# Impedance Matching in an Elastic Actuator

Tianyi Guo1 , Xiaoyu Zheng2, Peter Palffy-Muhoray1,2
_1 Advanced Materials and Liquid Crystal Institute, Kent State University, OH,
USA_
_2 Department of Mathematical Sciences, Kent State University, OH, USA_ Email:
<EMAIL_ADDRESS>Email<EMAIL_ADDRESS>author. Email:
<EMAIL_ADDRESS>
###### Abstract

We optimize the performance of an elastic actuator consisting of an active
core in a host which performs mechanical work on a load. The system, initially
with localized elastic energy in the active component, relaxes and distributes
energy to the rest of the system. Using the linearized Mooney-Rivlin
hyperelastic model in a cylindrical geometry and assuming the system to be
overdamped, we show that the value of the Young’s modulus of the impedance
matching host which maximizes the energy transfer from the active component to
the load is the geometric mean of Young’s moduli of the active component and
the elastic load. This is similar to the classic results for impedance
matching for maximizing the transmittance of light propagating through
dielectric media.

Keywords: impedance matching, elastic actuator, geometric mean

##  1 Introduction

When light propagates through a planar interface between two perfect
dielectrics, a portion of the light is reflected and the rest is transmitted.
To minimize the reflectance in medium 1 with refractive index $n_{1}$, or
equivalently, to maximize the transmitted light to medium 3 with refractive
index $n_{3}$, one can insert an index matching layer with refractive index
$n_{2}=\sqrt{n_{1}n_{3}}$ between the two media. Furthermore, the reflectance
is zero if the thickness of the index matching medium is one quarter of the
wavelength [1]. One can apply the same principle to achieve the perfect sound
transmittance by positioning a quarter wavelength impedance matching layer
with index $\sqrt{(\rho_{1}c_{1})(\rho_{3}c_{3})}$, with $\rho_{i}$ the mass
density and $c_{i}$ the speed of sound [2]. Impedance matching techniques are
widely used in applications involving elastic wave propagation as well as in
electronics [3, 4, 5].

The case of a head-on elastic collision of two rigid balls with masses $m_{1}$
and $m_{3}$ offers an interesting analogy. To maximize the energy transfer
from $m_{1}$, which has nonzero initial energy $E_{1}$, to $m_{3}$, which has
zero initial energy, one can position a rigid ball with mass
$m_{2}=\sqrt{m_{1}m_{3}}$ and zero initial energy, in between the two balls
[6].

The similarity of these very different physical phenomena is that energy is
conserved throughout the process: energy is either reflected or transmitted.
The transmitted energy can be increased when an impedance matching medium is
inserted. The fraction of reflected energy in case of normal
incidence/collision between two media/balls, is given by,

$R_{12}=\left(\frac{Z_{2}-Z_{1}}{Z_{2}+Z_{1}}\right)^{2},$ (1)

where $Z_{i}=n_{i}$ for the case of light propagation through an interface,
and $Z_{i}=m_{i}$ for elastic collision of rigid balls. The transmitted energy
is given by $T_{12}=1-R_{12}$. If there are three media in series, the
fraction of transmitted energy from medium $1$ to medium $3$, via medium $2$,
is given by

$T_{13}=T_{12}T_{23}.$ (2)

Here the interference due to multiple reflections has been neglected in Eq.
(2), and media $1$ and $3$ are assumed to be semi-infinite in the propagation
direction. Exact expression including the dependence on the thickness of layer
$2$ in the wave propagation case can be found in Refs. [1, 2]. Upon maximizing
$T_{13}$ in Eq. (2) with respect to $Z_{2}$, one arrives immediately at,

$Z_{2}=\sqrt{Z_{1}Z_{3}},$ (3)

thus the value of $Z_{2}$ which maximizes the energy transmission from $1$ to
$3$ is the geometric mean of $Z_{1}$ and $Z_{3}$. Mechanical impedance is a
measure of effectiveness of a force in producing velocity. Remarkably, one can
optimize certain energy transfer processes by inserting an index matching
component.

In this paper, we study a related problem of optimizing the transfer of
elastic energy from one elastic body to another via an impedance matching
element.

Specifically, we consider three elastic bodies: body $1$ is the active element
with stored elastic energy, contained in body $2$, the host, which is the
impedance matching element and body $3$ is the load to which we wish to
transfer elastic energy. For simplicity, we use cylindrical symmetry in our
example. All three bodies are isotropic, homogeneous and uniform and share the
same axis of symmetry; body $1$ is a cylinder, while bodies $2$ and $3$ are
annuli. The geometry is shown in Fig. 1.

Figure 1: A schematic showing the three bodies, the inner disk and two annuli,
each with different elastic modulus.

We begin with the host, body $2$, which is an annulus with a cylindrical
cavity. Initially it is stress free. We then take another elastic body, with a
different elastic modulus, which is too large (or too small) to fit fully into
the cavity of the host. We then compress (or stretch) this body until its
shape is the same as that of the cavity. This is the active body $1$. We then
place the active body, keeping its shape fixed, into the cavity of the host.
The third body is the load; an annulus whose cavity can perfectly accommodate
the host. Finally, we place the host with the active body into the cavity of
the load, as indicated in Fig. 1. The system is then allowed to relax.

When released, the internal stored elastic energy of the active medium will do
mechanical work on the load. The situation illustrated here is similar to a
light driven actuator, where the photoactive part of the system expands or
shrinks on illumination, distributing stress to the surrounding medium,
causing a deformation. The system can then do mechanical work, say expand
against a pressure. Given the properties of the actuator and the load, can we
maximize the work by choosing a suitable host material? Below, we present a
mathematical model of the deformation of elastic media in a cylindrical
geometry, and determine Young’s modulus of the impedance matching host which
maximizes the energy transferred to the load. The results suggest a strategy
for optimizing the performance of an elastic actuator.

We remark that the similar analysis cannot be carried out in a spherical
geometry with volume conserving materials, since an incompressible sphere
cannot be radially deformed.

##  2 Mathematical Model

We use Lagrangian mechanics to model the system. We start with the
incompressible Mooney-Rivlin’s hyperelastic model, in which the energy density
of an elastic material is a linear combination of invariants of the left
Cauchy-Green deformation tensor [7, 8]

$W=C_{1}(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3)+C_{2}(\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{1}^{2}\lambda_{3}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}-3),$
(4)

where $\lambda_{i},i=1,2,3$ are principal stretches, and
$\lambda_{3}=1/(\lambda_{1}\lambda_{2})$ due to incompressibility. Assuming
the deformations are small, expanding in terms of $\lambda_{1}-1$ and
$\lambda_{2}-1$, we get,

$W=4(C_{1}+C_{2})((\lambda_{1}-1)^{2}+(\lambda_{2}-1)^{2}+(\lambda_{1}-1)(\lambda_{2}-1)).$
(5)

We further assume that all deformations have cylindrical symmetry, and denote
the position of a point in body $R(r)\mathbf{\hat{r}}+Z(z)\mathbf{\hat{z}}$.
Here $r$ and $z$ are the Lagrangian coordinates denoting the position of mass
points in the undeformed system. Then the principal stretches are given by

$\lambda_{1}=\frac{\partial R}{\partial
r},\lambda_{2}=\frac{R}{r},\lambda_{3}=\frac{\partial Z}{\partial
z}=\frac{1}{\lambda_{1}\lambda_{2}},$ (6)

where $\lambda_{1}$ is along the radial, $\lambda_{2}$ along the azimuthal and
$\lambda_{3}$ along the $z-$ direction. In terms of $R(r)$, we can express the
elastic energy density in the linear regime as

$W_{E}=\frac{2}{3}E((R^{\prime}-1)^{2}+(\frac{R}{r}-1)^{2}+(R^{\prime}-1)(\frac{R}{r}-1)),$
(7)

where $E=6(C_{1}+C_{2})$ is Young’s modulus and $R^{\prime}=\partial
R/\partial r$.

The energy per length in the $z-$direction of the system consisting of the
elastic bodies $1$, $2$ and $3\,\ $is given by

$F=2\pi\left(\int_{0}^{M}W_{E_{1}}dr+\int_{1}^{r_{2}}W_{E_{2}}dr+\int_{r_{2}}^{r_{3}}W_{E_{3}}dr\right),$
(8)

where the radius of the central hole in the undeformed host is taken to be
unity, $M$ is the radius of the pre-strained active core, $r_{2}$ and $r_{3}$
are the outer radii of the host and the load, respectively.

Minimizing the total energy $F$ gives the Euler-Lagrange equation describing
the deformation. All three parts, active core, host, and load, share the same
form of the equation, which is given by

$R^{\prime\prime}r+R^{\prime}-\frac{R}{r}=0.$ (9)

It admits the solution

$R_{i}(r)=A_{i}r+\frac{B_{i}}{r},$ (10)

where $A_{i}$ and $B_{i}$, $i=1,2,3$, are determined by the interface and
boundary conditions, which are detailed below.

The continuity condition for displacements across the interfaces are given by,

$\displaystyle R_{1}(0)\text{ is finite, }$ $\displaystyle\text{or }B_{1}=0,$
(11) $\displaystyle R_{1}(M)=R_{2}(1),$ $\displaystyle\text{or
}A_{1}M=A_{2}+B_{2},$ (12) $\displaystyle R_{2}(r_{2})=R_{3}(r_{2}),$
$\displaystyle\text{or,
}A_{2}r_{2}+\frac{B_{2}}{r_{2}}=A_{3}r_{2}+\frac{B_{3}}{r_{2}}.$ (13)

In addition, the normal stresses are continuous across the two interfaces,
which are

$\displaystyle E_{1}(2R_{1}^{\prime}+\frac{R_{1}}{r}-3)r|_{r=M}$
$\displaystyle=$ $\displaystyle
E_{2}(2R_{2}^{\prime}+\frac{R_{2}}{r}-3)r|_{r=1},$ (14) $\displaystyle
E_{2}(2R_{2}^{\prime}+\frac{R_{2}}{r}-3)r|_{r=r_{2}}$ $\displaystyle=$
$\displaystyle E_{3}(2R_{3}^{\prime}+\frac{R_{3}}{r}-3)r|_{r=r_{2}}.$ (15)

We would need another boundary condition at the outmost boundary to complete
the set of equations to be solved.

###  2.1 Zero-strain boundary condition

If the outer boundary of the load is fixed, boundary condition at $r_{3}$
reads as,

$R_{3}(r_{3})=r_{3}.$ (16)

Together with the five interface conditions, these six linear equations
determine the six unknowns, $A_{i}$ and $B_{i}$ uniquely, and they are
functions of $E_{i}$ and $r_{i}$, $i=1,2,3$. Since the solutions are rather
lengthy algebraic expressions, we omit them and only report the final
optimization results.

We are interested in the transfer of elastic energy from the active core to
the outside load. We therefore ask: what value of Young’s modulus $E_{2}$ of
the host material will maximizes the transfer of energy from the active core
to the load? Maximizing the energy in the load transferred from the active
core is equivalent to maximizing the displacement of inner radius of the load
$R_{3}(r_{2})$. We note that $R_{3}(r_{2})=A_{3}r_{2}+B_{3}/r_{2}$. Taking the
derivative of $R_{3}(r_{2})$ with respect to $E_{2}$, we obtain,

$E_{2}=\sqrt{E_{1}E_{3}}\sqrt{\frac{3r_{2}^{2}+r_{3}^{2}}{r_{3}^{2}-r_{2}^{2}}}.$
(17)

In the case of $r_{3}\rightarrow\infty$, the load is infinitely large, we have
$E_{2}=\sqrt{E_{1}E_{3}}$. This results is a reminiscent of an equivalent
result for refractive indices in the case of impedance matching for light
propagation in 1D media.

###  2.2 Zero-stress boundary condition

In this case, the outer boundary of the load is free to move, and the boundary
condition at $r_{3}$ is given by,

$(2R_{3}^{\prime}+\frac{R_{3}}{r}-3)|_{r=r_{3}}=0.$ (18)

Together with the interface equations, the six unknowns, $A_{i}$ and $B_{i}$
are uniquely determined. Again, we are interested in the energy transfer from
the active core to the load, and we ask the same question as in the zero-
strain case: what value of Young’s modulus $E_{2}$ of the host material will
maximize the transfer of energy from the active core to the load? Although it
is not obvious, maximizing the energy in the load transferred from the active
core is equivalent to maximizing the displacement of outer radius of the load
$R_{3}(r_{3})$. Taking the derivative of $R_{3}(r_{3})$ with respect to
$E_{2}$, we obtain

$E_{2}=\sqrt{E_{1}E_{3}}\sqrt{\frac{3r_{3}^{2}-3r_{2}^{2}}{r_{2}^{2}+3r_{3}^{2}}}.$
(19)

Again the result is a geometric mean of Young’s modulus of medium 1 and 3,
multiplied with a geometric factor depending on radii of components. In the
case when $r_{3}\rightarrow\infty$, $E_{2}=\sqrt{E_{1}E_{3}}.$

Figure 2 demonstrates the maximum energy transferred to the load from the
active material occurs when the impedance matching modulus $E_{2}$ is given by
Eq. (17) or (19) at zero-strain or zero-stress boundary condition,
respectively.

(a) (b)

Figure 2: Transferred energy to the load $F_{3}$ normalized by initial energy
of active component $F_{1}^{0}$ as a function of $E_{2}$ with
$M=2,r_{2}=2,r_{3}=3$. The vertical dotdashed lines are located at the optimal
value of $E_{2}$ (a.u.), given by Eq. (17) and Eq. (19), respectively. (a)
zero-strain boundary condition, (b) zero-stress boundary condition.

###  2.3 Reflectance and transmittance

We finally look at the problem from the reflectance and transmittance point of
view and build a connection with the case of light propagation. Consider the
active core and the load only, with fixed outer boundary condition. The active
core initially has stored elastic energy; it is subsequently released and
transfers some of its stored energy to the load. We define the quantities

$R_{12}=\frac{F_{1}}{F_{1}^{0}},T_{12}=\frac{F_{2}}{F_{1}^{0}},$ (20)

as reflectance and transmittance, where $F_{i}$ is the final equilibrium
energy for each component and $F_{1}^{0}$ is the initial energy of the active
core. We remark that the total energy of the system in its final equilibrium
state is less than the initial energy of the active core due to dissipation.
In the limit that the outside radius of the load goes to infinity, we obtain

$T_{12}=\frac{3E_{1}E_{2}}{(3E_{1}+E_{2})^{2}}.$ (21)

Upon inserting an impedance matching host between the active core and the
load, the transmittance from the active core to the load becomes

$T_{13}=T_{12}T_{23}=\frac{9E_{1}E_{2}^{2}E_{3}}{(3E_{1}+E_{2})^{2}(3E_{2}+E_{3})^{2}}.$
(22)

Maximizing $T_{13}$ over $E_{2}$ gives

$E_{2}=\sqrt{E_{1}E_{3}}.$ (23)

We have recovered the results from above via an energy transfer point of view
in the limit when the size of the load goes to infinity. The main difference
between our case and light propagation case lies in that the energy is not
conserved in the former but is conserved in the latter case. It suggests that
energy conservation is not a key requirement in impedance matching mechanisms.

##  3 Conclusion

In this work, we are interested in the work done by an active material on the
materials surrounding it. Specifically, we have an initially nonequilibrium
elastic system with all the energy stored in one part, and the system is then
allowed to relax. We look for ways to improve the efficiency in transferring
energy to other parts of the system at equilibrium. To do so, we analyzed a
composite system consisting an active elastic material, a host, and a load. We
found that in the cylindrical geometry, the transferred energy from the active
material to the load can be maximized by tuning Young’s modulus of the
impedance matching host material. The analysis was done using the linearized
Mooney-Rivlin hyperelastic model and assuming incompressibility of all
components. We further assumed that the system was overdamped and elastic wave
propagation was not considered. The active material located at the center of
the host, when actuated, transfers stored energy to the load through the host.
We have considered two cases where the outer boundary of the load is fixed and
where it is free. Young’s modulus of the host material which maximizes the
energy transfer is found to be the geometric mean of the moduli of the active
material and the load, multiplied by a geometric factor which depends on the
radii of the components. In the limit when the size of the load goes to
infinity, Young’s modulus for the host is simply the geometric mean of the
moduli of the active material and the load. This coincides with the classical
result from impedance matching in the case of light propagating through
dielectric media. Although the model is simplified with idealized geometry and
is in the small strain limit, we anticipate the results will help optimize the
performance of photomechanical materials by using an impedance matching host
between the active material and the load.

## Acknowledgment

This work was supported by the Office of Naval Research [ONRN00014-18-1- 2624]

## References

  * [1] Stratton, J.A., 1941. Electromagnetic Theory, McGrow-Hill Book Company. Inc., New York, and London.
  * [2] Kim, Y.H., Sound propagation: an impedance based approach. John Wiley & Sons, (2010).
  * [3] Chen, S., Zhang, Y., Hao, C., Lin, S. and Fu, Z., 2014. Functionally graded materials for impedance matching in elastic media. Physics Letters A, 378(1-2), pp.77-81.
  * [4] Rahimzadeh, T., Arruda, E.M. and Thouless, M.D., 2015. Design of armor for protection against blast and impact. Journal of the Mechanics and Physics of Solids, 85, pp.98-111.
  * [5] Rathod, V.T., 2019. A review of electric impedance matching techniques for piezoelectric sensors, actuators and transducers. Electronics, 8(2), p.169.
  * [6] Santos, J., de Oliveira, B.P. and Nelson, O.R., Impedance of rigid bodies in one-dimensional elastic collisions. Revista Brasileira de Ensino de FÃ-sica, 34(1), 1-4 (2012).
  * [7] Mooney, M., 1940, A theory of large elastic deformation, Journal of Applied Physics, 11(9), pp. 582-592.
  * [8] Rivlin, R. S., 1948, Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), pp. 379-397.


# A Comparative Study of Relaying Schemes with Decode-and-Forward over
Nakagami-$m$ Fading Channels

George C. Alexandropoulos, Agisilaos Papadogiannis and Paschalis C. Sofotasios
G. C. Alexandropoulos is with the Department of Telecommunications Science and
Technology, University of Peloponnese, End of Karaiskaki Street, GR-22100
Tripolis, Greece (e-mail: alexandg@ieee.org).A. Papadogiannis is with the
Communications Research Group, Department of Electronics, University of York,
York, YO10 5DD, United Kingdom (e-mail: ap851@ohm.york.ac.uk).P. C. Sofotasios
is with School of Electronic and Electrical Engineering, University of Leeds,
Leeds, LS2 9JT, United Kingdom (e-mail: p.sofotasios@leeds.ac.uk).

###### Abstract

Utilizing relaying techniques to improve performance of wireless systems is a
promising avenue. However, it is crucial to understand what type of relaying
schemes should be used for achieving different performance objectives under
realistic fading conditions. In this paper, we present a general framework for
modelling and evaluating the performance of relaying schemes based on the
decode-and-forward (DF) protocol over independent and not necessarily
identically distributed (INID) Nakagami-$m$ fading channels. In particular, we
present closed-form expressions for the statistics of the instantaneous output
signal-to-noise ratio of four significant relaying schemes with DF; two based
on repetitive transmission and the other two based on relay selection (RS).
These expressions are then used to obtain closed-form expressions for the
outage probability and the average symbol error probability for several
modulations of all considered relaying schemes over INID Nakagami-$m$ fading.
Importantly, it is shown that when the channel state information for RS is
perfect, RS-based transmission schemes always outperform repetitive ones.
Furthermore, when the direct link between the source and the destination nodes
is sufficiently strong, relaying may not result in any gains and in this case
it should be switched-off.

###### Index Terms:

Bit error probability, cooperative diversity, Nakagami-$m$ fading, outage
probability, relay selection, repetitive transmission.

##  I Introduction

The significance of multiple-input multiple-output (MIMO) techniques for
modern wireless systems has been well appreciated. Multiple collocated
antennas can improve transmission reliability and the achievable capacity
through diversity, spatial multiplexing and/or interference suppression [1,
2]. However, the cost of mobile devices is proportional to their number of
antennas and this creates a serious practical limitation for the use of MIMO.
Cooperative diversity is a promising new avenue which allows cooperation
amongst a number of wireless nodes which effectively profit from MIMO
techniques without requiring multiple collocated antennas [3, 4].

Dual-hop cooperative diversity entails that the transmission of the source
node towards a destination node is assisted by one or more relay nodes which
can be seen to form a conceptual MIMO array [3]. Relay nodes can either be
fixed, being part of the system infrastructure, or mobile, i.e., mobile nodes
that relay signals intended for other mobile nodes [5]. Cooperative diversity
is shown to improve the transmission reliability and the achievable capacity
while it also extends coverage. Essentially, it can achieve diversity gains
and has the additional advantage over conventional MIMO that the remote
cooperating antennas experience very low or inexisting correlation [3, 6, 7,
8]. The performance of a system exploiting cooperative diversity depends on
the employed relaying protocol and scheme, i.e., the way of utilizing relay
nodes [3, 6, 7, 8]. Consequently, it is crucial to gain insights on which
relay scheme is most suitable for achieving a particular objective; some
common objectives are the minimization of the outage probability (OP) or the
average symbol error probability (ASEP) [9, 10].

In the present work, we consider the decode-and-forward (DF) relaying protocol
under the assumption that the message transmitted by the source is decoded and
retransmitted to destination by one or more relays in a dual-hop fashion. We
also take into account four relaying schemes; two based on repetitive
transmission and the other two based on relay selection (RS). According to
repetitive transmission, all relays that decode source’s message retransmit it
repetitively to the destination node which employs diversity techniques to
combine the different signal copies [11, 12]. One version of RS entails that
amongst relays decoding the source’s message only one is selected to
retransmit it to destination, the one with the strongest relay to destination
channel [6, 13]. Another version of RS utilizes the best relay only in the
case that it results in capacity gains over the direct source to destination
transmission [7, 10]. In the literature such schemes have been considered
partially and mainly assuming independent and identically distributed (IID)
Rayleigh fading channels [11, 6, 13]. Recently, the Nakagami-$m$ fading model
has received a lot of attention as it can describe more accurately the fading
process and helps in understanding the gains of cooperative diversity [14, 9,
15, 16, 10, 17, 18]. However, there has not been a complete study that sheds
light on the question of which relaying scheme is preferable and under which
channel conditions.

In this paper, we present a general analytical framework for modeling and
evaluating performance of relaying schemes with DF under independent and not
necessarily identically distributed (INID) Nakagami-$m$ fading channels.
Further to this, we obtain closed-form expressions for the OP and the ASEP
performance of the RS and repetitive schemes when maximal ratio diversity
(MRD) or selection diversity (SD) are employed at the destination node. We
conclude that the RS-based transmission always performs better in terms of OP
and ASEP than repetitive transmission when channel state information (CSI) for
RS is perfect. In addition, when the direct source to destination link is
sufficiently strong, relaying should be disabled when the objective is the
minimization of OP. Although RS requires only two time slots for transmission
(the repetitive scheme needs as many time slots as the number of decoding
relays) its performance heavily relies on the quality of CSI for RS.

The remainder of this paper is structured as follows: Section II outlines the
system and channel models. Section III presents closed-form expressions for
the statistics of the instantaneous output SNR of the considered DF relaying
schemes over INID Nakagami-$m$ fading channels. In Section IV, closed-from
expressions are derived for the OP and ASEP performance of all relaying
schemes. Section V contains numerical results and relevant discussion, whereas
Section VI concludes the paper.

Notations: Throughout this paper, $\left|\mathbb{A}\right|$ represents the
cardinality of the set $\mathbb{A}$ and ${\mathbb{E}}\langle\cdot\rangle$
denotes the expectation operator. $\Pr\left\\{\cdot\right\\}$ denotes
probability, $\mathbb{L}^{-1}\\{\cdot;\cdot\\}$ denotes the inverse Laplace
transform and $X\sim\mathcal{C}\mathcal{N}\left(\mu,\sigma^{2}\right)$
represents a random variable (RV) following the complex normal distribution
with mean $\mu$ and variance $\sigma^{2}$. $\Gamma\left(\cdot\right)$ is the
Gamma function [19, eq. (8.310/1)] and
$\mathsf{\Gamma}\left(\cdot,\cdot\right)$ is the lower incomplete Gamma
function [19, eq. (8.350/1)]. Moreover, $\delta\left(\cdot\right)$ is the
Dirac function, $u\left(\cdot\right)$ is the unit step function and
$\delta\left(\cdot,\cdot\right)$ is the Kronecker Delta function.

##  II System and Channel Model

We consider a dual-hop cooperative wireless system, as illustrated in Fig. 1,
consisting of $L+2$ wireless nodes: one source node $S$, a set $\mathbb{A}$ of
$L$ relay nodes each denoted by $R_{k}$, $k=1,2,\ldots,L$, and one destination
node $D$. All $R_{k}$’s are assumed to operate in half-duplex mode, i.e., they
cannot transmit and receive simultaneously, and node $D$ is assumed to possess
perfectly $S\rightarrow D$ and all $R_{k}\rightarrow D$ CSI. We consider
orthogonal DF (ODF) relaying [3] for which each $R_{k}$ that successfully
decodes $S$’s signal retransmits it to $D$; during each $R_{k}$’s transmission
to $D$ node $S$ remains silent. Assuming repetitive transmission [3], $L+1$
time slots are used to forward $S$’s signal to $D$ in a predetermined order,
whereas only two time slots are needed with RS-based transmission [6]. In
particular, during the first time slot for both transmission strategies, $S$
broadcasts its signal to all $R_{k}$’s and also to $D$. Considering quasi-
static fading channels, the received signal at $R_{k}$’s and $D$,
respectively, during the first time slot can be mathematically expressed as

$y_{D}^{(1)}=h_{SD}s+n_{SD}$ (1a) $y_{R_{k}}=h_{SR_{k}}s+n_{SR_{k}}$ (1b)

where $h_{SD}$ and $h_{SR_{k}}$ denote the $S\rightarrow D$ and $S\rightarrow
R_{k}$, respectively, complex-valued channel coefficients and $s$ is the
transmitted complex message symbol with average symbol energy $E_{s}$.
Moreover, the notations $n_{SD}$ and $n_{SR_{k}}$ in (1) represent the
additive white Gaussian noise (AWGN) of the $S\rightarrow D$ and $S\rightarrow
R_{k}$ channel, respectively, with
$n_{SD},n_{SR_{k}}\sim\mathcal{C}\mathcal{N}\left(0,N_{0}\right)$. For both
$n_{SD}$ and $n_{SR_{k}}$, it is assumed that they are statistically
independent of $s$.

Let us assume that set $\mathbb{B}\subseteq\mathbb{A}$ contains the relay
nodes that have successfully decoded $S$’s signal during the first time slot
of transmission. When repetitive transmission is utilized, $L$ more time slots
are used for $R_{k}$’s belonging to $\mathbb{B}$ to forward $s$ to $D$; each
$R_{k}$ retransmits $s$ during the $k$-th time slot111Note that the assignment
of each time slot to $R_{k}$’s is performed in a predetermined order [3].
Thus, due to different $R_{k}\rightarrow D$ $\forall\,k$ channel conditions
and ODF relaying, there might be some unused time slots.. Hence, for quasi-
static fading, the received signal at $D$ at the $k$-th time slot can be
expressed as

$y_{D}^{(k)}=h_{R_{k}D}s+n_{R_{k}D}$ (2)

with $h_{R_{k}D}$ representing the $R_{k}\rightarrow D$ complex-valued channel
coefficient and $n_{R_{k}D}\sim\mathcal{C}\mathcal{N}\left(0,N_{0}\right)$ is
AWGN of this channel that is assumed statistically independent of $s$.

When RS-based transmission is used, one time slot is needed for the relay node
$R_{\rm best}$ with the most favourable $R_{\rm best}\rightarrow D$ channel
conditions to forward $S$’s signal to $D$. Thus, for this transmission
strategy and during the second time slot, the received signal at $D$ for
quasi-static fading can be expressed as

$y_{D}^{(2)}=h_{R_{\rm best}D}s+n_{R_{\rm best}D}$ (3)

where $h_{R_{\rm best}D}$ and $n_{R_{\rm
best}D}\sim\mathcal{C}\mathcal{N}\left(0,N_{0}\right)$ denote the
$R_{k}\rightarrow D$ complex-valued channel coefficient and the AWGN for this
channel, respectively. As (1) and (2), it is assumed that $n_{R_{\rm best}D}$
is statistically independent of $s$.

The quasi-static fading channels $h_{SD}$, $h_{SR_{k}}$ and $h_{R_{k}D}$
$\forall\,k$, and $h_{R_{\rm best}D}$ are assumed to be modeled as INID
Nakagami-$m$ RVs [20]. Let
$\gamma_{0}=\left|h_{SD}\right|^{2}E_{\text{s}}/N_{0}$ and
$\gamma_{k}=\left|h_{R_{k}D}\right|^{2}E_{\text{s}}/N_{0}$, $k=1,2,\ldots,L$,
be the instantaneous received SNRs of the $S\rightarrow D$ and
$R_{k}\rightarrow D$ link, respectively, with corresponding average values
given by $\overline{\gamma}_{0}=\mathbb{E}\left|\langle
h_{SD}\right|^{2}\rangle E_{s}/N_{0}$ and
$\overline{\gamma}_{k}=\mathbb{E}\langle\left|h_{R_{k}D}\right|^{2}\rangle
E_{s}/N_{0}$, respectively. Clearly, each $\gamma_{\ell}$,
$\ell=0,1,\ldots,L$, is gamma distributed with probability density function
(PDF) given by [21, Table 2.2]

$f_{\gamma_{\ell}}\left(x\right)=\frac{C_{\ell}^{m_{\ell}}}{\Gamma\left(m_{\ell}\right)}x^{m_{\ell}-1}\exp\left(-C_{\ell}x\right)$
(4)

where $m_{\ell}\geq 1/2$ denotes the Nakagami-$m$ fading parameter and
$C_{\ell}=m_{\ell}/\overline{\gamma}_{\ell}$. Integrating (4), the cumulative
distribution function (CDF) of each $\gamma_{\ell}$ is easily obtained as

$F_{\gamma_{\ell}}\left(x\right)=\frac{\mathsf{\Gamma}\left(m_{\ell},C_{\ell}x\right)}{\Gamma\left(m_{\ell}\right)}.$
(5)

The PDFs and CDFs of the instantaneous received SNRs of the first hop,
$\gamma_{L+k}=\left|h_{SR_{k}}\right|^{2}E_{\text{s}}/N_{0}$ $\forall\,k$, are
given using (4) and (5) by $f_{\gamma_{L+k}}\left(x\right)$ and
$F_{\gamma_{L+k}}\left(x\right)$, respectively, with fading parameters and
average SNRs denoted by $m_{L+k}$ and
$\overline{\gamma}_{L+k}=\mathbb{E}\langle\left|h_{SR_{k}}\right|^{2}\rangle
E_{s}/N_{0}$, respectively.

##  III Statistics of ODF Relaying Schemes

Relay station nodes that are able to decode the transmitted signal from $S$
constitute the decoding set $\mathbb{B}$. Based on [3] and [13], for both
transmission strategies the elements of $\mathbb{B}$ are obtained as

$\mathbb{B}=\left\\{R_{k}\in\mathbb{A}:\log_{2}\left(1+\gamma_{L+k}\right)\geq\alpha\mathcal{R}\right\\}$
(6)

where $\mathcal{R}$ is $S$’s transmit rate and $\alpha=L+1$ for repetitive
transmission, whereas $\alpha=2$ for RS-based transmission. Hence, the
probability that $R_{k}$ does not belong to $\mathbb{B}$ is easily obtained as
$\mathcal{P}_{k}=\text{Pr}\left[\gamma_{L+k}<2^{\alpha\mathcal{R}}-1\right]$.
Substituting $F_{\gamma_{L+k}}\left(x\right)$ after using (5), a closed-form
expression for $\mathcal{P}_{k}$ in INID Nakagami-$m$ fading is given by

$\mathcal{P}_{k}=\frac{\mathsf{\Gamma}\left[m_{L+k},C_{L+k}\left(2^{\alpha\mathcal{R}}-1\right)\right]}{\Gamma\left(m_{L+k}\right)}.$
(7)

To analyze the performance of ODF relaying schemes, the $S\longrightarrow D$
direct channel plus the $S\longrightarrow R_{k}\longrightarrow D$ $\forall\,k$
RS-assisted channels are effectively considered as $L+1$ paths between $S$ and
$D$ [11, 15]. In particular, let the zero-th path represent the
$S\longrightarrow D$ direct link and the $k$-th path the $S\longrightarrow
R_{k}\longrightarrow D$ cascaded link. We define the instantaneous received
SNRs at $D$ of these paths as $g_{0}$ and $g_{k}$, respectively. By
substituting (4) into [11, eq. (4)], the PDFs of $g_{\ell}$’s,
$\ell=0,1,\ldots,L$, are given by

$f_{g_{\ell}}\left(x\right)=\mathcal{P}_{\ell}\delta\left(x\right)+\frac{\left(1-\mathcal{P}_{\ell}\right)C_{\ell}^{m_{\ell}}}{\Gamma\left(m_{\ell}\right)}x^{m_{\ell}-1}\exp\left(-C_{\ell}x\right)$
(8)

where $\mathcal{P}_{0}$ is the probability that $D$ belongs to $\mathbb{B}$.
Clearly, the direct $S\longrightarrow D$ path is not linked via a relay, i.e.,
$\mathcal{P}_{0}=0$, yielding
$f_{g_{0}}\left(x\right)=f_{\gamma_{0}}\left(x\right)$. Integrating (8) and
using (5), yields the following expression for the CDF of the $\ell$-th
cascaded path:

$F_{g_{\ell}}\left(x\right)=\mathcal{P}_{\ell}u\left(x\right)+\left(1-\mathcal{P}_{\ell}\right)\frac{\mathsf{\Gamma}\left(m_{\ell},C_{\ell}x\right)}{\Gamma\left(m_{\ell}\right)}.$
(9)

Note again that for the direct $S\longrightarrow D$ path yields
$F_{g_{0}}\left(x\right)=F_{\gamma_{0}}\left(x\right)$.

###  III-A Repetitive Transmission

The incoming signals at $D$ from $S$ and $R_{k}\in\mathbb{B}$ $\forall\,k$ may
be combined using a time-diversity version of MRD [3] and SD [12]. In
particular, $D$ combines $S$’s signal received at time slot one with the $S$’s
replicas received from all $R_{k}\in\mathbb{B}$ $\forall\,k$ at the $L$
subsequent slots using either the rules of MRD or SD.

####  III-A1 Repetitive with MRD

With MRD the instantaneous SNR at $D$’s output is expressed as

$g_{\text{end}}=g_{0}+\sum_{i=1}^{\left|\mathbb{B}\right|}g_{i}.$ (10)

Since $g_{\ell}$’s, $\ell=0,1,\ldots,L$, are independent, the moment
generating function (MGF) of $g_{\text{end}}$ can be easily obtained as the
product of the MGFs of $g_{\ell}$’s. As shown in [15] for INID Nakagami-$m$
fading, using (8) and the definition of the MGF of $g_{k}$, $k=1,2,\ldots,L$,

$M_{g_{k}}\left(s\right)=\int_{0}^{\infty}\exp\left(-sx\right)f_{g_{k}}\left(x\right)dx,$
(11)

yields
$\mathcal{M}_{g_{k}}\left(s\right)=\mathcal{P}_{k}+\left(1-\mathcal{P}_{k}\right)C_{k}^{m_{k}}\left(s+C_{k}\right)^{-m_{k}}$.
Similarly, using (4) the MGF of $g_{0}$ is easily obtained as
$\mathcal{M}_{g_{0}}\left(s\right)=C_{0}^{m_{0}}\left(s+C_{0}\right)^{-m_{0}}$.
Hence, the following closed-form expression for the MGF of $g_{\text{end}}$ in
INID Nakagami-$m$ is deduced

$\mathcal{M}_{g_{\text{end}}}\left(s\right)=C_{0}^{m_{0}}\left(s+C_{0}\right)^{-m_{0}}\prod_{k=1}^{L}\left[\mathcal{P}_{k}+\left(1-\mathcal{P}_{k}\right)C_{k}^{m_{k}}\left(s+C_{k}\right)^{-m_{k}}\right].$
(12)

Using the MGF-based approach [21], the CDF of $g_{\text{end}}$ can be obtained
as

$F_{g_{\text{end}}}\left(x\right)=\mathbb{L}^{-1}\left\\{\frac{\mathcal{M}_{g_{\text{end}}}\left(s\right)}{s};x\right\\}.$
(13)

Substituting (12) in (13) and similar to the analysis presented in [15], a
closed-form expression for $F_{g_{\text{end}}}\left(x\right)$ of repetitive
transmission with MRD over INID Nakagami-$m$ with integer $m_{\ell}$’s and
distinct $C_{\ell}$’s is given by

$\begin{split}F_{g_{\text{end}}}\left(x\right)=&\left(\prod_{\ell=0}^{L}\mathcal{P}_{\ell}\right)\left\\{1+\sum_{\left\\{\lambda_{k}\right\\}_{k=0}^{L}}\left(\prod_{n=0}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}C_{\lambda_{n}}^{m_{\lambda_{n}}}\right)\right.\\\
&\left.\times\sum_{p=0}^{k}\sum_{q=1}^{m_{\lambda_{p}}}\frac{\psi_{p}}{C_{\lambda_{p}}^{q}}\left[1-\exp\left(-C_{\lambda_{p}}x\right)\sum_{\ell=0}^{q-1}\frac{\left(C_{\lambda_{p}}x\right)^{\ell}}{\ell!}\right]\right\\}\end{split}$
(14)

The symbol $\sum_{\left\\{\alpha_{i}\right\\}_{i=\kappa}^{I}}$ is used for
short-hand representation of multiple summations $\sum_{i=\kappa}^{I}$
$\sum_{\alpha_{\kappa}=\kappa}^{I-i+\kappa}$
$\sum_{\alpha_{\kappa+1}=\alpha_{\kappa}+1}^{I-i+\kappa+1}$ $\cdots$
$\sum_{\alpha_{i}=\alpha_{i-1}+1}^{I}$ and
$\psi_{p}=\Psi_{p}\left(s\right)^{(m_{\lambda_{p}}-q)}|_{s=-C_{\lambda_{p}}}/(m_{\lambda_{p}}-q)!$
with
$\Psi_{p}\left(s\right)=(s+C_{\lambda_{p}})^{m_{\lambda_{p}}}\prod_{n=0}^{k}(s+C_{\lambda_{n}})^{-m_{\lambda_{n}}}$.
For $C_{\ell}=C$ $\forall\,\ell$ with arbitrary values for $m_{\ell}$’s and
following a similar analysis as for the derivation of (14), a closed-form
expression for $F_{g_{\text{end}}}\left(x\right)$ can be obtained as

$F_{g_{\text{end}}}\left(x\right)=\left(\prod_{\ell=0}^{L}\mathcal{P}_{\ell}\right)\left\\{1+\sum_{\left\\{\lambda_{k}\right\\}_{k=0}^{L}}\left(\prod_{n=0}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\mathsf{\Gamma}\left(\sum_{n=0}^{k}m_{\lambda_{n}},Cx\right)/\Gamma\left(\sum_{n=0}^{k}m_{\lambda_{n}}\right)\right\\}.$
(15)

####  III-A2 Repetitive with SD

Alternatively to MRD, $D$ may use a time-diversity version of SD to combine
the signals from $S$ and $R_{k}\in\mathbb{B}$ $\forall\,k$ [12]. With this
diversity technique the instantaneous SNR at $D$’s output is given by

$g_{\text{end}}=\max_{R_{k}\in\mathbb{C}}g_{k}$ (16)

where $\mathbb{C}=\\{S\\}\cup\mathbb{B}$. The $g_{\ell}$’s,
$\ell=0,1,\ldots,L$, are assumed independent, therefore
$F_{g_{\text{end}}}\left(x\right)$ of (16) can be easily obtained as the
product of the CDFs of $g_{\ell}$’s. Substituting (5) and (9) for $g_{0}$ and
$g_{k}$ $\forall\,k=1,2,\ldots,L$, respectively, a closed-form expression for
$F_{g_{\text{end}}}\left(x\right)$ of repetitive transmission with SD over
INID Nakagami-$m$ fading can be derived as

$F_{g_{\text{end}}}\left(x\right)=F_{\gamma_{0}}\left(x\right)F_{g_{\text{best}}}\left(x\right)$
(17)

where $F_{g_{\text{best}}}\left(x\right)$ is the CDF of the instantaneous SNR
of the $R_{\rm best}\rightarrow D$ channel, i.e., of

$g_{\text{best}}=\max_{R_{k}\in\mathbb{B}}g_{k},$ (18)

which is easily obtained using (9) for INID Nakagami-$m$ fading as

$F_{g_{\text{best}}}\left(x\right)=\prod_{k=1}^{L}\left[\mathcal{P}_{k}u\left(x\right)+\left(1-\mathcal{P}_{k}\right)\frac{\mathsf{\Gamma}\left(m_{k},C_{k}x\right)}{\Gamma\left(m_{k}\right)}\right].$
(19)

Differentiating (17), the PDF of $g_{\text{end}}$ is given by

$f_{g_{\text{end}}}\left(x\right)=f_{\gamma_{0}}\left(x\right)F_{g_{\text{best}}}\left(x\right)+F_{\gamma_{0}}\left(x\right)f_{g_{\text{best}}}\left(x\right)$
(20)

where $f_{g_{\text{best}}}\left(x\right)$ is the PDF of $g_{\text{best}}$. To
obtain an expression for $f_{g_{\text{best}}}\left(x\right)$, we first use
[19, eq. (8.352/1)] to obtain (19) for integer $m_{\ell}$’s yielding

$F_{g_{\text{best}}}\left(x\right)=\prod_{k=1}^{L}\left\\{\mathcal{P}_{k}u\left(x\right)+\left(1-\mathcal{P}_{k}\right)\left[1-\exp\left(-C_{k}x\right)\sum_{i=0}^{m_{k}-1}\frac{\left(C_{k}x\right)^{i}}{i!}\right]\right\\}.$
(21)

Then, differentiating (21) and using the formula

$\prod_{i=\kappa}^{I}\left(\chi_{i}+\psi_{i}\right)=\prod_{i=\kappa}^{I}\chi_{i}+\sum_{\left\\{\alpha_{i}\right\\}_{i=\kappa}^{I}}\prod_{s=\kappa}^{i}\psi_{\alpha_{s}}\prod_{\begin{subarray}{c}t=\kappa\\\
t\neq\left\\{\alpha_{u}\right\\}_{u=\kappa}^{i}\end{subarray}}^{I}\chi_{t},$
(22)

where symbol $\sum_{\left\\{\alpha_{i}\right\\}_{i=\kappa}^{I}}$ is used for
short-hand representation of multiple summations $\sum_{i=\kappa}^{I}$
$\sum_{\alpha_{\kappa}=\kappa}^{I-i+\kappa}\sum_{\alpha_{\kappa+1}=\alpha_{\kappa}+1}^{I-i+\kappa+1}\cdots\sum_{\alpha_{i}=\alpha_{i-1}+1}^{I}$,
we obtain after some algebraic manipulations the following closed-form
expression for $f_{g_{\text{best}}}\left(x\right)$ in INID Nakagami-$m$ fading
with integer $m_{\ell}$’s:

$\begin{split}&f_{g_{\text{best}}}\left(x\right)=\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{\delta\left(x\right)+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\\\
&\left.\times\left[-\left(\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)\exp\left(-x\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\\\
&\left.\left.\times
x^{\sum_{s=1}^{p}i_{\nu_{s}}-1}\exp\left(-x\xi_{p,j}\right)\left(1-\frac{\xi_{p,j}x}{\sum_{s=1}^{p}i_{\nu_{s}}}\right)\right]\right\\}.\end{split}$
(23)

In (23), parameters $\xi_{p,j}$’s, with $p$ and $j$ being positive integers,
are given by

$\xi_{p,j}=\sum_{s=1}^{p}C_{\lambda_{\mu_{\nu_{s}}}}+\sum_{\begin{subarray}{c}t=1\\\
t\neq\left\\{\nu_{u}\right\\}_{u=1}^{p}\end{subarray}}^{j}C_{\lambda_{\mu_{t}}}.$
(24)

Substituting (4) and (5) for integer $m_{\ell}$’s, i.e., after using [19, eq.
(8.352/1)] for expressing $\mathsf{\Gamma}\left(\cdot,\cdot\right)$’s, as well
as (21) and (23) to (20), a closed-form expression for
$f_{g_{\text{end}}}\left(x\right)$ over INID Nakagami-$m$ fading with integer
values of $m_{\ell}$’s and distinct $C_{\ell}$’s can be obtained after some
algebraic manipulations as

$\begin{split}&f_{g_{\text{end}}}\left(x\right)=\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{\frac{C_{0}^{m_{0}}}{\Gamma\left(m_{0}\right)}x^{m_{0}-1}\exp\left(-C_{0}x\right)\left\\{u\left(x\right)+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\right.\right.\\\
&\left.\left.\times\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\left[\exp\left(-x\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)-1+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\right.\\\
&\left.\left.\left.\times
x^{\sum_{s=1}^{p}i_{\nu_{s}}}\exp\left(-x\xi_{p,j}\right)\right]\right\\}+\left[1-\exp\left(-C_{0}x\right)\sum_{i=0}^{m_{0}-1}\frac{\left(C_{0}x\right)^{i}}{i!}\right]\right.\\\
&\left.\times\left\\{\delta\left(x\right)+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\right.\\\
&\left.\left.\times\left[-\left(\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)\exp\left(-x\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\right.\\\
&\left.\left.\left.\times
x^{\sum_{s=1}^{p}i_{\nu_{s}}-1}\exp\left(-x\xi_{p,j}\right)\left(1-\frac{\xi_{p,j}x}{\sum_{s=1}^{p}i_{\nu_{s}}}\right)\right]\right\\}\right\\}.\end{split}$
(25)

To obtain the MGF of $g_{\text{end}}$, we substitute the
$f_{g_{\text{end}}}\left(x\right)$ expression given by (25) to the definition
of the MGF given by (11), i.e., after replacing $g_{k}$ with $g_{\text{end}}$
in (11), and use[19, eq. (3.381/4)] to solve the resulting integrals. In
particular, by first deriving using (23) the following closed-form expression
for the MGF of $g_{\text{best}}$ in INID Nakagami-$m$ fading with integer
values of $m_{\ell}$’s and distinct $C_{\ell}$’s:

$\begin{split}&M_{g_{\text{best}}}\left(s\right)=\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{1+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\\\
&\times\left.\left\\{-\left(\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)\left(s+\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)^{-1}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)!\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\\\
&\left.\left.\times\left(s+\xi_{p,j}\right)^{-\sum_{s=1}^{p}i_{\nu_{s}}}\left[1-\xi_{p,j}\left(s+\xi_{p,j}\right)^{-1}\right]\right\\}\right\\},\end{split}$
(26)

a closed-form expression for the MGF of $g_{\text{end}}$ of repetitive
transmission with SD over INID Nakagami-$m$ fading channels with integer
values of $m_{\ell}$’s and distinct $C_{\ell}$’s is given by

$\begin{split}&\mathcal{M}_{g_{\text{end}}}\left(s\right)=\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{C_{0}^{m_{0}}\left\\{\left(s+C_{0}\right)^{-m_{0}}+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\right.\\\
&\left.\left.\times\left[\left(s+C_{0}+\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)^{-m_{0}}-\left(s+C_{0}\right)^{-m_{0}}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\right.\\\
&\left.\left.\left.\times\frac{\left(m_{0}+\sum_{s=1}^{p}i_{\nu_{s}}-1\right)!}{\left(m_{0}-1\right)!}\left(s+C_{0}+\xi_{p,j}\right)^{-\left(m_{0}+\sum_{s=1}^{p}i_{\nu_{s}}\right)}\right]\right\\}+\left\\{\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\right.\right.\\\
&\left.\left.\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\left\\{\left(\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)\sum_{q=0}^{m_{0}-1}C_{0}^{q}\left(s+C_{0}+\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)^{-(q+1)}\right.\right.\right.\\\
&\left.\left.\left.+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\sum_{q=0}^{m_{0}-1}\frac{C_{0}^{q}}{q!}\left(s+C_{0}+\xi_{p,j}\right)^{-\left(q+\sum_{s=1}^{p}i_{\nu_{s}}+1\right)}\right.\right.\right.\\\
&\left.\left.\left.\times\left(q+\sum_{s=1}^{p}i_{\nu_{s}}-1\right)!\left[q\xi_{p,j}-\left(s+C_{0}\right)\sum_{s=1}^{p}i_{\nu_{s}}\right]\right\\}\right\\}\right\\}+M_{g_{\text{best}}}\left(s\right).\end{split}$
(27)

For equal $C_{\ell}$’s, i.e., IID Nakagami-$m$ fading with $m_{\ell}=m$ and
$\overline{\gamma}_{\ell}=\overline{\gamma}$ $\forall\,\ell$, following a
similar procedure as for the derivation of (27) and using the binomial and
multinomial theorems [19, eq. (1.111)], we first obtain the following closed-
form expression for $M_{g_{\text{best}}}\left(s\right)$ for integer $m$

$\begin{split}&M_{g_{\text{best}}}\left(s\right)=L\mathcal{P}^{L}\left\\{1/L+\left(1-\mathcal{P}\right)C^{m}\left(s+C\right)^{-m}+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\right.\\\
&\left.\times\left\\{\mathcal{P}+\left(1-\mathcal{P}\right)C^{m}\left(s+C\right)^{-m}+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\right.\right.\\\
&\left.\left.\times\left[\mathcal{P}+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\frac{\left(1-\mathcal{P}\right)C^{m+\sigma}}{\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\frac{\prod_{i=1}^{m}n_{i}^{-1}}{\left(m-1\right)!}\left[s+(j+1)C\right]^{-\left(m+\sigma\right)}\right]\right\\}\right\\}\end{split}$
(28)

where $C=m/\overline{\gamma}$, $\mathcal{P}=\mathcal{P}_{k}$ $\forall\,k$,
$\sigma=\sum_{i=2}^{m}(i-1)n_{i}$ and symbol
$\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}$ is used for short-hand representation
of multiple summations
$\sum_{n_{1}=0}^{j}\sum_{n_{2}=0}^{j}\cdots\sum_{n_{m}=0}^{j}\delta(\sum_{i=1}^{m}n_{i},j)$.
Using (20) for IID Nakagami-$m$ fading, (28) and after some algebraic
manipulations, a closed-form expression for $M_{g_{\text{end}}}\left(s\right)$
of repetitive transmission with SD in IID Nakagami-$m$ fading channels with
integer $m$ is obtained as

$\begin{split}&M_{g_{\text{end}}}\left(s\right)=L\mathcal{P}^{L}\left\\{C^{m}\left\\{\left[1/L+\left(1-\mathcal{P}\right)\right]\left(s+C\right)^{-m}-(1-\mathcal{P})B_{0,1}\left(s\right)+\sum_{k=1}^{L-1}\binom{L-1}{k}\right.\right.\\\
&\left.\left.\times\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\left\\{\left[\mathcal{P}+\left(1-\mathcal{P}\right)\right]\left(s+C\right)^{-m}-(1-\mathcal{P})B_{0,1}\left(s\right)+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\left[\mathcal{P}\right.\right.\right.\right.\\\
&\left.\left.\left.\left.\times\left(s+C\right)^{-m}+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\frac{\left(1-\mathcal{P}\right)}{\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\frac{\prod_{i=1}^{m}n_{i}^{-1}}{\left(m-1\right)!}\left\\{\left(s+C\right)^{-m}-B_{\sigma,j+1}\left(s\right)\right\\}\right]\right\\}\right\\}\right.\\\
&\left.-C^{m}\left\\{\left(1-\mathcal{P}\right)B_{0,1}\left(s\right)+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\left\\{\left(1-\mathcal{P}\right)B_{0,1}\left(s\right)+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\right.\right.\right.\\\
&\left.\left.\left.\times\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\frac{\left(1-\mathcal{P}\right)C^{\sigma}}{\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\frac{\prod_{i=1}^{m}n_{i}^{-1}}{\left(m-1\right)!}\sum_{q=0}^{m-1}\frac{C^{q}}{q!(m+\sigma-1)!}(m+q+\sigma-1)!\right.\right.\right.\\\
&\left.\left.\left.\times[s+(\lambda+1)C]^{-(m+q)}\right\\}\right\\}\right\\}+M_{g_{\text{best}}}\left(s\right)\end{split}$
(29)

where function $B_{\kappa,\lambda}\left(s\right)$, with $\kappa$ and $\lambda$
being positive integers, is given by

$B_{\kappa,\lambda}\left(s\right)=\sum_{q=0}^{m+\kappa-1}\frac{(\lambda
C)^{q}}{q!(m-1)!}(m+q-1)![s+(\lambda+1)C]^{-(m+q)}.$ (30)

###  III-B RS-based Transmission

When RS-based transmission is utilized, RS [6] is first performed to obtain
$R_{\rm best}$. Relay node $R_{\rm best}$ is the one experiencing the most
favorable $R_{k}\rightarrow D$ $\forall\,k$ channel conditions, i.e., its
instantaneous SNR is given by (18). Using expressions derived in Section
III-A, we next obtain closed-form expressions for the statistics of a pure RS
scheme [13, 10] that combines at $D$ the signals from $S$ and $R_{\rm best}$
using a time-diversity version of MRD as well as of a rate-selective one [10]
that utilizes pure RS only if it is beneficial in terms of achievable rate
over the direct transmission.

####  III-B1 Pure RS

With pure RS $D$ utilizes MRD to combine the signals from $S$ and $R_{\rm
best}$ [13], therefore the instantaneous SNR at $D$’s output is given using
(18) by

$g_{\text{end}}=g_{0}+g_{\text{best}}.$ (31)

Similar to the derivation of (12), the MGF of $g_{\text{end}}$ of pure RS can
be obtained as the following product:

$\mathcal{M}_{g_{\text{end}}}\left(s\right)=C_{0}^{m_{0}}\left(s+C_{0}\right)^{-m_{0}}\mathcal{M}_{g_{\text{best}}}\left(s\right)$
(32)

where $\mathcal{M}_{g_{\text{best}}}\left(s\right)$ is given by (26) for INID
Nakagami-$m$ with integer $m_{\ell}$’s and distinct $C_{\ell}$’s, whereas by
(28) for IID Nakagami-$m$ with integer $m_{\ell}=m$ $\forall\,\ell$ and
$C_{\ell}=C$. Therefore substituting (26) to (32), a closed-form expression
for $\mathcal{M}_{g_{\text{end}}}\left(s\right)$ of pure RS for INID
Nakagami-$m$ fading with integer $m_{\ell}$’s and distinct $C_{\ell}$’s is
given by [10, eq. (7)]

$\begin{split}&\mathcal{M}_{g_{\text{end}}}\left(s\right)=C_{0}^{m_{0}}\left(s+C_{0}\right)^{-m_{0}}\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{1+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\\\
&\times\left.\left\\{-\left(\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)\left(s+\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)^{-1}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)!\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\\\
&\left.\left.\times\left(s+\xi_{p,j}\right)^{-\sum_{s=1}^{p}i_{\nu_{s}}}\left[1-\xi_{p,j}\left(s+\xi_{p,j}\right)^{-1}\right]\right\\}\right\\}.\end{split}$
(33)

For IID Nakagami-$m$ fading channels with integer $m$, substituting (28) to
(32) a closed-form expression for $\mathcal{M}_{g_{\text{end}}}\left(s\right)$
of pure RS is given by [10, eq. (10)]

$\begin{split}&\mathcal{M}_{g_{\text{end}}}\left(s\right)=L\mathcal{P}^{L}C^{m}\left(s+C\right)^{-m}\left\\{\frac{1}{L}+\left(1-\mathcal{P}\right)C^{m}\left(s+C\right)^{-m}+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\right.\\\
&\left.\times\left\\{\mathcal{P}+\left(1-\mathcal{P}\right)C^{m}\left(s+C\right)^{-m}+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\left[\mathcal{P}+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\frac{\left(1-\mathcal{P}\right)C^{m+\sigma}}{\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\right.\right.\right.\\\
&\left.\left.\left.\times\frac{\prod_{i=1}^{m}n_{i}^{-1}}{\left(m-1\right)!}\left[s+(j+1)C\right]^{-\left(m+\sigma\right)}\right]\right\\}\right\\}.\end{split}$
(34)

To obtain $F_{g_{\text{end}}}\left(x\right)$ of pure RS for INID Nakagami-$m$
fading with integer $m_{\ell}$’s and distinct $C_{\ell}$’s, we substitute (33)
to (13) and after some algebraic manipulations yields the following closed-
form expression [10, eq. (10)]

$\begin{split}&F_{g_{\text{end}}}\left(x\right)=\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{\mathcal{Z}_{m_{0}}\left(C_{0},x\right)+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\\\
&\left.\left.\left\\{-C_{0}^{m_{0}}\left\\{\mathcal{X}_{2,1}^{(1)}\mathcal{Z}_{0}\left[\chi_{2}^{(1)},x\right]+\chi_{2}^{(1)}\mathcal{Y}_{1}^{(1)}\left(x\right)\right\\}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\right.\\\
&\left.\left.\times
C_{0}^{m_{0}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)!\left[\mathcal{Y}_{2}^{(2)}\left(x\right)-\xi_{p,j}\mathcal{Y}_{2}^{(3)}\left(x\right)\right]\right\\}\right\\}\end{split}$
(35)

where
$\mathcal{Z}_{t}\left(c,x\right)=1-\exp\left(-cx\right)\sum_{i=1}^{t-1}\left(cx\right)^{i}/i!$,
with $c$ being positive real, and

$\mathcal{Y}_{\kappa}^{(u)}\left(x\right)=\sum_{r=1}^{\kappa}\sum_{t=1}^{b_{r}^{(u)}}\mathcal{X}_{r,t}^{(u)}\left[\chi_{r}^{(u)}\right]^{-t}\mathcal{Z}_{t}\left[\chi_{r}^{(u)},x\right]$
(36)

for $u=1,2$ and $3$. In the above two equations $\forall\,u$,
$\mathcal{X}_{r,t}^{(u)}=X_{r}^{(u)}\left(s\right)^{(b_{r}^{(u)}-t)}|_{s=-\chi_{r}^{(u)}}/(b_{r}^{(u)}-t)!$
and
$X_{r}^{(u)}\left(s\right)=(s+\chi_{r}^{(u)})^{b_{r}^{(u)}}\prod_{j=1}^{2}(s+\chi_{j}^{(u)})^{b_{j}^{(u)}}$.
Moreover, $\chi_{1}^{(u)}=C_{0}$ $\forall\,u$,
$\chi_{2}^{(1)}=\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}$ and
$\chi_{2}^{(2)}=\chi_{2}^{(3)}=\xi_{p,j}$ as well as $b_{1}^{(u)}=m_{0}$,
$\forall\,u$, $b_{2}^{(1)}=1$, $b_{2}^{(2)}=\sum_{s=1}^{p}i_{\nu_{s}}$ and
$b_{2}^{(3)}=b_{2}^{(2)}+1$. For $C_{\ell}=C$ $\forall\,\ell$ and integer $m$
and similar to the derivation of (35), substituting (34) to (13), a closed-
form expression for $F_{g_{\text{end}}}\left(x\right)$ of pure RS is given by
[10, eq. (11)]

$\begin{split}&F_{g_{\text{end}}}\left(x\right)=L\mathcal{P}^{L}\left\\{\frac{\mathcal{Z}_{m}\left(C,x\right)}{L}+\left(1-\mathcal{P}\right)\mathcal{Z}_{2m}\left(C,x\right)+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\right.\\\
&\left.\left\\{\mathcal{P}\mathcal{Z}_{m}\left(C,x\right)+\left(1-\mathcal{P}\right)\mathcal{Z}_{2m}\left(C,x\right)+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\left[\mathcal{P}\mathcal{Z}_{m}\left(C,x\right)\right.\right.\right.\\\
&\left.\left.\left.+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\left(1-\mathcal{P}\right)\frac{\prod_{i=1}^{m}n_{i}^{-1}C^{2m+\sigma}}{\left(m-1\right)!\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\mathcal{Y}_{2}^{(4)}\left(x\right)\right]\right\\}\right\\}\end{split}$
(37)

where $b_{1}^{(4)}=m$, $b_{2}^{(4)}=b_{1}^{(4)}+\sigma$, $\chi_{1}^{(4)}=C$
and $\chi_{2}^{(4)}=(j+1)\chi_{1}^{(4)}$.

####  III-B2 Rate-Selective RS

Dual-hop transmission incurs a pre-log penalty factor of $1/2$. To deal with
this rate loss, pure RS is considered only if it provides higher achievable
rate than that of the direct $S\longrightarrow D$ transmission [7, 22], i.e.,
higher than $\log_{2}\left(1+g_{0}\right)$. Using instantaneous CSI and (31),
rate-selective RS chooses between direct (non-relay assisted) and relay-
assisted transmission based on the following criterion [10, eq. (14)]

$\mathcal{R}_{\text{sel}}=\max\left\\{\frac{1}{2}\log_{2}\left(1+g_{\text{end}}\right),\log_{2}\left(1+g_{0}\right)\right\\}.$
(38)

As shown in [10], the MGF of ${g_{\text{sel}}}$ can be obtained using the
$M_{g_{\text{end}}}(s)$ of pure RS as

$M_{g_{\text{sel}}}(s)=M_{g_{0}}(s)F_{g_{\text{end}}}\left(\alpha\right)+M_{g_{\text{end}}}(s)[1-F_{g_{\text{end}}}\left(\alpha\right)]$
(39)

where $\alpha=g_{0}^{2}+2g_{0}$ is a RV with CDF given by
$F_{\alpha}(x)=F_{g_{0}}\left(\sqrt{x+1}-1\right)$ which can be obtained using
inverse sampling [23]. Substituting (33) and (35) to (39), a closed-form
expression for $M_{g_{\text{sel}}}\left(s\right)$ of rate-selective RS over
INID Nakagami-$m$ fading with integer $m_{\ell}$’s and distinct $C_{\ell}$’s
is derived as

$\begin{split}&M_{g_{\text{sel}}}\left(s\right)=\frac{C_{0}^{m_{0}}}{\left(s+C_{0}\right)^{m_{0}}}\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{\left\\{\mathcal{Z}_{m_{0}}\left(C_{0},\alpha\right)+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\right.\\\
&\left.\left.\left.\left\\{-C_{0}^{m_{0}}\left\\{\mathcal{X}_{2,1}^{(1)}\mathcal{Z}_{0}\left[\chi_{2}^{(1)},\alpha\right]+\chi_{2}^{(1)}\mathcal{Y}_{1}^{(1)}\left(\alpha\right)\right\\}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)C_{0}^{m_{0}}\right.\right.\right.\right.\\\
&\left.\left.\left.\times\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)!\left[\mathcal{Y}_{2}^{(2)}\left(\alpha\right)-\xi_{p,j}\mathcal{Y}_{2}^{(3)}\left(\alpha\right)\right]\right\\}\right\\}\right.+\left\\{1+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\\\
&\left.\times\left.\left\\{-\left(\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)\left(s+\sum_{p=1}^{j}C_{\lambda_{\mu_{p}}}\right)^{-1}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)!\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\right.\\\
&\left.\left.\left.\times\left(s+\xi_{p,j}\right)^{-\sum_{s=1}^{p}i_{\nu_{s}}}\left[1-\xi_{p,j}\left(s+\xi_{p,j}\right)^{-1}\right]\right\\}\right.\right.\\\
&\left.\left\\{1-\left(\prod_{k=1}^{L}\mathcal{P}_{k}\right)\left\\{\mathcal{Z}_{m_{0}}\left(C_{0},\alpha\right)+\sum_{\left\\{\lambda_{k}\right\\}_{k=1}^{L}}\sum_{\left\\{\mu_{j}\right\\}_{j=1}^{k}}\left(-1\right)^{j}\left(\prod_{n=1}^{k}\frac{1-\mathcal{P}_{\lambda_{n}}}{\mathcal{P}_{\lambda_{n}}}\right)\right.\right.\right.\\\
&\left.\left.\left.\left.\left\\{-C_{0}^{m_{0}}\left\\{\mathcal{X}_{2,1}^{(1)}\mathcal{Z}_{0}\left[\chi_{2}^{(1)},\alpha\right]+\chi_{2}^{(1)}\mathcal{Y}_{1}^{(1)}\left(\alpha\right)\right\\}+\sum_{\left\\{\nu_{i}\right\\}_{i=1}^{j}}\left(\prod_{s=1}^{p}\sum_{i_{\nu_{s}}=1}^{m_{\lambda_{\mu_{\nu_{s}}}}-1}\frac{C_{\lambda_{\mu_{\nu_{s}}}}^{i_{\nu_{s}}}}{i_{\nu_{s}}!}\right)\right.\right.\right.\right.\right.\\\
&\left.\left.\left.\left.\times
C_{0}^{m_{0}}\left(\sum_{s=1}^{p}i_{\nu_{s}}\right)!\left[\mathcal{Y}_{2}^{(2)}\left(\alpha\right)-\xi_{p,j}\mathcal{Y}_{2}^{(3)}\left(\alpha\right)\right]\right\\}\right\\}\right\\}\right\\}\end{split}$
(40)

while, substituting (34) and (37) to (39), yields the following closed-form
expression for $M_{g_{\text{sel}}}\left(s\right)$ of rate-selective RS over
IID Nakagami-$m$ fading channels with integer $m$:

$\begin{split}&\mathcal{M}_{g_{\text{sel}}}\left(s\right)=\frac{C^{m}L\mathcal{P}^{L}}{\left(s+C\right)^{m}}\left\\{\left\\{\frac{\mathcal{Z}_{m}\left(C,\alpha\right)}{L}+\left(1-\mathcal{P}\right)\mathcal{Z}_{2m}\left(C,\alpha\right)+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\right.\right.\\\
&\left.\left.\times\left\\{\mathcal{P}\mathcal{Z}_{m}\left(C,\alpha\right)+\left(1-\mathcal{P}\right)\mathcal{Z}_{2m}\left(C,\alpha\right)+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\left[\mathcal{P}\mathcal{Z}_{m}\left(C,\alpha\right)\right.\right.\right.\right.\\\
&\left.\left.\left.\left.+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\left(1-\mathcal{P}\right)\frac{\prod_{i=1}^{m}n_{i}^{-1}C^{2m+\sigma}}{\left(m-1\right)!\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\mathcal{Y}_{2}^{(4)}\left(\alpha\right)\right]\right\\}\right\\}+\left\\{\frac{1}{L}+\left(1-\mathcal{P}\right)C^{m}\right.\right.\\\
&\left.\left.\times\left(s+C\right)^{-m}+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\left\\{\mathcal{P}+\left(1-\mathcal{P}\right)C^{m}\left(s+C\right)^{-m}+\sum_{j=1}^{k}\binom{k}{j}\right.\right.\right.\\\
&\left.\left.\left.\times\left(-1\right)^{j}\left[\mathcal{P}+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\frac{\left(1-\mathcal{P}\right)C^{m+\sigma}}{\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\frac{\prod_{i=1}^{m}n_{i}^{-1}}{\left(m-1\right)!}\left[s+(j+1)C\right]^{-\left(m+\sigma\right)}\right]\right\\}\right\\}\right.\\\
&\times\left.\left\\{1-L\mathcal{P}^{L}\left\\{\frac{\mathcal{Z}_{m}\left(C,\alpha\right)}{L}+\left(1-\mathcal{P}\right)\mathcal{Z}_{2m}\left(C,\alpha\right)+\sum_{k=1}^{L-1}\binom{L-1}{k}\frac{\left(1-\mathcal{P}\right)^{k}}{\mathcal{P}^{k+1}}\right.\right.\right.\\\
&\left.\left.\left.\times\left\\{\mathcal{P}\mathcal{Z}_{m}\left(C,\alpha\right)+\left(1-\mathcal{P}\right)\mathcal{Z}_{2m}\left(C,\alpha\right)+\sum_{j=1}^{k}\binom{k}{j}\left(-1\right)^{j}\left[\mathcal{P}\mathcal{Z}_{m}\left(C,\alpha\right)\right.\right.\right.\right.\right.\\\
&\left.\left.\left.\left.\left.+\sum_{n_{1}+n_{2}+\cdots+n_{m}=j}^{j}\left(1-\mathcal{P}\right)\frac{\prod_{i=1}^{m}n_{i}^{-1}C^{2m+\sigma}}{\left(m-1\right)!\prod_{j=2}^{m-1}\left(j!\right)^{n_{j+1}}}\mathcal{Y}_{2}^{(4)}\left(\alpha\right)\right]\right\\}\right\\}\right\\}\right\\}.\end{split}$
(41)

##  IV Performance Analysis of ODF Relaying Schemes

In this section, the performance of ODF relaying schemes with repetitive and
RS-based transmission over INID Nakagami-$m$ fading channels will be analyzed.
We will present closed-form and analytical expressions, respectively, for the
following performance metrics: _i_) OP and _ii_) ASEP of several modulation
formats.

###  IV-A Repetitive Transmission

Using the closed-form expressions for $F_{g_{\text{end}}}(x)$ and
$M_{g_{\text{end}}}(s)$ presented in Section III-A, the OP and ASEP of
repetitive transmission with both MRD and SD are easily obtained as follows.

_OP:_ The end-to-end OP of repetitive transmission is easily obtained using
$F_{g_{\text{end}}}(x)$ as

$P_{\text{out}}=F_{g_{\text{end}}}\left[2^{(L+1)\mathcal{R}}-1\right].$ (42)

Substituting (14) and (15) to (42), closed-form expressions for the
$P_{\text{out}}$ of repetitive transmission with MRD over INID Nakagami-$m$
fading channels with integer $m_{\ell}$’s and distinct $C_{\ell}$’s as well as
with arbitrary $m_{\ell}$’s and $C_{\ell}=C$ $\forall\,\ell$, respectively,
are obtained. Similarly, substituting (17) with (5) and (19) to (42), yields a
closed-form expression for the $P_{\text{out}}$ of repetitive transmission
with SD over INID Nakagami-$m$ fading channels with arbitrary values for
$m_{\ell}$’s.

_ASEP:_ Following the MGF-based approach [21] and using the
$M_{g_{\text{end}}}(s)$ expressions given by (12) for repetitive transmission
with MRD and (27) and (29) for repetitive transmission with SD, the ASEP of
several modulation formats for both relaying schemes over INID Nakagami-$m$
fading channels can be easily evaluated. For example, the ASEP of non-coherent
binary frequency shift keying (NBFSK) and differential binary phase shift
keying (DBPSK) modulation schemes can be directly calculated from
$M_{g_{\text{end}}}(s)$; the average bit error rate probability (ABEP) of
NBFSK is given by $\overline{P}_{\rm b}=0.5\mathcal{M}_{g_{\text{end}}}(0.5)$
and of DBPSK by $\overline{P}_{\rm b}=0.5\mathcal{M}_{g_{\text{end}}}(1)$. For
other schemes, including binary phase shift keying (BPSK), $M$-ary phase shift
keying ($M$-PSK)222It is noted that, for modulation order $M>2$, Gray encoding
is assumed, so that $\overline{P}_{\rm s}=\overline{P}_{\rm b}\log_{2}(M)$.,
quadrature amplitude modulation ($M$-QAM), amplitude modulation ($M$-AM), and
differential phase shift keying ($M$-DPSK), single integrals with finite
limits and integrands composed of elementary functions (exponential and
trigonometric) have to be readily evaluated via numerical integration [21].
For example, the ASEP of $M$-PSK is easily obtained as

$\overline{P}_{\rm
s}=\frac{1}{\pi}\int_{0}^{\pi-\pi/M}\mathcal{M}_{g_{\text{end}}}\left(\frac{g_{\rm
PSK}}{\sin^{2}\varphi}\right){\rm d}\varphi$ (43)

where $g_{\rm PSK}=\sin^{2}\left(\pi/M\right)$, while for $M$-QAM, the ASEP
can be evaluated as

$\begin{split}\overline{P}_{\rm
s}=&\frac{4}{\pi}\left(1-\frac{1}{\sqrt{M}}\right)\left[\int_{0}^{\pi/2}\mathcal{M}_{g_{\text{end}}}\left(\frac{g_{\rm
QAM}}{\sin^{2}\varphi}\right){\rm d}\varphi\right.\\\
&\left.-\left(1-\frac{1}{\sqrt{M}}\right)\int_{0}^{\pi/4}\mathcal{M}_{g_{\text{end}}}\left(\frac{g_{\rm
QAM}}{\sin^{2}\varphi}\right){\rm d}\varphi\right]\end{split}$ (44)

with $g_{\rm QAM}=3/\left[2\left(M-1\right)\right]$.

###  IV-B RS-based Transmission

The closed-form expressions for the statistics of pure and rate-selective RS
presented in Section III-B can be easily used to obtain the OP and ASEP of
both RS-based schemes as follows.

_OP:_ Using the $F_{g_{\text{end}}}(x)$ expressions given by (35) and (37)
over INID and IID Nakagami-$m$ fading channels, respectively, with integer
fading parameters, the end-to-end OP of pure RS is easily obtained as

$P_{\text{out}}=F_{g_{\text{end}}}\left(2^{2\mathcal{R}}-1\right).$ (45)

For rate-selective RS, a closed-form expression for $P_{\text{out}}$ over INID
Nakagami-$m$ fading with arbitrary $m_{\ell}$’s can be easily obtained by
substituting (5) and (19) to [10, eq. (15)], yielding

$P_{\text{out}}=\frac{\mathsf{\Gamma}\left[m_{0},C_{0}\left(2^{\mathcal{R}}-1\right)\right]}{\Gamma\left(m_{0}\right)}\prod_{k=1}^{L}\left\\{\mathcal{P}_{k}u\left(2^{2\mathcal{R}}-1\right)+\left(1-\mathcal{P}_{k}\right)\frac{\mathsf{\Gamma}\left[m_{k},C_{k}\left(2^{2\mathcal{R}}-1\right)\right]}{\Gamma\left(m_{k}\right)}\right\\}.$
(46)

_ASEP:_ Following the MGF-based approach and using the $M_{g_{\text{end}}}(s)$
expressions for pure RS given by (33) and (34) over INID and IID Nakagami-$m$
fading channels, respectively, with integer fading parameters, the ASEP of
several modulation formats for pure RS can be easily calculated. Similarly,
the ASEP for rate-selective RS can be easily evaluated using the
$M_{g_{\text{sel}}}(s)$ expressions given by (40) and (41) over INID and IID
Nakagami-$m$ fading channels, respectively, with integer values for the fading
parameters.

##  V Numerical Results and Discussions

The analytical expressions of the previous section have been used to evaluate
the performance of ODF cooperative systems utilizing repetitive and RS-based
transmission over INID Nakagami-$m$ fading channels. Without loss of
generality, it has been assumed that $S$’s transmit rate $\mathcal{R}=1$
bps/Hz and that the fading parameters of the first and the second hop of all
$L$ links are equal, i.e., $m_{k}=m_{L+k}$ $\forall\,k$, as well as
$\overline{\gamma}_{k}=\overline{\gamma}_{L+k}$ $\forall\,k$. Moreover, for
the performance evaluation results we have considered the exponentially power
decaying profile
$\overline{\gamma}_{k}=\overline{\gamma}_{0}\exp\left(-\ell\delta\right)$ with
$\delta$ being the power decaying factor. Clearly, wherever ID fading is
considered, $\delta=0$ and
$\overline{\gamma}_{k}=\overline{\gamma}_{0}=\overline{\gamma}$ $\forall\,k$.
In all figures that follow, analytical results for the $P_{\text{out}}$ and
$\overline{P}_{\text{b}}$ match perfectly with equivalent performance
evaluation results obtained by means of Monte Carlo simulations, thus
validating our analysis.

Fig. 2 illustrates $P_{\text{out}}$ of the considered relaying schemes with
both repetitive and RS-based transmission as a function of $L$ for average
transmit SNR $\overline{\gamma}_{0}=5$ dB over IID Nakagami-$m$ fading
channels with different values of $m$. As clearly shown, $P_{\text{out}}$
improves with increasing $L$ for both RS-based transmission schemes whereas it
degrades severely for both repetitive ones. In particular, utilizing
repetitive transmission for $\overline{\gamma}_{0}=5$ dB is unavailing for
$L\geq 3$ irrespective of the fading conditions. As for RS-based transmission,
the gains from RS diminish as $m$ increases; the smaller the $m$, the greater
the $P_{\text{out}}$ gain from relaying. Furthermore, pure RS is inefficient
when $L$ is small and rate-selective RS leads to significant $P_{\text{out}}$
gains over the pure one as $m$ decreases. The impact of increasing $L$ in the
$P_{\text{out}}$ performance of all four relaying schemes is also demonstrated
in Fig. 3 for different values of $\overline{\gamma}_{0}$ and IID Rayleigh
fading conditions. As shown from this figure and Fig. 2 for both repetitive
and RS-based transmission, $P_{\text{out}}$ degrades with decreasing
$\overline{\gamma}_{0}$ and/or $m$. For example, although for
$\overline{\gamma}_{0}=5$ dB repetitive transmission does not benefit from
relaying for $L\geq 3$, this happens for $\overline{\gamma}_{0}=0$ dB when
$L\geq 2$. Moreover, for the latter transmit SNR, the improvement on the
$P_{\text{out}}$ of pure RS with increasing $L$ is very small, while
$P_{\text{out}}$ performance of rate-selective RS is irrespective of $L$; the
smaller the $\overline{\gamma}_{0}$, the smaller the gains from relaying. This
happens because as fading conditions become more severe and transmit SNR
reduces, the gains from RS eventually decrease.

Using (7) in Fig. 4, the decoding probability $1-\mathcal{P}_{k}$ is plotted
versus $L$ for both repetitive and RS-based transmission over IID Nakagami-$m$
fading conditions for the source to relay channels with different values of
$m$ and $\overline{\gamma}_{L+k}=\overline{\gamma}$ $\forall\,k,L$. As
expected, decreasing $m$ and/or $\overline{\gamma}$ decreases the decoding
probability. Moreover, this probability is severely degraded with increasing
$L$ for repetitive transmission, whereas it remains unchanged with increasing
$L$ for the RS-based one. The $P_{\text{out}}$ performance of pure and rate-
selective RS is depicted in Fig. 5 versus $\overline{\gamma}_{0}$ for $L=2$
relays over INID Nakagami-$m$ fading channels with different values of
$m_{0}$, $m_{1}$ and $m_{2}$. As expected, for both RS-based schemes,
$P_{\text{out}}$ improves with increasing $\overline{\gamma}_{0}$ and/or any
of the fading parameters. More importantly, it is shown that, as the fading
conditions of the relay to destination channels become more favorable than
those of the direct source to destination channel, RS-based transmission
improves $P_{\text{out}}$. On the contrary, whenever the fading conditions of
the relay to destination channels are similar to the direct one, non-relay
assisted transmission results in lower $P_{\text{out}}$ than that of the RS-
based one.

In Fig. 6, the $\overline{P}_{\text{b}}$ performance of square $4$-QAM is
plotted as a function of $L$ for average SNR per bit
$\overline{\gamma}_{\text{b}}=5$ dB for all second hop channels and
$\overline{\gamma}_{0}=0$ dB over IID Nakagami-$m$ fading channels with
different values of $m$. It is clearly shown that the
$\overline{P}_{\text{b}}$ of RS-based transmission improves with increasing
$L$ and/or $m$ whereas for both repetitive schemes, although
$\overline{P}_{\text{b}}$ improves with increasing $m$, it does not benefit
from increasing $L$. Interestingly, for RS-based transmission, as $L$ and/or
$m$ increase, the $\overline{P}_{\text{b}}$ improvement of pure compared with
rate-selective RS increases; as shown in (38), rate-selective RS might choose
the direct transmission even in cases where the received SNR from the best
relay is larger that from the source node. Assuming IID Rayleigh fading
conditions, Fig. 7 illustrates $\overline{P}_{\text{b}}$ of DBPSK versus $L$
for all considered relaying schemes and for different values of
$\overline{\gamma}_{0}$. As shown from this figure and Fig. 6 for both
repetitive and RS-based transmission, $\overline{P}_{\text{b}}$ for both
modulations improves with increasing $\overline{\gamma}_{0}$ and/or $m$. It
was shown in Fig. 4 that more favorable fading conditions and larger
$\overline{\gamma}_{0}$ increase the decoding probability, thus both
repetitive transmission schemes become unavailing for larger $L$, whereas the
gains from RS increase. The $\overline{P}_{\text{b}}$ performance of DBPSK of
pure and rate-selective RS is depicted in Fig. 8 versus
$\overline{\gamma}_{\text{b}}$ for all second hop channels with $L=1$, $2$ and
$3$ relays and for $\overline{\gamma}_{0}=0$ dB over INID Nakagami-$m$ fading
channels with different values for the fading parameters. Clearly, for both
RS-based transmission schemes, $\overline{P}_{\text{b}}$ improves with
increasing $\overline{\gamma}_{\text{b}}$ and/or $m$ and/or $L$. It is shown
that, as $\overline{\gamma}_{\text{b}}$ increases and as the fading conditions
of the relay to destination channels become more favorable than those of the
direct source to destination channel, RS-based transmission results in larger
$\overline{P}_{\text{b}}$ improvement. More importantly, even for strong
source to destination channel conditions, as $\overline{\gamma}_{\text{b}}$
increases the improvement on $\overline{P}_{\text{b}}$ with RS-based
transmission becomes larger than that with the non-relay assisted one.

##  VI Conclusion

Cooperative diversity is a very promising avenue for future wireless
communications. However, it is necessary to investigate how relays should be
utilized in order to achieve certain objectives. In this paper, we presented a
general analytical framework for modelling and evaluating performance of four
DF relaying schemes under INID Nakagami-$m$ fading channels. Moreover, we
obtained closed-form expressions for the OP and the ASEP performance of the RS
and repetitive schemes when MRD or SD are employed at the destination node. We
concluded that RS performs better than repetitive transmission when CSI for RS
is perfect. Furthermore, it was shown that relaying should be switched-off
when the source to destination direct link is sufficiently strong and the aim
is minimizing OP. However, in terms of ASEP, relaying was shown to be always
beneficial.

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FIGURES’ CAPTIONS

Fig. 1: Illustration of a dual-hop cooperative wireless system with $L+2$
wireless nodes: a source node $S$, $L$ relay station nodes $R_{k}$,
$k=1,2,\ldots,L$, and a destination node $D$.

Fig. 2: End-to-end OP, $P_{\text{out}}$, versus the number of relay nodes,
$L$, for average transmit SNR $\overline{\gamma}_{0}=5$ dB over IID
Nakagami-$m$ fading channels with different values of $m$: (A) Pure RS, (B)
Rate-Selective RS, (C) Repetitive transmission with MRD and (D) Repetitive
transmission with SD.

Fig. 3: End-to-end OP, $P_{\text{out}}$, versus the number of relay nodes,
$L$, for different average transmit SNRs over IID Rayleigh fading channels:
(A) Pure RS, (B) Rate-Selective RS, (C) Repetitive transmission with MRD and
(D) Repetitive transmission with SD.

Fig. 4: Decoding probability, $1-\mathcal{P}_{k}$, versus the number of relay
nodes, $L$, over IID Nakagami-$m$ fading channels with different values of $m$
and $\overline{\gamma}_{L+k}=\overline{\gamma}$ $\forall\,k,L$: (A) Repetitive
and (B) RS-based transmission.

Fig. 5: End-to-end OP, $P_{\text{out}}$, of RS versus the average transmit SNR
per bit, $\overline{\gamma}_{0}$, for $L=2$ relay nodes over INID Nakagami-$m$
fading channels: (A) $m_{0}=m_{1}=m_{2}=1$ and $\delta=0.3$, (B) $m_{0}=1$,
$m_{1}=m_{2}=6$ and $\delta=0.3$ and (C) $m_{0}=m_{1}=m_{2}=3$ and $\delta=0$.

Fig. 6: ABEP, $\overline{P}_{\text{b}}$, of square $4$-QAM versus the number
of relay nodes, $L$, for average transmit SNR $\overline{\gamma}_{\text{b}}=5$
dB and $\overline{\gamma}_{0}=0$ dB over IID Nakagami-$m$ fading channels with
different values of $m$: (A) Pure RS, (B) Rate-Selective RS, (C) Repetitive
transmission with MRD and (D) Repetitive transmission with SD.

Fig. 7: ABEP, $\overline{P}_{\text{b}}$, of DBPSK versus the number of relay
nodes, $L$, for different average transmit SNRs per bit and
$\overline{\gamma}_{0}=0$ dB, over IID Rayleigh fading channels: (A) Pure RS,
(B) Rate-Selective RS, (C) Repetitive transmission with MRD and (D) Repetitive
transmission with SD.

Fig. 8: ABEP, $\overline{P}_{\text{b}}$, of DBPSK for RS versus the average
relay SNR per bit, $\overline{\gamma}_{\text{b}}$, and for
$\overline{\gamma}_{0}=0$ dB over INID Nakagami-$m$ fading channels: (A)
$L=1$, $m_{0}=0.5$, $m_{1}=1$ and $\delta=0.1$, (B) $L=2$, $m_{0}=1$,
$m_{1}=m_{2}=2$ and $\delta=0$ and (C) $L=3$, $m_{\ell}=3$ for $\ell=0,1,2$
and $3$ and $\delta=0$.

Figure 1: Illustration of a dual-hop cooperative wireless system with $L+2$
wireless nodes: a source node $S$, $L$ relay station nodes $R_{k}$,
$k=1,2,\ldots,L$, and a destination node $D$.

Figure 2: End-to-end OP, $P_{\text{out}}$, versus the number of relay nodes,
$L$, for average transmit SNR $\overline{\gamma}_{0}=5$ dB over IID
Nakagami-$m$ fading channels with different values of $m$: (A) Pure RS, (B)
Rate-Selective RS, (C) Repetitive transmission with MRD and (D) Repetitive
transmission with SD.

Figure 3: End-to-end OP, $P_{\text{out}}$, versus the number of relay nodes,
$L$, for different average transmit SNRs over IID Rayleigh fading channels:
(A) Pure RS, (B) Rate-Selective RS, (C) Repetitive transmission with MRD and
(D) Repetitive transmission with SD.

Figure 4: Decoding probability, $1-\mathcal{P}_{k}$, versus the number of
relay nodes, $L$, over IID Nakagami-$m$ fading channels with different values
of $m$ and $\overline{\gamma}_{L+k}=\overline{\gamma}$ $\forall\,k,L$: (A)
Repetitive and (B) RS-based transmission.

Figure 5: End-to-end OP, $P_{\text{out}}$, of RS versus the average transmit
SNR per bit, $\overline{\gamma}_{0}$, for $L=2$ relay nodes over INID
Nakagami-$m$ fading channels: (A) $m_{0}=m_{1}=m_{2}=1$ and $\delta=0.3$, (B)
$m_{0}=1$, $m_{1}=m_{2}=6$ and $\delta=0.3$ and (C) $m_{0}=m_{1}=m_{2}=3$ and
$\delta=0$.

Figure 6: ABEP, $\overline{P}_{\text{b}}$, of square $4$-QAM versus the number
of relay nodes, $L$, for average transmit SNR $\overline{\gamma}_{\text{b}}=5$
dB and $\overline{\gamma}_{0}=0$ dB over IID Nakagami-$m$ fading channels with
different values of $m$: (A) Pure RS, (B) Rate-Selective RS, (C) Repetitive
transmission with MRD and (D) Repetitive transmission with SD.

Figure 7: ABEP, $\overline{P}_{\text{b}}$, of DBPSK versus the number of relay
nodes, $L$, for different average transmit SNRs per bit and
$\overline{\gamma}_{0}=0$ dB, over IID Rayleigh fading channels: (A) Pure RS,
(B) Rate-Selective RS, (C) Repetitive transmission with MRD and (D) Repetitive
transmission with SD.

Figure 8: ABEP, $\overline{P}_{\text{b}}$, of DBPSK for RS versus the average
relay SNR per bit, $\overline{\gamma}_{\text{b}}$, and for
$\overline{\gamma}_{0}=0$ dB over INID Nakagami-$m$ fading channels: (A)
$L=1$, $m_{0}=0.5$, $m_{1}=1$ and $\delta=0.1$, (B) $L=2$, $m_{0}=1$,
$m_{1}=m_{2}=2$ and $\delta=0$ and (C) $L=3$, $m_{\ell}=3$ for $\ell=0,1,2$
and $3$ and $\delta=0$.