exact error analysis and energy efﬁciency optimization of...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 7, JULY 2016 4973

Exact Error Analysis and Energy EfficiencyOptimization of Regenerative Relay Systems

Under Spatial CorrelationMulugeta K. Fikadu, Student Member, IEEE, Paschalis C. Sofotasios, Member, IEEE,

Qimei Cui, Member, IEEE, George K. Karagiannidis, Fellow, IEEE, and Mikko Valkama, Member, IEEE

Abstract—Energy efficiency and its optimization constitute crit-ical tasks in the design of low-power wireless networks. Thispaper is devoted to the error rate analysis and energy efficiencyoptimization of regenerative cooperative networks in the presenceof multipath fading under spatial correlation. To this end, ex-act and asymptotic analytic expressions are first derived for thesymbol error rate (SER) of M -ary quadrature amplitude andM -ary phase-shift keying modulations (M -QAM and M -PSK),respectively, assuming a dual-hop decode-and-forward (DF) relaysystem, spatially correlated Nakagami-m multipath fading, andmaximum ratio combining (MRC) at the destination. The derivedexpressions are subsequently employed in quantifying the energyconsumption of the considered system, incorporating both trans-mit energy and the energy consumed by the transceiver circuits,and in deriving the optimal power allocation (OPA) formula-tion for minimizing energy consumption under certain quality-of-service (QoS) requirements. A relatively harsh path-loss (PL)model, which also accounts for realistic device-to-device commu-nications, is adopted in numerical evaluations, and various usefulinsights are provided for the design of future low-energy wirelessnetworks deployments. Indicatively, it is shown that, dependingon the degree of spatial correlation, severity of fading, transmis-sion distance, relay location, and power allocation strategy, targetperformance can be achieved with much overall energy reductioncompared with direct transmission (DT) reference.

Manuscript received November 12, 2014; revised April 14, 2015; acceptedMay 30, 2015. Date of publication August 4, 2015; date of current versionJuly 14, 2016. This work was supported in part by the Finnish FundingAgency for Technology and Innovation (Tekes) under Project “Energy-EfficientWireless Networks and Connectivity of Devices-Systems (EWINE-S),” bythe Academy of Finland under Project 251138 “Digitally-Enhanced RF forCognitive Radio Devices” and Project 284694 “Fundamentals of Ultra Dense5G Networks with Application to Machine Type Communication,” and by theNational Natural Science Foundation of China under Grant 61471058. Thispaper was presented in part at the IEEE 26th Annual International Symposiumon Personal, Indoor, and Mobile Radio Communications (IEEE PIMRC’15),Hong Kong. The review of this paper was coordinated by Prof. N. Arumugam.

M. K. Fikadu and M. Valkama are with the Department of Electronicsand Communications Engineering, Tampere University of Technology, 33101Tampere, Finland (e-mail: [email protected]; [email protected]).

P. C. Sofotasios was with the School of Electronic and Electrical Engineer-ing, University of Leeds, Leeds LS2 9JT, U.K. He is now with the Depart-ment of Electronics and Communications Engineering, Tampere Universityof Technology, 33101 Tampere, Finland, and also with the Department ofElectrical and Computer Engineering, Aristotle University of Thessaloniki,54124 Thessaloniki, Greece (e-mail: [email protected]).

Q. Cui is with the Wireless Technology Innovation Institute, Beijing Uni-versity of Posts and Telecommunications, Beijing 100876, China (e-mail:[email protected]).

G. K. Karagiannidis is with the Department of Electrical and ComputerEngineering, Khalifa University, 127788 Abu Dhabi, UAE, and also with theDepartment of Electrical and Computer Engineering, Aristotle University ofThessaloniki, 54124 Thessaloniki, Greece (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2015.2464211

Index Terms—Asymptotic analysis, cooperative communica-tions, energy efficiency, error rate, maximum ratio combining(MRC), multipath fading, optimization, power allocation, qualityof service (QoS), regenerative relaying, spatial correlation.

I. INTRODUCTION

EMERGING communication systems are expected toprovide high-speed data transmission, efficient wireless

access, high quality of service (QoS), and reliable networkcoverage with reduced processing time and energy, as well aswidespread use of smartphones and other intelligent mobiledevices. However, the currently witnessed scarcity of the twocore fundamental resources, i.e., power and bandwidth, con-stitutes a significant challenge to satisfy these demands, whileit is known that wireless channel impairments, such as multi-path fading, shadowing, and interference, degrade informationsignals during wireless propagation. Furthermore, most energy-constrained devices, such as terminals of mobile cellular,ad-hoc, and wireless sensor networks, are typically poweredby small batteries where replacement is rather difficult andcostly [1], [2]. Therefore, finding a robust strategy for energy-efficient transmission and minimized energy consumption persuccessfully communicated information bit is essential in effec-tive design and deployment of wireless systems. This accountsfor example for cases such as low-energy sensor networks inecological environment monitoring and energy consumptionin infrastructure devices of cellular systems. In addition, it isin line with global policies and strategies on low energy con-sumption and awareness on environmental issues which, amongothers, has led to the rapid emerge of green communicationsand systems [3], [4].

It has also been shown that multiantenna systems consti-tute an effective method that can enhance spectral efficiency.However, this typically comes at a cost of complex transceivercircuitry and in massive systems with high energy consump-tion requirements. Furthermore, it is not currently feasible toembody large multiantenna systems at handheld terminals dueto spatial restrictions. As a result, cooperative communicationshave been proposed as an alternative solution that improvescoverage and performance under fading effects and have at-tracted significant attention due to their ability to overcome thelimitations of resource-constrained wireless access networks(see, e.g., [5]–[18] and the references therein). A distinct featureof cooperative communications is that wireless agents share

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4974 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 7, JULY 2016

resources, instead of competing for them, which ultimatelyenhances the overall system performance. In this context, var-ious resource allocation algorithms and techniques have beenproposed for improving the energy efficiency of resource-constrained wireless networks. Specifically, Cui et al. [19] ana-lyzed energy-efficient direct transmission (DT) adopting higherlevel modulation for short distances, where circuit power ismore dominant than transmission power. It was also suggestedthat high energy reduction can be achieved by optimizing thetransmission time and the modulation parameters, particularlyfor short transmission distances. Devarajan et al. [20] addressedthe optimal power allocation (OPA) and throughput transmis-sion strategy for minimizing the total energy consumption re-quired to transmit a given number of bits. In [21], minimizationof two-hop transmission energy with joint relay selection andpower control was proposed for two policies: 1) for minimizingthe energy consumption per data packet; and 2) for maximiz-ing the network lifetime. In the same context, in [22]–[26],the modulation optimization for minimizing the total energyconsumption for M -ary quadrature amplitude modulation (M -QAM) was addressed, whereas energy-efficient cooperativecommunication in clustered sensor networks was investigatedin [27]. Energy efficiency in cooperative networks was alsoanalyzed in [28]–[35] by optimizing energy consumption basedon the involved relay decoding strategy, modulation parame-ters, and the number of relay nodes and their distance fromthe source and the destination nodes. Likewise, an energy-efficient scheme was proposed in [36] by exploiting the wirelessbroadcast nature and the node overhearing capability, whereasan optimal energy-efficient strategy based on the cooperativenetwork parameters and transmission rate was reported in [37].Finally, realistic scenarios of energy-efficient infrastructure-to-vehicle communications were analyzed in [38].

It is also widely known that fading phenomena constitutea crucial factor of performance degradation in conventionaland emerging wireless communication systems. Based on this,numerous investigations have addressed the effect of differenttypes of fading conditions on the performance of coopera-tive communications [39]–[56]. However, the vast majorityof the reported investigations assume that the involved com-munication paths are statistically independent to each other.Nevertheless, this assumption is rather simplistic as, in realisticcooperative communication scenarios, the wireless channelsmay be spatially correlated, which should be taken into accountparticularly for deployments relating to low-energy consump-tion requirements. Based on this, the spatial correlation inrelay communications over fading channels was addressed in[57], whereas the performance of a decode-and-forward (DF)system with M -phase shift keying (PSK)-modulated signals intriple correlated branches over Nakagami-m fading channelsusing selection combining was investigated in [58]. In the samecontext, in [59], the performance of a multiple-input–multiple-output (MIMO) DF system with orthogonal space–time trans-mission over spatially correlated Nakagami-m fading channelsfor integer values of m was analyzed. The performance of atwo-hop amplify-and-forward (AF) relay network with beam-forming and spatial correlation for the case that the sourceand destination are equipped with multiple antennas while the

relay is equipped with a single antenna was investigated in [60].Likewise, spatial correlation in the context of indoor office en-vironments and multiantenna AF relaying with keyhole effectswas analyzed in [61] and [62], respectively, whereas the effectsof spatial correlation on the performance of similar relayingsystems were analyzed in [63]–[66]. However, to the best ofthe authors’ knowledge, a comprehensive exact and asymptoticerror rate analysis for regenerative systems over spatially corre-lated channels using maximum ratio combining (MRC), as wellas a detailed energy efficiency analysis and optimization, havenot been reported in the open technical literature.

Motivated by the above, the aim of this paper is twofold.First, we derive exact analytic expressions for the symbolerror rate (SER) of a two-hop DF relay system over spatiallycorrelated Nakagami-m fading channels for both M -QAMand M -PSK constellations along with simple and accurateasymptotic expressions for high signal-to-noise ratio (SNR)values. Second, we provide a comprehensive analysis of energyefficiency and the corresponding optimization in terms of powerallocation between cooperating devices. This is realized byminimizing the average total energy consumption of the DFrelay network over multipath fading conditions, for a givendestination bit error rate (BER) and maximum transmit powerconstraints.

In more detail, the technical contributions of this paper are asfollows.

• Exact closed-form expressions are derived for the end-to-end SER of M -QAM and M -PSK-based dual-hopregenerative relay networks with MRC reception at thedestination over Nakagami-m multipath fading channelswith arbitrary spatial correlation between source–destination (S–D) and relay–destination (R–D) links.

• Simple closed-form asymptotic expressions are derivedfor the given scenarios for high SNR values.

• The offered analytic results are employed in a comprehen-sive energy optimization analysis based on minimizingthe average total energy consumption of the overall regen-erative relay network under a given QoS target, in termsof (BER), and maximum transmit power constraints.

The remainder of this paper is organized as follows.Section II presents the considered relay system and channelmodel, whereas Sections III and IV are devoted to the derivationof the corresponding exact and asymptotic error rate expres-sions. The total power consumption models are presented inSection V, whereas the analysis of energy minimization andpower allocation optimization based on the given constraintsare provided in Section VI. Section VII presents the cor-responding numerical results along with extensive analy-sis and discussions, whereas closing remarks are providedin Section VIII.

II. SYSTEM AND CHANNEL MODEL

We consider a two-hop cooperative radio-access systemmodel consisting of source node (S), a relay node (R), anda destination node (D), where each node is equipped with asingle antenna, as shown in Fig. 1. Without loss of generality,

FIKADU et al.: EXACT ERROR ANALYSIS AND ENERGY EFFICIENCY OPTIMIZATION OF RELAY SYSTEMS 4975

Fig. 1. Dual-hop cooperative single-relay model.

the system can represent both conventional and emerging com-munication scenarios such as, for example, a mobile ad-hocnetwork or a vehicle-to-vehicle communication system. The co-operative strategy is based on a half-duplex DF relaying wheretransmission is performed using time-division multiplexing. Itis also assumed that the destination is equipped with MRCreception and that information signals are subject to multipathfading conditions that follow the Nakagami-m distribution.1

In phase I, the source broadcasts the signal to both destina-tion and relay nodes, and the corresponding received signalscan be expressed as

yS,D =

√PS

PLS,D

hS,Dx+ nS,D (1)

yS,R =

√PS

PLS,R

hS,Rx+ nS,R (2)

respectively, where PS is the transmit power, x is the transmit-ted symbol with normalized unit energy in the first transmissionphase, and PLS,D

and PLS,Rdenote the path-loss (PL) values

in the S–D and S–R paths, respectively. Moreover, hS,D andhS,R are the complex fading coefficients of the S–D and S–Rwireless links, respectively, whereas nS,D and nS,R are thecorresponding complex Gaussian noise terms with zero meanand variance N0. The relay then checks whether the receivedsignal can be decoded correctly, which can be, for example,realized by examining the included cyclic redundancy checkdigits or the received SNR levels [55], [57]. Based on this, ifthe signal is successfully decoded, the relay forwards it to thedestination during phase II with power PR = PR; otherwise,the relay does not transmit and remains idle with PR = 0.Hence, the signal at the destination during phase II can berepresented as follows:

yR,D =

√PR

PLR,D

hR,Dx+ nR,D (3)

1It is noted that the considered system requires the least resources in termsof bandwidth and power compared with multirelay assisted transmission; thus,it can be adequate for low-complexity and low-energy wireless networks.

where PR is the transmit power of the relay, PLR,Dis the PL of

the R–D path and hR,D and nR,D denote the channel coefficientand complex Gaussian noise term with zero mean and varianceN0, respectively. Finally, the destination combines the receiveddirect and relayed signals based on the MRC method where thecombined SNR can be expressed as follows [57], [67]:

γMRC =

(PS

PLS,D

)| hS,D |2 +

(PR

PLR,D

)| hR,D |2

N0. (4)

The fading effects between the devices are assumed to followthe Nakagami-m distribution, which is a widely used modelas the fading parameter m can easily account for both se-vere and moderate fading conditions. Thus, the correspondingchannel power gains | hS,D |2, | hS,R |2, and | hR,D |2 followthe gamma distribution [68] with different power parameters1/ΩS,D, 1/ΩS,R, and 1/ΩR,D, and fading parameters mS,D,mS,R, and mR,D, respectively.

In the considered regenerative system, arbitrary spatial cor-relation is assumed to exist between the S–D and R–D paths,as also adopted in the semi-analytical contribution of [57].To this effect, the corresponding moment-generating function(MGF) for the case of Nakagami-m fading is according to (5),shown at the bottom of the page, where γS,D and γR,D arethe corresponding average SNR values, s denotes the MGFparameter, and

ρ =Cov

(|hS,D|2, |hR,D|2

)√

Var (|hS,D|2)Var (|hR,D|2)(6)

represents the involved correlation coefficient [68], [69], withCov(·) and Var(·) denoting covariance and variance operations,respectively. It is noted here that other practical impairments,such as cochannel interference, are not considered in this paper.

III. SYMBOL ERROR RATE FOR M -QUADRATURE

AMPLITUDE MODULATION IN NAKAGAMI-mFADING WITH SPATIAL CORRELATION

This section is devoted to the error probability analysisof the dual-hop cooperative network over Nakagami-m fadingchannels with spatial correlation between the direct and theR–D paths [68]. To this effect and assuming MRC reception,the average end-to-end SER can be expressed according to(7), shown at the bottom of the next page, [57], where mc =mS,D = mR,D, gQAM = 3/2(M − 1), and

FQAM[v(θ)]=4π

(1− 1√

M

) π2∫

0

v(θ)dθ− 4π

(1− 1√

M

)2π4∫

0

v(θ)dθ.

(8)

The first two terms in (7) refer to the cases of incorrect andcorrect decoding of the received signal at the relay node,

M(s) =

⎛⎜⎝1 −

(γS,D

PLS,D

+γR,D

PLR,D

)m

s+

(1−ρ)γS,DγR,D

(PLS,DPLR,D

)

m2s2

⎞⎟⎠

−m

, s < 0 (5)


respectively, whereas the integral representation in (8) is used

for evaluating the SERCD numerically. In what follows, we first

derive a closed-form expression for the average SER in the caseof direct communication mode. This expression is subsequentlyemployed in the derivation of exact closed-form expressions forthe average SER of M -QAM and M -PSK-modulated regener-ative systems over Nakagami-m fading channels with spatialcorrelation. Furthermore, it is used in the analysis of the energyconsumption model and energy minimization in Section VIas it allows the derivation of an accurate expression for theenergy consumption in the DT, which acts as a benchmark in theevaluation of the energy reduction of the cooperative system.

A. Exact SER for the Direct Transmission

Theorem 1: For PS, PLS,D,ΩS,D, N0, gQAM ∈ R

+,M ∈ N,and mS,D ≥ (1/2), the SER of a M -QAM DT scheme canbe expressed according to (9), shown at the bottom of thepage, where 2F1(·) and F1(·) denote the Gauss hypergeometricfunction and the Appell hypergeometric function of the firstkind, respectively.

Proof: The proof is provided in Appendix A. �

B. Exact SER for the Cooperative Transmission

Capitalizing on the proof of Theorem 1, this part is devotedon the derivation of a novel closed-form expression for the av-erage SER of the cooperative transmission (CT) scenario whenthe involved relay node decodes and forwards successfullydecoded information signals to the destination. To this end, itis essential to first derive exact closed-form expressions for twoindefinite trigonometric integrals, which are generic and can

be particularly useful in various applications relating to naturalsciences and engineering, including wireless communications.

Lemma 1: For a, b,m ∈ R+ and 2m− (1/2) ∈ N, for the

infinite integral in (10), the closed-form expression in (11),shown below, is valid, i.e.,

J(a,b,m)=

∫1(

1 + asin2(θ)

+ bsin4(θ)

)m dθ (10)

= −2m− 1

2∑l=0

(2m−1

2

l

)(−1)l

(a+2 sin2(θ)−

√a2−4b

)m(2 + a−

√a2 − 4b

)m×F1

(l+ 1

2 ,m,m,l+ 32 ,

2 cos2(θ)

2+a−√a2−4b

, 2 cos2(θ)

2+a+√a2−4b

)(sin4(θ) + a sin2(θ) + b

)m×(a+ 2 sin2(θ) +

√a2 − 4b

)m(2 + a+

√a2 − 4b

)m cos1+2l(θ)

(1 + 2l)+ C.

(11)

Proof: The proof is provided in Appendix B. �Lemma 2: For a, b,m, n ∈ R

+ and m+ n− (1/2) ∈ N, theclosed-form expression in (13), below, is valid for the integralin (12):

K(a, b,m, n) =

∫1(

1 + asin2(θ)

)m (1 + b

sin2(θ)

)n dθ (12)

= −m+n− 1

2∑l=0

(m+n− 1

2

l

)(−1)l cos1+2l(θ)

(1+2l)(1+a)m(1+b)n

×F1

(l+

12,m, n, l+

32,cos2(θ)

1 + a,cos2(θ)

1 + b

)+C.

(13)

SERCD = FQAM

⎡⎢⎢⎢⎢⎢⎢⎣

1⎛⎝1 +

(PS

PLS,D

)ΩS,D gQAM

N0mc sin2 θ

⎞⎠

mc

⎤⎥⎥⎥⎥⎥⎥⎦FQAM

⎡⎢⎢⎢⎢⎢⎢⎣

1⎛⎝1 +

(PS

PLS,R

)ΩS,R gQAM

N0mS,R sin2 θ

⎞⎠

mS,R

⎤⎥⎥⎥⎥⎥⎥⎦

+FQAM

⎡⎢⎢⎢⎢⎢⎢⎣

1⎛⎝1+

[(PSΩS,DPLS,D

)+

(PRΩR,DPLR,D

)]gQAM

N0mc sin2 θ +(1−ρ)PSPRΩS,DΩR,Dg2

QAM

N20PLS,D

PLR,Dm2

c sin4 θ

⎞⎠

mc

⎤⎥⎥⎥⎥⎥⎥⎦×

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

1−FQAM

⎡⎢⎢⎢⎢⎢⎢⎣

1⎛⎝1+

(PS

PLS,R

)ΩS,RgQAM

N0mS,R sin2 θ

⎞⎠

mS,R

⎤⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(7)

SERDD=

2(√M−1)NmS,D

0 mmS,D

S,D PLS,D√πM(mS,DN0PLS,D

+PSΩS,DgQAM

)mS,D×

⎧⎨⎩Γ(mS,D+

12

)Γ(mS,D +1) 2F1

(mS,D,

12,mS,D+1,

mS,DN0PLS,D

mS,DN0PLS,D+PSΩS,DgQAM

)

+

√2(√M − 1)√π

F1

(12,

12−mS,D,mS,D,

32,

12,

mS,DN0PLS,D

2(mS,DN0PLS,D

+ PSΩS,DgQAM

))⎫⎬⎭ (9)


Proof: The proof is provided in Appendix C. �To the best of the authors’ knowledge, the generic solutions

in the given lemmas have not been previously reported in theopen technical literature. These results are employed in thesubsequent analysis.

Theorem 2: For {PS, PR, PLS,D, PLS,R

, PLR,D, N0} ∈ R

+,{|ΩS,D|, |ΩS,R|, |ΩR,D|} ∈ R, M ∈ N, mS,D ≥ (1/2), mS,R ≥(1/2), mR,D ≥ (1/2), 2mc − (1/2) ∈ N, and 0 ≤ ρ < 1, theSER of M -QAM based DF relaying over spatially correlatedNakagami-m fading channels, can be expressed as follows:

SERCD

=

⎧⎨⎩

2(mc − 1

2

)! 2F1

(mc,

12 ,mc + 1, 1

1+a1

)√π mc! M(

√M − 1)−1 (1 + a1)mc

+2√

2 F1

(12 ,

12 −mc,mc,

32 ,

12 ,

12+2a1

)π M (

√M − 1)−2 (1 + a1)mc

⎫⎬⎭

×

⎧⎨⎩

2(mS,R− 1

2

)! 2F1

(mS,R,

12 ,mS,R+1, 1

1+b1

)√π mS,R! M(

√M − 1)−1 (1 + b1)mS,R

+2√

2F1

(12 ,

12 −mS,R,mS,R,

32 ,

12 ,

12+2b1

)π M (

√M − 1)−2 (1 + b1)mS,R

⎫⎬⎭

+

⎧⎨⎩1−

(mS,R− 1

2

)! 2F1

(mS,R,

12 ,mS,R+1, 1

1+b1

)2−1√π mS,R!M(

√M−1)−1(1+b1)mS,R

−2√

2F1

(12 ,

12 −mS,R,mS,R,

32 ,

12 ,

12+2b1

)π M (

√M − 1)−2 (1 + b1)mS,R

⎫⎬⎭

×

⎧⎨⎩

2mc− 12∑

l=0

(−1)l4(√M − 1)22mc−l− 1

2

Mπ(1 + 2l) [(1 − 2A)(1 − 2B)]−mc

×(

2mc − 12

l

)F1

(l + 1

2 ,mc,mc, l +32 , 2A, 2B

)Mπ(1+2l)[1+2(a1+c1)4a1d1]

mc

+

2mc− 12∑

l=0

(2mc − 1

2

l

)(−1)l4(

√M − 1)

πM(1 + 2l)(a1d1)mc

×F1

(l + 1

2 ,mc,mc, l +32 ,A,B

)(1 −A)−mc(1 − B)−mc

⎫⎬⎭ (14)

where

a1 =PSΩS,DgQAM

PLS,DmS,DN0

(15)

b1 =PSΩS,RgQAM

PLS,RN0mS,R

(16)

c1 =PRΩR,DgQAM

PLR,DN0mR,D

(17)

d1 =(1 − ρ)PRΩR,DgQAM

PLR,DN0mR,D

(18)

{AB}=

1

2 + a1 + c1{−+

} √(a1 + c1)2 − 4a1d1

. (19)

Proof: The first term in (7) corresponds to the DT; thus, itcan be expressed in closed-form based on Theorem 1. Likewise,

the second and fourth terms in (7) have the same algebraic rep-resentation as (85) and (86) in Appendix A. Therefore, they canbe readily expressed in closed form by making the necessarychange of variables and substituting in (91) and (93), shown inAppendix A. As for the third term in (7), it is noticed that ithas the same algebraic representation with (11). As a result, aclosed-form expression is deduced by determining the follow-ing specific cases in (11), which practically evaluate (8), i.e.,

J(a, b,m, 0,

{π2π4

})=

{π2π4

}∫0

dθ(1 + a

sin2(θ)+ b

sin4(θ)

)m . (20)

Therefore, by carrying out some long but basic algebraicmanipulations and substituting in (7) along with the afore-mentioned closed-form expressions, one obtains (14), whichcompletes the proof. �

Remark 1: Equation (14) reduces to the uncorrelated sce-nario by setting ρ = 0. However, an alternative expressionfor this case that is valid for the case that {mS,D +mR,D −(1/2)} ∈ N can also be deduced by applying the derived ex-pressions in Theorem 1 and Lemma 2 in [57, Eq. (11)], namely

SERCD =

⎧⎨⎩1 +

2√

2F1

(12 ,

12 −mS,R,

32 ,

12 ,

12+2b1

)π(√M − 1)−2M(1 + b1)mS,R

−2Γ(mS,R+

12

)2F1

(mS,R,

12 ,1+mS,R,

11+b1

)√πM(

√M−1)−1Γ(1+mS,R)(1+b1)mS,R

⎫⎬⎭

×

⎧⎨⎩

mS,D+mR,D− 12∑

l=0

4(−1)l(√M − 1)

l!πM(1 + a1)mS,D

×F1

(l+ 1

2 ,mS,D,mR,D, l+32 ,

11+a1

, 11+c1

)(1+2l)(1+c1)mR,D

(mS,D +mR,D + 1

2

)−l

+

mS,D+mR,D− 12∑

l=0

(−1)l232−l(

√M − 1)2

l!πM(1 + a1)mS,D

×F1

(l+ 1

2 ,mS,D,mR,D, l+32 ,

12+2a1

, 12+2c1

)(1+c1)mR,D(1+2l)

(mS,D+mR,D + 1

2

)−l

⎫⎬⎭

+

⎧⎨⎩

2Γ(mS,D+

12

)2F1

(mS,D,

12 , 1+mS,D,

11+a1

)√πM(

√M−1)−1Γ(1+mS,D)(1+a1)mS,D

−2√

2(F1

(12 ,

12 −mS,D,

32 ,

12 ,

12+2a1

)π(√M − 1)−2M(1 + a1)mS,D

⎫⎬⎭

×

⎧⎨⎩

2Γ(mS,R+

12

)2F1

(mS,R,

12 , 1+mS,R,

11+b1

)√πM(

√M−1)−1Γ(1+mS,R)(1 + b1)mS,R

−2√

2F1

(12 ,

12 −mS,R,

32 ,

12 ,

12+2b1

)π(√M − 1)−2M(1 + b1)mS,R

⎫⎬⎭ (21)

where (x)n � Γ(x + n)/Γ(x) is the Pochhammer symbol [73].


C. Asymptotic SER for the Cooperative Transmission

Simple asymptotic expressions can be derived for the caseof high SNR at the three paths of the system. To this end, itis essential to first derive a closed-form expression for anothertrigonometric generic integral.

Lemma 3: For m ∈ R, the following generic closed-formexpression holds:∫

sin2m dθ = − cos(θ) 2F1

(12,

12−m,

32, cos2(θ)

)+ C.

(22)

Proof: The proof is provided in Appendix D. �Lemma 3 is subsequently employed in the derivation of the

following proposition.Proposition 1: For {PS, PR, PLS,D

, PLS,R, PLR,D

} ∈ R+,

{ΩS,D,ΩS,R,ΩR,D, N0}∈R+, M ∈N, mS,D≥(1/2), mS,R ≥

(1/2), mR,D≥(1/2), 2mc−(1/2) ∈ N, and 0 ≤ ρ < 1, theSER of M -QAM-based DF relaying over spatially correlatedNakagami-m fading channels in the high-SNR regime can beexpressed as follows:

SERCD

�(

PLS,DN0mc

PSΩS,DgQAM

)mc(

1 − 1√M

)

×{

2Γ(mc +

12

)√πΓ(1 +mc)

−(

1− 1√M

)2F1

(12 , 1,mc+

32 ,−1

)π2mc−2 (1 + 2mc)

}

×(PLS,R

N0mS,R

PSΩS,RgQAM

)mS,R(

1 − 1√M

)

×{

2Γ(mS,R+

12

)√π Γ(1+mS,R)

−(1− 1√

M

)2F1

(12 , 1,mS,R+

32 ,−1

)π 2mS,R−2(1+2mS,R)

}

+

(1− 1√

M

)(PLS,D

PLR,DN2

0m2c

(1−ρ) PSPRΩS,DΩR,Dg2QAM

)mc

×{

2Γ(2mc+

12

)√π Γ(1+2mc)

−(

1− 1√M

)2F1

(12 , 1, 2mc+

32 ,−1

)π 22mc−2(1 + 4mc)

}.

(23)Proof: In the high-SNR regime, it is realistic to as-

sume that PSΩS,D � PLS,DN0, PSΩS,R � PLS,R

N0, andPRΩR,D � PLR,D

N0. Based on this, an integral representationwas formulated in [57, Eq. (29)], i.e.,

SERCD�AcAS,R

(N0mcPLS,D

PSΩS,DgQAM

)mc(

N0mcPLS,R

PSΩS,RgQAM

)mS,R

+A2c

(N2

0m2cPLS,D

PLR,D

(1 − ρ)PSPRΩS,DΩR,Dg2QAM

)mc

(24)

where

{Ac

A2c

}=

4π

(1 − 1√

M

) π2∫

0

sin{2mc4mc} dθ

− 4π

(1 − 1√

M

)2π4∫

0

sin{2mc4mc} dθ (25)

AS,R =4π

(1 − 1√

M

) π2∫

0

sin2mS,R dθ

− 4π

(1 − 1√

M

)2π4∫

0

sin2mS,R dθ. (26)

Evidently, the terms Ac, A2c, and AS,R can be expressed inclosed form with the aid of Lemma 3. Based on this, by per-forming the necessary change of variables in (22) and substitut-ing in (24), (23) yields, which completes the proof. �

Remark 2: Using (23), the correlation coefficient for the caseof M -QAM modulation can be expressed in terms of the cor-responding source power and relay power, fading parameters,and average SER, according to (27), shown at the bottom of thepage, where

g = gQAM, (28)

C =(

1 − 1/√M)

(29)

K1 =N0mcPLS,D

PSΩS,Dg(30)

K2 =N0mS,RPLS,R

PSΩS,Rg(31)

K3 =N2

0m2cPLS,D

PLR,D

PSPRΩS,DΩR,Dg2. (32)

IV. SYMBOL ERROR RATE FOR M -PHASE SHIFT KEYING

MODULATION IN NAKAGAMI-m FADING WITH

SPATIAL CORRELATION

Having derived novel analytic expressions for the case ofM -QAM modulation, this section is devoted to the derivationof exact and asymptotic closed-form expressions for the case ofM -PSK constellations.

A. Exact SER for the Cooperative Transmission

Theorem 3: For {PS, PR, PLS,D, PLS,R

, PLR,D}∈R

+, {ΩS,D,ΩS,R,ΩR,D, N0}∈R

+, M ∈ N, mS,D ≥ (1/2), mS,R≥(1/2),

ρ = 1 −K3

{2CΓ(2mc+

12 )√

π Γ(1+2mc)− C2

2F1( 12,1,2mc+

32,−1)

π 22mc−2(1+4mc)

} 1mc

⎛⎜⎝SER

CD −

{2CΓ(mc+

12 )√

πΓ(1+mc)−

C22F1( 1

2,1,mc+3

2,−1)

π2mc−2 (1+2mc)

}{2CΓ(mS,R+1

2 )√π Γ(1+mS,R)

−C2

2F1( 12,1,mS,R+3

2,−1)

π 2mS,R−2

(1+2mS,R)

}

K−mc1 K

−mS,R2

⎞⎟⎠

1mc

(27)


mR,D ≥ (1/2), 2mc − (1/2) ∈ N, and 0 ≤ ρ < 1, the SER ofM -PSK-based DF relaying over spatially correlated Nakagami-m fading channels can be expressed according to (33), shownat the bottom of the page, where

a2 =PSΩS,DgPSK

PLS,DmS,DN0

(34)

b2 =PSΩS,RgPSK

PLS,RN0mS,R

(35)

c2 =PRΩR,DgPSK

PLR,DN0mR,D

(36)

d2 =(1 − ρ)PRΩR,DgPSK

PLR,DN0mR,D

. (37)

Proof: As a starting point, the average SER of M -PSK-modulated DF systems over Nakagami-m fading channels with

spatial correlation can be formulated according to [57, Eq. (23)]in (38), shown at the bottom of the page, where

FPSK [u(θ)] =1π

(M−1)πM∫0

u(θ)dθ. (39)

The involved four integrals have the same algebraic form as theintegrals in Theorem 1 and Lemma 1. Thus, the proof followsimmediately by performing the same necessary change of vari-ables and substituting in (38). �

B. Asymptotic SER for the Cooperative Transmission

Proposition 2: For {PS, PR, PLS,D, PLS,R

, PLR,D} ∈ R

+,{ΩS,D,ΩS,R,ΩR,D, N0}∈R

+ M ∈N, mS,D ≥ (1/2), mS,R ≥(1/2), mR,D≥(1/2), 2mc−(1/2) ∈ N, and 0 ≤ ρ < 1, theSER of M -PSK-based DF relaying over spatially correlated

SERCD =

sin2mc+1(

(M−1)πM

)(1 + 2mc)πa

mc2

F1

⎛⎝mc +

12,

12,mc,mc +

32, sin2

((M − 1)π

M

),sin2

((M−1)π

M

)a2

⎞⎠

×sin2mS,R+1

((M−1)π

M

)(1 + 2mS,R)πb

mS,R

2

F1

⎛⎝mS,R +

12,

12,mS,R,mS,R +

32, sin2

((M − 1)π

M

),sin2

((M−1)π

M

)b2

⎞⎠

+

⎧⎨⎩1 −

sin2mS,R+1(

(M−1)πM

)(1 + 2mS,R)πb

mS,R

2

F1

⎛⎝mS,R +

12,

12,mS,R,mS,R +

32, sin2

((M − 1)π

M

),sin2

((M−1)π

M

)b2

⎞⎠⎫⎬⎭

×

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2mc− 12∑

l=0

(2mc − 1

2

l

) (−1)lF1

(l + 1

2 ,mc,mc, l +32 ,

2

2+c2−√

c22−4d2

, 2

2+c2+√

c22−4d2

)

(1 + 2l)πdmc2

[(1 − 2

2+c2−√

c22−4d2

)(1 − 2

2+c2+√

c22−4d2

)]−mc

−2mc− 1

2∑l=0

(2mc − 1

2

l

) (−1)l cos1+2l(

(M−1)πM

)F1

(l+ 1

2 ,mc,mc, l+32 ,

2 cos2( (M−1)πM )

2+c2−√

c22−4d2

,cos2( (M−1)π

M )2+c2+

√c22−4d2

)

(1 + 2l)π(d2 sin

4(

(M−1)πM

)+ c2 cos2

((M−1)π

M

)+ c2

)mc

×

⎛⎝1 −

2 cos2(

(M−1)πM

)2 + c2 −

√c22 − 4d2

⎞⎠

mc ⎛⎝1 −

2 cos2(

(M−1)πM

)2 + c2 +

√c22 − 4d2

⎞⎠

mc

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(33)

SERCD = FPSK

⎡⎢⎣ 1(

1 +PSΩS,DgPSK

N0mcPLS,Dsin2(θ)

)mS,D

⎤⎥⎦FPSK

⎡⎢⎣ 1(

1 +PSΩS,RgPSK

N0mcPLS,Rsin2(θ)

)mS,R

⎤⎥⎦

+ FPSK

⎡⎢⎣ 1(

1 +(PSΩS,D+PRΩR,D)gPSK

N0mcPLS,DPLR,D

sin2(θ) +(1−ρ)PSPRΩS,DΩR,Dg2

PSK

N20m

2cPLS,D

PLR,Dsin4(θ)

)mc

⎤⎥⎦×

⎧⎪⎨⎪⎩1 − FPSK

⎡⎢⎣ 1(

1 +PSΩS,RgPSK

N0mcPLS,Rsin2(θ)

)mS,R

⎤⎥⎦⎫⎪⎬⎪⎭(38)


Nakagami-m channels in the high-SNR regime is expressed asfollows:

SERCD

�(

N0mcPLS,D

PSΩS,DgPSK

)mc

×{Γ(mc+

12

)2√π mc!

+cos(

πM

)2F1

(12 ,

12−mc,

32 , cos

2(

πM

))π

}

×(N0mS,RPLS,R

PSΩS,RgPSK

)mS,R

×{Γ(mS,R+

12

)2√π mS,R!

+cos(πM

)2F1

(12 ,

12−mS,R,

32 , cos

2(

πM

))π

}

+

(N2

0m2cPLS,D

PLR,D

(1 − ρ)PSPRΩS,DΩR,Dg2PSK

)mc

×{Γ(2mc+

12

)2√π(2mc)!

+cos(

πM

)2F1

(12 ,

12−2mc,

32 , cos

2(

πM

))π

}.

(40)

Proof: The asymptotic SER for high SNR values wasformulated in [57, Eq. (27)] as follows:

SERCD� AcAS,R

(N0mcPLS,D

PSΩS,DgPSK

)mc(N0mS,RPLS,R

PSΩS,RgPSK

)mS,R

+ A2c

(N2

0m2cPLS,D

PLR,D

(1 − ρ)PSPRΩS,DΩR,Dg2PSK

)mc

(41)

where

{Ac

A2c

}=

1π

(M−1)πM∫0

sin{2mc4mc} dθ (42)

AS,R =1π

(M−1)πM∫0

sin2mS,R dθ. (43)

Notably, the integrals in (42) and (43) have the same algebraicrepresentation as the integral in Lemma 3. As a result, byperforming the necessary change of variables and substitutingin (41), one obtains (40), which completes the proof. �

Remark 3: Based on (40), the corresponding correlationcoefficient can be expressed in terms of the involved sourcepower and relay power, fading parameters, and average SER,according to (44), shown at the bottom of the page, where

Fig. 2. Example SER performance over Nakagami-m fading channels withmS,D = mS,R = mR,D = m = {0.75, 1.25}, ΩS,D = ΩS,R = ΩR,D =0 dB for 4-QAM/QPSK constellations and different values of spatialcorrelation.

g = gPSK is set in the K1, K2, and K3 terms, which are givenin Remark 2.

Fig. 2 shows the SER performance as a function of SNR for4-QAM/QPSK modulations. The S–D transmission distance isindicatively considered at 600 m, whereas the relay is assumedlocated in the middle and the transmit power is shared equallyto the source and the relay. The corresponding PL effects areconsidered by adopting the PL model in [70], namely

PLi,j [dB] = 148 + 40 log10 (di,j [km]) (45)

which has been shown to characterize adequately harsh com-munication scenarios and is particularly applicable to mobilerelaying and device-to-device communications. It is clearlyobserved that the empirical simulated results are in excel-lent agreement with the respective analytical results. Further-more, the simple asymptotic results are also highly accurate athigher SNRs.

V. SYSTEM POWER CONSUMPTION MODEL AND ANALYSIS

Here, motivated by the general interests toward greencommunications and increasing incentives to save energy, wequantify the total energy consumption required to transmitinformation from the source to the destination. We assume thatthe transceiver circuitry operates on multimode basis: 1) Whenthere is a signal to transmit, the circuits are in active mode;

ρ = 1 −K3

{Γ(2mc+

12 )

2√π(2mc)!

+cos( π

M ) 2F1( 12, 12−2mc,

32,cos2( π

M ))π

} 1mc

⎛⎜⎝SER

CD −

{Γ(mc+

12 )

2√

π mc!+

cos( πM ) 2F1( 1

2, 12−mc,

32,cos2( π

M ))π

}{Γ(mS,R+1

2 )2√

π mS,R!+

cos( πM ) 2F1( 1

2, 12−mS,R, 3

2,cos2( π

M ))π

}

K−mc1 K

−mS,R2

⎞⎟⎠

1mc

(44)


Fig. 3. Elementary direct-conversion transmitter.

Fig. 4. Elementary direct-conversion receiver.

2) when there is no signal to transmit, the circuits operateon a sleep mode; and 3) the circuits are in transient modeduring the switching process from sleep mode to active mode.The elementary block diagrams of the assumed transmitter andreceiver are shown in Figs. 3 and 4, respectively. This model isbased on the energy-efficient and layout-area-efficient direct-conversion architecture that is commonly used in wirelesstransceivers. It is also assumed that all nodes are equippedwith similar transmitter and receiver circuit blocks and that thepower consumption of the active filters at the transmitter andreceiver is similar.

Considering a node that transmits L bits and total transmis-sion period T , the transient duration from active mode to sleepmode is short enough to be neglected. However, the startupprocess from sleep mode to active mode may be slower dueto the finite phase-locked loop settling time in the frequencysynthesizer. By denoting the duration of the sleep, transientand active modes as Tsp, Ttr, and Ton, respectively, the totaltransmission period is defined as

T = Tsp + Ttr + Ton (46)

with Ttr being equal to the frequency-synthesizer settling time.Based on this, the total energy required to transmit and receiveL information bits is expressed as

E = PonTon + PspTsp + PtrTtr (47)

where Pon, Psp, and Ptr denote the power consumption valuesduring the active, sleep, and transient modes, respectively. Inrealistic circuit designs, the power consumption in the sleepmode can be considered negligible compared with the activemode power [19]; thus, Psp � 0. It is also noted that powerconsumption during the transient mode practically refers tothe power consumption of the frequency synthesizers. Basedon this, it is assumed that Ptr = 2PLO; as a result, using thepower consumption values at both transmitter and receiver sidesduring the active mode, one obtains Pont, which is the totaltransmitter power consumption that accounts for the sum of sig-nal transmission and transmitter circuit power, and Ponr, whichis the total receiver power consumption. Hence, it follows that

Pont =Pt + Pamp + PCTx(48)

Pon =Pont + Ponr (49)

where Ponr = PCRx, Pt is the signal transmission power, Pamp

is the power consumption of the RF power amplifier (PA), andPCTx

and PCRxdenote the total transmitter circuit power and

the total receiver circuit power, i.e.,

PCTx=PDSPTx

+ PDAC + PFil + PMix + PLO (50)

PCRx=PADC+PVGA+2PFil+PMix+PLO+PLNA+PDSPRx

(51)

respectively. The PCTxmeasure consists of the following

power consumption entities: digital signal processor (DSP),PDSPTx

; digital-to-analog converter (DAC), PDAC; active filter,PFil; IQ mixer, PMix, and synthesizer, PLO. Likewise, theactive power consumption at the receiver comprises the powerconsumption values for DSP, PDSPRx

; ADC, PADC; variablegain amplifier (VGA),PVGA; active filter,PFil; IQ mixer,PMix;synthesizer, PLO; and low-noise amplifier (LNA), PLNA [75].Based on this, the total required circuit power consumption isgiven by

PTC = PCTx+ PCRx

. (52)

It is also noted that for signal transmission power Pt, the powerconsumption of the RF PA can be modeled by

Pamp = αPt (53)

where

α =ξ

η− 1 (54)

with η and ξ denoting the respective drain efficiency of theamplifier and the peak-to-average power ratio, which dependson the modulation order and the associated constellation size.Based on this, for the case of square uncoded M -QAM

ξ = 3

√M − 1√M + 1

(55)

Ton =LTS

b=

L

bB(56)

where b = log2 M is the constellation size, L is the transmis-sion block length in bits, and TS is the symbol duration thatrelates to the bandwidth B as TS ≈ 1/B [23].

VI. ENERGY OPTIMIZATION AND POWER ALLOCATION

Here, we deploy and combine the results of the previoussections and analyze the total energy required to transmitinformation efficiently from the source to the destination. Tothis end, we first quantify the total energy consumption in thedirect communication scenario. Hence, by applying (47)–(49)and recalling that Psp ≈ 0 and Ptr = 2PLO, the average energyconsumption per information bit is given by [76]

EDT = E

DT =

((1 + α)PS + PCTx+ PCRx

)Ton + 2PLOTtr

L(57)

where PS denotes the source transmit power. To determinethe average total energy consumption in the corresponding CT


system deploying the DF protocol, we formulate the total av-erage power consumption, which is a discrete random variablethat can be statistically expressed according to (58), shown atthe bottom of the page, where PR denotes the relay transmitpower. The first term of (58) refers to the absolute total powerconsumption by the nodes in the first transmission phase,whereas the second term represents the power consumption inthe second phase, subject to correct decoding of the receivedsignal by the relay, which is indicated by the probabilistic term(1 − SERS,R). Hence, the average total power consumption inthe CT mode can be expressed as

PCT = (PCTx

+ (1 + α)PR + PCRx)(1 − SERS,R

)+ PCTx

+ (1 + α)PS + 2PCRx. (59)

Based on this, the corresponding average energy consumptionper information bit is given by

ECT =

PCT Ton + 2PLOTtr

L. (60)

The achieved energy efficiency enhancement by the CT isdetermined with the aid of the cooperation gain (CG), whichis the ratio of the energy efficiency of CT over the energyefficiency of the DT, per successfully delivered bit, namely

CG =E

DT

(1 − BER

CD

)E

CT

(1 − BER

DD

) . (61)

Evidently, when the resulting ratio is smaller than one, itindicates that DT is more energy efficient; thus, the extra energyconsumption induced by cooperation outweighs its gain indecreasing the BER of the system.

In what follows, the given expressions are employed in for-mulating and solving the energy optimization problems aimingto guarantee certain QoS requirements, namely, target destina-tion BER. In this context, we also provide the OPA formulationfor the CT scenario under the maximum total transmit powerconstraint.

A. Direct Transmission

We first consider the energy optimization problem for mini-mizing the average total energy consumption in the direct com-munication scenario with the maximum transmission powerand target BER p∗ as constraints. We assume that the powerconsumption of the circuit components is fixed and independentof the optimization. Thus, the only variable in the optimizationis the transmit power of the source. To this effect and with the

aid of (57), the optimization problem for the DT mode can beformulated as follows:

minPS

EDT

subject to PS ≤ Pmaxt, PS ≥ 0

BERDD = p∗. (62)

Deriving the minimum average total energy required in thedirect communication scenario requires prior computation ofthe corresponding symbol error probability. This is realizedwith the aid of (9), which is expressed in closed form in termsof 2F1(m, (1/2);m+ 1; (1/(1 + a1))) and F1((1/2); (1/2)−m,m, (3/2); (1/2), (1/(2 + 2a1))) functions.2 It is recalledthat these functions are widely employed in natural sci-ences and engineering, and their computational implementationis rather straightforward as they are built-in functions inpopular software packages such as MATLAB, MAPLE, andMATHEMATICA. It is also noted that the representation ofthese functions in the present analysis allows the following use-ful approximative expressions: 2F1(m, (1/2);m+ 1; (1/(1 +a1))) � 1 and F1((1/2); (1/2)−m,m, (3/2); (1/2), (1/(2 +2a1)) � F1((1/2); (1/2)−m,m, (3/2); (1/2), 0). The accu-racy of these approximations is validated through extensivenumerical and simulation results, which indicate their tightnessfor random values of m and moderate and large values of a1.To this effect, one obtains the following accurate closed-formBER approximation for M -QAM constellations

BERDD �

2(√M − 1)Γ

(m+ 1

2

)√πMm!(1 + a1)m log2 M

+4 F1

(12 ;

12 −m,m, 3

2 ;12 , 0)

√2π(1 + a1)m log2 M

(1 − 1√

M

)2

. (63)

Importantly, the Appell function in (63) can be expressed interms of the Gauss hypergeometric function, namely

F1

(12;

12−m,m,

32;

12, 0

)= 2F1

(12,

12−m;

32,

12

).

(64)As a result, (63) becomes

BERDD �

2(√M − 1)Γ

(m+ 1

2

)√πMm!(1 + a1)m log2 M

+4 2F1

(12 ,

12 −m; 3

2 ,12

)√

2π(1 + a1)m log2 M

(1 − 1√

M

)2

. (65)

2For the sake of simplicity, we assume that m = mS,D.

PCT =

{PCTx

+ (1 + α)PS + 2PCRx, withPr = 1

PCTx+ (1 + α)PR+PCRx

, withPr = 1 − SERS,R

(58)


It is evident that (65) is a function of the modulation order, theseverity of multipath fading, and a1. Therefore, substituting thetargeted QoS p∗ in (65), recalling that

a1 =PSΩS,DgQAM

N0PLS,D

(66)

and carrying out some algebraic manipulations, one obtains

PS �mN0PLS,D

ΩS,DgQAM

[(C

p∗

) 1m

− 1

](67)

where

C =4(√M − 1)2 2F1

(12 ,

12 −m; 32 ;

12

)√

2Mπ log2 M

+2(√M − 1)Γ

(m+ 1

2

)√πm!M log2 M

. (68)

To this effect and with the aid of (57) and (67), it follows thatthe minimum total energy per information bit required for DTfor meeting the required QoS can be expressed in closed formas follows:

ED∗

T =(PCTx

+ PCRx) Ton

L+

2PLOTtr

L

+(1 + α)N0mS,DTonPLS,D

L ΩS,DgQAM

[(C

p∗

) 1m

− 1

]. (69)

Based on the total energy consumption in (69) and given theconstellation size, i.e.,

b =L

BTon(70)

it is shown that the proposed energy expression comprises thetransmission energy Et and circuit energy EC , namely

Et = PSTon =N0mPLS,D

ΩS,DgQAM

[(C

p∗

) 1m

− 1

]Ton

L(71)

whereC can be expressed as a function of the transmission timeTon as follows:

C =BTon

(2

L2BTon − 1

)2Lπ2

LBTon

− 32

2F1

(12,

12−m;

32;

12

)

+2BTon

2L

BTon mL√π

(12

)m

(2

L2BTon − 1

). (72)

Hence, by inserting (72) in (71), an analytic expression forthe transmission energy per information bit is deduced in (73),shown at the bottom of the page. Likewise, the circuit energy,EC , can be expressed as

EC = (PCTx+ PCRx

)Ton. (74)

Notably, (72) and (73) indicate that, for a fixed bandwidth Band packet length L, the transmission energy is a decreasingfunction with respect to the product TonB, whereas the circuitenergy increases monotonically with respect to Ton. In addition,it is shown that the transmission energy is dependent uponthe transmission distance dS,D and the severity of fading m,whereas the corresponding circuit energy remains fixed, regard-less of the value of dS,D and m.

B. Cooperative Transmission

Here, we present the energy optimization and power allo-cation problem when the involved relay forwards successfullydecoded signals, generally at different power than the powerof the source. Evidently, the respective optimization model isa 2-D problem; thus, we formulate the energy minimizationproblem with two optimization variables, namely, the sourcetransmit power PS and the relay transmit power PR. In this con-text, the aim is to minimize the total energy consumption of theoverall network instead of minimizing the energy consumptionat individual nodes. Based on this and with the aid of (60), theoptimization problem can be formulated as follows:

minPS,PR

ECT (PS, PR)

subject to (PS + PR) ≤ Pmaxt, PS ≥ 0, PR ≥ 0

BERCD(PS, PR) = p∗. (75)

The above optimization task is a nonlinear programming prob-lem since the objective function and the constraint BER areboth nonlinear functions of PS and PR. It is also recalledthat Karush–Kuhn–Tucker (KKT) conditions that handle bothequality and inequality constraints are in general the first-ordersufficient and necessary conditions for optimum solutions innonlinear optimization problems provided that certain regular-ity conditions are satisfied. To this end, using the Lagrangemultipliers λ1, λ2, λ3, and λ4, for the equality and inequalityconstraints, we set the corresponding Lagrangian equation thatdepends on the optimization variables and multipliers whilemeeting the KKT conditions in [77] for the nonlinear convexoptimization problem, namely

L = ECT + λ1

(BER

CD − p∗

)− λ2PS

− λ3PR + λ4 ((PS + PR)− Pmaxt) . (76)

Et =N0mPLS,DTon

L ΩS,DgQAM

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎣BTon

(2

L2BTon − 1

)2p∗Lπ2

LBTon

− 32

2F1

(12,12−m;

32;12

)+BTon

(2

L2BTon −1

)p∗mL

√π2

LBTon

−1

(12

)m

⎤⎥⎦

1m

−1

⎫⎪⎪⎬⎪⎪⎭ (73)


The proof for the convexity of the optimization problem isprovided in Appendix E.

Based on (76), the KKT conditions for the problem can beexpressed as follows:

∇ECT +λ1∇BER

CD−λ2∇PS−λ3∇PR+λ4∇(PS+PR)=0

(77)

whereas the associated complementary conditions are given by

BERCD = p∗

PS + PR ≤Pmaxt

λ1

(BER

CD − p∗

)= 0

λ2PS = 0λ3PR = 0

λ4(PS + PR − Pmaxt) = 0λ1, λ2, λ3, λ4 ≥ 0. (78)

In the above set of complementary KKT conditions, both λ2

and λ3 represent inactive constraints; therefore, they can beassumed zero. To this effect, by applying (77) and setting thederivatives with respect to PS and PR to zero, the followinguseful set of equations is deduced:

∂ECT

∂PS+ λ1

∂BERCD

∂PS+ λ4 = 0 (79)

∂ECT

∂PR+ λ1

∂BERCD

∂PR+ λ4 = 0. (80)

Solving for λ4 from (79) and substituting in (80) yield thefollowing relationship, which depends only on one of theLagrangian multipliers:

∂ECT

∂PS− ∂E

CT

∂PR+ λ1

(∂BER

CD

∂PS− ∂BER

CD

∂PR

)= 0 (81)

λ1 =

∂ECT

∂PS− ∂E

CT

∂PR(∂BER

CD

∂PR− ∂BER

CD

∂PS

) . (82)

Based on this and using the fact that λ1 ≥ 0, one obtains thefollowing necessary condition for minimizing the total averageenergy consumption of the CT mode at the optimal powervalues:

∂ECT (P ∗

S , P∗R)

∂PS≥ ∂E

CT (P ∗

S , P∗R)

∂PR. (83)

For a feasible set of optimal power, the BERCD = p∗ and PS +

PR ≤ Pmaxt constraints must be satisfied.Analytic solution for the optimal power in (83) is intractable

to derive in closed form. However, this can be alternativelyrealized with the aid of numerical optimization techniques,which can determine the optimal power at the source and relaynodes that minimize the average total energy consumption. Tothis end, we employ the MATLAB optimization tool box andits function fmincon in the respective numerical calculations

TABLE IASSUMED SYSTEM PARAMETERS

Fig. 5. Energy consumption per information bit versus S–D distance fordifferent relay locations over uncorrelated Nakagami-1.25 at target BER of10−2 for 4-QAM/QPSK constellation.

for allocating the available power optimally under the givenconstraints. Thus, the derived expressions and offered resultsprovide tools to understand, quantify, and analyze how muchenergy can in general be saved, per successfully communicatedbit in the system, if transmit power allocation and optimiza-tion beyond classical equal power allocation is pursued in thecooperative system on one side and how much energy can besaved against the classical noncooperative (DT) system on theother side. Furthermore, the considered values in this paperare indicative and are selected in the context of demonstratingthe validity of the proposed method. Therefore, the derivedoptimization flow can be readily extended to arbitrary designconstraints for the total network power consumption and targetdestination error rate in the presence of Nakagami-m multipathfading conditions.

VII. NUMERICAL RESULTS AND ANALYSIS

Here, we demonstrate and evaluate the average total energyconsumption of the considered regenerative system assum-ing that the S–D and S–R links are statistically independent,whereas the S–D and R–D paths are spatially correlated. As arealistic example, we assume M -QAM scheme over the S–D,S–R, and R–D links, in the case of CT mode, and over the S–Dlink in the case of only direct communication. For the sake ofsimplicity, it is also assumed that all wireless channels are sub-ject to Nakagami-m multipath fading conditions with ΩS,D =ΩS,R = ΩR,D = 0 dB. The involved PL effects are modeled byan example model of PLi,j

[dB] = 148 + 40 log10(di,j [km]),


TABLE IIOPTIMAL TRANSMIT POWER VALUES FOR SOURCE AND RELAY FOR DIFFERENT RELAY LOCATIONS OVER UNCORRELATED NAKAGAMI-1.25 LINKS

AT TARGET BER OF 10−2 FOR 4-QAM/QPSK MODULATION

which is also used in device-to-device-based communications[70] and thus applies to mobile relaying as well. Furthermore,to simplify the geometry-related calculations, we assume thatall nodes are located along a straight line, which satisfiesthe distance relationship dS,D = dS,R + dR,D. However, it isrecalled here that the PL and distance assumptions are onlyindicative in the context of the considered examples, whereasthe provided analysis and optimization frameworks are validmore generally. In this context, we further assume the fol-lowing system parameters: N0 = −174 dBm/Hz, Ttr = 5 μs,L = 2 kb, and PLO = 50 mW [19], [23]. We also use theconstant circuit power PCTx

= 100 mW and PCRx= 150 mW,

and the maximum transmission power Pmaxt = 1000 mW.The bandwidth of the system is assumed B = 200 kHz, andthe noise figure Nf = 6 dB. Due to the linearity requirementof the M -QAM signals, the value of the drain efficiency isassumed η = 0.35, which is a practical value for class-A andAB RF PAs. The considered system parameters are shown inTable I and are used unless otherwise stated.

We commence by analyzing the minimum energy per in-formation bit required for the DT and CT when the relaynode is taken into account and placed in different locations.The location of the relay node is represented with parameterf = dS,R/dS,D. Fig. 5 shows the total energy consumption perinformation bit as a function of the transmission distance fromsource to destination for 4-QAM/QPSK with fading parameterof m = 1.25, destination target BER of 10−2, and zero spatialcorrelation under the maximum transmit power constraint. Thetransmit power allocation is carried out by the derived OPAscheme resulting to the indicative values in Table II. It isobserved that distance thresholds separate the regions whereDT performs better than CT and vice versa. Furthermore, it isshown that, when the relay is located in the middle, i.e., theS–D distance equals the R–D distance (f = 0.5), it renders thebest energy efficiency among all relay locations. This indicatesthat the configuration is almost symmetric in the S–D and R–Ddistances, which assists the system to operate robustly in trans-mission over severe fading conditions. However, it is shownthat, at relatively small distances (here 0 ≤ dS,D ≤ 170 m),the exact location of the relay does not affect substantially theperformance of the cooperative system as it appears to remainalmost the same in all considered scenarios. This renders therelay positioning and planning rather simple when the relayfalls within this range, whereas it additionally provides insight,e.g., for relay selection algorithms in the case of randomlydistributed relays in emerging relay-based wireless networks.

Fig. 6. Energy consumption per information bit versus S–D distance whenthe relay is located in the middle over spatially correlated Nakagami-1.25 for4-QAM/QPSK constellations with different target BERs.

The corresponding energy efficiency is also analyzed fortarget BERs of 10−2 and 10−3. Fig. 6 shows the average totalenergy consumption per information bit for 4-QAM/QPSK inboth DT and CT in Nakagami-1.25 fading conditions underthe maximum transmit power constraint and the followingspatial correlation scenarios: ρ = {0, 0.5, 0.9}. Moreover, therelay node is located in the middle, and the transmit power isallocated optimally to the source and relay nodes in all cases.It is observed that, for the fixed target BERs of 10−2 and10−3, the direct scheme outperforms the CT only at averageS–D distances below 240 and 150 m, respectively. On thecontrary, for average distances greater than 240 and 150 m, CTbecomes more energy efficient as the transmit power constitutesa significant share of the average total energy consumptioneven under the worst spatial correlation scenario. Furthermore,it is shown that, for the given target BERs, the DT schemesattain maximum transmission distances of 390 and 250 m,respectively, under the given maximum transmission powerconstraint, whereas in both cases, the CT schemes extend tosubstantially longer distances. However, these advantages varyaccording to the level of the involved spatial correlation wherethe improvement in energy efficiency is inversely proportionalto ρ, in both scenarios. The reason is that for every step oftransmission distance, a greater proportion of power is as-signed to the source and relay nodes to overcome performance


TABLE IIIOPTIMAL TRANSMIT POWER VALUES FOR SOURCE AND RELAY FOR RELAY LOCATED IN THE MIDDLE OVER CORRELATED

NAKAGAMI-1.25 LINKS WITH TARGET BER OF 10−2 USING 4-QAM/ QPSK MODULATION

TABLE IVOPTIMAL TRANSMIT POWER VALUES FOR SOURCE WHEN THE RELAY IS LOCATED IN THE MIDDLE OVER CORRELATED

NAKAGAMI-1.25 LINKS WITH TARGET BER OF 10−3 USING 4-QAM/QPSK CONSTELLATIONS

losses incurred by the spatial correlation, within the givenresource constraints. Concrete examples are shown in Tables IIIand IV for some indicative transmission distances and the twotarget BERs; there, the energy savings using the CT scheme,

defined as 1 − EC/E

Dfor ρ = 0.9, ρ = 0.5, and ρ = 0 for

the target BER of 10−2 at dS,D = 390 m, are 65%, 67% and69.3%, respectively, whereas for a target BER of 10−3 atdS,D = 250 m, the energy reduction is 69%, 72%, and 73.8%,respectively. Interestingly, beyond a critical distance of 320 m,even the highly spatially correlated CT mode at the target BERof 10−3 exhibits better energy efficiency than the DT schemewith the target BER of 10−2.

In the same context, Fig. 7 shows the average total energyper information bit for both DT and CT for m = 0.75, whichcorresponds to severe fading conditions. The target BER is setto 10−2 under the given transmit power constraint while ρ ={0, 0.5, 0.9}. The transmit power is again allocated optimallyto the source and relay with the latter positioned in the centerof the network as shown in Table V. It is shown that DT outper-forms CT only when dS,D ≤ 170 m. However, as the distanceincreases beyond this point, the corresponding overall benefitsby CT are significant, even under the worst spatial correlationscenario. Indicatively, at a transmission distance of 280 m,which is the maximum distance that DT can operate with theavailable maximum transmission power, the energy gains byCT are 68%, 72%, and 74% ρ = {0.9, 0.5, 0}, respectively. Inaddition, it is shown that the advantage of the cooperation ismore significant in severe fading conditions.

Fig. 8 shows the average total energy consumption per in-formation bit required for CT and DT as a function of S–Ddistance for fading parameter, m = {0.75, 1.25, 1.75, 2.25} forCT and m = 2.25 for DT. The target BER is set to 10−3, thetransmit power is allocated optimally, the relay is located in themiddle, and a zero spatial correlation is assumed. It is observed

Fig. 7. Energy consumption per information bit versus S–D distance when therelay is located in the middle over spatially correlated Nakagami-0.75 fadingchannels at target BER of 10−2 for 4-QAM/QPSK constellation and differentspatial correlation values.

that the critical distances below which DT outperforms the CTin terms of energy efficiency are 250, 230, and 220 m form = 1.25, m = 1.75, and m = 2.25, respectively. Moreover,the analysis indicates that DT with nonsevere multipath fadingcondition (Nakagami-2.25) can operate only up to 360 m beforeutilizing the maximum transmit power, whereas the CT extendssignificantly even for moderate fading conditions, except for theworst-case scenario (m = 0.75). It is also shown that the gainfrom the cooperation is not uniform as the Nakagami parameterincreases from m = 0.75 to m = 1.25, from m = 1.25 to m =1.75, and then from m = 1.75 to m = 2.25.

Finally, Fig. 9 shows the CG, defined in (61), when therelay is located in the middle of the source and destination.


TABLE VOPTIMAL TRANSMIT POWER VALUES FOR SOURCE AND RELAY WHEN THE RELAY IS LOCATED IN THE MIDDLE OVER SPATIALLY CORRELATED

NAKAGAMI-0.75 LINKS WITH TARGET BER OF 10−2 USING 4-QAM/QPSK MODULATION

Fig. 8. Energy consumption per information bit versus S–D distance when therelay is located in the middle over uncorrelated Nakagami-m fading conditionsat target BER of 10−3 for 4-QAM/QPSK constellation.

Fig. 9. Cooperation gain versus S–D distance when the relay is located in themiddle over spatially correlated Nakagami-1.25 fading environment at targetBER of 10−2 for 4-QAM/QPSK constellations. After S–D distance of 390 m,DT runs out of power and can no longer reach the target BER.

The power allocation is again carried out by using the de-rived OPA scheme for target BER of 10−2 under the maxi-mum transmission power constraint with ρ = {0, 0.5, 0.9} form = 1.25. The transmission distance is limited to 390 m since,beyond this limit, it is only the CT mode that can transmit until

its maximum transmission distance, depending on the spatialcorrelation between S–D and R–D paths. When the CG is belowunity, the DT is actually more energy efficient than CT. Asaforementioned, the reason behind this is that, when CG ≤ 1,which corresponds to relatively small transmission distances,the actual transmit power constitutes only a small fraction ofthe total average power consumption. However, when CG >1, the system benefits significantly from cooperation, and ingeneral, CG increases proportionally as the transmission dis-tance increases for all scenarios of spatial correlation betweenthe S–D and R–D paths. Interestingly, the existence of suchefficiency threshold distance also implies that a hybrid system,where cooperation is only sought and deployed beyond certainminimum distance, can provide the most comprehensive so-lution to the energy efficiency optimization. The analysis andmodeling results and tools provided in this paper directly formthe basis for further development of such schemes in differentcommunication scenarios, which forms an important topic offuture work.

VIII. CONCLUSION

This paper was devoted to the end-to-end SER analysis andthe energy efficiency analysis and optimization of both directand regenerative CTs over Nakagami-m fading conditions inthe presence of spatial correlation. Novel closed-form expres-sions were first derived for the SER of both M -QAM andM -PSK constellations, which were subsequently employed informulating the constrained energy analysis and optimizationproblems under destination BER target and maximum transmitpower constraints considering both transmit energy and theenergy consumed by the transceiver circuits. The correspondingresults indicate that, depending on the severity of multipathfading, spatial correlation between the S–D and R–D paths andthe location of the relay node, the DT can be more energyefficient than CT, but only for rather short transmission dis-tances and up to a certain threshold value. Beyond this value,the system, as expected, benefits substantially from relaying,and the corresponding CG increases proportionally to the trans-mission distance. It is expected that the offered results can beuseful in the design, dimensioning, and deployment of low-costand energy-efficient cooperative communication systems in thefuture, particularly toward the green communications era wherethe requirements and incentives toward energy consumptionoptimization are considered critical.


APPENDIX APROOF OF THEOREM 1

The average SER for the DT can be expressed as

SERDD=FQAM

⎡⎢⎣⎛⎜⎝1+

(PS

PLS,D

)ΩS,DgQAM

N0mS,Dsin2(θ)

⎞⎟⎠

−mS,D⎤⎥⎦ . (84)

Evidently, a closed-form expression for (84) is subject to ana-lytic evaluation of the following integrals:

I(a,m; 0,

π

2

)=

π2∫

0

1(1 + a

sin2(θ)

)m dθ (85)

I(a,m; 0,

π

4

)=

π4∫

0

1(1 + a

sin2(θ)

)m dθ. (86)

By rewriting the indefinite form of the above class of integralsin the following form:

I(a,m) =

∫sin2m(θ)(

sin2(θ) + a)m dθ (87)

and setting u = cos2(θ), one obtains

I(a,m) = −∫

(1 − u)m− 12

2√u(1 − u+ a)m

du. (88)

The given integral can be expressed in closed form in terms ofthe Appell hypergeometric function of the first kind, namely

I(a,m)=− cos(θ)

(1+a)mF1

(12,

12−m,m,

32, cos2(θ),

cos2(θ)

1 + a

).

(89)

Equation (89) reduces to zero when θ = π/2. To this effect, itimmediately follows that

I(a,m, 0,

π

2

)=

1(1 + a)m

F1

(12,

12−m,m,

32, 1,

11 + a

)(90)

which, with the aid of the properties of F1(·) function, can beequivalently expressed as

I(a,m, 0,

π

2

)=

√πΓ(m+ 1

2

)2(1 + a)mΓ(m+ 1)

× 2F1

(12,m,m+ 1,

11 + a

). (91)

In the same context

I(a,m, 0,

π

4

)=

1(1 + a)2

F1

(12,

12−m,m,

32, 1,

11 + a

)

− 1√2(1 + a)2

F1

(12,

12−m,m,

32,

12,

12(1 + a)

)(92)

which can be alternatively expressed as follows:

I(a,m, 0,

π

4

)=

√πΓ(m+ 1

2

)2F1

(12 ,m,m+ 1, 1

1+a

)2(1 + a)mΓ(m+ 1)

−F1

(12 ,

12 −m,m, 32 ,

12 ,

12(1+a)

)√

2(1 + a)m. (93)

Performing the necessary change of variables in (91) and (93)and substituting in (84) yield (9).

APPENDIX BPROOF OF LEMMA 1

The J (a, b,m) integral can be re-written as

J (a, b,m) =

∫sin4m(θ)(

sin4(θ) + a sin2(θ) + b)m dθ. (94)

By setting u = cos2(θ), it follows that

J (a, b,m) = −12

∫(1 − u)2mdu

√u√

1 − u [(1 − u)2 + a(1 − u) + b]m.

(95)

By applying the binomial theorem in [74, Eq. (1.111)], oneobtains

J (a, b,m) = −2m− 1

2∑l=0

(2m− 1

2

l

)(−1)l

2

×∫

ul− 12

[1 − u(a+ 2) + u2 + a+ b]mdu. (96)

The above integral can be expressed in terms of the Appell func-tion of the first kind, yielding (97), shown at the bottom of thepage, which upon performing the necessary countersubstitutionand algebraic manipulations, yields (11).

J (a, b,m) = −2m− 1

2∑l=0

(2m− 1

2

l

) (−1)lul+ 12 F1

(l + 1

2 ,m,m, l+ 32 ,

2u2+a−

√a2−4b

, 2u2+a+

√a2−4b

)(1 + 2l) (a+ b+ (u − 1)2 − au)m

×(

1 − 2u

2 + a−√a2 − 4b

)m(1 − 2u

2 + a−√a2 − 4b

)m

(97)


APPENDIX CPROOF OF LEMMA 2

The K(a, b,m, n) integral can be alternatively rewritten as

K(a, b,m, n) =

∫sin2m+2n(θ)(

sin2(θ) + a)m (

sin2(θ) + b)n dθ (98)

which, by setting u = cos2(θ), can be expressed as

K(a, b,m, n) =

∫(1 − u)m+n− 1

2

2(1 − u+ a)m(1 − u+ b)n√udu. (99)

By applying the binomial theorem in [74, Eq. (1. 111)], itimmediately follows that

K(a, b,m, n) =

m+n− 12∑

l=0

(m+ n− 1

2

l

)(−1)l

2

×∫

ul− 12

(1 − u+ a)m(1 − u+ b)ndu. (100)

The above integral can be expressed in closed form in termsof the Appell function of the first kind. As a result, by makingthe necessary countersubstitution and performing some long butbasic algebraic manipulations, (13) is deduced.

APPENDIX DPROOF OF LEMMA 3

By setting u = sin2(θ), it follows that:∫sin2m(θ)dθ =

∫um− 1

2

2√

1 − udu. (101)

The given integral can be expressed in closed form in terms ofthe Gaussian hypergeometric function, namely∫

um− 12

2√

1 − udu= −

√1 − u 2F1

(12,

12−m,

32, 1+u

). (102)

Therefore, by performing the countersubstitution, (22) isdeduced.

APPENDIX EPROOF OF CONVEXITY OF THE OPTIMIZATION PROBLEM

In the following, we prove the existence of optimal powervalues, which are subsequently employed in minimizing theoverall energy consumption in the considered cooperative com-munication system. Based on (59), the average total powerconsumption can be rewritten as follows:

PCT = (C1 + (1 + α)PS) + (C2 + (1 + α)PR)

(1 − SERS,R

)(103)

where

C1 =PCTx+ 2PCRx

(104)C2 =PCTx

+ PCRx. (105)

The SER i.e., SERS,R can be expressed in closed form with theaid of Theorem 1. This expression is a function of 1+

PS ΩS,R gQAM/PLS,RN0 mS,R; therefore, for proving the

existence of the optimum values, it is sufficient to show that

∂2PCT

∂2P 2R

≥ 0 (106)

∂2PCT

∂2P 2S

≥ 0. (107)

To this end, it is straightforward to show that ∂2PCT /∂

2P 2R = 0.

Likewise, based on the optimal condition in (83) and taking thesecond-order partial derivative with respect to the variables PS

and PR, one obtains

∂2PCT

∂2P 2S

≥ ∂2PCT

∂2PSPR(108)

where

∂2PCT

∂PSPR=

(1 + α1)mS,RΩS,RgQAM

(1 + a1)N0PLS,R

K4 (109)

K4 =4Cπ

I(PSΩS,RgQAM

N0PLS,RmS,R

,mS,R; 0,π

2

)

− 4C2

πI(PSΩS,RgQAM

N0PLS,RmS,R

,mS,R; 0,π

4

)(110)

where it is recalled that C = 1 − 1/√M for the case of

M -QAM, whereas K4 denotes the SER representation withvalues in the range 0 ≤ SER ≤ 1 and with all other constantsbeing positive. Based on this, the second-order partial deriva-tives with respect to PS and PR are always greater than or equalto zero, which satisfies (107). Given the general second-orderconditions in [77], it immediately follows that (103) is convexwith respect to PS and PR and possesses a unique minimumvalue.

ACKNOWLEDGMENT

The authors would like to thank the editor and the anonymousreviewers for their constructive comments that improved thequality of this paper.

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Mulugeta K. Fikadu (S’12) received the M.Sc.degree in communication engineering from AddisAbaba University, Addis Ababa, Ethiopia, in 2007.He is currently working toward the Ph.D. degreewith the Department of Electronics and Communi-cations Engineering, Tampere University of Technol-ogy (TUT), Tampere, Finland.

In 2011, he worked as a Project Researcherwith the Communications and Systems EngineeringGroup, Department of Computer Science, Universityof Vaasa, Vaasa, Finland. He is currently a Project

Researcher with the Department of Electronics and Communications Engi-neering, TUT. His research interests include resource allocation in wirelesscooperative networks, space–time coding, and multiple-input–multiple-outputorthogonal frequency-division multiplexing systems.

Paschalis C. Sofotasios (M’11) was born in Volos,Greece, in 1978. He received the M.Eng. degreefrom the University of Newcastle upon Tyne, Tyneand Wear, U.K., in 2004; the M.Sc. degree fromthe University of Surrey, Guildford, U.K., in 2006;and the Ph.D. degree from the University of Leeds,Leeds, U.K., in 2011.

Until August 2013, he was a Postdoctoral Re-searcher with the University of Leeds and a Vis-iting Researcher with the University of California,Los Angeles, CA, USA; Aristotle University of

Thessaloniki, Thessaloniki, Greece; and Tampere University of Technology,Tampere, Finland. Since Fall 2013, he has been a Postdoctoral Research Fellowwith the Department of Electronics and Communications Engineering, TampereUniversity of Technology, and with the Wireless Communications SystemsGroup, Aristotle University of Thessaloniki. His research interests includefading channel characterization, cognitive radio, cooperative communications,optical wireless communications, and the theory and properties of specialfunctions and statistical distributions.

Dr. Sofotasios received the 2012 and 2015 Exemplary Reviewer Awardsfrom IEEE COMMUNICATION LETTERS and the IEEE TRANSACTIONS ON

COMMUNICATIONS, respectively, and the Best Paper Award at the FifthInternational Conference on Ubiquitous and Future Networks in 2013. HisMaster’s studies were funded by a scholarship from the U.K. Engineering andPhysical Sciences Research Council (UK-EPSRC), and his doctoral studieswere sponsored by UK-EPSRC and Pace plc.

Qimei Cui (M’09) received the B.E. and M.S.degrees in electronic engineering from HunanUniversity, Changsha, China, in 2000 and 2003,respectively, and the Ph.D. degree in telecommu-nications from Beijing University of Posts andTelecommunications, Beijing, China, in 2006. She iscurrently a Professor with the Wireless TechnologyInnovation Institute, Beijing University of Posts andTelecommunications. Her research interests includethe transmission theory and networking technologyfor the next-generation broadband wireless commu-

nications and green communications.Dr. Cui received the Best Paper Award at the 2012 IEEE International

Symposium on Communications and Information Technologies, the Best PaperAward at the 2014 IEEE Wireless Communications and Networking Con-ference, the “Honored Mentioned Demo Award” at the 2009 ACM AnnualInternational Conference on Mobile Computing and Networking, and theYoung Scientist Award in 2014 at the URSI General Assembly and ScientificSymposium.


George K. Karagiannidis (M’96–SM’03–F’14)was born in Pithagorion, Samos Island, Greece. Hereceived the University Diploma (five years) and thePh.D. degree in electrical and computer engineeringfrom the University of Patras, in 1987 and 1999,respectively.

From 2000 to 2004, he was a Senior Researcherwith the Institute for Space Applications and RemoteSensing, National Observatory of Athens, Athens,Greece. Since June 2004, he has been with AristotleUniversity of Thessaloniki, Thessaloniki, Greece,

where he is currently a Professor and the Director of the Digital Telecom-munications Systems and Networks Laboratory. In January 2014, he joinedKhalifa University of Science, Technology and Research, Abu Dhabi, UnitedArab Emirates, as a Professor with the Department of Electrical and ComputerEngineering and as a Coordinator of the ICT Cluster. He is the author orcoauthor of more than 300 technical papers published in scientific journalsand referred international conferences. He is also the author of the Greekedition of a book on telecommunications systems and a coauthor of the bookAdvanced Optical Wireless Communications Systems (Cambridge, 2012). Hisresearch interests include digital communications systems with emphasis oncommunications theory, energy-efficient multiple-input multiple-output andcooperative communications, cognitive radio, smart grid, and optical wirelesscommunications.

Dr. Karagiannidis has been a member of Technical Program Committeesfor several IEEE conferences, such as the IEEE International Conference onCommunications, the IEEE Global Communications Conference, the IEEEVehicular Technology Conference, and the IEEE Wireless Communicationsand Networking Conference. He served as an Editor for fading channels anddiversity for the IEEE TRANSACTIONS ON COMMUNICATIONS, as a SeniorEditor for IEEE COMMUNICATIONS LETTERS, and as an Editor for theEURASIP Journal of Wireless Communications & Networks. He served as theLead Guest Editor for the special issue on optical wireless communications ofthe IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and as aGuest Editor of the special issue on large-scale multiple antenna wireless sys-tems. He has served as Editor-in Chief of IEEE COMMUNICATIONS LETTERS

since 2012. He coreceived the Best Paper Award of the Wireless Communica-tions Symposium at the IEEE International Conference on Communications inJune 2007.

Mikko Valkama (S’00–M’02) was born in Pirkkala,Finland, on November 27, 1975. He received theM.Sc. and Ph.D. degrees (both with honors) in elec-trical engineering from Tampere University of Tech-nology (TUT), Tampere, Finland, in 2000 and 2001,respectively.

In 2003, he was working as a Visiting Researcherwith the Communications Systems and SignalProcessing Institute, San Diego State University,San Diego, CA, USA. He is currently a Full Profes-sor and the Department Vice Head with the Depart-

ment of Electronics and Communications Engineering, TUT. His main researchinterests include communications signal processing, estimation and detectiontechniques, signal processing algorithms for software-defined flexible radios,cognitive radio, full-duplex radio, radio localization, fifth-generation mobilecellular radio, digital transmission techniques such as different variants of multi-carrier modulation methods and orthogonal frequency-division multiplexing,and radio resource management for ad hoc and mobile networks.

Dr. Valkama has been involved in several organizing conferences. He servedas the Publications Chair for the IEEE International Workshop on SignalProcessing Advances in Wireless Communications in 2007. He received theBest Ph.D. Thesis Award from the Finnish Academy of Science and Lettersfor his dissertation entitled “Advanced I/Q signal processing for widebandreceivers: Models and Algorithms” in 2002.

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