JOURNAL OF TELECOMMUNICATIONS, VOLUME 1, ISSUE 1, FEBRUARY 2010 87

Dual-hop transmissions with fixed-gain relaysover Generalized-Gamma fading channels

Kostas P. Peppas, Akil Mansour and George S. Tombras

Abstract—In this paper, a study on the end-to-end performance of dual-hop wireless communication systems equipped with fixed-gainrelays and operating over Generalized-Gamma (GG) fading channels is presented. A novel closed form expression for the momentsof the end-to-end signal-to-noise ratio (SNR) is derived. The average bit error probability for coherent and non-coherent modulationschemes as well as the end-to-end outage probability of the considered system are also studied. Extensive numerically evaluated andcomputer simulations results are presented that verify the accuracy of the proposed mathematical analysis.

Index Terms—Dual-hop wireless communication systems, fixed-gain relays, Generalized-Gamma fading channels, average bit errorprobability, outage probability.

F

1 INTRODUCTION

R ECENTLY, research efforts have been focused on theinvestigation of multi-hop wireless communications

systems, which seem to extend the coverage withoutusing large power at the transmitter and increase connec-tivity and capacity in wireless networks [1]–[10]. Multi-hop wireless communications systems are able to pro-vide a potential for broader and more efficient coveragein bent pipe satellites and microwave links, as well asmodern ad-hoc, cellular, WLAN, and hybrid wirelessnetworks. In multi-hop networks, intermediate nodesoperate as relays between the source and the destina-tion terminal. Generally, there are two main categoriesof multi-hop wireless communication systems: Non-regenerative and regenerative systems. In the regener-ative systems, the relay re-encodes and retransmits thesignal towards the destination after demodulating anddecoding the received signal from the source. At the des-tination, the receiver can employ a variety of diversitycombining techniques to benefit from the multiple signalreplicas available from the relays and the source. Non-regenerative systems use less complex relays that justamplify and and re-transmit the information signal with-out performing any sort of decoding. Moreover, relaysin non-regenerative systems systems can in their turnbe classified into two subcategories, namely, channelstate information (CSI)-assisted relays and blind relays.Non-regenerative systems with CSI-assisted relays useinstantaneous CSI of the first hop to control the gainintroduced by the relay. On the other hand, systems with

• K. P. Peppas is with the Laboratory of Mobile Communications, Insti-tute of Informatics and Telecommunications, National Centre for Scien-tific Research–“Demokritos,” Patriarhou Grigoriou and Neapoleos, AgiaParaskevi, 15310, Athens, Greece.

• A. Mansour and G. S. Tombras are with the Department of Electronics,Computers, Telecommunications and Control, Faculty of Physics, Univer-sity of Athens 15784, Greece.

blind relays employ at the relaying nodes amplifiers withfixed gains. Although such systems are not expected toperform as well as systems equipped with CSI-assistedrelays, they are characterized by low complexity andease of deployment.

A versatile fading envelope distribution, which gen-eralizes many of the commonly used models for multi-path and shadow fading, is the generalized gamma (GG)distribution [11]. This fading model is quite general as itincludes the Nakagami-m and the Weibull distributionsas special cases and the log-normal distribution as a lim-iting case. Representative past works concerning the per-formance of dual-hop systems over fading channels canbe found in [6], [12], [13]. In [6], the authors have studiedthe end-to-end performance of dual-hop transmissionsystems with regenerative and non-regenerative relays,respectively, over Rayleigh fading channels. In [12], theperformance of dual-hop wireless communication sys-tems with fixed-gain relays over Nakagami-m fadingchannels was investigated. Also, in [13], lower bounds ofthe performance of dual-hop relaying over independentGG fading channels were given. In the view of theappropriateness of the GG distribution for characteriz-ing real-world communication links, it is appealing toinspect the performance of dual-hop systems operatingover these channels. However, to the best of the authors’knowledge, results concerning the performance of non-regenerative systems with fixed-gain relays operatingover GG fading are not available in the open technicalliterature.

In this paper a thorough performance analysis of dual-hop wireless communications systems with fixed gain re-lays is presented. A novel closed form expression for themoments of the end-to-end output SNR is derived. Basedon this formula, the outage performance and the averageerror probability for binary coherent and non-coherentmodulation schemes are studied, using the well-knownmoment-generating function (MGF) approach [14]. The

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proposed method for the evaluation of the MGF is basedon the Pade approximants theory. Moreover, an alter-native integral representation for the outage probabilityof the considered system is presented. A new closed-form expression is derived for the gain of previouslyproposed semi-blind relays. These formulae are used innumerical and computer simulations results, to verifythe correctness of the presented mathematical analysis.Our results incorporate similar others available in theopen technical literature, such as those for Nakagami-mfading channels.

The remainder of the paper is organized as follows:In Section 2, the system and channel model is describedin details. In Section 3, closed form expressions forthe moments of the end-to-end SNR are presented. InSection 4, the gain of a previously proposed class ofsemi-blind relays is derived in closed form. In Sections5 and 6, the error rate and outage performance of theconsidered system are addressed, respectively. Numer-ical and computer simulation results are presented inSection 7, while the paper concludes with a summarygiven in Section 8.

2 SYSTEM AND CHANNEL MODEL

We consider a wireless communication system where asource terminal A is communicating with a destinationterminal C through a terminal B which acts as a relay.The node B amplifies and forwards the received signalto the destination C without any sort of decoding. As-suming that the source is transmitting a signal with anaverage power normalized to unity, the end-to-end SNRis given as [12, Eq. 1]:

γend =(a21/N01)(a

22/N02)

(a22/N02) + (1/G2N01)(1)

where ai is the fading amplitude of the ith hop, i =1, 2, assumed to be GG distributed, G is the relay gain,and N0i is the single-sided power spectral density of theadditive white Gaussian noise (AWGN) at the i-th hop.When blind relays are used, the fixed gain G establishedin the connection is G2 = 1/(CN01), where C is a constant[6]. Thus (1) becomes:

γend =γ1γ2C + γ2

(2)

where γi = a2i /N0i is the instantaneous SNR of the ithhop. The probability density function (pdf) of γi is givenby [11]

fγi(γi) =βiγ

miβi/2−1i

2Γ(mi)(τiγi)miβi/2

exp

−( γiτiγi

) βi2

(3)

where βi > 0 and mi > 1/2 are parameters related tofading severity, γi = E⟨γi⟩ with E⟨·⟩ denoting expecta-tion, τi = Γ(mi)/Γ(mi + 2/βi) and Γ(x) ,

∫∞0e−ttx−1dt

is the Gamma function. For βi = 2, (3) reduces to theNakagami-m fading distribution whereas for m = 1 the

Weibull distribution is obtained. Moreover, the cumula-tive distribution function (cdf) of γi may be expressedas

Fγi(γi) = 1−Γ

(mi,

(γi

τiγi

) βi2

)Γ(mi)

(4)

where Γ(x, y) ,∫∞xe−tty−1dt is the upper incomplete

Gamma function.

3 MOMENTS OF THE END-TO-END SNR

In this section, a closed-form expression for the momentsof the end-to-end SNR is derived. The n-th moment ofγend is given by

E⟨γnend⟩ =∫ ∞

0

∫ ∞

0

(γ1γ2C + γ2

)n

fγ1(γ1)fγ2(γ2)dγ1dγ2 (5)

Using (3), E⟨γnend⟩ can be written as:

E⟨γnend⟩ =1

4

2∏i=1

βi

Γ(mi) (τiγi)βi2

×∫ ∞

0

γm1β1/2+n−11 exp

−( γ1τ1γ1

) β12

dγ1×∫ ∞

0

(γ2

C + γ2

)n

γm2β2/2−12 exp

−( γ2τ2γ2

) β22

dγ2(6)

The integral with respect to γ1, I1, can be evaluated by

applying the change of variables(

γ1

τ1γ1

) β12

= t and usingthe definition of the Γ function as

I1 = 2(τiγi)

m1β1/2+n

β1Γ

(m1 +

2n

β1

)(7)

The integral with respect to γ2, I2, can be evaluatedby expressing the exponential and the fraction in termsof Meijer-G functions i.e exp(−x) = G 1,0

0,1 [x |−0 ] [15, Eq.8.4.3.2] and (1+x)−ρ = 1

Γ(ρ)G1,11,1

[x∣∣1−ρ

0

][15, Eq. 8.4.2.5]

and with the application of [15, Eq. 2.24.1.1] as:

I2 =

√k2l

n−12 Cm2l2

Γ(n)(2π)l+k2−3

2

×G k2+l2,l2l2,k2+l2

[(τ2γ2)

−l2 Cl2

kk22

∣∣∣ ∆(l,1−m2l2−n)∆(k2,0),∆(l2,−m2l2)

] (8)

where k2 and l2 are the minimum integers that satisfyβ2 = 2l2/k2 and ∆(x, a) = { a

x ,a+1x , . . . a+x−1

x }. It isnoted that the Meijer-G function is a standard built-infunction available in the most popular software-basedmathematical packages such as Maple or Mathematica.Finally, the moments of the end-to-end SNR of the

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considered system are given in closed-form as

E⟨γnend⟩ =Γ(m1 +

2nβ1

)(τ1γ1)

nln2 Cm2l2

√k2Γ(m1)Γ(m2) (τ2γ2)

m2l2k2 Γ(n)(2π)l2+

k2−32

×G k2+l2,l2l2,k2+l2

[(τ2γ2)

−l2 Cl2

kk22

∣∣∣ ∆(l2,1−m2l2−n)∆(k2,0),∆(l2,−m2l2)

](9)

By substituting n = 1 to (9) a closed-form expression forthe average end-to-end SNR can be obtained. For thespecial case of Nakagami-m fading channels (β1 = β2 =2), it can be observed that (9) is reduced to a previouslyknown result [12, Eq. 9].

4 A CLASS OF ”SEMI-BLIND” RELAYS INGENERALIZED-GAMMA FADING CHANNELS

In this section, a new expression for the gain of a -previously published- class of ”semi-blind” relays ispresented in closed form for GG fading channels. In[6], the authors proposed a specific class of ”semi-blind”relays which consume the same average power withthe corresponding CSI-based relays. The proposed fixedgain relay, benefits from the knowledge of the first hopaverage fading power. In such a scenario, the fixed gainis considered equal to the average of CSI assisted gain,namely

G2 = E⟨

1

a21 +N201

⟩(10)

For GG fading, by performing the required statisticalaverage in (10) and following a process similar to theone for the evaluation of I2, G can be expressed in closedform as

G2 =l1

N01(2π)l1+

k1−32

√k1Γ(m1)(τ1γ1)

m1l1/k1

×G k1+l1,l1l1,k1+l1

[(τ1γ1)

−l1

kk11

∣∣∣ ∆(l1,1−m1l1)∆(k1,0),∆(l,1−m1l1)

] (11)

where k1 and l1 are the minimum integers that satisfyβ1 = 2l1/k1. Finally, the parameter C can be obtainedas C = 1/(G2N01). For Nakagami-m fading channels(l1 = k1 = 1, τ1 = 1/m1), by making use of the identityG 2,1

1,2

[x∣∣ 1−a0,1−b

]= Γ(a)Γ(a−b+1)Ψ(a, b, x) [15, Eq. 8.4.46.1]

where Ψ(·, ·, x) denotes the Tricomi hypergeometric func-tion [15, Eq. (7.2.2.7)] and Ψ(a, a, z) = ezΓ(1 − a, z) [15,Eq. (7.11.4.4)], we observe that G2 reduces to a previouslyknown result [12, Eq. (12)].

5 PADE APPROXIMANTS AND AVERAGE BITERROR PROBABILITY

In this section we address the error performance of theconsidered dual-hop system for different coherent andnon-coherent binary modulation schemes. The ABEP ofvarious digital modulation schemes over fading chan-nels can be evaluated by using the well-known MGF

based approach [14]. In this case, however, it is difficultto derive a closed form expression for the MGF of theoutput SNR, Mγend

(s). Instead, it is more convenient touse the Pade approximants method [16], a simple andefficient method to accurately approximate the MGF andin sequel to evaluate the ABEP. The main advantage ofthis method is that due to the form of the producedapproximation, the ABEP can be calculated directlyusing simple expressions for the non-coherent BinaryFrequency Shift Keying (BFSK) and Binary DifferentialPhase Shift Keying (BDPSK) modulation schemes, whilefor M-ary Quadrature Amplitude Modulation (QAM)and M-ary Phase Shift Keying (PSK), single integralswith finite limits and integrands composed of elemen-tary functions can be readily evaluated by numericalintegration.

A Pade approximant to the MGF is a rational functionof a specified order B for the denominator and A for thenominator, whose power series expansion agrees withthe (A+B)-order power expansion of the MGF, namely

Mγend(s) ≃ R[A/B](s) =

∑Ai=0 cis

i

1 +∑B

i=0 bisi≃

A+B∑n=0

E⟨γnend⟩sn

n!

(12)where bi and ci are real numbers. In order to obtainan accurate approximation of the MGF, we assume sub-diagonal Pade approximants (B = A+1) [16]. The coeffi-cients bi and ci may be numerically evaluated using anyof the most popular commercial software mathematicalpackages such as Maple or Mathematica.

Using (12), the ABEP of digital modulations for severalsignaling constellations may be efficiently evaluated. Forexample, the ABEP of BDPSK can be readily obtainedfrom (12) as P be = 0.5Mγend

(−1). Also, the ABEP forcoherent binary signals is given by [14]

P be =1

π

∫ π/2

0

Mγend

(− ψ

sin2 θ

)dθ (13)

where ψ = 1 for coherent binary phase shift keying(BPSK), ψ = 1/2 for coherent BFSK and ψ = 0.715 forcoherent BFSK with minimum correlation.

6 END-TO-END OUTAGE PROBABILITY (OP)

In this section the end-to-end Outage Probability (OP) ofthe considered system is addressed. The OP is defined asthe probability that the instantaneous output SNR, γend,falls below a specified threshold γth. Two methods forthe evaluation of the end-to-end OP are presented: Thefirst method is based of the Pade approximants theorywhere as the second one derives an easy-to-evaluateintegral representation of the OP.

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6.1 Evaluation of the end-to-end OP using Pade ap-proximants

The outage probability can be extracted from Mγend(s)

based on the following Laplace transformation

Pout (γth) = L−1

{Mγend

(s)

s; s; t

}∣∣∣∣t=γth

(14)

where L−1 {·, s; t; } denotes inverse Laplace transform.Using (12) and the residue inversion formula [17], theOP can be obtained as

Pout(γth) = 1−B∑i=1

λipiepiγth (15)

where pi are the poles of the of the Pade approximantsto the MGF, which must have negative real part, and λiare the residues.

6.2 An integral representation of the end-to-end OP

Using (2) the end-to-end OP may be obtained as

Pout (γth) = Pr(γend ≤ γth) =∫ ∞

0

Pr(

γ1γ2C + γ2

≤ γth

∣∣∣∣ γ2) fγ2(γ2)dγ2

=

∫ ∞

0

Pr(γ1 ≤ (C + γ2)γth

γ2

∣∣∣∣ γ2) fγ2(γ2)dγ2

= 1−∫ ∞

0

[1− Fγ1

((C + γ2)γth

γ2

)]fγ2(γ2)dγ2

= 1− β22Γ(m1)Γ(m2)(τ2γ2)

m2β2/2

×∫ ∞

0

Γ

m1,

(Cγth + γthγ2τ1γ1γ2

) β12

γm2β2/2−12

× exp

−( γ2τ2γ2

) β22

dγ2

(16)

where Pr(·) denotes the probability operator. A closedform expression for the previously defined integral isvery difficult, if not impossible to be obtained. However,after performing the change of variables γ2 = τ2γ2w

2β2 ,

the OP may be expressed after some algebraic manipu-lations as

Pout (γth) =1− 1

Γ(m1)Γ(m2)

∫ ∞

0

wm2−1e−w

× Γ

m1,

Cγth + γthτ2γ2w2β2

w2β2

∏2j=1 τjγj

β12

dw(17)

It can be observed that this integral can be accuratelyand efficiently evaluated by using the Gauss-Laguerrequadrature rule [18]. Thus, the end-to-end OP may be

0.0 0.5 1.0 1.5 2.0 2.5 3.01.0

1.5

2.0

2.5

3.0

First Hop Average SNR (dB)

m1 = 0.5

m1 = 2

m1 = 4

m1 = 6

m1 = 8

Fig. 1. Parameter C versus γ1 for β1 = 4/3 and for severalvalues of m1.

obtained as

Pout (γth) ≃ 1− 1

Γ(m1)Γ(m2)

×N∑i=1

Wixm2−1i Γ

m1,

Cγth + γthτ2γ2x2β2i

x2β2i

∏2j=1 τjγj

β12

(18)

where xi are the roots of the N -th order Laguerrepolynomial LN (x) and Wi are the corresponding weightsgiven by

Wi =xi

(N + 1)2[LN+1(xi)]2(19)

7 NUMERICAL AND COMPUTER SIMULATIONRESULTS

In this section, various performance evaluation resultsobtained by numerical and simulations techniques thatillustrate the formulations derived herein are presented.

In Fig. 1, the parameter C as a function of γ1 for severalvalues of m1 and β1 = 4/3 is depicted and as expected,C increases as m1 increases. In Fig. 2, the average end-to-end SNR as a function of γ1 for β1 = β2 = 3and m1 = m2 = 2 is depicted. In the same plot, theimpact of power imbalance between the two hops onthe considered metric is also illustrated. As expected[6], when γ2 > γ1, it is beneficial and, otherwise, it isdetrimental. In Fig. 3, the ABEP for BDPSK and BPSKis illustrated as a function of γ1 for balanced (γ2 = γ1)and unbalanced (γ2 = 2γ1) hops assuming β1 = β2 = 3and m1 = m2 = 2. As it is evident, ABEP decreases as γ1increases. Moreover, in Figs. 4 and 5 the impact of the

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0 5 10 15 200

20

40

60

80

100

5/12

3/12

12

12 5

simulation

Ave

rage

End

-To-

End

SNR

First Hop Average SNR (dB)

Fig. 2. Average end-to-end SNR versus γ1 for β1 = β2 =3 and m1 = m2 = 2 .

0 5 10 15 2010-5

10-4

10-3

10-2

10-1

100

BPSK

Balanced hops Unbalanced hops simulation

Ave

rage

Bit

Erro

r Pro

babi

lity

First Hop Average SNR (dB)

BDPSK

Fig. 3. Average Bit Error Probability versus γ1 for bal-anced (γ2 = γ1) and unbalanced (γ2 = 2γ1) hops(β1 = β2 = 3 and m1 = m2 = 2) .

fading parameters β and m on the ABEP is illustrated.More specifically, in Fig. 4 the ABEP of BDPSK and BPSKis illustrated for γ2 = 2γ1, assuming β1 = β2 = 2.5 andm1 = m2 = m, as a function of γ1 and for m = 1.5, 3and 3.5. One can observe that ABEP improves as m

0 5 10 15 2010-6

10-5

10-4

10-3

10-2

10-1

100

m = 1.5, = 2.5

m = 2.5, = 2.5

BDPSK BPSK simulation

Ave

rage

Bit

Erro

r Pro

babi

lity

First Hop Average SNR (dB)

m = 3.5, = 2.5

Fig. 4. Average Bit Error Probability versus γ1 for unbal-anced (γ2 = 2γ1) hops, β1 = β2 = 2.5, m1 = m2 = m andfor various values of m .

0 5 10 15 2010-6

10-5

10-4

10-3

10-2

10-1

100

m = 2, = 2

m = 2, = 1

BDPSK BPSK simulation

Ave

rage

Bit

Erro

r Pro

babi

lity

First Hop Average SNR (dB)

m = 2, = 3.5

Fig. 5. Average Bit Error Probability versus γ1 for unbal-anced (γ2 = 2γ1) hops, β1 = β2 = β, m1 = m2 = 2 andfor various values of β .

and/or γ1 increases. In Fig. 5, ABEP results for BDPSKand BPSK are presented for β1 = β2 = β, m1 = m2 = 2,β = 1, 2.5, 3.5 and as it is obvious the ABEP improves asβ and/or γ1 increases.

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0 5 10 15 2010-5

10-4

10-3

10-2

10-1

100

simulation

Out

age

Prob

abili

ty

First Hop Inverse Normalized Outage Threshold (dB)

Fig. 6. Outage Probability versus γ1/γth (β1 = β2 = 3and m1 = m2 = 2) .

In Fig. 6, the OP of the considered system is illustratedas a function of the first hop inverse normalized outagethreshold γ1/γth when β1 = β2 = 3 and m1 = m2 = 2.The impact of power imbalance between the two hops onthe OP is also illustrated. As far as the power imbalanceis concerned, one can verify similar findings to thatmentioned in Fig. 2. Finally, for all the considered testcases, our theoretical analysis is substantiated by meansof monte-carlo simulations and as it can be observed,the simulations are in perfect agreement with the ana-lytically obtained results.

8 CONCLUSION

In this paper, the end-to-end performance of dual-hopwireless communication systems with fixed-gain relaysoperating over GG fading channels was evaluated. Anovel closed-form expression for the moments of the out-put SNR was derived. Moreover, the average error andthe outage performance of the considered system werestudied using the MGF approach and the Pade approxi-mants method. An alternative integral representation forthe outage probability was also derived. This expressioncan be acurrately and efficiently evaluated by means ofthe Gauss-Laguerrre quadrature rule. Various numericaland computer simulations results were presented thatdemonstrated the proposed mathematical analysis.

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Kostas P. Peppas was born in Athens in 1975. He obtained his diplomain Electrical and Computer Engineering from the National technicalUniversity of Athens in 1997 and the Ph.D. degree in telecommunica-tions from the same department in 2004. His current research inter-ests include wireless communications, smart antennas, digital signalprocessing and system level analysis and design. He is a member ofIEEE and the National Technical Chamber of Greece.

Akil Mansour was born in Deir Ezzor, Syria, in 1969. He received theB.Sc. degree in physics from Aleppo University of Syria, in 1995, theM.Sc. degree in the field of Electronics and Telecommunication systemsat the department of Electronics and Telecommunications from Nationaland Kapodistrian University of Greece, Athens, in 2001. At the presenttime he is doing Ph.D. in the field of Wireless Mobile Communication.

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George S. Tombras was born in Athens, Greece, in 1956. He receivedthe B.Sc. degree in physics from Aristotelian University of Thessaloniki,Greece, the M.Sc. degree in electronics from University of Southampton,UK, and the Ph.D. degree from Aristotelian University of Thessaloniki,in 1979, 1981, and 1988, respectively. He is currently an Associate Pro-fessor of Electronics. His research interests include mobile communica-tions, analog and digital circuits and systems, as well as instrumentation,measurements, and audio engineering. Professor Tombras authored orcoauthored more than 70 journal and conference papers and manytechnical reports.