Optimal Resource Allocation in Full-Duplex Wireless-Powered Communication Network

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1 1 Optmal Resource Allocaton n Full-Duplex Wreless-owered Communcaton Network Hyungsk Ju and Ru Zhang, Member, IEEE arxv:143.58v3 [cs.it] 15 Sep 14 Abstract Ths paper studes optmal resource allocaton n the wreless-powered communcaton network WCN), where one hybrd access-pont H-A) operatng n full-duplex FD) broadcasts wreless energy to a set of dstrbuted users n the downlnk DL) and at the same tme receves ndependent nformaton from the users va tme-dvson-multple-access TDMA) n the uplnk UL). We desgn an effcent protocol to support smultaneous wreless energy transfer WET) n the DL and wreless nformaton transmsson WIT) n the UL for the proposed FD-WCN. We jontly optmze the tme allocatons to the H-A for DL WET and dfferent users for UL WIT as well as the transmt power allocatons over tme at the H-A to maxmze the users weghted sum-rate of UL nformaton transmsson wth harvested energy. We consder both the cases wth perfect and mperfect self-nterference cancellaton SIC) at the H-A, for whch we obtan optmal and suboptmal tme and power allocaton solutons, respectvely. Furthermore, we consder the half-duplex HD) WCN as a baselne scheme and derve ts optmal resource allocaton soluton. Smulaton results show that the FD-WCN outperforms HD-WCN when effectve SIC can be mplemented and more strngent power constrant s appled at the H-A. Index Terms Wreless-powered communcaton network WCN), wreless energy transfer WET), full-duplex FD) system, resource allocaton, convex optmzaton. I. INTRODUCTION Tradtonally, fxed energy sources e.g. batteres) have been used to power energy-constraned wreless networks, such as sensor networks, whch lead to lmted operaton tme. Although the lfetme of wreless networks can be extended by replacng or rechargng the batteres, t may be nconvenent, costly, and even dangerous e.g., n a toxc envronment) or nfeasble e.g., for sensors mplanted n human bodes). As an alternatve soluton, energy harvestng see e.g., [1], [] and the references theren) has recently receved a great deal of attenton snce t provdes more cost-effectve and truly perpetual energy supples to wreless networks through scavengng energy from the envronment. Among other commonly used energy sources e.g. solar and wnd), rado sgnal radated by ambent transmtters becomes a vable new source for energy harvestng. In partcular, harvestng energy from the far-feld rado-frequency RF) sgnal transmsson opens a new avenue for the unfed study of wreless power transmsson and wreless communcaton snce rado Ths work was done when H. Ju was wth the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore. He s now wth the Electroncs and Telecommuncatons Research Insttute, Korea e-mal: jugun@etr.re.kr). R. Zhang s wth the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore e-mal: elezhang@nus.edu.sg). He s also wth the Insttute for Infocomm Research, A*STAR, Sngapore. Hybrd A h D,1 h U,1 h D, h U, Energy transfer Informaton transfer U 1 h U, K h D, K Fg. 1. Wreless-powered communcaton network WCN) wth DL WET and UL WIT. sgnals carry energy and nformaton at the same tme. There are two man paradgms of research along ths drecton. One lne of work focuses on studyng the so-called smultaneous wreless nformaton and power transfer SWIT) by characterzng the achevable trade-offs n smultaneous wreless energy transfer WET) and wreless nformaton transmsson WIT) wth the same transmtted sgnal. SWIT has recently been nvestgated for varous channel setups, e.g., the pont-topont addtve whte Gaussan nose AWGN) channel [3], [4], the fadng AWGN channel [5], [6] the mult-antenna channel [7]-[9], the relay channel [1], [11], and the mult-carrer based broadcast channel [1]-[14]. Another lne of research s amed to desgn a new type of wreless network termed wreless-powered communcaton network WCN) n whch wreless termnals communcate usng the energy harvested from wreless power transmssons. The WCN has been studed under varous network setups, such as cellular network [15], random-access network [16], and mult-hop network [17]. In [18] and [19], another type of WCN was nvestgated, where moble chargng vehcles are employed to wrelessly power the user devces n the network. In addton, the wreless-powered cogntve rado network was consdered n [], where actve prmary users are utlzed as energy transmtters for chargng ther nearby secondary users. Furthermore, n our prevous work [1] we have studed one partcular WCN model, as shown n Fg. 1, where a hybrd access-pont H-A) operatng n tme-dvson half-duplex HD) mode coordnates WET/WIT to/from a set of dstrbuted users n the downlnk DL) and uplnk UL) transmssons, respectvely. It has been shown n [1] that there exsts a fundamental trade-off n allocatng DL tme for WET and UL U U K

2 tme for WIT n the HD WCN, snce ncreasng DL tme ncreases the amount of harvested energy and hence the UL transmt power at each user, but also decreases users UL tme for WIT gven a total DL and UL tme constrant. On the other hand, there has been recently a growng nterest n full duplex FD) based wreless systems, where the wreless node transmts and receves smultaneously n the same frequency band, thus potentally doublng the spectral effcency. However, due to the smultaneous transmsson and recepton at the same node, FD systems suffer from the selfnterference SI) that s part of the transmtted sgnal of a FD node receved by tself, thus nterferng wth the desred sgnal receved at the same tme. Self-nterference cancellaton SIC) s a key challenge for mplementng FD communcaton snce the power of SI typcally overwhelms that of the desred sgnal. Varous SIC technques have been proposed n the lterature see e.g., []-[5] and the references theren), whch are generally based on ether analog-doman SIC.e., SIC n the wreless propagaton channel or before the receved sgnal s processed by analog-to-dgtal converson ADC)), dgtal-doman SIC.e., SIC after ADC by dgtal sgnal processng technques), or ther assorted combnatons. By state-of-the-art SIC technques today, t has been reported that SIC up to 11dB hgher power of the desred sgnal can be mplemented [5]. In addton, varous effects of practcal hardware lmtatons on FD communcatons, e.g., the fnte dynamc range of transmt and/or receve flters [6], [7], have been nvestgated. Furthermore, full-duplex technques have been appled n varous wreless communcaton applcatons for performance enhancement, e.g., spectral effcency n relay network [8], mult-user cellular network [9], wreless physcal layer securty [3], and cogntve rado network [31]. In ths paper, we apply the FD technque to the WCN shown n Fg. 1, to further mprove throughput. We assume that the H-A operates n the FD mode to broadcast energy and receve nformaton to/from the dstrbuted users smultaneously over a gven frequency band, thus sgnfcantly savng the tme for separate DL WET and UL WIT as compared to the HD-WCN consdered n [1]. The proposed WCN wth the H-A operatng n FD mode s thus termed full-duplex WCN FD-WCN), to dffer from ts HD-WCN counterpart n [1]. It s worth notng that the FD technque appled at the H-A n our proposed FD-WCN s for hybrd energy/communcaton transmsson/recepton, whch s n sharp contrast to conventonal setups wth FD communcatons. However, smlar to FD communcaton systems, the proposed hybrd energy/communcaton FD system s also subject to the practcal ssue of mperfect SIC at the H-A,.e., the nterference due to the DL energy sgnal s not perfectly cancelled at the recever for decodng the UL nformaton. The man results of ths paper are summarzed as follows: For the proposed FD-WCN, we present a new protocol to enable smultaneous WET n the DL and WIT n the UL over the same band. It s assumed that the H-A operates n FD mode, whle the users all operate n tme dvson HD mode for the ease of mplementaton, whch transmt ndependent nformaton to the H-A by tmedvson-multple-access TDMA) n the UL and harvest energy n the DL when they do not transmt. We also compare the proposed protocol for FD-WCN wth the harvest-then-transmt protocol proposed n [1] for HD- WCN. Under the proposed protocol for FD-WCN, we characterze the maxmum weghted sum-rate WSR) of users n FD-WCN, by jontly optmzng the tme allocated to the H-A for DL WET and users for UL WIT and the transmt power of the H-A over tme subject to a gven total tme constrant as well as the average and transmt power constrants at the H-A. For the purpose of exposton, we frst consder the deal case of perfect SIC at the H-A n FD-WCN. It s shown that the WSR maxmzaton problem n ths case s convex, and hence we obtan the closed-form soluton for the optmal tme and power allocatons by applyng convex optmzaton technques. It s revealed that the optmal tme and power allocaton s algned wth maxmally explotng the opportunstc communcaton gan n the network,.e., more tme s assgned to the users wth stronger channels and/or hgher rate weghts prortes) n UL WIT, whle more power s broadcast by the H- A n DL WET over tme slots of the users wth weaker channels and/or lower weghts. We then study the practcal case wth mperfect SIC n FD-WCN. In ths case, the WSR maxmzaton problem s shown to be non-convex, and thus s dffcult to be solved optmally. Alternatvely, we propose an effcent algorthm to fnd at least one locally optmal soluton for ths problem based on the optmal soluton obtaned for the deal case wthout SI by teratvely optmzng the tme and power allocatons to maxmze the WSR. Furthermore, we nvestgate the WSR maxmzaton problem n a baselne HD-WCN system based on the harvest-then-transmt protocol proposed n [1]. It s worth pontng out the man dfference between the optmzaton problems consdered n ths work and our pror work [1] for HD-WCN. In ths paper, we study the jont power and tme allocatons subject to both the average and transmt power constrants n HD- WCN, whereas only tme allocaton s consdered n [1] by assumng constant transmt power at the H-A. By comparng the achevable rates of FD- versus HD- WCNs, t s revealed that the former s more benefcal than the latter when the SI can be more effectvely cancelled at the H-A and/or the transmt power constrant at the H-A s more strngent. The rest of ths paper s organzed as follows. Secton II ntroduces the FD-WCN model and the proposed transmsson protocol. Secton III presents the optmal tme and power allocaton solutons to the WSR maxmzaton problems n FD-WCN for both the deal case of perfect SIC and the practcal case wth mperfect SIC, respectvely. Secton IV addresses HD-WCN, and shows the optmal soluton for WSR maxmzaton n ths case. Secton V presents smulaton results for comparng the performance of FD-WCN aganst HD-WCN. Fnally, Secton VI concludes the paper.

3 3 II. SYSTEM MODEL As shown n Fg. 1, ths paper consders a WCN wth WET n the DL and WIT n the UL. The network conssts of one H- A and K users e.g., sensors) denoted by U, = 1,, K, operatng over the same frequency band. It s assumed that users are suffcently separated from each other. The H-A s equpped wth two antennas, and each user termnal s equpped wth one antenna. The H-A s assumed to have a stable energy supply, whereas each user termnal does not have any embedded energy sources. As a result, the users need to replensh energy from the receved sgnal from the H-A n the DL, whch s then used to power operatng crcut and transmt nformaton n the UL. In ths paper, we focus on the case of FD-WCN where the H-A operates n FD mode to broadcast energy for the DL WET and receve nformaton for the UL WIT at the same tme, whle the users are assumed to all operate n tme-dvson HD mode to harvest energy n the DL and transmt nformaton n the UL orthogonally over tme, for the ease of mplementaton. For the purpose of comparson, we also consder the case of HD-WCN where the H-A also operates n tme-dvson HD mode, as studed n our prevous work [1]. A. FD-WCN In FD-WCN, the H-A operates n FD mode by utlzng one antenna for transmttng energy to users n DL and the other antenna for recevng nformaton from users n UL smultaneously over the same bandwdth. The DL channel from the H-A to U and the correspondng reversed UL channel are denoted by complex coeffcents h D, and h U,, = 1,, K, respectvely. In addton, there exsts a loopback channel at the H-A, through whch the transmtted DL energy sgnal of one antenna s receved at the other antenna n addton to the UL nformaton sgnal receved from users. It s assumed that at the H-A, analog doman SIC [], [3] s frst performed, 1 g A thus denotes the effectve loopback channel after analog doman SIC at the H-A. We also assume that h D,, h U,, = 1,, K, and g A all follow quas-statc flat-fadng, and the channels reman constant durng each block transmsson tme, denoted by T, for the system of nterest. It s further assumed that the H-A knows perfectly h U, and h D,, = 1,, K. In FD-WCN, smultaneous WET n DL and WIT n UL can be acheved, as shown n Fg.. Each transmt block s dvded nto K + 1 slots each wth duraton of T, =, 1,, K. The th slot s a dedcated power slot for DL WET only and the th slot, = 1,, K, s used for both DL WET and UL WIT. Durng the th slot, the H- A broadcasts wreless energy wth transmt power, denoted 1 We refer to SIC before A/D converson as analog doman SIC, by e.g., offsettng the two transmt antennas by half a wavelength [], or usng an extra transmt RF chan to generate a reference RF sgnal [3], whle SIC after A/D converson, commonly referred to as dgtal doman SIC, wll be consdered later. Ths s to ensure that even for the case wth one user n the system,.e., K = 1, the user can stll receve energy from the dedcated power slot. T H-A U1 U T 1 T DL WET UL WIT U K T K Fg.. Smultaneous DL energy and UL nformaton transmsson n FD- WCN. by, to all users n the network. We thus have 1. 1) = In addton, the average transmt power of the H-A over K +1 slots n each block and the transmt power of the H-A at each slot are denoted by avg and, respectvely,.e., avg, ) =, =,, K. 3) In UL, we assume that the users transmt ndependent nformaton to the H-A usng TDMA,.e., the th slot s allocated to user U for WIT, = 1,, K. Specfcally, durng the th slot n each block, the H-A broadcasts energy usng one antenna and receves nformaton from U usng the other antenna smultaneously, snce the H-A operates n FD mode. On the other hand, each user U, = 1,, K, transmts ts own nformaton to the H-A n the th slot, but cannot receve energy from the H-A n the same slot, snce t operates n tme-dvson HD mode and has one sngle antenna. Instead, U harvests energy from the H-A durng the other K slots durng whch t s not scheduled for nformaton transmsson, and stores the energy harvested to be used n future. Consder the j th slot n one block of nterest, j =,, K, durng whch the transmtted sgnal of the H-A s denoted by x A,j. 3 We assume that x A,j s a pseudo-random sequence whch s a pror known at the H-A 4 satsfyng E[ x A,j ] = 1. Durng the j th slot, the receved sgnal at U, j, s then expressed as y,j = j h D, x A,j +z,j, 4) where y,j and z,j denote the receved sgnal and nose at U, respectvely. It s assumed that j s suffcently large such that the energy harvested due to the recever nose s neglgble. Furthermore, energy harvested due to the receved UL WIT 3 Note that x A,j s can also be used to send DL nformaton e.g., control sgnals from the H-A to users) at the same tme, whch s not consdered further n ths paper. Interested readers may refer to the lterature on SWIT [5]-[14]. 4 Ths facltes the mplementaton of SIC at the H-A, as wll be shown next.

4 4 sgnals from other users s also assumed to be neglgble snce users are suffcently separated from each other and ther transmt power level s much lower as compared to j n practce. Thus, the amount of energy harvested by U durng the j th slot can be expressed as E,j = ζ j h D, j T, 5) where < ζ < 1, = 1,, K, s the energy harvestng effcency at U. For convenence, we assume T = 1 n the sequel of ths paper wthout loss of generalty. From 5), the average harvested energy of U n each block s thus expressed as K E U = ζ h D, j j. 6) j= j On the other hand, durng the j th slot wth j =, U transmts ts own nformaton to the H-A usng a fxed porton of ts average harvested energy per block gven by 6). Wthn amount of tme allocated to U for UL WIT n slot, the average transmt power of U s thus gven by U = η E U, = 1,, K, 7) where < η < 1 denotes the porton of the average harvested energy used for WIT by U n steady state. In ths paper, we assume that η s are gven constants. We further denote x U, as the sgnal transmtted by U durng slot. For each user, x U, s assumed to be of zero mean and unt power,.e.,e[x U, ] = and E[ x U, ] = 1. The receved sgnal at the H-A n the th slot s then expressed as y A, = U h U, x U, + g A x A, +z A,, = 1,, K, 8) where y A,, g A x A,, and z A, denote the receved sgnal, the SI due to smultaneously transmtted energy sgnal x A,, and the nose at the H-A, respectvely. It s assumed that z A, CN,σ ), = 1, K, where CNν,σ ) stands for a crcularly symmetrc complex Gaussan CSCG) random varable wth mean ν and varance σ. Wthout loss of generalty, the effectve loopback channel after analog doman SIC, g A, can be expressed as g A = ϕĝ A, 9) where ϕ and ĝ A, respectvely, denote the power and normalzed complex coeffcent of the loopback channel after analog doman SIC, wth E[ ĝ A ] = 1. Even after analog doman SIC, however, the power of remanng SI can be stll much larger than that of receved sgnal n practce [7],.e., h U, U ϕ. Ths rases n the followng consderatons. Frst, the receved sgnal at the H-A gven by 8) s dstorted due to lmted transmtter and recever dynamc ranges due to large power of SI, whch ntroduces addtonal nose whose power s proportonal to transmt and/or receve power at the H-A see e.g., [6] and [7]). For the purpose of exposton, we assume the same sgnal processng mplemented at the H-A as n [7] wth deal automatc gan control AGC) and nfnte transmtter dynamc range. Thus, the addtonal recever nose, denoted by z Q,, s merely due to fnte recever dynamc range and the resultng quantzaton error after ADC [7], where z Q, CN,βσQ, ) wth β 1 and σ Q, s derved from 8) as [ σq, = E y A, ] = U, h U, +ϕ +σ. 1) Furthermore, although channel estmaton can be made suffcently accurate such that performance degradaton due to the estmaton errors n h U, and h D,, = 1,, K, s assumed to be neglgble, performance degradaton due to estmaton error of g A cannot be neglected n general due to more domnant power of SI than that of receved nformaton sgnal at the H-A. We thus denote ĝ A = ḡ A + ε g A wth ḡ A and ε ga denotng the estmaton of ĝ A and channel estmaton error, respectvely. It s assumed that g A CN,1) and ε 1. Gven ḡ A, we can apply dgtal doman SIC by subtractng known SI as n [6],.e., ϕ ḡ A x A, from the receved sgnal after ADC. After ADC and dgtal doman SIC, the receved sgnal can be expressed as ȳ A, = y A, +z Q, ϕ 1 ḡ A x A, = U h U, x U, + εϕ g A x A, +z Q, +z A,, 11) = 1,, K. It s worth notng that σq, ϕ n 1) snce the power of SI s n general much larger than those of receved nformaton sgnal fromu and recever nose, based on whch the receved sgnal-to-nterference-plus-nose rato SINR) of U after ADC and dgtal doman SIC can be expressed from 11) as ρ = θ H γ +σ 1 j= j j j, = 1,, K, 1) where θ = η ζ, H = h D, h U,, and γ denotes the power of effectve SI at the th slot wth γ = ϕε+β). 13) From 1), the achevable rate of U for the case of FD-WCN n bts/second/hz bps/hz) can be expressed as R F),) = log 1+ ρ ) Γ = log 1+ θ H 1 Γγ +σ j j ), 14) j= j where = [, 1, K ], = [, 1, K ], and Γ represents the SINR gap from the addtve whte Gaussan nose AWGN) channel capacty due to the practcal modulaton and codng scheme MCS) used. In 14), γ represents the resdual SI power 5 due to fnte recever dynamc range and mperfect channel estmaton. Furthermore, t s worth notng that swtchng transmt and receve antennas at the H-A does not change the achevable rate n 14) snce R F,) depends only on H, whch s the product of the UL and DL channel power gans. 5 Here, the resdual SI ncludes the quantzaton nose whose power s also proportonal to.

5 5 H-A maxmze the WSR n 1) for an deal FD-WCN assumng perfect SIC,.e., γ = n 14), and then study the general case wth fnte SI,.e., γ >. Fg. 3. WCN. T B. HD-WCN U1 U T 1 T DL WET UL WIT U K T K Orthogonal DL energy and UL nformaton transmssons n HD- In the case of HD-WCN, we assume that the harvest-thentransmt protocol proposed n [1] s employed to acheve orthogonal WET n DL and WIT of users n UL, as shown n Fg. 3. Snce nether H-A nor users can transmt and receve sgnals at the same tme, wth the harvest-then-transmt protocol, energy s broadcast n DL durng the th slot only whereas the th slot, = 1,, K, s used for UL WIT only, n whch users transmt ther ndependent nformaton to the H-A by TDMA. It then follows that 1. 15) = Denote as the transmt power of the H-A for DL WET n slot = n the case of HD-WCN. The achevable rate of U for UL WIT n bps/hz s then gven by [1] R H),) = log 1+ θ H Γσ ), = 1,, K, 16) where = [ 1,, K ]. Lke the case of FD-WCN, we have avg and as the average and transmt power constrants at the H-A, respectvely; thus we have = mn avg /, ). III. OTIMAL TIME AND OWER ALLOCATION IN FD-WCN In ths secton, we study the jont tme and power allocaton n FD-WCN to maxmze the throughput. Specfcally, from 14), we am to maxmze the WSR of all users n UL WIT, whch s formulated as the followng optmzaton problem. 1) : max, =1 ω R F),) s.t. 1), ), and 3), 1,, =, 1, K. Note thatω s the gven non-negatve rate weght foru. Let ω = [ω 1,, ω K ]. By changng the values of ω s n 1), we are able to characterze the maxmum throughput of FD- WCN wth dfferent rate trade-offs among the users. In the followng, we frst nvestgate tme and power allocatons to A. FD-WCN wth erfect SIC Wthout SI, the achevable rate of user U n the K-user FD-WCN s modfed from 14) as R F NoSI),) = log 1+ θ H 1 Γσ j j. 17) j= j Even wth each user s achevable rate gven by 17) assumng perfect SIC, problem 1) s stll non-convex due to the non-convexty shown n the average power constrant gven by ). Smlarly as n [14] and [3], we change the varables as E =, =, 1,, K, to make ths problem more analytcally tractable, where E denotes the energy broadcast by the H-A durng the th slot. From 17), the achevable rate of U can then be expressed as ˆR F NoSI),E) = log 1+α 1 E j, 18) j= j where E = [E, E 1,, E K ] and α = θh Γσ. Accordngly, problem 1) n the case of perfect SIC can be reformulated as ) : max,e s.t. =1 ω ˆRF NoSI),E) 1, = E avg, 19) = E, =, 1, K, ) 1, E, =, 1, K, where 19) and ) correspond to the orgnal constrants n ) and 3), respectvely. By ntroducng new varables n E, jont tme and power allocaton n problem 1) s converted to jont tme and energy allocaton n problem ). It s worth notng that for any gven, the objectve functon of problem ) s a monotoncally ncreasng functon of each ndvdual E, =, 1,, K, and thus the constrant n 19) should hold wth equalty at the optmal energy allocaton otherwse, the objectve functon can be further ncreased by ncreasng some E s). Therefore, the optmal tme and energy allocaton soluton for ) can be equvalently obtaned by solvng the followng problem: 3) : max,e =1 ω R F NoSI),E)

6 6 s.t. 1, 1) = E = avg, ) = E, =, 1, K, 3) 1, E, =, 1, K, 4) wth R F NoSI),E) gven by ) R F NoSI) 1,E) = log 1+α avg E ). 5) Lemma 3.1: R F NoSI,E) s a jontly concave functon of and E, =, 1,, K. roof: lease refer to Appendx A. From Lemma 3.1, t follows that the objectve functon of 3) s jontly concave over and E. Therefore, problem 3) s a convex optmzaton problem together wth the facts that the average power constrants n ) of 1), whch s nonconvex, s now transformed to a sum-energy constrant n ) of 3), whch s affne, and furthermore the constrant n 3) s an affne functon of both and E. roblem 3) can thus be solved by Lagrangan dualty, shown as follows. From 1), ), and 5), Lagrangan of 3) s gven by L,E,λ,µ) = =1 ω R F NoSI),E) K ) K ) λ 1 +µ E avg, 6) = = wth λ and µ denotng the Lagrange multplers assocated wth the constrants n 1) and ), respectvely. The dual functon of problem 3) s then gven by Gλ,µ) = max,e) D L,E,λ,µ), 7) where D s a feasble set of,e) specfed by 3) and 4). The dual problem of 3) s thus gven by mn Gλ,µ). λ,µ roposton 3.1: Gven λ, µ, and strctly postve weghts ω >, = 1,, K, the maxmzer of L,E,λ,µ) n 6) s gven by = [, 1, K ] and E = [E, E1, EK ], where { 1 =, f µ > and λ+µ ) >,, otherwse, 8) { E =, f µ >, 9), otherwse, [ α ) ] + = mn avg E ), 1, = 1,, K, 3) z [ E = mn avg + ω ) +, α µln = 1,, K, ], 31) TABLE I ALGORITHM TO SOLVE 1) WITH γ =. 1. Intalze λ and µ.. Repeat 1) Intalze and E, = 1,, K. ) Repeat ) Compute [ 1,, K ] by 3). ) Compute [E 1,, E K ] by 31). 3) Untl [ 1,, K ] and [E 1,, E K ] both converge. 4) Compute and E by 8) and 9). 5) Compute the sub-gradent of Gλ,µ) by 33) and 34). 6) Update λ and µ usng the ellpsod method. 3. Untl λ and µ converge to a predefned accuracy. 4. Set = and E = E, = 1,, K. 5. Obtan = 1 K =1 and E = avg K =1 E. 6. Set = E, = 1,, K. wth x) + = max,x), and z denotng the soluton of f z ) = λ ln ω where f z) = ln1+z) z 1+z. 3) roof: lease refer to Appendx B. Accordng to roposton 3.1, and E under whch Gλ,µ) n 7) s achved can be attaned as follows. We frst obtan [1,, K ] and [E 1,, EK ] by teratvely optmzng between [ 1,, K ] and [E 1,, E K ] usng 3) and 31), respectvely, wth one of them beng fxed at one tme untl they both converge. We then compute and E usng 8) and 9), respectvely. Wth Gλ,µ) obtaned for each gven par of λ and µ, the optmal dual varables λ and µ mnmzng Gλ,µ) can then be effcently found by sub-gradent based algorthms, e.g., the ellpsod method [34], wth the sub-gradent of Gλ,µ) gven by ν = [ν λ, ν µ ], where ν λ = 1, 33) = ν µ = avg E. 34) Denote the optmal tme and energy allocaton soluton for 3), and equvalently for problem ), as = [, 1, K ] and E = [E, E1, EK ], respectvely. It s then worth notng that the objectve functon of 3) s a monotoncally ncreasng functon of each ndvdual, = 1,, K for gven E, = 1,, K. Therefore, = 1 should hold at the optmal otherwse, the = objectve functon can be further ncreased by ncreasng some s). Furthermore, note that we obtan [1,, K ] and [E1,, EK ] at the optmal dual solutonλ and µ. Therefore, we have = 1 K, as well as E = avg K =1 = E =1 to satsfy the constrant n ). Once and E for 3) or equvalently for )) are obtaned, the optmal power allocaton soluton for 1) s obtaned as = E, =, 1,, K, n the case of perfect SIC. To summarze, one algorthm to solve 1) s gven n Table I.

7 7 The computaton tme of the algorthm gven n Table I s analyzed as follows. The tme complextes of steps.1)-.3) s OK), whle those of.4) and.5) are O1) and OK), respectvely. Note that only two dual varables, λ and µ, are updated by the ellpsod algorthm regardless of the number of users, K. The tme complexty of step.6) s thus O1) [34]. Therefore, the total tme complexty of the algorthm n Table I s OK). Next, to obtan more nsght to the soluton gven n roposton 3.1, we consder the specal case of =. From roposton 3.1, and E or ) for problem ) when = s gven n the followng corollary. Corollary 3.1: For problem ) wth =, the optmal tme and power allocaton solutons are gven by, ) = {, ) ), = α z avg,, otherwse, 35) where z denotes the soluton of fz) = λ ln ω wth λ > and fz) s gven by 3). roof: lease refer to Appendx C. Corollary 3.1 ndcates that gven rate-weght vector ω, the maxmum WSR of FD-WCN wthout SI as s acheved by choosng only the th slot for DL WET, whch has zero amount of tme.e., ), and sendng all avalable energy avg cf. 19)) n ths zero-tme slot.e.,, avg). It s worth notng that n ths case the WSR of the FD-WCN s equvalent to that of the conventonal K-user TDMA network for UL WIT only, where each user s equpped wth a constant energy supply to provde equal transmt energy consumpton at each block denoted by avg and the UL channel for each user s gven by α, = 1,, K. It s also worth notng that n ths specal case, the FD-WCN s n fact a HD-WCN wth and. When the sum-rate s maxmzed wth ω = 1, = 1,, K, t can be further shown that = and = α / K α j, = 1,, k. j=1 Note that when <, µ should be strctly postve snce E avg E f µ = and E > avg f µ <. lease see Appendx B for detals.). It s thus shown from 9) that E =. It thus follows that f E =, then = and thus =, and vce versa f =. For, = results n E = as shown from 31), whereas E = does not necessarly mply = as shown from 3). Ths s because E, = 1,, K, represents the amount of energy that the H-A broadcasts to U j s, j = 1,, K, j, durng the th slot, whle provded that >, U can transmt nformaton to the H-A even when the H-A does not broadcast any energy durng the slot,.e., E =. Therefore, even when <, t s possble to allocate zero transmt power to a slot wth at the H-A,.e., =. The only case where E = for gven can yeld = s when the correspondng ω = snce z should be zero for ths case, accordng to 3). It s also worth notng from Corollary 3.1 that n the case of =, the algorthm gven n Table I can be smlarly appled to update and E or equvalently ) * * * * * * * * * * = 5avg avg / = * * * * * * * * * * 4 Fg. 4. Tme and power allocatons to maxmze sum-rate n FD-WCN wth perfect SIC for K = 1, = avg and = 5 avg. from any arbtrary pont ), ) ) toward, ) gven n 35). Wth <, however,, = 1,, K, stops updatng when t reaches =, after whch,e ) s updated such that E =. Fg. 4 shows and to maxmze the sum-rate n a SI-free FD-WCN wth K = 1, avg = 1, σ = 1, 6 Γ = 1, and θ = 1,. We consder two dfferent values of maxmum power, = avg or 5 avg. Furthermore, the DL and UL channels are assumed to be drawn from Raylegh fadng,.e., h D, CN,1) and h U, CN,1),, and sorted n an ascendng order of n terms of α,.e., α 1 < α < < α 1. For both cases wth = avg and 5 avg, t s observed that no tme and power s allocated to the th slot dedcated power slot),.e.,, ) =,), to maxmze the sum-rate. When = avg, t s observed that 1 = = 7 =,.e., the H-A broadcasts energy n DL wth power of durng1st-7th slots n whch the users wth the weaker channel condtons transmt nformaton n UL. Furthermore, 97% of energy s broadcast from the H-A durng 1st-8th slots, where 8 =1 avg =.5. The remanng tme s allocated to the slots n whch the users wth better channel condtons transmt nformaton n UL, and U wth larger α s allocated more tme for UL WIT. In partcular, t s also observed that 1 =,.e., the H-A does not broadcast energy n DL when the user wth the best channel transmts nformaton n UL n order to maxmze the sum-rate. When = 5 avg, smlarly, 1 = = 3 = and the H-A broadcasts93% of energy durng the 1st-6th slots where 6 =1 = avg =.. In addton,7 < 8 < 9 < 1 and 1 =. From the above observatons, t can be nferred that to maxmze the WSR, the H-A does not broadcast energy n DL when the users wth larger α s and/or ω s transmt nformaton n UL, whereas t broadcasts energy only when 6 In Fg. 4, avg and σ are normalzed for the convenence of llustraton. In fact, ths corresponds to the case wth avg = dbm n the practcal smulaton setup of Secton V. lease refer to Secton V for the detaled smulaton setup. avg avg /

8 8 Achevable Rate of U bps/hz) Fg FD-WCN, nf =.4 FD-WCN, = 1 = avg FD-WCN,. = 5 = 5 avg FD-WCN, = avg Achevable Rate of U 1 bps/hz) Achevable rate trade-offs n a -user FD-WCN wth perfect SIC. the users wth weaker channel condtons and/or smaller rate weghts prortes) transmt nformaton. Fg. 5 shows the achevable rate regons of a -user FD- WCN wth perfect SIC. It s assumed that avg = 1, = avg, σ = 1, Γ = 1, and θ = 1, = 1,, K. The channels between the H-A and two users U 1 and U are set such that H 1 =.49 and H =.5. We consder four dfferent power values, = avg, 5 avg, 1 avg, and. When =, t s observed that the maxmum achevable rate of U slog 1+α avg ) and the rate regon of FD-WCN n ths case s equvalent to that of K-user TDMA network for WIT only, whch s obtaned wth 35). Wth fnte, the achevable rate regon s observed to be smaller as decreases. B. FD-WCN wth Fnte SI In the case of mperfect SIC wth γ >, problem 1) s n general non-convex, and thus cannot be solved optmally n an effcent way. However, we can fnd one locally optmal soluton for 1) by settng, ) for the case wthout SI from roposton 3.1 as an ntal pont, and then updatng and teratvely, shown as follows. Denote k) and k) as the tme and power allocaton obtaned at the k th teraton, respectvely. We then have ), ) ) =, ). At the k-th teraton, we frst update to obtan k) usng k 1). Then, s updated to obtan k) usng k) and k 1). The above teraton s repeated untl the WSR cannot be further mproved for problem 1). For smplfyng the above updates, R F),) gven by 14) s modfed as R F SI), ) = log 1+ θ H Γγ +σ ) avg ), 36) by assumng that the average power constrant n ) holds wth equalty. 7 From 36), k) at the k th teraton can be obtaned as the soluton of the followng problem. 4) : max s.t. ) ω R F SI), k 1) =1 1, 37) = = k 1) = avg, 38), =, K. Lemma 3.: Gven = k 1), R F SI), k 1) ) gven n 36) s a concave functon of, = 1, K. roof: lease refer to Appendx D. From Lemma 3., t follows that problem 4) s convex. Therefore, problem 4) can be solved by Lagrangan dualty. From 36)-38), the Lagrangan of 4) s gven by L F SI),λ,µ) = ) ω R F SI), k 1) =1 39) K ) K ) λ 1 µ k 1) avg, = where λ and µ denote the Lagrange multplers assocated wth the constrants n 37) and 38), respectvely. The dual functon of problem 4) s then gven by G F SI) λ,µ) = max = L F SI),λ,µ), and the dual problem of 4) s thus gven by mn λ,µ GF SI) λ,µ). roposton 3.: Gven k 1), λ and µ where λ+µ avg, the maxmzer of L F SI),λ,µ) s gven by = [ 1,, K ], where = { where C = θ H / C avg z +Ck 1) Γγ k 1) +σ ), =, = 1,, K, 4) ) and z s the soluton of fz ) = λ+ k) µ)ln/ω wth fz ) defned as f z) = ln1+z) z 1+z C k 1). 41) 1+z roof: lease refer to Appendx E. It can be shown that f z) s a monotoncally ncreasng functon of z snce f z z) = z+ck 1) 1+z) >. After obtanng G F SI) λ,µ) wth gven λ and µ, the mnmzaton of G F SI) λ,µ) over λ and µ can be effcently solved by the 7 The average power constrant n ) may not hold wth equalty at the optmal tme and power allocatons n general for 1), although ths s usually desrable n practce snce there s no energy wasted at the H-A. We make ths assumpton here n order to have the rate functon R F SI) n 36) only dependent on and, but no more on other j s and j s, j, as n 13).

9 9 TABLE II ALGORITHM TO SOLVE 1) WITH FINITE γ. 1. Intalze k =, ), ) ) =, ) obtaned by solvng 1).. Repeat 1) Set k k +1. ) Gven λ, µ, and k 1), solve 4) by roposton 3.. 3) Update λ and µ usng the ellpsod method and sub-gradent of G F SI) λ,µ) gven by 4) and 43), and obtan. 4) Set k). 5) Obtan k) usng k 1), k), 36), and 44)-47). 3. Untl W F SI k 1), k 1) ) W F SI k), k) ). ellpsod method [34], wth the sub-gradent of G F SI) λ,µ), denoted as υ = [υ λ, υ µ ] T, gven by υ µ = υ λ = = 1, 4) = k 1) avg, 43) k) ], where k) = [ k) 1,, k) K ]. =, k 1, as shown n 4), we can thus take any where, =,, K, are gven by 4). The optmal soluton of 4), denoted by, s then obtaned correspondng to the optmal dual soluton of λ and µ after the ellpsod method converges, and fnally we have k) = for gven k 1). Once k) s obtaned, k) at the k th teraton can be attaned usng the obtaned k). Denote k) at the k th teraton as k) = [ k) Snce k) k) value for k) such that k). Furthermore, can be obtaned by applyng the gradent projecton method based on 36), as follows: k) = E k 1) +s k) W F SI k), k 1))), 44) k) = k 1) +δ k) k) k 1)), 45) where δ k),1] and s k) are both small step szes. In addton, W F SI k), k 1)) = [q k) 1,, qk) K ]T denotes the gradent of W F SI k), k 1) ) = K =1 RF SI) k), k 1) ) wth q k), = 1,, K, gven by q k) = ω k) ln )) 46) θ H k) +γ avg k) k 1) ) ). Γ γ k 1) +σ k) +θ H avg k) k 1) At each teraton of the algorthm gven n Table II, the computatonal complexty of step.) s OK). Smlarly to the case wth perfect SIC n the prevous subsecton, OK) computatons are requred for step.3). In step.5), OK), OK ), and OK) computatons are requred for computng W F SI k), k 1)) n 46), k) n 44), and k) n 45), respectvely. Therefore, the total tme complexty of the algorthm n Table II s OK ). Remark 3.1: Wth, we can choose such that avg, resultng n =, = 1,, K. In ths specal case, the FD-WCN wth fnte SI s n fact equvalent to that n the prevous deal case of perfect SIC wth. IV. OTIMAL TIME AND OWER ALLOCATION IN HD-WCN In ths secton, we study the optmal tme and power allocaton n HD-WCN to maxmze the WSR. Gven = mn avg /, ), from 16) the WSR maxmzaton problem for HD-WCN s formulated as 5) : max, wth α defned n 18). As shown n [1], 5) s convex Furthermore, E denotes the feasble set of gven k) and and ts optmal tme allocaton soluton, denoted by = k), defned by [, 1,, K ], s gven n the followng lemma. { } Lemma 4.1: For problem 6), the optmal soluton s gven E = k) by = avg,, = 1,,K. =1 47) = 1 K 1+ j=1 α j/z j ), 51) Fnally, n 44) E x) denotes the operaton of projecton of x onto E. To summarze, one algorthm to solve problem 1) α /z = K wth fnte γ s gven n Table II. 1+ j=1 α, = 1,, K, 5) j/z j ) s.t. =1 ω R H),) 1, 48) = avg, 49). 5),, =, 1, K. roblem 4) s non-convex n general due to the nonconcave objectve functon defned n 16) and non-convex average power constrant n 49). To solve problem 4), we frst consder the followng WSR maxmzaton problem for HD-WCN when the transmt power of the H-A s fxed as =. 6) : max s.t. =1 ) ω log 1+α 1, =, =, 1, K,

10 1 where z, = 1,, K, s the soluton of the followng equatons: ln1+z ) z = ν ln, 53) 1+z ω =1 ω α = ν ln, 54) 1+z wth ν > beng a constant under whch K = = 1. roof: lease refer to [1]. Wth attaned from Lemma 4.1, we can obtan the optmal power and tme allocaton soluton for 5), denoted by and = [, 1,, K ], as gven n the followng proposton. roposton 4.1: The optmal transmt power soluton for 5) s =. In addton, the optmal tme allocaton soluton for 5) s gven by = = { avgα z { avg, f <, otherwse,, f < avg, otherwse avg 55), = 1,, K, 56) where, =, 1,, K, are gven n 51) and 5), and z, = 1,, K, and λ > are solutons of the followng K + 1) equatons: ln1+z ) z 1+z = λ ω ln, = 1,, K, 57) =1 α z = 1 avg 1. 58) roof: lease refer to Appendx F. From roposton 4.1, t s observed that the optmal transmt power of the H-A n HD-WCN s always =, regardless of the optmal tme allocatons. In addton, we can compute for 5) effcently as follows. Frst, we solve 6) to obtan one algorthm to solve 6) s gven n [1]). If avg, then the optmal tme allocaton soluton for 5) s =, =, 1,, K. Otherwse, f > avg, then = avg, and, = 1,, K, can be obtaned from 56) va solvng for z s by a bsecton search over λ untl both 57) and 58) hold wth equalty. Smlar to the case of FD-WCN wth perfect SIC, we further nvestgate the optmal tme and power allocaton for HD-WCN n roposton 4.1. Frst, consder the specal case of =. In ths case, t can be shown from 55) that = and thus such that avg. In addton, t can be shown from 5) that, = 1,, K, ncreases wth α or ω snce z decreases wth ω accordng to 53). Note that these observatons have been smlarly made for the FD-WCN n the specal case of. Next, consder the general case of < and furthermore < < avg. In ths case, ncreasng above reduces the WSR of HD-WCN, although the amount of harvested energy of each user n DL WET and hence ts transmt power n UL WIT ncreases wth ncreasng. Ths s because the Achevable Rate of U bps/hz) FD-WCN, nf = HD-WCN, nf =.4 FD-WCN, = 55 avg HD-WCN, = 55 avg. FD-WCN, = = avg HD-WCN, = avg Achevable Rate of U 1 bps/hz) Fg. 6. Rate regon comparson between FD-WCN wth perfect SIC versus HD-WCN. resultng decrease of transmsson tme for UL WIT reduces the WSR more substantally than the mprovement of WSR due to the ncrease of transmt power. At last, consder the case of < and > avg. In ths case, mnmzng,.e., by settng = avg, maxmzes the WSR snce the tme allocated for UL WIT s maxmzed for the gven total energy sent by the H-A, whch s avg. For both the above two cases wth <,, = 1,, K, ncreases wth α or ω, smlar to the case wth =. Fg. 6 compares the achevable rate regons of FD-WCN assumng perfect SIC) versus HD-WCN wth K = based on ropostons 3.1 and 4.1, respectvely, wth the same twouser channel setup as for Fg. 5 and three dfferent power values, = avg, 5 avg, and. It s worth notng that when =, the achevable rate regons of both FD-WCN and HD-WCN are dentcal, snce they have the same tme and power allocaton. However, when <, the achevable rate regon of HD-WCN s observed to be smaller than that of FD-WCN. Ths s because the H-A n the FD-WCN can broadcast all of ts avalable energy to users, whereas that n the HD-WCN n general cannot due to the total tme constrant. Furthermore, n the FD- WCN s n general smaller than that of HD-WCN snce the other K slots n the FD-WCN can also be used for DL WET whereas ths s not possble for the HD-WCN; as a result, more tme can be allocated to users for UL WIT n the case of FD-WCN, leadng to a larger rate-regon. artcularly, t s observed that when = avg, the achevable rate regon of FD-WCN s observed to be more notably larger than that of HD-WCN. Therefore, the beneft acheved by employng FD H-A over HD counterpart s more sgnfcant wth smaller, but s less evdent as ncreases. V. SIMULATION RESULTS In ths secton, we compare the achevable sum-rates of FD- WCN wth/wthout SI versus HD-WCN under a practcal system setup. The bandwdth s set as 1MHz. The dstance between the H-A and user U, denoted by D, s assumed

11 11 % $ "#! &')*+, +-./ &')*+, &13./ 4')*+ K =1 K = 5 9 = 8 X W V U < ST QR 7 NO M L K J ; I H G 6 FD-WCN, No SI FD-WCN, ϕ = 8dB FD-WCN, ϕ = 6dB FD-WCN, ϕ = 4dB HD-WCN YZ[\>]^_ ^` ab_ YZ[\>]^_ ab_ cccdeeeeee YZ[\>]^_ ab_ fgg [85CD YZ[\>]^_ ab_ fgg [75CD hz[\>]^ : K = :5 :6 :7 :8 :9 65 >?@A BCDEF Fg. 7. Average sum-rate vs. avg wth K = 1, = avg, and ϕ = 6dB. Fg. 8. Average sum-rate vs. avg for dfferent values of ϕ wth K = 1 and = avg. to be unformly dstrbuted wthn 5m D 1m, = 1,, K. For each user, the DL and UL channel power gans are modeled as h D, = 1 3 ρ D, D αd and h U, = 1 3 ρ U, D αu, respectvely, where the same pathloss exponents α D = α U = are assumed. In addton, ρ D, and ρ U, represent the channel short-term fadng n the DL and UL, respectvely, whch are both assumed to be Raylegh dstrbuted,.e., ρ D, and ρ U, are ndependent exponental random varables wth unt mean. Note that n the above channel model, a 3dB average sgnal power attenuaton s assumed at a reference dstance of 1m. The AWGN at the H-A recever s assumed to have a constant power spectral densty of 16dBm/Hz. For all users, t s assumed that θ =.5 8, = 1,, K. We set Γ = 9.8dB assumng that an uncoded quadrature ampltude modulaton QAM) s employed wth the requred bt-error-rate BER) of 1 7 [35]. Furthermore, to account for quantzaton error after ADC, we set β = 6dB as assumed n [6]. Fnally, t s assumed that the mean-square error MSE) of channel estmaton error of the loopback channel s set to be ε = 6dB due to the large power of the loopback sgnal. Fg. 7 shows the average sum-rates of FD-WCN versus HD-WCN for dfferent values of avg n dbm by averagng over 1 randomly generated fadng channel realzatons, wth K = 1, = avg, and ϕ = 6dB,.e., 6dB SIC n analog doman 9. Accordng to 13), the power of effectve SI n ths case s -13dB less than the power of the transmtted sgnal of the H-A at a gven tme slot. As shown n Fg. 7, the average sum-rate of FD-WCN s always larger than that of HD-WCN when SI s perfectly elmnated. In partcular, FD-WCN wth perfect SIC outperforms HD- WCN more consderably as the number of users n the 8 Assumng that practcal rectfer antenna s employed for wreless energy harvestng, the energy harvestng effcency s typcally n the range of 7-8% [36]. Gven that ζ = 75%, θ =.5 s obtaned by assumng that 67% of the harvested energy s used for WIT n the UL at U,.e., η = /3, = 1,, K. 9 By current technques, t has been reported that SI can be canceled up to 81dB n analog doman [4]. Average Sum-Rate Mbps) FD-WCN, No No SI SI FD-WCN, SI, eee=bbbbbb ϕ = 8dB data3 FD-WCN, ϕ = 6dB 1 data4 FD-WCN, ϕ = 4dB data5 HD-WCN Number of Users K) Fg. 9. Average sum-rate vs. K for dfferent values of ϕ wth avg = dbm and = avg. network, K, ncreases. However, when SI n FD-WCN s not perfectly cancelled, HD-WCN outperforms FD-WCN wth small number of users n the network, e.g., K =. As K ncreases, FD-WCN even wth fnte SI can outperform HD-WCN when avg s suffcently small or the FD- WCN s less nterference lmted, e.g., the sum-rate of FD- WCN wth fnte SI s larger than that of HD-WCN when dbm avg 5dBm wth K = 1. Next, by fxng K = 1, Fg. 8 shows the average sumrate comparson for dfferent values of ϕ, whch measures the effectve loopback channel power after analog doman SIC. It s observed that whenϕ = 8dB, the sum-rate of FD-WCN wth mperfect SIC converges to that wth perfect SIC, and also outperforms HD-WCN. When ϕ = 6dB, the sumrate of FD-WCN wth mperfect SIC s larger than that of HD-WCN when avg < 5dBm, but becomes smaller when avg 5dBm. Furthermore, the sum-rate of FD-WCN wth mperfect SIC s always smaller than that of HD-WCN when ϕ = 4dB.

12 1 Average Sum-Rate Mbps) data1 FD-WCN, No SI FD-WCN, SI, ϕ eee=bbbbbb = 8dB data3 FD-WCN, ϕ = 6dB data4 FD-WCN, ϕ = 4dB data5 HD-WCN / avg Fg. 1. Average sum-rate vs. / avg for dfferent values of ϕ wth K = 1 and avg = dbm. Fg. 9 shows the average sum-rates of FD- and HD-WCNs over the number of users, K, for dfferent values of ϕ. It s observed that the achevable sum-rates of both FD- and HD- WCNs ncrease wth K. In addton, when ϕ = 8dB, FD- WCN wth mperfect SIC s observed to have comparable achevable sum-rate to that wth perfect SIC and also have larger sum-rate than HD-WCN over all values of K. Furthermore, when ϕ = 6dB, t s observed that the average sum-rate of FD-WCN wth mperfect SIC s smaller than that of HD-WCN when K < 5, but becomes larger when K 5. However, the average sum-rate of FD-WCN wth mperfect SIC s observed to be always smaller than that of HD-WCN when ϕ = 4dB. At last, Fg. 1 shows the average sum-rate comparson for dfferent values of / avg wth K = 1 and avg = dbm. It s observed that the achevable sum-rate gan of FD- WCN wthout SI s more pronounced over HD-WCN when / avg s small, but the gap decreases wth ncreasng / avg. In addton, when / avg, t s also observed that the sum-rate of FD-WCN becomes more comparable to that of HD-WCN even wth mperfect SIC. VI. CONCLUSION Ths paper studed optmal resource allocaton n a new type of WCN where full-duplex H-A s employed, namely FD-WCN. We proposed a new transmsson protocol for the FD-WCN whch enables effcent smultaneous WET n DL and WIT n UL, over the same bandwdth. Wth the proposed protocol, we studed the jont tme and power allocaton n FD-WCN to maxmze the WSR n both the deal case assumng perfect SIC and the practcal case wth fnte resdue SI. It s shown that to maxmze the WSR of FD-WCN, the optmal tme and power allocaton should optmally explot the avalable multuser channel dversty n the hybrd network. We also studed the optmal jont tme and power allocaton for a baselne HD-WCN, and compared the achevable rates wth FD-WCN. Smulaton results revealed that the FD H-A s more benefcal than HD H-A n WCNs when the SI can be effectvely cancelled, the number of users n the network s suffcently large, and/or the transmt power constrant s more strngent as compared to the average transmt power at the H-A. AENDIX A ROOF OF LEMMA 3.1 For any = 1,, K, R F NoSI),E) gven n 5) s a perspectve of f E) [33], where f E) = log 1+α avg E )). 59) Note that f E) n 59) s a concave functon of E because f E) s a composton of a concave functon ˆf x) = log 1+α x) and an affne functon f E) = avg E,.e. f E) = ˆf ) f E). Snce the perspectve operaton preserves concavty [33], R F NoSI),E) s thus a jontly concave functon of and E, = 1,, k. Ths thus completes the proof of Lemma 3.1. AENDIX B ROOF OF ROOSITION 3.1 Lagrangan gven n 6) can be alternatvely expressed as L,E,λ,µ) = L,E,λ,µ)+λ µ avg, 6) = where L,E,λ,µ), =, 1,, K s gven by L,E,λ,µ) { λ +µe =, = ω ˆRF NoSI),E) λ µe, otherwse. 61) Gven λ and µ, Gλ, µ) n 7) can be obtaned by maxmzng ndvdual L,E,λ,µ), =, 1,, K, subject to,e ) D, sncel,e,λ,µ) depends only on ande,.e., by solvng the followng problem, =, 1,, K. max, E L,E,λ,µ) s.t. E, 1. 6) We frst consder the problem n 6) for the case wth = 1,, K. Gven λ, µ, and, we can obtan E that maxmzes L,E,λ,µ) n 61) by settng E =, from whch we have α ω = µln. 63) 1+ α avg E ) Snce E n D as shown n 3) and 4), E s gven from 63) as 31). It s shown from 63) that E avg f µ >, E f µ =, and E > avg f µ <. In addton, gven λ, µ, and E, we can also fnd that maxmzes L,E,λ,µ) n 61) by settng =, from whch we have ln1+z ) z = λln, 64) 1+z ω

13 13 where z = α avg E ). Snce 1 n D as shown n 4), s thus gven as 3) when z s the soluton of 64). Next, consder the case wth =. Snce E, t s easly shown from 61) that E maxmzng L,E,λ,µ) s gven as 9). Note that when µ >, E s gven by E = and thus L,E,λ,µ) = λ+µ ). Snce 1, to maxmze L,E,λ,µ) we have = 1 f λ + µ > and = otherwse. When µ, = to maxmze L,E,λ,µ) snce E = and thus L,E,λ,µ) = λ. Therefore, maxmzng L,E,λ,µ) s gven by 8). Ths completes the proof of roposton 3.1. AENDIX C ROOF OF COROLLARY 3.1 To prove Corollary 3.1, we consder the followng two cases for any = 1, K: E = and E >. Frst, when E =, t follows from 3) that = α z avg >. Next, when E >, from 3) and 31) we have = α z E = avg + avg E ), 65) 1 ω ) α µ, 66) ln where 65) s orgnated from the fact that < 1. By substtutng n 66) by 65), t follows that E 1+ α 1 z ω )) α µ ln = avg 1+ α 1 z ω )) α µ, ln from whch we have E = avg. Wth, therefore, we have,e ) =, avg) or α z avg,). If there s an {1,,K} such that,e ) =, avg ), t follows that Ej =, j =, 1,, K, j, to satsfy the sum-energy constrant n ). In addton, from 3) we have = and j = α z avg, j = 1,,K, j. In ths case, FD-WCN wth becomes equvalent to K 1)-user TDMA network consstng of U 1,,U 1,U +1,,U K, where each user consumes a constant energy for nformaton transmsson and the channel of U j s gven by α j. If,E ) = α z avg,), = 1,,K, t then follows that E = avg and =. In ths case, FD-WCN wth becomes equvalent to K-user TDMA network consstng of U s, = 1,,K, where each user consumes a constant energy for nformaton transmsson and the channel of U s gven by α. It s well known that the achevable rate regon of K 1)- user TDMA network s a subset of K-user TDMA network. Therefore,,E ) =, avg ) and,e ) = α z avg,), = 1,,K, s the optmal tme and energy allocaton soluton for problem ) wth. Corollary 3.1 s thus proved. AENDIX D ROOF OF LEMMA 3. From 36), the Hessan of R F SI), k 1) ), = 1,, K, s gven by R F SI), k 1) ) = [d ) j =,,K, k =,,K, where d ) j,k = C avg )) +C avg k 1), j = k =, otherwse, j,k ], 67) θ wth C = H ). Γ γ k 1) +σ We thus have v T R F SI), k 1) )v for any gven v = [v,, v K ] T. Ths thus proves Lemma 3.. AENDIX E ROOF OF ROOSITION 3. Gven λ, µ, and k 1), L F SI),λ,µ) can be expressed as L F SI),λ,µ) = L F SI),λ,µ), 68) = where L F SI),λ,µ) s gven by,λ,µ) ) λ+ k 1) µ = ω log 1+C avg k 1) ) λ+ k 1) µ L F SI) ), =, = 1,,K. 69) Smlar to the proof of roposton 3.1, L F SI),λ,µ) s maxmzed by maxmzng ndvdual L F SI),λ,µ), = 1,, K. We frst consder the case wth =. Snce 1, from 69) we have {, λ+ k 1) = µ 7) 1, otherwse. Next, consder the case wth = 1,, K. We can fnd that maxmzes L F SI),λ,µ) by settng L F SI),λ,µ) =, from whch we have = ) f z ) = ln1+z ) z +C k 1) λ+ k 1) µ = 1+z ln ω, 71) where z = C avg k 1). When z s the soluton of 71), we have 4) n roposton 3.. Note that fz ) gven n 71) also n 41)) s a monotoncally ncreasng functon of z, whch has the mnmum value C k 1) at z =. Therefore, there s no z whch s soluton of 71) f C k 1) ) λ+ k 1) µ ln > ) ω λ+ k 1) µ ln ω. In addton, f C k 1) <, we can fnd z beng a soluton of 71) and thus non-zero, but = 1 as shown n 7). Ths always volate the sum-tme constrant n 37). ) λ+ k 1) µ ln Therefore, t should be satsfed that ω,.e., λ+ k 1) µ)ln, wth =. Ths thus completes the proof of roposton 3..