Performance Analysis of Simultaneous Wireless Information and Power Transfer with Ambient RF Energy Harvesting
Xiao Lu, Ian Flint, Dusit Niyato, Nicolas Privault, Ping Wang
aa r X i v : . [ c s . I T ] J a n Performance Analysis of Simultaneous WirelessInformation and Power Transfer with Ambient RFEnergy Harvesting
Xiao Lu † , Ian Flint ‡ , Dusit Niyato † , Nicolas Privault ‡ , and Ping Wang †‡ School of Computer Engineering, Nanyang Technological University, Singapore † School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore
Abstract —The advance in RF energy transfer and harvestingtechnique over the past decade has enabled wireless energyreplenishment for electronic devices, which is deemed as apromising alternative to address the energy bottleneck of con-ventional battery-powered devices. In this paper, by using astochastic geometry approach, we aim to analyze the performanceof an RF-powered wireless sensor in a downlink simultaneouswireless information and power transfer (SWIPT) system withambient RF transmitters. Specifically, we consider the point-to-point downlink SWIPT transmission from an access point to awireless sensor in a network, where ambient RF transmitters aredistributed as a Ginibre α -determinantal point process (DPP),which becomes the Poisson point process when α approacheszero. In the considered network, we focus on analyzing the perfor-mance of a sensor equipped with the power-splitting architecture.Under this architecture, we characterize the expected RF energyharvesting rate of the sensor. Moreover, we derive the upperbound of both power and transmission outage probabilities.Numerical results show that our upper bounds are accurate fordifferent value of α . Index terms- RF energy harvesting, SWIPT, power splitting,determinantal point process, Poisson point process, Ginibremodel
I. I
NTRODUCTION
RF energy harvesting techniques have evolved as a promis-ing and cost-effective solution to supply energy for wirelessnetworks [1], [2]. The research efforts over the past decadehave advanced RF energy harvesting technique in circuitsensitivity, antenna efficiency, RF-to-DC conversion efficiency,and frequency range, etc [3]. The recent development has alsobrought commercial products into the market. For example,the Powercaster transmitter and Powerharvester receiver [4]allow a transmission of 1W or 3W isotropic wireless power,and reception of the power by converting the harvested RFwaves into electricity, respectively. In this context, RF signalshave been advocated to carry information as well as RF energyat the same time, which is referred to as the concept of simul-taneous wireless information and power transfer (SWIPT) [5].Recently, SWIPT has drawn great research attention and beenintensively investigated, for example, in SISO channel withoutand with co-channel interference, SISO relay channel, MISObroadcast system, MIMO broadcast system, and MIMO relaychannel [3].For performance analysis of large-scale RF energy har-vesting networks, stochastic geometry is a suitable tool thatcharacterizes random spatial patterns with point process. Poisson Point Process (PPP) modeling has been applied toanalyze RF energy harvesting performance in cellular net-work [6], cognitive radio network [7], relay network [8],and network-coded cooperative network [9]. The study in[6] investigates tradeoffs among transmit power and densityof mobiles and wireless charging stations which are bothmodeled as a homogeneous PPP. The authors in [7] studya cognitive radio network with energy harvesting secondaryusers, wherein both the primary and secondary networks aredistributed as independent homogeneous PPPs. The maximumthroughput of the secondary network has been characterizedunder the outage probability requirements for both primaryand secondary networks. Reference [8] focuses on the im-pact of cooperative density and relay selection in a large-scale network with transmitter-receiver pairs distributed asa PPP. In [9], the authors adopt PPP to model a two-waynetwork-coded cooperative network with energy harvestingrelays. The probability of successful data exchange and thenetwork lifetime gain are derived in closed-form expressions.Different from the above related work, our previous work in[10] adopts a more general analytical framework with Ginibre α -determinantal point process (DPP) modeling, wherein thePPP is a special case when α approaches zero. Consideringa stochastic network with ambient RF sources distributedfollowing a Ginibre α -DPP, we have investigated the uplinkperformance of an RF-powered sensor adopting separatedreceiver architecture, which equips the information receiverand RF energy harvester with independent antennas so thatthey function separately and observe different channel gains.In this work, we continue to adopt the Ginibre α -DPPmodeling approach, which is suitable for modeling randomphenomena where attraction/repulsion is observed. As at-traction (or clustering) and repulsion are common behaviorsin wireless communication systems, such as mobile cellularnetworks [11] and mobile social networks [12], we aimto analyze network performance by characterizing differentdegrees of repulsion with the Ginibre α -DPP. In particular,we focus on the downlink performance of a point-to-pointSWIPT system, where the receiver, i.e., an RF-powered sensor,performs information decoding and energy harvesting simul-taneously. The considered sensor adopts the power-splittingarchitecture [5], which allows the information receiver andRF energy harvester to share the same antenna. The sensor isassumed to be battery-free and operates based on the instant ig. 1. A network model of ambient RF energy harvesting. RF energy harvested from ambient RF transmitters. Based onthe considered model, we first characterize the expected RFenergy harvesting rate (in Watt), then derive the upper boundsof both power and transmission outage probabilities in closedforms. The performance analysis provides a useful insight intothe tradeoff among various network parameters.
Notations : Throughout the paper, we use E [ X ] to denote theprobabilistic expectation of a random variable X , and P ( A ) to denote the probability of an event A . A. Network Model
We consider a battery-free sensor node harvesting energyfrom an access point and ambient RF transmitters. The powersupply of the sensor solely comes from the instant harvestedRF energy. Figure 1 shows the considered network model,where the sensor node harvests RF energy and utilizes theinstantly harvested energy to power the circuit of the sensor.We assume that the ambient RF transmitters, e.g., wirelessrouters and cellular mobiles, which can be deemed as RFenergy sources for the sensor, are distributed as a generalclass of point processes, which will be specified in detail inSection I-B.The sensor is considered to adopt the power-splitting ar-chitecture [5], which enables the sensor to perform datatransmission and RF energy harvesting simultaneously. Asshown in Fig. 2, with the power-splitting architecture, thesensor is equipped with a single antenna. By adopting apower splitter, this architecture splits the received RF signalsinto two streams for the information receiver and RF energyharvester respectively. After the power splitting, the portionof RF signals split to the energy harvester is denoted by η (0 ≤ η ≤ ), and that to the information receiver is − η .The RF energy harvesting rate of the sensor node from theaccess point in a free-space channel P A H can be obtained basedon the Friis equation [13] as follows: P A H = ηβP A G A G H λ A (4 πd A ) , (1)where β is the RF-to-DC power conversion efficiency of thesensor node. P A , G A and λ A are the transmit power, transmit Other RF signal propagation models can also be used without loss ofgenerality in the analysis of this paper. Fig. 2. Power-splitting receiver architecture. antenna gain and transmitted wavelength of the access point,respectively. d A is the distance between the transmit antennaof the access point and the receiver antenna of the sensor node. G H is the receive antenna gain of the sensor node. Let x A ∈ R be the coordinates of the access point A in a referentialcentered at the sensor node. The distance can be obtained from d A = ǫ + k x A k , where ǫ is a fixed (small) parameter whichensures that the associated harvested RF power is finite inexpectation. Physically, ǫ is the closest distance that the accesspoint can locate near the sensor node.Let P k , G k and λ k denote the transmit power, transmitantenna gain and transmitted wavelength of the RF transmitter k ∈ K , respectively. As the focus of this paper is to analyzethe impact of the locations of ambient RF transmitters tothe performance of the sensor node, similar to the relatedwork [14], we intentionally make some other parameters tobe constants for ease of presentation and analysis. Specifically,we have P k = P S , G k = G S , and λ k = λ , for k ∈ K . Let x k ∈ R be the coordinates of the RF transmitter k . Similarto (1), we can calculate the RF energy harvesting rate fromeach RF transmitter k ∈ K . Then, the aggregated RF energyharvesting rate by the sensor node can be computed as follows: P P S H = ηβP A G A G H λ A (4 π k x A k ) + X k ∈K ηβP S G S G H λ (4 π ( ǫ + k x k k )) (2)where the second term represents the total energy harvestingrate from ambient RF transmitters. x k denotes the location ofRF transmitter k .For the considered SWIPT system, the scenarios of out-of-band transmission and in-band transmission need to beinvestigated. In the former, the access point transmits ona frequency band different from the one used for the RFenergy harvesting (without co-channel interference). In thelatter, the access point transmits on the same frequency band ofambient RF energy sources (with co-channel interference). Thedownlink information rate at the sensor can be computed asin (3) [15], where ξ in an indicator depending on whether weconsider an out-of-band (i.e., ξ = 0 ) or in-band transmissionscenario ( ξ = 1 ). σ and σ SP represent the additive whiteGaussian noise power and signal processing noise power,respectively. I P S denotes the interference from the ambientRF transmitters, which can be modeled as follows: I P S = X k ∈K (1 − η ) P S G S G H λ (4 π ( ǫ + k x k k )) . (4) B. Stochastic Modeling of Ambient RF Transmitters
We model the locations of RF transmitters using a pointprocess K on an observation window O := B (0 , R ) which is P S = ( W · log (cid:16) h A (1 − η ) P A ξI PS +(1 − η ) σ + σ SP (cid:17) if P P S H ≥ P C , if P P S H < P C , (3)the closed ball centered at the origin and of radius R > . Inother terms, K is an almost surely finite random collection ofpoints inside B (0 , R ) . We refer to [16] for the general theoryof point processes.We focus on the Ginibre α -DPP which is a type of α -DPP(see [17] for definitions and technical results). The Ginibreprocess is defined by the so-called Ginibre kernel given by K ( x, y ) = ρe πρx ¯ y e − πρ ( | x | + | y | ) , x, y ∈ O = B (0 , R ) . (5)We will write K ∼
Det( α, K, ρ ) when K is an α -DPPwith kernel K defined in (5) and density with respect to theLebesgue measure ρ . The spectral theorem for Hermitian andcompact operators yields the following decomposition for thekernel of K : K ( x, y ) = X n ≥ λ n ϕ n ( x ) ϕ n ( y ) , where ( ϕ i ) i ≥ is a basis of L ( O, λ ) , and ( λ i ) i ≥ the cor-responding eigenvalues. In e.g. [18], it is shown that theeigenvalues of the Ginibre point process on O = B (0 , R ) aregiven by λ n = Γ( n + 1 , πρR ) n ! , (6)where Γ( z, a ) := Z a e − t t z − d t, z ∈ C , a ≥ , (7)is the lower incomplete Gamma function. The eigenvectors of K are given by ϕ n ( z ) := 1 √ λ n √ ρ √ n ! e − πρ | z | ( √ πρz ) n , n ∈ N , z ∈ O. We refer to [18] for further mathematical details on the Ginibrepoint process.Lastly, we emphasize that the Ginibre α -DPP is stationary,in the sense that its distribution is invariant with respect totranslations, c.f. [18]. Hence, our choice of O = B (0 , R ) centered at the origin instead of x i is justified. C. Performance Metrics
We define the performance metrics of the sensor node asthe expected of RF energy harvesting rate, power outageprobability and transmission outage probability. The expectedRF energy harvesting rate is defined as: E P H , E (cid:2) P P S H (cid:3) . (8)Power outage occurs when the sensor node becomes inactivedue to lack of enough energy supply. The power outageprobability is then defined as follows: P po , P ( P H < P C ) , (9)where P C denotes the constant for power consumption of sensor node. Following practical models [19], the circuit powerconsumption of the sensor is assumed to be fixed.Let m ≥ denote the minimum transmission rate require-ment. If the sensor fails to meet this requirement, a transmis-sion outage occurs. The transmission outage probability canbe defined as follows: P to , P ( C < m ) . (10)Specifically, for the above two outage probabilities, wefocus on analyzing their upper bounds in this paper.II. P ERFORMANCE A NALYSIS
In this section we estimate the metrics defined in Section I-Cwhen
K ∼
Gin( α, ρ ) is the Ginibre α -DPP with parameter α = − /j , where j ∈ N ∗ , and density ρ > . A. RF Energy Harvesting Rate
The expected RF energy harvesting rate is evaluated asfollows, which is similar to the result obtained in Theorem 1in [10].
Theorem 1.
The expected RF energy harvesting rate in thepower-splitting architecture can be explicitly computed as E [ P P S H ] = ηβP A G A G H λ A (4 π k x A k ) + 2 πηβP S G S G H λ (4 π ) ρ (cid:18) ǫR + ǫ + ln( R + ǫ ) − − ln( ǫ ) (cid:19) (11) ≈ ǫ → ρηβP S G S G H λ π ln (cid:18) Rǫ (cid:19) . (12)We recall some remarks which were made in [10]. First, wenote that Theorem 1 implies that at the level of expectations,the Ginibre α -DPP behaves like a homogeneous PPP andin particular, the expectation of RF energy harvesting rateis independent of the repulsion parameter α . Therefore, onaverage, the harvested energy stays the same when α varies. Proof:
The proof is the same as that of Theorem 1 in[10], which we recall here for convenience. We have E [ P P S H ] = ηβP A G A G H λ A (4 π k x A k ) + ηβP S G S G H λ (4 π ) Z O ρ (1) ( x )( ǫ + k x k ) d x (13)by Campbell’s formula [16], where ρ (1) ( x ) = K ( x, x ) = ρ isthe intensity function of K given by [18]. We thus find E [ P P S H ] = ηβP A G A G H λ A (4 π k x A k ) + ηβP S G S G H λ (4 π ) π Z R ρ r ( ǫ + r ) d r, (14)y polar change of variable, and the integral on the r.h.s. iscomputed explicitly as follows: Z R r ( ǫ + r ) d r = (cid:18) ǫR + ǫ + ln( R + ǫ ) − − ln( ǫ ) (cid:19) , which yields the result. B. Power Outage Probability
Theorem 2.
Let us define γ P S := λ π s ηβP S G S G H P C − P A H , where P A H is defined in (1) .If P C ≥ P A H , then the following bound holds: P (cid:0) P P S H < P C (cid:1) ≤ Y n ≥ (cid:18) α Γ( n + 1 , πρ inf( R, γ
P S ) ) n ! (cid:19) − /α , (15) where Γ( z, a ) is the lower incomplete Gamma function definedin (7) .If P C < P A H , then P (cid:0) P P S H < P C (cid:1) = 0 .Proof: Note that P (cid:0) P P S H < P C (cid:1) = P X k ∈K ηβP S G S G H λ (4 π ( ǫ + k x k k )) < P C − P AH ! , (16)whence it suffices to apply Theorem 2 of [10] to conclude. C. Transmission Outage Probability
In contrast with what was obtained in Theorem 1, Theo-rem 2 shows that the power outage probability depends on therepulsion parameter α .We begin by studying the in-band transmission scenario inthe following theorem. Theorem 3.
Let us set T = h A P A m/W − − σ − σ SP − η . (17) Assume that we are in the in-band scenario, i.e. ξ = 1 . Thebound on the power outage probability is slightly differentdepending on the values of the parameters. Specifically, if P C − P A H > , then we obtain the bound in (18) : P (cid:0) C P S < m (cid:1) ≤ Y n ≥ (cid:18) α Γ( n + 1 , πρ inf( R, γ
P S ) ) n ! (cid:19) − /α + ρP S G S G H λ (cid:16) ǫR + ǫ + ln( R + ǫ ) − − ln( ǫ ) (cid:17) π max (cid:16) T, ηβ (cid:0) P C − P AH (cid:1)(cid:17) . (18) If P C − P AH ≤ , and T ≥ , then P (cid:0) C P S < m (cid:1) ≤ ρP S G S G H λ (cid:16) ǫR + ǫ + ln( R + ǫ ) − − ln( ǫ ) (cid:17) πT . (19) Lastly, if max (cid:0)
T, P C − P AH (cid:1) ≤ , then we have P (cid:0) C P S < m (cid:1) = 1 . Proof:
Using the definition of C P S given in (3), we find(20). By proceeding along the same lines as in the proof ofTheorem 2 of [10], we set f ( x k ) := P S G S G H λ (4 π ( ǫ + k x k k )) , for k ∈ K . Then, P (cid:0) C P S < m (cid:1) = P (cid:0) P P S H < P C (cid:1) + P X k ∈K f ( x k ) > max (cid:18) T, ηβ (cid:0) P C − P A H (cid:1)(cid:19)! (21)where P A H and T are defined in (1) and (17), respectively.From this, we conclude that if max (cid:16) T, ηβ (cid:0) P C − P A H (cid:1)(cid:17) ≤ ,then P (cid:0) C P S < m (cid:1) = 1 . Otherwise, if max (cid:16) T, ηβ (cid:0) P C − P A H (cid:1)(cid:17) > , then byMarkov’s inequality, we have P (cid:0) C P S < m (cid:1) ≤ P (cid:0) P P S H < P C (cid:1) + 1max (cid:16) T, ηβ (cid:0) P C − P A H (cid:1)(cid:17) E "X k ∈K f ( x k ) , (22)which can be computed as P (cid:0) C P S < m (cid:1) ≤ P (cid:0) P P S H < P C (cid:1) + ρP S G S G H λ (cid:16) ǫR + ǫ + ln( R + ǫ ) − − ln( ǫ ) (cid:17) π max (cid:16) T, ηβ (cid:0) P C − P A H (cid:1)(cid:17) , (23)by Theorem 1 of [10]. As for the first term of (23), it is upper-bounded by a straightforward application of Theorem 2, andwe find (24), where A is the indicator function of a set A ,i.e. the functional equal to on A and equal to elsewhere.Note that the right hand side of (24) might be larger than , and in that case we do better than the trivial inequality P (cid:0) C P S < m (cid:1) ≤ . Note also that (24) is a compact notationof the different cases discussed in Theorem 3.For the out-of-band transmission scenario, an upper boundof the transmission outage probability can also be derivedbased on Markov inequality, following similar steps similarto those in Theorem 3. Due to the space limit, we omit it inthis paper. III. N UMERICAL R ESULTS
We assume that all the ambient RF transmitters are LTE-enabled mobiles operating on the typical
M Hz fre-quency. The corresponding wavelength λ is . m . The (cid:0) C P S < m (cid:1) = P (cid:0) P P S H < P C (cid:1) + P (cid:0) C P S < m, P
P S H ≥ P C (cid:1) = P (cid:0) P P S H < P C (cid:1) + P (cid:16) h A (1 − η ) P A < (cid:0) (1 − η ) σ + σ SP + I P S (cid:1) (cid:16) m/W − (cid:17) , P P S H ≥ P C (cid:17) = P (cid:0) P P S H < P C (cid:1) + P max (cid:18) T, ηβ (cid:0) P C − P A H (cid:1)(cid:19) < X k ∈K P S G S G H λ (4 π ( ǫ + k x k k )) ! . (20) P (cid:0) C P S < m (cid:1) ≤ Y n ≥ (cid:18) α Γ( n + 1 , πρ inf( R, γ
P S ) ) n ! (cid:19) − /α { P C − P A H ≥ } + ρP S G S G H λ (cid:16) ǫR + ǫ + ln( R + ǫ ) − − ln( ǫ ) (cid:17) π max (cid:16) T, ηβ (cid:0) P C − P A H (cid:1)(cid:17) , (24) TABLE IP
ARAMETER S ETTING . Symbol G S , G A , G H β P S , P A W σ , σ SP Value 1.5 0.3 1W 10KHz -90dBmcircuit power consumption P C is fixed to be − dBm (i.e., . µW ) as in [10]. The other parameters adopted in thesimulations are shown in Table I unless specified otherwise.Note that the results for the PPP case are identical to that ofthe α -DPP, when α = 0 . The numerical results presented inthis section are averaged over simulation runs.We interpret the upper-bounds derived in the previoussection as worst-case scenarios. This leads us to performthe simulations of this section in a different regime, in anattempt to approach the upper bounds. The simulation underthis regime is known to perfectly approach the upper-boundof Theorem 2, whereas there is still a gap compared to theupper-bound of Theorem 3.Figure 3 shows the expected RF energy harvesting rateversus density of ambient RF transmitters. We can see thatthe simulation results, which were done in the general scenariodescribed in this paper, match the analytical expression (11)accurately over a wide range of transmitter densities ρ , i.e.,from . to . We observe that when ǫ = 0 . , the sensorachieves larger RF energy harvesting rate than that in the casewhen ǫ = 0 . . This result is expected since, from (2), thesmaller the distance ǫ (i.e, the RF transmitters can be locatednear the sensor) the more aggregated RF energy harvestingrate is available. We also find that the difference between theexact analytical results obtained from (11) and approximateresults obtained from (12) is also dependent on ǫ . Specifically,a smaller ǫ results in a more accurate approximation. As shownin Fig. 3, compared to when ǫ = 0 . , the approximate resultsmore closely approach the analytical results when ǫ = 0 . .Figure 4 shows the upper bound of the power outageprobability versus the density of ambient RF transmitters,when η = 0 . and η = 1 . We observe that P po is adecreasing function of the transmitter density ρ . The numericalresults, which were done in a worst-case scenario, are shownto approach the analytical expression in (15) accurately fordifferent settings of α . Moreover, a larger repulsion among −4 −3 Density of Ambient RF Transmitters R F E ne r g y H a r v e s t i ng R a t e RF Energy Harvesting Rate versus Density of Ambient RF Transmitters η =0.8, ε =0.1,Analysis η =0.8, ε =0.1,Approximation η =0.8, ε =0.1,Simulation η =0.8, ε =0.001,Analysis η =0.8, ε =0.001,Approximation η =0.8, ε =0.001,Simulation η =0.5, ε =0.1,Analysis η =0.5, ε =0.1,Approximation η =0.5, ε =0.1,Simulation Fig. 3. RF energy harvesting rate versus density of ambient RF transmitters. P o w e r O u t age P r obab ili t y Upper Bound of Power Outage Probability versus Density ofAmbient RF Transmitters DPP, α =−1,Upper boundDPP, α =−0.5,Upper boundPPP, Upper boundDPP, α =−1,SimulationDPP, α =−0.5,SimulationPPP, Simulation η =0.5 η =1 Fig. 4. Upper bound of power outage probability versus density of ambientRF transmitters. the location of the RF transmitters (i.e., smaller α ) results ina lower power outage probability.Figures 5 and 6 present the upper bound of the transmissionoutage probability versus the density of RF transmitters andversus the distance between the access point and the sensor,respectively, in an in-band transmission scenario. We set thetransmission rate requirement to be 0.02kbps and η to be 0.5.In Fig. 5, we see that, with the increase of the transmitterdensity, the upper bound of the transmission outage probabilityfirst decreases quickly, then begins to rebound slowly after acertain point. The rebound effect is caused by the increased T r an s m i ss i on O u t age P r obab ili t y Analysis,DPP, α =−1Analysis,PPPSimulation,DPP, α =−1Simulation,PPP Fig. 5. Upper bound of transmission outage probability versus density ofambient RF transmitters. A (in−band transmission)d A T r an s m i ss i on O u t age P r obab ili t y Analysis,DPP, α =−1Analysis,PPPSimulation,DPP, α =−1Simulation,PPP Fig. 6. Upper bound of transmission outage probability versus the distance d A ( ρ = 0 . ). interference due to the growth in transmitter density, whichlowers the achievable transmission rate. Similar to Fig. 5, weobserve in Fig. 6 that our analytical bound is tight when d A is small, and becomes more relaxed with the increase of d A .It can be seen that when d A is small, there is no transmissionoutage. This is because when the access point locates withina certain close range near the sensor, the sensor can receivenot only enough information rate but also sufficient powerfrom the SWIPT transmission alone, regardless of the ambienttransmitter density. Thus, the transmission outage probabilityequals zero in this case.IV. C ONCLUSION
We have analyzed the performance of an RF-poweredsensor network in a downlink SWIPT system with ambientRF transmitters. We have adopted a repulsive point process,called a Ginibre α -determintal point process, which allows tomodel a network where the locations of the RF transmittersdemonstrate repulsion. We have derived the expression of theexpected RF energy harvesting rate of the RF-power sensor.We have also characterized the worst-case performance ofthe sensor node in terms of the upper bounds of power andtransmission outage probabilities. The performance evaluationshows that the exact analytical results and simulation resultsare well matched with the simulation results. Therefore, theproposed analysis will be useful in practice. Our future workwill extend the performance analysis from a single-antenna network to a multi-antenna network. Another direction is toexplore the network performance in other metrics, such asdownlink coverage probability of the access point.A CKNOWLEDGEMENTS
This work was supported in part by Singapore MOE Tier 1(RG18/13 and RG33/12), MOE Tier 2 (MOE2013-T2-2-067)and MOE Tier 2 (ARC3/13).R
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