# Outage Probability in Network Coding Based Cooperative Wireless Networks
over Nakagami-$m$ Fading Channels

Elias Benamira1, Fatiha Merazka1, Güneş Karabulut Kurt2  1LISIC Laboratory,
Telecommunications Department. USTHB University,
P. O. Box 32 El Alia, Algiers, Algeria<EMAIL_ADDRESS>http://www.usthb.dz
2Department of Communications and Electronics Engineering,
Istanbul Technical University,34469, Maslak, Istanbul, Turkey,
<EMAIL_ADDRESS>http://www.itu.edu.tr

###### Abstract

In this paper, we develop an accurate closed-form analytic expression of the
outage probability for each source-destination (S-D) pair in the two S-D pairs
two relays wireless network, where cooperative network coding is applied over
Nakagami-$m$ fading channels. For different values of the shape parameter $m$,
our analytic relation is validated with Monte-Carlo simulations of the outage
probability using the overall equivalent signal-to-noise ratio (SNR) within
each source-destination pair. The outage probability is also provided for the
extended versions of the network consisting of multiple relays and multiple
S-D pairs. Furthermore, we investigate the role of varying the fading factor
($m$ values) on different links in the end-to-end outage probability
performance, which yields interesting results that may be crucial for relay
selection and power allocation procedures. Moreover, we derive the diversity
order for the generalized extended version of cooperative network coded
wireless networks.

###### Index Terms:

Cooperative communication, multiple S-D pairs, Nakagami-$m$ fading channel,
network coding, outage probability.

##  I Introduction

Developing highly reliable wireless networks is a challenge that compels
researchers to explore novel strategies combining network information theory
and signal processing for telecommunications principles for a better
optimization of wireless networks resource management and a higher efficiency
accomplishment. Cooperation and network coding in wired and wireless networks
has, since its proposal and first applications [1], been proven to provide
significant improvement to the network performance in terms of efficiency and
security. Besides the considerable throughput gain achieved [2, 3], which is
one of the main important targets of 5G technology wireless networks,
cooperative network coding increases the diversity order and security of data
transmission over the network [4, 5] while reducing the overall system power
consumption.

In network coded cooperation (NCC) paradigm, the relay in charge of network
coding (NC) function performs a linear combination of the $N$ symbols received
from $N$ source nodes and broadcasts the resulting symbol to different
destinations or other relays. Hence, the number of transmissions over the
network is reduced which increases the throughput gain. Furthermore, this
process enhances the security of each message $m_{k}$ being transmitted
considering that all other messages $m_{i}(i=1,...,N,i\neq k)$ contribute to
ensure its secrecy [6]. The use of exclusive OR (XOR) operator is
straightforward and practical for network coding application. The NC resulting
symbol at relay $R$ is given by $m_{NC}=m_{R}=\bigoplus_{i=1}^{N}m_{i}$ where
$\bigoplus$ represents the summation symbol over Galois field $GF(M)$. Each
destination will then recover its destined message by performing the same XOR
operation on its received signals. Cooperative communication, combined with
network coding technique, has the key benefits of combating the fading
inherent to wireless channels and increasing capacity and diversity gains [7].

In wireless environments, transmission of signals is subject to the effects of
various phenomena such as reflection, refraction, diffraction, absorption and
scattering which results in what we call multi-path received signal. In radio
channels, these effects are called flat multi-path fading and create small
scale fading alterations. These alterations are usually modeled by Rice,
Rayleigh and Nakagami-$m$ channel fading models. Rice and Rayleigh
distributions describe multi-path effects and Nakagami-$m$ is a general form
that can replace them both.
Several recent publications have suggested the use of Nakagami-$m$ and Weibull
fading models to provide a generalized description of fading in wireless
systems. One of the most versatile fading model is described by the
Nakagami-$m$ distribution. It has a greater flexibility and accuracy in
matching some experimental data than Rayleigh, log-normal, Weibull or Rician
distributions. The Nakagami-$m$ fading distribution was first used for
modeling ionospheric and tropospheric fast fading channels and has been widely
adopted for multi-path modeling in wireless communications. It provided the
best fit to some urban multi-path data [8].
Many previous works have been treating network coding and cooperation
application on different wireless networks and few of them investigated and
derived the outage probability of the end-to-end transmission over a NCC
wireless network. In [9], authors considered NC-based cooperative systems
where multiple sources are communicating with one destination via multiple
relays. Their system, based on code division multiple access (CDMA)
transmission technique, showed better performance over traditional cooperative
systems by reducing the outage probability. A two-way decode-and-forward (DF)
non-orthogonal multiple access (NOMA) scheme is proposed in [10] where the
relay performs digital network coding (DNC) in combination with NOMA. The
authors derived closed-form theoretical expression of the outage probability
over Rayleigh fading channels. Their proposed DNC based two-way NOMA (TWNOMA)
protocol outperforms the conventional DNC protocol when the relay is located
in some optimal positions between both sources. Authors in [11] derived
theoretical analytic expressions of outage probabilities for decode-and-
forward, selective DF and cooperative network coding (CNC) protocols in
multiple relay systems over Rayleigh fading channels. After verification of
analytic expressions of outage probabilities by simulations, they concluded
the CNC provides the best performance in increasing system throughput and
diversity gain of the relay-based cooperative system. Communication
performance in vehicular ad hoc network (VANET) is investigated in [12].
Authors assumed vehicles equipped with multiple antennas and applied non-
binary network coding scheme to prove that multiple input multiple output
(MIMO) based schemes give better performance over non-binary NC (NBNC) and
generalized dynamic NC (GDNC) schemes over Rayleigh fading model channels.
Outage behavior of a system composed of one source, multiple relays and one
destination under Nakagami-$m$ fading environment was investigated in [13].
Using instantaneous power-based approach, the authors derived the exact
closed-form expression of outage probability and found the optimal number of
relays required for a desired outage performance. Analysis of symbol error
rate (SER) performance of space-time network coding (STNC) over independent
but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ fading
channels is conducted in [14]. Authors assumed multiple sources communicating
with one destination via multiple amplify and forward (AF) relays. They
derived the exact expression of overall end-to-end received signal-to-noise
ratio (SNR) and its moment generating function (MGF). A closed-form expression
of the SER with $M$-PSK and $M$-QAM modulations is also presented in this
paper. Their simulations illustrated that the diversity order of STNC with
multiple AF relays is a sum function of the fading index of the direct link
and the minimal fading indices of the multiple two-hop links. Namdar et al.
[15] analysed the outage probability performance in the high SNR region along
with the average bit error rate (BER) performance using pulse position
modulation (PPM) with timing error rate for hybrid radio frequency/visible
light communication (RF/VLC) links in indoor relay-assisted systems. They
implemented differential evolution algorithm (DEA) to determine the optimal
PDF approximation value achieving the best outage probability performance.
An energy-constrained cooperative network with application of NOMA over
Nakagami-$m$ fading channels is studied in [16]. In their system model
composed of one base station, one relay and multiple users (destinations),
authors assumed the relay equipped with energy harvesting devices. They
derived a closed-form expression for the outage probability at each end user
along with the upper and lower bounds. Using analytic and simulated results,
they demonstrated that NOMA application yields significant improvement in
spectral efficiency in comparison to orthogonal multiple access (OMA)
strategy. Authors in [17] derived the exact outage probability expression of
their proposed signal space cooperative (SSC) scheme. Furthermore, the
asymptotic approximation of their closed-form outage probability shows a clear
gain in diversity and coding gain. Multiple relay selection problem in
cooperative communications is studied in [18]. Authors derived a tractable
bound for the outage probability constraint where the multiple relay selection
problem is formulated as a mixed integer optimization problem with the on-and-
off power mode at the relays.

Our paper offers the following main contributions: We derive the exact closed-
form expression of the outage probability in two S-D pairs two relays wireless
network over Nakagami-$m$ fading channels. The analytic expression obtained is
validated by the simulated version that uses the equivalent instantaneous SNR
over the end-to-end link between each source-destination pair. Furthermore, we
give an extended version of the outage probability for the general extensions
of the considered wireless network topology with multiple relays and multiple
source-destination pairs. Besides, we investigate how Nakagami $m$ factor
variation over the different links affects the outage probability behavior.
The rest of this paper is organized as follows: In Section 2, we present a
description of how the information is handled in our system model and the
channel fading model. Section 3 contains the detailed developments of outage
probability expressions investigated in this paper. The simulations carried
out along with the results obtained are provided in Section 4. A conclusion of
our work is finally given in Section 5.

##  II System Model

The system studied in this paper is the two S-D pairs two relays wireless
network illustrated in Fig. 1 where two source-destination pairs are
communicating via two relays in series. In such network, each source doesn’t
have direct radio access to its destination but has a direct link with the
other destination. The first relay $R_{1}$, working in half-duplex mode,
receives in time-slot $T_{1}$ message $x_{1}$ from source node $S_{1}$ and
then $x_{2}$ from $S_{2}$ in time-slot $T_{2}$ in a time division multiple
access (TDMA) transmission mode. We consider both sources are transmitting
binary symbols from the finite field $\mathbb{F}_{2}$ and hence we choose
binary phase-shift keying (BPSK) digital modulation technique for
transmission.

The general form of received signal over each link $(i\rightarrow j)$, is
given by

$y_{ij}=\sqrt{P_{i}}h_{ij}x_{i}+n_{ij}$ (1)

where $i\in\\{S_{1},S_{2},R_{1},R_{2}\\},j\in\\{R_{1},R_{2},D_{1},D_{2}\\}$,
$P_{i}$ is the transmit power of node $i$, $h_{ij}$ is the channel fading
coefficient of transmission from node $i$ to node $j$ and $n_{ij}$ represents
the complex additive white Gaussian noise (AWGN) with zero mean and variance
$\sigma_{n_{ij}}^{2}$ at receiver $j$,
$n_{ij}\sim\mathcal{C}\mathcal{N}(0,\sigma_{n_{ij}}^{2})$.

Figure 1: Considered system: Two S-D pairs two relays wireless network

We, first, define the following applications [19]:
1\. $\mathcal{M}_{2}:\mathbb{F}_{2}\rightarrow\mathbb{C}$ that maps each
symbol $m_{i}\in\mathbb{F}_{2}$ into the corresponding BPSK constellation
$x_{i}$, i.e. $x_{i}=\mathcal{M}_{2}(m_{i})$.
2\. $\mathcal{E}:\mathbb{C}\rightarrow\mathbb{C}$ that estimates the received
version $\tilde{x}_{ij}$ of the original one $x_{i}$ at the receiver $j$, i.e.
$\tilde{x}_{ij}=\mathcal{E}(y_{ij})=\mathcal{E}(\sqrt{P_{i}}h_{ij}x_{i}+n_{ij})=(\sqrt{P_{i}}h_{ij}x_{i}+n_{ij})h_{ij}^{*}$,
where $h_{ij}^{*}$ denotes the complex conjugate of $h_{ij}$.
In this paper, we assume that all transmitting nodes have the same transmit
power $P_{i}=P$ and perfect channel state information (CSI) available at each
receiver node.
The relay $R_{1}$ recovers both messages
$\tilde{m}_{S_{1}R}=\mathcal{M}_{2}^{-1}(\tilde{x}_{S_{1}R_{1}})=\mathcal{M}_{2}^{-1}(\mathcal{E}(y_{S_{1}R_{1}}))$
and
$\tilde{m}_{S_{2}R_{1}}=\mathcal{M}_{2}^{-1}(\tilde{x}_{S_{2}R_{1}})=\mathcal{M}_{2}^{-1}(\mathcal{E}(y_{S_{2}R_{1}}))$
before applying network coding operation XOR on them to generate
$m_{R_{1}}=m_{NC}=\tilde{m}_{S_{1}R_{1}}\oplus\tilde{m}_{S_{2}R_{1}}$ that it
modulates and transmits to relay $R_{2}$. The network coded (NC) symbol is
then amplified at $R_{2}$ (working in AF mode) with amplification factor
$\beta_{R_{1}R_{2}}=\sqrt{\frac{P_{R_{2}}}{P_{R_{1}}|h_{R_{1}R_{2}}|^{2}+\sigma_{R_{1}R_{2}}^{2}}}$
and forwarded (broadcast) to destination nodes $D_{1}$ and $D_{2}$.
Destinations $D_{1}$ and $D_{2}$ recover their respective messages by applying
the same operation on their received signals, i.e.

$\tilde{m}_{1}=\tilde{m}_{R_{2}D_{1}}\oplus\tilde{m}_{S_{2}D_{1}}=\mathcal{M}_{2}^{-1}(\mathcal{E}(y_{R_{2}D_{1}}))\oplus\mathcal{M}_{2}^{-1}(\mathcal{E}(y_{S_{2}D_{1}}))$
(2)

and

$\tilde{m}_{2}=\tilde{m}_{R_{2}D_{2}}\oplus\tilde{m}_{S_{1}D_{2}}=\mathcal{M}_{2}^{-1}(\mathcal{E}(y_{R_{2}D_{2}}))\oplus\mathcal{M}_{2}^{-1}(\mathcal{E}(y_{S_{1}D_{2}}))$

respectively.

All links in our network are subject to Nakagami-$m$ model fading channel with
the probability density function (PDF) [20]

$f(\mathcal{X})=\frac{2m^{m}\mathcal{X}^{2m-1}}{\Gamma(m)\mu^{m}}e^{\frac{-m\mathcal{X}^{2}}{\mu}}$
(3)

where

$\mu=E(\mathcal{X}^{2})\text{ and
}m=\frac{\mu^{2}}{(\mathcal{X}^{2}-\mu)^{2}}\geq\frac{1}{2}$

represents the shape factor of Nakagami-$m$ distribution and $\Gamma(.)$ is
the Gamma function defined by

$\Gamma(x)=\int_{0}^{\infty}a^{x-1}e^{-a}da$

In wireless networks links, $\mathcal{X}^{2}=|h_{ij}|^{2}$.

Let $\gamma_{ij}=\frac{|h_{ij}|^{2}P_{i}}{\sigma_{ij}^{2}}$ be the
instantaneous SNR over the link $(i\rightarrow j)$, then (3) becomes [21]

$f(\gamma_{ij})=\frac{m_{ij}^{m_{ij}}\gamma_{ij}^{m_{ij}-1}}{\bar{\gamma}_{ij}^{m_{ij}}\Gamma(m_{ij})}e^{\frac{-m_{ij}\gamma_{ij}}{\bar{\gamma}_{ij}}}$
(4)

where $\bar{\gamma}_{ij}=E(\gamma_{ij})$ is the average (mathematical
expectation) value and $m_{ij}$ is the shape factor of link $(i\rightarrow
j)$.
For the correspondence with Rice distribution $m=\frac{(K+1)^{2}}{2K+1}$, $K$
being the Rician fading factor. The special case of Nakagami shape factor
$m=1$ represents Rayleigh fading model.

##  III Outage Analysis

To derive the outage probability of the wireless network in question subject
to NCC application, we consider the source-destination pair $(S_{1}-D_{1})$
where the participating links are $(S_{1}\rightarrow R_{1}),(R_{1}\rightarrow
R_{2}),(R_{2}\rightarrow D_{1}),(S_{2}\rightarrow R_{1})$ and
$(S_{2}\rightarrow D_{1})$.
To the five links that intervene in message transmission from source node
$S_{1}$ to destination node $D_{1}$ correspond five outage events
$E_{S_{1}R_{1}}$, $E_{R_{1}R_{2}}$, $E_{R_{2}D_{1}}$, $E_{S_{2}R_{1}}$ and
$E_{S_{2}D_{1}}$ respectively. The resultant outage event $E_{S_{1}D_{1}}$ of
the end-to-end link $(S_{1}\rightarrow D_{1})$ has the following expression

$E_{S_{1}D_{1}}=E_{S_{2}D_{1}}\cap\bigg{(}E_{R_{2}D_{1}}\cup
E_{R_{1}R_{2}}\cup\Big{(}E_{S_{1}R_{1}}\cap E_{S_{2}R_{1}}\Big{)}\bigg{)}$ (5)

The probability of event $E_{S_{1}D_{1}}$ represents the NCC two S-D pairs two
relays network outage probability, i.e.

$\displaystyle P_{out}$ $\displaystyle=P_{r}\big{\\{}E_{S_{1}D_{1}}\big{\\}}$
$\displaystyle=P_{r}\Big{\\{}E_{S_{2}D_{1}}\cap\Big{(}E_{R_{2}D_{1}}\cup
E_{R_{1}R_{2}}\cup\Big{(}E_{S_{1}R_{1}}\cap
E_{S_{2}R_{1}}\Big{)}\Big{)}\Big{\\}}$ (6)

With the assumption of mutually independent links, (III) can be written as

$\displaystyle P_{out}$
$\displaystyle=P_{r}\big{\\{}E_{S_{2}D_{1}}\big{\\}}\Big{(}P_{r}\big{\\{}E_{R_{2}D_{1}}\big{\\}}+P_{r}\big{\\{}E_{R_{1}R_{2}}\big{\\}}$
$\displaystyle+P_{r}\big{\\{}E_{S_{1}R_{1}}\big{\\}}P_{r}\big{\\{}E_{S_{2}R_{1}}\big{\\}}\Big{)}$
(7)

or simply

$P_{out}=P_{out}^{S_{2}D_{1}}\Big{(}P_{out}^{R_{2}D_{1}}+P_{out}^{R_{1}R_{2}}+P_{out}^{S_{1}R_{1}}P_{out}^{S_{2}R_{1}}\Big{)}$
(8)

where $P_{out}^{ij}$ designates the outage probability of each link
$(i\rightarrow j)$.

The mutual information $I_{ij}$ between node $i$ and node $j$ is given by [22]

$I_{ij}=log_{2}(1+\gamma_{ij})$ (9)

An outage occurs on link $(i\rightarrow j)$ if $I_{ij}$ is lower than a target
information rate (spectral efficiency) $R_{t}$ in bits per second per Hertz
(bps/Hz), i.e.

$log_{2}(1+\gamma_{ij})<R_{t}$ (10)

or, equivalently, if

$\gamma_{ij}<2^{R_{t}}-1$ (11)

In other words, outage is the scenario when the instantaneous SNR, due to bad
channel realization, is unable to support the desired information rate [23].

In consequence, we can express the outage probability on link $(i\rightarrow
j)$ by

$P_{out}^{ij}=P_{r}\big{\\{}\gamma_{ij}<\gamma_{th}=2^{R_{t}}-1\big{\\}}$ (12)

where $\gamma_{th}$ is the lowest value of the instantaneous SNR (threshold)
that allows reliable transmission of information between node $i$ and node
$j$.
From (4) and (12), we have

$\displaystyle P_{out}^{ij}$
$\displaystyle=F(\gamma_{th})=\int_{0}^{\gamma_{th}}f(\gamma_{ij})d\gamma_{ij}$
$\displaystyle=\int_{0}^{\gamma_{th}}\frac{m_{ij}^{m_{ij}}\gamma_{ij}^{m_{ij}-1}}{\bar{\gamma}_{ij}^{m_{ij}}\Gamma(m_{ij})}e^{\frac{-m_{ij}\gamma_{ij}}{\bar{\gamma}_{ij}}}d\gamma_{ij}$
$\displaystyle=\Bigg{[}\frac{-\Gamma_{inc}\Big{(}m_{ij},\frac{m_{ij}\gamma_{ij}}{\bar{\gamma}{ij}}\Big{)}}{\Gamma(m_{ij})}\Bigg{]}_{0}^{\gamma_{th}}$
$\displaystyle=1-\frac{\Gamma_{inc}\Big{(}m_{ij},\frac{m_{ij}\gamma_{th}}{\bar{\gamma}{ij}}\Big{)}}{\Gamma(m_{ij})}$
(13)

where $F(\gamma_{th})$ is the cumulative density function (CDF) and
$\Gamma_{inc}(.,.)$ is the upper incomplete Gamma function defined by

$\Gamma_{inc}(x,y)=\int_{y}^{\infty}a^{x-1}e^{-a}da$

Substituting the probability of outage on each link by expression (III) in
relation (8), we obtain the general closed-form analytic expression of the
outage probability for our NCC based wireless network

$\displaystyle P_{out}$
$\displaystyle=\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{2}D_{1}},\frac{m_{S_{2}D_{1}}\gamma_{th}}{\bar{\gamma}_{S_{2}D_{1}}}\Big{)}}{\Gamma(m_{S_{2}D_{1}})}\Bigg{)}$
$\displaystyle\times\Bigg{[}\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{R_{2}D_{1}},\frac{m_{R_{2}D_{1}}\gamma_{th}}{\bar{\gamma}_{R_{2}D_{1}}}\Big{)}}{\Gamma(m_{R_{2}D_{1}})}\Bigg{)}$
$\displaystyle+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{R_{1}R_{2}},\frac{m_{R_{1}R_{2}}\gamma_{th}}{\bar{\gamma}_{R_{1}R_{2}}}\Big{)}}{\Gamma(m_{R_{1}R_{2}})}\Bigg{)}$
$\displaystyle+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{1}R_{1}},\frac{m_{S_{1}R_{1}}\gamma_{th}}{\bar{\gamma}_{S_{1}R_{1}}}\Big{)}}{\Gamma(m_{S_{1}R_{1}})}\Bigg{)}$
$\displaystyle\times\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{2}R_{1}},\frac{m_{S_{2}R_{1}}\gamma_{th}}{\bar{\gamma}_{S_{2}R_{1}}}\Big{)}}{\Gamma(m_{S_{2}R_{1}})}\Bigg{)}\Bigg{]}$
(14)

Replacing $\gamma_{th}$ by its expression from (12) we have

$\displaystyle P_{out}$
$\displaystyle=\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{2}D_{1}},\frac{m_{S_{2}D_{1}}(2^{R_{1}}-1)}{\bar{\gamma}_{S_{2}D_{1}}}\Big{)}}{\Gamma(m_{S_{2}D_{1}})}\Bigg{)}$
$\displaystyle\times\Bigg{[}\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{R_{2}D_{1}},\frac{m_{R_{2}D_{1}}(2^{R_{1}}-1)}{\bar{\gamma}_{R_{2}D_{1}}}\Big{)}}{\Gamma(m_{R_{2}D_{1}})}\Bigg{)}$
$\displaystyle+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{R_{1}R_{2}},\frac{m_{R_{1}R_{2}}(2^{R_{1}}-1)}{\bar{\gamma}_{R_{1}R_{2}}}\Big{)}}{\Gamma(m_{R_{1}R_{2}})}\Bigg{)}$
$\displaystyle+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{1}R_{1}},\frac{m_{S_{1}R_{1}}(2^{R_{1}}-1)}{\bar{\gamma}_{S_{1}R_{1}}}\Big{)}}{\Gamma(m_{S_{1}R_{1}})}\Bigg{)}$
$\displaystyle\times\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{2}R_{1}},\frac{m_{S_{2}R_{1}}(2^{R_{1}}-1)}{\bar{\gamma}_{S_{2}R_{1}}}\Big{)}}{\Gamma(m_{S_{2}R_{1}})}\Bigg{)}\Bigg{]}$
(15)

Finally, if we assume all links independent and identically distributed
(i.i.d.) affected by the same Nakagami fading factor $m$, i.e.
$\bar{\gamma}_{S_{2}D_{1}}=\bar{\gamma}_{R_{1}R_{2}}=\bar{\gamma}_{R_{2}D_{1}}=\bar{\gamma}_{S_{1}R_{1}}=\bar{\gamma}_{S_{2}R_{1}}=\bar{\gamma}$
and
$m_{S_{2}D_{1}}=m_{R_{1}R_{2}}=m_{R_{2}D_{1}}=m_{S_{1}R_{1}}=m_{S_{2}R_{1}}=m$,
then (III) reduces to

$\displaystyle P_{out}$
$\displaystyle=\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}^{2}$
$\displaystyle\times\Bigg{[}2+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}\Bigg{]}$
$\displaystyle=\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}^{2}$
$\displaystyle\times\Bigg{(}3-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}$
(16)

If we consider the common probability
$p=1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}$,
expression (III) yields

$P_{out}=p^{2}(2+p)=2p^{2}+p^{3}\sim p^{2}\sim
O\left(\frac{1}{SNR}\right)^{2}$ (17)

The diversity order achieved in high SNR values for this network is deduced
from (17) and equals 2. Actually, this result is valid only for $m=1$, in the
next section we will see how diversity order depends on $m$ value.

The previously derived outage probability expressions can be generalized to
more than two source-destination pairs and more than two relays in series.

Figure 2: Extended topology of the wireless network with $M$ relays and $N$
source-destination pairs
$\gamma_{S_{1}D1}^{eq}=\frac{P_{S_{2}}|h_{S_{2}D_{1}}|^{2}+P_{R_{1}}P_{R_{2}}|h_{R_{1}R_{2}}|^{2}|h_{R_{2}D_{1}}|^{2}\Big{(}P_{S_{1}}|h_{S_{1}R_{1}}|^{2}+P_{S_{2}}|h_{S_{2}R_{1}}|^{2}\Big{)}}{E\\{n_{S_{2}D_{1}}^{2}\\}+E\\{n_{R_{2}D_{1}}^{2}\\}+P_{R_{2}}|h_{R_{2}D_{1}}|^{2}E\\{n_{R_{1}R_{2}}^{2}\\}+P_{R_{1}}P_{R_{2}}|h_{R_{1}R_{2}}|^{2}|h_{R_{2}D_{1}}|^{2}\Big{(}E\\{n_{S_{1}R_{1}}^{2}\\}+E\\{n_{S_{2}R_{1}}^{2}\\}\Big{)}}$
(18)
$P_{out}^{sim}=P_{r}\Bigg{\\{}\frac{P_{S_{2}}|h_{S_{2}D_{1}}|^{2}+P_{R_{1}}P_{R_{2}}|h_{R_{1}R_{2}}|^{2}|h_{R_{2}D_{1}}|^{2}\Big{(}P_{S_{1}}|h_{S_{1}R_{1}}|^{2}+P_{S_{2}}|h_{S_{2}R_{1}}|^{2}\Big{)}}{E\\{n_{S_{2}D_{1}}^{2}\\}+E\\{n_{R_{2}D_{1}}^{2}\\}+P_{R_{2}}|h_{R_{2}D_{1}}|^{2}E\\{n_{R_{1}R_{2}}^{2}\\}+P_{R_{1}}P_{R_{2}}|h_{R_{1}R_{2}}|^{2}|h_{R_{2}D_{1}}|^{2}\Big{(}E\\{n_{S_{1}R_{1}}^{2}\\}+E\\{n_{S_{2}R_{1}}^{2}\\}\Big{)}}<\gamma_{th}\Bigg{\\}}$
(19)
$P_{out}^{sim}=P_{r}\Bigg{\\{}\frac{P_{S_{2}}|h_{S_{2}D_{1}}|^{2}+P_{R_{1}}P_{R_{2}}|h_{R_{1}R_{2}}|^{2}|h_{R_{2}D_{1}}|^{2}\Big{(}P_{S_{1}}|h_{S_{1}R_{1}}|^{2}+P_{S_{2}}|h_{S_{2}R_{1}}|^{2}\Big{)}}{E\\{n_{S_{2}D_{1}}^{2}\\}+E\\{n_{R_{2}D_{1}}^{2}\\}+P_{R_{2}}|h_{R_{2}D_{1}}|^{2}E\\{n_{R_{1}R_{2}}^{2}\\}+P_{R_{1}}P_{R_{2}}|h_{R_{1}R_{2}}|^{2}|h_{R_{2}D_{1}}|^{2}\Big{(}E\\{n_{S_{1}R_{1}}^{2}\\}+E\\{n_{S_{2}R_{1}}^{2}\\}\Big{)}}<2^{R_{t}}-1\Bigg{\\}}$
(20)

Figure 3: Decreasing analytic outage probability of (III) tightly with MC
simulated BER for increasing values of $m$ (0.5, 1, 2) and $R_{t}\approx
1.1~{}bps/Hz$.

Figure 4: Precise match of simulated with analytic decreasing outage
probabilities with increasing values of $m$ ($0.5,1,2,4$).

Figure 5: Outage probability of expression (III) for $m=2$ or $m=4$ on the
links and $R_{t}=1~{}bps/Hz$.

###### Theorem 1

_(Extended multiple S-D pairs multiple relays network)_
In an extended version of the network constituted of $N$ source-destination
pairs and two relays with $i.i.d.$ Nakagami-$m$ fading channel links and the
same value of fading factor $m$, the outage probability expression is given by

$\displaystyle P_{out}=$
$\displaystyle\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}^{N}$
$\displaystyle\times\Bigg{[}2+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}^{N-1}\Bigg{]}$
(21)

Moreover, if we have $M$ ($M>2$) relays $R_{1},R_{2},...,R_{M}$ in series
instead of two ($R_{1}$ being always the one in charge of network coding
operation and the rest working in AF mode) as illustrated in Fig. 2, the
outage probability of any source-destination pair is then given by

$\displaystyle P_{out}=$
$\displaystyle\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}^{N}$
$\displaystyle\times\Bigg{[}M+\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m,\frac{m(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m)}\Bigg{)}^{N-1}\Bigg{]}$
(22)

The diversity order achieved in high SNR region, using the result of (17), is
obtained from

$\displaystyle P_{out}$ $\displaystyle=p^{N}(M+p^{N-1})$
$\displaystyle=Mp^{N}+p^{2N-1}\sim p^{N}\sim O\left(\frac{1}{SNR}\right)^{N}$
(23)

and is equal to $N$.

In Theorem 1, we assume all links sharing the same Nakagami fading parameter
$m$. However, in real life wireless systems, this assumption is seldom true
and we have to deal with the generalized version of expression (2) that we
present in the following theorem.

###### Theorem 2

_(Generalized multiple S-D pairs multiple relays network extensions)_
Using the same assumptions of Theorem 1 with the exception of distinct value
of fading factor $m$ on each link, the outage probability expression has the
following general form:

$\displaystyle P_{out}=$
$\displaystyle\prod_{i=2}^{N}\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{i}D_{1}},\frac{m_{S_{i}D_{1}}(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m_{S_{i}D_{1}})}\Bigg{)}$
$\displaystyle\times\Bigg{[}\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{R_{M}D_{1}},\frac{m_{R_{M}D_{1}}(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m_{R_{M}D_{1}})}\Bigg{)}$
$\displaystyle+\sum_{j=1}^{M-1}\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{R_{j}R_{j+1}},\frac{m_{R_{j}R_{j+1}}(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m_{R_{j}R_{j+1}})}\Bigg{)}$
$\displaystyle+\prod_{k=1}^{N}\Bigg{(}1-\frac{\Gamma_{inc}\Big{(}m_{S_{k}R_{1}},\frac{m_{S_{k}R_{1}}(2^{R_{t}}-1)}{\bar{\gamma}}\Big{)}}{\Gamma(m_{S_{k}R_{1}})}\Bigg{)}\Bigg{]}$
(24)

The outage expression (III) is used for comparison with simulation results
assuming the hypothesis of having the same value of shape factor $m$ on all
links while the generalized outage expression (III) is used to investigate the
impact of different $m$ values over the different participating links on the
overall end-to-end outage probability.

The obtained outage probability is then compared with the simulated
probability of decoding erroneous symbols at the receiver $D_{1}$ considering
the equivalent instantaneous SNR between source node $S_{1}$ and destination
node $D_{1}$. Many previous papers have been treating similar problems of
outage probability computation for multiple relays cooperative or non
cooperative networks. In NCC based 2 S-D pairs 2 relays wireless network,
destination $D_{1}$ receives one signal from source node $S_{2}$ and another
from the relay $R_{2}$ that has received the NC message from $R_{1}$ that
combined its two received messages from $S_{1}$ and $S_{2}$. The general
expression that shows all transmissions over all involved links with their
respective attenuation effects and additive noises is given by

$\displaystyle\tilde{y}_{D_{1}}=$
$\displaystyle\sqrt{P_{S_{2}}}x_{S_{2}}h_{S_{2}D1}+n_{S_{2}D_{1}}$
$\displaystyle+\sqrt{P_{R_{2}}}\Big{[}\sqrt{P_{R_{1}}}\Big{(}\sqrt{P_{S_{1}}}x_{S_{1}}h_{S_{1}R_{1}}+n_{S_{1}R_{1}}$
$\displaystyle+\sqrt{P_{S_{2}}}x_{S_{2}}h_{S_{2}R_{1}}+n_{S_{2}R_{1}}\Big{)}h_{R_{1}R_{2}}+n_{R_{1}R_{2}}\Big{]}$
$\displaystyle\times h_{R_{2}D_{1}}+n_{R_{2}D_{1}}$ (25)

Expression (III) is employed to determine the equivalent instantaneous SNR
expression independently from the procedures of network coding at relay
$R_{1}$ and decoding at destination $D_{1}$ that have no influence on the
equivalent instantaneous receive SNR value.
Hence, we derive the general expression of the global SNR illustrated in (18).

Figure 6: Outage probability of expression (III) versus $R_{t}$ and average
SNR for $m=1,2$ and $4$.

A simpler version of (18) can result from the following assumptions:
independent and identically distributed fading channels with the same shape
factor $m$ on all links, i.e.
$h_{S_{1}R_{1}}=h_{S_{2}R_{1}}=h_{R_{1}R_{2}}=h_{R_{2}D_{1}}=h_{S_{2}D_{1}}=h$.
Both sources $S_{1}$ and $S_{2}$ and both relays $R_{1}$ and $R_{2}$ have
equal transmit powers $P_{S_{1}}=P_{S_{2}}=P_{R_{1}}=P_{R_{2}}=P$ and all
added Gaussian noise variances have the same value $N_{0}$ at all receiving
nodes, i.e.
$E\\{n_{S_{2}D_{1}}^{2}\\}=E\\{n_{R_{2}D_{1}}^{2}\\}=E\\{n_{R_{1}R_{2}}^{2}\\}=E\\{n_{S_{1}R_{1}}^{2}\\}=E\\{n_{S_{2}R_{1}}^{2}\\}=N_{0}$.
Therefore, (18) yields

$\gamma_{S_{1}D1}^{eq}=\frac{P|h|^{2}\Big{(}1+2P^{2}|h|^{4}\Big{)}}{N_{0}\Big{(}2+P|h|^{2}+2P^{2}|h|^{4}\Big{)}}$
(26)

Simulated outage probability is the probability that the equivalent
instantaneous SNR $\gamma_{S_{1}D1}^{eq}$ falls below average SNR threshold
value $\gamma_{th}$, i.e.

$\displaystyle P_{out}^{sim}$
$\displaystyle=P_{r}\Big{\\{}\gamma_{S_{1}D1}^{eq}<\gamma_{th}\Big{\\}}$
$\displaystyle=P_{r}\Bigg{\\{}\frac{P|h|^{2}\Big{(}1+2P^{2}|h|^{4}\Big{)}}{N_{0}\Big{(}2+P|h|^{2}+2P^{2}|h|^{4}\Big{)}}<2^{R_{t}}-1\Bigg{\\}}$
(27)

The simplified version of simulated outage probability given in (III) is used
to validate the analytic derived expression (III) in the scenario of unique
Nakagami fading factor $m$ in all links. However, for the most general case of
different $m$ factors, expression (20) of simulated outage probability serves
to validate the generalized analytic expression given in (III).

The diversity order deduced from outage probabilities of (17) and (1) in high
SNR regions can be computed using the symbolic calculus of MATLAB.
It is stated from [24] that the diversity order $d$ can be expressed as:

$d=-\lim_{SNR\to\infty}\frac{log(P_{out}(SNR))}{log(SNR)}$ (28)

We use our derived outage probability to compute the diversity order using
(28) for different values of $m$ factor. For $m=1,~{}2,~{}3$ and $4$, we get
the diversity values $d=2,~{}4,~{}6$ and $8$ respectively with expression
(III) of $P_{out}$ (two source-destination pairs) which leads to the result
$d=2m$ for NCC based wireless network. The result is the same for any value of
target rate $R_{t}$ and any number of relays $M$.
For an extension to $N$ ($S-D$) pairs, we use $P_{out}$ expression (1) with
different values of $N$ to confirm the following conclusion:
The diversity order achieved in high SNR regions of the extended $N$ S-D pairs
$M$ relays wireless network over Nakagami-$m$ fading channels, for any value
of $R_{t}$ and any number of relays $M$, is given by

$d=Nm$ (29)

This result is validated only for integer values of fading parameter $m$. The
result of expression (29) can be noticed, in case of $N=2$, on the graphs of
outage probability for different values of $m$ in Fig. 4. Actually, the
diversity order $d$ is the absolute value of the slope of the outage
probability at high SNRs.

##  IV Simulation Results

In this section, we use MATLAB software to perform SER simulations, analytic
and simulated outage probability computation and finally, to determine a
general reliable expression of the diversity order for the extended topology
of wireless networks. SER (or BER since we adopt binary symbols transmission)
is obtained from the average of $10^{3}$ realizations in each one $10^{4}$
symbols are randomly generated at each source node along with $10^{4}$
Nakagami-$m$ fading coefficients for each link among the active ones involved
in the end-to-end ($S_{1}-D_{1}$) pair communication. The received signal
$\tilde{s}_{1}$ at $D_{1}$ is then compared to the original symbol generated
at $S_{1}$. Figure 3 illustrates the results of BER simulations along with the
theoretical outage probability computed with expression (III). The perfect
match of simulated BER for $m=0.5,~{}1$ and $2$ with analytic outage
probability confirms the exactness of our derived closed-form expression for a
given value of target rate (spectral efficiency) $R_{t}$.
In Fig. 4, the curves of simulated outage probability of (III) are almost
completely fitting those of the analytic relation which confirms the high
reliability of our derived closed-form expression. In order to evaluate the
effect of Nakagami fading factor $m$ over the different links on the outage
probability, we present some results on Fig. 5. From these results, it can be
seen that links ($S_{1}\rightarrow R_{1}$) and ($S_{2}\rightarrow R_{1}$) are
not sensible to the variation of $m$. We notice almost the same conclusion for
links ($R_{1}\rightarrow R_{2}$) and ($R_{2}\rightarrow D_{1}$). However, the
deviation of $m$ value on link ($S_{2}\rightarrow D_{1}$) has an important
effect on the outage probability performance. These results can be crucial in
processes like relay selection and power allocation for a better optimization
of network resources management. A generalization for a wider range of SNR
values and target rates $R_{t}$ is illustrated by the evolution of the outage
probability for $m=1,~{}2$ and $4$ in Fig. 6 where it is shown how increasing
SNR allows to achieve higher spectral efficiency $R_{t}$.

##  V Conclusion

In this paper, an accurate closed-form expression of the theoretical outage
probability was derived for the network coding and cooperation based 2 S-D
pairs 2 relays wireless network over Nakagami-$m$ fading channels. We also
presented a generalization of the analytic outage probability expression for
an extended version of the considered wireless networks with $M$ relays and
$N$ S-D pairs. The perfect match of the simulated outage probability evolution
using the overall equivalent SNR confirmed the accuracy and reliability of our
derived expressions which can serve to get the precise outage behaviour for
such wireless networks and hence contribute in the network and resource
optimization. Interesting results were also obtained regarding the role of
Nakagami model fading parameter $m$ on different links in the outage
probability behavior, which can be exploited in network configuration
procedures like relay selection and power allocation. Moreover, we established
a reliable expression of the diversity order. In a future work, the outage
probability for the same wireless networks with application of error
correction codes on all links will be investigated.

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