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WIRELESS COMMUNICATIONS AND MOBILE COMPUTINGWirel. Commun. Mob.
Comput. (2011)
Published online in Wiley Online Library
(wileyonlinelibrary.com). DOI: 10.1002/wcm.1047
RESEARCH ARTICLE
Serial relaying communications overgeneralized-gamma fading
channelsChristos K. Datsikas1, Kostas P. Peppas2, Nikos C. Sagias3
and George S. Tombras1
1 Department of Electronics, Computers, Telecommunications, and
Control, Faculty of Physics, University of Athens, 15784
Athens,Greece2 Laboratory of Wireless Communications, Institute of
Informatics and Telecommunications, National Centre for Scientic
ResearchDemokritos, Agia Paraskevi, 15310 Athens, Greece3
Department of Telecommunications Science and Technology, School of
Applied Sciences and Technology, University of Peloponnese,End of
Karaiskaki Street, 22100 Tripoli, Greece
ABSTRACT
In this paper, a study on the end-to-end performance of
multi-hop non-regenerative relaying networks over
independentgeneralized-gamma (GG) fading channels is presented.
Using an upper bound for the end-to-end signal-to-noise ratio
(SNR),novel closed-form expressions for the probability density
function, the moments, and the moments-generating function ofthe
end-to-end SNR are presented. Based on these derived formulas,
lower bounds for the outage and the average bit errorprobability
(ABEP) are derived in closed form. Special attention is given to
the low- and high-SNR regions having practicalinterest as well as
to the Nakagami fading scenario. Moreover, the performance of the
considered system when employingadaptive square-quadrature
amplitude modulation is further analyzed in terms of the average
spectral efciency, the biterror outage, and the ABEP. Computer
simulation results verify the tightness and the accuracy of the
proposed bounds.Copyright 2011 John Wiley & Sons, Ltd.
KEYWORDS
adaptive square-QAM; amplify-and-forward; average bit error
probability (ABEP); bit error outage (BEO); generalized-gamma
fading;multi-hop relaying; Nakagami; outage probability;
Weibull
*CorrespondenceNikos C. Sagias, Department of Telecommunications
Science and Technology, School of Applied Sciences and Technology,
Universityof Peloponnese, End of Karaiskaki Street, 22100 Tripoli,
Greece.E-mail: [email protected]
1. INTRODUCTION
Recently, multi-hop networks technology has attractedgreat
interest as it is a promising solution to achieve highdata rate
coverage required in future cellular wireless localarea and hybrid
networks, as well as to mitigate wirelesschannels impairment. In a
multi-hop system, intermedi-ate nodes are used to relay signals
between the source andthe destination terminal. Relaying techniques
can achievenetwork connectivity when the direct transmission is
dif-cult for practical reasons, such as large path-loss or
powerconstraints. As a result, signals from the source to the
des-tination propagate through different hops/links. A specialclass
of multi-hop networks is the serial relaying networksfor which
various works have shown that they are able toachieve high
performance gains [1,2].
In the open technical literature, there are several
worksdealingwith theperformance analysis ofmulti-hop systems.
In Reference [3], the end-to-end outage probability andthe
average error rate for multi-hop wireless systems
withnon-regenerative relaying operating over Weibull fadingchannels
were evaluated. In References [4--7], the end-to-end outage
probability as well as the average error rate fordual-hop wireless
systems with non-regenerative relayingoperating over Rayleigh and
Nakagami-m fading channelswere presented. In References [8,9],
performance boundsfor multi-hop relaying transmissions with
xed-gain relaysover Rice, Hoyt, and Nakagami-m fading channels
weregiven using the moments-based approach. Moreover, inReference
[10], an extensive performance analysis for dual-hop
non-regenerative relaying communication systems
overgeneralized-gamma (GG) fading was presented. It is notedthat
the GG distribution is quite general as it includes theRayleigh,
the Nakagami-m, and the Weibull distribution asspecial cases,
aswell as the lognormal one as a limiting case.Furthermore, it is
considered to bemathematically tractable,
Copyright 2011 John Wiley & Sons, Ltd.
-
Serial relaying communications over generalized-gamma fading
channels C. K. Datsikas et al.
S
R1 R2 RN-2. . . RN-1
D
h2 h3 hN-2 hN-1
hNh1
Figure 1. The serial relaying communication system under
consideration.
as compared to lognormal-based models, and recently hasgained
increased interest in the eld of digital communica-tions over
fading channels [11].
Adaptive modulation techniques [12--14], where themodulation
index is chosen according to the instantaneouschannel conditions,
are considered as efcient means tocope with channel variations,
while keeping an accept-able quality of service (QoS). When channel
conditionsare favorable, the transmitter can use higher power,
largersymbol constellations, and reduced coding and error
cor-rection schemes, otherwise the transmitter switches tolow
power, smaller symbol constellations, and includesimproved coding
and error correction schemes. Althoughadaptive modulation
techniques have been extensivelyapplied for direct-link systems,
corresponding techniquesfor multi-hop systems have only recently
received somespecial attention. In Reference [15], the performance
ofadaptive modulation in a single relay network was stud-ied, while
Reference [16], generalizes Reference [15] formulti-hop
systems.
In this paper, we analyze the statistics and the
end-to-endperformance of non-regenerative
(amplify-and-forward)multi-hop systems, with relay nodes in series,
operatingover independent GG fading channels. By consideringa union
bound for the end-to-end SNR, closed-formexpressions for its
cumulative distribution function (CDF),probability density function
(PDF), moments, as well asits moments-generating function (MGF) are
derived. Usingthe CDF expression, lower bounds of the end-to-end
out-age probability (OP) are derived. Moreover, using thewell-known
MGF-based approach, lower bounds for theaverage bit error
probability (ABEP) of binary differ-ential phase shift keying
(BDPSK), binary phase shiftkeying (BPSK) and binary frequency shift
keying (BFSK)are also presented. For identically distributed hops,
theABEP expressions are in terms of a nite sum of
MeijerG-functions. An accurate analytical method for the
com-putation of the ABEP is given for arbitrarily distributedhops.
For the high- and the low-SNR region, having spe-cial practical
interest for multi-hop networks, our derivedABEP expressions are
signicantly simplied. The samealso occurs when Nakagami fading is
being considered as aspecial case of theGGdistribution. Finally,
the performanceof the considered system with fast adaptive
square-QAMand xed switching levels is analyzed in terms of the
aver-age spectral efciency (ASE), the bit error outage (BEO),and
the corresponding ABEP. Theoretical expressions forthe outage
probability the achievable spectral efciency and
ABEP are derived, while Monte Carlo simulation is used inorder
to verify the proposed analysis.
The paper is organized as follows: In Section 2 the systemand
channel models are described in details. In Section 3closed-form
expressions for the CDF, PDF, moments, andMGF of a bounded
end-to-end SNR are derived. Based onthese formulas, in Section 4,
an end-to-end performanceanalysis is presented for both xed
modulation and adap-tive QAM schemes. In Section 5 numerical and
computersimulation results are presented, demonstrating the
tight-ness of the proposed bound, while the paper concludes witha
summary given in Section 6.
2. SYSTEM AND CHANNEL MODEL
We consider a multi-hop system, as shown in Figure 1,with a
source node communicating with a destinationnode via N 1 relay
nodes in series. The fading channelcoefcients hi between
source-to-relay (S R), relay-to-relay (R R) and
relay-to-destination (R D)are considered as independent GG random
variables. Bydening as i = h2i Es/N0 the instantaneous receive
SNRof the ith hop, with i = 1, 2, . . . , N, Es being the
symbolsenergy, andN0 being the single sided power spectral
density,the PDF of i can be written as
fi () =i
mii/21
2(mi)(ii)mii/2
exp[(
ii
)i/2](1)
where i > 0 and mi > 1/2 are parameters related to fad-ing
severity, i = Ei with E denoting expectation,and i = (mi)/(mi +
2/i) where () is the gammafunction. For i = 2, Equation (1) reduces
to the squareNakagami-m fading distribution, whereas for mi = 1,
theWeibull distribution is obtained. Also the CDF of i is
givenby
Fi () = 1 1
(mi)
[mi,
(
ii
)i/2](2)
Note that for integer mi and using Reference [17,
equation(8.352.2)], the CDF of i can be rewritten as
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
-
C. K.Datsikas et al. Serial relaying communications over
generalized-gamma fading channels
Fi () = 1 exp[(
ii
)i/2]
mi1i=0
1i!
(
ii
)ii/2(3)
3. STATISTICS OF THE END-TO-ENDSNR
In this section bounds for the end-to-end SNR are proposed,while
associated closed-form lower bounds for the CDF, thePDF, the
moments, and the MGF are derived.
For the system in Figure 1 employing the amplify-and-forward
relaying protocol, the end-to-end SNR at thedestination node can be
written as Reference [5]
equ =[
Ni=1
(1 + 1
i
) 1]1
(4)
The above SNR expression is not mathematically tractablein its
current form due to the difculty in nding the statis-tics
associated with it. Similarly to Reference [10,18], thisform can be
upper bounded by
equ b = min{1, 2, . . . , N} (5)
Mathematically, this bound is obtained by observing thatthe
following inequalities are valid [18]
equ (
Ni=1
1i
)1 min{1, 2, . . . , N} (6)
A physical interpretation of this bound is that at high
SNRregion, the hop with the weakest SNR determines the end-to-end
system performance. This approximation, is adoptedin many recent
papers, e.g., References [18--20] and isshown to be accurate
enough, especially at medium andhigh SNR values.
3.1. Cumulative distribution function
Using Equations (2) and (5), the CDF of b can be expressedas
Fb () = 1 Ni=1
[1 Fi ()]
= 1 Ni=1
1(mi)
[mi,
(
ii
)i/2](7)
For independent and identically distributed (i.i.d) hops,(mi =
m, i = , i = , and i = i) and for integer
values of mi, the CDF of b can be expressed as
Fb () = 1 exp[N
(
)/2]
[
m1i=0
1i!
(
)i/2]N(8)
3.2. Probability density function
Using the multinomial identity [21, equation (24.1.2)],Equation
(8) can be reexpressed as
Fb () = 1 N! exp[N
(
)/2]
N
n0 ,n1 ,...,nm1=0n0+n1++nm1=N
An0,n1,...,nm1m1
i=1 ini/2
(9)
where
An0,n1,...,nm1 =m1i=0
[(i!)nini!()ini/2
]1 (10)The PDF of b can be found by taking the derivative
ofEquation (9) with respect to . After some straightfor-ward
algebraic manipulations, the PDF of b can be nallyexpressed as
fb () =N/21N!
2()/2 exp[N
(
)/2]
N
n0 ,n1 ,...,nm1=0n0+n1++nm1=N
An0,n1,...,nm1m1
i=1 ini/2
exp[N
(
)/2]N!2
N
n0 ,n1 ,...,nm1=0n0+n1++nm1=N
An0,n1,...,nm1
1+m1
i=1 ini/2m1i=1
ini (11)
Nn0 ,n1 ,...,nm1=0
n0+n1++nm1=Ndenotes multiple summation over n0, n1, . . . ,
nm1,
with n0 + n1 + . . . + nm1 = N.
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
-
Serial relaying communications over generalized-gamma fading
channels C. K. Datsikas et al.
3.3. Moments
The th order moment of b is dened as
b () Eb =
0fb ()d (12)
By substituting Equation (11) in Equation (12), making achange
of variables t = N[/()]/2, and using the deni-tion of the gamma
function, the th order moment of b canbe expressed in closed form
as
b () = N!N
n0, n1, . . . , nm1 = 0n0 + n1 + + nm1 = N
An0,n1,...,nm1
N2 2/ ( )2 (
2 1
)
N!N
n0, n1, . . . , nm1 = 0n0 + n1 + + nm1 = N
An0,n1,...,nm1
( )2
N2 2/
(2 2
) m1i=1
i ni (13)
where
1 = + 2m1i=1
ini + 2 (14a)
and
2 = 1 2 (14b)
3.4. Moments-generating function
The MGF of b dened asMb (s) Eexp(sb), can beextracting from the
CDF of b as
Mb (s) = sL{Fb (); s} (15)
whereL{; } denotes the Laplace transform.By substitutingEquation
(11) in the above equation and using Reference[22, equation
(2.2.1.22)], the MGF of b can be expressedin closed form as
Mb (s) = 1 N!N
n0 ,n1 ,...,nm1=0n0+n1++nm1=N
An0,n1,...,nm1
kl1+12 s1
(2)(k+l)/21 Gk, l
l, k
[Nkll/sl
()k/2kk(l,1)(k, 0)
](16)
whereGk,ll,k
[] is theMeijers G-function, see Reference [17,equation
(9.301)], k and l are two minimum integers thatsatisfy = 2l/k, (k,
) =
k, +1
k, . . . , +k1
k, and
1 = 2m1i=1
i ni (17)
It is noted that for Weibull fading channels (m = 1),Equation
(16) numerically coincides with a previouslyknown result, see
Reference [3, equation (11)]. Also, forNakagami-m fading channels,
( = 2, k = 1, l = 1), usingReference [23, equation
07.34.03.0271.01], Equation (16)simplies to
Mb (s) = 1 N!N
n0 ,n1 ,...,nm1=0n0+n1++nm1=N
Bn0,n1,...,nm1
s22!(
1 + Nms
)21(18)
where
Bn0,n1,...,nm1 =m1i=0
[(i!)nini!
(
m
)ini]1(19a)
and
2 = 21
(19b)
For non-identically distributed hops, the MGF of b canbe
obtained using Equations (15) and (7) as
Mb (s) = 1
0s exp(s )
Ni=1
1 (mi)
[mi,
(
i i
)i/2]d. (20)
Using the GaussLaguerre quadrature rule [21, pp. 890 and923], an
accurate approximation of the MGF is obtained as
Mb (s) 1 L
i=1
Nj=1
wi
(mj)
[mj,
(xi
sj j
)j/2]
(21)
whereL is the number of integration points,xis are the rootsof
the Laguerre polynomialLn(x) andwi the correspondingweights
with
wi = xi(L + 1)2 L2L+1(xi). (22)
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
-
C. K.Datsikas et al. Serial relaying communications over
generalized-gamma fading channels
Next, based on Equation (16), asymptotic MGF expres-sions for
high- and low-SNR regions are presented. Weshow that the MGF of b
can be signicantly simpliedexpressing it only in the form of
elementary functions.
3.4.1. Low-SNR region.By expressing the Meijer G-function as a
nite sum
of hypergeometric functions pFq(; ; z), see Reference[17,
equation (9.304)] and taking into consideration thatpFq(; ; z)
tends to unity as z 0, a simplied asymptoticexpression for the MGF
of b at low SNR may be obtainedas
Mb (s) = 1 N!N
n0, n1, . . . , nm1 = 0n0 + n1 + + nm1 = N
An0,n1,...,nm1
k l1+12 s1
(2) k+l2 1l
h=1
lj = 1j = h
(ah aj)
[
kj=1
(1 + bj ah)][
Nk ll
( )k/2 kk sl]ah1
(23)
where ai, bi denotes the ith element of the lists (l,1)and (k,
0), respectively. Note that the above expression isgiven in terms
of elementary functions only.
3.4.2. High-SNR region.A simplied expression for the MGF of equ
at high
SNR, dened asMequ (s) Eexp(sequ), is obtained byapplying the
analysis presented in Reference [24]. In thatwork it has been
proved that if Xis, i = 1, 2, , N, areN independent and
non-negative random variables and theseries expansion of the PDF of
Xi in the neighborhood zerocan be expressed as
PXi (x) = aixti + o(xti+) (24)
where ai > 0, ti 0, > 0, then the series expansion of
thePDF of Z = (N
i=1 1/Xi)1 is given by
fZ(z) =N
i=1,ti=tminaiz
tmin + o(ztmin+) (25)
it is noted that for two functions f and g of a real variable x,
we writef (x) = o(g(x)) when x x0 if limxx0 f (x)g(x) = 0.
where tmin = mini ti and is a constant that depends on ti
and
. Using exp(z) 1 when z 0 along with the previouslycited
theorem, the PDF of i can be approximated as
fi () i
2(mi)(ii)mii/2imi/21 (26)
Using the denition of the gamma function, the MGF ofequ may be
easily expressed as
Mequ (s) (tmin)2stmin
Ni=1
i
(mi)(ii)mii/2(27)
where tmin = mini{mii/2}. Note once again that the
aboveexpression is given in terms of elementary functions only.
4. END-TO-END PERFORMANCEANALYSIS
Using the previous analysis, lower bounds for the OP of
theend-to-end SNR are derived in this section. Also, using theMGF
approach, lower bounds for the ABEP for a variety ofmodulation
schemes are presented, while the performanceis further investigated
when employing fast adaptive QAM.
4.1. Fixed modulation scheme
Here we assume a xed modulation scheme with a constantmodulation
index.
4.1.1. Outage probability.The outage probability is dened as the
probability that
the end-to-end output SNR, falls below a specied thresholdth.
This threshold is a minimum value of the SNR abovewhich the quality
of service is satisfactory. For the consid-ered multi-hop system
the use of upper bound b leads tolower bounds for the outage
probability at the destinationterminal D expressed as Pout(th) Fb
(th). The outageprobability of the considered system can be
obtained basedon Equation (7) as
Pout (th) Fb (th) . (28)
It is noted that for identical parameters mi = m, i = ,i = , and
i = i and m integer, the outage probabilitycan be extracted based
on Equation (8).
4.1.2. Average bit error probability.The MGF of b can be
efciently used to evaluate lower
bounds for the ABEP of BDPSK, BPSK, and BFSK. TheABEP of BDPSK
can be readily obtained from Equation(16) or Equation (21) asPbe =
0.5Mb (1), while for coher-
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
-
Serial relaying communications over generalized-gamma fading
channels C. K. Datsikas et al.
ent binary signals as Reference [25]
Pbe = 1
/20
Mb[
sin2()
]d (29)
where = 1 for coherent BPSK, = 1/2 for coherentBFSK and = 0.715
for coherent BFSK with mini-mum correlation. An alternative ABEP
expression can beobtained following a technique described in
Reference [26]as
Pbe = 12
0
Fb
(t2
2
)exp
( t
2
2
)dt (30)
For non-identically distributed hops, theABEP for
coherentsignals can be evaluated using Equations (21) and (29)
or(7) and (30) by means of numerical integration.
For identically distributed hops, substituting Equation(16)
toEquation (29) and bymaking the change of variables,sin2() = x,
integrals of the form
I = 1
0x11/2(1 x)1/2
Gk, ll, k
[Nkllxl
()k/2kkl(l,1)(k, 0)
]dx (31)
need to be evaluated. Using Reference [27, equation(2.24.2.2)],
I can be evaluated in closed form as
I = l1/2
G k, 2l2l, k + l
[Nkll
()k/2kkl(l, 1/2 1),(l,1)(k, 0),(l,1)
](32)
Therefore, the ABEP at the destination node is lowerbounded
as
Pbe = 12 N!
Nn0 ,n1 ,...,nm1=0
n0+n1++nm1=N
An0,n1,...,nm1
kl11
(2) k+l2
G k, 2l2l, k + l
[Nkll
()k/2kkl(l, 1/2 1),(l,1)(k, 0),(l,1)
](33)
For Nakagami-m fading channels, ( = 2, k = 1, l = 1),using
References [23, equation 07.34.03.0400.01] and [27,equation
(7.3.1.27)], Equation (33) simplies to
Pbe = 12 N!2
Nn0 ,n1 ,...,nm1=0
n0+n1++nm1=N
Bn0,n1,...,nm1
(22 1)!!(2)2(
1 + Nm
)2 12(34)
where (22 1)!! 1 3 (22 1) is the double facto-rial.
Finally, at the high-SNR region and using Equations(27) and
(29), a simplied asymptotic expression for Pbeis obtained as
Pbe (
12 + tmin
)4tmin
tmin
Ni=1
i
(mi)(ii)mii/2(35)
4.2. Fast adaptive square-QAM
According to the fast adaptive QAM technique, the constel-lation
size is selected based on the instantaneous receivedSNR at the
destination. In particular, for a xed target biterror probability,
SNR thresholds for different constella-tion sizes are calculated.
By comparing the instantaneousSNR at the destination with these
thresholds, the size ofthe constellation M is adapted to provide
the best possiblethroughput while satisfyingQoS requirements.
Informationon which value ofM to be set is determined after
communi-cating the source with the destination through a reliable
andlow-delay feedback link. Note that an error-free feedbackfrom
the destination to source is being assumed.
4.2.1. Average spectral efciency.Let Mj and j , j = 0 J , be the
jth element from the
set of possible constellation sizes and corresponding
SNRthreshold respectively, to achieve a target bit error
probabil-ity Pb . The ASE is obtained using Reference [14,
Equation(19)] as
=J1j=0
Mj P{j < equ j+1}
+ MJ P{J < equ}
=J1j=0
Mj[Fequ
(j+1
) Fequ (j )]+ MJ
[1 Fequ
(J)]
(36)
where Mj = log2(Mj) and P{} denotes the probabilityoperator. To
evaluate the corresponding thresholds j , theanalytical expression
for the instantaneous bit error proba-bility is required. This
expression is given in Reference [28]as
Pb(e | equ) = 2M log2(
M)
log2(
M)
h=1
(12h)M1i=0
(1)i2h1/
M
(
2h1 i2h1
M+ 1
2
)
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
-
C. K.Datsikas et al. Serial relaying communications over
generalized-gamma fading channels
Q(
(2i + 1)
3equM 1
)(37)
where x denotes the largest integer less than or equal tox and
Q() is the well-known Gaussian Q-function. For agiven target bit
error probability Pb , j is obtained as thesolution of the
following equation
Pb(e | j ) = Pb (38)
Since the roots ofEquation (38) cannot be obtained in
closedform, any of the well-known root-nding techniques maybe used
for numerical evaluation.
4.2.2. Bit error outage.An important performance measure in
adaptive modula-
tion systems is the BEO, dened as the outage probabilitybased on
bit error probability, that is Reference [14]
Po(Pb ) = P{Pb(equ) Pb } (39)
where Pb(equ) is the bit error probability as a function ofthe
instantaneous SNR equ and Pb is the target bit errorprobability
dened above. Since the event Pb(equ) Pb isequivalent to the event
equ j , j = 0, 1, . . . , J 1, alower bound for BEO can be readily
obtained as
Po(Pb ) Fb (j ) (40)
4.2.3. Average bit error probability.The ABEP of an adaptive
M-QAM system is given by
Reference [29, equation (11)]
Pbe =J1
j=0 MjPjJ1j=0 Mjj
(41)
In the above equation j = P{j equ j+1} =Fequ (j+1) Fequ (j ) and
Pj is the bit error probability ofthe jth transmission mode, given
by
Pj =
j+1
j
PMj ()fequ ()d (42)
where PMj () is the bit error probability of Mj-QAM givenby
Equation (37). Since J = , it can be observed thatthe denominator
of Equation (41) is equal to the ASE, .Also, by performing partial
integration, Equation (42) canbe written as
Pj = Fequ (j+1)PMj (j+1) Fequ (j )PMj (j )
j+1
j
dPMj ()d
Fequ ()d (43)
In the above equation the probabilities PMj () are
linearcombinations of Gaussian Q-functions, the derivatives ofwhich
with respect to can be easily derived using theidentity dQ(x)/dx =
exp(x2/2)/2. Consequently,Pj can be easily evaluated using
numerical integration.
5. NUMERICAL AND COMPUTERSIMULATION RESULTS
In this section, numerical results for the derived lowerbounds
and computer simulation results for the correspond-ing exact OP and
ABEP performance criteria are presented.In Figure 2, lower bounds
curves for the OP of a three-hopsystem (N = 3) are plotted as a
function of the rst hopnormalized outage threshold th/1, for
non-identically dis-tributed hops, having = 2, 2 = 21, 3 = 31 and
fordifferent values of m. As expected, OP improves as th/1decreases
and/or m increases. Dashed curves for the exactoutage performance,
obtained via Monte Carlo simulationsbased on Equation (4), are also
included for comparisonpurposes. As it can be observed the
difference between theexact value of the OP and the obtained bound
gets tighter asth/1 decreases. However, at very high values of
th/1,the bounds get loose.
Moreover, in Figure 3 lower bounds for the OP for i.i.d.hops,
are plotted as a function of the normalized outagethreshold th/ ,
for = 1,m = 3 and for different values ofN. As it is evident, OP
improves as th/ and/orN decrease.Similar to the Figure 2, the
analytically obtained lowerbound results are compared to
corresponding exact outageperformance results, obtained viaMonte
Carlo simulations.It is obvious that the difference between the
exact value oftheOP and the obtained bound gets tighterwith the
decrease
Figure 2. End-to-end outage probability of a three-hop
wirelesscommunication system operating over non-identical GG
fadingchannels as a function of the rst hop normalized outage
thresh-old (N = 3, = 2, 2 = 21, 3 = 31, and m = 1.5,2.5,3.5).
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
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Serial relaying communications over generalized-gamma fading
channels C. K. Datsikas et al.
Figure 3. End-to-end outage probability of multi-hop
wirelesscommunication systems operating over i.i.d. GG fading
channelsas a function of the normalized outage threshold (m = 3, =
1,
and N = 2,3,5).
of th/ and/or N. Also, it can be observed that the boundsget
less tight at high values of th/ and as N increases.
In Figures 4 and 5 lower bound curves for the ABEP ofBDPSK and
BPSK modulations are plotted, respectively,for i.i.d. hops, as a
function of the average input SNRper bit , for various values of N,
= 1 and m = 3. Itis obvious that ABEP improves as N decreases.
Also, inFigure 6 the ABEP of BDPSK is plotted as a functionof for
various values of , m = 3 and N = 3. Asexpected, the ABEP improves
as increases. For the
Figure 4. End-to-end ABEP of BDPSK for multi-hop
wirelesscommunication systems operating over i.i.d. GG fading
channelsas a function of the average input SNR per bit (m = 3, = 1,
and
N = 2,3,5).
Figure 5. End-to-end ABEP of BPSK for multi-hop wireless
com-munication systems operating over i.i.d. GG fading channels asa
function of the average input SNR per bit (m = 3, = 1, and
N = 2,3,5).
low ( 2dB) and high ( 20dB) SNR regions, thesimplied expression
given by Equations (23) and (27)can be used, respectively, for the
numerical evaluationof the proposed ABEP bounds. Moreover, in
Figure 7lower bound curves for the ABEP of BDPSK and
BPSKmodulations are plotted as a function of the rst hop aver-age
input SNR per bit, 1, for non-identically distributedhops, N = 3, m
= [2.8, 1.3, 0.4], = [1.75, 3.2, 4.25],2 = 21, and 3 = 31.
Asymptotic ABEP curves are alsoplotted and as it can be observed,
Equations (27) and (35)
Figure 6. End-to-end ABEP of BDPSK for multi-hop
wirelesscommunication systems operating over i.i.d. GG fading
channelsas a function of the average input SNR per bit (m = 3, N =
3, and
= 1,2,3).
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
-
C. K.Datsikas et al. Serial relaying communications over
generalized-gamma fading channels
Figure 7. End-to-end ABEP of BDPSK and BPSK of a
three-hopwireless communication system operating over non identical
GGfading channels as a function of the rst hop average input SNRper
bit (m = [2.8,1.3,0.4], N = 3, = [1.75,3.2,4.25], 2 =
21, 3 = 31).
correctly predict the behavior of the ABEP at high
SNRregion.
For all the ABEP results in Figures 47, associate curvesfor the
exact error performance, obtained via Monte Carlosimulations based
on Equation (4) are also depicted forcomparison purposes. From all
comparisons one can verifysimilar ndings to that mentioned in
Figures 2 and 3.
Figure 8. End-to-end BEO of a multi-hop wireless communi-cation
system operating over non identical GG fading channelsversus 1, for
an adaptive QAM scheme with P
b
= 102, N = 3,m = 1.4, = 2.25, 2 = 21, 3 = 31 and different
constella-
tion sizes M .
Figure 9.End-to-end ASE of amulti-hopwireless
communicationsystem operating over non identical GG fading channels
versus1, for an adaptive QAM scheme with maximum constellationsize
16, 64, and 256, P
b= 102, N = 3, m = 1.4 = 2.25, 2 =
21 and 3 = 31.
In Figure 8 lower bound curves for theBEOof a three-hopsystem
using fast adaptive M-QAM are plotted versus 1,assuming Pb = 102, m
= 1.4, = 2.25, 2 = 21, 3 =31 and different constellation sizes.
Associate curves forthe exact BEO, obtained viaMonte Carlo
simulations basedon Equation (4) are also presented and as it can
be observed,similar ndings to that mentioned in Figures 2 and 3
maybe veried.
Figure 10. End-to-end ABEP of a multi-hop wireless
communi-cation system operating over non identical GG fading
channelsversus 1, for non-adaptive and adaptive
256-QAMschemeswithP
b= 102, N = 3, m = 1.4, = 2.25, 2 = 21 and 3 = 31.
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
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Serial relaying communications over generalized-gamma fading
channels C. K. Datsikas et al.
In Figure 9 lower bound curves for theASEof a three-hopsystem
using fast adaptive M-QAM are plotted versus 1,assuming Pb = 102, m
= 1.4, = 2.25, 2 = 21, 3 =31. We assume M0 = 4, M1 = 16, M2 = 64
and M3 =256. Different maximum constellation sizes are
considered.Curves for the exact ASE based on Equation (4) are
alsopresented and as one can observe the ASE bounds are tightat
high values of 1.
Finally, in Figure 10 curves for the ABEP of the samesystem
using non adaptive 256-QAM as well fast adaptive256-QAMare plotted
versus1. Clearly the adaptive systemoutperforms the non-adaptive
one and maintains the targetBER requirement, namely Pb = 102.
Curves for the exactABEP based on Equation (4) are also plotted and
as it isobvious the ABEP bounds are tight at high values of 1
forboth the adaptive and non-adaptive systems.
6. CONCLUSIONS
In this paper, we provided union tight bounds for multi-hop
transmissions with non-regenerative relays in series,operating over
independent GG fading channels. Using atight upper bound of the
end-to-end SNR, novel closed-form expressions for the MGF, PDF, and
CDF of thisupper bounded SNRwere derived. Additionally, tight
lowerbounds for the OP and the ABEP were presented. More-over the
performance of fast adaptive QAM was addressed,deriving tight
bounds for the ASE, the BEO, and the ABEP.Also, it is obvious that
all the derived bounds gets tighteras the number of relays
decreases. Numerical results werepresented and demonstrated the
accuracy and the tightnessof the proposed bounds. The obtained
results show that theproposed bounds gets tighter with the increase
of the SNRcorresponding to computer simulation resultswhich are
alsoincluded and verify the accuracy and the correctness of
theproposed analysis.
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AUTHORS BIOGRAPHIES
Christos K. Datsikas was born inAgrinio, Greece, in 1973. He
holdsa B.Sc. from the Department ofInformatics and
Telecommunicationsof the University of Athens andan M.Sc. in
Techno-Economics fromthe National Technical University ofAthens
(NTUA). He is currently aPh.D. candidate at the Physics Depart-
ment of the University of Athens. He is a highly skilledI.T.
Administrator and SW Engineer and has served as areviewer in
various journals and conferences. His researchinterests include
multihop networks and communicationtheory issues.
Kostas P. Peppas was born in Athens,Greece, in 1975. He obtained
hisdiploma from the School of Elec-trical and Computer Engineering
ofthe National Technical Universityof Athens in 1997 and the
Ph.D.degree in telecommunications from thesame department in 2004.
His cur-rent research interests include wireless
communications, smart antennas, digital signal processingand
system level analysis and design. He is a member ofIEEE and the
National Technical Chamber of Greece.
Nikos C. Sagias was born in Athens,Greece in 1974. He received
the BScdegree from the Department of Physics(DoP) of the University
of Athens(UoA), Greece in 1998. The M.Sc.and Ph.D. degrees in
Telecommunica-tion Engineering were received fromthe UoA in 2000
and 2005, respec-tively. Since 2001, he has been involved
in various National and European Research & Develop-ment
projects for the Institute of Space Applications andRemote Sensing
of the National Observatory of Athens,Greece. During 2006-2008, was
a research associate atthe Institute of Informatics and
Telecommunications ofthe National Centre for Scientic
Research-"Demokritos",Athens, Greece. Currently, he is an Assistant
Professor atthe Department of Telecommunications Science and
Tech-nology of the University of Peloponnese, Tripoli, Greece.
Dr. Sagias research interests are in the research areaof
wireless digital communications, and more specicallyin MIMO and
cooperative diversity systems, fading chan-nels, and communication
theory. In his record, he has over40 papers in prestigious
international journals and morethan 20 in the proceedings of world
recognized confer-ences. He has been included in the Editorial
Boards ofthe IEEE Transactions on Wireless Communications, Jour-nal
of Electrical and Computer Engineering, and the IETETechnical
Review, while he acts as a TPC member forvarious IEEE conferences
(GLOBECOM08, VTC08F,VTC09F, VTC09S, etc). He is a co-recipient of
the bestpaper award in communications in the 3rd
InternationalSymposium on Communications, Control and Signal
Pro-cessing (ISCCSP), Malta, March 2008. He is a member ofthe IEEE
and IEEE Communications Society as well as theHellenic Physicists
Association.
George S. Tombras was born inAthens, Greece. He received the
B.Sc.degree in Physics from AristotelianUniversity of Thessaloniki,
Greece, theM.Sc. degree in Electronics from Uni-versity of
Southampton, UK, and thePh.D. degree fromAristotelian Univer-sity
of Thessaloniki, in 1979, 1981, and1988, respectively. From 1981 to
1989
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm
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Serial relaying communications over generalized-gamma fading
channels C. K. Datsikas et al.
he was Teaching and Research Assistant and, from 1989to 1991,
Lecturer at the Laboratory of Electronics, PhysicsDepartment,
Aristotelian University of Thessaloniki. Since1991, he has been
with the Laboratory of Electronics, Fac-ulty of Physics, University
of Athens, where currently isan Associate Professor of Electronics
and Director of theDepartment of Electronics, Computers,
Telecommunica-
tions and Control. His research interests include
mobilecommunications, analog and digital circuits and systems,as
well as instrumentation, measurements and audio engi-neering.
Professor Tombras is the author of the textbookIntroduction to
Electronics (inGreek) and has authored orco-authored more than 90
journal and conference refereedpapers and many technical
reports.
Wirel. Commun. Mob. Comput. (2011) 2011 John Wiley & Sons,
Ltd.DOI: 10.1002/wcm