RonePage · 2018. 7. 6. · IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 1001 Energy-Efﬁcient and Low Signaling Overhead Cooperative Relaying With Proactive Relay

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016 1001

Energy-Efficient and Low Signaling OverheadCooperative Relaying With Proactive

Relay Subset SelectionBleron Klaiqi, Xiaoli Chu, and Jie Zhang

Abstract—Energy efficient wireless communications haverecently received much attention, due to the ever-increasingenergy consumption of wireless communication systems. In thispaper, we propose a new energy-efficient cooperative relayingscheme that selects a subset of relays before data transmission,through proactive participation of available relays using theirlocal timers. We perform theoretical analysis of energy efficiencyunder maximum transmission power constraint, using practicaldata packet length, and taking account of the overhead for obtain-ing channel state information, relay selection, and cooperativebeamforming. We provide the expression of average energyefficiency for the proposed scheme, and identify the optimalnumber and location of relays that maximize energy efficiencyof the system. A closed-form approximate expression for theoptimal position of relays is derived. We also perform overheadanalysis for the proposed scheme and study the impact of datapacket lengths on energy efficiency. The analytical and simulationresults reveal that the proposed scheme exhibits significantlyhigher energy efficiency as compared to direct transmission, bestrelay selection, all relay selection, and a state-of-the-art existingcooperative relaying scheme. Moreover, the proposed schemereduces the signaling overhead and achieves higher energy savingsfor larger data packets.

Index Terms—Cooperative communications, decode-and-forward relays, energy efficiency, relay selection, overhead.

I. INTRODUCTION

D UE TO THE rapid growth of energy-hungry wirelessmultimedia services, telecom energy consumption isincreasing at an extraordinary rate. Besides negative environ-mental impacts and higher energy bills for operators, it alsoaffects user experience as improvements in battery technolo-gies have not kept up with increasing mobile energy demands.Therefore, how to increase the energy efficiency of wirelesscommunications has gained a lot of attention [2]–[4]. Wirelesscooperative communications can significantly increase systemcapacity and reliability [5]–[8]. It is also widely recognized

Manuscript received July 3, 2015; revised October 16, 2015 and December12, 2015; accepted January 18, 2016. Date of publication January 26, 2016;date of current version March 15, 2016. This work was supported by the ECH2020 Project DECADE (MSCA-RISE-2014-645705). Part of this work waspresented at IEEE WCNC’15, New Orleans, USA, March 2015. The associateeditor coordinating the review of this paper and approving it for publication wasT. A. Tsiftsis.

The authors are with the Department of Electronic and ElectricalEngineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail:[email protected]; [email protected]; [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/TCOMM.2016.2521832

as a promising technique to improve the energy efficiency ofwireless networks [2]–[4].

One of the key challenges in wireless cooperative communi-cations is relay selection. Most existing cooperative commu-nication schemes select either the best relay [9]–[14] or allavailable relays [15]–[18] to cooperatively forward data to thedestination. Moreover, almost all existing cooperative relayingschemes have neglected the energy consumption and signalingoverhead needed for the acquisition of channel state informa-tion (CSI), relay selection, and coordination among selectedrelays. In [19], relay precoders and decoders were jointly opti-mized for amplify-and-forward (AF) cooperative systems undervarious CSI assumptions. In [20], energy efficient schemeswere proposed for AF relays in a single carrier frequencydivision multiple access system. Energy efficient best relayselection schemes for DF relays were studied in [21] and [22].In [23], energy efficiency was investigated for joint physicaland network layer cooperative relaying schemes. Nevertheless,none of these works have considered related signaling overheadin the performance analysis.

It has been shown that cooperative communications lead tohigher signaling overhead compared to direct transmission [24].Therefore, the signaling overhead and associated energy con-sumption have to be taken into account in the design of anenergy-efficient cooperative relaying system. The impact ofoverhead on the spectral efficiency has been investigated fordifferent relaying schemes in [24]. A timer based scheme wasproposed to reduce the overhead for best-relay selection [11],[25], where available relays each start a timer that is inverselyproportional to a relay-specific performance metric and therelay with the first expiring timer is selected. However, in prac-tical systems, such a timer based relay selection may fail incase more than one relays’ timers expire within a window ofvulnerability, leading to packet collisions at the receiver [11].In the timer-based relay selection schemes that maximize theprobability of successful selection [26]–[28], each relay needsto know either the exact number of available relays [26], [27]or the range within which that number lies [28] in order tooptimize their timers. That information has to be signalled toall available relays, hence increasing the signaling overhead ascompared to the best-relay selection in [11], [25]. Furthermore,each relay has to maintain a lookup table that needs to beupdated every time the number of available relays changes.

Selecting more than one relays may offer a higher energyefficiency than selecting only the best relay, but the overhead for

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1002 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016

CSI acquisition and feedback limits the number of relays thatcan be used for energy-efficient cooperative beamforming [29].The energy-efficiency oriented cooperative relaying scheme in[29] performs reactive relay selection, i.e., relay selection isperformed after data transmission from the source. In [30] and[31], the energy efficiency of clustered cooperative beamform-ing (where relays can overhear each other’s transmissions) wasanalysed considering the related overhead. However, none ofthese works have performed signaling overhead analysis orexplored the opportunity of further improving energy efficiencyof cooperative relaying by reducing the signaling overhead.

In addition to signaling overhead, other practical limitationssuch as maximum transmission power, length of data pack-ets, overhearing capabilities of relays, and relay location alsoaffect the energy efficiency of cooperative relaying. The maxi-mum transmission power constraint of practical communicationsystems was not considered in [29]. In [30] and [31], it wasassumed that relays can overhear each other’s transmissionsand very long data packets were used. Nevertheless, in practicalsystems hidden relay nodes that cannot hear and/or be heard byother relay nodes may exist [11] and the length of data packetsis restricted by the channel coherence time. The assumption ofextremely long data packets simplifies the analysis of energyefficiency, but may disguise the actual effect of overhead on theenergy efficiency of cooperative communications. Furthermore,none of the aforementioned works has investigated how thecooperating relays’ location and data packet length would affectthe optimal number of selected relays and the energy efficiencyof cooperative relaying.

In this paper, we provide a comprehensive study of the num-ber and location of relays that should be selected for cooperativecommunications to maximize energy efficiency, taking intoaccount the associated signaling overhead and practical con-straints such as maximum transmission power, practical datapacket lengths and the case that relays cannot overhear eachother’s transmissions. The main contributions of this work canbe summarized as follows:

• We propose a new energy-efficient and low signalingoverhead cooperative relaying scheme that proactivelyselects a subset of available relays before data trans-mission, using timers set at relays. Its performance interms of energy efficiency and required signaling over-head is compared to the cooperative relaying scheme in[29], best relay selection, all relay selection, and directtransmission.

• We perform theoretical analysis of energy efficiency con-sidering practical constraints such as maximum transmis-sion power, hidden relay nodes, and practical data packetlength. Furthermore, we carry out signaling overheadanalysis factoring in the costs for channel estimation,relay selection and cooperative beamforming.

• We study how the optimal number of relays that max-imizes the energy efficiency is affected by the num-ber of correctly decoding relays, relay location, anddata packet length. We identify the number and loca-tion of cooperating relays that maximize energy effi-ciency for given number of correctly decoding relays,source-to-destination distance, and data packet length.

The results can be used as a guideline for developingenergy-efficient transmission strategies that can dynam-ically switch between different communication modes:direct transmission, best-relay selection, and our pro-posed cooperative relaying scheme, depending on whichof them offers the highest energy efficiency for a givenscenario.

• We derive the expression of average energy efficiency forour proposed cooperative relaying scheme, and a closed-form approximate expression of the optimal location ofcooperating relays as a function of the numbers of cor-rectly decoding relays and selected relays that maximizesenergy efficiency. The accuracy of the expressions isevaluated through simulations.

The remainder of the paper is organized as follows. The sys-tem model and the proposed cooperative relaying scheme arepresented in Section II. Section III presents the energy effi-ciency analysis. In Sections IV and V, we derive the optimallocation of cooperating relays and perform signaling overheadanalysis, respectively. The simulation results are presented inSection VI. Finally, conclusions are given in Section VII.

Notations: |D| is the cardinality of the set D. Flooroperation is given by �.�. In order statistics [32], the i thlargest value among M values is denoted by gi :M , i.e.,g1:M ≥ g2:M ≥ . . . ≥ gM :M . E{X} and E{X |Y } denote theexpected value of X and the conditional expectation of X givenY , respectively, where X and Y are random variables.

II. SYSTEM MODEL AND COOPERATIVE RELAYINGSCHEME

We consider a wireless communication system consistingof one source-destination pair and N half-duplex decode-and-forward (DF) relays as shown in Fig. 1. Each node is equippedwith a single omni-directional antenna. The channel powergains between the source and relay i (i = 1, . . . , N ) and fromrelay i to the destination are given by hi and gi , respectively,which are independent and exponentially distributed randomvariables with the mean values, h̄i = (λc/4πd0)2 (dsi/d0)−ξand ḡi = (λc/4πd0)2 (did/d0)−ξ . Thereby, λc denotes the car-rier wavelength, d0 is the reference distance, ξ is the path-lossexponent, and dsi and did are the distances between sourceand relay i and between relay i and destination, respectively.It is assumed that inter-relay distances are much smaller thanthose between the source and relays and from relays to thedestination, i.e., we approximately have h̄i = h̄ and ḡi = ḡ(i = 1, . . . , N ), where h̄ and ḡ denote the mean channel powergains of all links between source and relays and all links fromrelays to destination, respectively. We assume that h̄ and ḡare known at source, relays and destination [29]. Furthermore,channel reciprocity is assumed, i.e., the forward and reverselinks between two nodes are identical and remain constantduring the time period for training, relay selection, and datatransmission [29]–[31]. It will be shown in Section VI that thetime required for training, relay selection and data transmissionby the proposed scheme is much shorter than the channel coher-ence time of low mobility scenarios (with typical pedestrianspeed of 3km/h). For higher mobility scenarios, data packets

KLAIQI et al.: ENERGY-EFFICIENT AND LOW SIGNALING OVERHEAD COOPERATIVE RELAYING 1003

Fig. 1. Proposed cooperative relaying scheme

can be split into smaller packets and more signaling overheadis necessary as channel changes much faster. Communicationsbetween any two nodes have a rate R (bits/symbol) and band-width B (Hz). Perfect channel estimation at each node is alsoassumed. We consider relatively long range transmissions, forwhich case it has been shown in [33] that the circuit energy con-sumption can be neglected as compared to the energy consumedfor signal transmission.

We propose an energy-efficient and low signaling overheadcooperative relaying scheme, which can be divided into threemain phases as illustrated in Fig. 1 and explained as follows.

A. Relay Channel Estimation Phase

Relays have to obtain first-hop CSI, in order to decode datafrom the source. To this end, source broadcasts training symbolsat the minimum power required to support the target rate R withoutage probability ptrout [29], i.e.,

P ST = N0 B1 − 2R

h̄ ln(1 − ptrout ), (1)

where N0 is the power spectral density of additive whiteGaussian noise (AWGN). Similarly, for relays to acquire theCSI on their links to the destination, the destination broadcaststraining symbols1 with the following power,

P DT = N0 B1 − 2R

ḡ ln(1 − ptrout ). (2)

1One way to ensure synchronisation between the source and the destinationis to let the source broadcast training symbol in the first time slot within achannel coherence time. In the second time slot the destination broadcasts itstraining symbol.

B. Relay Selection Phase

Step 1: Since we consider DF relays, only relays j ,1 ≤ j ≤ N , that can correctly decode the received data fromthe source, i.e., with received signal-to-noise ratio (SNR), γs j ,being able to support the rate R with the maximum allowedtransmission power, Pmax , are suitable to forward the receiveddata to the destination and hence become part of the decod-ing set D = {1 ≤ j ≤ N : γs j ≥ 2R − 1}. Furthermore, as therelays have estimated the channels from the source to them inthe relay channel estimation phase, under the assumption ofchannel reciprocity the relays would also know the channelsfrom themselves to the source. In Step 1 of the relay selectionphase, each correctly decoding relay transmits one bit “1” tothe source with channel inversion2, i.e., compensating the chan-nel effect before transmission so that the source can decode thetransmitted bits without CSI3. The source adds the received bitsup to obtain the number of correctly decoding relays (M), andthen based on M determines the optimal number of relays to beselected (K ) that maximizes energy efficiency (see Section VI).The overall relay transmission power for signaling the size ofthe decoding relay set, M = |D|, to the source is given by

PM = N0 B(2R − 1)M∑

j=1

1

h j. (3)

Each correctly decoding relay starts a timer once they havetransmitted the one bit “1” as follows

2Only relays that can decode the received data successfully (i.e., can sup-port rate R with Pmax ) perform channel inversion. This is known as truncatedchannel inversion that leads to finite average transmission power [34].

3Channel inversion at relays guarantees that the source can correctly decodeeach one-bit “1”.


t j :M =⌊

λ̃

g j :M�g

⌋�g, j ∈ D, g1:M ≥ g2:M ≥ . . . ≥ gM :M ,

(4)

where λ̃ = ḡλ, λ is a predefined constant parameter, and �g isa guard interval that depends on the processing delay, the prop-agation delay, and the transmitted symbol duration [11]. For theproposed scheme we set�g = NT TS , where NT and TS are thenumber of symbols used for training and the symbol duration,respectively. The processing delay and the propagation delayare negligible compared to the symbol duration. The correctlydecoding relays are ranked in descending order of their channelstrengths to the destination so that the timer of the relay withthe strongest channel in the second hop expires first, followedby the timer of the relay with the second strongest second-hopchannel and so on4.

Proposition 1: The time required for selecting the K bestrelays is obtained as

Tsel,K = �g M!(K − 1)!

nmax∑n=1

M−K∑i=0

(−1)i n(i + K )(M − i − K )!i!(

e−i+Kn+1 θ − e− i+Kn θ

), (5)

where θ = λ�g

, nmax =⌊

Tmax�g

⌋, and Tmax is the maximum

allowable relay selection time.

Proof: The proof is given in Appendix A. �Step 2: After the expiration of its timer, a relay transmits NT

training symbols with transmission power of

P RT = max{

P ST , PD

T

}. (6)

In this way, by exploiting the broadcast nature of the wire-less channel, the source and the destination can use the sametraining symbols to perform channel estimation and obtain thecorresponding CSI. The source will use the estimated first-hopCSI to adapt its data transmission power to the minimum levelrequired for reaching the selected relays (see Section II-C). Dueto the use of discrete relay timers in (4), collisions betweenrelay transmissions may occur if the timers of two or morerelays expire at the same time.

Proposition 2: The collision probabilities among the K bestrelays for K = 1 and K > 1 are given by

pcoll,K=1,nmax

= 1 − Mnmax∑n=0

(e−

θn+1 − e− θn

) (1 − e− θn+1

)M−1, (7)

pcoll,K>1,nmax

= 1 − M!(M − K )!(K − 1)!

nmax∑n=K−1

e−(K−1)θ

n

(e−

θn+1 − e− θn

)(

1 − e− θn+1)M−K (

1 − I{K≥3}(K )pcoll,K−2,n), (8)

4Note that as relays with the strongest second-hop channels among the cor-rectly decoding relays (i.e., relays with the first-hop link strength that satisfythe rate R with Pmax ) are selected, link strengths of both hops are consideredin the relay selection.

where the indicator function IA(x) = 1 if x ∈ A, 0 otherwise.Proof: The proof is given in Appendix B. �

Step 3: Once the source has received training symbols fromthe first K relays, using maximum transmission power Pmaxit informs the other M-K relays via a single bit “0”5to stoptheir timers, not to transmit training symbols, and not to partici-pate in the immediate data transmission phase6. We assume thatthe relay with timer tK+1:M receives the single bit “0” notifica-tion before it starts transmitting training symbols, given that thepropagation delay is negligible compared to the guard interval.

Prior to data transmission, the destination broadcasts the sumof the K estimated second-hop channel power gains, which willbe used by the selected K relays for cooperative beamform-ing with the optimal transmission power, using the followingtransmission power

PF B = N0 B(2R − 1)

gK :M, (9)

where the weakest second-hop channel power gain gK :Mamong the K selected relays is used, because this broadcastinformation has to reach all the K selected relays.

C. Data Transmission Phase

So far, the source and the K selected relays have obtainedall necessary information to perform data transmission in anenergy-efficient manner. In the first hop, the source trans-mits data with the minimum transmission power required forreaching the K selected relays, i.e.,

PAD = N0 B(2R − 1)

min{h1, . . . , hK } . (10)

To some extent, this may also prevent the other M-K relaysfrom unnecessarily decoding and buffering data packets.

In the second hop, the K selected DF relays perform coop-erative beamforming to transmit the decoded source data to thedestination, with the overall transmission power given by

PC B =K∑

i=1PiC B , (11)

where PiC B is the optimal transmission power at relay i and iscalculated as follows

PiC B = N0 B(2R − 1)⎛⎝ 1√

gi :M

K∑j=1

g j :M

⎞⎠

−2. (12)

5Note that only the M correctly decoding relays start their timers. As thechannel remains the same within channel coherence time, the M-K relaysshould be able to correctly decode the single bit notification from the sourceas long as the source transmits with maximum allowed power Pmax . Moreover,since the source notification signal serves only one purpose, namely stoppingthe M-K relay timers, it would be sufficient to just detect the arrival of such asignal (e.g., through energy detection), rather than having to correctly decodethe one bit message. We have used a “0” message just to indicate that a singlebit message would be sufficient to request the M-K relays to stop their timers.

6After the M correctly decoding relays each transmit a “1” message to thesource, they switch to idle mode waiting for expiration of their timers and arealso able to receive and process signals. Therefore, when the source transmitsthe one-bit “0” message, the not-selected M-K relays still have their timersrunning and are in idle mode.


The optimal beamforming weights have been calculated utiliz-ing the Langrangian multiplier technique [35].

The minimum channel coherence time required for the pro-posed cooperative relaying scheme is given by

Tmin−coh= ((K + 2)NT + (M + 1)/R + NF B + 2ND) TS + Tsel,K .

(13)

where NF B is the number of symbols used for destination feed-back, and ND is the number of symbols per data packet. Thefirst part in the summation represents the time needed for train-ing. The second part is the total time consumed for signalingthe size of decoding set M to the source and for invalidatingrelay timers of not selected relays. The third and fourth partsembody the time required for destination feeding back the sumof second-hop channel power gains to the K selected relays andthe time needed for cooperative data transmission, respectively.The last part is the time for selecting K relays.

III. ANALYSIS OF AVERAGE ENERGY EFFICIENCY

In this section, the average energy efficiency under maximumtransmit power constraint, Pmax , is analysed for both coop-erative communications and direct transmission, facilitating aquantitative comparison between them. Energy efficiency (inbits/Joule) is defined as the ratio of the number of successfullytransmitted data bits to the corresponding energy consumption.

A. Cooperative Communications

Without loss of generality, we assume that M ≥ 2 relaysdecode correctly the data transmitted from the source and that{hi }Mi=1 and {gi }Mi=1 are independent and identically distributed(i.i.d), i.e., h̄i = h̄ and ḡi = ḡ (i = 1, . . . ,M). The averageenergy efficiency of the proposed cooperative relaying schemeis given by

E ECC (K ,M, ψ) = (1 − pCCout )(1 − pcoll,K ,nmax )RNDE

{1

EO(K ,M, ψ)+ ED(K ,M, ψ)}

≈ (1 − pCCout )(1 − pcoll,K ,nmax )RND

E{EO(K ,M, ψ)} + E{ED(K ,M, ψ)} ,(14)

where the third line is obtained using the first-order Taylorapproximation, ψ is the location of the K selected cooperatingrelays, pCCout is the outage probability of cooperative commu-nications, EO(.) denotes the energy consumption caused bysignaling overhead, and ED(.) is the energy consumed for datatransmission.

Proposition 3: The outage probability of cooperative com-munications is given by

pCCout =M!

(K − 1)!M−K∑

j=0(−1) j

(1 − e− j+Kḡ μ

)( j + K )(M − K − j)! j! , (15)

where μ = N0 B(2R−1)Pmax .

Proof: The proof is given in Appendix C. �In (14), EO(.) is the total energy consumed for training

ET (.), for destination feedback EF B(.), for relays signaling Mto source EM (.), and for source telling non-selected relays toinvalidate their timers EI N V .

Proposition 4: The average energy consumption for thesignaling overhead is given by

E{EO(K ,M, ψ)} = ET (K ,M, ψ)+ E{EM (M, ψ)}+ I{2≤K≤M}(K )E{EF B(K ,M, ψ)} + EI N V , (16)

where

ET (K ,M, ψ) = NT N0 BTS(

1 − 2Rln(1 − ptrout )

)(

1

h̄+ 1

ḡ+ K max

(1

h̄,

1

ḡ

)), (17)

EI N V = NI N V TS Pmax , (18)E{EF B(K ,M, ψ)} = −NF B TS N0 B(2R − 1) M!

ḡ(K − 1)!⎛⎝M−K∑

j=0

(−1) j(M − K − j)! j! Ei

(− j + K

ḡμ

)⎞⎠(1 − M!(K − 1)!

M−K∑j=0

(−1) j(M − K − j)! j!( j + K )

(1 − e− j+Kḡ μ

)⎞⎠−1, (19)

E{EM (M, ψ)} = −M NM TS N0 B(2R − 1)

h̄eμ

h̄ Ei

(−μ

h̄

),

(20)

in which NI N V and NM are the numbers of symbols usedfor invalidating not-selected relays’ timers and relays signalingM to source, respectively, and Ei is the exponential integralfunction, defined as Ei(x) = ∫ x−∞ ett dt [36].

Proof: The proof is given in Appendix D. �The energy consumption for data transmission, ED(.), com-

prises the energy consumed in the first hop E ID(.) and that inthe second hop E I ID (.).

Proposition 5: The average energy consumption for the datatransmission is given by

E{ED(K ,M, ψ)}= E{E ID(K ,M, ψ)} + I{K=1}(K )E{E I ID (K = 1,M, ψ)}

+ I{2≤K≤M}(K )E{E I ID (K > 1,M, ψ)}, (21)where

E{E ID(K ,M, ψ)}

= −K NDTS N0 B(2R − 1)

h̄eμ

h̄K Ei

(−μ

h̄K

), (22)

E{E I ID (K = 1,M, ψ)} = −NDTS N0 B(2R − 1)M!

ḡ⎛⎝M−1∑

j=0

(−1) j(M − j − 1)! j! Ei

(− j + 1

ḡμ

)⎞⎠


⎛⎝1 − M! M−1∑

j=0

(−1) j(M − j − 1)!( j + 1)!

(1 − e− j+1ḡ μ

)⎞⎠−1,

(23)

E{E I ID (K > 1,M, ψ)} =NDTS N0 B(2R − 1)

ḡ

(MK

)⎛⎝(

K − 1, K μḡ)

(K − 1)! +M−K∑i=1

(−1)i+K−1(

M − Ki

)(

K

i

)K−1 (Ei

(−K μ

ḡ

)− Ei(

− (K + i) μḡ

)

−K−2∑j=1

(− iK ) jj!

(j, K

μ

ḡ

)⎞⎠⎞⎠(1 − M!

(K − 1)!M−K∑

j=0

(−1) j( j + K )(M − K − j)! j!

(1 − e− j+Kḡ μ

)⎞⎠−1, (24)

with (α, x) = ∫∞x tα−1e−t dt being the upper incompletegamma function [36].

Proof: The proof is given in Appendix E. �Lemma 1: The average energy efficiency in (14) can be

upper bounded as follows

E ECC (K ,M, ψ) ≤ (1 − pCCout )(1 − pcoll,K ,nmax )RND

EL BO (K ,M, ψ)+ E L BD (K ,M, ψ)

, (25)

where EL BO (K ,M, ψ) and E

L BD (K ,M, ψ) denote the lower

bound of average energy consumption for signaling overheadand for data transmission, respectively, and can be calculatedas follows

EL BO (K ,M, ψ) = ET (K ,M, ψ)+ E L BM (M, ψ)+ EI N V

+ I{2≤K≤M}(K )E L BF B(K ,M, ψ), (26)E

L BD (K ,M, ψ)

= E I,L BD (K ,M, ψ)+ I{K=1}(K )E I I,L BD (K = 1,M, ψ)+ I{2≤K≤M}(K )E I I,L BD (K > 1,M, ψ), (27)

where

EL BM (M, ψ) =

NM MTS N0 B(2R − 1)μ+ h̄ , (28)

EL BF B(K ,M, ψ) = NF B TS N0 B(2R − 1)

(K − 1)!M!

⎛⎝1−

M!

(K − 1)!M−K∑

j=0

(−1) j(M − K − j)! j!( j + K )

(1 − e− j+Kḡ μ

)⎞⎠⎛⎝M−K∑

j=0

(−1) j(M − K − j)! j!( j + K )e

− j+Kḡ μ(μ+ ḡ

j + K)⎞⎠

−1,

(29)

EI,L BD (K ,M, ψ) = NDTS N0 B(2R − 1)eK

μ

h̄

(K

h̄ + μK),

(30)

EI I,L BD (K = 1,M, ψ) =

NDTS N0 B(2R − 1)M!⎛

⎝1 − M! M−1∑j=0

(−1) j(M − j − 1)!( j + 1)!

(1 − e− j+1ḡ μ

)⎞⎠⎛⎝M−1∑

j=0

(−1) j(M − j − 1)!( j + 1)!e

− j+1ḡ μ(μ+ ḡ

j + 1)⎞⎠

−1,

(31)

EI I,L BD (K > 1,M, ψ) = NDTS N0 B(2R − 1)⎛⎝ K∑

i=1

⎛⎝M−i∑

j=0

(−1) j(M − i − j)! j!( j + i)

(1 − e− j+iḡ μ

)⎞⎠⎛⎝M−i∑

j=0

(−1) j(M − i − j)! j!( j + i)e

− j+iḡ μ(μ+ ḡ

j + i)⎞⎠

−1⎞⎟⎠−1

.

(32)

Proof: The proof is given in Appendix F. �

B. Direct Transmission

For energy efficiency analysis, we consider two transmissionstrategies for the direct communication between the source andthe destination.

In the first strategy, source transmits training symbols at theminimum power required to satisfy the target R with outageprobability ptrout , i.e.,

P SDT = N0 B1 − 2R

h̄0 ln(1 − ptrout ), h̄0 =

(λc

4πd0

)2 (dsdd0

)−ξ,

(33)

where h̄0 and dsd denote the mean channel power gain andthe distance of the direct link from source to destination,respectively. Subsequently, data is transmitted using the maxi-mum allowed transmission power, Pmax . The resulting averageenergy efficiency is given by

E EM AXDT = (1 − pDTout )

RNDTS

(NT N0 B

1 − 2Rh̄0 ln(1 − ptrout )

+ND Pmax)−1

, pDTout = 1 − e− μ

h̄0 . (34)

In the second strategy, during the channel estimation phase,source sends training symbols with transmission power as in(33) and destination estimates the channel gain. Thereafter, thedestination feedbacks CSI to the source. This enables the sourceto transmit data with the minimum power required to meet thetarget rate R. The corresponding average energy efficiency andits upper bound are given, respectively, by

E EAD PDT ≈ (1 − pDTout )

(RND

N0 BTS

)(h̄0

1 − 2R)

(NT

ln(1 − ptrout )+ (ND + NF B)e

μ

h̄0 Ei

(− μ

h̄0

))−1, (35)


E EAD P,U BDT = (1 − pDTout )

(RND

N0 BTS(1 − 2R))

(NT

h̄0 ln(1 − ptrout )− ND + NF B

μ+ h̄0

)−1. (36)

IV. OPTIMAL LOCATION OF RELAYS

In this section, we derive the optimal location of cooperatingrelays that maximizes the average energy efficiency. Withoutloss of generality, we assume that source is located at the ori-gin (0, 0), destination is located at (dsd , 0), and the selectedrelays are relatively close to one another so that their distancesto the source are approximately the same. Furthermore, it isassumed that diversity gains offered by relays are sufficientlyhigh to keep the outage probability very low, i.e., pCCout ≈ 0.In this case, the expressions in (28)-(32) can be simplifiedby replacing conditional expectations with unconditional ones.Since maximizing the average energy efficiency while main-taining the target rate R, is equivalent to minimizing the lowerbound of average energy consumption, the optimal location ofcooperating relays is given by

ψopt (K ,M) ≈ argminψ

(E

L BO (K ,M, ψ)+ E L BD (K ,M, ψ)

),

(37)

where ψ denotes the distance from the source along the directline connecting source and destination.

Proposition 6: The optimal position of cooperating relays isapproximately given by

ψopt (K ,M) ≈(

1 +(α(K ,M)

β(K ,M)

) 1ξ−1)−1

dsd , (38)

where

α(K ,M)

β(K ,M)> 1, ξ > 1,

α(K ,M) = MR

+ K ND − NTln(1 − ptrout )

,

β(K ,M) = I{K=1}(K )ND⎛⎝ M∑

j=1

1

j

⎞⎠

−1+ I{2≤K≤M}(K )

⎛⎜⎝ND

K

⎛⎝1 + M∑

j=K+1

1

j

⎞⎠

−1+ NF B

⎛⎝ M∑

j=K

1

j

⎞⎠

−1⎞⎟⎠− (1 + K ) NT

ln(1 − ptrout ).

Proof: The proof is given in Appendix G. �From (38) we can see that the optimal source-to-relay dis-

tance increases with dsd and the path-loss exponent ξ . Theaccuracy of (38) will be evaluated through simulation inSection VI.

V. OVERHEAD ANALYSIS

The signaling overhead for the cooperative relaying systemin Fig. 1 can be calculated as

�pro = (K + 2)NT + M + 1R

+ I{2≤K≤M}(K )NF B . (39)

It consists of three main parts. The first part is the overheadfor transmitting NT training symbols from the source to relaysand from the destination to relays as well as the overhead forbroadcasting NT training symbols from the K selected relays.The second part represents the overhead for the M correctlydecoding relays to inform the source of the number of correctlydecoding relays (each of them uses one bit message, i.e., intotal M /R symbols, where R is the data rate in bits/symbol)and for the source to stop the M-K relay timers via a single-bit message, i.e., 1/R symbols. The last part is the overheadneeded for the destination to use NF B symbols to feedback thesum of K best second-hop channel power gains to the selectedrelays. Since the feedback from the destination is only requiredfor cooperative beamforming (K ≥ 2), the feedback overheadis multiplied by the indicator function I{2≤K≤M}(K ).

The signaling overhead for the cooperative relaying schemein [29], which is referred to as the reference scheme hereafter,can be calculated as

�re f = M NT +(Kref + I{2≤Kre f ≤M}(Kref )

)NF B, (40)

where the number of selected relays, Kref , is given by

Kref =⎧⎨⎩

1, M ≤ 22, 3 ≤ M ≤ 63, 7 ≤ M ≤ 15

.

Compared to the reference scheme, the signaling overheadreduction achieved by the proposed scheme is given by

�red =(�re f −�pro

�re f

)100%

=((M − K − 2) NT − M + 1

R

+ (Kref + I{2≤Kref ≤M}(Kref )− I{2≤K≤M}(K )) NF B)(M NT +

(Kref + I{2≤Kre f ≤M}(Kref )

)NF B)−1 100%. (41)

When the number of correctly decoding relays approachesinfinity, the overhead reduction converges to

limM→∞�red =

(1 − 1

RNT

)100%, (42)

which depends only on the data rate (R) and the number oftraining symbols (NT ) used for channel estimation. IncreasingR and/or NT for both schemes would lead to more significantoverhead reduction by the proposed scheme.

VI. SIMULATION RESULTS

The performance of the proposed cooperative relayingscheme and the accuracy of the analytical results are evaluated


TABLE ISYSTEM PARAMETERS

Fig. 2. Collision probability and relay selection time versus θ .

through simulation. In the simulation, source and destinationare located at (0, 0) and (dsd , 0), respectively. The M(> 1)relays that can correctly decode messages from the source, aresituated close to one another with approximately the same dis-tance ψ from the source. System parameters as listed in Table Iconform to 3GPP LTE-A [37]. For illustration purposes we con-sider a single subcarrier with 16-QAM modulation, i.e., R = 4.During training, one OFDM symbol (NT = 1) is transmitted atthe target rate R with outage probability ptrout = 0.12. The desti-nation utilizes two OFDM symbols (NF B = 2) to feedback thesum of second-hop channel power gains to the selected relays.

Fig. 2 shows both the analytically calculated and simu-lated collision probability (pcoll,K ,nmax ) and relay selectiontime (Tsel,K ) versus θ(= λ/�g) for two different numbersof selected relays and M = 10. The results calculated using(5), (7), and (8) are in close agreement with those obtainedfrom simulation. We can see that with increasing θ , the col-lision probability decreases, whereas the relay selection timeincreases. There exists a trade-off between pcoll,K ,nmax andTsel,K that is controlled by λ for given �g(= NT TS). For agiven θ , selecting one more relay leads to a higher collisionprobability and a higher relay selection time. In the following,we set θ = 70 as it provides a good trade-off between colli-sion probability and relay selection time, both of which willbe included in the evaluation of energy efficiency and spec-tral efficiency. With the parameter values in Table I and forM = 10, it can be calculated using (5) and (13) that the mini-mum channel coherence time required for the proposed schemeis 22ms, which is significantly shorter than the channel coher-ence time Tcoh = 76.1ms for low mobility scenario (with speedof 3km/h).

Fig. 3 plots the simulation results of average energy effi-ciency forψ = 50m over different values of M and K . It can be

Fig. 3. Average energy efficiency versus the number of selected relays (K ), fordifferent numbers of correctly decoding relays (M) and ψ = 50m.

Fig. 4. Optimal number of cooperating relays versus their location for differentvalues of M .

seen that the maximum average energy efficiency is achieved byselecting the K = 2 best relays. We also observe that deploy-ing all decoding relays, i.e., K = M (M > 2), for cooperativebeamforming exhibits the lowest energy efficiency, becausethe energy consumption for signaling overhead outweighs theenergy savings from cooperative beamforming. For a given K , alarger number of correctly decoding relays (M) leads to a higherenergy efficiency due to increased diversity gain.

Fig. 4 plots the optimal number of selected relays thatmaximizes the average energy efficiency obtained through sim-ulations versus the source-to-relay distance. We can see that forM = 3 and M = 5 (the two curves overlap with each other),selecting the best two relays is optimal for source-to-relaydistances up to 150m, beyond which the best relay selection(K = 1) maximizes the energy efficiency. This is because forlong source-to-relay distances, the overhead energy consump-tion required to select one additional relay plus the extra sourcetransmission power required to reach the additional relay in thefirst hop outweighs the energy savings from cooperative beam-forming in the second hop. In the case of M = 10, the thresholdsource-to-relay distance reduces to 130m due to increased relaytransmission collision probability. The results may change withdifferent sizes of data packets (see Fig. 9 and Fig. 10).

In Fig. 5, we evaluate the accuracy of the approximate opti-mal location of cooperative relays in (38) by comparing it withsimulation results. There is a good match between the theo-retically calculated optimal location of relay(s) and that foundthrough simulation for both the best relay selection and the


Fig. 5. Approximate optimal location of cooperative relays versus M .

Fig. 6. Overhead reduction of the proposed cooperative relaying scheme overthe reference scheme [29] for different numbers of training symbols NT .

proposed scheme. Conforming to the observation in Fig. 4,the optimal location of relays is closer to the source for theproposed scheme than for the best relay selection. For bothschemes, as M increases (e.g., due to better first-hop channelconditions), the optimal location of relays gets only slightlycloser to the source. This indicates that the optimal locationof relay(s) can be predicted using (38) for both the proposedcooperative relaying scheme and the best relay selection, andthe prediction does not need to be updated frequently.

In Fig. 6, the overhead reduction offered by the proposedscheme as compared to the reference scheme [29] calculatedusing (41) is depicted versus M for three different numbersof training symbols (NT ). The reduction in signaling overheadincreases with increasing M for all considered NT , due to thestronger dependence on M of the reference scheme than theproposed scheme, as shown in (40). For M < 6, a smaller NTleads to a higher reduction in signaling overhead; while forM > 8, a larger NT leads to a higher overhead reduction. Asit can be seen from (41), for small M , e.g., M = 3, M-K -2< 0,and increasing NT decreases the overhead reduction. Accordingto (42) for large M , the signaling overhead reduction increaseswith NT for given R. Significant increase of �red occurs fromM = 6 to M = 7 because the reference scheme increases thenumber of selected relays from 2 to 3 as M increases from 6 to7 (see (40)).

Table II shows the signaling overhead reduction achievedby the proposed scheme with respect to the reference scheme[29] for different modulation orders and numbers of training

TABLE IISIGNALING OVERHEAD REDUCTION �red (%) COMPARED TO [29] FOR

DIFFERENT MODULATION ORDERS, NT = 1, 2, AND M = 10

symbols. The results in the table conform to (42), i.e.,increasing modulation order and/or number of training sym-bols leads to a higher reduction in signaling overhead. Forinstance, increasing modulation order from 4-QAM to 64-QAMfor NT = 1, increases the overhead reduction from 36.11% to56.48%.

In Fig. 7, the simulated average energy efficiency of the pro-posed scheme is compared to that of the reference scheme [29]for three different locations of cooperative relays. In [29], thesource transmits data packets with a fixed transmission power.The M correctly decoding relays each transmit a training sym-bol to the destination, which performs channel estimation andselects the Kref relays (as shown in Section V) with the highestsecond-hop channel power gains. The destination feeds backfirst the corresponding channel power gain to each selectedrelay and then the sum of the Kref channel power gains toall of them. We can see that the performance of the referencescheme with fixed source transmit power (Pmax ) is nearly inde-pendent of the relay location and the value of M . For a morecomprehensive comparison, we assume that the source knowsthe minimum power required to reach all M correctly decodingrelays, so that the reference scheme is also able to use adaptivesource transmission power. We can see that the energy effi-ciency of the reference scheme is significantly improved dueto the use of adaptive source transmission power. For M > 2,the proposed scheme offers higher energy efficiency than thereference scheme (with adaptive source transmit power) for allthree cases, and the gap between the two schemes increaseswith M for each given relay location. This is mainly becauseof two reasons. First, the proposed scheme enables the sourceto adapt its transmission power to reach only the K selectedrelays (K ≤ M), while the reference scheme requires a sourcetransmission power that can reach all the M correctly decodingrelays. Second, the energy consumption for signaling overheadis reduced in the proposed scheme. In contrary to the referencescheme that loses energy efficiency with increasing M for largevalues of M , the proposed scheme is able to maintain a stableenergy efficiency at large values of M , indicating a much betterscalability.

Comparison of average energy efficiency between the pro-posed cooperative relaying scheme, best relay selection, anddirect transmission using adaptive transmission power isdepicted in Fig. 8, where the position of cooperating relays is setat ψ = dsd /10 for different dsd . Fig. 8 presents both simulationresults and theoretical results calculated using (25) and (36) forcooperative and direct transmissions, respectively. We can seethat the theoretical results closely match the simulation results.Direct transmission is more energy efficient than the proposedscheme and best relay selection for dsd < 300 m, as it requiresless signaling overhead. As dsd increases, the energy efficiency


Fig. 7. Average energy efficiency for the proposed cooperative relaying schemeand the reference scheme [29] for three different locations of cooperatingrelays.

Fig. 8. Energy efficiency comparison between direct transmission and cooper-ative communications for ψ = dsd/10.

of cooperative communications decreases much slower thandirect transmission, leading to a higher energy efficiency fordsd ≥ 300 m. This is because cooperative communicationshave lower outage probability and can use lower transmissionpower than direct transmission for long source-to-destinationdistances, due to the cooperative gains. The proposed schemeachieves higher energy efficiency than the best relay selection,because deploying one more relay offers higher cooperativegains.

Fig. 9 shows the simulation results of the average energyefficiency over different data packet sizes ND for the pro-posed scheme and best relay selection. In all considered cases,increasing data packet size leads to higher energy efficiency, asdata transmission becomes the dominant part in overall energyconsumption and the impact of overhead diminishes. As shownin Fig. 3, for ND = 140 OFDM symbols, the optimal number ofrelays for cooperative beamforming is limited to K = 2 by therelated signaling overhead. For ND > 200 OFDM symbols, theoptimal number of relays selected for cooperative beamform-ing increases to 3, because the impact of overhead on energyefficiency is mitigated by long data packets. The increase of Kleads to a higher cooperative beamforming gain, which furtherimproves the energy efficiency.

Fig. 9. Average energy efficiency versus data packet length for different M andK , and ψ = 50m.

Fig. 10. Optimal number of cooperating relays versus data packet size fordifferent values of M and ψ = 50m.

Fig. 10 plots the optimal number of selected relays (K ) thatmaximizes the average energy efficiency obtained through sim-ulation versus data packet size (ND) for three different valuesof M . Due to the same reason as explained for Fig. 9, the opti-mal number of cooperating relays increases with the data packetlength for each given M . Moreover, for a large data packetsize (e.g., ND > 200 OFDM symbols), K also increases withM , because increasing M offers a higher diversity gain, thusallowing the recruiting of more relays.

In the following, we include a comparison of spectral effi-ciency (SE) to make the performance evaluation more compre-hensive. The SE of direct transmission is given by [24]

SEDT = (1 − pDTout )R

B

(Tcoh − T DTO

Tcoh

), (43)

where pDTout and Tcoh are the outage probability of direct trans-mission and channel coherence time, respectively, and T DTO =(NT + NF B) TS denotes the overhead transmission time (i.e.,source training and destination feedback for CSI) of directtransmission. With the half-duplex DF relays, the SE of theproposed cooperative relaying scheme can be calculated as

SECC

= (1 − pCCout )(1 − pcoll,K ,nmax )R

2B

(Tcoh − Tsel,K − T CCO

Tcoh

),

(44)


Fig. 11. Normalized spectral efficiency comparison between best relay selec-tion, the proposed scheme and the reference scheme for M = 10, 16-QAM and64-QAM.

where the factor 1/2 results from the two-hop half-duplex transmission, pCCout is given in (15), and

T CCO =((K + 2)NT + NF B + M+1R

)TS is the overhead

transmission time of cooperative relaying.Fig. 11 shows the SE of the proposed scheme, the best

relay selection, and the reference scheme [29] normalized withrespect to that of direct transmission (i.e, SECC/SEDT ) versusdsd for M = 10 and two different modulation orders. The nor-malized SE of the best relay selection and the proposed schemein the ideal case without any relay transmission collision ordelay due to relay selection (i.e., pcoll,K ,nmax = 0, Tsel,K = 0)is also plotted. In the ideal case, the proposed scheme and thebest relay selection achieve the same SE, which is the high-est SE that can be expected for cooperative communicationsin theory, because there is no loss of SE due to relay trans-mission collisions or relay selection time. We can see that withrelay transmission collisions and relay selection time taken intoaccount, the SE of the proposed scheme is reasonably close tothat of the ideal case. This shows that the loss of SE causedby the proactive relay subset selection in the proposed schemeis reasonably low. In most cases considered in Fig. 11, thenormalized SE is less than one, i.e., cooperative communica-tions are less spectral efficient than direct transmission. Thisis mainly due to the factor 1/2 in (44) of half-duplex relay-ing. For each considered modulation, the normalized SE of theproposed scheme and the best relay selection is much higherand increases much faster with dsd than that of the referencescheme. This indicates that while the SE of direct transmis-sion decreases with dsd , the proposed scheme and the best relayselection achieve much higher SE than the reference schemeat long source-to-destination distances. The reason is that thereference scheme requires the relays to transmit on orthogo-nal subcarriers in order to ensure the orthogonality betweenrelay transmissions during the training phase [29], while in theproposed scheme relays contend with each other for the samesubcarrier. For 64-QAM and dsd > 860m, the proposed schemeis more spectral efficient than direct transmission. The proposedscheme exhibits slightly lower SE than the best relay selec-tion owing to the higher collision probability and longer relayselection time for deploying more relays.

VII. CONCLUSIONS

In this paper, we have proposed an energy-efficient and lowsignaling overhead cooperative relaying scheme that selects asubset of DF relays for cooperative beamforming in a proactivemanner by relays using their local timers. We have carried outtheoretical analysis of energy efficiency under maximum trans-mission power constraint, with practical data packet length, andconsidering the overhead for obtaining CSI, relay selection, andcooperative beamforming. The accuracy of our derived expres-sion of average energy efficiency and closed-form approximateexpression for the optimal location of relays that maximizesenergy efficiency has been verified by simulation results. Theanalytical and simulation results have shown that the proposedscheme not only reduces the signaling overhead significantly,but also exhibits higher energy efficiency compared to theexisting energy-efficient cooperative relaying scheme [29], bestrelay selection, all relay selection, and direct transmission,especially for relays located close to the source. We have alsodemonstrated that energy efficiency of cooperative relayingincreases with data packet size under the constraint of chan-nel coherence time. Our results can be used as a guidelinefor developing dynamic energy-efficient cooperative transmis-sion strategies that can adapt to different channel and systemconditions.

APPENDIX APROOF OF PROPOSITION 1

Probability density function (pdf) for the K -th best channelpower gain gK :M is given by [32]

pgK :M (x) =M!

ḡ(K − 1)!M−K∑i=0

(−1)i(M − K − i)!i!e

− i+Kḡ x . (45)

It follows then for the average relay selection time

Tsel,K = E{tK :M } = �gE{⌊

λ̃

gK :M�g

⌋}

= �gnmax∑n=1

n Pr

{λ̃

(n + 1)�g ≤ gK :M ≤λ̃

n�g

}

= �gnmax∑n=1

n

λ̃/n�g∫λ̃/(n+1)�g

pgK :M (x)dx . (46)

Using (45) and evaluation of the integral in (46) leads to (5).

APPENDIX BPROOF OF PROPOSITION 2

Let K be the set containing (K -1) best relays andR = D \ (K ∪ { j}). For collision-free K best relay selection,the following conditions have to be satisfied: (1) for (K -1)best relays λ̃/gi∈K < n�g and no collisions between relaysin this interval, (2) for the K th best relay n�g ≤ λ̃/g j �=i <(n + 1)�g , and (3) for the remaining (M-K ) relays λ̃/gr∈R ≥


(n + 1)�g . For the best relay selection (K = 1) only conditions(2) and (3) are relevant.

Using multinomial distribution, the probability that all thethree conditions (for K > 1) are fulfilled is given by

pno−coll,K>1,nmax

= M!(M − K )!(K − 1)!

nmax∑n=K−1

(∏i∈K

Pr{λ̃/gi < n�g

})

Pr{

n�g ≤ λ̃/g j �=i < (n + 1)�g}

(P∏

r∈Rr{λ̃/gr ≥ (n + 1)�g

})(1 − I{K≥3}(K )pcoll,K−2,n

)= M!(M − K )!(K − 1)!

nmax∑n=K−1

(1 − Fg(λ̃/n�g)

)K−1(

Fg(λ̃/n�g)− Fg(λ̃/(n + 1)�g))

F M−Kg (λ̃/(n + 1)�g)(1 − I{K≥3}(K )pcoll,K−2,n

), (47)

while the probability that only conditions (2) and (3) are sat-isfied for best relay selection (K = 1) can be calculated asfollows

pno−coll,K=1,nmax = Mnmax∑n=0

(Fg(λ̃/n�g)

−Fg(λ̃/(n + 1)�g))

F M−1g (λ̃/(n + 1)�g), (48)where Fg(x) = 1 − e−x/ḡ is cumulative distribution function(cdf) of channel power gain g. The collision probability can becalculated using pcoll,K ,nmax = 1 − pno−coll,K ,nmax .

APPENDIX CPROOF OF PROPOSITION 3

As we have assumed that M ≥ 2, outage occurs only inthe second hop. For the best relay selection (K = 1), outageis declared if channel power gain g1:M cannot support thetarget rate R under maximum transmission power constraint,Pmax . For cooperative beamforming (K ≥ 2) outage occurs ifthe destination transmit power to feedback the sum of second-hop channel power gains does not meet the target rate R withPmax . It follows then for the outage probability of cooperativecommunications

pCCout = I{K=1}(K )Pr{g1:M < μ}

+ I{2≤K≤M}(K )Pr{gK :M < μ} =μ∫

0

pgK :M (x)dx, (49)

using (45) in (49) and integration leads to (15).

APPENDIX DPROOF OF PROPOSITION 4

Expressions (16)-(18) can be obtained easily from the Fig. 1and discussions in Section II.

Average energy consumption for the destination feedback iscalculated as follows

E{EF B(K ,M, ψ)}= NF B TS N0 B(2R − 1)E

{1

gK :M|gK :M ≥ μ

}

= NF B TS N0 B(2R − 1)∞∫μ

1

xpgK :M (x)dx(1 − FgK :M (μ))−1,

(50)

where FgK :M (μ) =μ∫0

pgK :M (x)dx is cdf of the K -th best chan-

nel power gain. Using (45) in (50) and performing the integra-tion results in (19).

Average energy consumed to signal M to the source is givenby

E{EM (M, ψ)} = NM TS N0 B(2R − 1)M∑

i=1E

{1

hi|hi ≥ μ

}

= NM TS N0 B(2R − 1)M∑

i=1

∞∫μ

1

xphi (x)dx(1 − Fhi (μ))−1,

(51)

where phi (x) = 1h̄ e− x

h̄ for i = 1, . . . ,M . Evaluation of (51)leads to (20).

APPENDIX EPROOF OF PROPOSITION 5

Summation of the average energy consumption for the firstand second hop data transmission as well as considering bothcases best relay selection (K = 1) and cooperative beamform-ing (K ≥ 2) leads to (21).

The average energy consumption for data transmission fromthe source to the K selected relays is given by

E{E ID(K ,M, ψ)}= NDTS N0 B(2R − 1)E

{1

Hmin|Hmin ≥ μ

}

= NDTS N0 B(2R − 1)∞∫μ

1

xpHmin (x)dx(1 − FHmin (μ))−1,

(52)

where Hmin = min{h1, . . . , hK } and [32]

pHmin (x) =M!

ḡ

M−1∑i=0

(−1)i(M − i − 1)!i!e

− i+1ḡ x ,

Calculation of the integral in (52) yields (22).Average energy consumed in the second hop for the data

transmission for the best relay selection (K = 1) can be cal-culated as follows

E{E I ID (K = 1,M, ψ)}= NDTS N0 B(2R − 1)E

{1

g1:M|g1:M ≥ μ

}


= NDTS N0 B(2R − 1)∞∫μ

1

xpg1:M (x)dx(1 − Fg1:M (μ))−1,

(53)

Using (45) with K = 1 and evaluation of (53) results in (23).Average energy consumed in the second hop for the data

transmission for cooperative beamforming (K ≥ 2) is given byE{E I ID (K > 1,M, ψ)}

= NDTS N0 B(2R − 1)E{

1∑Ki=1 gi :M

|gK :M ≥ μ}

= NDTS N0 B(2R − 1)∞∫

μK

1

xp∑K

i=1 gi :M(x)dx

(1 − FgK :M (μ))−1. (54)It is shown in [38] that using statistical independence prop-

erty of spacings between consecutive exponentially distributedordered random variables, calculation of pdf for sum of the Klargest ordered statistics can be simplified. Let dm = gm:M −gm+1:M , 1 ≤ m ≤ M , be spacing between two adjacent orderedrandom variables, then

gM :M = dM ,gM−1:M = dM + dM−1,

...

gK :M = dM + dM−1 + · · · + dK ,...

g1:M = dM + dM−1 + · · · + dK + · · · + d1,and for the sum of K largest channel power gains

K∑i=1

gi :M =K∑

j=1jd j + K

M∑j=K+1

d j . (55)

Spacing pdf is given by [38]

pdm (x) =m

ḡe−m

xḡ , x ≥ 0. (56)

It follows for moment generating function (MGF)

M∑Ki=1 gi :M

(s) = (1 − ḡs)−KM∏

j=K+1

(1 − ḡK

js

)−1. (57)

Using partial fraction for simple roots [36] leads to

M∑Ki=1 gi :M

(s) = M!K K !ḡ(1 − ḡs)K

M∑j=K+1

(−1) j+K( j − K − 1)!(M − j)!

(s − j

ḡK

)−1, (58)

and pdf can be computed as

p∑Ki=1 gi :M

(x) = L−1{M∑K

i=1 gi :M(−s)}

= M!K K !ḡK+1

M∑j=K+1

(−1) j+K−1( j − K − 1)!(M − j)!

L−1{(

s + 1ḡ

)−K}∗ L−1

{(s + j

ḡK

)−1}, (59)

where L−1 is inverse Laplace transformation and ’*’ denotesconvolution operator. Using Laplace transform table [36] andperforming convolution leads to

p∑Ki=1 gi :M

(x) = M!K K !ḡK+1

M∑j=K+1

(−1) j+K−1( j − K − 1)!(M − j)!

(K ḡ

K − j)K e− jḡK x

(K − 1)!γ(

K ,

(1 − j

K

)x

ḡ

),

(60)

where γ (α, x) = ∫ x0 tα−1e−t dt [36] is the lower incompletegamma function and for the special case above is

γ

(K ,

(1 − j

K

)x

ḡ

)

= (K − 1)!⎛⎜⎝1 − e−

(1− jK)

xḡ

K−1∑k=0

((1 − jK

)xḡ

)kk!

⎞⎟⎠ .

Insertion of (60) in (54) and integration yields (24).

APPENDIX FPROOF OF LEMMA 1

Using Jensen’s inequality for conditional expectations,

E

{1X |Y}

≥ 1E{X |Y } , where X and Y are random variables, the

average energy consumption for the signaling overhead anddata transmission can be lower bounded as follows

E{EO(K ,M, ψ)} ≥ ET (K ,M, ψ)+ E L BM (M, ψ)+ EI N V+ I{2≤K≤M}(K )E L BF B(K ,M, ψ),

E{ED(K ,M, ψ)} ≥ E I,L BD (K ,M, ψ)+ I{K=1}(K )E I I,L BD (K = 1,M, ψ)+ I{2≤K≤M}(K )E I I,L BD (K > 1,M, ψ),

where

EL BM (M, ψ) = NM TS N0 B(2R − 1)

M∑i=1

1

E{hi |hi ≥ μ} , (61)

EL BF B(K ,M, ψ) =

NF B TS N0 B(2R − 1)E{gK :M |gK :M ≥ μ} , (62)

EI,L BD (K ,M, ψ) =

NDTS N0 B(2R − 1)E{Hmin|Hmin ≥ μ} , (63)

EI I,L BD (K = 1,M, ψ) =

NDTS N0 B(2R − 1)E{g1:M |g1:M ≥ μ} , (64)

EI I,L BD (K > 1,M, ψ) =

NDTS N0 B(2R − 1)K∑

i=1E{gi :M |gi :M ≥ μ}

. (65)

Evaluations of E{.} in (61)–(65) can be done similar toAppendix E and result in (28)–(32).


APPENDIX GPROOF OF PROPOSITION 6

The optimal location of cooperating relays is given by

ψopt (K ,M) ≈ argminψ

(E

L BO (K ,M, ψ)+ E L BD (K ,M, ψ)

)

= argminψ

((CT + CID + CM )ψξ +

(CT + CI ID

+I{2≤K≤M}(K )CF B)(dsd − ψ)ξ

+KCT max(ψξ , (dsd − ψ)ξ )), (66)

where

CT = − NTln(1 − ptrout )

,CID = K ND,CM =M

R,

CI ID = I{K=1}(K )ND⎛⎝ M∑

j=1

1

j

⎞⎠

−1+ I{2≤K≤M}(K )ND

K

⎛⎝1 + M∑

j=K+1

1

j

⎞⎠

−1,CF B = NF B

⎛⎝ M∑

j=K

1

j

⎞⎠

−1,

In order to find ψopt (K ,M), two different cases have to beinvestigated.

Case I: 0 ≤ ψ ≤ dsd2Using the following substitutions in (66),

αI = CT + CID + CM ,βI = (1 + K )CT + CI ID + I{2≤K≤M}(K )CF B,

we rewrite the optimization problem as follows

minψαIψ

ξ + βI (dsd − ψ)ξ

s.t.

ψ ≥ 0, ψ ≤ dsd2. (67)

It can be solved using Karush-Kuhn-Tucker (KKT) condi-tions [39]

ξ(αIψ

ξ−1 − βI (dsd − ψ)ξ−1)

+ λ1 − λ2 = 0,

λ1

(ψ − dsd

2

)= 0,

λ2ψ = 0,λ1 ≥ 0, λ2 ≥ 0.

The above KKT conditions are only fulfilled for

λ1 = λ2 = 0,

ψI =(

1 +(αI

βI

) 1ξ−1)−1

dsd . (68)

Case II: dsd2 < ψ ≤ dsdAnalogous to case I, it can be shown that

ψI I =(

1 +(αI I

βI I

) 1ξ−1)−1

dsd . (69)

As αI I > αI and βI I < βI

ψI I <

(1 +(αI

βI

) 1ξ−1)−1

dsd , (70)

i.e., ψI I < ψI and this violates ψI I >dsd2 . Therefore, the

optimal solution is ψopt = ψI .

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Bleron Klaiqi received Dipl.-Ing. degree in elec-trical engineering and information technology fromRWTH Aachen University, Aachen, Germany. He iscurrently pursuing the Ph.D. degree at the Universityof Sheffield, Sheffield, U.K. From 2006 to 2012,he worked as a Software Engineer for UMTS/HSPALayer 1 and for interworking between multiple RATs(GSM/UMTS/LTE) at Intel Corporation, Nuremberg,Germany. His research interests include cooperativecommunications, energy-efficient wireless communi-cations, multihop D2D communications, and energy

harvesting. He served as a Session Chair for the IEEE ICC’15 and as a reviewerfor several major IEEE journals and conferences.

Xiaoli Chu received the B.Eng. degree in elec-tronic and information engineering from Xi’an JiaoTong University, Xi’an, China, in 2001, and thePh.D. degree in electrical and electronic engineer-ing from the Hong Kong University of Science andTechnology, Clear Water Bay, Hong Kong, in 2005.She is a Lecturer with the Department of Electronicand Electrical Engineering, University of Sheffield,Sheffield, U.K. From September 2005 to April 2012,she was with the Centre for TelecommunicationsResearch, King’s College London, London, U.K. She

has authored more than 80 peer-reviewed journal and conference papers. Sheis the leading Editor/Author of the book Heterogeneous Cellular NetworksTheory, Simulation and Deployment (Cambridge University Press, 2013) andthe book 4G Femtocells: Resource Allocation and Interference Management(Springer USA, 2013). She was a Guest Editor of the IEEE TRANSACTIONSON VEHICULAR TECHNOLOGY Special Section on Green Mobile MultimediaCommunications (June 2014) and the ACM/Springer Journal of MobileNetworks and Applications Special Issue on Cooperative Femtocell Networks(October 2012). She was a Co-Chair of the Wireless CommunicationsSymposium for the IEEE International Conference on Communications 2015(ICC15), Workshop Co-Chair for the IEEE International Conference on GreenComputing and Communications 2013 (GreenCom13), and has been technicalprogram committee Co-Chair of several workshops on heterogeneous networksfor the IEEE GLOBECOM, WCNC, PIMRC, etc.

Jie Zhang received the Ph.D. degree in 1995. Hebecame a Lecturer, Reader, and Professor, in 2002,2005, and 2006, respectively. He and his students andcolleagues have pioneered research in femto/smallcell and HetNets and published some of the mostcited publications in these topics. He has been aFull Professor and has held the Chair in wirelesssystems at the Department of EEE, University ofSheffield, Sheffield, U.K., since 2011. He co-foundedRANPLAN Wireless Network Design Ltd., whichproduces a suite of world leading in-building DAS,

indoor–outdoor small cell/HetNet network design and optimisation toolsiBuildNet. Since 2005, he has been received over 20 projects by the EPSRC, theEC FP6/FP7/H2020 and industry, including some of earliest research projectson femtocell/HetNets.

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