Research Article Characterization of Energy Availability in RF Energy Harvesting Networks

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1 Mathematica Probems in Engineering Voume 206, Artice ID , 9 pages Research Artice Characterization of Energy Avaiabiity in RF Energy Harvesting Networks Daniea Oiveira and Rodofo Oiveira,2 Instituto de Teecomunicações (IT), Avenida Rovisco Pais, isboa, Portuga 2 Dep. de Eng. Eectrotécnica, Facudade de Ciências e Tecnoogia (FCT), Universidade Nova de isboa, CostadaCaparica,Portuga Correspondence shoud be addressed to Rodofo Oiveira; rado@fct.un.pt Received 2 June 206; Accepted 8 September 206 Academic Editor: Babak Shotorban Copyright 206 D. Oiveira and R. Oiveira. This is an open access artice distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. The mutipe nodes forming a Radio Frequency (RF) Energy Harvesting Network (RF-EHN) have the capabiity of converting received eectromagnetic RF signas in energy that can be used to power a network device (the energy harvester). Traditionay the RF signas are provided by high power transmitters (e.g., base stations) operating in the neighborhood of the harvesters. Admitting that the transmitters are spatiay distributed according to a spatia Poisson process, we start by characterizing the distribution of the RF power received by an energy harvester node. Considering Gamma shadowing and Rayeigh fading, we show that the received RF power can be approximated by the sum of mutipe Gamma distributions with different scae and shape parameters. Using the distribution of the received RF power, we derive the probabiity of a node having enough energy to transmit a packet after a given amount of charging time. The RF power distribution and the probabiity of a harvester having enough energy to transmit a packet are vaidated through simuation. The numerica resuts obtained with the proposed anaysis are cose to the ones obtained through simuation, which confirms the accuracy of the proposed anaysis.. Introduction The nodes forming a Radio Frequency (RF) Energy Harvesting Network (RF-EHN) have the capabiity of converting received eectromagnetic RF signas in energy. The RF energy is converted by an energy harvester device, which is composedofanrfantenna,abandpassfiterparametrizedto the RF signas, and a rectifying circuit abe to convert RF to DC power []. In this way the converted RF signas are used to charge a battery (usuay a supercapacitor) with finite capacity [2]. The harvested energy accumuated in the battery can then be used to transmit a packet. However, the transmission is ony possibe if the eve of accumuated energy is higher than a given threshod representing the minimum eve of energy required to compete a packet transmission. Recenty, RF energy harvesting has attracted much attention and many efforts are being dedicated to deveop innovative RF energy harvesting technoogies as we as to investigate the performance of the networks formed by the harvesting devices. The RF energy harvesting iterature dedicated to the efficient design of RF harvesting devices (see [3 7] for a few exampes) is mainy focused on the minimization of the oss effects due to the RF-to-DC conversion and battery charging process. A different focus is aso found in the iterature, where the main goa is the study and characterization of RF-EHNs. Adopting a generic mode for the RF energy harvesting devices, the goas are usuay reated with the scheduing of the harvesting devices in order to maximize the utiization of the RF energy and the frequency band constrained by specific throughput fairness poicies [8]; the optimization of the harvester communication task to dea with the mutipe tradeoffs associated with the physica and MAC ayers [9, 0]; the characterization of the RF-EHN performance (throughput) and stabiity when RF energy harvesting is adopted []. Reference [2] investigates the performance (throughput) of a sotted Aoha random access wireess network consisting of two types of nodes: with unimited energy suppy and soey powered by an RF energy

2 2 Mathematica Probems in Engineering harvesting circuit. To iustrate the design considerations of RF-based harvesting networks, [3] points out the primary chaenges of impementing and operating such networks, incuding nondeterministic energy arriva patterns, energy harvesting mode seection, and energy-aware cooperation among base stations. Reference [4] adopts a stochastic geometry framework based on the Ginibre mode to anayze the performance of sef-sustainabe communications over ceuar networks with genera fading channes. The expectation of the RF energy harvesting rate, the energy outage probabiity, and the transmission outage probabiity are evauated over Nakagami-m fading channes. RF-EHNs may aso act as cognitive radio networks (CRNs), that is, using the spectrum in an opportunistic way without being icensed. Severa works have expored these kinds of networks. Reference [5] provides an overview of the RF-EHNs incuding system architecture, RF energy harvesting techniques, and existing appications. The authors aso expore various key design issues in the deveopment of RF- EHNs, incuding cognitive radio networks. The work in [6] provides a comprehensive overview of recent deveopment and chaenges regarding the operation of cognitive radio networks powered by RF energy. Spectrum efficiency and energy efficiency are two critica issues in designing cognitive radio RF-EHNs. Reference [7] provides an overview of the RF-powered CRNs and discusses the chaenges that arise for dynamic spectrum access in these networks. Focusing on the trade-off among spectrum sensing, data transmission, and RF energy harvesting, the authors discuss the dynamic channe seection probem in a mutichanne RF-powered CRN. Reference [8] proposes a nove method for wireess networks coexisting where ow-power mobies in a secondary network, harvest ambient RF energy from transmissions by nearby active transmitters, whie opportunisticay accessing the spectrum icensed to the primary network. The authors anayze the transmission probabiity of harvesting terminas and the resuting spatia throughput. The optima transmission power and terminas density are aso derived for maximizing the throughput. The work in [9] considers an RF-powered green cognitive radio network, where a centra node harvests energy from ambient sources and wireessy deivers random harvested energy to cognitive users. The work evauates the performance of such a network, showing the feasibiity of the behavior if the energy transmission rate is beow a certain threshod. Reference [20] considers a network where the unicensed users can perform channe access to transmit a packet or to harvest RF energy when the seected channe is ide or occupied by the primary user, respectivey. The work is mainy focused on finding the channe access poicy that maximizes the throughput of the secondary user. Reference [2] anayzes an energy harvestingbased cognitive radio system to find the optima spectrum sensing time, which maximizes the harvested energy. The work in [22] anayzes a cognitive and energy harvestingbased device-to-device (D2D) communication in ceuar networks. The authors empoy toos from stochastic geometry to evauate the performance of the proposed communication system mode with genera path-oss exponent in terms of outage probabiity for D2D and ceuar users. One of the work concusions is that energy harvesting can be a reiabe aternative to power cognitive D2D transmitters, whie achieving acceptabe performance. In this work we are particuary focused on the characterization of the RF power received by each harvester and its impact in terms of the probabiity of accumuating enough energy to transmit a packet. A generaized radio propagation environment is considered. Assuming that the sources of high power RF signas (e.g., base stations) are distributed according to a spatia Poisson process, we characterize the distribution of the received RF power from the mutipe transmitters. Path oss, shadowing, and fading effects are considered. The distribution of the RF power is then used to derive the probabiity of a harvester node having enough energy to transmit a packet after a given period of time. A soft computationa mode (Gaussian approach) and a more compex mode (non-gaussian approach) are presented to compute the probabiity of a harvester node having enough energy to transmit. These are the main contributions of the paper. Considering mutipe spatia and propagation scenarios, we vaidate the distribution of the RF power andtheprobabiityofaharvesterhaveenoughenergyto transmit a packet. The numerica resuts obtained with the proposed anaysis are cose to the ones obtained through simuation, which confirms the accuracy of the proposed anaysis.inthisway,weprovideacharacterizationofthe battery charging time considering innovative assumptions, incuding the spatia distribution of the RF transmitters, the propagation effects, and the osses associated with the RF-to- DC conversion and battery charging process. The proposed mode can thus be adopted to determine the probabiity of a harvester accumuating enough energy after a given period of time, which is a determinant condition to compute the throughputofrf-ehns.asfarasweknown,thisisthe first work to derive such a probabiity when the mutipe RF signas received by the harvesters are differenty affected by mutipe propagation effects. The rest of the paper is organized as foows. The system description is presented in Section 2. The characterization of the received RF power is characterized in Section 3. Section 4 derives the probabiity of accumuating enough energy to transmit a packet through the Gaussian and non- Gaussian approaches. Finay, vaidation resuts are presented in Section 5 and concusions are drawn in Section System Description This work considers a RF energy harvesting network, where each node accumuates energy from the base stations and other RF transmitters ocated in the neighborhood. A harvester node is abe to initiate a packet transmission whenever the eve of accumuated energy is above a transmission threshod. 2.. Spatia Distribution of the RF Transmitters. We consider the scenario iustrated in Figure, where the node N H (the harvester node) accumuates energy from the transmitters that might be ocated in the area A = π((r o )2 (R i )2 ).The

3 Mathematica Probems in Engineering 3 The PDF of R canbewrittenastheratiobetweenthe perimeterofthecircewithradiusx and the tota area A, being represented as foows: N H R i R o R2 i R 2 o R3 i Figure : The harvester node N H receives RF power from the transmitters ocated in the area A = π((r o )2 (R i )2 ). R i R o f R (x) = { 2πx R i { A <x<r o, { 0, otherwise. To characterize the distribution of Ψ i the sma-scae fading (fast fading) and shadowing (sow fading) effects must be considered. The ampitude of the sma-scae fading effectisassumedtobedistributedaccordingtoarayeigh distribution, which is represented by (4) f ζ (x) = x σ 2 e x2 /2σ 2 ζ, (5) ζ area A can be obtained via cacuus by dividing the annuus upintoaninfinitenumberofannuiofinfinitesimawidthdχ and area 2πχ dχ and then integrating from χ=r i to χ=r o ; that is; A= R o 2πχ dχ. UsingtheRiemannsum,A can be Ri approximated by the sum of the area of a finite number ()of annui of width ρ, A A, () where A = π((r o )2 (R i )2 ) denotes the area of the annuus. R o =(R i +ρ)and R i =(R i +( )ρ)represent the radius of the arger and smaer circes of the annuus,respectivey. The number of transmitters ocated in a specific annuus {,...,}, represented by the random variabe (RV) X,is approximated by a Poisson process, being its Probabiity Mass Function (PMF) for a finite domain given by [23] ((β A ) k /k!) e β A P(X =k)= n i=0 ((β A ) i, /i!) e β A k=0,,...,n, where β is the spatia density of the RF nodes transmitting in the annuus and n is the tota number of mobie nodes Propagation Assumptions. We consider that the RF power I i received by the harvester N H from the RF transmitter i is given by (2) I i =P Tx ψ i r α, (3) where P Tx is the transmitted power eve of the ith RF transmitter (P Tx = mw is assumed for each node) and ψ i is an instant vaue of the fading and shadowing gain observed in the channe between the receiver N H and the transmitter node i. r represents the distance between the ith transmitter and the receiver. The vaues r and ψ i represent instant vaues of the random variabes R and Ψ i,respectivey. α represents the path-oss coefficient. where x istheenveopeampitudeofthereceivedsigna.2σ 2 ζ is the mean power of the mutipath received signa. 2σ 2 ζ = isadoptedinthisworktoconsiderthecaseofnormaized power. Regarding the shadowing effect, we have assumed that it foows a og-norma distribution f ξ (x) = 2 2πσ ξ x e (n(x) μ) /2σ 2 ξ, (6) where σ ξ is the shadow standard deviation when μ = 0. The standard deviation is usuay expressed in decibes and is given by σ ξdb = 0σ ξ / n(0). Forσ ξ 0,noshadowing resuts. Athough (6) appears to be a simpe expression, it is often inconvenient when further anayses are required. Consequenty, [24] has shown that the og-norma distribution can be accuratey approximated by a Gamma distribution, defined by f ξ (x) Γ (θ) ( θ θ ) x θ e x(θ/ωs), (7) ω s where θ is equa to /(e σ2 ξ )and ωs is equa to e μ (θ + )/θ. Γ( ) represents the Gamma function. The probabiity density function of Ψ i is thus represented by f Ψi (x) f ζ 2 (x) f ξ (x) 2 Γ (θ) ( θ (θ+)/2 ) x (θ )/2 K ω θ ( 4θx ), s ω s which is the Generaized-K distribution, where K θ ( ) is the modified Besse function of the second kind. Due to the anaytica difficuties of the Generaized- K distribution, an approximation of the PDF (8) by a more tractabe PDF is needed. Reference [25] provides an approximation of the Generaized-K distribution by using the moment matching method to determine the parameters of the approximated Gamma distribution. With this method, (8)

4 4 Mathematica Probems in Engineering [25] shows that the scae (θ ψ )andshape(k ψ ) parameters of the Gamma distribution are given by θ ψ =( 2 (θ+) )ω θ s, (9) respectivey. k ψ = 3. RF Received Power 2 (θ+) /θ, (0) 3.. RF Received Power due to the Transmitters ocated within the Annuus. The amount of RF power received by the harvester node N H ocated in the centre of an annuus is given by Σ= n i= I i, () where I i is the RF power received from the ith transmitter and n is the tota number of transmitters in the annuus. et M i I (s) represent the MGF of the ith transmitter ocated within the annuus (i=,...,n )givenby M i I (s) = E I i [e si i ]=E Ψi [E R [e si i ]]. (2) Using the PDF of the distance given in (4) and the PDF of the sma-scae fading and shadowing effects in (8), the MGF of the power received by the node N H from the ith transmitter in(2)canbewrittenasfoows: M i + R I (s) = o e si i f R (r ) f Ψi (ψ i ) dr dψ i, (3) 0 Ri which using (3), (9), (0), and (4) can be simpified to M i I (s) = 2π A (2 + k ψ α) (P Tx θ ψ s) k ψ (4) ((R o )2+k ψα (R o ) (R i )2+k ψα (R i )), where (x) = 2 F (k ψ,k ψ +2/α,+k ψ +2/α, x α /P Tx θ ψ s) and 2F represents the Gauss Hypergeometric function [26]. Departing from the fact that the individua power I i is independent and identicay distributed when compared to the other transmitters, the PDF of the aggregate RF power I given a tota of k active transmitters is the convoution of the PDFs of each I i. Foowing this rationae, the MGF of I is given by M I/k (s) =M I (s) M2 I (s) Mk I (s) =(M i I (s))k. (5) Using the aw of tota probabiity, the PDF of the RF power I canbewrittenas f I (j) = n k=0 f I (j X =k)p (X =k), (6) eading to the MGF of the aggregate power, I, whichcanbe written as E [e si ]= = n k=0 n k=0 + P (X =k) P (X =k)m I/k (s). e sj f I (j X =k)dj Using (5), the MGF of I is given as foows: E [e si ]= n k=0 (7) P (X =k)e k n(mi I (s)). (8) Using the MGF of the Poisson distribution in (8), the MGF of I is finay given by E [e si ]=e β A (M i I (s) ). (9) The first- and second-order statistics of the aggregate RF power received by N H from the transmitters ocated within the annuus are an important too. E[I], the expected vaue of the aggregate RF power, can be determined by using the awoftotaexpectation.itcanbeshownthat E [I] = E [E [I X ]] =2πβ P Tx e σ 2 ξ ( (R o )2 α (R 2 α i )2 α ). (20) Making simiar use of the aw of Tota Variance, the variance of the aggregate RF power can be described as Var [I] = Var [I i ] E [X ]+E [I i ] 2 Var [X ]. (2) Since X is given by a Poisson distribution (with mean β A ), the variance of the RF power is given as foows: Var [I] =β A ( 2 M i I (0) s 2 ) =πβ P 2 Tx k ψθ 2 ψ ( + k ψ)( (R o )2 2α (R α i )2 2α ). (22) Thefirstandsecondmomentscanbematchedwiththe respective moments of a given distribution to obtain a cosedform approximation for the aggregate received RF power. Asshownin[27],theaggregateRFpowerduetopathoss, fast fading, and shadowing effect can be approximated by a Gammadistribution.Consequenty,theshapeandthescae parameters of the Gamma distribution, denoted by k and θ, are, respectivey, given by k E [I]2 Var [I], (23) θ Var [I] E [I]. (24)

5 Mathematica Probems in Engineering RF Received Power due to the Transmitters ocated within Annui. As shown in the previous subsection, the RF power I received from the transmitters ocated within the th annuusisapproximatedbyagammadistribution,withmgf M I (s) = ( θ s) k.sincetheannuusofwidthr o R i where the transmitters are ocated can be expressed as a summation of annui of width ρ, the MGF of the aggregate power received from the transmitters ocated within the annui is given by M Iagg (s) = ( θ s) k. (25) Finay the expectation of the aggregate RF power can be computed as foows: E [I agg ]= M I agg (0). (26) s 3.3. Distribution of the Aggregate RF Power. The aggregate RF power may be stated as being the summation of the individua aggregated RF powers received from the transmitters ocated within each annuus. Expressions for the PDF and the CDF of the summation of independent Gamma random variabes were initiay derived by Mathai in [28]. Those were simpified in [29] in order to be computed more efficienty. et {Z } be independent but not necessariy identicay distributed Gamma variabes with parameters k (shape) and θ (scae). The PDF of the aggregate RF power is written as I agg = Z,whichcanbeapproximatedby[29] f Iagg (x) ( θ θ ) k + w=0 δ w x ( k+w ) exp ( x/θ ), θ ( k +w) Γ( k +w) (27) where θ = min {θ }, δ w coefficients are computed recursivey, δ w+ = (/(w+)) w+ i= [ k ( θ /θ ) i ]δ w+ i,andδ 0 =. Γ( ) is the Gamma function. Finay, the CDF of I agg, F Iagg (x) = x f I agg (z)dz,iscomputedasfoows[29]: F Iagg (x) ( θ k ) θ w=0 δ w θ k +w Γ( k +w) x z k+w exp ( z )dz. 0 θ 4. Probabiity of Transmission (28) 4.. Gaussian Approach. Departing from the fact that the RF power received from the transmitters ocated in the annuus canbeapproximatedbyagammadistribution, Gamma (k,θ ), (29) with k and θ given by (23) and (24), respectivey, the enveope signa (ampitude) received from the transmitters is given by the square root of a Gamma distributed random variabe, which is given by a Generaized Gamma distribution with the foowing parameters, A I = GG ( θ,2k,2). (30) Since a Gamma distribution, with shape k and scae θ,isthe sum of k Exponentia (/θ ) distributions, using the Centra imittheorem(ct),whenk is arge, the Generaized Gamma distribution can be approximated by a norma distribution [30]. In these conditions the ampitude of the aggregatesignasreceivedbytheharvestern H from the transmitters ocated in the annuus canbeasoapproximated by a norma distribution represented by A I N (μ A I,σ2 A I η 2 θ ( Γ(k +) Γ(k ) Γ(k ) N(η θ +/2), Γ(k ) Γ(k +/2) 2 Γ(k ) 2 )), (3) where the oss factor 0<η<represents the osses associated with the RF-to-DC conversion and battery charging efficiency. During the battery charging period, the received power (A I )2 isaccumuatedinadiscreteperiodoftimeδ t.the amount of energy stored in the battery of the harvester node during n t time intervas is given by n t ε =Δ t n= 2 A I. (32) Considering the unit variance random variabe ε = n t n= A I /σ A I 2,with ε =ε σ2 A I, (33) and considering Δ t =forthe sake of simpicity, ε foows a noncentra Chi-squared distribution with noncentraity parameter λ = n t k= (μ A /σ I A )2.Whenn I t is arge enough, it is possibe to use the Centra imit Theorem to approximate the Chi-square distribution to a Gaussian distribution [3], and the foowing approximation hods ε N (n t +λ,2(n t +2λ )). (34) Using (33), the energy accumuated in the battery of the harvester node N H due to the transmitters ocated in the annuus foows the foowing Gaussian distribution: ε N (μ,σ 2 ) N (σ 2 A (n I t +λ ),σ 4 A [2 (n I t +2λ )]). (35)

6 6 Mathematica Probems in Engineering Because annui are considered, the energy accumuated in the battery of the harvester node N H due to the transmitters ocated in the annui is given by ε= and ε foows the foowing distribution ε N ( μ, ε, (36) σ 2 ) N (μ Σ,σ 2 Σ ). (37) Therefore, denoting γ as the eve of battery charge (accumuated energy) required to transmit a packet, the probabiity of reaching a γ eve of accumuated energy after n t units of time is given by P C = Q ( γ μ Σ σ 2 Σ ), (38) where Q(x) = (/ 2π) /2 du is the compementary x e u2 distribution function of the standard norma Non-Gaussian Approach. In the ast subsection the CT was used to approximate the ampitude of the aggregate signas received by the harvester N H from the transmitters ocated in the annuus, A I,asrepresentedin(3).However, when k issma,thectdoesnothodand,consequenty,the Gaussian approach is not vaid. In what foows we present a formuation when CT does not hod. Whie the formuation is more computationay compex, it exhibits higher accuracy for ow k vaues. Departing from the MGF in (25), the Characteristic Function (CF) of the RF power received from the annuus is written as φ (t) = ( ηθ it) k, (39) where 0<η<is the oss factor. When the aggregate power from the annui is considered the CF is written as foows: φ Iagg (t) = ( ηθ it) k. (40) Because the amount of energy received in n t sampes is expressed as the CF of ε is as foows: φ ε (t) = = n t b= ε= (φ Iagg (t)) = n t n= I agg (t), (4) n t b= ( [( ηθ it) κ ] n t. ( ηθ it) κ ) (42) Tabe : Parameters adopted in the vaidation and simuations. ρ {400, 200, 00, 4} m {2, 4, 00} β {,2,3,4} 0 7 nodes/m 2 α 2 η 0.5 Δ t minute R i 0 m R o 40 m P Tx 20 W σ ξdb 4.5 db γ 0 mah n t 20 Using the Fourier Transform, the PDF of ε is written as f ε (x) = + 2π e itx φ ε (t) dt. (43) Again, the probabiity of reaching a γ eve of accumuated energy after n t unitsoftimeisgivenby γ P C (ε>γ)= P(ε γ)= f ε (x) dx. (44) 5. Vaidation Resuts This section describes a set of simuations and numerica resuts to vaidate the anaytica methodoogy proposed in the paper. The simuated scenario considered a spatia circuar area A as described in Section 2 with R i = 0mand R o = 40 m. The mutipe nodes were spread over the area A according to the spatia Poisson process and 4 different spatia densities were simuated, β = {, 2, 3, 4} 0 4 nodes/m 2. In each simuation the RF propagation scenario described in Section 2 was parametrized with α=2,andσ ξdb = 4.5 db. Finay, we have considered the battery operation votage equa to V, the oss factor η = 0.5, and the required energy threshod to transmit a packet (γ) equato0mah.the parameters used in the vaidation are summarized in Tabe. The first resuts, presented in Figure 2, compare the CDF of the RF received power computed with (28). The simuated resuts were obtained for the spatia density vaue β = 0 4. In the mode, a different number ( = {2, 4, 00}) of annui were adopted to compute the mode and compare the accuracyofthemodefordifferentnumberofannui.as can be seen, the accuracy of the mode increases with the number of annui considered in the mode. This is because as more annui are considered for the same circuar area A = π((r o )2 (R i )2 ),thewidthofeachannuusρ decreases, eading to a more accurate vaue of the mean and variance (E[I] and Var[I], resp.) of the RF power received from the transmitters ocated in a singe annui. This fact increases the accuracy of the conditions in (23) and (24), eading to a more accurate characterization of the distribution of the received RF power. From the resuts potted in Figure 2, we observe that for = 00 the numerica resuts are cose to the resuts

7 Mathematica Probems in Engineering 7 CDF I agg (mw) Simuation Mode, =4 Mode, =2 Mode, = 00 Figure 2: CDF of the RF power received by the node N H from the transmitters ocated in the area A = π((r o )2 (R i )2 ). CDF β =e 4 β =2e 4 I agg (mw) β =3e 4 β =4e 4 Figure 3: CDF of the RF power received by the node N H considering different densities of transmitters (β in nodes per square meter) ocated in the area A = π((r o )2 (R i )2 ). obtained through simuation, confirming the accuracy of the proposed mode. The numerica resuts presented in Figure 3 compare the CDF of the RF received power (computed with (28)) for different spatia density vaues (β = { 0 4,2 0 4,3 0 4,4 0 4 }). The numerica resuts were obtained considering = 00 (consequenty ρ = 4). As expected, the resuts confirm that the average of the RF power received by the harvester increases with the spatia density of the transmitters. In Figure 4 we compare the probabiity of having enough energy accumuated in the harvester battery to transmit a P C Time (minutes) Simuation Mode, = Mode, =4 Mode, =40 Figure 4: Gaussian approach mode: probabiity of having enough energy accumuated in the harvester battery to transmit a packet (P C ) for different density β=4 0 4 of transmitters (β in nodes per square meter) ocated in the area A = π((r o )2 (R i )2 ). packet (P C ). The probabiity was computed using (38); that is, the Gaussian approach was adopted. The simuated vaues were obtained for the spatia density vaue β = nodes/m 2.Thechargingthreshodγ was defined to 0 mah and we have considered a battery votage of V. The mode was computed for =(ρ = 400), =2(ρ = 200), and = 40 (ρ =0).Ascanbeobserved,theresutscomputedwith the mode do not match with the ones obtained by simuation. This fact was intentionay expoited to show that whie for different parameterizations the mode and the simuation resuts match, for the specific parameterization adopted in the vaidation scenario the CT does not hod because the k vaues are too sma. Consequenty, (30) is not an accurate approximation and a arge deviation of the P C s mode is observed.inthiscase,itwoudbebettertoadoptthenon- Gaussian approach, because it eads to more accurate mode resuts. In Figure 5 we compare the probabiity of having enough energy accumuated in the harvester battery to transmit a packet (P C ). The probabiity was computed using (44), that isthenon-gaussianapproach,for=2(ρ = 200), =4 (ρ = 00), and =40(ρ =0). The simuated vaues are the same as depicted in Figure 4; that is, the considered scenario is the same. As can be observed, the resuts computed with the mode do not match with the ones obtained by simuation for =2and =4.However,ifmoreannuiareused,themode accuratey characterizes P C,asisthecasefor=40in the figure.thisfactisduetotheapproximationofk and θ in (23) and (24), respectivey. As the number of annui () increases, E[I] and Var[I] in (23) and (24) become more accurate. As can be observed, the resuts computed with the mode for =40arecosetotheonesobtainedbysimuation.Moreover, the probabiity of reaching a battery charging eve equa

8 8 Mathematica Probems in Engineering P C Time (minutes) Simuation Mode, =4 Mode, =2 Mode, =40 Figure 5: Non-Gaussian approach mode: probabiity of having enough energy accumuated in the harvester battery to transmit a packet (P C ) for different density β=4 0 4 of transmitters (β in nodes per square meter) ocated in the area A = π((r o )2 (R i )2 ). to the γ threshod increases over time, as expected. The resuts confirm the accuracy of the proposed characterization, which may be easiy adopted to evauate the probabiity of charging over time. Finay, we highight that the mean aggregate RF power (I agg ) considered in the vaidation scenario is ow to show the error of the Gaussian approach. For higher I agg vaues, the error of the Gaussian approach becomes smaer and the mode becomes more accurate. The non-gaussian approach is generay a better soution (because it does not depend on the CT); however it exhibits a higher computationa compexity. 6. Fina Remarks In this paper we have characterized the battery charging time of a harvester node that accumuates the received RF energy in a battery. Admitting that the transmitters are spatiaydistributedaccordingtoaspatiapoissonprocess, we use the distribution of the received RF power from mutipe transmitters to derive the probabiity of a harvester having enough energy to transmit a packet after a given amount of charging time. The distribution of the RF power and the probabiity of a harvester node having enough energy to transmit a packet are vaidated through simuation. The numerica resuts obtained with the proposed anaysis are cose to the ones obtained through simuation, which confirms the accuracy of the proposed anaysis. Competing Interests The authors decare that they have no competing interests. Acknowedgments This work was partiay supported by the Portuguese Science and Technoogy Foundation (FCT/MEC) under the project UID/EEA/50008/203. References [] T. Soyata,. Copeand, and W. Heinzeman, RF energy harvesting for embedded systems: a survey of tradeoffs and methodoogy, IEEE Circuits and Systems Magazine,vo.6,no., pp , 206. [2]X.u,P.Wang,D.Niyato,D.I.Kim,andZ.Han, Wireess networks with RF energy harvesting: a contemporary survey, IEEE Communications Surveys and Tutorias, vo.7,no.2,pp , 205. [3] Y.Yao,J.Wu,Y.Shi,andF.F.Dai, Afuyintegrated900-MHz passive RFID transponder front end with nove zero-threshod RF-DC rectifier, IEEE Transactions on Industria Eectronics, vo.56,no.7,pp ,2009. [4] T.Sater,K.Choi,M.Peckerar,G.Metze,andN.Godsman, RF energy scavenging system utiising switched capacitor DC-DC converter, Eectronics etters,vo.45,no.7,pp ,2009. [5]G.Papotto,F.Carrara,andG.Pamisano, A90-nmCMOS threshod-compensated RF energy harvester, IEEE Soid-State Circuits,vo.46,no.9,pp ,20. [6] S. Scorcioni,. archer, A. Bertacchini,. Vincetti, and M. Maini, An integrated RF energy harvester for UHF wireess powering appications, in Proceedings of the IEEE Wireess Power Transfer (WPT 3), pp , IEEE, Perugia, Itay, May 203. [7] B. R. Franciscatto, V. Freitas, J. M. Duchamp, C. Defay, and T. P. Vuong, High-efficiency rectifier circuit at 2.45 GHz for ow-input-power RF energy harvesting, in Proceedings of the European Microwave Conference (EuMC 3), pp , IEEE, Nuremberg, Germany, October 203. [8] H. Ju and R. Zhang, Throughput maximization in wireess powered communication networks, IEEE Transactions on Wireess Communications, vo. 3, no., pp , 204. [9] R. Zhang and C. K. Ho, MIMO broadcasting for simutaneous wireess information and power transfer, IEEE Transactions on Wireess Communications,vo.2,no.5,pp ,203. [0] X.Zhou,R.Zhang,andC.K.Ho, Wireessinformationand power transfer: architecture design and rate-energy tradeoff, IEEE Transactions on Communications, vo. 6, no., pp , 203. [] N. Pappas, J. Jeon, A. Ephremides, and A. Traganitis, Optima utiization of a cognitive shared channe with a rechargeabe primary source node, Communications and Networks, vo.4,no.2,pp.62 68,202. [2] A. M. Ibrahim, O. Ercetin, and T. EBatt, Stabiity anaysis of sotted aoha with opportunistic RF energy harvesting, IEEE Journa on Seected Areas in Communications,vo.34,no.5,pp , 206. [3] A. Ghazanfari, H. Tabassum, and E. Hossain, Ambient RF energy harvesting in utra-dense sma ce networks: performance and trade-offs, IEEE Wireess Communications, vo.23, no. 2, pp , 206. [4]X.u,I.Fint,D.Niyato,N.Privaut,andP.Wang, Sefsustainabe communications with RF energy harvesting: ginibre point process modeing and anaysis, IEEEJournaonSeected Areas in Communications,vo.34,no.5,pp ,206.

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