source
messages, i.e., $\\{S_{1},S_{2},\cdots,S_{M}\\}$.

2) The Second Hop: Relay-Destination Transmission

The following notations will be used in our following description.

${\Theta}$: the index set of all the relays.

${{{\Phi_{n}}}}$: Suppose in the first hop, $n$ relays succeed in receiving
and decoding all the user messages. Their index set is ${{{\Phi_{n}}}}$. Note
that ${n=0}$ means that no relay receives and decodes all the user messages.

$\psi_{\tau}$: Suppose in the second hop, $\tau$ relays manage to forward
messages to the destination. ${\tau}=0$ means no relay forwarding messages to
the destination.

Clearly, $\tau\leq n\leq N$ and
$\psi_{\tau}\subseteq{{{\Phi_{n}}}}\subseteq{\Theta}$.

If $R_{j}$ fails to decode any user message, it will not forward messages.
Otherwise, if it can decode all user messages, a network coding scheme based
on pre-defined MDNC coding coefficients will be applied. A network codeword
${W_{j}}$ is generated at $R_{j}$ by the linear combination of
${S_{1}},\,{S_{2}},\,\cdots\,\,,{S_{M}}$ over a finite field, i.e.,

${W_{j}}=\mathop{\boxplus}\limits_{i=1}^{M}\gamma_{ij}{{S_{i}}},\forall
i\in\\{1,2,\cdots,M\\},\forall j\in{{{\Phi_{n}}}},$

where “$\boxplus$” is the addition operation in the finite field;
$\gamma_{ij}$ is the global encoding kernel for ${{S_{i}}}$ at relay $R_{j}$.
$\gamma_{ij}$ constitutes the transfer matrix ${\mathbf{G}_{M\times N}}$
corresponding to MDNC333Since the design of MDNC encoding and decoding are not
our main points, we skip their design details.

${\mathbf{G}_{M\times
N}}=\left({\begin{array}[]{*{20}{c}}{{\gamma_{11}}}&{{\gamma_{12}}}&\ldots&{{\gamma_{1N}}}\\\
{{\gamma_{21}}}&{{\gamma_{22}}}&\ldots&{{\gamma_{2N}}}\\\
{}\hfil&\ldots&\ldots&{}\hfil\\\
{{\gamma_{M1}}}&{{\gamma_{M2}}}&\ldots&{{\gamma_{MN}}}\end{array}}\right).$
(7)

${\mathbf{G}_{M\times N}}$ is row full rank [21].

Before being forwarded to the destination, ${W_{j}}$ ($\forall
j\in{{{\Phi_{n}}}}$) is first protected by channel coding and then modulated
into a unit-power signal, denoted as $X({W_{j}})$. Note that with network
coding, we have $|{W_{j}}|=|{S}|$. Correspondingly, every transmission in the
second hop also takes $T$ seconds.

At the destination, the signal from $R_{j}$ is received as

$F({W_{j}})={g_{j}}\sqrt{{{p^{\prime}_{j}}}}X({W_{j}})+z_{j},\forall{j}\in{\psi_{\tau}},$
(8)

where ${{{p^{\prime}_{j}}}}$ is the transmitting power at relay $R_{j}$;
$z_{j}\sim{\mathcal{N}}(0,{N_{0,j}}B)$ is the noise term; ${N_{0,j}}$ is the
power spectral density of noise and ${g_{j}}$ is the channel coefficient.

Finally, the destination obtains ${S_{1}}$, ${S_{2}}$, $\cdots$, ${S_{M}}$
jointly from $\\{F({W_{j}}),j\in{\psi_{\tau}}\\}$ by network decoding. In
effect, the row full rank property of ${\mathbf{G}_{M\times N}}$ guarantees
that $\\{{S_{1}},{S_{2}},\cdots,{S_{M}}\\}$ can be recovered at the
destination as long as

$\tau\geq M,$ (9)

otherwise, none of the user messages can be obtained and we claim an outage
event happens for all user message transmissions.

A transmission period is defined as the duration in which all the $M$ users
complete one cycle broadcasting in the TDMA scheme, which lasts $MT$ seconds.
In total, $K$ consecutive transmission periods are considered. Let $k$
($k=0,1,\cdots,K$) represent the index of the transmission period. Take
$U_{1}$ as an example, as illustrated in Fig. 2, the $k$th transmission period
corresponds to $t\in[(k-1)MT,kMT)$. The second hop in the $k$th transmission
period can simultaneously proceed with the first hop of the $(k+1)$th
transmission period. Thus, the duration for one cycle “first hop + second hop”
transmission is equivalently calculated as $MT$ seconds.

###  II-C Energy Harvesting and Cooperation Model

As shown in Fig. 2, in any transmission period, the incoming energy at one
user is either harvested externally or obtained from other users via wireless
energy transfer (e.g., in the $2$nd transmission period, $U_{1}$ obtains
energy from other users). Correspondingly, the user can either consume energy
for the data transmission or transfer energy to the other users (e.g., in the
$K$th transmission period, $U_{1}$ transfers energy to other users).

Figure 2: Energy harvesting and transfer models for $U_{1}$. The thin arrow
marks that energy is harvested from the external environment while “$\times$”
means no harvested energy. The bold and upward arrow records that energy is
transferred to other users during the corresponding transmission period, while
the bold and downward arrow represents that energy is provided by other users.

In our energy harvesting model, we consider the off-line policy [15], in the
sense that the harvested energy amount and harvesting time are known (or can
be precisely predicted) to all nodes in advance. We assume that $U_{i}$
($\forall i$) harvests $Eu_{i,k}$ ($\forall k$) joules of energy from the
external environment when $t\in[(k-2)MT,(k-1)MT)$. Obviously, $Eu_{i,k}$ can
be consumed in the $k$th and later transmission periods. Note the users do not
necessarily harvest energy at the same time. That is, $Eu_{i,k}$ ($\forall i$,
$\forall k$) can be zero.

We further describe the energy transfer morel. Suppose that in the $k$th
transmission period, $U_{i}$ transfers ${{E_{i\to i^{\prime},k}}}$ joules
energy to ${U_{i^{\prime}}}$. We define an energy transfer matrix
${\mathbf{E}_{I\to I^{\prime},k}}$ ($\forall k$), as

${\mathbf{E}_{I\to I^{\prime},k}}=\left({\begin{array}[]{*{20}{c}}0&{{E_{1\to
2,k}}}&\ldots&{{E_{1\to M,k}}}\\\ {{E_{2\to 1,k}}}&0&\ldots&{{E_{2\to
M,k}}}\\\ {}\hfil&\ldots&\ldots&{}\hfil\\\ {{E_{1\to M,k}}}&{{E_{2\to
M,k}}}&\ldots&0\end{array}}\right).\vspace{+0.7em}$

Note that all the diagonal elements of ${\mathbf{E}_{I\to I^{\prime},k}}$ are
zero, since one user does not transfer energy to itself. The set of
${\mathbf{E}_{I\to I^{\prime},k}}$ ($\forall k$) is denoted as
${\mathbf{E}_{I\to I^{\prime}}}$.

The wireless energy transmission efficiency is denoted as $\eta<1$. Then,
$U_{i}$ receives $\eta\sum_{i^{\prime}=1}^{M}{E_{i^{\prime}\to i,{k}}}$ joules
energy from the other cooperative source nodes during the $k$th transmission
period. Correspondingly, $U_{i}$ totally transfers
$\sum_{i^{\prime}=1}^{M}{E_{i\to i^{\prime},{k}}}$ joules energy to other
users. We claim that either $\eta\sum_{i^{\prime}=1}^{M}{E_{i^{\prime}\to
i,{k}}}$ or $\sum_{i^{\prime}=1}^{M}{E_{i\to i^{\prime},{k}}}$ must be zero.
This is because if one user simultaneously obtains energy from other users and
transfers energy to other users, unnecessary energy loss will happen due to
the wireless energy transmission inefficiency.

The energy evolution process is depicted as below. Let ${Eu_{i,k}^{ava}}$
denote the available energy for _data transmission_ in the $k$th transmission
period. The available energy for the $1$st transmission period is described as

$\displaystyle
Eu_{i,1}^{ava}=E{u_{i,0}}+E{u_{i,1}}+\eta\sum\limits_{i^{\prime}=1}^{M}{E_{i^{\prime}\to
i,1}}-\sum\limits_{i^{\prime}=1}^{M}{E_{i\to i^{\prime},1}},\forall i,$ (10)

where $Eu_{i,0}$ stands for the initial energy storage, which is set as $0$
without loss of generality; $E{u_{i,1}}$ is the energy harvested externally
before the first transmission period.

For $k\geq 2$, ${Eu_{i,k}^{ava}}$ evolves as

$\displaystyle Eu_{i,k}^{ava}=$ $\displaystyle
Eu_{i,k-1}^{ava}-p_{i,{k-1}}T+E{u_{i,k}}\hfill$
$\displaystyle+\eta\sum\limits_{i^{\prime}=1}^{M}{E_{i^{\prime}\to
i,{k}}}-\sum\limits_{i^{\prime}=1}^{M}{E_{i\to i^{\prime},{k}}},\hfill$ (11)

where $\eta\sum_{i^{\prime}=1}^{M}{E_{i^{\prime}\to i,{k}}}$ is the possibly
obtained energy from the other cooperative sources during the $k$th
transmission period while $\sum_{i^{\prime}=1}^{M}{E_{i\to i^{\prime},{k}}}$
indicates the energy transferred to other users and $p_{i,{k-1}}$ denotes the
data transmission power during the $(k-1)$th transmission period,
respectively.

Without loss of generality, we assume that ${d_{ij}}$,
$\Omega_{{h_{ij}}}^{2}$, ${\beta_{ij}}$ and $N_{0,ij}$ and their corresponding
parameters in the second hop are the same in all $K$ transmission periods.
Thus, we can drop the index $k$ in the above mentioned parameters to ease the
following notations.

##  III EE Maximization Problem

In this section, we respectively formulate the expressions for the EE, total
consumed energy and outage probability. Following that, the EE maximization
problem is finally presented.

###  III-A Energy Efficiency

EE is evaluated as the expected number of successfully transmitted information
bits, $\mathbb{E}[L]$, divided by the total consumed energy ${E_{tot}}$, i.e.,

$U_{EE}=\frac{\mathbb{E}[L]}{{{E_{tot}}}}.$ (12)

As we have illustrated in (9), the network decoder at the destination either
recovers all source messages or cannot decode any of them, the outage
probabilities of all users are the same. Let ${{\Pr}_{out,k}}$ denote the
outage probability for all the users in the $k$th transmission period. Thus,
over $K$ consecutive transmission periods, $\mathbb{E}[L]$ can be expressed as
[20]

$\mathbb{E}[L]=\sum\limits_{k=1}^{K}{M{{\alpha_{0}}T{(1-{{\Pr}_{out,k}})}}}.$
(13)

In the following, we give the expressions for ${{\Pr}_{out,k}}$,
${{{E_{tot}}}}$ and their corresponding constraints in different transmission
periods, respectively.

###  III-B Total Consumed Energy

The total consumed energy includes the energy used for data transmission and
wasted during the energy cooperation, which is denoted by

${E_{tot}}=\sum\limits_{k=1}^{K}\left({\sum\limits_{i=1}^{M}{{p_{i,k}T}+(1-\eta)\sum\limits_{i=1}^{M}{\sum\limits_{i^{\prime}=1}^{M}{{E_{i\to
i^{\prime},k}}}+}\sum\limits_{j=1}^{N}{p^{\prime}_{j,k}}T}}\right),$ (14)

where the term $(1-\eta)\sum_{i=1}^{M}(\cdot)$ in (14) represents the overall
energy loss incurred by the wireless energy transmission inefficiency during
the wireless energy transfer.

In our model, an infinite-sized battery capacity at the user is assumed, which
has also been adopted in [12], [13]. Specifically, a super-capacitor can be
implied to store the incoming energy. Since the energy that has not yet
arrived cannot be consumed ahead of time due to the energy causality, it is
required that the consumed energy amount cannot exceed the available amount.
Correspondingly, we formulate the power control constraints as

$\displaystyle 0<p_{i,k}T\leq{Eu_{i,k}^{ava}},\forall i,k,\hfill$ (15)
$\displaystyle 0<p_{i,k}\leq p_{max},\forall i,k,\hfill$ (16) $\displaystyle
0\leq p^{\prime}_{j,k}\leq p_{max},\forall j,k,$ (17)

where $p_{max}$ is the maximum transmitting power.

###  III-C Outage Probability

We first give the exact and approximated outage probability expression for one
individual channel, based on which the outage probability of the whole network
is derived.

1) Outage Probability of One Individual Channel

For the Nakagami-$m$ fading channel, take the $U_{i}$-$R_{j}$ channel as an
example, the outage probability can be calculated according to (5) and (6).
Specifically, we have

$\displaystyle{{\Pr}_{e,{ij,k}}}$
$\displaystyle={\Pr}\\{C_{ij}<{\alpha_{0}}\\}={\Pr}\\{|{h_{ij}|^{2}}<\frac{({2^{{{\alpha_{0}}\mathord{\left/{\vphantom{RB}}\right.\kern-1.2pt}B}}}-1){N_{0,ij}}B}{p_{i}}\\}\hfill$
$\displaystyle=1-\int_{\frac{({2^{{{\alpha_{0}}\mathord{\left/{\vphantom{RB}}\right.\kern-1.2pt}B}}}-1){N_{0,ij}}B}{p_{i}}}^{+\infty}f(x_{ij}){d{x_{ij}}}\hfill$
$\displaystyle=\Gamma\left({m,\frac{{m(2^{\alpha_{0}/B}-1)N_{0,ij}B}}{{d_{ij}^{-\beta_{ij}}\Omega_{h_{ij}}p_{i,k}}}}\right)\Gamma(m)^{-1},$
(18)

where $\Gamma(a,b)=\int_{0}^{b}{x^{a-1}}\exp(-x)dx$ is the upper incomplete
gamma function and $\Gamma(a)=\int_{0}^{+\infty}{x^{a-1}}\exp(-x)dx$ is the
complete gamma function.

The outage probability of one individual channel in (18) is not tractable
mathematically since $p_{i,k}$ is not isolated but contained in the gamma
function. However, the incomplete gamma function can be well approximated as
[25]

$\Gamma(a,b)\approx(1/a)b^{a}$

for small $b$. This approximation offers one method in isolating $p_{i,k}$
from the gamma function. Specifically, (18) can be approximated as

$\displaystyle{{\Pr}_{e,{ij,k}}}$
$\displaystyle\approx\left({\frac{{m(2^{\alpha_{0}/B}-1)N_{0,ij}B}}{{d_{ij}^{-\beta_{ij}}\Omega_{h_{ij}}p_{i,k}}}}\right)^{m}\Gamma(m+1)^{-1}$
$\displaystyle=c_{ij}p_{i,k}^{-m},$ (19)

where

$c_{ij}=\left({\frac{{m(2^{\alpha_{0}/B}-1)N_{0,ij}B}}{{d_{ij}^{-\beta_{ij}}\Omega_{h_{ij}}}}}\right)^{m}\Gamma(m+1)^{-1}>0.$
(20)

Similarly, the outage probability of the $R_{j}$-D channel can be given as

${{\Pr}_{e,j,k}}\approx c_{j}{(p^{\prime}_{j,k})}^{-m},$ (21)

where

$c_{j}=\left({\frac{{m(2^{\alpha_{0}/B}-1)N_{0,j}B}}{{d_{j}^{-\beta_{j}}\Omega_{g_{j}}}}}\right)^{m}\Gamma(m+1)^{-1}>0.$
(22)

As can be seen from (20) and (22), $c_{ij}$ and $c_{j}$ combine all the
channel paramenters. Increasing $c_{ij}$ or $c_{j}$ will lead to larger outage
probability of one indiviual channel. Thus, larger $c_{ij}$ and $c_{j}$
represent worse channel conditions.

2) Outage Probability of the Whole Network

As we illustrated in (9), an outage event happens when $\tau\leq M$. In the
following, we focus on deriving the probability that $\tau\leq M$.

Suppose in the $k$th transmission period, $n$ relays succeed in receiving all
the source messages. An outage event happens in the following two cases in
terms of $n$. In case ${A_{k}}$, $n<M$. User messages cannot be recovered no
matter how the second hop proceeds. In case $B_{k}$, $n\geq M$. An outage
event happens when the number of relays forwarding the codewords to the BS in
the second hop is smaller than $M$.

We denote the probability that case ${A_{k}}$ and ${B_{k}}$ repectively
happening as $\Pr\\{A_{k}\\}$ and $\Pr\\{B_{k}\\}$. Since cases ${A_{k}}$ and
${B_{k}}$ are independent, then the outage probability for the whole network
can be calculated as

${\Pr}_{out,k}=\Pr\\{A_{k}\\}+\Pr\\{B_{k}\\}.$ (23)

We have respectively formulated $\Pr\\{A_{k}\\}$ and $\Pr\\{B_{k}\\}$ as (24)
and (25) in [20], where the nodes are not capable of harvesting or
transferring energy and the channels follow Rayleigh fading.

$\Pr\\{A_{k}\\}=\sum\limits_{n=0}^{M-1}\sum\nolimits_{{\Phi_{n,k}}}{\left({\prod\limits_{{j}\in{\Phi_{n,k}}}{{\rho_{j,k}}}{\prod_{{j}\in{\Theta}\backslash{\Phi_{n,k}}}}(1-{\rho_{j,k}})}\right)}.$
(24)

$\displaystyle\Pr\\{B_{k}\\}=\sum\limits_{n=M}^{N}\left({\sum\nolimits_{{\Phi_{n,k}}}{({\prod\limits_{{j}\in{\Phi_{n,k}}}{{\rho_{j,k}}}{\prod_{{j}\in{\Theta}\backslash{\Phi_{n,k}}}}(1-{\rho_{j,k}})})}\cdot\sum\limits_{\tau=0}^{M-1}\sum\nolimits_{{\psi_{\tau,k}}}{({\prod\limits_{{j}\in{\psi_{\tau,k}}}{(1-{{\Pr}_{e,j,k}})}{\prod_{{j}\in{\Phi_{n,k}}\backslash{\psi_{\tau,k}}}}{{\Pr}_{e,j,k}}})}}\right).$
(25)

We note that $\sum\nolimits_{{\Phi_{n,k}}}{(\beta)}$ in (24) and (25)
represents the sum of $\beta$ when ${{\Phi_{n,k}}}$ is in different cases.
${{\Phi_{n,k}}}$ consists of $n$ relays randomly chosen from $N$ relays in the
$k$th transmission period, including $C_{N}^{n}$ cases. ${{\psi_{\tau,k}}}$
consists of $\tau$ relays randomly chosen from $n$ relays in the $k$th
transmission period, including $C_{n}^{\tau}$ cases. In (23) and (24),
${\rho_{j,k}}$ measures the probability that $R_{j}$ manages to receive all
the $M$ user messages in the $k$th transmission period. It is evaluated by
[20]

${\rho_{j,k}}=\prod\limits_{i=1}^{M}{(1-{{\Pr}_{e,ijk}})}.$ (26)

For the energy harvesting and cooperation scenario, where the channels follow
Nakagami-$m$ fading distributions, we can obtain ${{\Pr}_{out,k}}$ by
substituting ${\Pr}_{e,ij,k}$ and ${\Pr}_{e,j,k}$ into (23)-(25).

Our objective is to maximize the EE across $K$ transmission periods by jointly
optimizing $p_{i,k}$, $p^{\prime}_{j,k}$ and ${{E_{i\to i^{\prime},k}}}$
($\forall i,j,k$) according to the harvested energy and the channel
parameters, including $Eu_{i,k}$, $d_{ij}$, $\beta_{ij}$, $\Omega_{{h_{ij}}}$,
$d_{j}$, $\beta_{j}$ and $\Omega_{{g_{j}}}$ ($\forall i,j,k$). The
optimization problem can be formulated as

$\displaystyle\mathbf{P1}:\mathop{\max}\;U_{EE}\hfill$ $\displaystyle
s.t.\;\;\eqref{const_ps}-\eqref{const_pr},\hfill$
$\displaystyle\quad\;\;{{\Pr}_{out,k}}\leq{{\Pr}_{out,0}},\forall k,$ (27)

where ${{\Pr}_{out,0}}$ is the predefined outage probability threshold for
every transmission period. We note that the outage probability threshold may
vary in different transmission periods. To ease the notations, we set them as
the same value, denoted as ${{\Pr}_{out,0}}$.

##  IV Problem Transformation and Solving

The key challengings in solving $\mathbf{P1}$ stem from the following facts.

Although $\Pr_{out}$ in $\mathbf{P1}$ represents the exact outage probability,
it consists of multiple exponential items. Note the coefficients of
exponential items are positive and negative constants that alternately appear.
This makes the outage probability constraint and the objective function in
$\mathbf{P1}$ neither in their concave nor convex forms.

On the other hand, the widely adopted Lagrange duality method [26] in [5],
[6], [8], [10], [12]-[16] is not applicable in our network coding scenario
since the product forms of $p_{i,k}$ ($\forall i,k$) and $p^{\prime}_{j,k}$
($\forall j$) make the equations obtained via KKT conditions very complicated
to be solved. Hereby, the closed-form solutions are hard to achieve. The
Brute-force algorithm is also infeasible even for small $M$ and $N$. That is
because for the network with $M$ users and $N$ relays, there are
$Num_{Var}=(M-1)(M-1)K+MK+NK$ variables to be determined in total, including
$(M-1)(M-1)K$ energy transferring variables, $MK$ power allocation variables
of $M$ users and $NK$ power allocation variables of $N$ relays.

In the sequel, we exploit the relaxation and approximation methods, which
alleviate the optimization difficulties substantially. First, the outage
probability is converted into its geometric programming form. Then, we covert
the objective function and energy causality constraints into their convex
forms, thereby finally converting the primal optimization problem into a
standard convex one. The details are given in the following.

###  IV-A Transformation of the Outage Probability

In the case of low outage probability threshold ${{\Pr}_{out,0}}$, both
${{c_{ij}}/p_{i,k}^{m}}$ and $c_{j}/(p^{\prime}_{j,k})^{m}$ are required to be
small according to (23)-(25). The outage probability constraint can be
satisfied if the transmitting power is allocated appropriately and $c_{ij}$
and $c_{j}$ (specifically, $\alpha_{0}/B$ and noise power) are small. With
small values of ${{c_{ij}}/p_{i,k}^{m}}$ and $c_{j}/(p^{\prime}_{j,k})^{m}$,
we can derive the following approximations

$1-{\Pr}_{e,ij,k}=1-c_{ij}p_{i,k}^{-m}\approx\exp(-c_{ij}p_{i,k}^{-m})\sim 1,$
(28)

and

${\rho_{j,k}}\mathop{\sim}\limits^{\eqref{ro}}1.$ (29)

Moreover, since $\mathop{\lim}\limits_{x\to 0}\;1-\exp(-x)=x$, we have

$\displaystyle 1-\rho_{j,k}$
$\displaystyle=1-\prod\limits_{i=1}^{M}{(1-{\Pr}_{e,ij,k})}$
$\displaystyle\mathop{=}\limits^{\eqref{1_Pr}}1-\prod\limits_{i=1}^{M}{\exp(-c_{ij}p_{i,k}^{-m})}$
$\displaystyle\approx\sum\limits_{i=1}^{M}{c_{ij}p_{i,k}^{-m}}.\vspace{+0.7em}$
(30)

By substituting (28)-(30) into (24) and (25), we obtain the tight
approximations for $\Pr\\{A_{k}\\}$ and $\Pr\\{B_{k}\\}$, which are
repectively given in (31) and (32).

$\Pr\\{A_{k}\\}\approx\sum\limits_{n=0}^{M-1}{\sum\nolimits_{{\Phi_{n,k}}}{\left({\prod\limits_{{j}\in{\Theta}\backslash{\Phi_{n,k}}}{\sum\limits_{i=1}^{M}{\frac{{{c_{ij}}}}{{{p_{i,k}^{m}}}}}}}\right)}}$
(31)

$\displaystyle\Pr\\{B_{k}\\}\approx\sum\limits_{n=M}^{N}{\left({(\sum\nolimits_{{\Phi_{n,k}}}{(\prod\limits_{{j}\in{\Theta}\backslash{\Phi_{n,k}}}{(\sum\limits_{i=1}^{M}{\frac{{{c_{ij}}}}{{{p_{i,k}^{m}}}}}})}))\cdot(\sum\limits_{\tau=0}^{M-1}{\sum\nolimits_{{\Phi_{n,k}}}{(\prod\limits_{{j}\in{\Phi_{n,k}}\backslash{\psi_{\tau}}}{\frac{{{c_{j}}}}{(p^{\prime}_{j,k})^{m}}}}))}}\right)}.$
(32)

Furthermore, we introduce two new variables, i.e., ${{\tilde{p}}_{i,k}}$ and
${{\tilde{p}^{\prime}}_{j,k}}$ as below

${p_{i,k}}={e^{{{\tilde{p}}_{i,k}}}},\quad\;{{{p^{\prime}_{j,k}}}}={e^{{{\tilde{p}^{\prime}}_{j,k}}}}.$
(33)

By substituting ${{{\tilde{p}}_{i,k}}}$ and ${{\tilde{p}^{\prime}}_{j,k}}$
into (31) and (32), we have (34) and (35).

$\Pr\\{A_{k}\\}\approx\sum\limits_{n=0}^{M-1}{\sum\nolimits_{{\Phi_{n,k}}}{\left({\prod\limits_{{j}\in{\Theta}\backslash{\Phi_{n,k}}}{\sum\limits_{i=1}^{M}{{c_{ij}}{e^{-m{{\tilde{p}}_{i,k}}}}}}}\right)}}.$
(34)

$\displaystyle\Pr\\{B_{k}\\}\approx\sum\limits_{n=M}^{N}{\left({(\sum\nolimits_{{\Phi_{n,k}}}{(\prod\limits_{{j}\in{\Theta}\backslash{\Phi_{n,k}}}{(\sum\limits_{i=1}^{M}{{{c_{ij}}e^{-m{{\tilde{p}}_{i,k}}}}}})}))\cdot(\sum\limits_{\tau=0}^{M-1}{\sum\nolimits_{{\Phi_{n,k}}}{(\prod\limits_{{j}\in{\Phi_{n,k}}\backslash{\psi_{\tau}}}{{e^{-m{{\tilde{p}^{\prime}}_{j,k}}}}}}))}}\right)}.$
(35)

As can be seen, both (31) and (32) are given in the geometric programming
forms of $p_{i,k}$ and $p^{\prime}_{j,k}$ while (34) and (35) are in the sum-
exponential forms of ${{{\tilde{p}}_{i,k}}}$ and
${{\tilde{p}^{\prime}}_{j,k}}$. With all the coefficients being positive,
${{\Pr}_{out,k}}$ is finally approximated to its convex form. Then the
objective function can be dealt with parametric method based on the fractional
programming theory.

###  IV-B Transformation of the Objective Function

In the sequel, we apply Dinkelbach’s method [27] to transform the fractional
problem into its subtractive form. The following proposition is provided.

######  Proposition 1.

The PA and energy cooperation policies can achieve the maximum energy
efficiency

$q^{*}=\max\\{{U_{EE}}\\},$

if and only if

$\displaystyle V(q^{*},\mathcal{\tilde{P}}^{*},{\mathbf{E}_{I\to
I^{\prime}}^{*}})$ $\displaystyle=\max_{q\geq
0}\\{{M{\alpha_{0}}T\sum\limits_{k=1}^{K}(1-{{\Pr}_{out,k}})}-{q}{E_{tot}})\\}$
$\displaystyle=0,$ (36)

where $q^{*}$ is the maximum EE, ${\mathbf{E}_{I\to I^{\prime}}^{*}}$ is the
optimal energy cooperation policy, $\mathcal{P}^{*}$ is the set of optimum
solutions of ${p_{i,k}}$, ${p^{\prime}_{j,k}}$ ($\forall i,j,k$), while
$\mathcal{\tilde{P}}^{*}$ is the set of optimum solutions of
${{\tilde{p}}_{i,k}}$, ${{\tilde{p}^{\prime}}_{j,k}}$ ($\forall i,j,k$).

According to Proposition 1, we reformulate $\mathbf{P1}$ as $\mathbf{P2}$.

$\displaystyle\mathbf{P2}:\max
V={{M{\alpha_{0}}T\sum\limits_{k=1}^{K}(1-{{\Pr}_{out,k}})}}-{q}{E_{tot}}$
(37) $\displaystyle s.t.\;\;\eqref{Poutconstr},\hfill$
$\displaystyle\quad\;\;0<p_{i,k}T=e^{{{\tilde{p}}_{i,k}}}T\leq{Eu_{i,k}^{ava}},\forall
i,k,\hfill$ (38) $\displaystyle\quad\;\;0<p_{i,k}=e^{{{\tilde{p}}_{i,k}}}\leq
p_{max},\forall i,k,\hfill$ (39)
$\displaystyle\quad\;\;0\leq{{p^{\prime}_{j,k}}}={e^{{{\tilde{p}^{\prime}}_{j,k}}}}\leq
p_{max},\forall j,k.$ (40)

We focus on finding $q^{*}$, $\mathcal{\tilde{P}}^{*}$ and ${\mathbf{E}_{I\to
I^{\prime}}^{*}}$ to satisfy

$\max\\{V(q^{*},\mathcal{\tilde{P}}^{*},{\mathbf{E}_{I\to
I^{\prime}}^{*}})\\}=0.$ (41)

In Dinkelbach’s method, $q$ is iteratively updated in every iteration;
meanwhile, $V(q^{*},\mathcal{\tilde{P}}^{*},{\mathbf{E}_{I\to
I^{\prime}}^{*}})$ is judged whether it converges to a given tolerance. If
not, $q$ is updated and we repeat the maximization problem until it converges
or reaches the maximal iterations.

Note that with given $q$ in every iteration, we have

$\max\;V\Leftrightarrow\min\;V^{\prime}=-V+MK{\alpha_{0}}{T},$ (42)

Correspondingly, $\mathbf{P2}$ is equivalently transformed into $\mathbf{P3}$
as shown below,

$\displaystyle\mathbf{P3}:\min\;V^{\prime}(q^{*},\mathcal{\tilde{P}}^{*},{\mathbf{E}_{I\to
I^{\prime}}^{*}})$ $\displaystyle
s.t.\quad\eqref{Poutconstr},\eqref{eq63eu}-\eqref{eq631a}.$

According to (36) and (42), the optimum solution of $\mathbf{P3}$ must satisfy

$\min\\{V^{\prime}(q^{*},\mathcal{\tilde{P}}^{*},{\mathbf{E}_{I\to
I^{\prime}}^{*}})\\}=MK{\alpha_{0}}T.$ (43)

We first provide a proposition for $\mathbf{P3}$.

######  Proposition 2.

Given $q$, $\mathbf{P3}$ is jointly convex with respect to (_w.r.t_)
${{{\tilde{p}}_{i,k}}}$, ${{{\tilde{p}^{\prime}}_{j,k}}}$ and ${E_{i\to
i^{\prime},k}}$, $\forall i,i^{\prime},j,k$. Efficient interior-point method
can be applied to obtain its optimum solution.

###### Proof.

Proof is provided in the Appendix. ∎

We summarize the overall procedure to solve $\mathbf{P1}$ in Algorithm $1$.

_Complexity Analysis:_ With the Brute-force algorithm, the complexity is
$\mathcal{O}(\upsilon^{Num_{Var}})$, where $\upsilon$ is the iteration time
for one variable and determined by the step size. As we can see, the
complexity with the Brute-force algorithm increases exponentially with $M$,
$N$ and $K$. In contrast, with Dinkebach’s method, the iteration time for $q$
is limited [27]. Furthermore, with the interior-point method applied, the
complexity will be $\mathcal{O}(\mathcal{C}_{1}\mathcal{C}_{2})$, where
$\mathcal{C}_{1}={(Num_{Var}+Z+1)^{1/2}}$,
$\mathcal{C}_{2}=(Num_{Var}+1){{Z}^{2}}+{{Z}^{3}}+{Num_{Var}^{3}}$ and $Z$ is
the total number of exponential terms in the objective and constraints [29].
It can be found that $\mathcal{O}(\mathcal{C}_{1}\mathcal{C}_{2})$ is a
polynomial in $M$, $N$, and $K$.

_Model Extension:_ Our energy harvesting and transferring cooperative networks
model can also be extended to more general scenarios. For example, relays may
also be capable of harvesting energy externally and transferring energy to
users or other relays. In this case, when formulate the EE maximization
problem, we can regard the relays as users and let $U_{M+j}$ represent
$R_{j}$. In other words, $U_{i}$ denotes one relay rather than an user if
$i\in\\{M+1,M+2,\cdots,M+N\\}$. Specifically, the following two minor changes
are needed. First, we rewrite (17), i.e., the power constraint at relay
$R_{j}$ as

$\displaystyle 0<p_{i,k}T\leq\min\\{{Eu_{i,k}^{ava}},p_{max}T\\},\hfill$ (44)
$\displaystyle\quad\quad\quad i\in\\{M+1,M+2,\cdots,M+N\\},\forall k,\hfill$

where $p_{M+j,k}$ and ${Eu_{M+j,k}^{ava}}$ are the transmitting power and
available energy at $R_{j}$ in the $k$th transmission period, respectively.
Second, ${Eu_{i,k}^{ava}}$ ($i\in\\{1,2,\cdots,M+N\\}$) is not only related
with the incoming/outgoing energy transferred from/to other users but also the
relays. Then the expressions for available energy at sources and relays can be
simply obtained by changing “$M$” in (10) and (11) into “$M+N$”. Algorithm $1$
is still feasible for the extended scenarios. On the other hand, if a part of
users are not energy harvesting nodes or cannot transfer energy to other
users/relays, then we only need to delete the energy causality constraint for
them. Algorithm $1$ is also applicable to such kind of scenarios.

##  V Numerical Results

In what follows, we will present numerical results. Energy arrivals are
generated randomly and independently. Their specific values are shown in Fig.
3. We simulate the process for $10$ consecutive transmission periods. We
assume that $M=2$, $N=4$, $B=125K$Hz, $\alpha_{0}=10^{5}$ bits per second and
$p_{max}=20$ watts. We normalize $T$ as $1$. The following randomly generated
values are also assumed,

$\displaystyle{{\mathbf{\Omega}}_{\mathbf{h}}}=\left[{\begin{array}[]{*{20}{c}}{2.5646}&{1.7520}&{2.1684}&{0.5798}\\\
{2.3024}&{0.4753}&{3.5462}&{0.3904}\end{array}}\right],\hfill$ (47)
$\displaystyle{{\mathbf{\Omega}}_{\mathbf{g}}}=\left[{\begin{array}[]{*{20}{c}}{3.3120}&{1.1286}&{0.3284}&{0.7821}\end{array}}\right],\hfill$
(49)
$\displaystyle{{\mathbf{d}}_{\mathbf{h}}}=\left[{\begin{array}[]{*{20}{c}}{1565}&{765.2}&{1704.2}&{1530.4}\\\
{1979.6}&{1831}&{720.6}&{173.6}\end{array}}\right],\hfill$ (52)
$\displaystyle{{\mathbf{\beta}}_{\mathbf{h}}}=\left[{\begin{array}[]{*{20}{c}}{2.5570}&{2.9150}&{2.3152}&{3.0143}\\\
{3.0938}&{2.1298}&{2.6412}&{2.9708}\end{array}}\right],\hfill$ (55)
$\displaystyle{{\mathbf{d}}_{\mathbf{g}}}=\left[{\begin{array}[]{*{20}{c}}{471.7}&{1045.7}&{902.2}&{1079.4}\end{array}}\right],\hfill$
(57)
$\displaystyle{{\mathbf{\beta}}_{\mathbf{g}}}=\left[{\begin{array}[]{*{20}{c}}{2.6103}&{3.2838}&{1.8435}&{2.3515}\end{array}}\right],\hfill$
(59)
$\displaystyle{{\mathbf{N}}_{{\mathbf{0}},{\mathbf{h}}}}=10^{-15}\left[{\begin{array}[]{*{20}{c}}{0.126}&{0.07}&{0.54}&{0.006}\\\
{0.002}&{0.002}&{0.429}&{1.096}\end{array}}\right],\hfill$ (62)
$\displaystyle{{\mathbf{N}}_{{\mathbf{0}},{\mathbf{g}}}}=10^{-15}\left[{\begin{array}[]{*{20}{c}}{0.3799}&{0.7243}&{0.0265}&{0.1225}\end{array}}\right].\hfill$
(64)

Note that ${{\mathbf{\Omega}}_{\mathbf{h}}}$ and
${{\mathbf{\Omega}}_{\mathbf{g}}}$ denote the variance matrices of the average
channel gain; ${\mathbf{{N_{0,h}}}}$ and ${\mathbf{{N_{0,g}}}}$ represent the
power spectrum density matrices, which are measured in Watts/Hz.
${\mathbf{{d_{h}}}}$ and ${\mathbf{{d_{g}}}}$ represent the distance matrices
which are measured in meter. ${\mathbf{{\beta_{h}}}}$ and
${\mathbf{{\beta_{g}}}}$ represent the path-loss exponent matrices.
Specifically, elements at the $i$th row and the $j$th column of
${\mathbf{{\Omega_{h}}}}$, ${\mathbf{{d_{h}}}}$, ${\mathbf{{\beta_{h}}}}$ and
${\mathbf{{N_{0,h}}}}$ correspond to the parameters for the $S_{i}-R_{j}$
channel. Element at the $j$th columns in ${\mathbf{{\Omega_{g}}}}$,
$\mathbf{{d_{g}}}$, ${\mathbf{{n_{g}}}}$ and ${\mathbf{{N_{0,g}}}}$
corresponds to the parameters for the $R_{j}$-destination channel.

###  V-A Optimal Policy Illustration

We first take the scenario when $m=1$ (i.e., Rayleigh fading channel),
$\eta=0.6$ and $\Pr_{out,0}=6\times 10^{-7}$ as example. The cumulative
harvested energy and the optimal power policies are depicted in Fig. 3.

For the cumulative harvested energy curves, the rising height at the beginning
of every transmission period represents the amount of newly harvested energy
that can be consumed in that period. Zero-rising height implies that no energy
is harvested. From Fig. 3, we observe that $U_{2}$ harvests sufficient energy
from the external environment while $U_{1}$ harvests small amount of energy
and suffers from energy deficiency. Especially in the $2$nd and $3$rd
transmission periods, as can be seen, no energy is harvested at $U_{1}$.

Figure 3: Optimal policy and cumulative harvested energy; $\eta=0.6$; $m=1$.

To compensate for $U_{1}$ such that NC cooperative transmission can be carried
out, the energy cooperation policy is adopted. Fig. 3 shows that $U_{2}$
respectively transfers $1.8240$, $1.0642$ and $0.08$ joules energy to $U_{1}$
during the $2$nd, $3$rd and $4$th transmission periods. Take the second
transmission period as an example. Since $U_{2}$ transfers energy to $U_{1}$
during the $2$nd transmission period, the available energy at $U_{2}$ is
smaller than the cumulative harvested amount. In contrast, due to the
additional incoming energy from $U_{2}$, the available energy at $U_{1}$
exceeds the cumulative harvested amount.

For the optimal policy curve, the slope of one line segment represents the
transmitting power in the corresponding transmission period. Zero-slope
represents that no energy is consumed and no transmission proceeds. We can
observe that due to the TDMA transmission scheme, the zero-slope line segments
in the optimal policy curves of $U_{1}$ and $U_{2}$ alternately appear.
Moreover, as can be seen, the optimal power policy curves of $U_{1}$ and
$U_{2}$ in Fig. 3 are not higher than the available energy curves due to the
energy causality constraint.

Additionally, in some transmission periods, the optimal consumed energy amount
is not necessarily the same with the available amount. In other words, the
energy is not depleted and some is saved and will be consumed later for the
sake of maximizing the EE. Take $U_{1}$ as an example, the available energies
are not used up till the end of the $3$rd and $9$th transmission periods.
Similar conclusions can be obtained for $U_{2}$. Specially, though no
harvested energy or cooperation energy from $U_{1}$ in the $10$th transmission
period, data transmission still proceeds at $U_{2}$ and the outage probability
threshold is satisfied, which benefits from its cumulative harvested energy in
the prior transmission periods, which is not achievable with the policy in
[9].

###  V-B Impacts of the Relay Locations

To investigate the impact of relay locations on the EE, we fix the distance
between the users and destinations but move the relays between the users and
the destination. To be specific, the distance between $U_{i}$ ($\forall i$),
and $R_{j}$ ($\forall j$), is changed into $({d_{{ij}}}+\Delta)$ meters while
the distance between $R_{j}$ and the destination is reduced into
$({d_{j}}-\Delta)$ meters, where $\Delta$ is the shifting distance of one
relay.

Figure 4: Effects of relay locations; $\eta=0.6$;
$\Pr_{out,0}=[10^{-4},5\times 10^{-5},10^{-5},10^{-6},6\times 10^{-7}]$;
$m=1$.

In Fig. 4 and Fig. 5, the tradeoff curves between ${U_{EE}}$ and the
predefined outage threshold, $\Pr_{out,0}$ for $\Delta=0,150,300$ cases are
depicted. The channels are assumed to be either LOS (e.g., $m=3$) or NLOS
(e.g., $m=1$) ones. The numerical results are obtained by Algorithm $1$, the
Brute-force algorithm and simulations, respectively. The simulation results
are obtained by respectively averaging the outage probability, total consumed
energy and EE over $10^{9}$ random realizations of the fading channels. As can
be seen, for both NLOS and LOS channel scnearios, their analytical results
obtained from Algorithm $1$ closely match the results from the Brute-force
algorithm and simulation, especially in the low $\Pr_{out,0}$ region where
higher SNR is needed. All these show that the analytical results obtained by
Algorithm $1$ are valid.

We can also observe that EE decreases with the pre-defined outage probability
threshold, which implies that the decrease in the outage probability threshold
can cost significant EE penalty.

Figure 5: Impacts of relay locations; $\eta=0.6$;
$\Pr_{out,0}=[10^{-4},5\times 10^{-5},10^{-5},10^{-6},6\times 10^{-7}]$;
$m=3$.

In Fig. 4, the gaps among the tradeoff curves demonstrate the EE loss
resulting from increasing $\Delta$ in the NLOS channel environment. It can be
noticed that $12\%$ and $25\%$ EE losses are respectively generated for the
cases when $\Delta=150$ and $300$ meters. $\Pr_{out,0}=6\times 10^{-7}$ is
even not achievable when $\Delta=300$. Similar conclusions can also be
obtained for the LOS scenario, as shown in Fig. 5. It is because the first hop
transmission is dominant in the two-hop transmission scheme. The increase in
the transmission distance of the first hop deteriorates the outage probability
performance, which needs more energy in the second hop to compensate and
results in lower EE.

In Fig. 5, we obtian the optimum EE when $\eta=0.6$ in the LOS channel
environment (e.g., $m=3$). As can be observed, its EE is around 20 times that
of NLOS scenarios. Additionally, in contrast to the NLOS scenarios, the LOS
results show that, even for the strictest outage probability requirement
(i.e., when $\Pr_{out,0}=6\times 10^{-7}$), no energy cooperation is needed
among the users. Thus, energy loss is avoided during the wireless energy
transferring. This advocates the rationale since if $m$ increases, the
channels become more advantageous for data transmission. In other words, less
power is needed to meet a specific target outage probability level, which is
also clearly revealed in (35). Hereby, the increase of $m$ results in a higher
EE.

###  V-C Impacts of the Energy Transmission Efficiency

In Fig. 6, the EE curves for the scenario when $\Delta=150$ and $\eta=0.2$,
$0.6$ and $1$ are plotted. It is clear that for the same $\Pr_{out,0}$, more
EE losses are caused when $\eta$ takes a smaller value. Moreover,
$\Pr_{out,0}=6\times 10^{-7}$ is not achievable when $\eta=0.2$ due to the
significant energy losses during the energy cooperation. Note that three
curves overlap when $\Pr_{out,0}=1\times 10^{-4}$ due to the fact that no
energy cooperation is needed to satisfy the outage probability threshold. Such
numerical results give references on the system parameter settings.

Figure 6: Impacts of wireless energy transmission efficiency; $\Delta=150$;
$\Pr_{out,0}=[10^{-4},5\times 10^{-5},10^{-5},10^{-6},6\times 10^{-7}]$;
$m=1$.

###  V-D Performance comparison of different transmission schemes

For comparison, in Fig. 7, we provide close-to-optimal results obtained by our
proposed algorithm for the scenario without network coding (NoNC) [28]. In the
NoNC scenario, decode-and-forward (DF) relaying protocol is adopted at $N$
relays. It is shown that the EE of the NC scenario is more than $30\%$ higher
than that of the NoNC scenario, which demonstrates that considerable EE gains
can be achieved with NC.

Figure 7: EE performance comparison for different transmission strategies;
$\eta=0.6$; $\Delta=0$; $m=1$.

Moreover, for the coded scenario, EE obtained with another three transmission
strategies is also provided. First, in the “No Energy Transferring” scheme,
there is no energy cooperation among the users but the energy can be schuduled
over different transmission periods. The EE gap between “No Energy
Transferring” and our proposed algorithm $1$ indicates the EE gain obtained
from the energy cooperation. Specifically, $46.5\%$, $18\%$, $11\%$ and
$3.7\%$ EE gains can be obtained when $\Pr_{out,0}=10^{-6}$, $10^{-5}$,
$5\times 10^{-5}$, $10^{-4}$, respectively. It can be observed that
$\Pr_{out}\leq 6\times 10^{-7}$ is even not achievable if no energy
cooperation. Additionally, as can be seen, with the increase of $\Pr_{out,0}$,
the above EE gains decrease. This is because less energy from other users is
needed if $\Pr_{out,0}$ increases. Correspondingly, the energy cooperation
advantages fade.

For “Depleted Energy” scheme adopted in [9], the harvested energy at every
user is used up within every transmission period. The EE gaps between “No
Energy Transferring” and “Depleted Energy” schemes show the gains from energy
scheduling among different transmission periods. Specifically, around $37\%$,
$153\%$, $360\%$ and $484\%$ EE gains can be obtained when
$\Pr_{out,0}=10^{-6}$, $10^{-5}$, $5\times 10^{-5}$, $10^{-4}$, respectively.
It is clear that the gains increase with ${\Pr}_{out,0}$ since in larger
${\Pr}_{out,0}$ case, less energy is needed for data transmission and more
energy shall be saved. Depleting energy will definitely lead to a lower EE.

On the other hand, in the “Uniform Power policy”, all sources transmit with
the same power obtained by averaging all the harvested energy in $K=10$
transmission periods among the two users. Note for comparison, in the “Uniform
Power policy”, the power at relays refers to the results obtained with our
proposed algorithm $1$. Moreover, the outage probability threshold cannot be
guaranteed. Thus the outage probability requirement is removed. It is shown
that our algorithm outperforms the “Uniform Power policy” scheme.

To conclude, network coding, energy scheduling among different transmission
periods and energy cooperation among different users can provide a notable EE
improvement.

##  VI Conclusions

We have studied the energy harvesting and wireless energy transferring
networks that was coded over finite field. Energy management including
determining the optimal power and energy cooperation policies was conducted
over consecutive transmission periods and under the independent but not
necessarily identically distributed (i.n.i.d.) Nakagami-$m$ channel
environment. The energy efficiency was maximized under the constraints of the
energy causality and outage probability constraints. With the geometric and
fractional programming, the optimization problem was converted into a convex
one. The efficient interior-point method was applied to achieve close-to-
optimal solutions. The gap between our optimal policy and the decode-and-
forward relaying scenario showed the notable energy efficiency gains from the
network coding. Additionally, for the network coding scenario, our suboptimal
policy outperformed the “No Energy Transferring”, “Depleted Energy” and
“Uniform Power” policies. It was shown that the harvested energy was not
necessarily depleted and part of the energy was saved for usage in the later
transmission periods or transferred to its cooperative partners. Finally, it
was revealed that both the increase of the transmission distance in the first
hop and wireless power transmission inefficiency resulted in a degraded energy
efficiency.

## Appendix

We first prove the convexity of the objective function. In every iteration of
$q$,

$\displaystyle V^{\prime}$
$\displaystyle=M{\alpha_{0}}{\sum\limits_{k=1}^{K}}{{\Pr}_{out,k}}+q{E_{tot}}+q{K}T\sum\limits_{j=1}^{N}{{c_{j}}}\hfill$
$\displaystyle=M{\alpha_{0}}{}{\sum\limits_{k=1}^{K}}{{\Pr}_{out,k}}+q\cdot\hfill$
$\displaystyle\sum\limits_{k=1}^{K}{\left({\sum\limits_{i=1}^{M}{{e^{{{\tilde{p}}_{i,k}}}T}+(1-\eta)\sum\limits_{i=1}^{M}\sum\limits_{i^{\prime}=1}^{M}{{E_{i\to
i^{\prime},k}}}+\sum\limits_{j=1}^{N}{{c_{j}}{e^{{{\tilde{p}^{\prime}}_{j,k}}}}T}}}\right)}$
$\displaystyle>0.$ (65)

The first item in (65) is the sum of multiple exponential terms multiplied by
positive constants and thus convex [26]. Meanwhile, the second item is obvious
convex. Then $V^{\prime}$ is convex. The proof of the convexity property of
(27) follows the same approach.

For constraint (39), we separate it into two inequations, i.e.,

$\sum\limits_{l=1}^{k}{{e^{{{\tilde{p}}_{i,l}}}T}}+\sum\limits_{l=1}^{k}{\sum\limits_{i^{\prime}=1}^{M}{{E_{i\to
i^{\prime},l}}}}-\eta\sum\limits_{l=1}^{k}{\sum\limits_{i^{\prime}=1}^{M}{{E_{i^{\prime}\to
i,l}}}}\leqslant\sum\limits_{l=1}^{k}{E{u_{i,l}}},$ (66)

$0<{{e^{{{\tilde{p}}_{i,k}}}}}\leq p_{max}.$ (67)

(66) is convex _w.r.t_ to $\mathcal{P}^{*}$ and ${\mathbf{E}_{I\to
I^{\prime},k}}$ ($\forall k$), due to the fact that the first item in the left
side of (66) is convex and the other items are linear. It is obvious that (40)
and (67) are convex.

The convexity of $\mathbf{P3}$ is proved.

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