Analysis and Optimization for Weighted Sum Rate in Energy Harvesting Cooperative NOMA Systems

Transcription

1 Analysis and Optimization for Wightd Sum Rat in Enrgy Harvsting Cooprativ NOMA Systms Binh Van Nguyn, Quang-Doanh Vu, and Kison Kim arxiv:86.264v [cs.it] 6 Jun 28 Abstract W considr a cooprativ non-orthogonal multipl accss systm with radio frquncy nrgy harvsting, in which a usr with good channl harvsts nrgy from its rcivd signal and srvs as a dcodand-forward rlay for nhancing th prformanc of a usr with poor channl. W hr aim at maximizing th wightd sum rat of th systm by optimizing th powr allocation cofficint usd at th sourc and th powr splitting cofficint usd at th usr with good channl. By xploiting th spcific structur of th considrd problm, w propos a low-complxity on-dimnsional sarch algorithm which can provid optimal solution to th problm. As a bnchmark comparison, w driv analytic xprssions and simpl high signal-to-nois ratio SNR approximations of th rgodic rats achivd at two usrs and thir wightd sum with fixd valus of th powr allocation and th powr splitting cofficints, from which th scaling of th wightd sum in th high SNR rgion is rvald. Finally, w providd numrical rsults to dmonstrat th validity of th optimizd schm. Indx Trms Cooprativ NOMA, RF-nrgy harvsting, wightd sum rat analysis and optimization. I. Introduction Non-orthogonal multipl accss NOMA transmission is mrging as a promising multipl accss tchniqu for th nxt gnration of wirlss ntworks []. Th cornrston of NOMA is to xploit th powr domain and channl quality diffrnc among usrs to achiv multipl accss. An issu rising in a NOMA systm is that usrs with good channl conditions can significantly strngthn thir prformanc, whil th prformanc of usrs with bad channl conditions ar rlativly poor [2]. A possibl solution for this problm is combining cooprativ communication with NOMA to gnrat a cooprativ NOMA C-NOMA transmission schm in which usrs with good channl conditions oprat as rlays to strngthn th transmission rliability for usrs suffring from bad channl conditions [3]-[6]. Rcntly, radio frquncy nrgy harvsting RF-EH has bcom an fficint solution to prolong th liftim Binh Van Nguyn and Kison Kim ar with th School of Elctrical Enginring and Computr Scinc, Gwangju Institut of Scinc and Tchnology, Rpublic of Kora. {binhnguyn, kskim}@gist.ac.kr. Quang-Doanh Vu is with th Cntr for Wirlss Communications, Univrsity of Oulu, Finland. doanh.vu@oulu.fi. Th authors gratfully acknowldg th support from Elctronic Warfar Rsarch Cntr at Gwangju Institut of Scinc and Tchnology GIST, originally fundd by Dfns Acquisition Program Administration DAPA and Agncy for Dfns Dvlopmnt ADD. of nrgy-constraint wirlss communication systms [7]. Th advantag of RF-EH is from th fact that RF signals carry both information and nrgy at th sam tim, i.. RF-EH allows limitd-powr nods to scavng nrgy and procss information simultanously [8]. Thr xist two main RF-EH tchniqus, namly, tim switching TS and powr splitting PS. With TS, a rcivr switchs btwn nrgy harvstr and data dcodr. With PS, a rcivr sparats th RF signals into two parts on for EH and th othr for dcoding by a PS cofficint. Hr, w mainly focus on PS, sinc PS is considrd to b mor gnral compard to TS [9]. Clarly, RF-EH provids mor incntivs for usr coopration, thus it is natural to us RF-EH in C-NOMA systms. Rprsntativ xampls for this approach ar []-[] whr th systms with on sourc and multipl usrs ar considrd. Ths two works proposd usr-pair slction schms and analyz th prformanc in trms of outag probability. In this papr, w invstigat th impact of powr allocation and PS cofficints on th prformanc of C- NOMA systms. Diffrnt from [] and [], w focus on wightd sum rat of th systms which has bn still rlativly opn. It is worth mntioning that wightd sum rat finds many practical applications sinc it is hlpful for prioritizing usrs [2]. Spcifically, our main contributions ar as follows. W considr a C-NOMA with RF-EH systm whr a sourc communicats with two usrs. W first formulat th problm of wightd sum rat maximization in which powr allocation and PS cofficints ar th dsign paramtrs. Th problm is non-convx whos optimal solution can b found by th xhaustiv twodimnsional 2D sarch. Towards a mor fficint solution, w dvlop an on-dimnsional D sarch algorithm by xploiting th spcific structur of th problm. For a comparison bnchmark, w driv closd-form xprssions and high signal-to-nois ratio SNR approximations of th rgodic rats achivd at th two usrs and thir wightd sum with fixd powr allocation and PS cofficints. W numrically dmonstrat that optimizd powr allocation and PS cofficint can significantly improvs th systm prformanc in trms of wightd sum rat, i.. 45% nhancmnt whn th avrag SNR is db and th wight ratio is 5. On th othr hand, th analysis rsults rval that th scaling of th wightd sum rat is w 2 log 2 SNR, whr w is

2 2 th priority wight of th usr with good channl. II. Systm Modl W considr a wirlss communication systm consisting of a sourc, dnotd by S, and two usrs which ar associatd with diffrnt channl conditions; w dnot th usr with good channl by U, and th on with bad channl by U 2. All nods ar quippd with a singl-antnna and oprat in th half-duplx mod. Lt h, h 2, and h 3 dnot th complx channl cofficint btwn S and U, S and U 2, U and U 2, rspctivly. All channls ar assumd to b indpndnt and idntically distributd Rayligh block fading. From th assumption about channl quality, w hav g > g 2 whr g i = h i 2. W focus on th transmission from S to th usrs. Th transmission protocol includs two phass, ach of lngth T in tim unit. In particular, lt x i, i {, 2}, b th normalizd complx signal for U i, and P S b th transmit powr at S. In th first phas, S gnrats a suprimposd signal givn by x S = P S x + P S x 2, whr dnots th powr allocation cofficint, and broadcasts x S to th usrs. Th rcivd signal at U i during this phas is y i = h i x S + n i whr n i is th additiv whit Gaussian nois AWGN with varianc N. Usr U uss its rcivd signal for dcoding x, harvsting nrgy, and dcoding x 2. In particular, U divids y into two parts with a PS cofficint ρ [, ]. Th first part givn by y h = ρy is for harvsting nrgy, and th scond part givn by y ip = ρy is for dcoding information. Consquntly, th nrgy harvstd at U is [] E = T ηρp S g 2 whr η dnots th nrgy convrsion fficincy. U dcods x 2 basd on y ip, thn applis succssiv intrfrnc cancllation SIC bfor dcoding x. Thrfor, th signal-to-intrfrnc-plus-nois ratios SINRs for dcoding x 2 and x at U ar γ x2, ρ = ρ P S g ρ P S g + ρ N + µn, 3 γ x, ρ = ρ P Sg ρ N + µn 4 rspctivly. Hr, th last trm in th dnominator of γ x2, ρ and γx, ρ ar du to th convrsion nois which is assumd to b AWGN with varianc µn [3]. In th scond phas, U uss th harvstd nrgy E to transmit x 2 to U 2. Th signal rcivd at U 2 during this phas is ỹ 2 = ρηp S g h 3 x 2 + n 2. 5 W suppos that th maximal ratio combining MRC rcivr is usd at U 2 [4]. Thn th SINR for dcoding x 2 at U 2 is 2 log 2 γ2 MRC P S g 2, ρ = + ρηp Sg g 3. 6 P S g 2 + N + µn N + µn In summary, th instantanous achivd rat at U and U 2 ar C, ρ { = 2 log 2 + γ x, ρ and C 2, ρ = + min γ x 2, ρ, γmrc 2, ρ }, rspctivly. III. Wightd Sum Rat Optimization Our aim is to maximiz th wightd sum rat of th systm. Particularly, th optimization problm is formulatd as maximiz w C, ρ + w 2 C 2, ρ,ρ subjct to < <, ρ, 7a 7b whr w > and w 2 > ar th priority wights. Hr w focus on th cas w 2 > w. A practical xampl for th considrd scnario is that in cllular ntwork, th usr at cll-dg suffring bad channl conditions for a long tim will b assignd a largr wight compard to th on in nar bas station ara for fairnss and/or stability [2]. Objctiv function 7a is non-convx with rspct to th rlatd variabls. For achiving an optimal solution, a xhaustiv 2D sarch procdur ovr and ρ can b usd. Clarly, doing this is highly complx and infficint. In th following, by looking insid th problm, w dvlop a low-complxity D sarch algorithm which solvs 7 optimally. W start with an usful rsult statd as follows. Lmma. Lt, ρ b an optimal of 7, thn C 2, ρ = 2 log 2 + γ MRC 2, ρ. 8 Proof: Th lmma can b provd by contradiction. Spcifically, suppos that thr xists an optimal point, ρ such that log 2 + γ x2, ρ < log 2 + γ MRC 2, ρ. 9 Sinc γ2 MRC, ρ and γ x2, ρ ar incrasing and dcrasing functions of ρ, w can always find ρ > such that C 2, ρ ρ > C 2, ρ. Morovr, C, ρ ρ > C, ρ bcaus γ x, ρ is a dcrasing function of ρ. Consquntly, w hav C, ρ ρ + C 2, ρ ρ > C, ρ + C 2, ρ, which contradicts th assumption that, ρ is an optimal. This complts th proof. From Lmma and th monotonicity of th logarithmic function, w can rwrit 7 as maximiz,ρ f, ρ a subjct to γ x2, ρ γmrc 2, ρ b < <, ρ < c Th optimal solution for th cas w 2 w is trivial, i.. it is not difficult to justify that th optimal solution for this cas is =, ρ =.

3 3 whr f, ρ + γ x, ρ + γ MRC 2, ρ w 2, and w 2 = w 2 /w. As a furthr stp, w quivalntly rwrit as maximiz f, ρ,ρ subjct to < <, ρ ρ, a b whr ρ = b b 2 4ac 2a, γ = P S /N, a = η γgg3 γg+, b = η γgg3 γg+ γg2 γg+ γg 2+µ+ + γg, and c = γg γg2 γg+ γg 2+µ+. Th quivalnc can b provd as follows. W first not that th lft handsid LHS of b monotonically incrass whil th right hand-sid RHS of b monotonically dcrass with ρ. In addition, whn ρ =, th RHS is largr than th LHS du to th assumption g > g 2. Morovr, th RHS whn ρ. Thus, givn,, thr xists an uniqu ρ, such that b is satisfid if and only if ρ [, ρ. It is noting that b can b writtn as aρ 2 bρ + c, from which w yild ρ. W now focus on objctiv function a. For a givn, a rducs to a function of ρ givn as f ρ d ρ t ρ p + qρ w2 2 whr d = + µ + γg, = + γg, t = + µ, p = + γg2 γg, q = η γgg3 2+. W also introduc a function of givn as θ = β 2 q w 2 dp pt + q w 2 td 3 whr β =.5qd w 2 +.5qt w 2 +. W hav an usful proprty of f ρ statd as follows. Proposition. If θ > and ρ = β θ q w 2N v,, whn ρ incrass, f ρ incrass until rachs a maximum at ρ thn dcrass. If θ > and ρ, f ρ is dcrasing ovr ρ,. Othrwis, f ρ is incrasing ovr ρ,. Th proof of th proposition can b asily obtaind via th gradint of f ρ givn as f ρ ρ = [d t p + qρ + q w 2 t ρ d ρ] p + qρ w2 t ρ 2. 4 Th algbraic stps ar skippd for th sak of brvity. Th proprty allows us to find th optimal valu of ρ whn th optimal valu is givn as follows. ρ = ρ if θ > and < ρ < ρ. If θ > and ρ <, ρ =. Othrwis ρ = ρ. In summary, w outlin th proposd D sarch procdur in Algorithm which outputs th optimal solution of 7. IV. Ergodic Rat Analysis In this sction, w driv th rgodic and thir corrsponding wightd sum rats achivd at th usrs with fixd valus of and ρ, which can b usd as a bnchmark in valuating Algorithm. Algorithm Th D sarch for solving 7 optimally. : For ach,, calculat ρ and θ. 2: if θ > thn 3: Calculat ρ = β θ q w 2 4: if ρ, ρ thn ρ = ρ, 5: lsif ρ, thn ρ =, 6: ls ρ = ρ, nd if. 7: ls 8: ρ = ρ 9: nd if : Output:, ρ = arg max f, ρ,ρ A. Ergodic Rat of U Lt us first driv th rgodic rat of th U, which can b xprssd as follows [4] C = 2 ln 2 F X x dx, 5 + x whr X = ρ γg ρ+µ, and F X x dnots th cumulativ distributd function CDF of X which is givn by ρ + µ x F X x = xp ρ γδ 2, 6 whr δi 2 is th powr of th channl h i. Plugging 6 into 5 givs C = ρ + µ 2 ln 2 xp ρ + µ ρ γδ 2 Γ, ρ γδ 2, 7 whr Γ x, y is th incomplt uppr Gamma function. B. Ergodic Rat of U 2 Similar to 5, w hav C 2 = 2 ln 2 F Z z dz, 8 + z whr Z = min { γ x2, ρ, γmrc 2, ρ } = min {Y, W} and F Z z can b approximatd as F Z z Pr [Y > z] Pr [W > z], 9 whr th corrlation btwn Y and W is ignord. It can b radily vrifid that th corrlation btwn Y and W vanishs in th high SNR rgion implying that th approximation is tight whn th avrag SNR gos larg. Th probability trm Pr [Y > z] is first drivd as {, if z Pr [Y > z] = xp, ρ+µz γδ 2 ρ z, if z <. 2 Scondly, Pr [W > z] can b xprssd as follows Pr [W > z] = F W z y f W2 y dy, 2

4 Achivabl Rats 4 whr W = γg2 γg, W 2+ 2 = ρη γgg3, and {, if z, F W z = xp z, if z <, 22 γδ 2 2 z + µ + µ z f W2 z = 2 K ρη γδ 2 δ2 3 ρη γδ 2 δ2 3 whr K i x dnots th modifid Bssl function of th scond kind of ordr i th. W not that z y is always lss than whn z <. On th othr hand, whn z, z y if y z and z y < if z y z. Bas on this fact, w can furthr xtnd 2 as follows Pr [W > z] = F W2 z + µ z y + xp Lz γδ 2 2 f W2 y dy, 24 z + y whr L z = if z < /, L z = z / othrwis, and + µ z + µ z F W2 z = 2 ρη γδ 2 K 2 δ2 3 ρη γδ δ2 3 Plugging 24 and 2 into 9 and 8, w obtain C2 2 + x xp ρ + µ z γδ 2 ρ z + µ z + µ z K 2 ρη γδ 2 dz δ2 3 ρη γδ 2δ2 3 + xp ρ + µ z xp γδ 2 ρ z + µ z y γδ 2 2 z + y + µ z 2 ρη γδ 2δ µ ρη γδ 2 δ2 3 + xk dydz. 26 It is worthy noting that 26 can b radily valuatd by using standard mathmatical programs such as Matlab and Mathmatica. In addition, from 7 and 26, w can straightforwardly obtain th systm wightd sum rat, i.. C sum = w C + w 2 C 2, with fixd valu of and ρ. C. High SNR Analysis To gain novl insights from our afor-prsntd analytic rsults, w now invstigat th rgodic rats in th high SNR rgion. Proposition 2. In th high SNR rgion, th rgodic rats of U and U 2 can b approximatd as follows [ C ρ + µ χ ln 2 ln 2 ρ δ γ 2 + ρ + µ ] ρ δ γ 2, 27 C2 2 log 2 +, 28 whr χ dnot th Eulr constant. 2 - =. =.3 w = Simulatd rats C, analytic C 2, analytic w = 5 2 C, analytic sum Scaling of C -2 sum Figur. Achivabl rats with fixd valus of and ρ. Proof: For C, w first not that Γ, x = Ei x, whr Ei x dnots th xponntial intgral function. x Thn using th th facts that xpx and x Ei x χ + ln x + x, w can obtain 27. For C2, lt s first rcall its instantanous xprssion C 2 = 2 log { 2 + min γ x 2, ρ, γmrc 2, ρ }. Thn, in th high rgion of γ, w can radily show that γ x2, ρ < γmrc 2, ρ, from which 28 can + ρη γgg3 b obtaind. Proposition 2 implis that as th avrag SNR γ incrass, th rgodic rat of U monotonically incrass, howvr, that of U 2 is saturatd. This is rasonabl bcaus as γ incrass, th SNR usd for dcoding x at U also incrass, and thus, th rgodic rat of U incrass. On th othr hand, th actual SINR usd for dcoding x 2 is limitd by th minimum of th SINRs usd for dcoding x 2 at U and U 2. In addition, whn γ incrass, th SINR usd for dcoding x 2 at U quickly convrgs to and limits th actual SINR usd for dcoding x 2, which maks th rgodic rat of U 2 saturatd. From Proposition 2, w hav C sum = w C + w 2C 2 w 2 log 2 γ + w 2 2 log 2 w 2 log 2 γ, 29 which rvals that whn γ, th scaling of th systm wightd sum rat is w 2 log 2 γ. V. Numrical Rsults and Discussions In this sction, w provid rprsntativ simulatd and analytical rsults to validat our analysis and dmonstrat th nhancmnt of th systm prformanc achivd by th proposd D sarch algorithm. Th simulation stup follows th systm modl givn in Sction II with η = and δ 2 = δ2 2 = δ3 2 =. Figur plots th rgodic rats of th considrd systm with fixd valus of and ρ. Th first obsrvation is that th analytic curv of C follows th corrsponding simulatd on xcllntly, whil th analytic curvs of C2 and Csum quickly convrg to th corrsponding simulatd curvs in th mdium and high SNR rgions. This rsult

5 Achivabl Rats Avrag valus of and with and with =. and =.3 with =. and =. with =.3 and =.5 with =.3 and = = 2 db = 2 db = db = db = 5 db = 5 db w 2 / w Figur 2. ρ. Wightd sum rat with optimal and fixd valus of and Figur 3. Avrag valu of ρ and vrsus w 2. implis that our analyss on th systm s rgodic rats ar valid. Clarly, th figur confirms our finding on th scaling of th wightd sum rat in th high SNR rgion. Th othr intrsting obsrvation is that th rgodic rat of U 2 is saturatd as th avrag SNR gts larg, rvaling that incrasing th avrag SNR or qually incrasing th transmit powr P S cannot nhanc th prformanc of th usr with poor channl. Figur 2 plots th systm wightd sum rats with optimal and fixd valus of and ρ as functions of th avrag SNR. W tak w 2 = {2, 5}. Th figur clarly shows that using Algorithm rmarkably nhancs th wightd sum rat prformanc of th systm. Particularly, at γ = db, optimal valus of and ρ provids 45% and 29.3% wightd sum rat nhancmnts with w 2 = 5 and w 2 = 2. Thus, th rsults strongly suggst that paramtr and ρ should b optimizd. In Fig. 3, w illustrat th avrag of th optimal valus of and ρ i.. E {a } and E {ρ }, rspctivly vrsus w 2. An obsrvation is that as w 2 incrass, E {a } rducs and approachs zro. This is du to th fact that whn w 2 nlargs, U 2 has a highr priority compard to U, and thus, mor powr should b allocatd to th transmission of x 2. On th othr hand, w can also obsrv that E {ρ } incrass and tnds to a crtain valu. This is bcaus th rat of U 2 providd in Lmma is an incrasing function with ρ, and ρ should b small nough so that constraint b is satisfid. VI. Conclusion W considrd a C-NOMA systm with RF-EH including a sourc and two usrs. W first dvlopd a D sarch algorithm to optimally solv th problm of wightd sum rat maximization rspct to powr allocation and powr splitting cofficint ρ. Thn, w drivd closd-form xprssions and high SNR approximations of th rgodic rats achivd at th two usrs with fixd valus of and ρ. Th numrical rsults dmonstratd that using th optimal valus of and ρ significantly nlargs th systm wightd sum rat, i.. 45% nhancmnt whn th avrag SNR is db and th wight ratio is 5. In addition, from analytic rsults, w rvald that th scaling of th wightd sum with fixd valu of and ρ is w 2 log 2 γ. Rfrncs [] L. Zhang, J. Liu, M. Xiao, G. Wu, Y. Liang, and S. Li, Prformanc Analysis and Optimization in Downlink NOMA Systms with Cooprativ Full-Duplx Rlaying, IEEE J. Slct. Aras Commun., vol. 35, no., pp , Oct. 27. [2] Z. Yang, Z. Ding, Y. Wu, and P. Fan, Novl Rlay Slction Stratgis for Cooprativ NOMA, IEEE Trans. Vh. Tchnol., vol. 66, no., pp. 4-23, Nov. 27. [3] Z. Ding, M. Png, and H. V. Poor, Cooprativ Non-orthogonal Multipl Accss in 5G Systms, IEEE Commun. Ltt., vol. 9, no. 8, pp , Aug. 25. [4] J. Kim and I L, Capacity Analysis of Cooprativ Rlaying Systms using Non-orthogonal Multipl Accss, IEEE Commun. Ltt., vol. 9, no., pp , Nov. 25. [5] D. Wan, M. Wn, H. Yu, Y. Liu, F. Ji, and F. Chn, Nonorthogonal Multipl Accss for Dual-Hop Dcod-and-Forward Rlaying, in Proc. IEEE Global Communication Confrnc, Washington, USA, Dc. 26. [6] J. Mn, J. G, and C. Zhang, Prformanc Analysis of Non-orthogonal Multipl Accss For Rlaying Ntworks ovr Nakagami-m Fading Channls, IEEE Trans. Vh. Tchnol., vol. 66, no. 2, pp. 2-28, Apr. 26. [7] X. Fafoutis, A. D. Mauro, C. Orfanidis, and N. Dragoni, Enrgy-Efficint Mdium Accss Control for Enrgy Harvsting Communications, IEEE Trans. Consum. Elctron., vol. 6, no. 4, pp. 42-4, Nov. 25. [8] F. Wang, S.Guo, Y. Yang, and B. Xiao, Rlay Slction and Powr Allocation for Cooprativ Ntwork With Enrgy Harvsting, IEEE Sys. J., vol. pp, no. 99, pp. -2, April 26. [9] Q. Shi, L. Liu, W. Xu and R. Zhang, Joint Transmit Bamforming and Rciv Powr Splitting for MISO SWIPT Systms, IEEE Trans. Wirlss Commun., vol. 3, no. 6, pp , Jun 24. [] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, Cooprativ Non-orthogonal Multipl Accss with Simultanous Wirlss Information and Powr Transfr, IEEE J. Sl. Aras Commun., vol. 34, no. 4, pp , April 26. [] N. T. Do, D. B. da Costa, T. Q. Duong, and B. An, A BNBF Usr Slction Schm for NOMA-basd Cooprativ Rlaying Systms with SWIPT, IEEE Commun. Ltt., vol. 2, no. 3, pp , Mar. 27. [2] S. S. Christnsn, R. Agarwal, E. D. Carvalho and J. M. Cioffi, Wightd sum-rat maximization using wightd MMSE for MIMO-BC bamforming dsign, IEEE Trans. Wirlss Commun., vol. 7, no. 2, pp , Dc. 28. [3] B. V. Nguyn, H. Jung, D. Har, and K. Kim, Prformanc Analysis of a Cognitiv Radio Ntwork With an Enrgy Harvsting Scondary Transmittr Undr Nakagami-m Fading, IEEE Accss, vol. 6, pp , Jan. 28. [4] X. Yu, Y. Liu, S. Kang, A. Nallanathan, Z. Ding, Exploiting Full/Half-Duplx Usr Rlaying in NOMA Systms, IEEE Trans. Commun., vol. 66, no. 2, pp , Sp. 27.