DC Programming for Power Minimization in a Multicell Network with RF-Powered Relays

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1 nd Internatonal Conference on Telecommuncatons ICT 015 DC Programmng for Power Mnmzaton n a Multcell etwork wth RF-Powered Relays Al A. asr, Duy T. go and Salman Durran Research School of Engneerng, The Australan atonal Unversty, Canberra, Australa Emal: {al.nasr, salman.durran}@anu.edu.au School of Electrcal Engneerng and Computer Scence, Unversty of ewcastle, Australa Emal: duy.ngo@newcastle.edu.au Abstract We consder a multcell network where an amplfyand-forward relay s deployed n each cell to help the base staton BS serve ts cell-edge user. We assume that each relay scavenges energy from all receved rado sgnals to process and forward the nformaton data from the BS to the correspondng user. For ths, a power spltter and a wreless energy harvester are mplemented n the relay. Our am s to mnmze the total power consumpton n the network whle guaranteeng mnmum data throughput for each user. To ths end, we develop a resource management scheme that jontly optmzes three parameters, namely, BS transmt powers, power splttng factors for energy harvestng and nformaton processng at the relays, and relay transmt powers. As the formulated problem s hghly nonconvex, we devse a successve convex approxmaton algorthm based on dfferenceof-convex-functons DC programmng. The proposed teratve algorthm transforms the nonconvex problem nto a sequence of convex problems, each of whch s solved effcently n each teraton. We prove that ths path-followng algorthm converges to an optmal soluton that satsfes the Karush-Kuhn-Tucker KKT condtons of the orgnal nonconvex problem. Smulaton results demonstrate that the proposed jont optmzaton soluton substantally mproves the network performance. I. ITRODUCTIO In a multcell network, users at the cell edges are the vctms of strong ntercell nterference whle recevng weak sgnals from ther servng base statons BSs. A vable soluton s to deploy relay nodes to provde network coverage to the celledge users [1]. Whle connectng to the BSs va wreless lnks, the opportunstc nature of relay deployments may restrct ther access to a man power supply. Ths problem can be solved by mplementng wreless energy harvestng technques at the relays, where energy s scavenged from the ambent propagatng electromagnetc waves n the rado frequency RF [] [4]. Wreless energy harvestng solutons are feasble for relays, whch only requre sgnfcantly low transmt power due to ther restrcted network coverage. Wth RF-powered relays, the network performance s determned by the transmt powers allocated to the BSs and the relays, as well as the use of RF energy harvested at the relays. The work of [5] proposes coordnated schedulng and power control algorthms only for the BSs of a multcell network. The study of [6] develops resource allocaton schemes specfcally for the remote rado heads. Reference [7] studes the power control problem for multuser broadband wreless systems wthout relays, where smultaneous wreless nformaton and The work of A. A. asr and S. Durran was supported n part by the Australan Research Councl s Dscovery Project fundng scheme project number DP The work of D. T. go was supported n part by the Unversty of ewcastle s Early Career Researcher Grant G power transfer s assumed. An optmal power splttng rule s devsed n [8] for energy harvestng and nformaton processng at the RF-powered relays of a multuser nterference network. Ths paper consders a multcell network where an energyconstraned relay node asssts the BS of each cell to communcate wth ts cell-edge user. Equpped wth a wreless energy harvester, the relay s capable of scavengng part of the RF energy n the receved sgnal. A power spltter s ncluded n the relay to help decde how much receved sgnal energy s to be harvested. The nformaton transcever at the relay wll use the harvested energy to amplfy and then forward the BS sgnal to ts correspondng user. We am to devse an optmal scheme to manage the rado resources at the BSs and RF-powered relays. The objectve s to mnmze the total transmt power of all BSs, subject to meetng a mnmum data rate for each user. Ths problem s of partcular nterest for green communcatons, where one wshes to reduce the envronmental mpacts of largescale wreless network deployment whle provdng qualty of servce to the most vulnerable users. Dfferent from exstng works, we jontly optmze the transmt powers at the BSs and the relays, together wth fndng an optmal power splttng rule for energy harvestng and sgnal processng at the relays. The strongly-coupled optmzng varables render the formulated problem hghly nonconvex, and thus challengng to solve. In ths work, we employ dfference-of-convex-functons DC programmng and apply the successve convex approxmaton SCA method to transform the nonconvex problem to a seres of convex programs [9], [10]. Specfcally, we express the nonconvex throughput functon as the dfference of two convex functons and apply the frst-order Taylor seres approxmaton to convexfy t. We then propose an teratve algorthm where only one smple convex problem s to be solved at each teraton. We prove that the DC-based SCA algorthm generates a sequence of feasble and mproved solutons, whch monotoncally converges to solutons satsfyng the KKT condtons of the orgnal nonconvex problem. Whle a true globally optmal method does not exst for the formulated problem, t has been shown that the SCA-based solutons often emprcally acheve the global optmum n most practcal scenaros [11], [1]. II. SYSTEM MODEL AD PROBLEM FORMULATIO We consder the downlnk of a multcell network wth unversal frequency reuse, as llustrated n Fg. 1. Let = {1,...,} denote the set of all cells. We assume that the /15/$ IEEE 174

2 nd Internatonal Conference on Telecommuncatons ICT 015 Cell 1 Base Staton Relay User CP h1,1 h 3,1 g 1,1 g 3,1 g,1 h,1 g, h, Cell specfes how much receved sgnal energy s dedcated to each of the two purposes [], [8]. Fg. shows that the power spltter at relay dvdes the power of y R nto two parts n the proporton of α : 1 α, where α 0,1 s the power splttng factor. The frst part α y R s processed by the energy harvester and stored as energy e.g., by chargng a battery at relay for use n the second tme slot. The amount of energy harvested at relay s gven by E = ηα T P j h, Cell 3 h 3,3 g 3,3 ntended sgnal nterferng sgnal Fg. 1. Illustraton of a multcell network wth 3 cells. Whle only the nterferng scenaro s shown for cell 1, the nterference actually occurs at the recevers of all three relays and three users. BSs are connected to a central processng CP unt whch coordnates the multcell transmssons and rado resource management. Whle the BS of cell attempts to communcate wth ts sngle user, ths user s located at the cell edge and hence out of reach. A relay node s deployed n each cell to provde network coverage, assstng the communcaton from the BS to ts user. Denote the channel coeffcent from BS to relay j as h,j, and that from relay j to user k as g j,k. We assume that all BSs send the avalable channel state nformaton to the CP unt va a dedcated control channel. We assume that each of relays s equpped wth a wreless energy harvester that scavenges energy n the receved RF sgnals from all BSs. The harvested energy s used by an nformaton transcever at the relay to forward the message sgnal to the ntended user. We dvde the total transmsson block tme T nto two equal tme slots as llustrated n Fg.. A. BS-to-Relay Transmssons and Wreless Energy Harvestng at Relays The frst tme slot [0, T/] ncludes BS-to-relay transmssons and RF energy harvestng at the relays. Ths s done whle all the relays do not transmt. Let x denote the normalzed nformaton sgnal sent by BS,.e., E{ x } = 1. Let P be the transmt power of BS, d h,j the dstance between BS and relay j, and β the path-loss exponent. Assume that n a s the zero-mean addtve whte Gaussan nose AWG wth varance σ a at the receve antenna of relay. The receved sgnal at relay s expressed as y R = h P x h Pj x j n a, 1,j where h h d h β/ s the effectve channel gan from BS j to relay. To mplement dual energy harvestng and sgnal processng at the relays, each relay s equpped wth a power spltter that where η 0,1 denotes the effcency of energy converson. The second part 1 α y R of the receved sgnal s passed to an nformaton transcever. In Fg., n r denotes the AWG wth zero mean and varance σ r ntroduced by the baseband processng crcutry. Snce antenna nose power σ a s very small compared to the crcut nose power σ r n practce [13], n a has a neglgble mpact on both the energy harvester and the nformaton transcever of relay. For smplcty, we wll thus gnore the effect of n a by settng σ a = 0. The sgnal at the nput of the nformaton transcever of relay s then expressed as y I R = 1 α y R n r = 1 α h, P x 1 α,j h Pj x j n r where the frst term n 3 s the desred sgnal from BS, and the second term s the total nterference from all other BSs. B. Sgnal Processng at Relays and Relay-to-User Transmssons The second tme slot [T/,T] ncludes sgnal processng at the relays and relay-to-user transmssons. Ths s done whle all the BSs suppress ther transmssons. Here, the nformaton transcever amplfes the sgnaly I R before forwardng t to user. Let p denote the transmt power of relay transcever. From, t s clear that 3 p E T/ = ηα P j h. 4 The transmtted sgnal from relay to user s gven by x R = ζ p y I R, 5 [ where ζ 1 α 1/ P j h σ] r s the amplfyng factor that ensures power constrant n 4 be met. The receved sgnal at user s then y U = ḡ, x R,j ḡ x Rj n u, 6 where ḡ g d g β/ s the effectve channel gan from BS j to relay, d g,j the dstance between relay and user j, and n u the AWG wth zero mean and varance σ u at the 175

3 nd Internatonal Conference on Telecommuncatons ICT 015 nterference P BS h, n a y R Power Spltter α y R 1 α y R Wreless Energy Harvester y R I E AF Informaton Transcever nterference p g, User Relay n r T Frst tme slot: [ 0, T ] T Second tme slot: [ T,T] Fg.. BS-to-user communcaton asssted by an RF-powered relay. recever of user. Usng x R n 5 and y I R n 3, we can rewrte 6 as y U = ζ ḡ, h, p P 1 α x ζ ḡ, p 1 α ζ ḡ, p n r,j,j h Pj x j ζ j ḡ pj y I R j n u. 7 The frst term n 7 s the sgnal from BS to ts user, and the remanng terms are the total ntercell nterference and nose. Wthout loss of generalty, let us assumeσ r = σu = σ,. From 7, the sgnal-to-nterference-plus-nose rato SIR at the recever of user s gven n 8 at the bottom of ths page, where we defne 1 ḡ, h σ ; h σ ; 3 ḡ σ ; φ,j,k 4 ḡ h k, σ. 9 For notatonal convenence, let us also defne P [P 1,...,P ] T,p [p 1,...,p ] T, and α [α 1,...,α ] T. From 8, the acheved throughput of cell s gven as τ P,p,α = 1 log 1γ. 10 It s observed from 8 and 10 that f we ncrease α.e., dedcate more receved power for energy harvestng at relay, the end-to-end throughput τ of cell mght be degraded. Ths can be verfed upon dvdng both the numerator and the denomnator of γ n 8 by 1 α. If we nstead decrease α, the transmt power avalable at the nformaton transcever of relay n 4 wll be further lmted, thus potentally reducng τ. Smlarly, ncreasng P or p does not necessarly enhance τ because these two parameters appear n the postve terms at both the numerator and the denomnator of γ. In ths paper, we am to develop a resource allocaton scheme that delvers an optmal tradeoff among transmt power P at the BSs, transmt power p at the relays, and power splttng factor α at the relays. Our objectve s to mnmze the total transmt power from the BSs, whle assurng a mnmum data throughput τ mn for each user. Other resource allocaton problems are consdered n [14]. The desgn problem s formulated as follows: mn P,p,α =1 P 11a s.t. τ τ mn, 11b 0 α 1, 11c P mn P P max, 0 p ηα P j h,, 11d 11e where P max denotes the maxmum power avalable for transmsson at each BS and P mn s the mnmum transmt power requred at each BS to ensure the actvaton of energy harvestng crcutry at the relay. Constrant 11b s mposed to protect the throughput performance of all cell-edge users. Constrant 11c s for the power splttng factors at all relays, whereas constrants 11d and 11e ensure that the transmt powers at the BSs and relays do not exceed the maxmum lmt. Problem 11 s nonconvex n P, p, α because the throughput τ n 11b s nonconvex n these three varables. Even f we fx p and α and try to optmze P alone, τ would stll be hghly nonconvex n P due to the cross-cell nterference terms. Smultaneously optmzng P, p and α s much more challengng due to the nonlnearty ntroduced by the crossmultplyng terms, e.g., P k p j α n 8 and α P j n 11e. III. PROPOSED SCA SOLUTIO USIG DC PROGRAMMIG We propose to adopt the successve convex approxmaton SCA approach [11], [1], [14] combned wth DC programmng [9], [10] to transform the nonconvex problem 11 nto a γ =,j 1 P jp 1 α φ, 1 P p 1 α P j1 α 3 p j,j k=1,k 4 P k p j 1 α

4 nd Internatonal Conference on Telecommuncatons ICT 015 l 1 v x = v xln e P l p t φ,l 1 e P l t φ,l e p φ, 3 e p l φ,l 3,j e P j p t 1, f l = k=1 e P j p t 1 e P j t 0, otherwse. e P l p j t,l 4, f l {1,...,} e P k p l t φ,l,k 4, f l { 1,...,}\{ },j k=1 e P k p j t,k 4, f l = 19 sequence of convex subproblems. Frst, we wll convexfy the nonconvex constrants 11b and 11e. Let us start wth 11b and rewrte the throughput expresson τ n 10 as: τ = 1 v x u x, 1 where x = [P T,p T,t T ] T R 3,t = 1 α R, and u x log 1 P jp t P jt 3 p j,j v x log,j k=1 4 P k p j t 1, 13,k 1 P jp t P jt 3 p j,j k=1 4 P k p j t 1. 14,k Usng the followng logarthmc change of varables: P lnp ; p lnp ; t lnt ; φ,j 1 ln 1 ; ln ; φ,j 3 ln 3 ; φ,j,k 4 ln,k 4, 15 for all,j,k, we can further wrte u and v n terms of the sums of exponentals n x: u x = log v x = log,j e P j p t 1 e P j t e pj 3,j k=1 e P k p j t,k 4 1, 16 e P j p t 1 e P j t e pj 3,j k=1 e P k p j t,k 4 1, 17 where x [ P T, p T, t T ] T, P [ P1,..., P ] T, p [ p 1,..., p ] T, and t [ t 1,..., t ] T. Snce the log-sum-exp functon s convex [15], both u x and v x are convex n x. However, v x u x n 1 s not necessarly concave. Usng the frst-order Taylor seres expanson around a gven pont x [m 1], we propose to approxmate v x by an affne functon as follows [10]: T v x v x [m 1] v x [m 1] x x [m 1], 18 where the l-th element of gradent v x s gven by 19, at the top of ths page. Wth the affne approxmaton 18 and the convex functon u x, t s clear that the constrant 11b can now be approxmated by a concave functon as: T v x [m 1] v x [m 1] x x [m 1] u x τ mn. 0 ext, we convexfy constrant 11e. By the varable change ᾱ lnα, 1 and upon denotng ᾱ [ᾱ 1,...,ᾱ ] T, 11e s rewrtten as: e p e P j h. ηeᾱ Applyng the arthmetc-geometrc nequalty, we have that: e P j h where P [m 1] s a fxed pont and e P j h e P [m 1] j h k=1 λ [m 1], 3 e P [m 1] j hk,. 4 As such, can be replaced by a strcter constrant: e p w ᾱ, P ηeᾱ e P j h λ [m 1] whch s equvalent to the followng affne constrant: p ᾱ, 5 P j c 0, 6 where c lnη λ[m 1] ln h ln s a constant. From the convex approxmatons 0 and 6, we now have the followng convex optmzaton problem whch 177

5 nd Internatonal Conference on Telecommuncatons ICT 015 Algorthm 1 Proposed DC-based SCA Algorthm 1: Intalze m := 1. : Choose a feasble pont x [0] P [0],p [0],t [0] ;α [0] and evaluate x [0] P[0], p [0], t [0] ;ᾱ [0] usng 15 and 1. 3: Compute v x [0] [0], log v x and λ [0],,j usng 17, 19 and 4, respectvely. 4: repeat 5: Gven v x [m 1] [m 1], log v x and, form a convex problem 7. 6: Use nteror-pont method to solve 7 for an approxmated soluton x [m] P[m], p [m], t [m] ;ᾱ [m] of problem 11 at the m-th teraton. 7: Update v x [m] [m], log v x and λ [m],,j usng 17, 19 and 4, respectvely. 8: Set m := m1. 9: untl Convergence of x, ᾱ or no further mprovement n the objectve value 7a 10: Recover the optmal soluton x ;α from x ;ᾱ va 15 and 1. gves an approxmated soluton to problem 11 at the m-th teraton: 7a mn x,ᾱ =1 P s.t. v x [m 1] e t eᾱ 1, eᾱ 1, v x [m 1] T x x [m 1] u x τ mn, 7b e P P max, p ᾱ P j c 0, 7c 7d 7e 7f where x [m 1] s known from the m 1-th teraton. In Algorthm 1, we propose an SCA algorthm n whch a convex problem 7 s optmally solved at each teraton. Proposton 1: Algorthm 1 generates a sequence of mproved feasble solutons that converges to a pont satsfyng the KKT condtons of the orgnal problem 11. Proof: Snce the gradent of the convex functon v x s ts subgradent [15], t follows that: T v x v x [m 1] v x [m 1] x x [m 1]. 8 Ths means that the approxmated concave constrant 7b s strcter than the orgnal nonconvex / constrant 11b. Moreover, from 3 we have that e p ηeᾱ e P j h e p / w ᾱ, P. Ths means that the approxmated affne constrant 7f s strcter than the orgnal nonconvex constrant 11e. Due to the strcter constrants, the optmal soluton of problem 7 wll belong to the feasble set of problem 11. If we ntalze Algorthm 1 wth a feasble pont x [0] ;ᾱ [0], the presence of strcter constrants guarantees that the optmal soluton x [m 1] ;ᾱ [m 1] to problem 7 at the m 1- th teraton s a feasble soluton to the same problem 7 y coordnate m Cell 1 Cell 3 Base Staton BS Relay R User U Cell x coordnate m Fg. 3. The example multcell network used n the smulaton. at the m-th teraton. As such, the optmal value of problem 7 at the m-th teraton ether mproves or, at least, remans unchanged by takng the optmal soluton x [m 1] ;ᾱ [m 1] from the prevous teraton. Algorthm 1 therefore converges to a pont x ;ᾱ. Fnally, t can be verfed that v x u x = v x [m 1] x= x [m 1] T v x [m 1] x x [m 1] u x 9 x= x [m 1] e p ηeᾱ e P j h ᾱ=ᾱ [m 1] P= P [m 1] = e p P. w ᾱ, ᾱ=ᾱ [m 1] P= P [m 1] 30 The results n 9-30 mply that the KKT condtons of the orgnal problem 11 wll be satsfed after the seres of approxmatons nvolvng convex problem 7 converges to x ;ᾱ. Ths completes the proof. IV. ILLUSTRATIVE RESULTS Fg. 3 shows an example heterogeneous network where all cells have an equal cell radus of 100m. In cells 1 and 3, we set the BS-relay and relay-user dstances as 4m and 51m, respectvely. In cell, the correspondng dstances are4m and 48m. ote that the BS, relay and user n each cell do not le on a straght lne. We set the path loss exponent as β = 3 and assume that the channel coeffcents h,j and g j,k,,j,k are crcularly symmetrc complex Gaussan random varables wth zero mean and unt varance. Wth a block fadng model, the randomly-generated values of h,j and g j,k reman unchanged for the duraton of resource allocaton. We fx BS transmt power budget at P max = 46dBm. Recent research has shown that energy harvestng effcency s practcal n the range and mnmum nput power requred to actvate energy harvestng crcut s practcal n the range 10 dbm to 30 dbm [16]. Hence we set η = 0.5 and P mn = 6dBm. These values ensure that for the assumed cell radus, the energy harvestng crcut s properly actvated. Wth the channel bandwdth of 0kHz and the nose power densty of 174dBm/Hz, the total nose power s σ = 131dBm. We 178

6 nd Internatonal Conference on Telecommuncatons ICT 015 Total BS transmt power, =1 P dbm τ mn = τ mn = 0.16 τ mn = ς = 0.1 ς = Iteratons Fg. 4. Convergence process of Algorthm 1. ntalze Algorthm 1 wth P [0] = ςp max ; α [0] = ς; t [0] = 1 α [0] ; p [0] = ςηα [0] P[0] j h,, where ς 0,1. To solve each convex problem n Algorthm 1, we resort to CVX, a package for specfyng and solvng convex programs [17]. Fg. 4 plots the convergence of the total BS power by Algorthm 1. Here, each teraton corresponds to solvng of a problem 7 by CVX. It s clear from the fgure that the proposed algorthm quckly converges n 3 teratons and the total requred transmt power drops to the possble mnmum,.e., P mn = 30.7 dbm for dfferent values of τ mn. It s not practcal to compare the performance of Algorthm 1 wth that of a globally optmal soluton. There s no global optmzaton approach avalable n the lterature to solve the hghly nonconvex problem 11. A drect exhaustve search would ncur a prohbtve computatonal complexty. However, Fg. 4 shows that ntalzng the algorthm wth dfferent values of ς does not affect the fnal soluton. Ths means that the acheved throughput s nsenstve to the ntal ponts, further suggestng that the attaned soluton corresponds to the global optmum n our specfc example [10], [11]. Fg. 5 demonstrates the advantage of jontly optmzng P, p, α as n the proposed Algorthm 1 over ndvdual optmzaton of BS transmt power P. In the latter approach, we fx p = ηα P j and α = 0.5,. To obtan the results presented n the fgure, we setς = 0.5 and use the target mnmum data rate τ mn = {0.161,...,0.165}. As seen, the proposed Algorthm always outperforms the sole optmzaton approach and db mprovement s observed. V. COCLUSIOS Ths paper has consdered the problem of mnmzng the total BS transmt power n a multcell network wth RFpowered relays. We have attempted to jontly optmze the BS transmt powers, the relay power splttng factors and the relay transmt powers. To resolve the hghly nonconvex problem formulaton, we have proposed a successve convex approxmaton algorthm based on DC programmng. We have proven that the devsed algorthm converges to a locally optmal soluton that satsfes the KKT condtons of the orgnal nonconvex problem. umercal examples have demonstrated the advantage of our approach. Total BS transmt power, =1 P dbm Optmzng P Optmzng P,p,α by Alg τ mn bps/hz Fg. 5. Performance comparson of jont optmzaton Algorthm 1 and the sole optmzaton of P. REFERECES [1] Y. A. Sambo, M. Z. Shakr, K. A. Qaraqe, E. Serpedn, and M. A. Imran, Expandng cellular coverage va cell-edge deployment n heterogeneous networks: Spectral effcency and backhaul power consumpton perspectves, IEEE Commun. Mag., vol. 5, no. 6, pp , Jun [] Z. Dng, S. M. Perlaza, I. Esnaola, and H. V. 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