Energy Coverage in Millimeterwave Energy Harvesting Networks

Transcription

1 Energy Coverage in Millimeterwave Energy Harvesting Networs Talha Ahmed Khan, Ahmed Alhateeb and Robert W. Heath Jr. Abstract Wireless energy harvesting in millimeter wave mmwave) cellular networs is attractive, thans to the large antenna arrays and the anticipated dense deployment of these systems. The signal propagation at mmwave frequencies, however, shows peculiar propagation characteristics such as extreme sensitivity to building blocages. This wor analyzes the energy harvesting performance at receivers powered by a mmwave cellular networ. Leveraging tools from stochastic geometry, closed-form analytical expressions are derived to characterize the energy coverage probability at a typical receiver in terms of the cellular networ density, beamforming beamwidth, and channel parameters. Results show that there is an optimum transmit antenna beamwidth that maximizes the networ-wide energy coverage probability for many operating scenarios. Simulation results further suggest that mmwave energy harvesting could provide a substantial performance boost compared to lower frequency solutions. I. INTRODUCTION Millimeterwave mmwave) communications is a candidate technology for future 5G cellular networs. This is mainly due to the availability of large spectrum resources at higher frequencies which could enable ultra high data rates. Recent research suggests that mmwave systems will typically feature i) large-dimensional antenna arrays with directional beamforming at the transmitter/receiver which is motivated by the small wavelength that allows pacing a large number of antenna elements into small form-factors; and ii) a dense deployment of base-stations BSs) to ensure comparable coverage to ultra high frequency ) networs. These mmwave design features are also attractive for RF radio frequency) energy harvesting where a harvesting device may extract energy from the incident RF signals 2. This could potentially power the massive number of low-power wireless devices in future paradigms such as the Internet of Things 3. The signal propagation at mmwave frequencies, however, is prone to blocage by buildings 4. It is unclear if mmwave cellular networs will be more favorable for RF energy harvesting compared to the frequencies. Further, the networ-level design principles for mmwave energy harvesting systems are not well exposed. Hence, it is relevant to provide a networ viewpoint on energy harvesting in a mmwave cellular networ. The authors are with the Wireless Networing and Communications Group at The University of Texas at Austin. {talhahan, aalhateeb, rheath}@{utexas.edu}. This wor was supported by the National Science Foundation under Grant No and gifts from Mitsubishi Electric Research Labs, Cambridge and Noia. Wireless energy harvesting is becoming increasingly feasible due to the reduction in the power consumption requirements of wireless sensors and the improvements in harvesting technologies 5 7. This has motivated research in developing networ/system level insights for different wireless energy harvesting networs 2, 8 2. In 8, stochastic geometry tools have been used to characterize the performance of ambient RF energy harvesting in large-scale networs. In 9,, cognitive radio networs have been considered, and opportunistic wireless energy harvesting was proposed and analyzed. Simultaneous information and power transfer in a relay-aided networ was considered in, while 2 studied a hybrid networ consisting of cellular BSs overlaid with power beacons to charge mobiles. None of this wor, however, has considered wireless energy harvesting in a mmwave networ. In another line of wor, the performance of mmwave cellular networs in terms of signal-to-interference-and-noise ratio SINR) coverage and rate has been analyzed in 3 using stochastic geometry tools. In this paper, we provide an analytical framewor to characterize the performance of RF energy harvesting at mmwave frequencies using tools from stochastic geometry. In particular, closed-form expressions are derived for the energy coverage probability at a typical mmwave energy harvester in terms of the BS networ density, the antenna beam pattern and the mmwave channel parameters. The derived results are used to investigate the interplay between ey design parameters such as the transmit antenna beamwidth and the energy harvesting performance. Numerical results suggest that there is an optimum beamwidth that maximizes the energy coverage probability for many operating scenarios. Moreover, a comparison with networ reveals that mmwave energy harvesting could provide a superior performance. II. SYSTEM MODEL A. Networ Model We consider a large-scale cellular networ consisting of mmwave BSs and a population of energy harvesting nodes or users) that operate by harvesting energy in the mmwave band. We focus on energy harvesting performance leaving joint information and power transfer for future wor. The mmwave BSs are drawn from a homogeneous Poisson point process PPP) Φλ) of density λ. The user population is located according to another PPP Φ u λ u ) of density λ u, independently of Φ. The propagation at mmwave frequencies is prone to blocage by buildings 4, 3. To model this blocage effect,

2 we leverage the results in 4 and define a line-of-sight LOS) probability function pr) =e βr for a lin of length r, where β is a constant that depends on the building geometry: a BSreceiver lin of length r is declared LOS with a probability pr), independently of other lins. We now describe the channel model for a typical user located at origin. Empirical evidence suggests that mmwave frequencies exhibit different propagation characteristics for the LOS/NLOS lins 4. We let α L and α N be the path loss exponents for the LOS and NLOS lins respectively. We can now define g l r l ), the distance-dependent path loss for a user located a distance r l from the l-th BS: g l r l )= C L r αl l when the lin is LOS, where C L is the path loss intercept; and g l r l ) = C N r αn l for the NLOS case. We can similarly define h l to be the small-scale fading coefficient corresponding to a BS l Φ. Assuming independent Naagami fading for each lin, the small-scale fading power H l = h l 2 can be modeled as a normalized Gamma random variable, i.e., H l ΓN L, /N L ) when the lin is LOS and H l ΓN N, /N N ) for the NLOS case, where the fading parameters N L and N N are assumed to be integers for simplicity. We allow the user population to consist of two types of users, namely connected and nonconnected. A connected user is assumed to be tagged with the BS, either LOS or NLOS, that maximizes the average received power at that user. Moreover, we assume perfect beam alignment between a BS and its tagged users. For a nonconnected user, however, we do not assume any prior beam alignment with a BS. This allows us to model a wide range of scenarios. For instance, due to limited resources, the mmwave networ may directly) serve only a fraction of the user population as connected users, leaving the rest in the nonconnected mode. Another interpretation could be that due to the challenges associated with channel acquisition, not all the users could be served in the connected mode. We let ɛ be the probability that a randomly selected node is a connected user, independently of other nodes. With this assumption, we can thin the user PPP Φ u into two independent PPPs Φ u,con and Φ u,ncon, with respective densities ɛλ u and ɛ)λ u. Note that an arbitrary user, either connected or nonconnected, may experience an energy outage if the received power falls short of a required threshold. This threshold would depend on the power consumption as well as the sensitivity requirements of the harvesting circuit. We define P con λ, ) to be the energy coverage probability given an outage threshold for a connected user, while P ncon λ, ) denotes the same for the nonconnected case. With these definitions, we define the overall energy coverage Λɛ, λ, ) of the networ as Λɛ, λ, ) =ɛp con λ, )+ ɛ)p ncon λ, ) ) where the energy coverage probability is a function of several parameters such as the BS density, the channel propagation parameters, as well as the antenna beam patterns at the transmitter/receiver Section III). B. Antenna Model To compensate for higher propagation loss, mmwave BSs will use large directional antennas arrays. We assume that the BSs and users are equipped with N t and N r antenna elements. To simplify the analysis while capturing the ey antenna characteristics, we use a sectored antenna model inspired by the one in 3, 5. We use G M,m,θ, θφ) to characterize the antenna beam pattern, where φ gives the angle from the boresight direction, M denotes the directivity gain and θ the half power beamwidth for the main lobe, while m and θ give the corresponding parameters for the side lobe. With this notation, G Mt,m t,θ t, θ t ) denotes the antenna beam pattern at an arbitrary BS in Φ, and G Mr,m r,θ r, θ r ) denotes the same for an energy harvesting user in Φ u. We further define δ l = G Mt,m t,θ t, θ t φ l t )G Mr,m r,θ r, θ r φ l r ), the total directivity gain for the lin between the l-th BS and the typical user; φ l t and φ l r give the angle of arrival and angle of departure of the signal. Without any further assumptions about the beam alignment, we model the directivity gain δ l as a random variable. We assume the angles φ l t and φ l r are uniformly distributed in, 2π). Due to the sectored antenna model, the random variable δ l = D i with a probability p i i {, 2, 3, 4, 5}), where D i {M t M r,m t m r,m t M r,m t m r, } and p i {q t q r,q t q r, q t q r, q t q r,q o } in respective order; the constants q t = θt 2π, q t = θ t 2π, q r = θr 2π, q r = θ r 2π, and q o =2 q t q t q r q r. Note that for the connected mode, since we assume perfect beam alignment between the typical user and its serving BS hereby denoted by subscript ), the directivity gain δ = M t M r due to the sectored antenna model. III. MMWAVE ENERGY HARVESTING In this section, we provide analytical expressions to compute the energy coverage probability in a mmwave networ for both connected and nonconnected cases. We first provide some lemmas before stating the main analytical results. Lemma From 4): The probability density function PDF) of the distance from an energy harvesting user to its nearest LOS BS, given that the user observes at least one LOS BS, is given by τ L x) =2πλB L xpx)e 2πλ x vpv)dv 2) where x> and B L = e 2πλ vpv)dv is the probability that the receiver observes at least one LOS BS. Similarly, the distance distribution of the lin between the user and its nearest NLOS BS, given that the user observes at least one NLOS BS, is given by τ N x) =2πλB N x px))e 2πλ x v pv))dv 3) where x > and B N = e 2πλ v pv))dv is the probability that the user observes at least one NLOS BS. Lemma 2 From 3 ): Let ϱ L and ϱ N denote the probability that the energy harvesting user is connected to a LOS and a NLOS BS respectively, then ϱ L is given by ϱ L = B L e 2πλ ρ L x) pv))vdv τ L x) dx 4)

3 ) α where ρ L x) = CN N C L x α L α N and ϱ N = ϱ L. Lemma 3 From 3): Given that the energy harvesting user is connected to a LOS mmwave BS, the PDF of the lin distance is given by ˆτ L x) = B Lτ L x) e 2πλ ρl x) pv))vdv 5) ϱ L where x>. Given that the user is connected to a NLOS mmwave BS, the PDF of the lin distance is given by ˆτ N x) = B Nτ N x) e 2πλ ρn x) pv)vdv 6) ϱ N ) α for x> and ρ N x) = CL L C N x α N αl. Stochastic Geometry Analysis: Leveraging Slivnya s theorem 6, we consider a typical energy harvesting user located at origin. If the BSs transmit with power P t, the energy harvested at a typical receiver in a given time-slot) can be expressed as γ = ξ P t δ l H l g l r l )+σ 2 7) l Φλ) where the rectifier efficiency ξ is assumed to be unity without loss of generality, σ 2 gives the receiver noise power in the mmwave band, while the remaining parameters follow from Section II. The following theorem provides an analytical expression for the energy coverage probability P con λ, ) = Pr{γ >} at a connected user, where γ is given in 7) and is the energy outage threshold. Note that P con λ, ) can also be interpreted as the complementary cumulative distribution function CCDF) of the instantaneous harvested energy. Theorem : In a mmwave networ with density λ, the energy coverage probability P con λ, ) for the connected case given an energy outage threshold can be evaluated as P con λ, ) =P con,l λ, ˆ ) ϱ L + P con,n λ, ˆ ) ϱ N 8) where ˆ = σ 2, ϱ L = ϱ N is given in Lemma 2, while P con,l ) and P con,n ) are the conditional energy coverage probabilities given the serving BS is LOS and NLOS respectively. Further, P con,l λ, ) ) N = ) r g ζ L r) e Υ λ,,r) Υ 2 λ,,ρ Lr))ˆτ L r) dr ) NL, where ζ + L x) = aptmtmrcl N Lx α L the approximation constant a = NN!) N where N denotes the number of terms in the approximation, while r g defines the minimum lin distance and is included to avoid unbounded path loss at 9) the receiver. Similarly, P con,n λ, ) ) N = ) r g ζ N r) e Υ λ,,ρ 2 Nr)) Υ λ,,r)ˆτ N r) dr ) NN, where ζ + N x) = aptmtmrcn N Nx α N Υ λ,, x) =2πλ 2 Υ λ,, x) =2πλ 4 i= 4 i= p i p i x x ) + ap ) NL td i C L N L t αl pt)tdt, ) + ap ) NN td i C N N N t αn pt)) tdt, 2) and ˆτ L ) and ˆτ N ) follow from Lemma 3. Proof: See Appendix A. Recall that pt) = e βt is the LOS probability function defined in Section II-A, and captures the effect of building blocages. In 9), ζ L ) models the contribution from the LOS serving lin, Υ ) accounts for other LOS lins, and Υ 2 ) captures the effect of the NLOS lins. Similarly, ζ N ) in ) models the case where the serving BS is NLOS. Note that these terms further depend on the channel propagation conditions α L, α N, N L, N N, C L, C N ), the networ density λ as well as the antenna design parameters via D i, p i ). Furthermore, the outage threshold may depend on the sensitivity of the harvesting circuit, as well as the power requirements at a particular user. The following theorem characterizes the energy coverage probability at a typical user for the nonconnected case. Theorem 2: In a mmwave networ of density λ, the energy coverage probability for the nonconnected case P ncon λ, ) given an outage threshold can be evaluated using ) P ncon λ, ) ) N e Υ λ, ˆ,r g) Υ 2 λ, ˆ,r g) = 3) where Υ ) and 2 Υ ) are given by ) and 2) respectively, ˆ = σ 2, and r g is the minimum lin distance. Proof: The proof follows from that of Theorem and is therefore omitted. Similar to the connected case, the energy coverage probability for this case is also a function of the propagation conditions, networ density and antenna design parameters. IV. RESULTS AND DESIGN INSIGHTS In this section, we verify the accuracy of the analytical expressions provided in Section III using numerical simulations. We also study how ey design parameters such as the antenna beam pattern affects the energy coverage probability in purely

4 mmwave sim) 3, 3, 9, 27 mmwave sim),, 3, 33 mmwave sim) 5, 5,, 35. mmwave anlt) Energy Outage Threshold db).2 mmwave sim) 3, 3, 9, 27 mmwave sim),, 3, 33 mmwave sim) 5, 5,, 35. mmwave anlt) Energy Outage Thresold db) Fig.. Energy coverage probability Λɛ,, λ) for different transmit antenna beam patterns parameterized by M t,m t,θ t, θ t in a purely connected networ ɛ =, λ = /m 2 ). The performance improves with narrower beams for this case. P t =3dB, W = MHz, α L =2, α N =4, N L =2, N N =3, and r g =m. A good agreement can be observed between simulation and analytical anlt) results obtained using Theorem with N =5terms. Fig. 3. Energy coverage probability Λɛ,, λ) for different transmit antenna beam patterns in a nonconnected networ ɛ =, λ = /m 2 ). The performance improves with wider beams for this case. Other simulation parameters are same as given in Fig.. Analytical anlt) results are obtained using Theorem 2 with N =5terms mmwave sim) λ=5/m 2 mmwave sim) λ=/m 2 mmwave sim) λ=2/m 2 mmwave sim) λ=4/m 2 mmwave anlt) Energy Outage Thresold db) Fig. 2. Energy coverage probability Λ,,λ) for different networ densities for connected users. Transmit beam pattern is fixed to,, 3, 33. Other parameters are same as given in Fig.. connected ɛ ) and nonconnected ɛ ) networs. We also compare the performance of mmwave energy harvesting with lower frequency solutions. After developing ey insights for purely connected/nonconnected scenarios, we then provide energy coverage results for the general case < ɛ < ), where the networ serves both types of users. In the following plots, a user is assumed to be equipped with a single omnidirectional receive antenna, the mmwave carrier frequency is set to 28 GHz, and blocage constant β =.7 3. Connected case ɛ ): In Fig., we plot the energy coverage probability with three distinct transmit beam patterns for a given networ density. There is a nice agreement between analytical based on Theorem ) and Monte Carlo simulation results. Further, we can observe that the energy harvesting performance improves with narrower beams i.e., smaller beamwidths and larger directivity gains). As the beamwidth decreases, relatively fewer beams from the neighboring BSs would be incident on a typical user. But the beams that do reach, will have larger directivity gains, which results in an overall performance improvement. This is possible due to the use of potentially large antenna arrays at the mmwave BSs. Note that this performance boost will possibly be limited due to the ensuing EIRP equivalent isotropically radiated power) or other safety regulations on future mmwave systems 7. For the purpose of comparison, we also plot the energy coverage probability for energy harvesting under realistic assumptions: Given the current state-of-the-art, the BSs are assumed to have 8 transmit antennas each. For a fair comparison, they are assumed to employ maximal ratio transmit beamforming to serve a connected user. For the channel model, we assume an IID Rayleigh fading environment and a path loss exponent of 3.6 no blocage is considered). The networ density is set to 25 nodes/m 2, which corresponds to an average distance of about 3m to the closest BS. The carrier frequency is set to 2. GHz while the signal bandwidth is MHz. As can be seen from Fig., mmwave energy harvesting could provide considerable performance gain over its lower frequency counterpart. Moreover, the anticipated dense deployments of mmwave networs would further widen this gap. This effect is illustrated in Fig. 2, where we plot the energy coverage probability for different mmwave networ densities for a given transmit antenna beam pattern. Nonconnected case ɛ ): We now analyze the energy harvesting performance when the harvesting devices operate in the nonconnected mode. In a star contrast to the connected case, Fig. 3 shows that for the nonconnected case, mmwave energy harvesting could in fact benefit from using wider beams. This is because BS connectivity is critical for the nonconnected case. With wider beams, it is more liely that a mmwave BS gets aligned with a receiver, though sacrificing the beamforming gains. Moreover, a comparison with energy harvesting shows that mmwave energy harvesting gives a comparable performance to solutions. Similarly, Fig. 4 plots the energy coverage probability for different deployment densities. We can observe that performance can be further improved with denser deployments, which would be a ey feature in future mmwave cellular systems. General case <ɛ<): Having presented the energy coverage trends for the two extreme networ scenarios, we

5 ε=.25 ε=.55 ε= mmwave sim) λ = 5/m 2 mmwave sim) λ = /m 2.2 mmwave sim) λ = 2/m 2 mmwave sim) λ = 4/m 2. mmwave anlt) Energy Outage Thresold db) ε,,λ) Transmit array size N t Fig. 4. Energy coverage probability Λ,,λ) for different networ densities for nonconnected users. Transmit beam pattern is fixed to,, 3, 33. Other parameters are same as given in Fig.. Fig. 5. The fraction of users under energy coverage for different values of ɛ. is 7 db for Φ u,con and 85 db for Φ u,ncon. P t =3dB, λ = 2/m 2. Other parameters are same as given in Fig.. now consider the general case where the user population consists of both connected and nonconnected users. We expect this to be the liely scenario for reasons explained in Section II-A. As described in Section II-B, an antenna beam pattern can be characterized by the half power beamwidth and directivity gain for both the main and side lobes. By tuning these parameters, the beam pattern can be particularized to a given antenna array. As an example, we assume that uniform linear arrays are deployed at the mmwave BSs. We use the following relations to approximate the main and side lobe beamwidths as a) function of the transmit array size: θ t 36 π arcsin.892 N t and θ ) t = 72 2 π arcsin N t 8. Further, we use M t = log N t ) and m t = M t 2 for the directivity gains of the main and side lobes 8. We further assume the resulting sectorized transmit beam pattern are θ normalized over the parameter space, i.e., t 2π M t + θ t 2π m t =. In Fig. 5, we plot the overall energy coverage probability Λɛ,, λ) against transmit array size N t for different values of parameter ɛ. We find that the optimal transmit array size depends on the type of user population. For example, when ɛ is large, it is desirable to use large antenna arrays at the BSs. When ɛ is small, it is favorable to use small antenna arrays to improve the overall coverage probability. Depending on the networ load or the user population mix) captured via ɛ, the energy coverage probability can be substantially improved by intelligent antenna switching schemes. Since the parameter ɛ would typically vary over large time-scales, such schemes would be practically feasible. V. CONCLUSION We analyzed the energy harvesting performance at nodes powered by a mmwave cellular networ. Leveraging tools from stochastic geometry, we derived the energy coverage probability for mmwave energy harvesting in terms of important system and channel parameters. For the connected case when the transmitter and receiver beams are aligned, results show that the energy coverage improves with narrower beams. In contrast, wider beams result in better energy coverage when the receivers are not connected to a particular transmitter. This trade-off is evident in the more general scenario having both types of receivers, where there typically exists an optimal beamforming beamwidth. Moreover, the results show that mmwave cellular networs could potentially provide better energy coverage than lower frequency solutions. APPENDIX A: THEOREM We first provide an inequality to approximate the tail probability of a normalized Gamma distribution. Lemma 4 From 9): For a normalized Gamma random variable u with parameter N, the inequality Pr u <x) < e ax ) N holds, where x> and a = NN!) N. We can write P con λ, ) = Prγ> = Pr S + I + σ 2 >, where S = P t M t M r H g r ) and I = l>,l Φλ)\Br P g) tδ l H l g l r l ). We can derive the result in Theorem by finding the conditional distributions P con,l λ, ) and P con,n λ, ). To proceed, first consider the conditional distribution P con,l λ, ) = PrS + I> L) given the receiver is aligned with a LOS BS which is indicated by the subscript L in the following notation). P con,l λ, ) =E S,I L Pr u< S + I ) a) ) S+I N a E S,I L e N ) = E S,I L ) N S+I a e = = ) N = )E S,I L e âs+i) 4) where we have included a dummy random variable u Γ ) N, N in the first equation. Note that u converges to as N. Therefore, this substitution is in fact an approximation when N is finite. The introduction of u allows leveraging the inequality in Lemma 4, which leads to a), where the constant a = NN!) N. The last equation follows from the Binomial series expansion of b), and by further substituting â = a. To evaluate the expectation in 4),

6 consider E S,I L e âs+i) = E S L e âs E I S,L e âi. 5) The inner expectation in 5) can be simplified by applying the thinning theorem for a PPP 6. Note that Φ can be independently thinned into two PPPs Φ L and Φ N, where the former comprises the LOS BSs whereas the latter consists of NLOS BSs. We can further thin Φ L into four independent PPPs {Φ i L }4 i=, where each resulting PPP Φi L contains BSs that correspond to a nonzero directivity gain D i with p i being the thinning probability. This follows because the beam orientations are assumed to be independent across lins. Thus, a lin can have a directivity gain of D i with probability p i independently of other lins. We let the received power due to the transmission from the BSs in Φ i L be Ii L. Liewise, Φ N can be split into {Φ i N }4 i= with the corresponding received powers denoted by {IN i }4 i=. Since the resulting PPPs are independent, 5) can be simplified as E I S,L e âi 4 4 = E I S,L e âii L E I S,L e âij N 6) i= where E I S,L e âii L = E Φ i L,H r o e a) = E Φ i L r o b) = E Φ i L r o l Φ i L \Bro) E Hl l Φ i L \Bro) ro j= α â P th l D ic Lr l L l Φ i L \Bro) e âpth α ld ic Lr l L ) NL +âp t D i C L r α l LNL +âptd i C L t α L N L ) ) NL pt)tdt = e 2πλpi 7) where a) follows by conditioning on the length r o of the serving LOS lin, and by further noting that small-scale fading is independent across lins. Here, B r o ) denotes a circular disc of radius r o centered at the typical user. b) is obtained by using the moment generating function of a normalized Gamma random variable, while the last equation follows by invoing the probability generating functional 6 of the PPP Φ i L. Substituting 7) in the first left) product term of 6) yields ). Similarly, we can obtain α â P th l D ic Nr l N E I S,L e âii l Φ N = E Φ i L,H r o e i N \Bρ L ro)) = e ) ) NN 2πλp i +âptd ρ L ro) i C N x α N N N pt))tdt. 8) By substituting 8) in the second right) product term of 6) yields 2). Using the expressions in 6) 8) in 5), and by further evaluating the expectation of the resulting expression with respect to S, we obtain ) NL e Υ λ,,r) Υ 2 λ,,ρ Lr)) +âp t M t M r C L r αl N L r g ˆτ L r)dr 9) where we have again used definition of the moment generating function of a normalized Gamma distribution. Υ ) and 2 Υ ) are given in ) and 2) respectively, r g denotes the minimum lin distance, while the distance distribution is provided in Lemma 3. Using 4) and 5), we can thus retrieve the expression in 9). We can similarly derive the conditional distribution P con,n λ, ) =PrS + I> N) in ) for the NLOS case. REFERENCES T. S. Rappaport et al., Millimeter Wave Wireless Communications. Pearson Education, S. Uluus et al., Energy harvesting wireless communications: A review of recent advances, IEEE J. Sel. Areas Commun., vol. 33, pp , Mar A. Zanella et al., Internet of Things for smart cities, IEEE Internet Things J., vol., pp , Feb S. Rangan, T. S. Rappaport, and E. Erip, Millimeter wave cellular wireless networs: Potentials and challenges, arxiv preprint arxiv:4.256, S. Gollaota et al., The emergence of RF-powered computing, Computer, vol. 47, pp , Jan M. Pinuela, P. Mitcheson, and S. Lucyszyn, Ambient RF energy harvesting in urban and semi-urban environments, IEEE Trans. Microw. Theory Techn., vol. 6, pp , Jul C. Valenta and G. Durgin, Harvesting wireless power: Survey of energy-harvester conversion efficiency in far-field, wireless power transfer systems, IEEE Microw. Mag., vol. 5, pp. 8 2, Jun I. Flint et al., Performance analysis of ambient RF energy harvesting with repulsive point process modeling, IEEE Trans. Wireless Commun., vol. PP, no. 99, pp., S. Lee, R. Zhang, and K. Huang, Opportunistic wireless energy harvesting in cognitive radio networs, IEEE Trans. Wireless Commun., vol. 2, pp , Sep. 23. A. Hamdi and E. Hossain, Cognitive and energy harvesting-based D2D communication in cellular networs: Stochastic geometry modeling and analysis, IEEE Trans. Commun., vol. PP, no. 99, pp., 25. I. Kriidis, Simultaneous information and energy transfer in largescale networs with/without relaying, IEEE Trans. Commun., vol. 62, pp. 9 92, Mar K. Huang and V. Lau, Enabling wireless power transfer in cellular networs: Architecture, modeling and deployment, IEEE Trans. Wireless Commun., vol. 3, pp , Feb T. Bai and R. Heath, Coverage and rate analysis for millimeter-wave cellular networs, IEEE Trans. Wireless Commun., vol. 4, pp. 4, Feb T. Bai, R. Vaze, and R. Heath, Analysis of blocage effects on urban cellular networs, IEEE Trans. Wireless Commun., vol. 3, pp , Sep A. Hunter, J. Andrews, and S. Weber, Transmission capacity of ad hoc networs with spatial diversity, IEEE Trans. Wireless Commun., vol. 7, pp , Dec M. Haenggi, Stochastic geometry for wireless networs. Cambridge University Press, T. Wu, T. S. Rappaport, and C. M. Collins, The human body and millimeter-wave wireless communication systems: Interactions and implications, arxiv preprint arxiv: , H. L. Van Trees, Detection, estimation, and modulation theory, optimum array processing. John Wiley & Sons, H. Alzer, On some inequalities for the incomplete gamma function, Mathematics of Computation of the American Mathematical Society, vol. 66, no. 28, pp , 997.