WileySerial relaying communications over generalized-gamma fading channels

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.



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)



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-



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

)



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).



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).



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.



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



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.


Documents

WileySerial relaying communications over generalized-gamma fading channels