SIMULTANEOUS wireless information and power transfer. Joint Optimization of Power and Data Transfer in Multiuser MIMO Systems

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1 Jont Optmzaton of Power and Data ransfer n Mutuser MIMO Systems Javer Rubo, Antono Pascua-Iserte, Dane P. Paomar, and Andrea Godsmth Unverstat Potècnca de Cataunya UPC, Barceona, Span ong Kong Unversty of Scence and echnoogy KUS, ong Kong Stanford Unversty, CA, USA Emas:{javer.rubo.opez, antono.pascua}@upc.edu, paomar@ust.hk, andrea@ws.stanford.edu arxv:64.44v [cs.i] Apr 6 Abstract We present an approach to sove the nonconvex optmzaton probem that arses when desgnng the transmt covarance matrces n mutuser mutpe-nput mutpe-output MIMO broadcast networks mpementng smutaneous wreess nformaton and power transfer SWIP. he MIMO SWIP probem s formuated as a genera mut-objectve optmzaton probem, n whch data rates and harvested powers are optmzed smutaneousy. wo dfferent approaches are apped to reformuate the nonconvex mut-objectve probem. In the frst approach, the transmtter can contro the specfc amount of power to be harvested by power transfer whereas n the second approach the transmtter can ony contro the proporton of power to be harvested among the dfferent harvestng users. he computatona compexty w aso be dfferent, wth hgher computatona resources requred n the frst approach. In order to sove the resutng formuatons, we propose to use the majorzatonmnmzaton MM approach. he dea behnd ths approach s to obtan a convex functon that approxmates the nonconvex objectve and, then, sove a seres of convex subprobems that w converge to a ocay optma souton of the genera nonconvex mut-objectve probem. he souton obtaned from the MM approach s compared to the cassca bock-dagonazaton BD strategy, typcay used to sove the nonconvex mutuser MIMO network by forcng no nterference among users. Smuaton resuts show that the proposed approach mproves over the BD approach both the system sum rate and the power harvested by users. Addtonay, the computatona tmes needed for convergence of the proposed methods are much ower than the ones requred for cassca gradent-based approaches. I. INRODUCION SIMULANEOUS wreess nformaton and power transfer SWIP s a technque by whch a transmtter actvey feeds a recever or a set of recevers wth power that s sent through rado frequency RF sgnas and, smutaneousy, sends usefu nformaton to the same or dfferent recevers []. By harvestng ths transmtted energy, battery-constraned mobe termnas are abe to recharge ther batteres and, thus, proong ther operaton tme []. Athough there are many dfferent harvestng technques used to power devces, such as soar or wnd, SWIP technoogy provdes an appeang souton he research eadng to these resuts has receved fundng from the European Commsson n the framework of the FP7 Network of Exceence n Wreess COMmuncatons NEWCOM# Grant agreement no. 86, from the Spansh Mnstry of Economy and Compettveness Mnstero de Economía y Compettvdad through the project EC-96-C- GREN-LINK- MAC, project EC-45-R DISNE, and FPI grant BES--585, from the Cataan Government AGAUR through the grant 4 SGR 6, from the ong Kong Government through the research grant ong Kong RGC 6784, and from the NSF Center for Scence of Informaton CSoI: NSF- CCF-997. snce the transmtter can contro the amount of energy that the mobe termnas need to keep ave. storcay, due to the hgh attenuaton of mcrowave sgnas over dstance, SWIP technques were ony consdered n ow-power devces, such as RFID tags []. Nevertheess, recent advances n antenna technooges and RF harvestng crcuts have enabed energy to be transferred and harvested much more effcenty [], []. he concept of SWIP was frst studed from a theoretca pont of vew by Varshney [4]. e showed that, for the snge-antenna addtve whte Gaussan nose AWGN channe, there exsts a nontrva trade-off n maxmzng the data rate versus the power transmsson. In [5], the authors consdered a mutpe-nput mutpe-output MIMO scenaro wth one transmtter capabe of transmttng nformaton and power smutaneousy to one recever. Later, n [6], the authors extended the work n [5] by consderng that mutpe users were present n the broadcast MIMO system. owever, the mut-stream transmt covarance optmzaton that arses n broadcast MIMO systems s a very dffcut nonconvex optmzaton probem. In order to overcome that dffcuty, authors n [6] consdered a bock-dagonazaton BD strategy [7], n whch nterference s pre-canceed at the transmtter. he BD technque aows for a smpe souton but wastes some degrees of freedom and, thus, degrades the overa performance. Works [8] and [9] consdered a MIMO network consstng of mutpe transmtter-recever pars wth co-channe nterference. he study n [8] focused on the case wth two transmtter-recever pars whereas n [9], the authors generazed [8] by consderng that k transmtterrecevers pars were present. he work n [] consdered a MIMO system wth snge-stream transmsson. In contrast to prevous works where the system rate was optmzed, the objectve was to mnmze the overa power consumpton wth per-user sgna to nterference and nose rato SINR constrants and harvestng constrants. he desgn of mutuser broadcast networks under the framework of mutpe-nput snge-output MISO beamformmg optmzaton has aso been addressed n works such as [] and []. here exst two approaches n the terature that dea wth the nonconvex optmzaton of the transmt covarance matrces n mutuser mut-stream MIMO networks. he frst s based on the duaty prncpe []. In [4], authors apped that prncpe to obtan the beamformng optmzaton souton for the mutuser MIMO SWIP broadcast channe. owever, that work consdered an overa sum harvestng constrant nstead of ndvdua per-user harvestng constrants. he second approach s based on the mnmzaton of the mean square

2 error MSE [5]. owever, ths technque cannot be apped to the SWIP framework due to fact that the resutng probem remans nonconvex. he man dfference of our work wth respect to the prevous works descrbed above s that we assume a broadcast mutuser mut-stream non BD-based MIMO SWIP network, n whch per-user harvested power and nformaton transfer must be optmzed smutaneousy. We mode our transmtter desgn as a mut-objectve probem n whch the scenaros studed n [5] and [6] are shown to be partcuar soutons of the proposed framework. Addtonay, we assume that nterference s not pre-canceed.e., the BD approach s not apped and, thus, both arger nformaton transfer and harvested power can be acheved smutaneousy. he resutng probem s nonconvex and very dffcut to sove. In order to obtan oca soutons, we derve dfferent methods based on majorzaton-mnmzaton MM technques. By means of ths strategy, we are abe to reformuate our orgna nonconvex probem nto a seres of convex subprobems that are easy soved.e., through agorthms that have a very ow computatona compexty and whose soutons converge to a ocay optma souton of the orgna nonconvex probem. he remander of ths paper s organzed as foows. In Secton II, we ntroduce a summary of the mathematca technques empoyed n ths paper. In Secton III we present the system and sgna modes and the probem formuaton. In Secton IV we derve the mathematca modeng requred to reformuate the orgna nonconvex probem nto convex subprobems that are soved usng the MM approach. In Secton V, we evauate the performance of the proposed methods and, fnay, n Secton VI, we draw some concusons. Notaton: We adopt the notaton of usng bodface ower case for vectors x and upper case for matrces X. he transpose, conjugate transpose hermtan, and nverse operators are denoted by the superscrpts,, and, respectvey. r and det denote the trace and the determnant of a matrx, respectvey. vecx s a coumn vector resutng from stackng a coumns of X. We use X to denote the N tupe X X N = = X,..., X N and F to denote the matrx Frobenus norm. II. MAEMAICAL PRELIMINARIES A. Mut-Objectve Optmzaton Mut-objectve optmzaton aso known as mut-crtera optmzaton or vector optmzaton s a type of optmzaton that nvoves mutpe objectve functons that are optmzed smutaneousy [6]. For a nontrva mut-objectve probem, n genera, there does not exst a snge souton that smutaneousy optmzes each objectve. In that case, the objectve functons are sad to be confctng, and there exsts a possby nfnte number of Pareto optma soutons. A souton s caed Pareto optma f none of the objectve functons can be mproved n vaue wthout degradng some of the other objectve vaues. Defntons Defnton [6]. A mut-objectve probem can be formay expressed as maxmze fx = f x,..., f K x x subject to x X, where f k : C N R for k =,..., K and X s the feasbe set that represents the constrants. Let Y be the set of a attanabe ponts for a feasbe soutons,.e., Y = fx. Effcent Soutons Defnton [6], Defnton.. A pont x X s caed Pareto optma f there s no other x X such that fx fx, where refers to the component-wse nequaty,.e., f x f x, =,..., K. Sometmes, ensurng Pareto optmaty for some probems s dffcut. Due to ths, the condton of optmaty can be reaxed as foows. Defnton [6], Defnton.4. A pont x X s caed weaky Pareto optma or weaky effcent f there s no other x X such that fx fx, where refers to the strct component-wse nequaty,.e., f x > f x, =,..., K. A Pareto optma soutons are aso weaky Pareto optma. Fndng Pareto Optma Ponts here are severa methods for fndng the Pareto ponts of a mut-objectve probem. In the seque, we present three dfferent scaarzaton technques.. Weghted sum method: the smpest scaarzaton technque s the weghted sum method whch coapses the vectorobjectve nto a snge-objectve component sum: maxmze x X K β k f k x, k= where β k are rea non-negatve weghts. he foowng resuts present the reaton between the optma soutons of and the Pareto optma ponts of the orgna probem. Proposton [6], Proposton.9. Suppose that x s an optma souton of. hen, x s weaky effcent. Proposton [6], Proposton.. Let X be a convex set, and et f k be concave functons, k =,..., K. If x s weaky effcent, there are some β k such that x s an optma souton of. As as resut, convexty s apparenty requred for fndng a weaky Pareto optma ponts wth the weghted sum method, whch means that f the orgna probem s not convex, a the Pareto optma ponts may not be found by usng the weghted sum method. owever, there are other weghted sum technques n the terature see, for exampe, the adaptve weghted sum method [7] that are abe to fnd a Pareto optma ponts for nonconvex probems at the expense of a hgher computatona compexty.. Epson-constrant method: n ths method, ony one of the orgna objectves s maxmzed whe the others are trans-

3 formed nto constrants: maxmze f j x x X subject to f k x ɛ k, k =,..., K, k j. Let us ntroduce the foowng resuts. Proposton [6], Proposton 4.. Let x be an optma souton of for some j. hen x s weaky Pareto optma. Proposton 4 [6], Proposton 4.5. A feasbe souton x X s Pareto optma f, and ony f, there exsts a set of ˆɛ k, k =,..., K such that x s an optma souton of for a j =,..., K. Contrary to the weghted sum method, convexty s not needed n the prevous two propostons but convexty s st typcay requred to sove probems ke.. ybrd method: ths method combnes the prevous two methods,.e., the weghted sum method and the epsonconstrant method. In ths case, the scaarzed probem to be soved has a weghted sum objectve and constrants on a or some objectves. maxmze β k f k x 4 x X k K subject to f k x ɛ k, k K, where K K and K K, beng A the cardnaty of set A, and β k are rea non-negatve weghts. B. Majorzaton-Mnmzaton Method he MM s an approach to sove optmzaton probems that are too dffcut to sove n ther orgna formuaton. he prncpe behnd the MM method s to transform a dffcut probem nto a sequence of smpe probems. Interested readers may refer to [8] and references theren for more detas. he method works as foows. Suppose that we want to maxmze f x over X. In the MM approach, nstead of maxmzng the cost functon f x drecty, the agorthm optmzes a sequence of approxmate objectve functons that mnorze f x, producng a sequence {x k } accordng to the foowng update rue: x k+ = arg max x X ˆf x, x k, 5 where x k s the pont generated by the agorthm at teraton k and ˆf x, x k known as surrogate functon s the mnorzaton functon of f x at x k,.e., t has to be a goba ower bound tght at x k. Probem 5 w be referred as surrogate probem. In addton, the surrogate functon must aso be contnuous n x and x k. he ast condton that the surrogate functon must fuf s that the drectona dervatves of tsef and of the orgna objectve functon f x must be equa at the pont Let f : C N R. hen, the drectona dervatve of fx n the drecton of vector d s gven by f x; d m λ fx+λd fx λ. x k. A n a, the four condtons are as foows: A : ˆf x k, x k = f x k, x k X, 6 A : ˆf x, x k f x, x, x k X, 7 A : ˆf x, x k ; d x=x k = f x k ; d, d wth x k + d X, 8 A4 : ˆf x, x k s contnuous n x and x k. 9 Under assumptons A A4, every mt pont of the sequence {x k } s a ocay optma pont of the orgna probem gobay optma f the probem s convex see [8] for detas. III. PROBLEM FORMULAION Let us consder a wreess broadcast mutuser system consstng of one base staton BS transmtter equpped wth n antennas and a set of K recevers, denoted as U = {,,..., K}, where the k-th recever s equpped wth n Rk antennas. We assume that a gven user s not abe to decode nformaton and to harvest energy smutaneousy, and that a user beng served wth nformaton by the BS uses a the energy to decode the sgna. hus, the set of users s parttoned nto two dsjont subsets. One that contans the nformaton users, denoted as U I U wth U I = N, and the other subset that contans harvestng users, denoted as U E U wth U E = M. herefore, U I U E = and U I + U E = N + M = K. Wthout oss of generaty w..o.g., et us ndex users as U I = {,..., N} and U E = {N +,..., N + M}. he equvaent baseband channe from the BS to the k-th recever s denoted by k C n R k n. It s aso assumed that the set of matrces { k } s known to the BS and to the correspondng recevers the case of mperfect CSI s out of the scope of the paper. As far as the sgna mode s concerned, the receved sgna for the -th nformaton recever can be modeed as y = B x + B k x k + n, U I. k In the prevous notaton, B x represents the transmtted sgna for user U I, where B C n n S s the precoder matrx and x C n S represents the nformaton symbo vector. It s aso assumed that the sgnas transmtted to dfferent users are ndependent and zero mean. n S denotes the number of streams assgned to user U I and we assume that n S = mn{n R, n } U I. he transmt covarance matrx s S = B B f we assume w..o.g. that E [ ] x x = InS. n C n R denotes the recever nose vector, whch s consdered Gaussan wth E [ ] n n = InR. Note that the mdde term of s an nterference term. he covarance matrx of the nterference pus nose s wrtten as Ω S = S + I, U I, In ths paper, we assume for smpcty n the formuaton that a user beongs to ether the harvestng set or the nformaton set and that both sets are known and fxed. hs assumpton coud be generazed by consderng that some users are not seected n ether set as we as by defnng whch partcuar users are schedued n each partcuar set.e., user groupng strateges. owever, ths fas out of the scope of ths paper. We assume that nose power σ = w..o.g., otherwse we coud smpy appy a scae factor at the recever and re-scae the channes accordngy.

4 4 where S = S k. Let x = Bx denote the sgna vector k transmtted by the BS, where the jont precodng matrx s defned as B = [B... B N ] C n n S, beng n S = n S the tota number of streams of a nformaton users, and the data vector as x = [ x... xn] C n S, that must satsfy the power constrant formuated as E[ x ] = rs P, where P represents the tota avaabe transmsson power at the BS. he tota RF-band power harvested by the j-th user from a recevng antennas, denoted by Q j, s proportona to that of the equvaent baseband sgna,.e., j U E, we have: [ j Q j = ζ j E B x ] = ζ j E[ j B x ], where ζ j s a constant that accounts for the oss for convertng the harvested RF power to eectrca power. Notce that, for smpcty, n we have omtted the harvested power due to the nose term snce t can be assumed neggbe. he transmtter desgn that we propose n ths paper s modeed as a nonconvex mut-objectve optmzaton probem. he goa s to maxmze, smutaneousy, the ndvdua data rates and the harvested powers of the nformaton and harvestng users, respectvey. Gven ths and the prevous system mode, the optmzaton probem s wrtten as maxmze {S } subject to R n S n UI, E m S m UE C : rs P C : S, U I, where S S UI, the data rate expresson s gven by R n S = og det I + n S n n Ω n S n 4 = og det Ω n S n + n S n n og det Ω n S n 5 = og det I + n Sn og det Ω n S n, 6 }{{}}{{} s ns g nω ns n wth S = S k, and the harvested power s gven by E m S = r m S m. 7 he prevous probem n s not convex due the objectve functons n fact, due to Ω S and s dffcut to sove. In order to fnd Pareto optma ponts, we can reformuate t by usng any of the technques presented n Secton II-A. In the foowng, we propose two approaches based on the weghted sum method and on the hybrd method. For convenence, we start wth the hybrd method as t s the one that has receved the most attenton n the terature [5], [9]. owever n that terature, the nterference n s assumed to be removed by the transmsson strategy. hs assumpton makes the probem convex and hence easer to sove. A. ybrd-based Formuaton to Sove In the hybrd approach, some of the objectve functons are coapsed nto a snge objectve by means of scaarzaton and some of the objectve functons are added as constrants. In partcuar, the data rates are eft n the objectve whereas the harvestng constrants are ncuded as ndvdua harvestng constrants. Wth ths partcuar formuaton, we are abe to guarantee a mnmum vaue for the power to be harvested by the harvestng users. hus, probem s formuated as max {S } ω og det I + S ω og det Ω S s. t. C : r j S j Q j, j U E 8 C : rs P C : S, U I, Q mn j Q mn j where Q j = ζ j, beng { } the set of mnmum power harvestng constrants, and ω are some rea non-negatve weghts. For smpcty n the notaton, et us defne the feasbe set S as { S S : r j S j Q j, j U E, rs P, S, U I }. 9 For a set of fxed harvestng constrants, the convex hu of the rate regon can be obtaned by varyng the vaues of ω. In addton, we can use the vaues of the weghts to assgn prortes to some users f user schedung s to be mpemented, foowng, for exampe, the proportona far crteron [], []. Notce that constrant C s assocated wth the mnmum power to be harvested for a gven user. Note aso the smartes of probem 8 wth the snge user case presented n [5] and ts extenson to the mutuser case presented n [6]. As commented before, the novety s that we do not force the transmtter to cance the nterference generated among the nformaton users as opposed to BD approaches [7] and, thus, we aow the system to have more degrees of freedom to mprove the system throughput and the harvested power smutaneousy. Later n Secton IV-A, we w present a method based on MM to sove the nonconvex probem n 8. B. Weghted Sum-Based Formuaton to Sove In stuatons where the exact amount of power to be harvested by harvestng users s not needed, we can aso obtan Pareto optma ponts by means of the smper weghted-sum method. In ths case, we can assgn prortes so that some users tend to harvest more power than others, athough the exact amounts cannot be controed. As we w see ater, the overa probem based on ths new formuaton s much easer to sove. he transmtter desgn s obtaned through the foowng nonconvex

5 5 optmzaton probem: max ω og det I + S ω og det Ω S {S } + α j r j S j j U E s. t. C : rs P C : S, U I, where α j are some rea non-negatve weghts. For smpcty n the notaton, et us defne the feasbe set S as } {S : UI rs P, S, U I S. As we w show ater n Secton IV-B, the agorthm to sove s easer than the agorthm to sove 8. ence, there s a trade-off n terms of speed of convergence of the agorthms and n terms of the harvested power contro snce, as we ntroduced before, n 8 the transmtter can fuy contro the amount of power to be harvested by the users whereas n the transmtter can ony contro the proporton of the power to be harvested among the users. IV. MM-BASED ECNIQUES O SOLVE PROBLEM In ths secton, we present a method based on the MM phosophy to sove probems 8 and. Snce the orgna probems 8 and are nonconvex, we reformuate them and make them convex before appyng the MM method. hs reformuaton w foow two steps. In the frst step, probems 8 and w be convexfed by usng a near approxmaton of the nonconvex terms. hs s the approach taken n papers such as [], [], and [4]. Instead of sovng the reformuated convex probem, n the second step, we desgn a quadratc approxmaton of the remanng convex terms n order to fnd a surrogate probem easer to sove. Fnay, we appy the MM method to the quadratc reformuaton. As benchmarks for comparson, we w consder the case of just convexfyng the nonconvex terms, whch s an approach taken n the prevous terature, and aso consder a gradent method apped drecty to the nonconvex probems 8 and. Athough the mathematca deveopments of the proposed MM approaches are more tedous than the approaches usuay taken n the terature, the resutng agorthms are faster. A. Approach to Sove the ybrd Formuaton n 8 As we ntroduced before, we need to reformuate the orgna nonconvex probem 8 and make t convex. hs w be done n two steps. Motvated by the work n [], n ths frst step, we derve a near approxmaton for the nonconcave rghthand sde part of the objectve functon of 8,.e., f S = ω s S ω g Ω S, n such a way that the modfed probem s convex 4. In order to fnd a concave ower bound of 4 In fact, by appyng the approxmaton, the overa objectve functon becomes concave. f S, g can be upper bounded neary at pont Ω S k + I as k g Ω S g Ω + r = constant + r Ω Ω Ω S Ω S Ω = ĝ Ω S, Ω. Even though probem 8 reformuated wth the prevous upper bound ĝ Ω S, Ω s convex, we want to go one step further and appy a quadratc ower bound for the eft hand sde of f S,.e., s S n a way that the overa ower bound fufs condtons A A4 presented before n Secton II-B and the MM method can be nvoked. Note that the upper bound ĝ Ω S, Ω aready fufs the four condtons A A4. he dea of mpementng ths quadratc bound s to fnd a surrogate probem that s much smper and easer to sove than the one obtaned by just consderng the near bound ĝ Ω S, Ω. 5 We now focus attenton on dervng the surrogate functon for the eft hand sde of f S,.e., s S. In order for the surrogate probem to be easy soved, we force the surrogate functon of s S around S to be quadratc, where S = S k and s the souton of the agorthm at the prevous teraton. By dong ths, as w be apparent ater, the overa surrogate probem can be formuated as an SDP optmzaton probem. S k Proposton 5. A vad surrogate functon, ŝ S, S, for the functon s S = og det I + n S n that satsfes condtons A A4 s ŝ S, S r J S + r S M S + κ, S, S S n +, wth matrces J = G S, M M S, G = I + S and M = γ I, beng γ λ max, κ contans some terms that do not depend on S, and S n + denotes the set of postve semdefnte matrces. Proof: See Appendx B. Let us now reformuate the optmzaton probem n 8 wth the surrogate functon ŝ S, S ĝ Ω S, Ω : r E S + r S M S + r R S + κ, 4 where R = Ω C n n, E = J R, and κ contans some terms that do not depend on S. hus, probem 5 he surrogate probem obtaned by just appyng the bound ĝ Ω S, Ω w be used as benchmark. he specfc mathematca detas of the optmzaton probem and the agorthm w be descrbed n App. A.

6 6 8 can be reformuated as max {S } ω ρ S S s. t. S S, r E S + r S M S + r R S F 5 where we have added a proxma quadratc term to the surrogate functon n whch ρ s any non-negatve constant that can be tuned by the agorthm. hs term provdes more fexbty n the agorthm desgn stage and may hep to speed up the convergence. By performng some mathematca manpuatons, we are abe to obtan the foowng resut: Proposton 6. he optmzaton probem presented n 8 can be soved based on MM method by sovng recursvey the foowng SDP probem: mn {S }, s, t t 6 ti C s c s. t. C : C s c C : s = vec S, C : S S, U I where s = [ vecs vecs... vecs N ] C n n U I, t s a dummy varabe, and C,, and c are some constant matrces and vectors computed as shown n Appendx C. Vector c depends on matrx S. Proof: See Appendx C. he fna agorthm s presented n Ag.. B. Approach to Sove the Sum Method Formuaton n Let us start the deveopment by reformuatng probem : max ω s S ω g Ω S + rr S {S } s. t. S S, 7 where R = j U E α j j j. he rght hand sde of the objectve functon of 7 s convex n fact t s near whereas the eft hand sde s not convex. Let us appy the same steps that we apped before but wth a sght modfcaton. Prevousy n, we found that g Ω S coud be approxmated by ĝ Ω S, Ω = r Ω Ω S omttng the constant term. Now, as the objectve functon s dfferent than the one from probem 8, the goa s to fnd a surrogate functon for the functon s S that aows us to fnd effcenty a souton for the surrogate probem. Proposton 7. A vad surrogate functon, ŝ S, S, for the functon s S that satsfes condtons A A4 s ŝ S, S U I r J S + U I r S M S + κ, S, S S n +, 8 Agorthm Agorthm for Sovng Probem 8 : Intaze S S. Set k = : Repeat : Compute c wth S k, gven n 6 4: Generate the k + -th tupe S UI by sovng the SDP n 6 5: Set S k+ = S, U I, and set k = k + 6: Unt convergence s reached wth matrces J = G S, M M S, G = I +, and M = ξ I, beng S k ξ U I λ max, and κ contans the constant terms that do not depend on S. Proof: See Appendx D. Remark. Note that the two surrogate functons and 8 have the same form but wth a dfference n the quadratc term. Notce that surrogate functon 8 s tghter than and wth cross-products. As w be shown ater, ths w aow us to decoupe the optmzaton probem for each nformaton user and, thus, sove a probems n parae. On the other hand, thanks to the fact that surrogate functon s ooser than 8, a faster convergence can be obtaned than f surrogate 8 were to be apped n probem 8. Let us now reformuate probem 7 wth the ower bound that we just found omttng the constant terms: ˇJ S + S ˇMS max {S } r r R s. t. S S, where ˇJ = Ǧ S, r k S k + rr S 9 ˇM ˇMS, wth ˇM = ω k M k and Ǧ = ω k G k. Note that we have arranged the ndces to make the notaton easer to foow and consstent wth the orgna notaton. We can further smpfy the objectve functon by groupng terms consderng that matrx ˇM s dagona,.e., ˇM = βi, beng β = U I ω k λ max k k : where mn {S } β r s. t. S S, S S r F S F = ˇJ R k + R. k Note that we have changed the sgn of the objectve and reformuated the probem as a mnmzaton one. he dea s to fnd a cosed-form expresson for the optmum covarance matrces {S }. If we duaze constrant C and form a parta

7 7 Agorthm Agorthm for Sovng Probem : Intaze S S. Set k = : Repeat : Compute F wth matrx S k, U I, gven n 4: Compute EVD of F = U F Λ F U F, U I 5: Compute µ such that r [Λ F µ I] + = βp 6: Compute S µ = β [F µ I] +, U I 7: Set S k+ = S µ, U I, and set k = k + 8: Unt convergence s reached Lagrangan, we obtan the foowng optmzaton probem: β r S S r W µs mn {S } s. t. S, U I, where W µ = F µi, for µ the Lagrange mutper assocated wth constrant C of probem 7. he prevous probem s ceary separabe for each user. hus, for each nformaton user, probem s equvaent to sovng the foowng projecton probem: mn βs ˇW µ S F s. t. S, where ˇW µ = W β µ = F β µi. he prevous resut s very nce as the souton of s smpe and eegant, thanks to the fact that probem s a projecton onto the semdefnte cone and has a cosed-form souton [5]. Let the egenvaue decomposton EVD of matrx F be F = U F Λ F U F. he expresson of S µ s, thus, gven by S µ = [ ˇW µ] + = β β U F [Λ F µi] + U F, U I, 4 where λ k [X] + = mn, λ k X, wth λ k X the k-th egenvaue of matrx X. Now t remans to compute the optma Lagrange mutper µ. hs can be found by means of the smpe bsecton method fufng r [Λ F µi] + = βp. It turns out that, at each nner teraton, we need to compute a snge EVD per nformaton user, that s, the EVD of F, and a few teratons to fnd the optma mutper µ. Note that the surrogate probem can be soved straghtforwardy wth the prevous steps. he fna agorthm s presented n Ag.. C. Approaches Used as Benchmarks for Performance Comparson As the probem ntroduced n has not been addressed before n the terature, there are not specfc benchmarks to compare our approaches wth. For ths reason, n ths secton, we propose some benchmark agorthms that w be used n the smuaton secton to compare the performance of the proposed MM approaches. hese benchmarks are: Gradent-based agorthms based on [6, Sec. 7] apped drecty to the nonconvex probems 8 and. he gradents are not presented due to space mtatons. MM approaches consderng just the near approxmaton presented n,.e., ĝ Ω S, Ω, apped to probems 8 and. he specfc optmzaton probems and agorthms can be found n App. A. V. NUMERICAL EVALUAION In ths secton, we evauate the performance of the prevous agorthms. In the frst part of ths secton, we present some convergence and computatona tme resuts. For the smuatons, we consder a system composed of transmtter wth 6 antennas, and nformaton users and harvestng users wth antennas each. In the second part of the secton, we show the performance of the proposed methods compared to the cassca BD approach. In ths case, for ease of presentng the nformaton, we assume a system composed of transmtter wth 4 antennas, and nformaton users and harvestng users wth antennas each. he smuaton parameters common to both scenaros are the foowng. he maxmum radated power s P = W. he channe matrces are generated randomy wth..d. entres dstrbuted accordng to CN,. he weghts ω are set to. A. Convergence Evauaton In ths subsecton, we evauate the convergence behavor and the computatona tme of the methods presented n Sectons IV-A and IV-B and the benchmark approach presented n App. A. he benchmark method for probem presented n App. A w not be evauated as t s ceary worse 6 than the one presented n Secton IV-B. In the fgures, the egend s nterpreted as foows: MM-L for 8 refers to the method deveoped n App. A for probem 8, MM-Q for 8 refers to the method n Secton IV-A, and MM-Q for refers to the method n Secton IV-B. In order to compare a methods, we set the vaues of α j and the vaues of Q j so that the same system sum rate s acheved. hese vaues are: α = [, 5, ], and Q = [.8, 7., 6.4] power unts. Software package CVX s used to sove probem 5 [7], and SeDuM sover s used to sove probem 6 [8]. Fgure presents the sum rate convergence as a functon of teratons. he three approaches converge to the same sum rate vaue but requre a dfferent number of teratons. In fact, the requred number of teratons depends on how we the surrogate functon approxmates the orgna functon. Note that the surrogate functon used n the MM-L for 8 approach s the one that best approxmates the objectve functon and, thus, fewer teratons are needed. Fgure shows the computatona tme requred by the three prevous methods. We see that the MM-Q for method converges much faster than the other two approaches, as expected. he MM-Q for 8 approach requres more teratons than the MM-L for 8 approach but each teraton s soved faster snce a specfc agorthm can be empoyed to sove the convex optmzaton probem. ence, the MM-Q for 8 agorthm s the best opton. For the sake of comparson and competeness, we aso show n Fgures and 4 the convergence and the computatona tme 6 owever, t was ncuded n the paper for the sake of competeness

8 8 Sum rate bts/s/z.5.5 f f k best GRAD for 8 a ones GRAD for 8 dentty ones GRAD for dentty ones GRAD for a ones MM-Q for MM-Q for 8 MM-L for Iteratons Fg. : Convergence of the system sum rate vs number of teratons for three dfferent approaches Iteratons 5 Fg. : Convergence of the system sum rate vs teratons for a gradent approach for constraned optmzaton..5 Sum rate bts/s/z MM-Q for MM-Q for 8 MM-L for Computatona tme s Fg. : Convergence of the system sum rate vs computatona tme for three dfferent approaches. of a gradent-ke benchmark approach. he pot egend reads as foows: GRAD for 8 and GRAD for refers to a gradent approach apped to probems 8 and, respectvey. a ones and dentty mean that covarance matrces are ntazed usng an a ones matrx and the dentty matrx, respectvey. Resuts show that the proposed MM approaches are one to two orders of magntude faster than the gradent-based methods. B. Performance Evauaton In ths secton, we evauate the performance of the MM approach as compared to the cassca BD strategy consdered n the terature see, for exampe, [6], [9]. In order to show how harvestng users at dfferent dstances affect the performance, we have generated channe matrces wth dfferent norms. We woud ke to emphasze that, as the nose and channes are normazed, we w refer to the powers harvested by the recevers n terms of power unts nstead of Watts. Fgures 5 and 6 show the rate-power surface, that s, the mutdmensona trade-off between the system sum rate and Sum rate bts/s/z.5.5 GRAD for 8 a ones.5 GRAD for 8 dentty ones GRAD for dentty ones GRAD for a ones Computatona tme s Fg. 4: Convergence of the system sum rate vs computatona tme for a gradent approach for constraned optmzaton. Sum rate bts/s/z Q Power unts 9 Fg. 5: Rate-power surface for the MM method. 4 Q Power unts

9 9 9 Sum rate bts/s/z.5 Q Power unts Q Power unts 9 4 Q Power unts 4 Q Power unts Fg. 6: Rate-power surface for the BD method. Fg. 8: Contour of rate-power surface for the BD method. Q Power unts 9 6 R bts/s/z.5.5 Q j = Q j =.5 Q j = Q j =.5 Q j =.5 4 Q Power unts Fg. 7: Contour of rate-power surface for the MM method. the powers to be coected by harvestng users see [6] for a forma defnton of the rate-power surface. As we see, the MM approach outperforms the BD strategy n both terms, sum rate and harvested power. he maxmum system sum rate obtaned wth the MM approach when Q and Q are set to s 4.5 bt/s/z, whereas the sum rate obtaned wth the BD approach s.75 bt/s/z. he rate-power surfaces are generated by varyng the vaues of {Q j } n probem 8 or, equvaenty, by varyng the vaues of {α j } n probem. A way to reduce the computatona compexty assocated wth the generaton of the rate-power surface s to use as an ntazaton pont the souton that was obtaned for the prevous vaues of {Q j } or {α j } to generate the new vaue of the curve []. Note, however, that the whoe rate-power surface need not be generated for each transmsson as t s just the representaton of the exstng ratepower tradeoff. In order to ceary see the benefts n terms of coected power, Fgures 7 and 8 show the contour pots of the prevous D pots. We observe that users n the MM approach coect roughy 5% more power than the power coected by users when appyng the BD strategy R bts/s/z Fg. 9: Rate regon for dfferent vaues of Q j n power unts. Fnay, Fgure 9 presents the rate-regon of the MM approach for dfferent vaues of {Q j }. he same vaue of Q j s set to the two harvestng users. In ths case, we vary the vaues of ω to acheve the whoe contour of the rate regons. We observe that, the arger the harvestng constrants, the smaer the rate-regon, as expected. owever, the reaton between the harvestng constrants and the rate-regon s not near. As the harvestng constrants ncrease, a sma change n the {Q j } produces a arge reducton of the rate-regon. hs s because the D rate-power surfaces presented before are not panes. VI. CONCLUSIONS We have presented a method to sove the dffcut nonconvex probem that arses n mutuser mut-stream broadcast MIMO SWIP networks. We formuated the genera SWIP probem as a mut-objectve optmzaton probem, n whch rates and harvested powers were to be optmzed smutaneousy. hen, we proposed two dfferent formuatons to obtan soutons of the genera mut-objectve optmzaton probem dependng on the desred eve of contro of the power to be harvested. In the frst approach, the transmtter was abe to contro the specfc amount of power to be harvested by each user whereas n the

10 second approach ony the proportons of power to be harvested among the dfferent users coud be controed. Both nonconvex formuatons were soved based on the MM approach. We derved a convex approxmaton for two nonconvex objectves and deveoped two dfferent agorthms. Smuaton resuts showed that the proposed methods outperform the cassca BD n terms of both system sum rate and power coected by users by a factor of approxmatey 5%. Moreover, the computatona tme needed to acheve convergence was shown to be reay ow for the approach n whch the transmtter coud ony contro the proporton of powers to be harvested around two orders of magntude ower than a gradent-ke approach. APPENDIX A BENCMARK FORMULAIONS AND ALGORIMS In ths appendx, we are gong to descrbe the benchmarks based on the works n [], [], and [4]. We start wth the benchmark for probem 8. Note that the upper bound ĝ Ω S, Ω can be used to bud a ower bound of f S that fufs the four condtons A A4 presented before n Secton II-B. By appyng a successve approxmaton of f through the appcaton of the prevous surrogate functon,.e., ˆf S, S k = ω s S ω ĝ Ω S, Ω k ρ S S k, where F S k S k UI, for dfferent evauaton ponts, we obtan an teratve agorthm based on the MM approach that converges to a statonary pont or oca optmum of the orgna probem 8. Note that we have consdered a proxma-ke term. Gven ths, the convex optmzaton probem to sove s max ω s S ω ĝ Ω S, Ω k ρ S S k {S } F s. t. S S. 5 We must proceed teratvey unt convergence s reached. he procedure s presented n Ag.. Let us now contnue wth the benchmark for probem. If we appy the bound from,.e., ĝ Ω S, Ω, probem can be soved by sovng consecutvey the foowng probem: max ω s S ω ĝ Ω S, Ω k + rr S {S } ρ S S k 6 s. t. S S. F As probem 6 s convex, the MM method can be nvoked to obtan a oca optmum of probem, foowng the same procedure as we dd before for probem 5. Agorthm Agorthm for Sovng Probem 8 : Intaze S S. Set k = : Repeat : Generate the k + -th tupe S UI by sovng 5 4: Set S k+ = S, U I, and set k = k + 5: Unt convergence s reached APPENDIX B PROOF OF PROPOSIION 5 he proposed quadratc surrogate functon of s S has the foowng form: ŝ S, S og det I + S { } +Re r G S S + r S S M S S og det I + S, S, S S n +, 7 where matrces G C n n and M C n n need to be found such that condtons A through A4 are satsfed, and Re{x} denotes the rea part of x. Note that A and A4 are aready satsfed. Ony A and A must be ensured. Let us start by provng condton A. Let S and S be two postve semdefnte matrces,.e, S, S S n +. hen, the drectona dervatve of the surrogate functon ŝ S, S n 7 at S wth drecton S S s gven by: { S } Re r G S. 8 Now, et us compute the drectona dervatve of the term og det I + S : r I + S S S, 9 where we have used d og detx = rx dx []. ence, by appyng condton A, the two drectona dervatves 8 and 9 must be equa, from whch we are abe to dentfy matrx G as G = I + S, G = G. 4 Note that as matrx G s hermtan, the rea operator s no onger needed snce the trace of the product of two hermtan matrces s rea. In order to prove condton A, t suffces to show that for each near cut n any drecton, the surrogate functon s a ower bound. Let S = S S + µ S, µ [, ]. hen, t suffces to show 4. Snce the eft hand sde of 4 s concave wth respect to µ, a suffcent condton s that the second dervatve of the eft hand sde of 4 must be ower than or equa to the second dervatve of the rght hand sde of 4 for any µ [, ] and any S, S S n +, thus, 4 must hod. Let us compute the second dervatve of the rght hand sde of 4. he frst dervatve s gven by 4 and the second dervatve s gven by 44, where we have used the dentty

11 og det I + S og det + µ r G S S + µ r I + S + µ S S S S S M S, S, S S n +, µ [, ]. 4 r S S M S S µ og det I + S + µ S S µ og det I + S + µ S S = r S S I + + µ S S S S, S S n +, µ [,].4, 4 S S µ og det I + + µ S = r A S S A S S, 44 dx = X dxx [] and matrx A C n R n R s defned as A = I + S + µ S S. We need to manpuate the prevous expressons. o ths end, et us defne matrx P = A C n n and et us vectorze the resut found n 44: P S S S P S r S =vec S I P S P vec S, 45 where we have used the foowng propertes: rab = veca vecb, vecab = veca I B, vecab = I AvecB, and A BC D = AC BD. Let us now vectorze the eft hand sde of 4: r S S M S S S = vec S S I M vec S, 46 where n 46 we have used the fact that S S s hermtan and rabc = veca I BvecC. Fnay, we end up wth the reaton from forcng that 46 must be ower than or equa to 45. hs reaton can be expressed as gven by 47. A suffcent condton for expresson 47 s: I M + I P P = I M + P P, 48 whch means that M + P P. 49 Now, f we set M = αi note that ths s a partcuar smpe souton, we have that α λ max P P, 5 where λ max X s the maxmum egenvaue of matrx X. Now, et us ntroduce the foowng resut: heorem []. Let A, B C n n, assume that A s postve defnte, and assume that B s postve defnte. Let λ A be the -th egenvaue of matrx A such that λ A λ A λ n A. hen, for a, j, k {,..., n} such that j +k +, In partcuar, for a =,..., n, λ AB λ j Aλ k B. 5 λ Aλ n B λ AB λ Aλ B. 5 hanks to the prevous resut, α λ max P. Now, et the snguar vaue decomposton of be = U Σ V. From ths, we can upper bound λ max P = λ max A = V Σ σ max λ mn A, where σ max X s the maxmum snguar vaue of matrx X. Because matrx A s postve defnte wth λ mn A, we can concude that λ max Σ V A α σ4 max, 5 and thus, a possbe matrx M satsfyng condtons A A4 s fnay M = σ4 max I = λ max I. 54 APPENDIX C PROOF OF PROPOSIION 6 Let us start by vectorzng the surrogate functon n 4: ˆR S, S = ŝ S, S ĝ Ω S, Ω = vec S I M vec S + e vec S +r vec S + κ, 55 where e = vec E C n n, r = vec R C n n, and κ contans some constant terms that do not depend on {S }. Let s = [ vecs vecs... vecs UI ] C n n U I. Note that vec S = s, where C n n n n U I s composed of U I dentty matrces of sze

12 S vec S [ I M + ] I P S P vec S. 47 n n n n,.e., = [I I... I]. Now, we can rewrte 55 as omttng the constant terms ˆR S, S =s I M s + e s + r vec S. 56 We know proceed to formuate the objectve functon denoted by f S, S of probem 8 but substtutng the bound that we just computed and consderng the proxma term. If we ncorporate a the terms but omttng the constant ones we have f S, S = s I M s + e s + r vec S ω ρ S S F = s Ms + ẽ s + ˆr s ρs s + ρs, s 57 +ρs s ρs, s, 58 where M = ω I M C n n n n, ẽ = [ ω e, ˆr = r r... r U I ] C n n U I, and s = [ ] vecs vecs... vecs U I C n n U I. Now takng nto account that the objectve functon f S, S must be rea and combnng terms omttng terms that do not depend on s we obtan f S, S = s Cs + b s + s b, 59 where b = ẽ + ˆr + ρs, C n n U I and matrx C s C = M ρi C n n U I n n U I. For convenent purposes, et us change the sgn of f S, S such that f S, S = f S, S = s Cs b s s b, where C = C. Fnay, we can equvaenty rewrte the objectve functon as the foowng expresson wth ths new reformuaton, the objectve s to mnmze f S, S nstead of maxmzng t: where f S, S = C s c, 6 c = C b C n n U I. 6 Note that the term c c does not affect the optmum vaue of the optmzaton varabes as ths term does not depend on s. Now, we can reformuate the optmzaton probem presented n 8 as mnmze {S }, s C s c 6 subject to C : s = vec S, U I C : S S, where = [,,...,, I,,..., ] R }{{} n n n n U I s composed of zero matrces of dmenson n n n n wth an dentty matrx at the -th poston. Probem 6 can be further reformuated as mnmze {S }, s, t subject to t 6 C : C s c t C : s = vec S, C : S S, U I and, fnay, as the foowng SDP optmzaton probem mnmze t 64 {S }, s, t ti C s c subject to C : C s c C : s = vec S, C : S S. APPENDIX D PROOF OF PROPOSIION 7 U I he proposed quadratc surrogate functon of s S has the foowng form: ŝ S,S og det I { Re r U I G S k S S } + r S S M S S U I og det I + S k, S, S S n +, where matrces G C n n and M C n n need to be found such that condtons A through A4 are satsfed. Note that A and A4 are aready satsfed. Ony A and A must be ensured. Let us start wth condton A. Let S S, S n +,. hen, the drectona dervatve of the surrogate functon ŝ S, S n 65 at S wth drecton S S s gven by { } r G S S, 66 U I Re and the drectona dervatve of the rght hand sde of 65 at S wth drecton S S s gven by 67. From 66 and 67, we dentfy the matrces G as G = I +, G = G, 68 S k where we fnd that a matrces G for a gven user can be the same, G = G.e., they do not depend on.

13 r I + S k U I = U I r S S I + S k S S 67 vec U I I S S P P vec S S, 7 U I Now, we seek to fnd matrces {M } based on condton A. o ths end, we foow the same procedure presented before. We make near cuts n each possbe drecton and appy the condton over the second dervatve see 4. he second dervatve of the eft hand sde of 65 s gven by U I r S S M S S = 69 vec S S I M vec S S, U I and the second dervatve of the rght hand sde s gven by 7, where P = I + S UI + µ S S, beng constant µ [, ]. Now, et s = [ ] vec S S vec S U I S U I and et us ntroduce the foowng bock dagona matrx I M... M = I M I M UI hen we have that the foowng condton shoud be fufed: whch means that s M s + s I P P s, 7 M + I P P. 7 Note that the partcuar structure of matrx I P P s gven by I P P... I P P I P I P P P =....., 74 I P P... I P P From the prevous condtons we can see that a matrces M w be the same for user,.e., M = M,. Now f we choose the partcuar structure M = α I, then condton 7 s equvaent to α I + I P P. 75 Now, condton 75 s equvaent to α g g g I P P g, g = 76 α g g g λ max I P P, g = 77 α g g g λ max P P, g. 78 Now, the term g can be further smpfed. Based on the structure of matrx, we have that g = n n g + g +n n g +n n U I + 79 = n n = n n = U I max{g,..., g +n n U I + } 8 n n U I = U I U I g g +n n U I + 8 = g = U I g. 8 hus, a suffcent condton to fuf 78 s and, fnay, α g U I g λ max P P, g, 8 α U I λ max P P U I λ max. 84 ence, a possbe matrx M satsfyng assumptons A A4 s, fnay, M = U I λ max I. 85 REFERENCES [] X. Lu et a., Wreess networks wth RF energy harvestng: A contemporary survey, IEEE Commun. Surveys and utoras, vo. 7, no., pp , Secondquarter 5. [] J. Paradso and. Starner, Energy scavengng for mobe wreess eectroncs, IEEE Computng Pervasve, vo. 4, no., pp. 8 7, Jan. 5. [] S. B, C. K. o, and R. Zhang, Wreess powered communcaton: opportuntes and chaenges, IEEE Commun. Mag., vo. 5, no. 4, pp. 7 5, Apr. 5. [4] L. R. Varshney, ransportng nformaton and energy smutaneousy, n Int. Symp. Inform. heory, oronto, Ju. 8. [5] R. Zhang and C. K. o, MIMO broadcastng for smutaneous wreess nformaton and power transfer, IEEE rans. Wreess Commun., vo., pp. 989, May.

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