# Symbol Error Rate of Space-Time Network Coding in Nakagami-$m$ Fading

Ang Yang, Zesong Fei, Nan Yang, Chengwen Xing, and Jingming Kuang A. Yang, Z.
Fei, C. Xing, and J. Kuang are with the School of Information and Electronics,
Beijing Institute of Technology, Beijing, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>JMKuang@bit.edu.cn).N. Yang is
with the Wireless and Networking Technologies Laboratory, CSIRO ICT Centre,
Marsfield, NSW 2122, Australia (email: jonas.yang@csiro.au).

###### Abstract

In this paper, we analyze the symbol error rate (SER) of space-time network
coding (STNC) in a distributed cooperative network over independent but not
necessarily identically distributed (i.n.i.d.) Nakagami-$m$ fading channels.
In this network, multiple sources communicate with a single destination with
the assistance of multiple decode-and-forward (DF) relays. We first derive new
exact closed-form expressions for the SER with $M$-ary phase shift-keying
modulation ($M$-PSK) and $M$-ary quadrature amplitude modulation ($M$-QAM). We
then derive new compact expressions for the asymptotic SER to offer valuable
insights into the network behavior in the high signal-to-noise ratio (SNR)
regime. Importantly, we demonstrate that STNC guarantees full diversity order,
which is determined by the Nakagami-$m$ fading parameters of all the channels
but independent of the number of sources. Based on the new expressions, we
examine the impact of the number of relays, relay location, Nakagami-$m$
fading parameters, power allocation, and nonorthogonal codes on the SER.

###### Index Terms:

Space-time network coding, symbol error rate, Nakagami-$m$ fading.

##  I Introduction

Cooperative communication has been recognized as a promising low-cost solution
to combat fading and to extend coverage in wireless networks [1, 2]. The key
idea behind this solution is to employ relays to receive and transmit the
source information to the destination, which generates a virtual multiple-
input and multiple-output (MIMO) system to provide spatial cooperative
diversity [3, 4]. Apart from diversity, throughput enhancement is another
critical challenge for wireless networks. Against this background, network
coding (NC) is proposed as a potentially powerful tool to enable efficient
information transmission, where data flows coming from multiple sources are
combined to increase throughput [5, 6, 7].

Recently, the joint exploitation of cooperative diversity and NC has become a
primary design concern in distributed networks with multiple users and
multiple relays [8, 9, 10, 11]. Motivated by this, [8] investigated various
sink network decoding approaches for the network with intermediate node
encoding. In [9], linear network coding (LNC) was applied in distributed
uplink networks to facilitate the transmission of independent information from
multiple users to a common base station. In [10], low-density parity-check
(LDPC) code and NC was jointly designed for a multi-source single-relay FDMA
system over uniform phase-fading Gaussian channels. In [11], cooperative
network coding strategies were proposed for a relay-aided two-source two-
destination wireless network with a backhaul connection between the sources.

In order to increase the capacity and transmission reliability of wireless
cooperative networks, multiple antennas are deployed to gain the merits of
MIMO processing techniques [12, 13, 14, 15, 16, 17]. In this strategy,
distributed space-time coding (DSTC) was proposed to further boost network
performance, where the antennas at the distributed relays are utilized as
transmit antennas to generate a space-time code for the receiver [18, 19]. A
differential DSTC was proposed in [20] to eliminate the requirement of channel
information at the relays and the receiver. The combined benefits of maximum-
ratio combining (MRC) and DSTC were investigated in [21]. In [22], the impact
of DSTC in two-way amplify-and-forward relay channels was characterized.

One of the principal challenges in distributed cooperative networks is to
leverage the benefits from both NC and DSTC. A promising solution that
addresses this challenge is space-time network coding (STNC), which was
proposed in [23]. Fundamentally, STNC combines the information from different
sources at a relay, which involves the concept of NC. Moreover, STNC transmits
the combined signals in several time slots using a set of relays, which
involves the concept of DSTC. Based on these, STNC achieves spatial diversity
with low transmission delay under the impact of imperfect frequency and timing
synchronization. We note that in [23], the performance of STNC was evaluated
for Rayleigh fading channels, where no closed-form expression was presented.

In this paper, we consider a distributed cooperative network using STNC over
independent but not necessarily identically distributed (i.n.i.d.)
Nakagami-$m$ fading channels, which generalizes the result in [23]. In this
network, multiple sources communicate with a single destination with the
assistance of multiple relays. Here, we focus on decode-and-forward (DF)
protocol at the relays, which arises from the fact that this protocol has been
successfully deployed in practical wireless standards, e.g., 3GPP Long Term
Evolution (LTE) and IEEE 802.16j WiMAX [24]. Different from [23], we examine
two fundamental questions as follows: “ _1) What is the impact of STNC on the
symbol error rate (SER) in general Nakagami- $m$ fading channels?_” and “ _2)
Can we provide closed-form expressions for the SER in Nakagami- $m$ fading to
alleviate the burden of Monte Carlo simulations?_” The rationale behind these
questions is that Nakagami-$m$ fading covers a wide range of typical fading
scenarios in realistic wireless applications via the $m$ parameter. Notably,
Nakagami-$m$ fading encompasses Rayleigh fading ($m=1$) as a special case
[25]. To tackle these questions, we first derive new closed-form expressions
for the exact SER, which are valid for multiple phase shift-keying modulation
($M$-PSK) and $M$-ary quadrature amplitude modulation ($M$-QAM). To further
provide valuable insights at high signal-to-noise ratios (SNRs), we derive new
compact expressions for the asymptotic SER, from which the diversity gain is
obtained. Specifically, it is demonstrated that the diversity order is
determined by the Nakagami-$m$ fading parameters of all the channels, but
independent of the number of sources. Various numerical results are utilized
to examine the impact of the number of relays, relay location, Nakagami-$m$
fading parameters, power allocation, and nonorthogonal codes on the SER.
Importantly, it is shown that nonorthogonal codes provide higher throughput
than orthogonal codes, while guaranteeing full diversity over Nakagami-$m$
fading channels. Our analytical expressions are substantiated via Monte Carlo
simulations.

##  II System Model

Fig. 1 depicts a distributed cooperative network where $L$ sources,
$U_{1},U_{2},\ldots,U_{L}$, transmit their own information to a common
destination $D$ with the aid of $Q$ relays, $R_{1},R_{2},\ldots,R_{Q}$. In
this network, each node is equipped with a single antenna. We consider a
practical and versatile operating scenario where the source-relay, the relay-
destination, and the source-destination channels experience i.n.i.d.
Nakagami-$m$ fading. As such, we denote the Nakagami-$m$ fading parameter
between $U_{l}$ and $R_{q}$ as $m_{lq}$, the Nakagami-$m$ fading parameter
between $R_{q}$ and $D$ as $m_{qd}$, and the Nakagami-$m$ fading parameter
between $U_{l}$ and $D$ as $m_{ld}$. We further denote the channel coefficient
between $U_{l}$ and $R_{q}$ as $h_{lq}$, the channel coefficient between
$R_{q}$ and $D$ as $h_{qd}$, and the channel coefficient between $U_{l}$ and
$D$ as $h_{ld}$, where $1\leq{}l\leq{}L$ and $1\leq{}q\leq{}Q$. Throughout
this paper, we define the variances of these channel coefficients as
$h_{\phi\varphi}\sim\mathcal{CN}\left(0,d_{\phi\varphi}^{-\alpha}\right)$,
where $\phi\in\left\\{l,q\right\\}$, $\varphi\in\left\\{q,d\right\\}$, and
$\phi\neq\varphi$. Here, we incorporate the path loss in the signal
propagation such that $d_{\phi\varphi}$ denotes the distance between $\phi$
and $\varphi$ and $\alpha$ denotes the path loss exponent.

In this network, the STNC transmission between the sources and the destination
is divided into two consecutive phases: 1) source transmission phase and 2)
relay transmission phase. In the source transmission phase, the sources
transmit their symbols in the designated time slots. In this phase, the relays
receive a set of overheard symbols from the sources. In the relay transmission
phase, each relay encodes the set of overheard symbols to a single signal and
then transmits it to the destination in its designated time slot. As
illustrated in Fig. 2, $(L+Q)$ time slots are required to complete the STNC
transmission to eliminate the detrimental effects of imperfect synchronization
on any point-to-point transmission in this network at any time slot.

We proceed to detail the transmission in the two phases, as follows:

In the source transmission phase, the signals received at the destination from
$U_{l}$ in the time slot $l$ is given by

$y_{ld}\left(t\right)=h_{ld}\sqrt{P_{l}}x_{l}s_{l}\left(t\right)+w_{ld}\left(t\right),$
(1)

where $P_{l}$ denotes the transmit power at $U_{l}$, $x_{l}$ denotes the
symbol transmitted by $U_{l}$, $s_{l}\left(t\right)$ denotes the spreading
code of $x_{l}$, and $w_{ld}$ is the additive white Gaussian noise (AWGN) with
zero mean and the variance of $N_{0}$. The cross correlation between
$s_{p}\left(t\right)$ and $s_{q}\left(t\right)$ are expressed as
$\rho_{pq}=\left\langle{s_{p}\left(t\right),s_{q}\left(t\right)}\right\rangle$,
where
$\left\langle{f\left(t\right),g\left(t\right)}\right\rangle\triangleq\frac{1}{T}\int_{0}^{T}{f\left(t\right)g^{\ast}\left(t\right)dt}$
is the inner product between $f\left(t\right)$ and $g\left(t\right)$ during
the symbol interval $T$. Moreover, we assume that
$\rho_{ll}=\left\|{s_{l}\left(t\right)}\right\|^{2}=1$. The signals received
at $R_{q}$ from $U_{l}$ is given by

$y_{lq}\left(t\right)=h_{lq}\sqrt{P_{l}}x_{l}s_{l}\left(t\right)+w_{lq}\left(t\right),$
(2)

where $w_{lq}\left(t\right)$ is AWGN with zero mean and the variance of
$N_{0}$.

In the relay transmission phase, the signal received at the destination from
$R_{q}$ is given by

$y_{qd}\left(t\right)={h_{qd}}\underbrace{\sum_{l=1}^{L}{{\beta_{ql}}\sqrt{{P_{ql}}}{x_{l}}{s_{l}}\left(t\right)}}_{{f_{q}}\left(x\right)}+{w_{qd}}\left(t\right),$
(3)

where $P_{ql}$ denotes the transmit power at $R_{q}$ and $w_{qd}$ is AWGN with
zero mean and the variance of $N_{0}$. In (3), the scalar $\beta_{ql}$ denotes
the state whether $R_{q}$ decodes $x_{l}$ correctly. Specifically,
$\beta_{ql}$ is equal to $1$ if $R_{q}$ decodes $x_{l}$ correctly, but $0$
otherwise.

For the detection of the received signals at the destination, we assume that
the full knowledge of the channel state information are available at the
receivers with the aid of a preamble in the transmitted signal. We also assume
that the destination has the detection states at the relays, which can be
obtained via an indicator in the relaying signal. At the destination, the
spreading codes $s_{l}$ is employed such that the information symbols $x_{l}$
is separated from $y_{ld}$ and $y_{qd}$, where $l\in\\{1,\cdots,L\\}$. For any
desired symbol $x_{l}$, the destination combines the information of $x_{l}$
from $U_{l}$ and the $Q$ relays using maximum ratio combining (MRC).
Therefore, the instantaneous signal-to-noise ratio (SNR) of $x_{l}$ is
expressed as [23]

$\gamma_{l}=\frac{{P_{l}\left|{h_{ld}}\right|^{2}}}{{N_{0}}}+\sum_{q=1}^{Q}{\frac{{\beta_{ql}P_{lq}\left|{h_{qd}}\right|^{2}}}{{N_{0}\varepsilon_{l}}}},$
(4)

where $\varepsilon_{l}$ is the $l$th diagonal element of matrix
${\bf{R}}^{-1}$ associated with symbol $x_{l}$ and $\bf{R}$ is given by

${\bf{R}}=\left[{\begin{array}[]{*{20}c}1&{\rho_{21}}&\cdots&{\rho_{Q1}}\\\
{\rho_{12}}&1&\cdots&{\rho_{Q2}}\\\ \vdots&\vdots&\ddots&\vdots\\\
{\rho_{1Q}}&{\rho_{2Q}}&\cdots&1\\\ \end{array}}\right].$ (5)

To facilitate the performance analysis in the following section, we re-express
(4) as a unitary expression given by

$\gamma_{l}=c_{0}\left|{h_{0}}\right|^{2}+\sum_{q=1}^{Q}\beta_{ql}c_{q}\left|{h_{q}}\right|^{2},$
(6)

where $c_{0}=P_{l}d_{dl}^{-\alpha}/N_{0}$ denotes the equivalent SNR at $D$
received from $U_{l}$, $h_{0}$ denotes the unitary Nakagami-$m$ fading
coefficient between $U_{l}$ and $D$ with variance one,
$c_{q}=P_{l}d_{ql}^{-\alpha}/N_{0}\varepsilon_{l}$ denotes the $q$th
equivalent SNR received at $D$ from $R_{q}$, and $h_{q}$ denotes the unitary
Nakagami-$m$ fading coefficient between $R_{q}$ and $D$ with variance one.

##  III SER Analysis over Nakagami-$m$ Fading Channel

In this section, we first derive new closed-form expressions for the exact SER
with $M$-PSK and $M$-QAM. We then derive new compact expressions for the
asymptotic SER, which will allow us to examine the network behavior in the
high SNR regime.

###  III-A Exact SER

In DF protocol, $\beta_{ql}$ denotes the decoding state at $R_{q}$ associated
with $x_{l}$. Based on the values of all $\beta_{ql}$’s, we define a decimal
number as
$S_{l}=\left[\beta_{1l}~{}\beta_{2l}~{}\cdots~{}\beta_{Ql}\right]_{2}$ to
represent one of $2^{Q}$ network decoding states at $Q$ relays associated with
$x_{l}$. Since all the channels in this network are mutually independent, the
events that whether $R_{q}$ correctly decodes the received signal are
independent. It follows that $\beta_{ql}$’s are independent Bernoulli random
variables, the distribution of which is written as

$G\left({\beta_{ql}}\right)=\left\\{{\begin{array}[]{ll}{1-{\rm{SER}}_{ql},}&{{\rm{if}}~{}{\beta_{ql}=1}}\\\
{{\rm{SER}}_{ql},}&{{\rm{otherwise,}}}\\\ \end{array}}\right.$ (7)

where ${\rm{SER}}_{ql}$ denotes the SER of detecting $x_{l}$ at $R_{q}$.
Therefore, the joint probability of a particular combination of $x_{l}$ in
$S_{l}$ is written as

${\Pr}\left(S_{l}\right)=\prod_{q=1}^{Q}{G\left(\beta_{ql}\right)}.$ (8)

Applying Bayesian rule, the SER of detecting $x_{l}$ at $D$ is derived as

$\displaystyle{\rm{SER}}_{l}=\sum_{S_{l}=0}^{2^{Q}-1}{{\rm{SER}}_{\gamma_{l|S_{l}}}\Pr\left({S_{l}}\right)}=\sum_{S_{l}=0}^{2^{Q}-1}{{\rm{SER}}_{\gamma_{l|S_{l}}}\prod_{q=1}^{Q}{G\left({\beta_{ql}}\right)}},$
(9)

where ${{\rm{SER}}_{\gamma_{l|S_{l}}}}$ denotes the SER of detecting $x_{l}$
at $D$ conditioned on $S_{l}$. To facilitate the calculation of (9), we
present the exact closed-form results for ${{\rm{SER}}_{\gamma_{l|S_{l}}}}$
and ${G\left({\beta_{ql}}\right)}$, as follows.

####  III-A1 Exact Results for ${\rm{SER}_{\gamma_{l|S_{l}}}}$

We commence the derivation of ${\rm{SER}_{\gamma_{l|S_{l}}}}$ by presenting
the PDF of $\gamma_{l|S_{l}}$, $f_{\gamma_{l|S_{l}}}\left(v\right)$. If there
are $N$ “$1$” elements in one set $S_{l}$, let
$a_{1},a_{2},\cdots,a_{N}\in\left\\{c_{1},c_{2},\cdots,c_{Q}\right\\}$ denote
the equivalent SNRs of the $N$ relays which decode $x_{l}$ successfully, where
$N\leq Q$. We then define $a_{0}=c_{0}$ to make the source equivalent to the
zeroth relay. As such, the SNR at $D$ to one desired symbol $x_{l}$ can be
rewritten as

$\gamma_{l|S_{l}}=a_{0}\left|{h_{0}}\right|^{2}+\sum_{n=1}^{N}{a_{n}\left|{h_{n}}\right|^{2}}=\sum_{n=0}^{N}{\underbrace{a_{n}\left|{h_{n}}\right|^{2}}_{Y_{n}}}.$
(10)

According to [25], the PDF of $Y_{n}$ in Nakagami-$m$ fading is given by

$f_{Y_{n}}\left(y\right)=\frac{{{m_{n}}^{m_{n}}y^{m_{n}-1}}}{{{a_{n}}^{m_{n}}\Gamma\left(m_{n}\right)}}\exp\left({-\frac{{m_{n}y}}{{a_{n}}}}\right),$
(11)

where $m_{n}$ is the Nakagami-$m$ fading parameter between the $n$th
successful relay and the destination. For example, if the first and the second
relays out of three relays successfully decode the information from $U_{l}$,
we have $m_{1}=m_{1d}$ and $m_{2}=_{2d}$. Using Fourier transform together
with [26, eq. (3.351.3)], the characteristic function (CF) of $Y_{n}$ is
calculated as

$\begin{split}C_{Y_{n}}\left(u\right)&=\int_{-\infty}^{\infty}{f_{Y_{n}}\left(y\right)e^{juy}dy}=\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}.\end{split}$
(12)

Given that $\gamma_{l|S_{l}}$ is the sum of $Y_{n}$’s, the CF of
$\gamma_{l|S_{l}}$ is obtained as

$\begin{split}C_{\gamma_{l|S_{l}}}\left(u\right)&=\prod\limits_{n=0}^{N}{C_{Y_{n}}\left(u\right)}=\prod\limits_{n=0}^{N}{\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}}.\end{split}$
(13)

Applying inverse Fourier transform, the PDF of $\gamma_{l|S_{l}}$ is derived
as

$\begin{split}f_{\gamma_{l|S_{l}}}\left(v\right)&=\frac{1}{{2\pi}}\int_{-\infty}^{\infty}{C_{\gamma_{l|S_{l}}}\left(u\right)e^{-juv}du}\\\
&=\frac{1}{{2\pi}}\int_{-\infty}^{\infty}{\underbrace{\left({\prod\limits_{n=0}^{N}{\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}}}\right)e^{-juv}}_{g\left(u\right)}du}.\end{split}$
(14)

We next seek the solution for $g\left(u\right)$. Due to the randomicity of the
wireless channels, we note that $g(u)$ has $N+1$ different poles
$z_{0}=-jm_{0}/a_{0}$, $z_{1}=-jm_{1}/a_{1}$, $\cdots$, $z_{N}=-jm_{N}/a_{N}$
in complex field. As such, based on residue theorem [27], the residue of
$k^{\rm{th}}$ pole of $g(u)$ in complex field can be expressed as

$\displaystyle{\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]=$
$\displaystyle\frac{1}{{\left({m_{k}-1}\right)!}}\mathop{\lim}\limits_{u\to
z_{k}}\frac{{d^{m_{k}-1}}}{{du^{m_{k}-1}}}\left[{\left({u-z_{k}}\right)^{m_{k}}g\left(u\right)}\right]$
$\displaystyle=$
$\displaystyle\frac{{{m_{k}}^{m_{k}}}}{{(-j)^{m_{k}}{a_{k}}^{m_{k}}\left({m_{k}-1}\right)!}}\mathop{\lim}\limits_{u\to
z_{k}}\frac{{d^{m_{k}-1}}}{{du^{m_{k}-1}}}\left[{e^{-juv}\left({\prod\limits_{n=0,n\neq
k}^{N}{\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}}}\right)}\right].$
(15)

As per the general Leibniz’s rule, we derive
${\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]$ in (III-A1) as

$\displaystyle{\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]$
$\displaystyle=$
$\displaystyle\frac{{{m_{k}}^{m_{k}}}}{{(-j)^{m_{k}}a_{k}^{m_{k}}\left({m_{k}-1}\right)!}}\mathop{\lim}\limits_{u\to
z_{k}}\sum_{i=0}^{m_{k}-1}{\left[{\left({-jv}\right)^{m_{k}-1-i}e^{-juv}}\right]}\sum_{i_{0}=0}^{i}{\sum_{i_{1}=0}^{i_{0}}{\cdots\sum_{i_{k-1}=0}^{i_{k-2}}{\sum_{i_{k+1}=0}^{i_{k-1}}{\cdots\sum_{i_{N-1}=0}^{i_{N-2}}}}}}$
$\displaystyle\times{{m_{k}-1}\choose i}{i\choose i_{0}}{i_{0}\choose
i_{1}}\cdots{i_{k-2}\choose i_{k-1}}{i_{k-1}\choose
i_{k+1}}\cdots{i_{N-2}\choose
i_{N-1}}\left[{\frac{{d^{i_{N-1}}}}{{du^{i_{N-1}}}}\left({1-\frac{{jua_{N}}}{m_{N}}}\right)^{-m_{N}}}\right]$
$\displaystyle\times\left[{\frac{{d^{i_{N-2}-i_{N-1}}}}{{du^{i_{N-2}-i_{N-1}}}}\left({1-\frac{{jua_{N-1}}}{m_{N-1}}}\right)^{-m_{N-1}}}\right]\cdots\left[{\frac{{d^{i_{k-1}-i_{k+1}}}}{{du^{i_{k-1}-i_{k+1}}}}\left({1-\frac{{jua_{k+1}}}{m_{k+1}}}\right)^{-m_{k+1}}}\right]$
$\displaystyle\times\left[{\frac{{d^{i_{k-2}-i_{k-1}}}}{{du^{i_{k-2}-i_{k-1}}}}\left({1-\frac{{jua_{k-1}}}{m_{k-1}}}\right)^{-m_{k-1}}}\right]\cdots\left[{\frac{{d^{i_{0}-i_{1}}}}{{du^{i_{0}-i_{1}}}}\left({1-\frac{{jua_{1}}}{m_{1}}}\right)^{-m_{1}}}\right]\left[{\frac{{d^{i-i_{0}}}}{{du^{i-i_{0}}}}\left({1-\frac{{jua_{0}}}{m_{0}}}\right)^{-m_{0}}}\right].$
(16)

Upon close observation, we simplify (III-A1) as

${\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]=\sum_{i=0}^{m_{k}-1}{jB_{N,k,i}}v^{m_{k}-1-i}\exp\left({-\frac{{m_{k}v}}{{a_{k}}}}\right),$
(17)

where $B_{N,k,i}$ is defined as

$\displaystyle B_{N,k,i}$ (18) $\displaystyle=$
$\displaystyle\frac{{{m_{k}}^{m_{k}}}{(-1)}^{-i}}{{a_{k}^{m_{k}}\left({{m_{k}}-1}\right)!}}\sum_{i_{0}=0}^{i}{\sum_{i_{1}=0}^{i_{0}}{\cdots\sum_{i_{k-1}=0}^{i_{k-2}}{\sum_{i_{k+1}=0}^{i_{k-1}}{\cdots\sum_{i_{N-1}=0}^{i_{N-2}}}}}}{{m_{k}-1}\choose
i}{i\choose i_{0}}{i_{0}\choose i_{1}}\cdots{i_{k-2}\choose
i_{k-1}}{i_{k-1}\choose i_{k+1}}\cdots{i_{N-2}\choose i_{N-1}}$
$\displaystyle\times\left({\frac{{a_{N}}}{{m_{N}}}}\right)^{i_{N-1}}\left({\frac{{a_{N-1}}}{{m_{N-1}}}}\right)^{i_{N-2}-i_{N-1}}\cdots\left({\frac{{a_{k+1}}}{{m_{k+1}}}}\right)^{i_{k-1}-i_{k+1}}\left({\frac{{a_{k-1}}}{{m_{k-1}}}}\right)^{i_{k-2}-i_{k+1}}\cdots\left({\frac{{a_{1}}}{{m_{1}}}}\right)^{i_{0}-i_{1}}\left({\frac{{a_{0}}}{{m_{0}}}}\right)^{i-i_{0}}$
$\displaystyle\times\left(m_{N}\right)_{i_{N-1}}\left(m_{N-1}\right)_{i_{N-2}-i_{N-1}}\cdots\left(m_{k+1}\right)_{i_{k-1}-i_{k+1}}\left(m_{k-1}\right)_{i_{k-2}-i_{k-1}}\cdots\left(m_{1}\right)_{i_{0}-i_{1}}\left(m_{0}\right)_{i-i_{0}}$
$\displaystyle\times\left({1-\frac{{m_{k}a_{N}}}{{m_{N}a_{k}}}}\right)^{-m_{N}-i_{N-1}}\left({1-\frac{{m_{k}a_{N-1}}}{{m_{N-1}a_{k}}}}\right)^{-m_{N-1}-i_{N-2}+i_{N-1}}\cdots\left({1-\frac{{m_{k}a_{k+1}}}{{m_{k+1}a_{k}}}}\right)^{-m_{k+1}-i_{k-1}+i_{k+1}}$
$\displaystyle\times\left({1-\frac{{m_{k}a_{k-1}}}{{m_{k-1}a_{k}}}}\right)^{-m_{k-1}-i_{k-2}+i_{k-1}}\cdots\left({1-\frac{{m_{k}a_{1}}}{{m_{1}a_{k}}}}\right)^{-m_{1}-i_{0}+i_{1}}\left({1-\frac{{m_{k}a_{0}}}{{m_{0}a_{k}}}}\right)^{-m_{0}-i+i_{0}},$
(19)

and
$\left(m_{n}\right)_{i}=\Gamma\left({m_{n}+i}\right)/\Gamma\left(m_{n}\right)$
is the Pochmann symbol. Based on the residues of the poles, we confirm that
$g(u)$ is available for the residue theorem. Specifically, using the residue
of $k^{\rm{th}}$ pole of $g(u)$ in complex field, we obtain

$\int_{-\infty}^{\infty}{g\left(u\right)}du=-2\pi j\sum_{k=0}^{N}{{\mathop{\rm
Res}\nolimits}\left[{g\left({z_{k}}\right),z_{k}}\right]}.$ (20)

The proof of (20) is shown in Appendix A. Based on (17) and (20), the PDF of
$\gamma_{l|S_{l}}$ in (14) is derived as

$f_{\gamma_{l|S_{l}}}\left(v\right)=\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}v^{m_{k}-1-i}\exp\left({-\frac{{m_{k}v}}{{a_{k}}}}\right)}.$
(21)

With the aid of $f_{\gamma_{l|S_{l}}}\left(v\right)$ in (21), we are capable
to derive ${\rm{SER}}_{\gamma_{l|S_{l}}}$ for $M$-PSK and $M$-QAM. First,
${\rm{SER}}_{\gamma_{l|S_{l}}}$ for $M$-PSK is derived as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\int_{0}^{\infty}{\alpha\exp\left({-\frac{{b\gamma}}{{\sin^{2}\theta}}}\right)f_{\gamma_{l|S_{l}}}\left(\gamma\right)}d\gamma
d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}}\int_{0}^{\left({M-1}\right)\pi/M}{\int_{0}^{\infty}{\gamma^{m_{k}-1-i}\exp\left({\left({-\frac{b}{{\sin^{2}\theta}}-\frac{m_{k}}{{a_{k}}}}\right)\gamma}\right)}d\gamma
d\theta},$ (22)

where $\alpha$ and $b$ are modulation specific constants. For $M$-PSK,
$\alpha=1$ and $b=\sin^{2}(\pi/M)$. With the aid of [26, eq. (3.351.3)] and
[28, eq. (5A.17)], (III-A1) is derived as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}}\Gamma\left({m_{k}-i}\right)\int_{0}^{\left({M-1}\right)\pi/M}{\left({\frac{b}{{\sin^{2}\theta}}+\frac{m_{k}}{{a_{k}}}}\right)^{-m_{k}+i}d\theta}$
$\displaystyle=$
$\displaystyle\alpha\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}}\Gamma\left({m_{k}-i}\right)\left({\frac{{a_{k}}}{m_{k}}}\right)^{m_{k}-i}\left[\frac{{M-1}}{M}-\frac{1}{\pi}\sqrt{\frac{{a_{k}b}}{{a_{k}b+m_{k}}}}\left(\left({\frac{\pi}{2}+\tan^{-1}\omega}\right)\sum_{p=0}^{m_{k}-i-1}{2p\choose
p}\right.\right.$
$\displaystyle\left.\left.\times\left(4\left({1+\frac{{a_{k}b}}{m_{k}}}\right)\right)^{-p}+\sin\left({\tan^{-1}\omega}\right)\sum_{p=1}^{m_{k}-i-1}\sum_{t=1}^{p}T_{p,t}\left(1+\frac{a_{k}b}{m_{k}}\right)^{-p}\left({\cos\left({\tan^{-1}\omega}\right)}\right)^{2\left({p-t}\right)+1}\right)\right],$
(23)

where $\omega=\sqrt{\frac{{a_{k}b}}{{a_{k}b+m_{k}}}}\cot\frac{\pi}{M}$ and
$T_{p,t}=\frac{{2p\choose
p}}{{2(p-t)\choose{p-t}}4^{t}\left[{2\left(p-t\right)+1}\right]}$.

We then derive ${\rm{SER}}_{\gamma_{l|S_{l}}}$ for $M$-QAM as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}=$
$\displaystyle\frac{4}{\pi}\int_{0}^{\pi/2}{\int_{0}^{\infty}{\alpha\exp\left({-\frac{{b\gamma}}{{{{\sin}^{2}}\theta}}}\right){f_{{\gamma_{l|{S_{l}}}}}}\left(\gamma\right)}d\gamma
d\theta}$
$\displaystyle-\frac{4}{\pi}\int_{0}^{\pi/4}{\int_{0}^{\infty}{\alpha^{2}\exp\left({-\frac{{b\gamma}}{{{{\sin}^{2}}\theta}}}\right){f_{{\gamma_{l|{S_{l}}}}}}\left(\gamma\right)}d\gamma
d\theta},$ (24)

where $\alpha=4\left({1-1/\sqrt{M}}\right)$ and
$b=3/\left(2\left({M-1}\right)\right)$. Calculating the integrals in (III-A1),
${\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}$ is derived as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}$ $\displaystyle=$
$\displaystyle
4\alpha\sum\limits_{k=0}^{N}{\sum\limits_{i=0}^{{m_{{}_{k}}}-1}{{B_{k,i}}}}\Gamma\left({{m_{k}}-i}\right){\left({\frac{{{a_{k}}}}{{{m_{k}}}}}\right)^{{m_{i}}-i}}\left[\frac{1}{2}-\frac{1}{\pi}\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\left(\left({\frac{\pi}{2}+{{\tan}^{-1}}{\omega_{1}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}\right.\right.$
$\displaystyle\times\left.\left.{\left({4\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}+\sin\left({{{\tan}^{-1}}{\omega_{1}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\omega_{1}}}\right)}\right)^{2\left({p-q}\right)+1}}\right)\right]$
$\displaystyle-4\alpha^{2}\sum\limits_{k=0}^{N}{\sum\limits_{i=0}^{{m_{{}_{k}}}-1}{{B_{k,i}}}}\Gamma\left({{m_{k}}-i}\right){\left({\frac{{{a_{k}}}}{{{m_{k}}}}}\right)^{{m_{i}}-i}}\left[\frac{1}{4}-\frac{1}{\pi}\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\left(\left({\frac{\pi}{2}-{{\tan}^{-1}}{\omega_{2}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}\right.\right.$
$\displaystyle\times\left.\left.{\left({4\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}-\sin\left({{{\tan}^{-1}}{\omega_{2}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\omega_{2}}}\right)}\right)^{2\left({p-q}\right)+1}}\right)\right],$
(25)

where
${\omega_{1}}=\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\cot\frac{\pi}{2}$
and
${\omega_{2}}=\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\cot\frac{\pi}{4}$.

####  III-A2 Exact Results for $G\left(\beta_{ql}\right)$

We now analyze $G\left(\beta_{ql}\right)$ for $M$-PSK and $M$-QAM,
respectively. According to (7), it is equivalent to analyze ${\rm{SER}}_{ql}$.
Using (2), the received SNR at $R_{q}$ is written as

$\displaystyle\gamma_{ql}=\frac{{P_{l}\left|{h_{ql}}\right|^{2}}}{{N_{0}}}=\frac{{P_{l}d_{ql}^{-\alpha}}}{{N_{0}}}\left|h\right|^{2}=c_{ql}\left|h\right|^{2},$
(26)

where $c_{ql}=P_{l}d_{ql}^{-\alpha}/N_{0}$ denotes the equivalent SNR at
$R_{q}$ received from $U_{l}$, and $h$ denotes the unitary Nakagami-$m$ fading
coefficients between $U_{l}$ and $R_{q}$ with variance one.

We first derive ${\rm{SER}}_{ql}$ for $M$-PSK as

$\displaystyle{\rm{SER}}_{ql,{\textrm{MPSK}}}=$
$\displaystyle\frac{{\alpha(M-1)}}{M}-\frac{\alpha}{\pi}\sqrt{\frac{{c_{ql}b}}{{c_{ql}b+m_{lq}}}}\left[\left({\frac{\pi}{2}+\tan^{-1}\varpi}\right)\sum_{p=0}^{m_{lq}-1}{2p\choose
p}\left({4\left({1+\frac{{c_{ql}b}}{m_{lq}}}\right)}\right)^{-p}\right.$
$\displaystyle+\left.\sin\left({\tan^{-1}\varpi}\right)\sum_{p=1}^{m_{lq}-1}{\sum_{t=1}^{p}{T_{p,t}}}\left({1+\frac{{c_{ql}b}}{m_{lq}}}\right)^{-p}\left({\cos\left({\tan^{-1}\varpi}\right)}\right)^{2\left({p-t}\right)+1}\right],$
(27)

where $\varpi=\sqrt{\frac{{c_{ql}b}}{{c_{ql}b+m_{lq}}}}\cot\frac{\pi}{M}$.

We then derive ${\rm{SER}}_{ql}$ for $M$-QAM as

$\displaystyle{\rm{SER}}_{ql,{\textrm{MQAM}}}=$ $\displaystyle
2\alpha-\frac{4\alpha}{\pi}\sqrt{\frac{{{c_{ql}}b}}{{{c_{ql}}b+{m_{k}}}}}\left[\left({\frac{\pi}{2}+{{\tan}^{-1}}{\varpi_{1}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}{\left({4\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}\right.$
$\displaystyle+\left.\sin\left({{{\tan}^{-1}}{\varpi_{1}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\varpi_{1}}}\right)}\right)^{2\left({p-q}\right)+1}}\right]$
$\displaystyle-\alpha^{2}+\frac{{{4\alpha^{2}}}}{\pi}\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\left[\left({\frac{\pi}{2}-{{\tan}^{-1}}{\varpi_{2}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}{\left({4\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}\right.$
$\displaystyle\left.-\sin\left({{{\tan}^{-1}}{\varpi_{2}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\varpi_{2}}}\right)}\right)^{2\left({p-q}\right)+1}}\right],$
(28)

where
${\varpi_{1}}=\sqrt{\frac{{{c_{ql}}b}}{{{a_{k}}b+{m_{lq}}}}}\cot\frac{\pi}{2}$
and
${\varpi_{2}}=\sqrt{\frac{{{c_{ql}}b}}{{{a_{k}}b+{m_{lq}}}}}\cot\frac{\pi}{4}$.

Substituting (III-A2) and (III-A2) into (7), we obtain
${G\left({\beta_{ql}}\right)}$ for $M$-PSK and $M$-QAM, respectively.
Therefore, we insert (7) and (III-A1) into (9), which yields the exact closed-
form SER for $M$-PSK, and substitute (7) and (III-A1) into (9), which gives
the exact SER for $M$-QAM. Observing (III-A1), (III-A1), (III-A2), and
(III-A2), we see that the exact SER expressions for $M$-PSK and $M$-QAM are
given in closed-form and are valid to arbitrary numbers of sources and relays.

###  III-B Asymptotic SER

We now provide useful insights into the network behavior in the high SNR
regime. In doing so, new compact closed-form expressions are presented for the
asymptotic SER.

We first focus on $M$-PSK. Based on (10), ${\rm{SER}}_{\gamma_{l|S_{l}}}$ for
$M$-PSK can be alternatively written as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\int_{0}^{\infty}{\exp\left({-\frac{{b\gamma_{l|S_{l}}}}{{\sin^{2}\theta}}}\right)f_{{}_{\gamma_{l|S_{l}}}}\left({\gamma_{l|S_{l}}}\right)}d\gamma_{l|S_{l}}d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\left({\int_{0}^{\infty}{\int_{0}^{\infty}{\cdots\int_{0}^{\infty}{\exp\left({-\frac{{b\sum_{i=0}^{N}{y_{i}}}}{{\sin^{2}\theta}}}\right)\left({\prod\limits_{i=0}^{N}{f_{Y_{i}}\left({y_{i}}\right)}}\right)\prod\limits_{i=0}^{N}{dy_{i}}}}}}\right)d\theta}.$
(29)

Substituting (11) into (III-B) and using [26, eq. (3.351.3)], we obtain
${\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$ as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\prod\limits_{i=0}^{N}{\left({\int_{0}^{\infty}{\frac{{{m_{i}}^{m_{i}}y_{i}^{{m_{i}}-1}}}{{{a_{i}}^{m_{i}}\Gamma\left({m_{i}}\right)}}\exp\left({-\left({\frac{b}{{\sin^{2}\theta}}+\frac{{m_{i}}}{{a_{i}}}}\right)y_{i}}\right)dy_{i}}}\right)}d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}^{m_{i}}}}{{a_{i}^{m_{i}}}}\left({\frac{b}{{\sin^{2}\theta}}+\frac{{m_{i}}}{{a_{i}}}}\right)^{-{m_{i}}}}\right)}d\theta}.$
(30)

We next use [26, eq. (2.513.1)] to develop an asymptotic expression for
${\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$ as

$\displaystyle{\rm{SER}}_{{\gamma_{l|{S_{l}}}},{\rm{MPSK}}}^{\infty}$
$\displaystyle\leq\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}^{{m_{i}}}}}{{a_{i}^{{m_{i}}}}}{{\left({\frac{b}{{{{\sin}^{2}}\theta}}}\right)}^{-{m_{i}}}}}\right)}d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\left({\prod\limits_{i=0}^{N}{{{\left({\frac{{{m_{i}}}}{{{a_{i}}b}}}\right)}^{{m_{i}}}}}}\right)\int_{0}^{\left({M-1}\right)\pi/M}{{{\sin}^{2\sum\nolimits_{i=0}^{N}{{m_{i}}}}}\theta
d\theta}$
$\displaystyle=\frac{\alpha}{{\pi{b^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}}{A_{M,N}}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}}}{{{a_{i}}}}}\right)}^{{m_{i}}}}$
(31)

where

$\displaystyle A_{M,N}=$
$\displaystyle\frac{1}{{2^{2\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}{{2\sum\nolimits_{i=0}^{N}{{m_{i}}}}\choose{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}\frac{\left({M-1}\right)\pi}{M}+\frac{{\left({-1}\right)^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}{{2^{2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-1}}}$
$\displaystyle\times\sum_{k=0}^{{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-1}{\left({-1}\right)^{k}{2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}\choose
k}\frac{{\sin\left({\left({2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-2k}\right)\frac{\left({M-1}\right)\pi}{M}}\right)}}{{2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-2k}}}.$
(32)

Similarly, an asymptotic expression for ${\rm{SER}}_{ql,{\textrm{MPSK}}}$ is
obtained as

${\rm{SER}}_{ql,{\textrm{MPSK}}}^{\infty}\leq\frac{\alpha}{\pi}\left({\frac{{{m_{lq}}}}{{b}}}\right)^{{m_{lq}}}\frac{A_{M,{m_{lq}},0}}{{c_{ql}^{m_{lq}}}}.$
(33)

Correspondingly, the asymptotic $G\left({\beta_{ql}}\right)$ for $M$-PSK is
given as

$G\left({\beta_{ql}}\right)_{\textrm{MPSK}}^{\infty}=\left\\{{\begin{array}[]{ll}{1,}&{{\rm{if}}~{}{\beta_{ql}=1}}\\\
{{\rm{SER}}_{ql,{\textrm{MPSK}}}^{\infty},}&{{\rm{otherwise.}}}\\\
\end{array}}\right.$ (34)

Based on (III-B), (33), and (34), the asymptotic SER for $M$-PSK is derived as

${\rm{SER}}_{l,{\textrm{MPSK}}}^{\infty}=\sum_{S_{l}=0}^{2^{Q}-1}{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}^{\infty}\prod_{q=1}^{Q}G\left({\beta_{ql}}\right)_{\textrm{MPSK}}^{\infty}.$
(35)

Following the same procedure outlined for $M$-PSK, we derive the asymptotic
SER for $M$-QAM as

${\rm{SER}}_{l,{\textrm{MQAM}}}^{\infty}=\sum_{S_{l}=0}^{2^{Q}-1}{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}^{\infty}\prod_{q=1}^{Q}G\left({\beta_{ql}}\right)_{\textrm{MQAM}}^{\infty},$
(36)

where the asymptotic ${\rm{SER}}_{\gamma_{l|S_{l}},}$ for $M$-QAM is derived
as

$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}^{\infty}\leq\frac{4\alpha}{{\pi{b^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}}{A_{2,N}}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}}}{{{a_{i}}}}}\right)}^{{m_{i}}}}-\frac{{4{\alpha^{2}}}}{{\pi{b^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}}{A_{4/3,N}}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}}}{{{a_{i}}}}}\right)}^{{m_{i}}}},$
(37)

with ${A_{2,{m_{i}},N}}$ and ${A_{4/3,{m_{i}},N}}$ being defined in (III-B),
and the asymptotic $G\left({\beta_{ql}}\right)$ for $M$-QAM is derived as

$G\left({\beta_{ql}}\right)_{\textrm{MQAM}}^{\infty}=\left\\{{\begin{array}[]{ll}{1,}&{{\rm{if}}~{}{\beta_{ql}=1}}\\\
{{\rm{SER}}_{ql,{\textrm{MQAM}}}^{\infty},}&{{\rm{otherwise,}}}\\\
\end{array}}\right.$ (38)

with

${\rm{SER}}_{ql,{\textrm{MQAM}}}^{\infty}\leq\frac{{4\alpha}}{\pi}{\left({\frac{{{m_{lq}}}}{b}}\right)^{{m_{lq}}}}\frac{{{A_{2,{m_{lq}},0}}}}{{c_{ql}^{{m_{lq}}}}}-\frac{{4{\alpha^{2}}}}{\pi}{\left({\frac{{{m_{lq}}}}{b}}\right)^{{m_{lq}}}}\frac{{{A_{4/3,{m_{lq}},0}}}}{{c_{ql}^{{m_{lq}}}}}.$
(39)

Based on (35) and (36), we next examine the diversity order of the network,
which represents the slope of the SER against average SNR in a log-log scale.
According to (35) and (36), the asymptotic SER of $U_{l}$ can rewritten as

${\rm{SER}}_{l}^{\infty}=\sum\limits_{{S_{l}}=0}^{{2^{Q}}-1}{{\Theta_{{S_{l}}}}\left[{\prod\limits_{i=0}^{N}{\frac{1}{{a_{i}^{{m_{i}}}}}}}\right]}\prod\limits_{q=1}^{Q}{\frac{1}{{c_{ql}^{{m_{lq}}}}}},$
(40)

where $\Theta_{{S_{l}}}$ denotes the coefficient independent of $a_{i}$ and
$c_{ql}$.in which the diversity order can be confirmed as

${\rm{div}}={m_{ld}}+\sum\nolimits_{q=1}^{Q}{\min\left({{m_{qd}},{m_{lq}}}\right)}.$
(41)

It is evident from (41) that the full diversity order is achieved, which is
determined by the Nakagami-$m$ fading parameters of all the channels. Notably,
the diversity order is independent of the number of sources. In particular,
this full diversity order is preserved even non-orthogonal STNC codes are
employed.

##  IV Simulation and Numerical Results

In this section, simulation and numerical results are presented to examine the
impact of network parameters with STNC (e.g., the number of relays, the relay
location, Nakagami-$m$ fading parameters, and power allocation) on the SER of
$U_{l}$. In the figures, we consider a practical scenario where the relays are
placed at different distances from $D$ and $U_{l}$ with $c_{j}\neq c_{i}$ and
$c_{jl}\neq c_{il}$ for $j\neq i$. We set the distance between $U_{l}$ and $D$
as $d_{ld}=1$. The cross correlations between different spread codes, defined
in (1), are set to be zero. We also assume equal transmit power at each node.
Further, our results concentrate on the practical example of a highly shadowed
area with the path loss exponent as $\alpha=3.5$ [29]. In the figures, the
exact SER for $M$-PSK is evaluated by substituting (7), (III-A1), and (III-A2)
into (9), and the exact SER for $M$-QAM is evaluated by substituting (7),
(III-A1), and (III-A2) into (9). The asymptotic SER for $M$-PSK and $M$-QAM is
calculated from (35) and (36), respectively.

###  IV-A Impact of Number of Relays and Equal Nakagami-$m$ Fading Parameters

In this subsection, we focus on equal Nakagami-$m$ fading parameters with
$m_{i}=m$. The average received SNRs are set as
$c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$ and
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Fig. 3 plots the exact and asymptotic SER
with 4QAM. Fig. 4 plots the exact and asymptotic SER with 8PSK. From Figs. 3
and 4, we see that the asymptotic SER curves accurately predict the exact ones
in the high SNR regime. By observing these asymptotic curves, it is evident
that the diversity order increases with $Q$, which indicates that increasing
the number of relays brings an improved performance. It is also seen that the
diversity order increases with $m$, which indicates that the improvement in
fading channels leads to a reduction in the SER. Moreover, we see that the
simulation points are in precise agreement with our exact analytical curves,
which demonstrates the correctness of our analysis in Section III. Comparing
the SER in Fig. 3 with that in Fig. 4, we further see a poorer network
performance is achieved by higher order modulation schemes.

###  IV-B Impact of Relay Location

In this subsection, we consider ${d_{lq}}\neq{d_{qd}}$, which leads to
${c_{q}}\neq{c_{qd}}$, and consider equal Nakagami-$m$ fading parameters with
$m_{i}=m=2$. We further normalize ${d_{ld}}$ to unity with ${d_{ld}}=1$. Fig.
5 plots the exact SER with BPSK for $Q=2$. In this figure, _Cases 1_ , _2_ ,
_3_ represent the scenario where the relays are located close to the source,
while _Cases 4_ , _5_ , _6_ represent the scenario where the relays are
located close to the destination.

We first consider _Cases 1_ , _2_ , and _3_. We see that _Case 1_ offers a
prominent SNR advantage relative to _Case 2_. This indicates that the
reduction in the distance between the relay and the destination brings a
substantial SER improvement. We also see that _Case 1_ and _Case 3_ achieve
almost the same SER across the entire SNR range. This indicates that the SER
improvement from the reduced distance between the source and the relay is
negligible. These observations are due to the fact that the network
performance is dominant by the relay-destination link when the relays are
close to the source. As such, the quality improvement of the relay-destination
link has a higher positive impact on the SER than that of the source-relay
link.

We next consider _Cases 4_ , _5_ , and _6_. It is seen that _Case 4_ provides
a substantial SNR advantage compared to _Case 5_. It is also seen that _Case
4_ achieves a slight SNR advantage compared to _Case 6_. These observations
are explained by the fact that the network performance is dominant by the
source-relay link when the relays are close to the destination.

###  IV-C Impact of Unequal Nakagami-$m$ Fading Parameters

We concentrate on unequal Nakagami-$m$ fading parameters and set the average
received SNRs as $c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$
and $c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Fig. 6 plots the exact SER with 4QAM
for $Q=2$. This figure clearly shows that the diversity order in (41) is
accurate. For example, it is evident that the asymptotic SER curves of _Cases
1_ , _2_ , and _3_ are in parallel, which indicates that they achieve the same
diversity order. As indicated in (41), _Cases 1_ , _2_ , and _3_ achieve
identical diversity order of $3$. Moreover, we see that the diversity order of
_Case 4_ increases to $4$, and the diversity order of _Case 5_ increases to
$6$. This is predicted by (41), which shows that the diversity order is
determined by the Nakagami-$m$ fading parameters of all the channels.

###  IV-D Impact of Power Allocation

We now focus on arbitrary transmit power at each node. We consider equal
Nakagami-$m$ fading parameters with $m_{i}=m=2$, set the relay location as
${d_{l1}}=0.8$, ${d_{l2}}=1$, ${d_{1d}}=0.9$, and ${d_{2d}}=0.7$, and
normalize ${d_{ld}}$ as ${d_{ld}}=1$. We denote the transmit powers at
$U_{l}$, $R_{1}$, and $R_{2}$ as $P_{0}$, $P_{1}$, and $P_{2}$, respectively.
Under the total power constraint, we have $P_{0}+P_{1}+P_{2}=3P$. Fig. 7 plots
the exact SER versus $\xi=P_{1}/(P_{1}+P_{2})$ with 4QAM for $Q=2$ and
$P/N_{0}=12~{}{\rm{dB}}$. We see that the optimal value of $P_{1}/P_{2}$
depends on $P_{0}$. For example, the optimal power allocation is at
$P_{1}=1.75P$ and $P_{2}=0.75P$ when $P_{0}=0.5P$. Moreover, the optimal power
allocation is at $P_{1}=P_{2}=0.5P$ when $P_{0}=2P$. Using our SER expressions
with different relay locations, iterative search method can be used to find
the optimal power allocation that minimizes the SER.

###  IV-E Impact of Nonorthogonal Codes

We now turn our attention to the impact of nonorthogonal codes. We consider
$\rho_{pq}=\rho\neq 0$ for all $p,q$ and equal Nakagami-$m$ fading parameters
with $m_{i}=2$. Fig. 8 plots the exact SER with 4QAM for $Q=2$ and $N=3$. The
case of $\rho=0$ represents orthogonal codes. We see a reduction in the SER as
$\rho$ increases. We also see that the diversity order is not affected by
cross correlation. As such, the nonorthogonal codes which permit broader
applications can be used for higher throughput without sacrificing the error
rate significantly.

##  V Conclusions

In this paper, we analyzed the SER of STNC in a distributed cooperative
network where $L$ sources communicate with a single destination with the
assistance of $Q$ relays. For $M$-PSK and $M$-QAM modulation, new exact
closed-form expressions of SER over independent but not necessarily
identically distributed Nakagami-$m$ fading channels were derived. Moreover,
the asymptotic SER was derived to reveal the network performance in the high
SNR regime. Specifically, the asymptotic SER reveals that the diversity order
of STNC was determined by the Nakagami-$m$ fading parameters of all the
channels. Simulation results were used to validate our analytical expressions
and to examine the impact of Nakagami-$m$ fading parameters, relay location,
power allocation, and nonorthogonal codes on the SER.

##  Appendix A Proof of (20)

According to Fig. 9, all poles are located in a closed curve. According to the
residue theorem, the integral over a closed curve can be expressed as the
linear combination of the residues of the poles in the curve. Mathematically,
we have

$\int_{R}^{-R}{g\left(u\right)}du+\int_{C_{R}}{g\left(z\right)dz}=2\pi
j\sum_{k=0}^{N}{\rm Res}\left[{g\left({z_{k}}\right),z_{k}}\right],$ (42)

where $C_{R}$ is the counterclockwise semicircular curve in Fig. 3. The
absolute value of $\int_{C_{R}}{g\left(z\right)dz}$ in (42) can be upper
bounded as

$\displaystyle\left|{\int_{C_{R}}{g\left(z\right)dz}}\right|$
$\displaystyle\leq\int_{C_{R}}{\left|{g\left(z\right)}\right|dz}$
$\displaystyle=\int_{C_{R}}{\left|{\left({\prod\limits_{n=0}^{N}{\left({1-\frac{{jza_{n}}}{m_{n}}}\right)^{-m_{n}}}}\right)e^{-jzv}}\right|dz}$
$\displaystyle\leq\int_{C_{R}}{\left|{\prod\limits_{n=0}^{N}{\frac{1}{{\left({1-ja_{n}z}\right)^{m_{n}}}}}}\right|\left|{e^{-jzv}}\right|dz}$
$\displaystyle=\int_{C_{R}}{\prod\limits_{n=0}^{N}{\left|{\frac{1}{{\left({1-ja_{n}z}\right)^{m_{n}}}}}\right|}dz}.$
(43)

When $R\to\infty$, which indicates that the integral range of
$g\left({u}\right)$ is from $\infty$ to $-\infty$, we find the property of (A)
as

$\displaystyle\left|{\int_{C_{R}}{g\left(z\right)dz}}\right|$
$\displaystyle=\int_{C_{R}}{\prod\limits_{n=0}^{N}{\frac{1}{{{a_{n}}^{m_{n}}\left|z\right|^{m_{n}}+O\left({z^{m_{n}}}\right)}}}dz}$
$\displaystyle=\left({\prod\limits_{n=0}^{N}{\frac{1}{{{a_{n}}^{m_{n}}}}}}\right)\int_{C_{R}}{\frac{1}{{\left|z\right|^{\sum\nolimits_{n=0}^{N}{{m_{n}}}}}}dz}$
$\displaystyle=\left({\prod\limits_{n=0}^{N}{\frac{1}{{{a_{n}}^{m_{n}}}}}}\right)\frac{\pi}{{R^{\sum\nolimits_{n=0}^{N}{{m_{n}}}-1}}}$
$\displaystyle\to 0.$ (44)

Substituting (A) into (42), we obtain (20) and thus complete this proof.

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Figure 1: System model. Figure 2: A framework of space-time network coding.
Figure 3: Exact and asymptotic SER with 4QAM for $Q=1,2,3$,
$c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$, and
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Figure 4: Exact and asymptotic SER with
8PSK for $Q=1,2,3$,
$c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$, and
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Figure 5: Exact SER with BPSK for $Q=2$,
$m=2$ and 6 cases: _Case 1_ : $d_{l1}=0.2,d_{l2}=0.3,d_{1d}=1.1,d_{2d}=1.2$;
_Case 2_ : $d_{l1}=0.2,d_{l2}=0.3,d_{1d}=2,d_{2d}=2.2$; _Case 3_ :
$d_{l1}=0.8,d_{l2}=0.9,d_{1d}=1.1,d_{2d}=1.2$; _Case 4_ :
$d_{l1}=1.1,d_{l2}=1.2,d_{1d}=0.2,d_{2d}=0.3$; _Case 5_ :
$d_{l1}=2,d_{l2}=2.2,d_{1d}=0.2,d_{2d}=0.3$; and _Case 6_ :
$d_{l1}=1.1,d_{l2}=1.2,d_{1d}=0.8,d_{2d}=0.9$. Figure 6: Exact SER with 4QAM
for $Q=2$, $c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$,
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$ and 6 cases: _Case 1_ : $m_{i}=m=1$; _Case
2_ : $m_{l1}=1,m_{l2}=1,m_{ld}=1,m_{1d}=2,m_{2d}=2$; _Case 3_ :
$m_{l1}=2,m_{l2}=2,m_{ld}=1,m_{1d}=1,m_{2d}=1$; _Case 4_ :
$m_{l1}=1,m_{l2}=2,m_{ld}=1,m_{1d}=1,m_{2d}=2$; _Case 5_ : $m_{i}=m=2$. Figure
7: Exact SER with 4QAM for $Q=2$, $m=2$ with ${d_{l1}}=0.8$, ${d_{l2}}=1$,
${d_{1d}}=0.9$, ${d_{2d}}=0.7$, and different power allocation $P_{1}$,
$P_{2}$, $P_{3}$. Figure 8: Exact SER with 4QAM for $Q=2$, $N=3$, $m_{i}=2$
and different cross correlation $\rho$ with ${d_{l1}}=0.6$, ${d_{l2}}=0.8$,
${d_{1d}}=0.6$ and ${d_{2d}}=0.8$, where ${d_{ld}}$ is normalized as
${d_{ld}}=1$. Figure 9: The distribution of the poles.