IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014 2183
Serial AmplifyandForward Relay TransmissionSystems in Nakagami
m
Fading ChannelsWith a Poisson Interference Field
Valentine A. Aalo,
Senior Member, IEEE
, Kostas P. Peppas,
Member, IEEE
, George P. Efthymoglou,
Member, IEEE
,Mohammed M. Alwakeel, and Sami S. Alwakeel,
Member, IEEE
Abstract
—In this paper, the endtoend performance of a wireless relay transmission system that employs amplifyandforward(AF) relays and operates in an interferencelimited Nakagami
m
fading environment is studied. The wireless links from one relaynode to another experience Nakagami
m
fading, and the number of interferers per hop is Poisson distributed. The aggregateinterference at each relay node is modeled as a shotnoise processwhose distribution follows an
α
stable process. For the consideredsystem, analytical expressions for the moments of the endtoendsignaltointerference ratio (SIR), the endtoend outage probability (OP), the average biterror probability (ABEP), and the averagechannelcapacityareobtained.Generalasymptoticexpressionsfor the endtoend ABEP are also derived. The results provideuseful insights regarding the factors affecting the performance of the considered system. Monte Carlo simulation results are furtherprovidedtodemonstratethevalidityoftheproposedmathematicalanalysis.
Index Terms
—Amplifyandforward (AF), average biterrorprobability (ABEP), channel capacity, cochannel interference,Fox’s Hfunction, Meijer’s Gfunction, multihop relaying, Poissoninterference ﬁeld.
I. I
NTRODUCTION
M
ULTIHOP relaying has recently received considerableattention in the literature because of its potential toprovide more efﬁcient and broader coverage in microwave andbentpipe satellites links, as well as cellular, modern ad hoc,wireless local area, and hybrid wireless networks [1]. Therefore, multihop relaying, which is designed for extended cover
Manuscript received November 13, 2012; revised July 5, 2013 andOctober 12, 2013; accepted October 17, 2013. Date of publicationNovember 14, 2013; date of current version June 12, 2014. This work wassupported by the Sensor Networks and Cellular Systems Research Centerof the University of Tabuk. The review of this paper was coordinated byProf. J. Y. Chouinard.V. A. Aalo is with the Department of Computer and Electrical Engineeringand Computer Science, Florida Atlantic University, Boca Raton, FL 33431USA , and also with SNCS Research Center, University of Tabuk, Saudi Arabia(email: aalo@fau.edu).K. P. Peppas is with the Institute of Informatics and Telecommunications,National Centre for Scientiﬁc Research “Demokritos,” 15310 Athens, Greece(email: kpeppas@iit.demokritos.gr).G. P. Efthymoglou is with the Department of Digital Systems, University of Piraeus, 18534 Piraeus, Greece (email: gefthymo@unipi.gr).M. M. Alwakeel is with the Sensor Networks and Cellular Systems ResearchCenter, University of Tabuk, Tabuk 71491, Saudi Arabia (email: malwakeel@ut.edu.sa).S. S. Alwakeel is with the Department of Computer Engineering, King SaudUniversity, Riyadh 11543, Saudi Arabia, and also with the Sensor Networksand Cellular Systems Research Center, University of Tabuk, Tabuk 71491,Saudi Arabia (email: swakeel@ksu.edu.sa).Digital Object Identiﬁer 10.1109/TVT.2013.2291039
age and throughput enhancement, has been adopted in severalwireless standards [2]–[4]. In a multihop relaying system,intermediateidlenodesthatareclosertothetransmitterthanthedestination operate as relays between the source node and thedestination node when the direct link between the source nodeand the destination node is deeply faded or highly shadowed.Various protocols have been proposed to achieve the beneﬁtsof multihop transmission. One of them is the socalled amplifyandforward (AF) protocol, in which the received signal issimply ampliﬁed and forwarded to the receiver without performing any decoding [5]. The performance of multihop AFrelaying systems in series has been addressed in many pastworks based on the assumption that the system performanceis limited by Gaussian [6], [7] or generic noise [8]. On the otherhand, practical relaying systems generally employ frequencyreuse, which results in cochannel interference. The impact of cochannel interference on the performance of AF relay systemsin a fading environment has been studied in many recent works,assumingaﬁxednumberofinterferingsignalsthatareRayleighor Nakagami
m
distributed at each relay node and at the destinationnode.(See[9]–[18]andreferencesthereinforexamples.)However, in a practical wireless environment, the numberof interfering signals at each relay may be a random variableas well. Moreover, in many wireless networks, the interfering signals also experience attenuation due to path loss andshadowing, whereas their location and activity around thereceiving node may vary randomly [19]. Speciﬁcally, in theemergingheterogeneouscellularsystems,thepositionsofmanyinfrastructure elements are unknown
a priori,
and the presenceof unplanned network deployments should be considered bysystemdesigners.Moreover,duetotherandomspatialpositionsof the interferers, it is more insightful to provide performancemetrics that are averaged over fast fading and spatial positionof each interferer.Inspired by the seminal work in [20], in the emerging heterogeneous cellular systems, a promising approach to model interference is to treat the locations of some network elements (e.g.,cognitive radios and femto base stations) as points distributedaccordingtothespatialPoissonpointprocess(PPP).Thismodeloffers analytical ﬂexibility and can provide insightful informationontheeffectofcriticalstatistical–physicalparametersonthesystem performance. For these reasons, network interferencemodeling based on PPP spatial models has attracted the interestof many researchers. Speciﬁcally, in [21] and [22], PPPs have
00189545 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
2184 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
been used to model cochannel interference from macrocellularbase stations. Crosstier interference from femtocells was discussed in [23] and [24], whereas in [25], cochannel interferencein ad hoc networks was investigated. Cochannel interference asa generic source of interference was addressed in [26] and [27].In [28], a uniﬁed framework for interference characterizationsand analysis in the unlicensed frequency bands was presented,assuming that interferers can have any power spectral densityand are distributed according to a Poisson process in spaceand frequency domains. The performance of diversity receiversin a Rayleigh fading environment and network interferencefrom a Poisson ﬁeld of interference sources was addressed in[29]. In [30], a simpliﬁed interference model for heterogeneousnetworks to analyze downlink performance in a ﬁxed size cellin a Poisson ﬁeld of interferers was proposed. In [31], theperformance of multiantenna systems in a Poisson ﬁeld of interferers was addressed.To the best of our knowledge, although the PPP interference model has been used extensively in a variety of wirelessnetworks to account for the randomness in the number, thelocation, and the activity of the interferers, this model hasrarely been applied to the relay and destination nodes of amultihop relay network. Two recent examples include [32]and [33]. Speciﬁcally, in [32], analytical expressions for theoutage probability (OP) and average biterror rate of dualhop AF relaying, using the best relay in a 2D Poisson ﬁeldof relays, were derived. In [33], the random access transportcapacity of multihop AF relaying in a Poisson interference ﬁeldwas addressed. In this paper, we analyze the performance of multihop AF systems in the presence of a Poisson distributedinterference ﬁeld, in which the relay is assumed to possess perfect channel state information. The interference model adoptedin this paper is based on the assumption that the number of interferers is a Poisson distributed random variable, whereasthe terminals are randomly distributed over the network areaand undergo Nakagami
m
fading. Moreover, the wireless linksbetween relay nodes are assumed to be subject to Nakagami
m
fading as well. It is also noted that the interference model underconsideration takes into account the randomness in the numberand location of the interferers and the effect of path loss for theinterfering signals. The main contributions of this paper are asfollows.ã Using the theory of Fox’s Hfunctions and Mellin–Barnesintegrals, a novel closedform expression for the probability density function (pdf) of the aggregate interference isﬁrst derived. This result is afterward used to obtain closedform expressions for important statistics of the signaltointerference ratio (SIR) of each hop, i.e., the pdf, thecumulative distribution function (cdf) of the SIR, and themomentgenerating function (MGF) of the inverse SIR.ã Exact analytical expressions for the
u
th moment of theendtoend SIR are derived.ã Exact analytical expressions and closedform lowerbounds for the OP are derived. These bounds become tightat high values of SIR.ã Exactanalyticalexpressionsandtightlowerboundsfortheaverage biterror probability (ABEP) of digital modulationschemes and exact analytical expressions for the average
TABLE IM
ATHEMATICAL
O
PERATORS AND
F
UNCTIONS
channel capacity are derived in terms of single integrals.Such integrals can be efﬁciently evaluated by employingGauss quadrature techniques.ã An asymptotic error rate performance analysis is presented.Thisprovidesinsightsintotheparametersaffectingsystem performance under the presence of interference.The proposed analysis is tested and veriﬁed by numericallyevaluated results accompanied with Monte Carlo simulations.The remainder of this paper is structured as follows. InSection II, some important properties of the Fox’s Hfunction,which are frequently used throughout this paper, are summarized. Section III outlines the system and the interferencemodels. In Section IV, the statistical properties of the endtoend SIR are investigated. In Section V, analytical expressionsfor the
u
th moment of the endtoend SIR, the OP, the ABEP,and the average channel capacity are presented. In Section VI,the various performance results and their interpretationsare presented. Finally, concluding remarks are presented inSection VII.For the convenience of the reader, a comprehensivelist of the mathematical operators and functions used in thispaper is presented in Table I.II. M
ATHEMATICAL
P
RELIMINARIES
Throughout this paper, Fox’s Hfunction is used to obtainanalytical expressions for the statistics of the endtoend SIRand for the important performance metrics of interest, suchas the OP, the ABEP, and the average channel capacity. Here,known results on Fox’s Hfunction are summarized to make thispaper more accessible.
Deﬁnition 1:
The Fox’s Hfunction is deﬁned as[36, Eq. (1.2)]
H
k,n p,q
x
(
a
p
,A
p
)(
b
q
,B
q
)
=
12
πı
C
kj
=1
Γ(
b
j
+
B
j
s
)
pj
=
n
+1
Γ(
a
j
+
A
j
s
)
×
nj
=1
Γ(
1
−
a
j
−
A
j
s
)
qj
=
k
+1
Γ(
1
−
b
j
−
B
j
s
)
x
−
s
ds
(1)
AALO
et al.
: SERIAL AF RELAY TRANSMISSION SYSTEMS IN NAKAGAMI
m
FADING CHANNELS 2185
where
C
is a suitable contour separating the poles of
Γ(
b
j
+
B
j
s
)
from the poles of
Γ(
1
−
a
j
−
A
j
s
)
.Note that, for
A
j
=
B
j
=
1, the Fox’s HFunction reducesto the more familiar Meijer’s Gfunction [35, Eq. (8.2.1)]. Thefollowing identities presented serve as a direct consequence of the deﬁnition of the Hfunction by the application of certainproperties of gamma functions.
Property 1:
There holds the formula [36, Eq. (1.58)]
H
k,n p,q
x
(
a
p
,A
p
)(
b
q
,B
q
)
=
H
n,kq,p
1
x
(
1
−
b
q
,B
q
)(
1
−
a
p
,A
p
)
.
(2)
Property 2:
The following reduction formulas are valid[36, Eqs. (1.56, 1.57)]:
H
k,n p,q
x
(
a
1
,A
1
)
,...,
(
a
p
,A
p
)(
b
1
,B
1
)
,...,
(
b
q
−
1
,B
q
−
1
)
,
(
a
1
,A
1
)
=
H
k,n
−
1
p
−
1
,q
−
1
x
(
a
2
,A
2
)
,...,
(
a
p
,A
p
)(
b
1
,B
1
)
,...,
(
b
q
−
1
,B
q
−
1
)
(3)provided that
n
≥
1 and
q > k
and
H
k,n p,q
x
(
a
1
,A
1
)
,...,
(
a
p
−
1
,A
p
−
1
)
,
(
b
1
,B
1
)(
b
1
,B
1
)
,...,
(
b
q
,B
q
)
=
H
k
−
1
,n p
−
1
,q
−
1
x
(
a
1
,A
1
)
,...,
(
a
p
−
1
,A
p
−
1
)(
b
2
,B
2
)
,...,
(
b
q
,B
q
)
(4)provided that
k
≥
1 and
p > n
.The
r
thorder derivative of Fox’s Hfunction can be obtainedusing the following property.
Property 3:
Identity (5), shown at the bottom of the page,holds [36, Eq. (1.83)], where
h >
0.Throughout this paper, integral transforms of Fox’sHfunction are used to derive the main results. The Mellin andLaplace transforms of the Fox’s Hfunction are of particularinterest. An important property of the Hfunction states thatthe Mellin transform of the product of two Fox’s Hfunctionsis also a Fox’s Hfunction, as summarized in the followingtheorem.
Theorem 1:
The following integral identity is valid[36, p. 60], [35, Eq. (2.25.1.1)]:
∞
0
x
α
−
1
H
s,tu,v
σx
(
c
u
,C
u
)(
d
v
,D
v
)
H
k,n p,q
ωx
r
(
a
p
,A
p
)(
b
q
,B
q
)
dx
=
σ
−
α
H
k
+
t,n
+
s p
+
v,q
+
u
ωσ
r
(˜
a
p
+
v
,
˜
A
p
+
v
)(˜
b
q
+
u
,
˜
B
q
+
u
)
(6)where
α
,
σ
, and
ω
are complex numbers;
r >
0; and
(˜
a
p
+
v
,
˜
A
p
+
v
) =
{
(
a
n
,A
n
)
,
(
1
−
d
v
−
αD
v
,rD
v
)(
a
n
+1
,A
n
+1
)
,...,
(
a
p
,A
p
)
}
(˜
b
q
+
u
,
˜
B
q
+
u
) =
{
(
b
k
,B
k
)
,
(
1
−
c
u
−
αC
u
,rC
u
)(
b
k
+1
,B
k
+1
)
,...,
(
b
q
,B
q
)
}
provided that the following conditions are satisﬁed:
a
∗
∆
=
n
j
=1
A
j
−
p
j
=
n
+1
A
j
+
k
j
=1
B
j
−
q
j
=
k
+1
B
j
>
0
b
∗
∆
=
t
j
=1
C
j
−
u
j
=
t
+1
C
j
+
s
j
=1
D
j
−
v
j
=
s
+1
D
j
>
0
r >
0

arg
σ

< b
∗
π/
2

arg
ω

< a
∗
π/
2
{
α
}
+
r
min
1
≤
j
≤
k
{
b
j
/B
j
}
+ min
1
≤
h
≤
s
{
d
h
/D
h
}
>
0
{
α
}
+
r
max
1
≤
j
≤
n
{
(
a
j
−
1
)
/A
j
}
+ max
1
≤
h
≤
t
{
(
c
h
−
1
)
/C
h
}
<
0
.
Proof:
See [36, p. 60].
Another interesting property of the Fox’s Hfunction statesthat its inverse Laplace transform is also a Fox’s Hfunction.Speciﬁcally, the following theorem holds [36, Eq. (2.21)].
Theorem 2:
The following inverse Laplace transform pair isvalid:
L
−
1
t
−
ρ
H
k,n p,q
at
σ
(
a
p
,A
p
)(
b
q
,B
q
)
;
t
;
x
=
x
ρ
−
1
H
k,n p
+1
,q
ax
−
σ
(
a
p
,A
p
)
,
(
ρ,σ
)(
b
q
,B
q
)
(7)provided that
{
s
}
>
0,
σ>
0,
{
ρ
}
+
σ
max
1
≤
i
≤
n
[(
1
/A
i
)
−
(
{
a
i
}
/A
i
)]
>
0, and

arg
a

< π
(
a
∗
−
σ
)
, where
a
∗
is deﬁnedin Theorem I.
Proof:
See [36, p. 51].
Finally, power series expansion of the Fox’s Hfunction,which are useful in deriving asymptotic results for importantperformance metrics of interest, are discussed. Speciﬁcally, thefollowing theorem holds [37, Eq. (3.4)].
Theorem 3:
Let us assume that the poles of
Γ(
1
−
a
j
−
A
j
s
)
,
j
=
1
,...,n
, and
Γ(
b
j
+
B
j
s
)
,
j
=
1
,...,k
do not
r
i
=1
x ddx
−
c
i
x
s
H
k,n p,q
zx
h
(
a
1
,A
1
)
,...,
(
a
p
,A
p
)(
b
1
,B
1
)
,...,
(
b
q
,B
q
)
=
x
s
×
H
k,n
+
r p
+
r,q
+
r
zx
h
(
c
1
−
s,h
)
,...,
(
c
r
−
s,h
)
,
(
a
1
,A
1
)
,...,
(
a
p
,A
p
)(
b
1
,B
1
)
,...,
(
b
q
,B
q
)
,
(
c
1
−
s
+
1
,h
)
,...,
(
c
r
−
s
+
1
,h
)
(5)
2186 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 2014
coincide. Then, for
qj
=1
B
j
−
pj
=1
A
j
>
0,
x
=
0, or for
qj
=1
B
j
−
pj
=1
A
j
=
0, 0
<

x

<
pi
=1
A
−
A
i
i
qj
=1
B
B
j
j
,the Hfunction has the following power series expansion:
H
k,n p,q
x
(
a
p
,A
p
)(
b
q
,B
q
)
=
k
j
=1
∞
=0
h
∗
j
x
bj
+
Bj
(8)where the constants
h
∗
j
are given by [37, Eq. (3.3)]
h
∗
j
= (
−
1
)
!
B
j
ki
=1
,i
=
j
Γ
b
i
−
[
b
j
+
]
B
i
B
j
pi
=
n
+1
Γ
a
i
−
[
b
j
+
]
A
i
B
j
×
ni
=1
Γ
1
−
a
i
+ [
b
j
+
]
A
i
B
j
qi
=
k
+1
Γ
1
−
b
i
+ [
b
j
+
]
B
i
B
j
.
(9)
Proof:
See [37].
Finally, as far as the computational implementation of theFox’s Hfunction is concerned, it is noted that the Fox’sHfunction is still not available in standard mathematical software packages such as Mathematica and Maple. Nevertheless,in two recent works, numerically efﬁcient methods to evaluatethis function have been developed using Matlab [38, Table 2]and Mathematica [39, Appendix A]. Both methods are basedon the deﬁnition of the Fox’s Hfunction, which is given by (1).III. S
YSTEM AND
I
NTERFERENCE
M
ODELS
A. System Model
An
N
hop wireless network is considered, which consists of source terminal
S
, several AF relay nodes
R
n
1
and destinationterminal
D
. The distance between the source and destinationnodes is assumed to be too long for a reliable direct communication link to be established, given the power constraints andchannel fading effects. Therefore,
N
−
1 relay terminals areemployed via which the source and destination terminals cancommunicate. Each relay node and the destination terminal areinterrupted by a random number of cochannel interferers in aNakagami
m
fading environment.It is also assumed that the number of interfering signals at the
n
th relay terminal or at the destination is a Poisson distributeddiscrete random variable [12], [28], [40] and belongs to the PPP
K
n
. The received signal at the
n
th relay node is given by
y
n
=
P
R
n
−
1
d
−
v
n
/
2
n
a
n
x
n
−
1
+
P
I,n
i
∈K
n
r
−
v
n
/
2
i,n
ξ
i,n
x
i,n
+
w
n
(10)where
P
R
n
−
1
is the transmit power of the
(
n
−
1
)
th node;
d
n
isthe distance between nodes
R
n
−
1
and
R
n
;
a
n
is the Nakagami
m
distributed fading amplitude for the direct channel betweenthe said nodes;
ξ
i,n
is the Nakagami
m
distributed fadingamplitude for the channel from the
i
th interfering transmitter to
1
Throughout this analysis and without loss of generality, index
n
=
1
,
2
,...,N
−
1.
the
n
th relay node, respectively;
x
n
−
1
is the signal transmittedfrom the
(
n
−
1
)
th node; and
x
i,n
are the interfering signals tothe
n
th node. Moreover,
r
i,n
is the random distance from the
i
th interferers location to the
n
th relay, and
v
n
is the pathlossexponent in the environment surrounding the
n
th relay terminal with 2
≤
v
n
≤
5. All the interfering signals are assumedtransmitted with the same power
P
I,n
[25], [41] but experiencemutually independent path loss and Nakagami
m
fading.The pathloss model used in this paper is unbounded andunrealistic in practice as the received power is inﬁnite at
d
n
=
0 or
r
i,n
=
0 [42]; thus, it may cause the moments of theaggregate interference to become inﬁnite. However, this modelis commonly used in the literature as it leads to more tractablemathematical formulation (see, e.g., [20], [25], [28], [40], etc.).In practice, a small exclusion region may be placed aroundthe
n
th relay, from within which interfering transmissions areprohibited [43].The average power of the desired signal on the link betweenrelays
R
n
−
1
and
R
n
is given by
Ω
s,n
=
P
R
n
−
1
d
n
−
v
n
, whereasthe average power of the signal from the
i
th interferer torelay node
n
is given by
Ω
i,n
=
P
I,n
r
−
v
n
i,n
. Throughout thisanalysis, an interferencelimited system is considered, in whichthe effect of additive white Gaussian noise (AWGN) on systemperformance can usually be neglected. For an AF transmissionscheme, the
n
th relay ampliﬁes its received signal by gain
G
n
.One choice for the relay gain is proposed in [6] to be
G
n
=
P
R
n
/
(
P
R
n
−
1
a
2
n
)
, which corresponds to the ideal gain at the
n
th AF relay that inverts the channel gain in the
n
th link.Such a relay serves as a benchmark for all practical relayingsystems employing the AF protocol [6]. Then, the endtoend instantaneous received SIR is related to the (normalizedby
N
) harmonic mean of the individual perhop SIRs [6], [13],[14], i.e.,
γ
eq
=
N
n
=1
1
γ
n
−
1
(11)where
γ
n
=
a
2
n
Z
n
(12)is the instantaneous SIR of the
n
th hop for interferencelimitedrelay systems. In (12),
Z
n
=
i
∈K
n
ξ
2
i,n
r
−
v
n
i,n
κ
n
is the totalinstantaneous interference power at the
n
th relay, where
κ
n
isa lognormal random variable that represents the effect of shadowing. Note that the expression for the SIR in (12) accountsnot only for multipath fading but for the effect of locationdependentpathlossaswell,whereitisassumedthatthenumberof interferers
i
is a Poisson distributed random variable thatbelongs to PPP
K
n
.
B. Distribution of the Aggregate Interference
Throughout this analysis, it is assumed that the PPPs at eachrelay and the destination node are independent homogeneouspoint processes. Therefore, an independent interference modelis adopted [31], in which the relay and destination nodes haveno interferers in common. The statistical characteristics of the