This paper studies a system where a set of $N$ relay nodes harvest energy
from the signal received from a source to later utilize it when forwarding the
source's data to a destination node via distributed beamforming. To this end,
we derive (approximate) analytical expressions for the mean SNR at destination
node when relays employ: i) time-switching based energy harvesting policy, ii)
power-splitting based energy harvesting policy. The obtained results facilitate
the study of the interplay between the energy harvesting parameters and the
synchronization error, and their combined impact on mean SNR. Simulation
results indicate that i) the derived approximate expressions are very accurate
even for small $N$ (e.g., $N=15$), ii) time-switching policy by the relays
outperforms power-splitting policy by at least $3$ dB.Comment: 4 pages, 3 figures, accepted for presentation at IEEE VTC 2017 Spring
  conferenc

Abbasi, Qammer H.

Iqbal, Muhammad Ozair

Mahmood, Ammar

Rahman, Muhammad Mahboob Ur

English

arXiv

Muhammad Ozair Iqbal

Ammar Mahmood

Muhammad Mahboob Ur Rahman

Qammer H. Abbasi

Crossref

Distributed Beamforming with Wirelessly Powered Relay Nodes

This paper studies a system where a set of N relay nodes harvest energy from the signal received from a source to later utilize it when forwarding the source's data to a destination node via distributed beamforming. To this end, we derive (approximate) analytical expressions for the mean SNR at destination node when relays employ: i) time-switching based energy harvesting policy, ii) power- splitting based energy harvesting policy. The obtained results facilitate the study of the interplay between the energy harvesting parameters and the synchronization error, and their combined impact on mean SNR. Simulation results indicate that i) the derived approximate expressions are very accurate even for small N (e.g., N=15), ii) time-switching policy by the relays outperforms power-splitting policy by at least 3 dB

Enlighten

     Iqbal, M. O., Mahmood, A., Rahman, M. M. U. and Abbasi, Q. H. (2017) Distributed Beamforming with Wirelessly Powered Relay Nodes. In: IEEE 85th Vehicular Technology Conference: VTC2017-Spring, Sydney, Australia, 4-7 June 2017, ISBN 9781509059324 (doi:10.1109/VTCSpring.2017.8108316)  This is the author’s final accepted version.  There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.     http://eprints.gla.ac.uk/150044/            Deposited on:  18 October 2017             Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk Distributed Beamforming with Wirelessly PoweredRelay NodesMuhammad Ozair Iqbal∗, Ammar Mahmood∗, Muhammad Mahboob Ur Rahman∗, Qammer H. Abbasi†∗Electrical Engineering department, Information Technology University (ITU), Lahore, Pakistan{ozair.iqbal,ammar.mahmood,mahboob.rahman}@itu.edu.pk†Department of Electrical and Computer Engineering, Texas A&M University at Qatarqammer.abbasi@qatar.tamu.eduAbstract—This paper studies a system where a set of 𝑁relay nodes harvest energy from the signal received from asource to later utilize it when forwarding the source’s data toa destination node via distributed beamforming. To this end,we derive (approximate) analytical expressions for the meanSNR at destination node when relays employ: i) time-switchingbased energy harvesting policy, ii) power-splitting based energyharvesting policy. The obtained results facilitate the study ofthe interplay between the energy harvesting parameters andthe synchronization error, and their combined impact on meanSNR. Simulation results indicate that i) the derived approximateexpressions are very accurate even for small 𝑁 (e.g., 𝑁 = 15), ii)time-switching policy by the relays outperforms power-splittingpolicy by at least 3 dB.I. INTRODUCTIONDistributed transmit beamforming is a technique wherebymultiple transmitters cooperate in a way that their signals (car-rying a common message) combine coherently, over-the-air, atthe intended receiver. For the unit-gain channels between thetransmit nodes and the receiver, distributed beamforming leadsto an 𝑁2-fold increase in mean SNR at the receiver (where𝑁 is the number of cooperating transmitters) [1]. However,the energy-efficiency advantage of distributed beamformingcomes at a cost, the carrier synchronization cost. Specifically,the individual passband signals sent from cooperating transmitnodes combine constructively at the receiver only when trans-mit nodes are frequency, time and phase synchronized [1].Quite recently, wireless power transfer where a transmitnode lets its receive counterpart harvest energy from theradio frequency (RF) signal it transmits, has attracted a lot ofattention [2]. In the literature, two energy harvesting scenarioshave been widely studied: i) time-switching (TS) based energyharvesting (EH) where the receiver spends a (time) fractionof every symbol it receives for energy harvesting, ii) power-splitting (PS) based energy harvesting where the receiverspends a fraction of the received power for energy harvesting.This paper studies a system where a set of 𝑁 relay nodesharvest energy from the signal received from a source to laterutilize it when forwarding the source’s data to a destinationnode via distributed beamforming. Specifically, the paper de-rives (approximate) analytical expressions for the mean SNR atdestination node when relays employ: i) TS based EH scheme,ii) PS based EH scheme. The obtained results facilitate us tostudy the interplay between the energy harvesting parametersand the synchronization error, and their influence on meanSNR. Simulation results indicate that the derived approximateexpressions are very accurate even for small 𝑁 (e.g., 𝑁 = 15).The related works closest to this work are [3],[4]. [3] con-siders a single multi-antenna relay which harvests energy froma source (and external interferences) to later forward its data(via maximum ratio transmission) to the destination; authors of[3] then derive closed-form expressions for outage probabilityand ergodic capacity of the system. In [4], multiple transmitnodes do (received-assisted) distributed beamforming towardsa receiver node where the receiver node harvests energy fromthe received sum signal; [4] then studies the trade-off betweenfeedback rate and amount of energy harvested at the receiver.Nevertheless, to the best of authors’ knowledge, the interplaybetween energy harvesting parameters and synchronizationerror, and their collective impact on mean SNR (presentedin this work) has not been studied before.Outline. The rest of this paper is organized as follows.Section-II introduces the system model. Section-III (Section-IV) provides an approximate analytical expression for meanSNR at the destination node when the relays employ time-switching (power-splitting) based energy harvesting policy.Section-V provides some numerical results. Section-VI con-cludes.II. SYSTEM MODELA system consisting of a source node 𝑆, a destinationnode 𝐷 and 𝑁 relay nodes (𝑅1, ...,𝑅𝑁 ) is studied (see Fig.1(a)). Following assumptions are in place: direct link between𝑆 and 𝐷 is not available; the relay nodes operate in half-duplex mode and employ decode-and-forward (DF) strategy;the relays do distributed beamforming towards 𝐷; the relaysare fully, wirelessly powered by the 𝑆; the channels on bothhops are quasi-static (i.e., each channel stays constant for aslot duration 𝑇 and channel realizations are 𝑖.𝑖.𝑑 between theslots), frequency-flat, block fading with Rayleigh distribution.III. TIME SWITCHING BASED ENERGY HARVESTING ATRELAYSLet 𝑇 denote the block time during which source 𝑆transmits a certain amount of information to destination 𝐷.Then, under time-switching (TS) based energy harvesting (EH)Fig. 1. System modelpolicy, the relays harvest energy from source’s transmission fora duration 𝛼𝑇 , where 0 ≤ 𝛼 ≤ 1 (see Fig. 1(b)).Specifically, on the first hop, source 𝑆 transmits message 𝑥(with power 𝑃𝑠) to the relays. Then, relay 𝑅𝑛 (𝑛 = 1, ...,𝑁 )receives 𝑦𝑛 = √𝑃𝑆𝑔𝑛𝑥+𝑤𝑛, where 𝑔𝑛 is the channel betweensource and relay 𝑅𝑛, and 𝑤𝑛 is the noise at relay 𝑅𝑛. Then, theamount of energy harvested by 𝑅𝑛 is 𝐸𝐻𝑛 = 𝜂∣√𝑃𝑆𝑔𝑛∣2×𝛼𝑇where 0 ≤ 𝜂 ≤ 1 is the (RF to DC) energy conversion efficiency.Since the relay 𝑅𝑛 uses all of the energy harvested to relay the(perfectly recovered) message 𝑥 to the destination, the transmitpower of 𝑅𝑛 is 𝑃𝑛 = 𝐸𝐻𝑛(1−𝛼)𝑇 /2 = 2𝜂∣√𝑃𝑆𝑔𝑛∣2𝛼(1−𝛼) . Next, on thesecond hop, each of the 𝑁 relays simultaneously forwardsthe precoded message 𝑟𝑛 = 𝑎𝑛𝑥 to the destination 𝐷 (𝑎𝑛 isthe precoding weight applied by 𝑅𝑛). The net (sum) signalreceived at 𝐷 is:𝑧 = 𝑁∑𝑛=1√𝑃𝑛ℎ𝑛𝑟𝑛 +𝑤𝐷 (1)where ℎ𝑛 = ∣ℎ𝑛∣𝑒𝑗𝜙𝑛 is the channel between the relay 𝑅𝑛 anddestination 𝐷, and 𝑤𝐷 is the noise at 𝐷; 𝑤𝐷 ∼ CN(0, 𝜎2𝐷).Let 𝑔𝑛, ℎ𝑛 ∼ CN(0,1), ∀ 𝑛 = 1, ...,𝑁 . Then, one canverify that 𝑃𝑛 ∼ exp (𝜆𝑝) where 𝜆𝑝 = 1−𝛼2𝜂𝛼𝑃𝑆 , and √𝑃𝑛 ∼Rayleigh(𝜎) where 𝜎 = √ 𝜂𝛼𝑃𝑆(1−𝛼) . When relays do distributedbeamforming, relay 𝑅𝑛 chooses 𝑎𝑛 ≜ 1∣ℎ𝑛∣𝑒−𝑗(𝜙𝑛−𝜃𝑛) (thiscould be achieved by running the protocols proposed in, e.g.,[1],[5],[6]). Then, Eq. (1) can be rewritten as:𝑧 = 𝑁∑𝑛=1√𝑃𝑛𝑒𝑗𝜃𝑛𝑥 +𝑤𝐷 (2)where 𝜃𝑛 models the channel phase estimation error forthe channel ℎ𝑛. However, we note that 𝜃𝑛 could very wellrepresent the net phase difference between 𝑅𝑛 and 𝐷, i.e., itcould assimilate the effects of channel phase estimation error,frequency and phase offsets etc.). Indeed, in this work, weassume that 𝜃𝑛 denotes the effective phase difference between𝑅𝑛 and 𝐷. Moreover, we assume that 𝜃𝑛 are 𝑖.𝑖.𝑑 with∼ N(0, 𝜎2𝜃). Next, assuming that 𝑥 ∈ 𝑀 -PSK constellation(for any 𝑀 ) and that 𝜎2𝐷 = 1, the instantaneous SNR at thedestination 𝐷 is:𝛾𝐷({𝜃𝑛},{𝑃𝑛}) = 𝛾𝐷({𝜃𝑛}, 𝛼) = ∣ 𝑁∑𝑛=1√𝑃𝑛𝑒𝑗𝜃𝑛 ∣2 (3)Then, an (approximate) expression for mean SNR 𝐸[𝛾𝐷] isprovided in Corollary 1.Corollary 1: Let 𝛾𝐷 ≜ lim𝑁→∞ 𝛾𝐷. Then, the followingholds:𝐸[𝛾𝐷({𝜃𝑛}, 𝛼)] = 𝑎2 + 𝑏2 + 𝑐 + 𝑑 (4)where𝑎 = 𝑁√ 𝜋𝜂𝛼𝑃𝑆2(1 − 𝛼)𝑒−𝜎2𝜃/2; 𝑏 = 0;𝑐 = 𝑁[ 𝜂𝛼𝑃𝑆(1 − 𝛼)(1−𝑒−𝜎2𝜃)2+ 2𝜂𝛼𝑃𝑆(1 − 𝛼)𝑒−𝜎2𝜃/2]−𝑁(√𝜋𝜂𝛼𝑃𝑆2(1 − 𝛼) .𝑒−𝜎2𝜃/2)2;𝑑 = 𝑁𝜂𝛼𝑃𝑆(1 − 𝛼) (1 − 𝑒−2𝜎2𝜃).Proof: See Appendix A.IV. POWER SPLITTING BASED ENERGY HARVESTING ATRELAYSUnder power-splitting based energy harvesting policy, relay𝑅𝑛 harvests 𝐸𝐻𝑛 = 𝜂𝜌∣√𝑃𝑆𝑔𝑛∣2 × 𝑇 /2 amount of energyfrom source’s transmission for a duration 𝑇 /2 (see Fig.1(c)). Since the relay 𝑅𝑛 uses all of the energy harvestedto relay the message 𝑥 to 𝐷, the transmit power of 𝑅𝑛 is𝑃𝑛 = 𝐸𝐻𝑛𝑇 /2 = 2𝜂𝜌∣√𝑃𝑆𝑔𝑛∣2. In this case, 𝑃𝑛 ∼ exp (𝜆𝑝) where𝜆𝑝 = 12𝜂𝜌𝑃𝑆 , and √𝑃𝑛 ∼ Rayleigh(𝜎) where 𝜎 = √𝜂𝜌𝑃𝑆 .Then, one can verify that the sum signal 𝑧 received at 𝐷 andinstantaneous SNR 𝛾𝐷 are once again given by Eq. (2) and Eq.(3) respectively. Then, an (approximate) expression for meanSNR 𝐸[𝛾𝐷] is provided in Corollary 2.Corollary 2: Let 𝛾𝐷 ≜ lim𝑁→∞ 𝛾𝐷. Then, the followingholds:𝐸[𝛾𝐷({𝜃𝑛}, 𝜌)] = 𝑝2 + 𝑞2 + 𝑟 + 𝑠 (5)where𝑝 = 𝑁√𝜋𝜂𝜌𝑃𝑆2𝑒−𝜎2𝜃/2; 𝑞 = 0;𝑟 = 𝑁[𝜂𝜌𝑃𝑆(1−𝑒−𝜎2𝜃)2+2𝜂𝜌𝑃𝑆𝑒−𝜎2𝜃/2]−𝑁(√𝜋𝜂𝜌𝑃𝑆2.𝑒−𝜎2𝜃/2)2;𝑠 = 𝑁𝜂𝜌𝑃𝑆(1 − 𝑒−2𝜎2𝜃).Proof: See Appendix A.V. SIMULATION RESULTSIn all plots, solid lines represent analytical predictions byEqs. (4), (5) while dotted lines represent Monte-Carlo simu-lation results. Figs. 2, 3 show the following: i) the analyticalapproximations of Eqs. (4), (5) are indeed very accurate for𝑁 as low as 15 (while the approximations degrade for 𝑁 = 2);ii) the mean SNR degrades as the variance of the net phaseerror increases (due to poorer oscillators, poor synchronizationprotocol etc.); iii) for a given system state (of phase errorvariance), the mean SNR can be improved by doing moreenergy harvesting at the relays; iv) the TS based EH schemeoutperforms PS based EH scheme by at least 3 dB, for a givenphase error variance.20 40 60 80 100 12005101520253035σθ2 (Degrees)Mean SNR (dB)  → N= 2→ N = 15→ N = 30 α = 0.3α = 0.5α = 0.7Fig. 2. TS based EH scheme.0 20 40 60 80 100 12005101520253035σθ2 (Degrees)Mean SNR (dB)  → N = 2→ N = 15→ N = 30ρ = 0.3ρ = 0.5ρ = 0.7Fig. 3. PS based EH scheme.VI. CONCLUSIONThis preliminary work studied a system where a set of 𝑁relay nodes harvest energy from the signal received from asource to later utilize it when forwarding the source’s datato a destination node via distributed beamforming. Monte-Carlo simulation results showed that the derived approximateexpressions for the mean SNR at the destination are veryaccurate for 𝑁 as low as 15. Last but not the least, TS basedEH scheme outperformed PS based EH scheme by at least3 dB. Immediate future work will investigate the coupling(dependence) between energy harvesting parameters and thephase error (due to clock drift) and their combined impact onmean SNR (and Ergodic capacity) at the destination.APPENDIX AAPPENDIX A: AN APPROXIMATE EXPRESSION FOR THEMEAN SNR 𝐸[𝛾𝐷]We can rewrite 𝛾𝐷 from Eq. (3) as:𝛾𝐷 = ( 𝑁∑𝑛=1√𝑃𝑛 cos 𝜃𝑛)2 + ( 𝑁∑𝑛=1√𝑃𝑛 sin 𝜃𝑛)2 (6)Let 𝑋𝑛 = √𝑃𝑛 cos 𝜃𝑛 and 𝑌𝑛 = √𝑃𝑛 sin 𝜃𝑛. Then, 𝛾𝐷 =(∑𝑁𝑛=1𝑋𝑛)2 + (∑𝑁𝑛=1 𝑌𝑛)2. Note that even though √𝑃𝑛 ∼Rayleigh(𝜎) and 𝜃𝑛 ∼ N(0, 𝜎2𝜃), the distribution of each of𝑋𝑛 and 𝑌𝑛 is not easy to obtain. However, note that both√𝑃𝑛 and 𝜃𝑛 are i.i.d; therefore, if one knows the means𝐸[𝑋𝑛],𝐸[𝑌𝑛] and variances 𝑉 𝑎𝑟[𝑋𝑛],𝑉 𝑎𝑟[𝑌𝑛] of 𝑋𝑛 and𝑌𝑛 respectively, then (for large 𝑁 ) one can invoke CentralLimit Theorem to get a step closer towards obtaining expectedvalue of 𝛾𝐷. To this end, we have:𝐸[𝑋𝑛] = 𝐸[√𝑃𝑛 cos 𝜃𝑛] = 𝐸[√𝑃𝑛].𝐸[cos 𝜃𝑛] = 𝜎.√𝜋/2.𝑒−𝜎2𝜃/2(7)where we have used the fact that√𝑃𝑛 and cos 𝜃𝑛 are inde-pendent of each other. And𝐸[𝑌𝑛] = 𝐸[√𝑃𝑛 sin 𝜃𝑛] = 𝐸[√𝑃𝑛].𝐸[sin 𝜃𝑛] = 0 (8)Similarly, we have:𝑉 𝑎𝑟[𝑋𝑛] = 𝐸[𝑋2𝑛] − (𝐸[𝑋𝑛])2= 1𝜆𝑝(12(1 − 𝑒𝜎2𝜃)2 + 𝑒−𝜎2𝜃/2) − (𝜎.√𝜋2.𝑒−𝜎2𝜃/2)2(9)𝑉 𝑎𝑟[𝑌𝑛] = 𝐸[𝑌 2𝑛 ] − (𝐸[𝑌𝑛])2 = 12𝜆𝑝 (1 − 𝑒−2𝜎2𝜃) (10)Let 𝐼 = ∑𝑁𝑛=1𝑋𝑛 and 𝑄 = ∑𝑁𝑛=1 𝑌𝑛. Then, 𝛾𝐷 = 𝐼2 +𝑄2. Let𝛾𝐷 ≜ lim𝑁→∞ 𝛾𝐷. Then, according to Central Limit Theo-rem, the following relations hold: lim𝑁→∞ 𝐼 ∼ N(𝑚𝐼 , 𝜎2𝐼);lim𝑁→∞𝑄 ∼ N(𝑚𝑄, 𝜎2𝑄) where 𝑚𝐼 = 𝑁𝐸[𝑋𝑛] =𝑁𝜎√𝜋2𝑒−𝜎2𝜃/2;𝑚𝑄 = 𝑁𝐸[𝑌𝑛] = 0;𝜎2𝐼 = 𝑁.𝑉 𝑎𝑟[𝑋𝑛]= 𝑁[ 12𝜆𝑝(1 − 𝑒−𝜎2𝜃)2 + 1𝜆𝑝𝑒−𝜎2𝜃/2] −𝑁(𝜎.√𝜋2.𝑒−𝜎2𝜃/2)2;(11)𝜎2𝑄 = 𝑁.𝑉 𝑎𝑟[𝑌𝑛] = 𝑁2𝜆𝑝 (1 − 𝑒−2𝜎2𝜃). (12)Then,𝐸[𝛾𝐷] = 𝐸[ lim𝑁→∞𝐼2] +𝐸[ lim𝑁→∞𝑄2] = 𝜎2𝐼 +𝑚2𝐼 + 𝜎2𝑄 +𝑚2𝑄(13)ACKNOWLEDGEMENTSThis publication was made possible by NPRP grant #7 −125−2−061 from the Qatar National Research Fund (a memberof Qatar Foundation). The statements made herein are solelythe responsibility of the authors.REFERENCES[1] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility of dis-tributed beamforming in wireless networks,” Wireless Communications,IEEE Transactions on, vol. 6, no. 5, pp. 1754–1763, May 2007.[2] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relayingprotocols for wireless energy harvesting and information processing,”IEEE Transactions on Wireless Communications, vol. 12, no. 7, pp. 3622–3636, July 2013.[3] G. Zhu, C. Zhong, H. A. Suraweera, G. K. Karagiannidis, Z. Zhang,and T. A. Tsiftsis, “Wireless information and power transfer in relaysystems with multiple antennas and interference,” IEEE Transactions onCommunications, vol. 63, no. 4, pp. 1400–1418, April 2015.[4] R. Wang, R. David, and D. R. Brown, “Feedback rate optimization inreceiver-coordinated distributed transmit beamforming for wireless powertransfer,” in 2015 49th Annual Conference on Information Sciences andSystems (CISS), March 2015, pp. 1–6.[5] F. Quitin, M. M. U. Rahman, R. Mudumbai, and U. Madhow, “A scal-able architecture for distributed transmit beamforming with commodityradios: Design and proof of concept,” IEEE Transactions on WirelessCommunications, vol. 12, no. 3, pp. 1418–1428, 2013.[6] F. Quitin, A. T. Irish, and U. Madhow, “A scalable architecture for dis-tributed receive beamforming: Analysis and experimental demonstration,”IEEE Transactions on Wireless Communications, vol. 15, no. 3, pp. 2039–2053, March 2016.

Enlighten: Publications

http://eprints.gla.ac.uk/150044/7/150044.pdf

Distributed Beamforming with Wirelessly Powered Relay Nodes

Abstract

Similar works

Full text

Available Versions

Crossref

Enlighten

Enlighten: Publications