arX

iv:1

310.

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v2 [

cs.IT

] 11

Feb

201

41

Simultaneous Information and Energy Transfer inLarge-Scale Networks with/without Relaying

Ioannis Krikidis,Senior Member, IEEE

Abstract—Energy harvesting (EH) from ambient radio-frequency (RF) electromagnetic waves is an efficient solutionfor fully autonomous and sustainable communication networks.Most of the related works presented in the literature are based onspecific (and small-scale) network structures, which although giveuseful insights on the potential benefits of the RF-EH technology,cannot characterize the performance of general networks. Inthis paper, we adopt a large-scale approach of the RF-EHtechnology and we characterize the performance of a networkwith random number of transmitter-receiver pairs by usingstochastic-geometry tools. Specifically, we analyze the outageprobability performance and the average harvested energy,whenreceivers employ power splitting (PS) technique for “simultane-ous” information and energy transfer. A non-cooperative scheme,where information/energy are conveyed only via direct links, isfirstly considered and the outage performance of the system aswell as the average harvested energy are derived in closed formin function of the power splitting. For this protocol, an int erestingoptimization problem which minimizes the transmitted power un-der outage probability and harvesting constraints, is formulatedand solved in closed form. In addition, we study a cooperativeprotocol where sources’ transmissions are supported by a randomnumber of potential relays that are randomly distributed into thenetwork. In this case, information/energy can be received at eachdestination via two independent and orthogonal paths (in caseof relaying). We characterize both performance metrics, whena selection combining scheme is applied at the receivers andasingle relay is randomly selected for cooperative diversity.

Index Terms—RF energy harvesting, stochastic geometry, Pois-son point process, relay channel, power consumption, outageprobability.

I. I NTRODUCTION

Energy efficiency is of paramount importance for futurecommunication networks and is a main design target for all5G radio access solutions. It refers to an efficient utilizationof the available energy and consequently extends the networklifetime and/or reduces the operation cost. Specifically, con-ventional battery-powered communication systems suffer fromshort lifetime and require periodic replacement or rechargingin order to maintain network connectivity. On the other hand,communication systems that are supported by a continuouspower supply such as cellular networks require a power gridinfrastructure and may result in large energy consumption thatwill further increase due to the increasing growth of data

Manuscript received October 20, 2013; revised January 7, 2014. The editorcoordinating the review of this paper and approving it for publication was E.Larsson.

I. Krikidis is with the Department of Electrical and Computer Engineer-ing, Faculty of Engineering, University of Cyprus, Nicosia1678 (E-mail:krikidis@ucy.ac.cy).

This work was supported by the Research Promotion Foundation, Cyprusunder the project KOYLTOYRA/BP-NE/0613/04 “Full-Duplex Radio: Mod-eling, Analysis and Design (FD-RD)”.

traffic. The investigation of energy-aware architectures as wellas transmission techniques/protocols that prolong the lifetimeof the networks or provide significant energy savings has beena hot research area over several years, often under the umbrellaof the green radio/communications [1], [2].

Due to the limited supply of non-renewable energy re-sources, recently, there is a lot of interest to integrate theenergy harvesting (EH) technology to power communicationnetworks [3]. Energy harvesting is a new paradigm and allowsnodes to harvest energy from natural resources (i.e., solarenergy, wind, mechanical vibrations etc.) in order to maintaintheir operation. Related literature concerns the optimization ofdifferent network utility functions under various assumptionson the knowledge of the energy profiles. The works in [4], [5]assume that the EH profile is perfectly known at the transmit-ters and investigate optimal resource allocation techniques fordifferent objective functions and network configurations.Onthe other hand, the works in [6], [7] adopt a more networkingpoint of view and maximize the performance in terms ofstability region by assuming only statistical knowledge oftheEH profile. Although energy harvesting from natural resourcesis a promising technology towards fully autonomous and self-sustainable communication networks, it is mainly unstable(i.e., weather-dependent) and thus less efficient for applicationswith critical quality-of-service (QoS) requirements.

An interesting solution that overcomes the above limitationis to harvest energy from man-made electromagnetic radiation.Despite the pioneering work of Tesla, who experimentallydemonstrated wireless energy transfer (WET) in late 19thcentury, modern wireless communication systems mainly fo-cus on the information content of the radio-frequency (RF)radiation, neglecting the energy transported by the signals.Recently, there is a lot of interest to exploit RF radiationfrom energy harvesting perspective and use wireless energytransfer in order to power communication devices. The fun-damental block for the implementation of this technologyis the rectifying-antenna (rectenna) which is a diode-basedcircuit that converts the RF signals to DC voltage. Severalrectenna architectures and designs have been proposed in theliterature for different systems and frequency bands [8], [9].An interesting rectenna architecture is reported in [10], wherethe authors study a rectenna array in order to further boost theharvesting efficiency. Although information theoretic studiesideally assume that a receiver is able to decode informationandharvest energy independently from the same signal [11], [12],this approach is not feasible due to practical limitations.In theseminal work in [13], the authors introduce two practical RFenergy harvesting mechanisms for “simultaneous” informationand energy transfer: a) time switching (TS) where dedicated

2

time slots are used either for information transfer or energyharvesting, b) power splitting (PS) where one part of thereceived signal is used for information decoding, while theother part is used for RF energy harvesting.

The employment of the above two practical approaches indifferent fundamental network structures, is a hot researchtopic and several recent works appear in the literature. In[13], the authors study the problem of beamforming designfor a point-to-point multiple-input multiple-output (MIMO)channel and characterize the rate-energy region for both TSand PS techniques. This work is extended in [14] for thecase of an imperfect channel information at the transmitterbyusing robust optimization tools. The work in [15] investigatesthe optimal PS rule for a single-input single-output (SISO)channel in order to achieve different trade-offs between er-godic capacity and average harvested energy. An interestingproblem is discussed in [16], where the downlink of an accesspoint broadcasts energy to several users, which then use theharvested energy for time-division multiple access (TDMA)uplink transmissions. In [17], the authors study a fundamentalmulti-user multiple-input single-output (MISO) channel wherethe single-antenna receivers are characterized by both QoSand PS-EH constraints. On the other hand, cooperative/relaynetworks is a promising application area for RF energy har-vesting, since relay nodes have mainly limited energy reservesand may require external energy assistance. The works in [18]–[21] deal with the integration of both TS and PS techniquesin various cooperative topologies with/without batteriesforenergy storage. The simultaneous information/energy transferfor a MIMO relay channel with a separated energy harvestingreceiver is discussed in [22].

Although several studies deal with the analysis of communi-cation networks with RF energy harvesting capabilities, mostof existing work refers to specific (fixed) single/multiple usernetwork configurations. Since harvesting efficiency is asso-ciated with the interference and thus the geometric distancebetween nodes, a fundamental question is to study RF energyharvesting for large-scale networks by taking into accountrandom node locations. Stochastic-geometry is a useful the-oretical tool in order to model the geometric characteristics ofa large-scale network and derive its statistical properties [23]–[25]. Several works in the literature adopt stochastic-geometryin order to analyze the outage probability performance orthe transmission capacity for different conventional (withoutharvesting capabilities) networks e.g., [26]–[29]. Large-scalenetworks with energy harvesting capabilities are studied in[30]–[33] for different network topologies and performancemetrics. These works model the energy harvesting operationasa stochastic process and mainly refer to energy harvesting fromnatural resources e.g., solar, wind, etc. However, few studiesanalyze the behavior of a RF energy harvesting network froma stochastic-geometry standpoint. In [34], the authors studythe interaction between primary and cognitive radio networks,where cognitive radio nodes can harvest energy from theprimary transmissions, by modeling node locations as Poissonpoint processes (PPPs). A cooperative network with multipletransmitter-receiver pairs and a single energy harvestingrelayis studied in [35] by taking the spatial randomness of user

locations into consideration. The analysis of large-scaleRFenergy harvesting networks with practical TS/PS techniques,is an open question in the literature.

In this paper, we study the performance of a large-scalenetwork with multiple transmitter-receiver pairs, where trans-mitters are connected to the power grid, while receiversemploy the PS technique for RF energy harvesting. By usingstochastic-geometry, we model the randomness of node loca-tions and we analyze the fundamental trade-off between outageprobability performance and average harvested energy. Specif-ically, we study two main protocols: a) a non-cooperativeprotocol, and b) a cooperative protocol with orthogonal relayassistance. In the non-cooperative protocol, each transmittersimultaneously transfers information and energy at the as-sociated receiver via the direct link. The outage probabilityof the system as well as the average harvested energy arederived in closed form in function of the power splitting ratio.In addition, an optimization problem which minimizes thetransmitted power under some well-defined outage probabilityand average harvesting constraints, is discussed and closedform solutions are provided.

The cooperative protocol is introduced to show that relayingcan significantly improve the performance of the system andachieve a better trade-off between outage probability perfor-mance and energy harvesting transfer. Relaying cooperation isintegrated in several systems and standards in order to providedifferent levels of assistance (i.e., cooperative diversity, energysavings, secrecy etc); in this work, relays are used in ordertofacilitate the information/energy transfer. For the cooperativeprotocol, we introduce a set of potential dynamic-and-forward(DF) relays, which are randomly distributed in the networkaccording to a PPP; these relays have similar characteristicswith the transmitters and are also connected to the power grid.In this case, information and energy can be received at eachdestination via two independent paths (in case of cooperation).For the relay selection, we adopt a random selection policybased on a sectorized selection area with central angle atthe direction of each receiver. The outage performance of thesystem for a selection combining (SC) scheme as well as theaverage harvested energy are analyzed in closed form and val-idate the cooperative diversity benefits. Numerical results fordifferent parameter set-up reveal some important observationsabout the impact of the central angle and relay density on thetrade-off between information and energy transfer. It is the firsttime, to the best of the authors’ knowledge, that stochastic-geometry is used in order to analyze a PS energy harvestingnetwork with/without relaying.

The remainder of this paper is organized as follows. SectionII describes the system model and introduces the consideredperformance/harvesting metrics. Section III presents thenon-cooperative protocol and analyzes its performance in termsofoutage probability and average harvested energy. Section IVintroduces the cooperative protocol and analyzes both perfor-mance metrics considered. Simulation results are presented inSection V, followed by our conclusions in Section VI.

Notation: Rd denotes thed-dimensional Euclidean space,1(·) denotes the indicator function,| · | is the Lebesguemeasure,b(x, r) denotes a two dimensional disk of radiusr

3

centered atx, ‖x‖ denotes the Euclidean norm ofx ∈ Rd,

P(X) denotes the probability of the eventX and E(·) rep-resents the expectation operator. In addition, the number ofpoints inB is denoted byN(B).

II. SYSTEM MODEL

We consider a 2-D large-scale wireless network consistingof a random number of transmitter-receiver pairs. The trans-mitters form an independent homogeneous PPPΦt = xkwith k ≥ 1 of intensityλ on the planeR2, wherexk denotesthe coordinates of the nodek. Each transmitterxk has a uniquereceiver r(xk) (not a part ofΦt) at an Euclidean distanced0 in some random direction [26]. All nodes are equippedwith single antennas and have equivalent characteristics andcomputation capabilities. The time is considered to be slottedand in each time slot all the sources are active withoutany coordination or scheduling process. In the consideredtopology, we add a transmitterx0 at the origin[0 0] and itsassociated receiverr(x0) at the location[d0 0] without loss ofgenerality; in this paper, we analyze the performance of thistypical communication link but our results hold for any nodein the processΦt ∪ x0 according to Slivnyak’s Theorem [23].

We assume apartial fading channel model, where desireddirect links are subject to both small-scale fading and large-scale path loss, while interference links are dominated by thepath-loss effects. According to the literature [36], [37],thischannel model is denoted as “1/0 fading” and serves as auseful guideline for more practical configurations e.g., all linksare subject to fading [38], [39]. More specifically, the fadingbetweenxk andr(xk) is Rayleigh distributed so the power ofthe channel fading is an exponential random variable with unitvariance. The path-loss model assumes that the received poweris proportional tod(X ,Y)−α whered(X ,Y) is the Euclideandistance between the transmitterX and the receiverY, α > 2denotes the path-loss exponent and we defineδ , α/2. TheEuclidean distance between two nodes is defined as

d(X ,Y) =

‖X − Y)‖−α If ‖X − Y‖ > r0r−α0 elsewhere,

(1)

where the parameterr0 > 1 refers to the minimum possiblepath-loss degradation and ensures the accuracy of our path-loss model for short distances [29]. The instantaneous fadingchannels are known only at the receivers in order to performcoherent detection. In addition, all wireless links exhibit ad-ditive white Gaussian noise (AWGN) with varianceσ2.

The transmitters are continuously connected to a powersupply (e.g., battery or power grid) and transmit with thesame powerPt. On the other hand, each receiver has RFenergy harvesting capabilities and can harvest energy fromthereceived electromagnetic radiation. The RF energy harvestingprocess is based on the PS technique and therefore eachreceiver splits its received signal in two parts a) one part isconverted to a baseband signal for further signal processingand data detection (information decoding) and b) the otherpart is driven to the rectenna for conversion to DC voltageand energy storage. Letνd ∈ (0, 1) denote the power splittingparameter for each receiver; this means that100νd% of the

received power is used for data detection while the remainingamount is the input to the RF-EH circuitry. We assume an idealpower splitter at each receiver without power loss or noisedegradation, and that the receivers can perfectly synchronizetheir operations with the transmitters based on a given powersplitting ratioνd [40]. During the baseband conversion phase,additional circuit noise,v, is present due to phase-offsets andcircuits’ non-linearities and which is modeled as AWGN withzero mean and varianceσ2

C [13]. Based on the PS techniqueconsidered, the signal-to-interference-plus-noise ratio (SINR)at the typical receiver can be written as

SINR0 =νdPth0d

−α0

νd(σ2 + PtI0

)+ σ2

C

, (2)

where I0 ,∑

xk∈Φt

d(xk)−α denotes the total (normalized)

interference at the typical receiver withd(xk) , d(xk, r(x0)

)

and hk denotes the channel power gain for the linkxk →r(xk). A successful decoding requires that the received SINRis at least equal to a detection thresholdΩ. On the otherhand, RF energy harvesting is a long term operation1 andis expressed in terms of average harvested energy [13], [15].Since100(1− νd)% of the received energy is used for recti-fication, the average energy harvesting at the typical receiveris expressed as

E0 = ζ · E

((1− νd)Pt

[h0d

−α0 + I0

]), (3)

where ζ ∈ (0, 1] denotes the conversion efficiency fromRF signal to DC voltage; for the convenience of analysis,it is assumed thatζ = 1. It is worth noting that the RFenergy harvesting from the AWGN noise is considered to benegligible.

III. N ON-COOPERATIVE PROTOCOL FOR SIMULTANEOUS

INFORMATION/ENERGY TRANSFER

The first investigated scheme does not enable any coopera-tion between the nodes and thus communication is performedin a single time slot; all the sources simultaneously trans-mit towards their associated receivers. Fig. 1 schematicallypresents the network topology for the non-cooperative case.The information decoding process is mainly characterized bythe outage probability which denotes the probability that theinstantaneous SINR is lower than the predefined thresholdΩ. By characterizing the outage probability for the typicaltransmitter-receiver linkx0 → r(x0), we also characterize theoutage probability for whole network (∀ xk ∈ Φt ∪ x0).

Proposition 1. The outage probability for the non-cooperativeprotocol is given by

ΠNC(νd, Pt) = PSINR0 < Ω

= 1− exp

(−Ωdα0σ

2

Pt

−Ωdα0σ

2C

νdPt

)Ξ(λ, d0, r0),

(4)

1In most real-world applications, the received power is verylow (scale ofdBm); therefore instantaneous harvesting has not practical interest.

4

r0

d0

xi

r(xj)

Fig. 1. Network topology for the non-cooperative protocol;d0 is theEuclidean distance between a transmitter and its associated receiver andr0is the minimum path-loss distance.

where

Ξ(x, y, z) , exp

(− πx

[ (exp

(−Ωyαz−α

)− 1)z2

+Ωδy2γ(1− δ,Ωyαz−α

) ])

× exp(−2πΩxyαz2−α

). (5)

Proof: See Appendix A.For high transmitted powers i.e.,Pt → ∞ andνd > 0 the

system becomes interference limited and the outage probabil-ity converges to a constant error floor given by

Π∞NC→ 1− Ξ(λ, d0, r0). (6)

As for the average harvested energy, by expanding (3) we havethe following proposition:

Proposition 2. The average harvested energy for the non-cooperative protocol is given by

ENC = (1− νd)Pt

[d−α0 +Ψ(λ)

], (7)

where

Ψ(x) , πxr2−α0

α

α− 2. (8)

Proof: From (3), we have:

ENC = E

((1− νd)Pt

[h0d

−α0 + I0

])

= (1− νd)Pt

(d−α0 E(h0) + E(I0)

)

= (1− νd)Pt

[d−α0 +Ψ(λ)

], (9)

whereE(hk) = 1 for all k, and the proof ofE(I0) = Ψ(λ)can be found in Appendix B.

1) Optimization problem- minimum transmitted power:Aninteresting optimization problem is formulated when energybecomes a critical issue for the network and each receiveris characterized by both QoS and RF energy harvesting con-straints. Due to the symmetry of the nodes, the minimizationof the transmitted power for the typical transmitter, it alsominimizes the total energy consumption for whole network.The optimization problem considered can be written as

minPt,νd

Pt

subject toΠNC ≤ CI ,

ENC ≥ CH ,

0 ≤ νd ≤ 1,

Pt ≥ 0, (10)

where the QoS constraint ensures an outage probability lowerthan a thresholdCI , while the RF energy harvesting constraintrequires an average harvested energy at least equal toCH

(i.e., it represents the minimum required energy to maintainsoperability at each device). For the case where the powersplitting ratio is constant i.e.,νd = ν0, the solution to theoptimization problem in (10) is simplified as follows:

P ∗t =

max

[G1

(1−ν0),(G2 +

G3

ν0

)/G0

]If Π∞

NC ≤ CI

Infeasible, elsewhere(11)

where G0 , ln(

Ξ(λ,d0,r0)1−CI

), G1 , CH/[d−α

0 + Ψ(λ)],

G2 , Ωd20σ2 andG3 , Ωd20σ

2C . The asymptotic expression in

(6) is involved in the optimization problem and determines itsfeasibility. More specifically, if the outage probability floor in(6) is higher than the outage probability constraintCI , there isnot any transmitted power that can satisfyCI and therefore theoptimization problem becomes infeasible. The constant powersplitting case corresponds to a low implementation complexityand is appropriate for (legacy) systems where the rectenna’sdesign is predefined and the power splitting parameter is notadaptable. For the general case, where bothPt and νd areadjustable, it can be easily seen that the two main constraintsare binding at the solution2. In this case, the optimizationproblem is transformed to the solution of a standard quadraticequation and forΠNC∞ ≤ CI the solution is given by

P ∗t =

G1

(1 − ν∗d),

ν∗d =−(G0G1 +G2 −G3) +

√(G0G1 +G2 −G3)2 + 4G2G3

2G2.

(12)

We note that the optimization problem is infeasible forP∞NC >

CI . The general optimization problem requires adaptive anddynamic RF power splitting and therefore refers to a higherimplementation complexity.

2Problem in (10) requires at least one of the constraints to bebinding,otherwise the value ofPt can further be reduced. By examining the caseswhere one constraint is binding and the other holds with inequality, we showthat at the optimal solution the inequality constraint holds with equality.

5

r

r0

η

d0

c

θ

r(xk)Bk

yk

Fig. 2. Network topology for the cooperative protocol;η is the radius of theselection sector.

The implementation problem can be solved either by acentral controller or in a distributed fashion. In the first case, acentral unit that controls the network, solves the problem andbroadcasts the common solution (transmitted power, powersplitting ratio) to all nodes; the transmitters and the receiversadjust their transmitted power and the power splitting ratio,respectively. In the distributed implementation, each nodecan locally solve the optimization problem without requiringexternal signaling (but with the cost of a higher computationalcomplexity). The optimization problem involves only deter-ministic and average system parameters such as geometricdistances, network density, path-loss exponent and channelstatistics; these parameters are estimated at the beginning ofthe communication and remain constant for a long operationtime.

IV. COOPERATIVE PROTOCOL FOR SIMULTANEOUS

INFORMATION/ENERGY TRANSFER

The cooperative scheme exploits the relaying/cooperativeconcept in order to combat fading and path-loss degradationeffects. The network topology considered is modified byadding a group of single-antenna DF relays, which have nottheir own traffic and are dedicated to assist the transmitters.Fig. 2 schematically presents the network topology for the co-operative protocol. The location of all relay nodes are modeledas a homogeneous PPP denoted byΦr = yk with densityλr; this assumption refers to mobile relays where their positionas well as their “availability” changes with the time. The relaynodes are also continuously connected to a power supply (e.g.,battery) and have equivalent computation/energy capabilities.We adopt an orthogonal relaying protocol where cooperationisperformed in two orthogonal time slots [26], [28]. It is worthnoting that although several cooperative schemes have beenproposed in the literature e.g., [41], the orthogonal relayingprotocol has a low complexity and is sufficient for the purposesof this work. The cooperative protocol operates as follows:

1) The first phase of the protocol is similar to the non-cooperative scheme and thus all transmitters simulta-neously broadcast their signals towards the associatedreceivers. Each transmitterxk defines a 2-D relayingareaBk around its location and each relay node located

inside this area is dedicated to assist this transmitter;this means that all relaysyi ∈ Bk consider the signalgenerated byxk as a useful information and all the othersignals as interference. In accordance to the generalsystem model, we assume that direct links suffer fromboth small-scale fading and path-loss, while interferencelinks are dominated by the path-loss attenuation (1/0partial fading [36]). By focusing our study on the typicaltransmitterx0, we define asgk the power fading gain forthe link x0 → yk with yk ∈ B0. In this case, the directlink x0 → r(x0) is characterized by (2), (3), while theSINR at the relayyk is written as

SINRk =Ptgkd(x0, yk)

−α

σ2 + PtIk, (13)

whereIk =∑

x∈Φtd(x, yk)

−α denotes the total (nor-malized) interference received atyk. If the relay nodeyk can decode the transmitted signal, which means thatSINRk ≥ Ω, it becomes a member of the transmitter’spotential relay. It is worth noting that the relay nodes useall the received signal for information decoding, sincethey have not energy harvesting requirements.

2) In the second phase of the protocol, one relay node(if any) that successfully decoded the transmitted signalaccesses the channel and retransmits the source’s signal.We assume arandom selectionprocess which selectsa single relay out of all potential relays with equalprobability. The random relay selection does not requireany instantaneous channel feedback or any instanta-neous knowledge of the geometry and is appropriatefor low complexity implementations with strict energyconstraints [28]. More sophisticated relay selection poli-cies, which take into account the instantaneous channelconditions [26], [42], can also be considered in orderto further improve the cooperative benefits. We defineas y∗ the selected relay for the typical transmitterx0,f is the channel power gain for the linky∗ → r(x0)and Φ∗

r is the homogeneous PPP that contains all theselected relays for whole network. If the potential relayset is empty for a specific transmitter (no relay was ableto decode the source’s transmitted signal), its messageis not transmitted during the second phase of the pro-tocol and therefore does not enjoy cooperative diversitybenefits. The relaying link for the (typical) receiver ischaracterized by the following equations

SINR′0 =

bνrPrfd(y∗, r(x0)

)−α

νr(σ2 + PrI ′0

)+ σ2

C

, whenb = 1, (14)

E′0 = ζ · E

((1 − νr)Pr

[bfd(y∗, r(x0))

−α + I ′0])

,

(15)

where I ′k =∑

y∈Φ∗

r

d(y, r(xk))−α denotes the total

(normalized) interference at the receiverr(xk), νr ∈(0, 1) is the power splitting ratio used in the secondphase of the protocol,Pr is the transmitted power foreach active relay and the binary variableb ∈ 0, 1 is

6

equal to one in case of a relaying transmission, whileit takes the value zero when the relay set is empty. Aperfect synchronization between the selected relay andthe associated receiver is assumed for a given powersplitting ratio νr. As for the decoding process at thereceivers, we assume that the two copies of the trans-mitted signal are combined with a simple SC technique;this means that information decoding is based on thebest path between direct/relaying links [42]. It is wellknown that SC only requires relative SINR measure-ments and thus it is simpler than maximum ratio com-biner, which requires exact knowledge of the channelstate information for each diversity branch. In addition,SC significantly reduces power consumption becausecontinuous estimates of the channel state informationare not necessary; this is beneficial for the consideredRF energy harvesting system, where energy saving isa critical requirement e.g., [43]–[45]. Regarding thepotential use of the non-selected branch for RF energyharvesting, here, we assume a simple/conventional im-plementation and the received energy allocated to the SCcannot be used for RF energy harvesting purposes. Onthe other hand, the energy harvesting process exploitsboth transmission phases and the total average energyharvested becomes equal to

ECO = E0 + E′0. (16)

The considered random relay selection process does notrequire any instantaneous channel feedback and is appropriatefor scenarios with critical energy/computation constraints.However, the definition of the selection areaBk has a sig-nificant impact on the system performance. In this work, weassume thatBk is a circular sector3 with centerxk, radiusη > r0 and central angle with orientation at the direction ofthe receiverr(xk).

Remark 1. By appropriately adjusting the central angle of thesector, we can ensure that the relaying paths are shorter thanthe direct distanced0; this parameterization avoids scenarioswhere the selected relay experiences more serious path-losseffects than the direct link. It is proven in Appendix C that aselection areaBk =

r ∈ [0 η], θ ∈ [−θ0 θ0]

with θ0 ≤

cos−1(η/(2d0)) satisfies this requirement.

A. Outage probability and average harvested energy

For the cooperative protocol, an outage event occurs when(a) the direct link is in outage and no relay is able to decodethe source’s message or (b) a relay node is able to decodethe transmitted signal but both direct and relaying link arein outage. Based on these two cases, we have the followingproposition:

3The considered cooperative protocol assumes that the selection sectorshave not overlaps and therefore a relay can be inside into a single selectionsector. Although this assumption simplifies our analysis, the work in [34,Sec. II.B] shows that for practically smallλ andη, circular discs around thedifferent transmitters do not overlap at most of the time. Inour case, we havecircular sectors and therefore the probability of overlapping becomes muchlower.

Proposition 3. The outage probability for the cooperativeprotocol is given by

ΠCO(νd, νr, Pt, Pr) = ΠNC(νd, Pt) · Πc(Pt)︸ ︷︷ ︸case (a)

+(1−Πc(Pt)

)· ΠNC(νd, Pt) · Πr(νr, Pr)︸ ︷︷ ︸

case(b)

(17)

where

Πc(Pt)

= exp

(−λr

[∫ +θ0

−θ0

∫ η

r0

exp

(−σ2Ωrα

Pt

)Ξ(λ, r, r0)rdrdθ

+θ0r20 exp

(−σ2Ωrα0Pt

)Ξ(λ, r0, r0)

]),

→ exp

(−λr

[∫ +θ0

−θ0

∫ η

r0

Ξ(λ, r, r0)rdrdθ + θ0r20Ξ(λ, r0, r0)

])

︸ ︷︷ ︸Π∞

c, for Pt→∞

,

(18)

Πr(νr, Pr) = 1−1

θ0(η2 − r20)

∫ θ0

−θ0

∫ η

r0

exp

(−σ2Ωcα

Pr

)

× exp

(−σ2CΩc

α

νrPr

)Ξ(λ[1 − Πc], c, r0)rdrdθ,

→ 1−1

θ0(η2 − r20)

∫ θ0

−θ0

∫ η

r0

Ξ(λ[1 −Πc], c, r0)rdrdθ

︸ ︷︷ ︸Π∞

r, for Pr→∞

,

(19)

with c =√r2 + d20 − 2rd0 cos(θ).

Proof: See Appendix D for the outage probability of thefirst hop (Πc(·)) and Appendix E for the outage probability ofthe second (relaying) hop (Πr(·)).

We note that for the case wherePt, Pr →∞ or σ2, σ2C →

0 with νd, νr > 0, the system becomes interference limitedand the outage performance converges to a constant outageprobability floor given by

Π∞CO→ Π∞

NCΠ∞c + (1−Π∞

c )Π∞NCΠ

∞r . (20)

On the other hand, each receiver harvests energy from bothphases of the cooperative protocol. In contrast to the in-formation decoding, which highly depends on the relayingtransmission and thus becomes inactive in case of an emptyrelay set, the RF harvesting process is active in all cases. Morespecifically, in case where the relay set is empty (no relayreforwards the source’s message), the corresponding receiverdoes not employ a PS technique and uses all the receivedenergy (e.g., interference) for RF energy harvesting. Based onthis fundamental remark, we have the following proposition

Proposition 4. The average harvested energy for the cooper-ative protocol is given by

ECO = (1− νd)Pt

[d−α0 +Ψ(λ)

]+Πc · PrΨ

(λ[1 −Πc]

)

+ (1−Πc) · (1 − νr)Pr

[Z +Ψ(λ[1−Πc])

], (21)

7

whereZ ≈

((r0+η2

)2+ d20 − 2d0

r0+η2 · sin(θ0)

θ0

)−δ

denotes

the average attenuation for the relaying link.

Proof: From (16), we have

ECO = E0 + E′0

= ENC︸︷︷︸direct link

+ Πc · E(PrI′0)︸ ︷︷ ︸

relaying link is inactive

+(1−Πc

)· E(Prd(y

∗)−α + PrI′0

)︸ ︷︷ ︸

relaying link is active

(22)

= (1− νd)Pt

[d−α0 +Ψ(λ)

]+Πc · PrΨ

(λ[1−Πc]

)

+ (1−Πc) · (1− νr)Pr

[Z +Ψ(λ[1−Πc])

], (23)

where the proof ofE(d(y∗)−α) ≈ Z is reported in AppendixF; it is worth noting that Appendix F provides both the exactvalue of the average attenuation as well as the above simplifiedapproximation.

It is worth noting that a similar optimization problem with(10) can be formulated for the relaying case; the objectivefunction could be the minimization of the total transmittedpower i.e.,minPt + Pr. However, as it can be seen fromProposition 3, the expressions of the outage probability arecomplicated in the relaying case and do not allow elegantclosed form solutions for the optimization problem.

V. NUMERICAL RESULTS

Computer simulations are carried out in order to evaluatethe performance of the proposed schemes. The simulationenvironment follows the description in Sections II, IV withparameters4 d0 = 20 distance units (we will use meters (m)for the sake of presentation),r0 = 4 m, νd = νr = 0.3,σ2 = σ2

C = 1, Ω = −30 dB, α = 4 and ζ = 1; theaverage harvested energy is measured in Watts. For the sakeof simplicity, we assume that the relay nodes have similarcomputational/complexity characteristics with the transmitters(e.g., they could be inactive transmitters of the network) andtherefore transmit with a powerPr = Pt. For the cooperativeprotocol we assumeη = 8 m and thus Remark 1 correspondsto θ0 ≤ cos−1(1/5) = 0.4359π. The presented results concernthe typical linkx0 → r(x0) but refer to any link of the networkΦt ∪ x0 (according to the Slivnyak’s Theorem [29, Sec. 8.5]).

A. Non-cooperative protocol

Fig.’s 3(a), 3(b) deal with the performance of the non-cooperative scheme for different network densities e.g.,λ =10−5, 5×10−5, 10−4, 10−3. Specifically, Fig. 3(a) plots theoutage probability of the system versus the transmitted powerPt. The first main observation is that the outage performanceconverges to a constant floor for high transmitted powersPt →∞. This behavior is due to the fact that all nodes transmitwith the same power without any coordination (scheduling)and therefore the system becomes interference limited as

4Unless otherwise defined.

20 30 40 50 60 70 80

10−3

10−2

10−1

100

Pt [dB]

Out

age

Pro

babi

lity

λ=10−3

λ=10−4

λ=5*10−5

λ=10−5

(a) Outage probability.

0 10 20 30 40 50 60 70 8010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

Pt [dB]

Ave

rage

Har

vest

ed E

nerg

y

λ=10−3

λ=10−4

λ=5*10−5

λ=10−5

Direct: 61.5%

Direct: 1.5%

Direct: 24.1%

Direct: 13.7%

(b) Mean harvested energy.

Fig. 3. Performance of the non-cooperative protocol versusPt for differentnetwork densitiesλ; σ2 = σ2

C= 1, Ω = −30 dB, r0 = 4 m, d0 = 20 m,

νd = 0.3 andα = 4. Analytical results are shown with dashed lines.

Pt increases. As for the impact of the network density onthe outage performance, it can be seen that as the densityincreases, the outage probability of the system increases;forλ = 10−5 the outage probability converges to6×10−4, whilefor λ = 10−3 it converges to6×10−2. This observation showsthat the network density and the related multi-user interferencesignificantly affects the decoding ability of the receivers. In thesame Figure, we plot the analytical results given by (4), whichmatch with the simulation results and validate our analysis.On the other hand, Fig. 3(b) plots the RF average harvestedenergy versus the transmitted powerPt. It can be seen thatthe RF average harvested energy is a linear function of thetransmitted power (as it can be observed by (7)) and increasesas the transmitted power increases. By comparing the differentcurves, we can see that as the network density increases, theaverage harvested energy increases; e.g., ifPt = 45 dB, wehaveENC = 1 Watt for λ = 10−4 and ENC = 9 Watt for

8

λ = 10−3. This remark shows that a dense network facilitatesthe RF energy harvesting process and thus interference isbeneficial from an energy harvesting standpoint. In Fig. 3(b),we also show the percentage of the harvested energy whichis from the direct link. As it can be seen, for small networkdensities, the direct component significantly contributesto thetotal average harvested energy while becomes less importantas the network density increases. For high network densi-ties i.e., λ = 10−3, interference dominates the RF energyharvesting process and the percentage of the direct link isalmost negligible i.e.,1.5 %. The theoretical curves perfectlymatch with the simulation results and validate our analysisin Proposition 2. It is worth noting that these two figuresdemonstrate the fundamental trade-off between informationdecoding and RF energy harvesting; interference significantlydegrades the achieved outage performance, while it becomeshelpful for the RF energy harvesting process.

Fig.’s 4(a), 4(b) show the impact of the power splitting ratioνd on the outage performance and the RF average harvestedenergy, respectively. We assumePt = 45 dB and the otherparameters are similar to the previous simulation example.A higher Pt affects (decreases) the outage probability inaccordance with Fig. 3(a) but does not change the observedbehavior of the outage probability versusνd curves; thereforefurther simulation results with otherPt parameters do notadd value to the main remarks of the paper. It can beseen that the power splitting ratio significantly affects theperformance of the system and defines the balance between thetwo conflicting objectives i.e., outage probability Vs energyharvesting. Specifically, asνd increases the outage perfor-mance is improved while the harvesting process becomesless efficient, since most of the received energy is used forinformation decoding; whenνd decreases we have the inversebehavior, since most of the received energy is used for RF-to-DC rectification. Regarding the network’s densityλ, ourobservations confirm the previous main remarks.

In Table I, we deal with the optimization problem discussedin Section III-1. Specifically, we present the optimal solutionfor the case of a constant splitting power ratioν0 = 0.5(Optimization I) as well as for the case where both parametersPt, νd can be adjusted (Optimization II). The first observa-tion is that Optimization II gives a solutionP ∗

t which ismuch lower than this one ofOptimization I, since the powerspitting parameter is optimized accordingly. The solutionofOptimization II satisfies both constraints with equality, asit has been discussed in Section III-1. On the other hand,for the first set-up(CI = 10−3, CH = 103), the solutionof the Optimization Isatisfies the harvesting constraint withequality, sinceCH is the dominant constraint. For the setup(CI = 10−2, CH = 10−1), the optimal solution satisfies theoutage constraint with equality, sinceCI becomes the dominantconstraint for this case.

B. Cooperative protocol

Although our analysis is general and concerns anyλ, λr, inthe simulations results, we assume thatλr is much higher thanthe network densityλ in order to demonstrate the potential

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−2

10−1

100

νd

Out

age

Pro

babi

lity

λ=10−3

λ=10−4

λ=5*10−5

λ=10−5

(a) Outage probability.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−2

10−1

100

101

νd

Ave

rage

Har

vest

ed E

nerg

y

λ=10−3

λ=10−4

λ=5*10−5

λ=10−5

Direct: 1.5%

Direct: 13.7%

Direct: 24.1%

Direct: 61.5%

(b) Mean harvested energy.

Fig. 4. Performance of the non-cooperative protocol versusνd for differentnetwork densitiesλ; σ2 = σ2

C= 1, Ω = −30 dB, r0 = 4 m, d0 = 20 m,

Pt = 45 dB andα = 4. Analytical results are shown with dashed lines.

gains from relaying at the outage/harvesting performance.More specifically, with a smallλr (e.g.,λr = 10−4 or 10−5)the probability of decoding at the relays (according to (18)) be-comes almost zero and therefore we cannot show the impact ofcooperation. On the other hand, asλ increases, the multi-userinterference significantly increases and the achieved outageprobability of the system has not practical interest. Therefore,in order to reveal the potential benefits of cooperation, weassume a smallλ which ensures a low (non-cooperative)probability outage floor as well as a higherλr which providesa non-empty relay set and therefore cooperative diversity.Afurther optimization of the network densities is an interestingproblem that could be considered for future work [34]; here,we assume that network densities are fixed and can not betaken into account in the design. This network density setupwith λ < λr could refer to a bursty network with a smalltransmission probability, where relays are part of the same

9

TABLE IOPTIMAL TRANSMITTED POWER(IN WATT) FOR THE NON-COOPERATIVE PROTOCOL; λ = 10−5 , σ2 = σ2

C= 1, d0 = 20 M , r0 = 4 M , Ω = −30 DB

AND α = 4.

CI = 10−3

CH = 103Transmitted powerPt Power splittingνd Outage Probability Average harvested energy

Optimization I 1.9652× 108 0.5 6.0153 × 10−4 103

Optimization II 9.8661× 107 0.0041 10−3 103

CI = 0.01CH = 0.1

Transmitted powerPt Power splittingνd Outage Probability Average harvested energy

Optimization I 5.0788× 104 0.5 0.01 0.2584Optimization II 3.9470× 104 0.7511 0.01 0.1

10 20 30 40 50 60 70 80

10−3

10−2

10−1

100

Pt [dB]

Out

age

Pro

babi

lity

λ=10−3

λ=10−4

λ=5*10−5

λ=10−5

(a) Outage probability.

0 10 20 30 40 50 60 70 80

10−4

10−3

10−2

10−1

100

101

102

103

104

Pt [dB]

Ave

rage

Har

vest

ed E

nerg

y

λ=10−3

λ=10−4

λ=5*10−5

λ=10−5

(b) Mean harvested energy.

Fig. 5. Performance of the cooperative protocol versusPt for differentnetwork densitiesλ; Pr = Pt, λr = 10−2, σ2 = σ2

C= 1, Ω = −30 dB,

r0 = 4 m, η = 8 m, θ0 = π/3, d0 = 20 m, νd = νr = 0.3 andα = 4.Analytical results are shown with dashed lines.

network and correspond to the inactive nodes [28].Fig.’s 5(a), 5(b) show the performance of the cooperative

protocol in terms of outage probability and average harvestedenergy, for a simulation setup withPr = Pt, νd = νr = 0.3,λr = 10−2, η = 8 m and θ0 = π/3 with θ0 ≤ 0.4359π

(Remark 1); the other parameters are defined as before.Specifically, Fig. 5(a) plots the outage probability versusthetransmitted powerPt. As it can be seen the main observationsare similar to the non-cooperative protocol and thus the outageprobability converges to a constant outage floor for highPt,since there is not any coordination/scheduling in both phasesof the protocol. As for the network densityλ, we can see thatit significantly affects the outage performance of the system;the associated multi-user interference degrades the decoder’sperformance at both the receivers and the relays in the firstphase of the protocol. In the same figure, we plot the theoret-ical expressions given by (17); we can see that the theoreticalresults provide a near-perfect match to the simulations resultsand validate our analysis for the cooperative case.

A direct comparison between Fig.’s 3(a), 5(a) for highPt (i.e., Pt → ∞)5, shows that the cooperative protocolimproves the outage probability of the system and achievesa lower outage probability floor e.g., forλ = 10−5, theoutage probability converges to an outage probability equalto 6 × 10−4 and 3 × 10−4, for the non-cooperative protocoland the cooperative protocol, respectively, (a more significantgain can be observed for another simulation setup as it isreported in the following discussion). In case of a successfuldecoding at the relay nodes (the relay set is not empty), thecooperative protocol provides a retransmission of the source’smessage from a shorter distance than the direct link as well asvia an independent fading channel and therefore improves theachieved outage probability due to the cooperative diversity.In Fig. 5(b), we plot the RF average harvested energy versusthe transmitted powerPt. As it can be seen the relayingoperation further improves the average harvested energy sincethe receivers can harvest energy from the relaying links. Forλ = 10−3 and Pt = 60 dB, the average harvested energyincreases from275 Watt to 450 Watt due to cooperation. Itis worth noting that when a receiver has not relay assistance,it uses all the received power during the second phase of thecooperative protocol for energy harvesting.

In Fig. 6, we show the impact of the central angleθ0and the relay densityλr on the outage performance of thecooperative protocol; we assumeλr = 10−1, 10−2, 10−3,θ0 = π/3, π/2, π, while the other simulation parameters fol-lows the previous example. It can be seen that the combination(λr, θ0) = (10−1, π/2) achieves the best outage performance

5For the casePt → ∞, the comparison between non-cooperative andcooperative protocol is fair and the their performance gap is due to thecooperative diversity associated with the cooperative scheme.

10

0 10 20 30 40 50 60 70 8010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Pt [dB]

Out

age

Pro

babi

lity

λr=0.1,θ

0=π/3

λr=0.1,θ

0=π/2

λr=0.1,θ

0=π

λr=0.01,θ

0=π/3

λr=0.01,θ

0=π/2

λr=0.01,θ

0=π

λr=0.001,θ

0=π/3

λr=0.001,θ

0=π/2

λr=0.001,θ

0=π

λr=0.1 λ

r=0.01

λr=0.001

Fig. 6. Outage probability versusPt for differentλr andθ0; Pr = Pt, λ =

10−5, λr = 10−1, 10−2, 10−3, σ2 = σ2

C= 1, θ0 = π/3, π/2, π,

Ω = −30 dB, r0 = 4 m, η = 8 m, d0 = 20 m, νd = νr = 0.3 andα = 4.

for highPt (it converges to the lowest outage floor). This resultreveals a very interesting relation between these two param-eters as well as a multidimensional trade-off. Specifically, ahigh λr ensures a non-empty relay set during the first phaseof the cooperative protocol and provides cooperative diversitybenefits. In this case, a smaller angle i.e.,θ0 = π/2 (Remark 1)can further protect the system from large-distance relays andtherefore achieves a better outage probability performance thanθ0 = π. On the other hand, asλr decreases, the probabilityto have a non-empty relay set increases and a larger angleis required in order to still have a potential relay at the areaof the transmitter; the condition in Remark 1 becomes lessimportant, since successful relay decoding is the priorityforthe system. The combination(λr, θ0) = (10−1, π/2) seemsto provide the best balance between successful relay decodingand protection from large-distance relays. In comparison tothe non-cooperative protocol, the considered setting revealsa significant gain of the cooperative protocol against thenon-cooperative scheme e.g., forλr = 0.1 and θ0 = π/2,the outage probability converges to10−7 in comparison to6 × 10−4 reported in Fig. 3(a). Finally, Fig. 7 depicts theaverage harvested energy versusPt. We observe that a smallangle is beneficial for the energy harvesting operation since anempty relay set allows the receiver to use all the received signalfor energy harvesting in the second phase of the protocol. Inaddition, asλr increases, the probability of relaying increaseswhich is beneficial for the energy harvesting process.

VI. CONCLUSION

This paper has dealt with the PS harvesting technique inlarge-scale networks with multiple transmitter-receiverpairs,where receivers are characterized by both QoS and RF energyharvesting requirements. A non-cooperative scheme where alltransmitters simultaneously communicate with their associatedreceivers without any coordination, is analyzed in terms ofoutage performance and average harvested energy by using

0 10 20 30 40 50 60

10−5

10−4

10−3

10−2

10−1

100

101

Pt [dB]

Ave

rage

Har

vest

ed E

nerg

y

λ

r=0.1,θ

0=π/3

λr=0.1,θ

0=π/2

λr=0.1,θ

0=π

λr=0.01,θ

0=π/3

λr=0.01,θ

0=π/2

λr=0.01,θ

0=π

λr=0.001,θ

0=π/3

λr=0.001,θ

0=π/2

λr=0.001,θ

0=π

38 39 40 41

10−1

Fig. 7. Average harvested energy versusPt for different λr and θ0;Pr = Pt, λ = 10−5, λr = 10−1, 10−2, 10−3, σ2 = σ2

C= 1,

θ0 = π/3, π/2, π, Ω = −30 dB, r0 = 4 m, η = 8 m, d0 = 20 m,νd = νr = 0.3 andα = 4.

stochastic-geometry. We show that network density and powersplitting ratio significantly affects the fundamental trade-offbetween outage performance and energy harvesting. For thiscase, an optimization problem that minimizes the transmittedpower under outage probability and harvesting constraintsisformulated and solved in closed form. In addition, a cooper-ative scheme where sources’ transmissions are assisted by arandom set of orthogonal relays is analyzed. A random relayselection policy is considered with a sectrorized selection areaat the direction of the receivers. Analytical and simulationresults reveal the impact of relay density and selection areaon the achieved outage-probability/average harvested perfor-mance. An extension of this work is to integrate a coordination(scheduling) between the different transmissions and study thetrade-off between energy harvesting and potential diversitygains. In addition, more sophisticated cooperative protocolsand diversity combining schemes can also be considered inorder to further boost the simultaneous information/energytransfer.

APPENDIX AOUTAGE PROBABILITY FOR THE NON-COOPERATIVE

PROTOCOL: PROOF OFPROPOSITIONI

In order to calculate the outage probability for the non-cooperative protocol, we need to calculate the Laplace trans-form of the normalized interference termI0 =

∑x∈Φt

d(x)−α.

11

We have

LI0(s) = E(exp(−sI0)

)

= E

(exp

(−s

∑

x∈Φt

d(x)−α

))

= E

(∏

x∈Φt

exp(− sd(x)−α

))

= E

∏

x∈Φt,‖x−r(x0)‖>r0

exp(−sd(x)−α)

× E

∏

x∈Φt,‖x−r(x0)‖≤r0

exp(−sr−α0 )

= L′I0(s)[exp

(−sr−α

0

) ]E(N(b(0,r0))

)

= L′I0(s) exp(−sπλr2−α

0

), (24)

whereE(N(b(0, r0))

)= λπr20 denotes the average number of

pointsxk ∈ Φt falling in a disk of radiusr0 [29, 2.4.2]. Forthe computation ofL′I0(s), we have

L′I0(s) = E

∏

x∈Φt

‖x−r(x0)‖>r0

exp(−sd(x)−α

)

= exp

(− λ

∫

R

(1− exp

(−sr−α

))dr

)(25a)

= exp

(−λ

∫ π

−π

∫ ∞

rα0

1

α

(1− exp

(−s

y

))yδ−1dydθ

)

(25b)

= exp

(−2λπ

∫ r−α

0

0

1

α

(1− exp(−su)

)u−δ−1du

)

(25c)

= exp

(− λπ

[(exp

(−sr−α

0

)− 1

)r20

+ sδγ(1− δ, sr−α

0

)])

, (25d)

whereR , r0 ≤ r, θ ∈ [−π, π] denotes the integrationarea, (25a) follows from the probability generating functionalof a PPP [29, Sec. 4.6], (25b) by using the transformationy ← rα, (25c) by using the transformationu← y−1, and (25d)from integration by parts;γ(n, β) ,

∫ β

0 yn−1 exp(−y)dy isthe lower incomplete gamma function [46].

The outage probability for the typical transmitter-receiver

link x0 → r(x0) can be written as

Pout = 1− P

(νdPth0d

−α0

νd(σ2 + PtI0) + σ2C

≥ Ω

)

= 1− P

(h0 ≥

Ωdα0 σ2

Pt

+Ωdα0 σ

2C

νdPt

+Ωdα0 I0

)

= 1− E exp

(−Ωdα0σ

2

Pt

−Ωdα0 σ

2C

νdPt

− Ωdα0 I0

)(26a)

= 1− exp

(−Ωdα0σ

2

Pt

−Ωdα0σ

2C

νdPt

)E exp(−Ωdα0 I0)︸ ︷︷ ︸

LI0(Ωdα

0)

,

(26b)

where (26a) follows from the cumulative distribution func-tion of an exponential random variable with unit varianceFX(x) = 1 − exp(−x) and the Laplace transform in (26b)is given by (24). It is worth noting that although the aboveanalytical method is similar to several stochastic geometryworks e.g., [25], [29], our analysis/result concerns a differentproblem and is based on different system assumptions.

APPENDIX BMEAN OF THE INTERFERENCE TERMI0

LetΦt be a PPP with densityλ and letI0 =∑

x∈Φtd(x)−α;

by using Campbell’s theorem for the expectation of a sum overa point process [29, 4.2], we have:

E(I0) = E

∑

x∈Φt,‖x−x0‖>r0

d(x)−α

+ E

∑

x∈Φt

‖x−x0‖≤r0

r−α0

= λ

∫ π

−π

∫ ∞

r0

r−αrdrdθ + E

(N(b(0, r0))

)r−α0

=2πλr2−α

0

α− 2+ λπr2−α

0

= πλr2−α0

α

α− 2, (27)

whereE

(N(b(0, r0))

)= λπr20 .

APPENDIX CSELECTION SECTORBk- CENTRAL ANGLE

We define asθ , ∠ yxkr(xk) the angle which is formedby the relay nodey, the transmitterxk and the receiverr(xk),r , d(xk, y) and c , d(y, r(xk)), as depicted in Fig. 2. Byusing the cosine rule, the requirement that the relay-receiverdistance should be shorter thand0 gives:

r2 + d20 − 2rd0 cos θ ≤ d20

⇒ θ ∈

[− cos−1

(r

2d0

), cos−1

(r

2d0

)]. (28)

In the case where the selection area is a sector with a constantcentral angle, by applying the above condition to the borderof the sector (i.e., for a distanceη), we have

θ ∈

[− cos−1

(η

2d0

), cos−1

(η

2d0

)]. (29)

12

It is worth noting that the above condition gives the maximumrange of the angle; any angle defined in this range, it alsosupports the distance requirement.

APPENDIX DOUTAGE PROBABILITY FOR THE FIRST HOP(EMPTY RELAY

SET)- Πc(Pt)

Let r be the distance between transmitter and relay. Therelay nodes that are able to successfully decode the source’ssignal form the point processΦ′

r, which is generated bythe homogeneous PPP processΦr by applying a thinningprocedure [29, 2.7.3]; thereforeΦ′

r is a PPP with intensity

λr′(x) = λrE

(1(x0 → yk|Φt

))(30)

=

λr exp(−σ2Ωrα

Pt

)Ξ(λ, r, r0) If r > r0

λr exp(−

σ2Ωrα0

Pt

)Ξ(λ, r0, r0) If r ≤ r0,

(31)

where for the above expression we have used the expressionin Proposition 1 for a direct distance equal tor andσ2

C = 0.If we focus on the typical transmitter, the mean ofΦ′

r insidethe areaB0 is equal to

µr′(B0) =

∫

B0

λr′(x)dx

=

∫

B0

λrexp

(−σ2Ωrα

Pt

)Ξ(λ, r, r0)dx

+ E[N(B0)

]λr exp

(−σ2Ωrα0Pt

)Ξ(λ, r0, r0)

=

∫ +θ0

−θ0

∫ η

r0

λr exp

(−σ2Ωrα

Pt

)Ξ(λ, r, r0)rdrdθ

+ λrθ0r20 exp

(−σ2Ωrα0Pt

)Ξ(λ, r0, r0) (32)

By using fundamental properties of a PPP process [29,2.4.3], the probability to have an empty relaying set is equalto

Πc(Pt) = PN(B0) = 0 = exp(− µr′(B0)

). (33)

APPENDIX EOUTAGE PROBABILITY FOR THE RELAYING HOP- Πr(νr, Pr)

By using the cosine rule, the distance relay-receiver can beexpressed asc ,

√r2 + d20 − 2rd0 cos(θ), wherer denotes

the distance transmitter-relay (see also Fig. 2). In the case ofa relaying transmission, the interference at each receiverisgenerated by all selected relays which form a PPPΦr∗ withdensityλ(1 − Πc) (i.e., one relay is selected for each trans-mitter with probability(1 − Πc)). For the outage probabilityof the relaying link, we can apply the derived expressions for

the direct link as follows

Πr(νr, Pr) = 1−1

|Bk|

∫

Bk

exp

(−σ2Ωcα

Pr

)exp

(−σ2CΩc

α

νrPr

)

× Ξ(λ(1 −Πc), c, r0)dx

= 1−1

θ0(η2 − r20)

∫ θ0

−θ0

∫ η

r0

exp

(−σ2Ωcα

Pr

)

× exp

(−σ2CΩc

α

νrPr

)Ξ(λ(1 −Πc), c, r0)rdrdθ,

(34)

where |Bk| = θ0η2 − θ0r

20 gives the area ofBk. We note

that the above expression takes into account that the smallestdistance between a communication pair isr0; the points ofBkwith r < r0 are considered to have a distancer0 according tothe considered radio propagation model in (1).

APPENDIX FAVERAGE RELAYING ATTENUATION

Let c = d(y, r(xk)) the distance relay-receiver andr =d(xk, y) the distance transmitter-relay; by using the cosine rule(see also Fig. 2), we havec2 = r2 + d20 − 2rd0 cos(θ). For aselection areaBk = r ∈ [0 η], θ ∈ [−θ0, θ0], the averagerelay-receiver attenuation can be expressed as

E(c−α) = E(r2 + d20 − 2rd0 cos(θ)

)−δ

=1

|Bk|

∫

Bk

(r2 + d20 − 2rd0 cos(θ)

)−δdx

=1

θ0(η2 − r20)

∫ θ0

−θ0

∫ η

r0

(r2 + d20 − 2rd0 cos(θ)

)−δrdrdθ,

(35)

where the above expression takes into account the radio prop-agation model in (1). In order to have a simple expression forthe average relaying attenuation, we apply Jensen’s inequality:

E(c−α) = E(r2 + d20 − 2rd0 cos(θ)

)−δ

≥(E(r2 + d20 − 2rd0 cos(θ))

)−δ(36a)

≥

((E(r)

)2+ d20 − 2d0E(r)E

(cos(θ)

))−δ

(36b)

=

((r0 + η

2

)2

+ d20 − 2d0r0 + η

2·sin(θ0)

θ0

)−δ

,

(36c)

where (36a) holds due to the convexity of the functionsf(x, θ) = (x2 + d20 − 2xd0 cos(θ))

−δ, (36b) holds due to theconvexity off(x) = x2 andE[cos(θ)] = sin(θ0)/θ0 in (36c).

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Ioannis Krikidis (S’03-M’07-SM’12) received thediploma in Computer Engineering from the Com-puter Engineering and Informatics Department(CEID) of the University of Patras, Greece, in 2000,and the M.Sc and Ph.D degrees from Ecole Na-tionale Superieure des Telecommunications (ENST),Paris, France, in 2001 and 2005, respectively, allin electrical engineering. From 2006 to 2007 heworked, as a Post-Doctoral researcher, with ENST,Paris, France, and from 2007 to 2010 he was aResearch Fellow in the School of Engineering and

Electronics at the University of Edinburgh, Edinburgh, UK.He has held alsoresearch positions at the Department of Electrical Engineering, Universityof Notre Dame; the Department of Electrical and Computer Engineering,University of Maryland; the Interdisciplinary Centre for Security, Reliabilityand Trust, University of Luxembourg; and the Department of Electricaland Electronic Engineering, Niigata University, Japan. Heis currently anAssistant Professor at the Department of Electrical and Computer Engineering,University of Cyprus, Nicosia, Cyprus. His current research interests includeinformation theory, wireless communications, cooperative communications,cognitive radio and secrecy communications.

Dr. Krikidis serves as an Associate Editor for the IEEE WIRELESSCOMMUNICATIONS LETTERS, IEEE TRANSACTIONS ON VEHICU-LAR TECHNOLOGY and Elsevier TRANSACTIONS ON EMERGINGTELECOMMUNICATIONS TECHNOLOGIES. He was the Technical Pro-gram Co-Chair for the IEEE International Symposium on Signal Processingand Information Technology 2013. He received an IEEE COMMUNICA-TIONS LETTERS and an IEEE WIRELESS COMMUNICATIONS LET-TERS exemplary reviewer certificate in 2012. He was the recipient of theResearch Award Young Researcherfrom the Research Promotion Foundation,Cyprus, in 2013.