Transmit Power Optimization and Feasibility Analysis of Self-backhauling Full-Duplex Radio Access Systems

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1 Tampere Universiy of Technology Transmi Power Opimizaion and Feasibiliy Analysis of Self-backhauling Full-Duplex Radio Access Sysems Ciaion Korpi, D., Riihonen, T., Sabharwal, A., & Valkama, Transmi Power Opimizaion and Feasibiliy Analysis of Self-backhauling Full-Duplex Radio Access Sysems. IEEE Transacions on Wireless Communicaions. DOI: /TWC Year 018 Version Peer reviewed version pos-prin Link o publicaion TUTCRIS Poral hp:// Published in IEEE Transacions on Wireless Communicaions DOI /TWC Copyrigh 018 IEEE. Personal use of his maerial is permied. Permission from IEEE mus be obained for all oher uses, in any curren or fuure media, including reprining/republishing his maerial for adverising or promoional purposes, creaing new collecive works, for resale or redisribuion o servers or liss, or reuse of any copyrighed componen of his work in oher works. Take down policy If you believe ha his documen breaches copyrigh, please conac ucris@u.fi, and we will remove access o he work immediaely and invesigae your claim. Download dae:

2 1 Transmi Power Opimizaion and Feasibiliy Analysis of Self-backhauling Full-Duplex Radio Access Sysems Dani Korpi, ember, IEEE, Taneli Riihonen, ember, IEEE, Ashuosh Sabharwal, Fellow, IEEE, and ikko Valkama, Senior ember, IEEE Absrac We analyze an inband full-duplex access ha is serving mobile users while simulaneously connecing o a core nework over a wireless backhaul link, uilizing he same frequency band for all communicaion asks. Such wireless selfbackhauling is an inriguing opion for he nex generaion wireless sysems since a wired backhaul connecion migh no be economically viable if he access s are deployed densely. In paricular, we derive he opimal ransmi power allocaion for such a sysem in closed form under Qualiy-of-Service QoS requiremens, which are defined in erms of he minimum daa raes for each mobile user. For comparison, he opimal ransmi power allocaion is solved also for wo reference scenarios: a purely half-duplex access, and a relay-ype full-duplex access. ased on he obained expressions for he opimal ransmi powers, we hen show ha he sysems uilizing a full-duplex capable access have a fundamenal feasibiliy boundary, meaning ha here are circumsances under which he QoS requiremens canno be fulfilled using finie ransmi powers. This fundamenal feasibiliy boundary is also derived in closed form. The feasibiliy boundaries and opimal ransmi powers are hen numerically evaluaed in order o compare he differen communicaion schemes. In general, uilizing he purely full-duplex access resuls in he lowes ransmi powers for all he communicaing paries, alhough here are some nework geomeries under which such a sysem is no capable of reaching he required minimum daa raes. In addiion, he numerical resuls indicae ha a full-duplex capable access is bes suied for relaively small cells. Index Terms Self-backhauling, full-duplex wireless, massive IO, ransmi power opimizaion. I. INTRODUCTION WIRELESS inband full-duplex communicaions is widely considered o be one of he key enabling echnologies in achieving he required hroughpu gains of he fuure 5G neworks. Is basic idea is o allow a radio o ransmi and anuscrip received February 0, 017; revised Sepember 14, 017 and January 19, 018; acceped arch 5, 018. This work was suppored in par by he Academy of Finland under he projecs #30180, #304147, and #310991, in par by he Finnish Funding Agency for Technology and Innovaion Tekes, under he TAKE-5 projec, in par by Tampere Universiy of Technology Graduae School, in par by Nokia Foundaion, in par by Tuula and Yrjö Neuvo Research Fund, and in par by Emil Aalonen Foundaion. Corresponding auhor: Dani Korpi. D. Korpi, T. Riihonen, and. Valkama are wih he Laboraory of Elecronics and Communicaions Engineering, Tampere Universiy of Technology, Tampere, Finland, dani.korpi@u.fi. A. Sabharwal is wih he Deparmen of Elecrical and Compuer Engineering, Rice Universiy, Houson, TX 77005, USA. Color versions of one or more of he figures in his paper are available online a hp://ieeexplore.ieee.org Digial Objec Idenifier /TWC.018.XXXXXXX receive daa signals simulaneously using he same cener frequency, and hence i has he capabiliy o double he specral efficiency of he exising sysems as long as is full poenial can be harnessed properly [1] [5]. any real-world demonsraions of inband full-duplex radios have already been developed by various research groups, which indicaes ha he concep is indeed feasible [1], [5] [8]. In addiion, he framework and heoreical boundaries of inband full-duplex radios have been sudied exensively in he recen years [], [9] [15]. In erms of a pracical implemenaion, he fundamenal issue for inband full-duplex devices is he coupling of he own ransmi signal o he receiver. In paricular, since he ransmission and recepion occur simulaneously over he same frequency channel, he ransceiver will inherenly receive is own ransmi signal. Wha makes his especially problemaic is he exremely high power level of he own ransmission a his sage, which means ha i will compleely drown ou he inended received signal. This phenomenon is ypically referred o as self-inerference SI, and reducing is effec has been one of he main research areas in his field. The various proposed SI cancellaion soluions [7], [16] [0] and acual implemenaions and measuremens already show ha solving he problem of SI is no far from realiy [1], [6] [8], [0], [1]. In addiion o SI cancellaion, a large porion of he research has also focused on how o bes make use of he full-duplex capabiliy on a nework level [4], [] [4]. This is a edious issue since in many applicaions he raffic requiremens are highly asymmeric beween he wo communicaion direcions, such as in mobile neworks [5]. ecause he inband full-duplex principle requires compleely symmeric raffic o realize he doubling of specral efficiency a radio link level, his will compromise he poenial hroughpu gains i can provide in pracice. Thus, more advanced schemes are likely needed in order o realize he full poenial of inband full-duplex radios in pracical nework scenarios. One such opion is employing a full-duplex access in an oherwise legacy halfduplex mobile cell [4], [3], [4], [6], hereby allowing he o simulaneously serve he uplink UL and he downlink DL using he very same frequency resources. Wih proper muliplexing and acive scheduling, such a scheme enables he o fully exploi is full-duplex capabiliy in boh direcions [4]. In his paper, he above ype of a scheme will be analyzed and developed furher under a scenario where insalling wired backhaul links for all he cells is no feasible. This means ha

3 wireless self-backhauling, where he same frequency band is also used o backhaul he UL and DL daa, is required [4], [7] [33]. This ype of a siuaion can occur, for insance, due o densely deployed cells, a probable scenario in he fuure 5G neworks [34], [35]. Hence, in addiion o communicaing wih he user equipmens s, also he backhaul daa is ransferred inband wih a wireless poin-o-poin link beween he and a so-called backhaul N. The N hen furher connecs o he acual core nework using eiher a wired or a wireless link. As he self-backhauling is performed on he same frequency band as he DL and UL daa ransfer, no addiional specral resources are needed, which furher improves he applicabiliy of such a soluion. This ype of inband self-backhauling has been also invesigaed in he earlier lieraure. Therein, mos works have considered a relay-ype ha is direcly forwarding he signals ransmied by he UL s o he N, or vice versa [7] [30], [3], [36], [37]. The reason for his is likely he fac ha such a relay-ype is more or less direcly compaible wih he exising neworks, as i would essenially jus exend he range of he N or macro base saion S. In paricular, in [7], [36], he power conrol of such a relay-ype is invesigaed, and he performance of boh half-duplex and full-duplex operaion modes is hen compared. The findings obained in [7], where he ransmi powers are numerically opimized, indicae ha he full-duplex can obain higher hroughpus han he corresponding half-duplex sysem. The same conclusion is reached in [36], where he specral efficiency of a similar sysem is maximized by solving he opimal power allocaion for boh full-duplex and half-duplex s, wih he power allocaion of he former being solved using an ieraive algorihm. oreover, he effec of radio resource managemen RR on he performance of he relay-ype full-duplex is invesigaed in [8], where he resuling soluion is shown o ouperform he half-duplex benchmark scheme. The work in [9], on he oher hand, invesigaes differen beamforming soluions for a N wih massive anenna arrays, alhough no full-duplex operaion is assumed in any of he s herein. The DL coverage of a relay-ype self-backhauling is hen analyzed in [30], [37]. The findings in [37] indicae ha, while he hroughpu of he nework wih full-duplex-capable s is almos doubled in comparison o he half-duplex sysems, he increased inerference levels resul in a somewha smaller coverage. The resuls obained in [30] sugges, on he oher hand, ha on a nework level i may be beer o have also some s ha perform he self-backhauling on a differen frequency band, in order o reduce he inerference levels. The work in [3] analyzes he hroughpu and ouage probabiliy of a relay-ype full-duplex under an anenna selecion scheme where individual ransmi and receive anennas are chosen in he based on a given crierion. Again, he full-duplex is shown o usually ouperform he corresponding half-duplex. All in all, even hough differen inband self-backhauling soluions for small cells have been invesigaed in he earlier lieraure, none of he above works have considered a scenario where also he UL and DL ransmissions are performed simulaneously on he same cener frequency. Considering he promising findings regarding he relay-ype scenario where he DL and UL are separaed eiher in ime or in frequency, his means ha he purely full-duplex scheme analyzed in his aricle is an inriguing opion for furher improving he specral efficiency of hese ypes of neworks. Furhermore, o properly evaluae he full-duplex self-backhauling soluion for he, is performance is compared o wo reference schemes, one of which relies on radiional half-duplex communicaion while in he oher he acs as a one-direcional full-duplex relay. The laer reference scheme corresponds o he soluion mosly invesigaed in he earlier works. In addiion, in his aricle i is also assumed ha he has large arrays of anennas a is disposal. Therefore, in he full-duplex soluion, he same ime-frequency resource can be used for all he individual UL and DL ransmissions, as well as for he wireless backhaul link, since such massive anenna arrays allow for efficien beamforming, which can be used o preven he inerference beween he various spaial sreams [4], [30], [33]. The massive arrays also faciliae efficien aenuaion of he SI by zero-forcing ZF beamforming [9], [38]. Namely, he ransmi signals will be precoded such ha nulls are formed in he posiions of all he receive anennas, which will significanly decrease he SI power coupled back o he receivers. To suppress he residual SI remaining afer he ZF procedure, addiional SI cancellaion can also be performed, e.g., in he digial domain [3], [6], [8], [17]. The differen communicaion schemes are hen analyzed under a scenario where a minimum Qualiy-of-Service QoS requiremen is given for each, defined in erms of minimum DL and UL daa raes. This definiion ensures uniform QoS for all he s, which makes i a reasonable choice. The problem is hen o deermine he minimum ransmi powers for each communicaing pary under he consrain ha each achieves he minimum required daa rae. Furhermore, since wireless self-backhauling is assumed, he and he N mus also allocae some ransmi power for he backhaul link o ensure sufficien backhauling capabiliy. A similar sysem was considered by he auhors already in [33], where he sum-rae was opimized under a grealy simplified sysem model, while he ransmi power minimizaion under QoS consrains was preliminarily considered in [4]. The curren aricle complees and archives our research work in he mos comprehensive form under a generic seing by presening closed-form soluions for he opimal ransmi powers in hree differen communicaion schemes: a full-duplex scheme, a half-duplex scheme, and a hybrid relay scheme. To he bes of our knowledge, his is somehing ha has no been solved before for any selfbackhauling radio access sysem. The major conribuions of his paper can be deailed as follows: We derive closed-form soluions for he opimal ransmi powers of all he considered communicaion schemes ha fulfill he QoS requiremens. We show ha he full-duplex and hybrid relay schemes canno always fulfill he QoS requiremens, even if he ransmi powers end owards infiniy. In oher words, hese wo schemes are feasible only under some circumsances,

4 3 meaning ha here is a fundamenal limi for he daa raes ha hey can achieve. The condiion for his is derived in closed form, while accurae approximaions for he feasibiliy boundary are also provided. We provide exensive numerical resuls o illusrae differen aspecs of he considered communicaion schemes. In paricular, he numerical resuls show ha in mos cases he full-duplex scheme is indeed he mos ransmi power efficien soluion. However, he resuls also indicae ha he schemes uilizing a full-duplex capable are fundamenally limied o relaively small cell sizes. The res of his aricle is organized as follows. The sysem model is firs presened in Secion II, alongside wih he achievable DL and UL daa rae expressions of he hree differen communicaion schemes. The opimal QoS-fulfilling ransmi powers are hen derived in Secion III. Afer his, he feasibiliy of he full-duplex and hybrid relay schemes is invesigaed in Secion IV, he feasibiliy boundaries being derived in closed form. The numerical resuls are hen given and analyzed in Secion V, while he conclusions are drawn in Secion VI. II. SYSTE ODEL, COUNICATION SCHEES, D SU-RATE EXPRESSIONS Le us consider a wireless cell wih a large-array ha is communicaing wih a muliple-inpu and muliple-oupu IO N and half-duplex single-anenna s, he s being furher divided ino UL and DL s. The is assumed o have N ransmi and N r receive anennas, while he amoun of ransmied and received signal sreams is assumed o be significanly lower. oreover, he same anenna arrays are used for serving he DL and UL s as well as for communicaing wih he N, he forwarding he daa beween he s and he N in a decode-and-forward manner. As for he number of N anennas, he analysis can be inerpreed such ha he number of backhaul signal sreams in eiher direcion is equal o he number of corresponding N anennas. Three differen communicaion schemes are analyzed in his aricle, each of hem depiced in Fig. 1. elow, we describe he differen communicaion schemes in deail, and also derive he expressions of he achievable daa raes for each scheme. These can hen be used in deermining he opimal ransmi power allocaions. A. Full-Duplex Scheme In he full-duplex scheme, he ransmis signals simulaneously o he N and o he DL s while also receiving signals from he UL s and he N, all of he ransmissions occurring on he same cener frequency. Consequenly, boh he and he N mus be full-duplex capable, while he s are legacy half-duplex devices, as already menioned. This ype of a full-duplex sysem suffers from he SI, he IUI beween he UL and he DL s, as well as from he inerference beween he N and he s. Even hough here are also advanced mehods for aenuaing he UL-o-DL IUI [39] [41], in his work we assume ha is power level is only affeced by he ransmi power of he UL s and he pah losses beween he UL and DL s. Denoing he number of DL s by D and he number of ransmied backhaul signal sreams by, he overall sacked spaial signal received by he s and he N can be represened as a vecor, whose firs D elemens conain he samples received by he DL s, while he las elemens conain he samples received by he N consising of he parallel sreams of backhauled UL daa. This oal received signal vecor can be wrien as follows: y = L H x + z, 1 where L = diag L d 1 L,..., d D,..., L,..., L is a D + D + diagonal marix, L d i is he pah loss normalized fading variance beween he and he ih DL, L is he pah loss normalized fading variance beween he and he N, H C D+ N is he normalized channel marix beween he and all he inended receivers, x C N 1 is he ransmi signal of he and z C D+ 1 represens he differen noise and inerference sources. In his aricle, Rayleigh fading beween all communicaing paries is assumed, which means ha he enries of H are independen and idenically disribued i.i.d. zero-mean complex Gaussian random variables wih uni variance. In he coninuaion, o simplify he lierary presenaion, he pah loss normalized fading variances are simply referred o as pah losses. Also noe ha, while he pah losses beween he and he s are differen, he pah losses of he backhaul signals are idenical as hey all correspond o he link beween he and he N. The precoded ransmi signal x is formed from he DL and backhaul ransmi daa as follows: x = WΓq, where W C N D+ is he precoding marix, q C D+ 1 conains all he ransmi daa symbols, Γ = diag p d 1 p,..., d D,..., P u /,..., P u / is a D+ D + diagonal marix, p d i is he ransmi power allocaed for he ih DL signal sream, and Pu is he oal ransmi power allocaed for backhauling he UL daa. The power [ of he daa symbols is assumed o be normalized as E q i ] = 1 where q i is he ih elemen of q. Even hough he ransmier s power amplifier induced nonlinear disorion is ypically a significan issue in full-duplex devices [3], in his analysis we are using a linear signal model for simpliciy. In fac, in a massive IO ransmier, he powers of he individual ransmiers are ypically small, which somewha alleviaes he effecs of he nonlineariies. The precoding is performed using ZF beamforming since i ypically performs well under high signal-o-noise raios [4]. Assuming ha also he effecive SI channel experiences Rayleigh fading, he SI channel marix beween he ransmiers and receivers can be expressed as L s H s C Nr N, where L s is a diagonal marix conaining he amouns of SI suppression beween he ransmier and receiver pairs wihou any ZF nulling assuming ha he amouns of SI suppression are equal for all ransmier and receiver pairs and he elemens

5 4 Full-duplex Downlink ransmission Uplink ransmission Access Cell radius ackhaul ackhaul ransmissions Unsuppressed inerference Suppressed inerference Half-duplex Time slo 1 Time slo Access Cell radius ackhaul Access Cell radius ackhaul Hybrid relay Time slo 1 Time slo Access Cell radius ackhaul Access Cell radius ackhaul Fig. 1: An illusraion of he hree considered communicaion schemes: he full-duplex scheme, he half-duplex scheme, and he hybrid relay scheme. of H s are i.i.d. zero-mean complex Gaussian random variables wih uni variance. Noe ha assuming he SI channel o experience Rayleigh fading can be expeced o be relaively accurae when here is a cerain level of acive SI suppression before he oal received signal is decoded [1]. For he same reason, he overall SI suppression beween each ransmier and receiver pair can be expeced o be roughly he same, as more acive SI cancellaion is ypically obained in he receivers wih sronger SI coupling in he anenna inerface [1]. Under he condiion ha he has full knowledge of he channel sae informaion CSI of he links used for daa ransfer ha is, he channel marix H is fully known and assuming ha N > N r +D+, he ransmier ZF precoding marix in full-duplex mode can hen be wrien as [4] W = H H HH H 1 Λ, 3 where H H = [ H H Ĥ H s ], Ĥs is an imperfec esimae of he effecive SI coupling channel, and Λ C D+ +Nr D+ is a non-square diagonal normalizaion marix conaining he normalizaion facor N D N r in each diagonal enry [38], [4]. oreover, H denoes he Hermiian ranspose. The purpose of he normalizaion marix is simply o ensure ha precoding does no affec he expeced effecive powers of he daa symbols. I should be noed ha assuming he o have full knowledge of he CSI of he daa ransfer links is obviously no fully pracical, bu i allows he derivaion of analyical daa rae expressions ha provide informaion abou he ulimae performance limis of he considered sysem. Namely, his assumpion means ha, apar from SI, none of he signals received or ransmied by he inerfere wih each oher,

6 5 Variable N / N r Definiion TALE I: The mos imporan symbols used in he aricle. Number of ransmi/receive anennas in he / r Number of backhaul signal sreams ransmied/received by he D / U Number of DL/UL s in he cell Λ X / Λ X r The degrees-of-freedom of he ransmier/receiver, X = {FD, HD, RL} L d i / Lu j Pah loss beween he and he ih DL / jh UL L ud ij L d i L σ n / L u j /α N p d i p u j P d Pu η ρ d / ρ u Pah loss beween he ih DL and he jh UL Pah loss beween he N and he ih DL / jh UL Pah loss beween he and he N Noise floor in all he receivers Amoun of SI cancellaion in he /N Transmi power used for he ih DL signal sream Transmi power of he jh UL Toal amoun of ransmi power used by he N Toal amoun of ransmi power allocaed for self-backhauling in he Proporion of ime spen in he DL ime slo in he half-duplex and hybrid relay schemes DL/UL rae requiremen of an individual represening a bes-case scenario in his respec. Neverheless, he effec of residual SI is sill considered in he sysem, as no full knowledge of he effecive SI coupling channel is assumed. Furhermore, in order o simplify he sysem models and derivaions, only he beamforming performed by he is explicily considered, meaning ha analysis of any poenial spaial domain processing in he oher s is omied. Now, we can rewrie he signals received by he DL s and he N as y = L H x + z = L H WΓq + z = L ΛΓq + z, 4 where Λ C D+ D+ denoes Λ wih all he zero rows removed. To express he individual received daa sreams, 4 can alernaively be wrien componen wise using he elemens of he various vecors and marices. Then, we ge L d i N D y i = N r p d i qi + zi, i = 1,..., D L N D N r P u q i+z i, i=d+1,...,d+ 5 Semming from 5 and assuming a large ransmi anenna array [38], [4], he signal-o-inerference-plus-noise raio SINR of he ih DL signal can hen be expressed as follows for he full-duplex scheme: E SINR d,fd i = E = [ y i z i ] E [ z i ] [ ] L d i N D N r p d i q i E [ z i ] L d i = pd i σn + L d i Pd +, i = 1,..., D U j=1 Lud ij pu j 6 where he power of he noise-plus-inerference erm z i has been expanded o reflec he various componens, = N N r D, and he res of he symbols are as defined in Table I. To furher illusrae he many symbols and parameers used hroughou his work, Fig. provides also a visual depicion of heir meaning wihin he considered sysem. Similarly, he SINR of he backhaul signal sreams ransmied by he, used for backhauling he UL daa, can be wrien as SINR u,fd = L Pu σn + α N Pd + U j=1 Lu j p u j, 7 wih he symbols again defined in Table I and illusraed in Fig.. The SINRs of he signals received by he can be derived in an essenially similar manner as hose of he ransmi signals cf. [38], and hence heir deailed derivaion is omied for breviy. In paricular, he SINR of he jh UL signal can be shown o read: SINR u,fd j = r L u j pu j σ n + P u + D i=1 pd i, j = 1,..., U, where again he symbols are as defined in Table I, and r = N r U r. Correspondingly, he SINR of he backhaul signals received by he, backhauling he DL daa, is as follows: SINR d,fd = r r L P d [σ n + P u + D i=1 pd i 8 ]. 9 The hereby obained SINR expressions can hen be used o deermine he achievable raes of he full-duplex scheme. In paricular, using 6, he DL sum-rae of his communicaion

7 6 N /N r anennas P u sreams L α N p d i r L sreams P d L d i L u j p u j L d i L u j N L ud ij jh UL U in oal ih DL D in oal Fig. : An illusraion depicing he relevan symbols. scheme can be expressed as follows: R d = D Ri d, i=1 Ri d = log 1 + SINR d,fd i Λ = log 1 FD L d i + pd i σn + L d i Pd +, 10 U j=1 Lud ij pu j I can be observed ha, in his communicaion scheme, he DL daa rae is degraded by he UL-o-DL IUI and by he inerference produced by he N ransmission. Similarly, using 8, he oal UL daa rae can be wrien as: R u = U Rj u, j=1 Rj u = log 1 + SINR u,fd j = log 1 + r L u j pu j σ n + P u + D i=1 pd i, 11 where he SINR is now degraded by he residual SI wihin he. Noe ha his work does no assume any specific SI cancellaion performance since all he derivaions are done for an arbirary amoun of SI suppression, consising of passive anenna isolaion, ZF beamforming a he ransmi side o form nulls a he receive anennas, and oher possible SI cancellaion mehods. Since he mus be capable of backhauling all he daa, he backhaul daa raes mus also be aken ino accoun in he analysis. Wih he help of 7, he daa rae of he backhaul signal ransmied by he for backhauling UL daa can be expressed as follows: R u, = log 1 + SINR u,fd k=1 = Λ log 1 FD L Pu / + σn + α N Pd + U j=1 Lu j p u j, 1 which is affeced by he residual SI wihin he N, and also by he inerference caused by he UL ransmissions. In a similar fashion, using 9, he daa rae of he received backhaul signal sreams in he for backhauling DL daa can be wrien as follows: r R d, = log 1 + SINR d,fd l=1 = r log 1 + σn + r L Pd / r P u + D i=1 pd i. 13 The daa rae of he received backhaul signals is decreased by he residual SI wihin he, similar o he UL signals. Pu ogeher, he daa rae expressions in can be used o deermine he opimal ransmi powers for he considered sysem under some given daa rae requiremens, as will be done in Secion III.. Half-duplex Scheme Perhaps he mos obvious alernaive o he aforemenioned full-duplex scheme is for all he s o operae in half-duplex mode. This can be done by uilizing ime-division duplexing TDD where each wihin he considered sysem eiher ransmis or receives a any given ime, using all of he available specrum. In erms of he analyzed wih a wireless backhaul link, one possible TDD scheme is shown in he middle par of Fig. 1. There, he sysem has wo differen ime slos: one where he ransmis daa o he N and o he s, and one where

8 7 i receives daa from hem. As can easily be observed from Fig. 1, his ype of a scheme requires only half-duplex capable s since none of hem have o engage in simulaneous ransmission and recepion. This removes he problems of SI and UL-o-DL IUI a he cos of decreased specral efficiency since he now requires more emporal resources o carry ou he same asks in comparison o he full-duplex scheme. Alhough he deailed derivaions mus be omied for breviy, he SINRs for his ype of a half-duplex scheme can be obained in a similar manner as done for he full-duplex scheme above, and he corresponding DL sum-rae of he half-duplex sysem can be expressed as follows: D R d = Ri d, i=1 Ri d = η log 1 + SINR d,hd i = η log 1 + ΛHD L d i pd i, 14 where he symbols are as defined in Table I and Λ HD = N D. The relaive lenghs of he wo ime slos are conrolled by he duplexing parameer η, which defines he proporion of ime spen in he DL ime slo he relaive lengh of he UL ime slo being 1 η. The corresponding UL sum-rae can hen be wrien as: U R u = Rj u, j=1 Rj u = 1 η log 1 + SINR u,hd j = 1 η log 1 + ΛHD r L u j pu j, 15 where Λ HD r = N r U r. Hence, as can be observed, he daa raes are decreased due o ime division, bu he DL and UL ransmissions in he half-duplex scheme do no suffer from any form of inerference. Furhermore, no degrees-of-freedom are los due o having o null he receive anennas. The backhaul daa raes in half-duplex mode can hen be expressed as σ n R u, = η log 1 + SINR u,hd k=1 = η log 1 + ΛHD L Pu, 16 σn r R d, = 1 η log 1 + SINR d,hd l=1 σ n = 1 η r log 1 + ΛHD r L Pd r σn. 17 ased on 14 17, an inuiive inerpreaion regarding he relaionship beween he daa raes of he full-duplex and halfduplex schemes is ha i highly depends on he level of he oal inerference. Wih low pah loss beween he UL and DL s and/or poor SI cancellaion performance, he halfduplex scheme is likely o be he beer opion due o i being immune o he inerference. On he oher hand, if he is capable of efficienly suppressing he SI signal and he s do no srongly inerfere wih each oher or wih he N, he full-duplex scheme will likely provide he higher performance. These aspecs are invesigaed furher in Secion V wih he help of numerical resuls. C. Hybrid Relay Scheme In addiion o he above exreme cases of purely full-duplex and half-duplex sysems, an ineresing scheme is a full-duplex relay-ype, which simply relays he UL signal o he N during one ime slo, and hen in he oher ime slo relays he signal from he N o he DL s. The boom par of Fig. 1 illusraes his ype of a soluion. The benefi of his scheme is ha he problem of UL-o-DL IUI is compleely avoided, similar o he half-duplex scheme, while he fullduplex capabiliy of he is sill uilized o some exen as he relaying is performed inband. The obvious drawback is, however, ha now everyhing canno be done simulaneously, which will inherenly decrease he achievable rae. Furhermore, he relay scheme sill suffers from he inerference beween he N and he s. Also he SINRs of his ype of a hybrid relay scheme can be derived in a similar manner as hose of he full-duplex scheme in Secion II-A, he DL sum-rae being now R d = D Ri d, i=1 Ri d = η log 1 + SINR d,rl i = η log 1 + ΛRL σn + L d L d i pd i i Pd, 18 where Λ RL = N D N r. The UL sum-rae can correspondingly be wrien as: R u = U Rj u, j=1 Rj u = 1 η log 1 + SINR u,rl j = 1 η log 1 + ΛRL r L u j pu j σn + Pu, 19 where Λ RL r = N r U. Now, here is sill some inerference, which degrades he DL and UL SINRs, bu his scheme can be considered a rade-off beween oleraing inerference and duplexing he ransmissions and recepions in ime. Finally, he backhaul daa raes of he hybrid relay scheme can be expressed as follows: R u, = 1 η log 1 + SINR u,rl k=1 = 1 η log 1 + ΛRL, L Pu / σn + U j=1 Lu j p u j, 0

9 8 r R d, = η log 1 + SINR d,rl l=1 = ηr log 1 + ΛRL r, L Pd / r σn D, 1 + i=1 pd i where Λ RL, = N N r and Λ RL r, = N r r are he degrees-of-freedom of he ransmier and receiver for backhauling daa in he hybrid relay scheme, respecively. Again, a crucial aspec of he considered cell is ha he backhaul link should be able o mach he daa raes of UL and DL. Oherwise he sysem will be bolenecked by he backhaul connecion. III. TRSIT SU-POWER INIIZATION UNDER RATE CONSTRAINTS Nex, he problem of minimizing he ransmi powers of he sysem under some given daa rae requiremens for he DL and he UL is invesigaed. In paricular, le us define he per- QoS requiremens in erms of he minimum daa raes as ρ d and ρ u for he DL and he UL, respecively. This resuls in he following opimizaion problem. Problem Transmi Sum-Power inimizaion: 1 T D+U p + Pd + Pu minimize p, Pd, P u subjec o C1: Ri d ρ d, i = 1,..., D, C: Rj u ρ u, j = 1,..., U, D C3: R d, Ri d, C4: R u, i=1 U Rj u, j=1 where p = [ ] p T d p T T u, pd and p u are column vecors conaining he DL and UL ransmi powers p d i and pu j sacked, respecively, and 1 N is a column vecor consising of N ones. The consrains C1 and C ensure he QoS of he s, while he consrains C3 and C4 ensure sufficien backhauling capabiliy in he. This opimizaion problem will nex be solved separaely for he hree considered communicaion schemes, while he associaed infeasible sysem scenarios and QoS requiremens ha manifes hemselves as negaive ransmi power values in he following heorems are characerized laer in Secion IV. A. Full-Duplex Scheme Theorem 1: The opimal DL and UL ransmi powers for he full-duplex scheme are [ ] p p = d p u σ n γ d q d + σ = n 1+Sd γ d α +γ θ1 γ σ 1+Sd n γu γ d α +γ θ1 γ γ d γ r q /d + γ dγ u q u L d ud q u, 3 when each elemen of p is posiive and finie. Oherwise he QoS requiremens canno be fulfilled and he sysem is infeasible. Here, γ d = αρ d 1, γ u = αρu 1, Dρ d / r 1 γr = r r L, γ = Uρu/ 1 L, γ = α N γ γr, {q d } i = 1/L d i, {q u} j = 1/L u j, { q /d }i = /L d i, {L u} j = L u j, {L ud } ij = L ud ij, Ld ud = L ud q d 1 T U, L d i r and denoes he Hadamard produc. Noe ha q d, q u, q /d, and L u are column vecors. Furhermore, he parameer θ is defined as θ = 1 γ dγ r S /d 1 γ γ uγ S /u 1 γ γ dγ u S ud 1 γ, 4 where S d = 1 T D q d, S /d = 1 T D q /d, S /u = L T u q u, and S ud = 1 T D Ld ud q u. The opimal backhauling powers Pd and Pu hen follow direcly from p as shown below, viz. 9, 30. Proof: In order o arrive a he above closed-form soluion, le us firs rewrie Consrains C1 and C from in erms of he individual UL and DL ransmi powers as follows: ρ d 1 σ p d n + L d i Pd + U j=1 Lud ij pu j i L d, 5 [ i σn + Pu + ] D i=1 pd i ρu 1 p u j r L u j. 6 inimizing p d i and p u j by seing hem equal o heir lower bounds, we can wrie Ri d = ρ d i and Rj u = ρ u j. Hence, he backhauling consrains become R d, Dρ d and R u, Uρ u. Uilizing 1 and 13, he following lower bounds for he backhaul-relaed ransmi powers are obained: ] D Pd γr [σ n + Pu + p d i, 7 P u γ σ n + α N P d + i=1 U j=1 L u j p u j. 8 These ransmi powers are also minimized by seing hem equal o heir lower bounds. Solving hen for Pd and P u in erms of p d i and pu j from 7 and 8, we ge: Pd = γr D 1 γ p d i + γ γ U r 1 γ L u j p u j i=1 j=1 1 + α γ γ + r σn 1 γ, 9 D U Pu = γ 1 γ p d i + γ 1 γ L u j p u j i=1 j=1 1 + αn γr γ + σn 1 γ. 30 Then, by subsiuing 9 ino 5 and 30 ino 6, we ge he following sysem of D + U equaions wih D + U unknown ransmi powers: W FD p = v FD, 31

10 9 where, by denoing a N N ideniy marix by I N, W FD can be wrien in blockwise form as W FD = I D γ dγr q /d1 T D 1 γ γ r γ γ dq /d L T u 1 γ γ dl d ud, γuqu1t D 1 γ I U γuγ qult u 1 γ 3 while he vecor v FD is defined as follows: γ d σ n α v FD = q d + γ dγ r σ n 1 1 γ + γ γ uσ n 1 + γ q u 1 γ q /d The soluion o 31 can hen simply be obained by. 33 p = W 1 FD v FD. 34 To express he inverse of he marix W FD, le us firs wrie i as a sum of hree marices as follows: where W FD = I D+U + FG + H, 35 F = γ dγ G = r 1 γ q /d γu 1 γ q u [ 1 T D 0 T U 0 T D LT u ] γ r γ γ d 1 γ [ ] 0D D γ d α H = L d ud. 0 U D 0 U U q /d γuγ 1 γ q u Here, 0 N refers o a column vecor consising of N zeros, while 0 N refers o an N marix consising of all zeros. Now, he inverse can be wrien as follows: W 1 FD = I D+U + H + FG 1 = I D+U + H 1 I D+U + H 1 F 1 I + G I D+U + H F 1 G ID+U + H 1, 36 where he laer form is obained using he Kailah Varian [43]. The inverse of he marix I D+U + H can easily be obained as: [ ] I D+U + H 1 ID D γ 1 d α = L d ud 0 U D I U U [ ] γ ID D d α = L d ud. 0 U D I U U Furhermore, since I + G I D+U + H 1 F is in fac a marix, is inverse can also be calculaed in a sraighforward manner. In paricular, we ge 1 I + G I D+U + H 1 F = 1 1 γuγ θ LT u qu 1 γ γ ul T u qu 1 γ γ r γ γ d1 T D q /d 1 γ 1 γ dγ r 1T D q /d 1 γ + γ dγ uγ 1T D Ld ud qu 1 γ γ dγ u1 T D Ld ud qu 1 γ where θ is he deerminan of he invered marix, and i is defined as shown in 4., Having now calculaed all he inverses in 36, he opimal ransmi powers can be expressed by using only vecor/marixmuliplicaions as follows: [ p = I D+U + H 1 I D+U + H 1 F I + G I D+U + H F ] 1 1 G ID+U + H 1 v FD, which, afer subsiuing he marices wih he corresponding expressions and manipulaing he equaion, resuls in 3. These DL and UL ransmi powers can hen be subsiued ino 9 and 30 in order o obain he corresponding opimal backhaul-relaed ransmi powers.. Half-Duplex Scheme Theorem : For he half-duplex scheme, he opimal ransmi powers in closed form are [ ] p p = d = p u P d = P u = ρ dη 1 σ n q Λ HD d 1 η 1 σ n Λ q HD u r Dρ d ρu, 37 r 1 η 1 r σn Λ HD, 38 r L Uρu η 1 σn Λ HD. 39 L The QoS requiremens can always be fulfilled and he sysem is always feasible. Proof: This analyical soluion is again obained by firs rewriing he Consrains C1 and C in erms of he DL and UL ransmi powers wih he help of 14 and 15 as follows: ρ d η 1 p d i p u j σ n Λ HD L d i ρu 1 η 1 Λ HD r L u j σ n, Again, hese ransmi powers are minimized by seing hem equal o he lower bounds, and consequenly he backhauling requiremens become R d, Dρ d and R u, Uρ u. These, ogeher wih 16 and 17, yield he following bounds for he backhaul-relaed ransmi powers: Dρ d r 1 η 1 Pd r σn Λ HD, 4 r L Uρu η 1 Pu σn Λ HD, 43 L which are also minimized by seing hem equal o heir respecive lower bounds.

11 10 Opimizing he Duplexing Parameer for he Half-duplex Scheme: In addiion, for he half-duplex scheme, also he duplexing parameer η should be opimized, since i direcly affecs he overall ransmi power. Having solved he opimal ransmi powers as shown above, hey can be used o formulae he opimizaion problem in erms of η as follows: minimize η S HD η, where S HD η = 1 T D+U p +Pd +Pu. 44 Using 37 39, he objecive funcion can be wrien as follows: ρ d η 1 σns d ρu 1 η 1 σns u S HD η = Λ HD + Λ HD r Dρ d r 1 η 1 r σn Uρu η 1 σn + Λ HD + r L Λ HD, L where S u = 1 T U q u. I is easy o show ha his is a convex funcion in erms of η, and hence is global minimum is found a he zero-poin of is derivaive. Hence, 44 is in fac equivalen o solving he following equaion: d dη SHD η = ln σ n Dρd + Λ HD r L ρ dη Sd ρ d Λ HD η + Dρ d r 1 η 1 η Uρu ρu Su ρ u 1 η Λ HD r 1 η Λ HD L Uρu η η = In principle, his can be inerpreed as he join opimizaion problem for he ransmi powers and he duplexing parameer. Namely, as he objecive funcion in 44 is consruced from he closed-form soluions of he minimum ransmi powers, which represen he opimal soluion for any given duplexing parameer, subsiuing he opimal value of η ino he expressions in gives direcly he ransmi power allocaions ha have been opimized wih respec o all he parameers. However, as can be easily observed, he opimal duplexing parameer does no have a closed-form soluion and herefore i is solved numerically for he resuls of Secion V. C. Hybrid Relay Scheme Theorem 3: The opimal DL and UL ransmi powers for he hybrid relay scheme are [ ] γ d σ n p p α = d p = q d + γ r 1+S dγ d 1 γ d γr S q /d /d, 46 u 1+αγ q u σ n γu 1 γ uγ S /u when each elemen of p is posiive and finie. Oherwise he QoS requiremens canno be fulfilled and he sysem is infeasible. Here, γ d =α ρ d /η 1, γ Λ RL u =α ρu/1 η 1 Λ, γ RL r = r r Dρ d / r η 1, and γ Λ RL r, L = Uρu/ 1 η 1. The Λ RL, L opimal backhaul-relaed ransmi powers again direcly follow from p as shown below, viz. 49, 50 wih equaliies. Proof: Following a similar procedure as in he full-duplex and half-duplex schemes, he firs sep in obaining he above closed-form soluion is rewriing he QoS consrains in as boundaries for he DL and UL ransmi powers using 18 and 19 as follows: ρ d σ η 1 n + L d p d i Pd i Λ RL L d, 47 i σ ρu 1 η 1 n + P p u u j Λ RL r L u. 48 j inimizing again hese ransmi powers by seing hem equal o heir lower bounds, he self-backhauling consrains become R d, Dρ d and R u, Uρ u. Hence, by using 0 and 1, we can wrie: P d P u γ r γ σ n + σ n + U j=1 D p d i i=1 L u j p u j, Seing also hese backhaul-relaed ransmi powers equal o heir lower bounds and subsiuing hem ino 47 and 48, we obain he following expressions for he individual ransmi powers: D p d i = γ dγr L d i L d i U p u j = γ uγ L u j l=1 k=1 σ nγ d p d k + L d i 1 + L d i γr, 51 L u l p u l + σ nγ u 1 + L u α γ. 5 j These can easily be rearranged ino a sysem of equaions for he unknown DL and UL ransmi powers as follows: W RL p = v RL, 53 where W RL can be wrien in blockwise form as [ ID γ d γr q /d 1 T D 0 D U W RL = 0 U D I U γ u γ q u L T u ], 54 and he vecor v RL is defined as follows: [ σ n γ ] d α qd v RL = + γr q /d σ. 55 n γu 1 + α γ qu The opimal ransmi powers are hen obained similar o he full-duplex scheme, i.e., from p = W 1 RL v RL, 56 which, due o he block diagonal naure of he marix W RL, can in fac be solved separaely for he DL and UL ransmi powers. Hence, he opimal DL ransmi powers are as follows: p d = I D γ d γr q /d 1 T 1 σnγ d D qd + γ r q /d = I D + γ dγr q /d 1 T D σnγ d 1 γ d γr 1 T D q qd + γ r q /d /d = σ nγ d q d + γ r 1 + S d γ d 1 γ d γr q /d, 57 S /d

12 11 where he marix inverse has been calculaed by using he Sherman-orrison formula [43]. The opimal UL ransmi powers are obained in an idenical manner, and hey read: p u = I U γ u γ q u L T 1 σnγ u u = σ nγ u 1 + α γ 1 γ u γ S /u q u 1 + α γ qu. 58 The opimal backhaul-relaed ransmi powers can be solved by subsiuing he opimal DL and UL ransmi powers ino he expressions in 49 and 50 wih equaliies. Opimizing he Duplexing Parameer for he Hybrid Relay Scheme: Similar o he half-duplex scheme, he soluion in 56 is for a given duplexing parameer η, and hus he ransmi powers of he hybrid relay scheme should be furher minimized also wih respec o η. This resuls in he following opimizaion problem: minimize η S RL η, where S RL η = 1 T D+U p +Pd +Pu. 59 In order o obain he expression of he objecive funcion, he overall DL ransmi power can firs be wrien as follows, based on 57: 1 T Dp d = σ nγ d = σ nγ d 1 T Dq d + γ r 1 + S d γ d 1 γ d γr 1 T S Dq /d /d Sd + γr S /d 1 γ d γr. 60 S /d Following a similar procedure, he sum UL ransmi power is obained using 58 as follows: 1 T U p u = σ nγ u Su 1 + α γ 1 γ u γ. 61 S /u Furhermore, in order o express he ransmi power used for backhauling he uplink daa Pu, he erm L T u p u mus be calculaed. Using an idenical procedure as in 61, i can be derived as follows: L T up u = σ nγ u S/u 1 + α γ 1 γ u γ. 6 S /u Having obained he expressions for he sum DL and UL ransmi powers, as well as for L T u p u, hey can be subsiued ino he expressions in 49 and 50 o solve he corresponding backhaul-relaed ransmi powers. Afer his, he objecive funcion can be wrien as follows: Sd + γ r S /d S RL η = σ nγ d α 1 + γ r 1 γ d γr + γr σn S /d Su + γ S /u + σ nγ u α 1 + γ 1 γ u γ + γ σ S n, 63 /u where he duplexing parameer η is conained in he erms γ d, γ u, γ r, and γ, as defined earlier. As is shown in Secion IV below where he feasibiliy of he hybrid relay scheme is discussed in more deail, he sysem is in fac feasible when 1 γ d γ r S /d > 0 and 1 γ u γ S /u > 0. Solving hese inequaliies in erms of η resuls in an open inerval wihin which he minimum poin is locaed, and i can be easily observed ha he funcion S RL η is also coninuous wihin his inerval. The above opimizaion problem wih respec o η can also in his case be inerpreed as he join opimizaion problem for he ransmi powers and he duplexing parameer since he resuling opimal value of η gives direcly he opimal values of he ransmi powers wih 46, 49, and 50. As here is again no closed-form soluion for he opimal η, in he forhcoming numerical resuls he opimal duplexing parameer is deermined by numerically opimizing 63 over he open inerval defined by he feasibiliy condiions. This can be done by uilizing any one-dimensional opimizaion procedure. IV. FEASIILITY ALYSIS OF FULL-DUPLEX D HYRID RELAY SCHEES The feasibiliy of he considered communicaion schemes can be deermined by invesigaing he resuling required ransmi powers. In paricular, if heir values are posiive and finie, he sysem is capable of fulfilling he QoS requiremens, while infinie or negaive ransmi powers in he above heorems naurally indicae ha he required daa raes canno be achieved. This sems from he physical inerpreaion of a ransmi power, which obviously canno be negaive. The half-duplex scheme does no suffer from any inerference sources, and hence i is feasible under all circumsances. In oher words, i can fulfill any QoS requiremens wih appropriaely high ransmi powers. However, boh he fullduplex and hybrid relay schemes have various inerference sources, which resul in a fundamenal upper bound for he achievable daa raes. We refer o his as he feasibiliy boundary, since i deermines wheher he whole sysem is feasible in he firs place. Essenially, his means ha he full-duplex and hybrid relay schemes have an upper bound for he DL and/or UL daa raes, which can be expressed as follows for he kh : R x k R x k,max p d 1,..., p d D, p u 1,..., p u U, P d, P u 0, where x = d and/or x = u. This means ha, if he DL/UL daa rae requiremen is higher han Rk,max x, he QoS requiremens canno be fulfilled for he kh, and consequenly he sysem is infeasible. Noe ha essenially his ype of a feasibiliy analysis considers a case where all he ransmi powers end owards infiniy, meaning ha he derived boundary condiions are very fundamenal in naure. Hence, he corresponding feasibiliy limis for resriced ransmi powers are somewha sricer. A. Feasibiliy of he Full-duplex Scheme Theorem 4: The feasibiliy condiion of he full-duplex scheme can be expressed as follows: γ d γr S /d 1 γ + γ uγ S /u 1 γ + γ dγ u S ud 1 γ < 1, 64 γ < 1,

13 1 where he firs condiion is simply θ > 0 rewrien in a slighly differen form. Proof: These feasibiliy condiions sem from he fac ha all he ransmi powers in 3 are posiive and finie under hese condiions. In paricular, if γ < 1, all he erms in 3, apar from θ, are always posiive. Then, when also θ > 0, all he ransmi powers are clearly posiive and he sysem is feasible. I is also eviden from 3 ha θ < 0 and γ < 1 resul in a leas he UL ransmi powers being negaive, while γ > 1 resuls in θ > 0, meaning ha he UL ransmi powers are negaive also in his case. This proves ha he sysem is infeasible if and only if he condiions in 64 do no hold. Corollary 1: For any ypical sysem parameers, he erm α N is exremely small, meaning ha usually θ > 0 is a sufficien feasibiliy condiion since γ 1. In fac, i can be assumed wih high accuracy ha γ 0, and he feasibiliy condiion in erms of he physical sysem parameers can consequenly be approximaed as: ρu 1 ρ d 1S ud r Dρ d + ρ d 1 r r L Uρu + ρu 1 1 S /u 1 r S /d r L < Since 65 is clearly a monoonic funcion of ρ d, he maximum suppored DL daa rae requiremen is obained by solving he roo of he above expression wih respec o ρ d. Due o he muliplicaions of he exponenial rae erms, here is no closed-form soluion for he roo, even if assuming ha ρ d 1 ρ d and Dρ d/ r 1 Dρ d/ r. Hence, he highes feasible DL daa rae requiremen of he full-duplex scheme is obained by solving he roo numerically. On he oher hand, when considering he required amoun of SI cancellaion o make he sysem feasible, a closed-form soluion can be easily obained from 65. In paricular, he minimum amoun of required SI cancellaion in decibels is as follows: α d < 10 log 10 Λ FD r L [ Uρu 10 log 10 ρu 1 1 S /u Dρ d + ρ d 1 r 1 r S /d + ρu 1 ρ d 1 L S ud ]. 66 In he numerical resuls, hese feasibiliy boundaries obained from he simplified expression in Corollary 1 are compared o he exac soluions defined in Theorem 4. The approximaed boundaries are shown o be highly accurae, which means ha Corollary 1 can be used o obain reliable informaion regarding he feasibiliy of a sysem uilizing he full-duplex scheme.. Feasibiliy of he Hybrid Relay Scheme Theorem 5: The hybrid relay scheme is feasible under he following condiions: ρ Dρ d d η 1 r η 1 r S /d ρu 0 < η < 1. Λ RL < 1, r, ΛRL L Uρu 1 η 1 1 η 1 S /u Λ RL < 1,, ΛRL r L 67 Proof: These condiions are obained by observing from 46 ha all he ransmi powers are posiive and finie when 1 γd γ r S /d > 0 and 1 γu γ S /u > 0, because all he variables hemselves are posiive. Furhermore, since he sum DL and UL ransmi powers in 60 and 61 are negaive when 1 γd γ r S /d < 0 and 1 γu γ S /u < 0, 67 represens indeed he exac feasibiliy condiion. When opimizing he duplexing parameer η, hese condiions can be used o deermine is upper and lower bound. In paricular, i can easily be shown ha he firs condiion is monoonically decreasing wih respec o η, while he second condiion is monoonically increasing. Hence, he firs inequaliy resuls in a lower bound for η, while he second inequaliy defines is upper bound. The sysem is hen feasible if here exiss a value for η which fulfills all of hese inequaliies. Since i is no possible o obain closed-form soluions for he upper and lower boundaries of η using he exac form of 67, he problem can be made analyically racable by assuming ha ρ d η 1 ρ d η, ρu 1 η 1 ρu Dρ d 1 η, r η 1 Dρ d Uρu Uρu r η, and 1 η 1 1 η. This approximaion is raher accurae wih any reasonable rae requiremens, and i represens a pessimisic esimae of he feasibiliy boundary, which is asympoically approaching he rue boundary when ρ d, ρ u. Now, he boundaries for η can be expressed as follows: ρ d + Dρ d r Λ RL r, log ΛRL L r S /d < η < 1 Noe ha we have assumed here ha Λ RL r Λ RL, L S /u ρ u + Uρu Λ. RL r Λ log RL, L S /u Λ RL r, ΛRL L r S /d > 1 and > 1, since his ensures ha he hird condiion, i.e., 0 < η < 1, is fulfilled. ecause hese inequaliies can be expeced o hold when considering any realisic sysem parameers, hey are no explicily analyzed in his aricle. Corollary : Noing ha, for a feasible sysem, he lower bound of η mus be sricly less han is upper bound, an approximaive feasibiliy condiion for he hybrid relay scheme can be expressed as ρ d + Dρ d r Λ RL r, log ΛRL L r S /d + ρ u + Uρu Λ < RL r Λ log RL, L S /u

14 13 TALE II: The essenial defaul sysem parameers. any of he parameer values are also varied in he evaluaions. Parameer Value No. of TX/RX anennas N /N r 00/100 No. of DL and UL s D = U 10 No. of DL/UL backhaul sreams r / 1/6 Receiver noise floor σ n SI cancellaion in he /N /α N Per- DL/UL rae requiremen ρ d /ρ u Cell radius Disance beween he and he N -90 dm 10/ 10 d 8/ bps/hz 50 m 75 m No. of one Carlo simulaion runs If he lower bound is equal o he upper bound, his means ha 1 γd γ r S /d 0 and 1 γu γ S /u 0, indicaing ha he required ransmi powers end o infiniy, and hence his condiion represens he feasibiliy boundary. Using 68, we can easily derive he boundary for he DL daa rae requiremen wih respec o he oher sysem parameers, and i is as follows: r, ΛRL L r S /d Λ RL log ρ d < 1 + D r 1 ρ u + Uρu Λ RL r Λ log RL, L S /u. 69 The minimum requiremen for SI cancellaion in he can also be wrien in closed form using 68. Expressing in decibels, i reads as follows: Λ RL α d r Λ RL Λ RL, < 5 log ΛRL r, L 10 r S /d S /u [ 5 ρ d + ρ u + Dρ d log 10 r + Uρ u [ + ρ d ρ u + Dρ d r + log Λ RL, ΛRL r Λ RL + 4 ρ d + Dρ d r Uρ u r S /d Λ RL r, S /u ρ u + Uρ u ]1 /]. 70 Noe ha solving 68 for requires solving he roos of a nd-degree polynomial, bu one of he wo soluions can easily be shown o resul in he duplexing parameer being ouside he open inerval 0, 1. Hence, here is only one valid soluion for he inequaliy. In Secion V, he above feasibiliy boundaries given by Corollary are shown o be very close o he exac feasibiliy boundaries given by Theorem 5. Hence, hese approximaive closed-form boundaries provide highly accurae resuls when deermining he feasibiliy of he hybrid relay scheme. V. NUERICAL RESULTS Nex, he proposed sysem is numerically evaluaed wih one Carlo simulaions. In paricular, we consider a cell of Cumulaive probabiliy RL, approximaive RL, exac FD, approximaive FD, exac ρ d /ρ u = {8/, 6/1.5, 4/1} inimum SI cancellaion requiremen a he, d Fig. 3: CDFs of he minimum SI cancellaion requiremen in he in he full-duplex and hybrid relay schemes, shown for differen DL/UL daa rae requiremens. a given size where he specified amoun of DL and UL s are randomly posiioned. y calculaing he opimal ransmi powers and he feasibiliy condiions for a large number of random posiions, he cumulaive disribuion funcions CDFs of he corresponding quaniies can hen be obained. The defaul sysem parameers, which are used unless oherwise menioned, are shown in Table II. The pah losses beween he differen paries are calculaed based on he disances in each random realizaion, using he measuremen-based pah loss model for a cener frequency of 3.5 GHz presened in [44] o reflec a concree pracical example; he line-of-sigh LOS model is adoped for he link beween he and he N, while he non-line-of-sigh NLOS model is adoped in all he oher cases. To ensure a pracical sysem, he scheduled DL and UL s are chosen from he opposie sides of he cell, which resuls in a smaller level of UL-o-DL IUI [41]. The s can hen alernae beween DL and UL modes a regular inervals, by which each ges served boh in he DL and in he UL, regardless of heir posiion in he cell. Furhermore, in order o faciliae a fair comparison beween he differen schemes, in he forhcoming figures he ransmi powers of he half-duplex and hybrid relay scheme are weighed by he proporion of ime spen in he corresponding ime slo deermined by he duplexing parameer η. For breviy, he full-duplex, half-duplex and hybrid relay schemes are referred o as FD, HD, and RL, respecively, in all he figures. A. Feasibiliy In order o firs analyze he feasibiliy of he full-duplex and he hybrid relay schemes, Fig. 3 shows he CDFs of he SI cancellaion performance required in he o make he sysem feasible. The figure shows boh he approximaed closed-form soluions given in 66 and 70 as well as he exac soluions obained from 64 and 67. Firsly, i can be observed ha he approximaed feasibiliy boundaries mach he exac boundaries very closely, indicaing ha he approximaions do no compromise he accuracy of he derived equaions. Furhermore, Fig. 3 indicaes ha he required SI cancellaion performance of he full-duplex scheme is less affeced by he daa rae requiremens han ha of he

15 Cumulaive probabiliy ρ u = {6, 4, } RL, approximaive RL, exac FD, approximaive FD, exac Cumulaive probabiliy FD, daa rae raio: 0.1 FD, daa rae raio: 0.5 FD, daa rae raio: 0.5 RL, daa rae raio: 0.1 RL, daa rae raio: 0.5 RL, daa rae raio: aximum DL daa rae requiremen ρ d, bps/hz aximum sum daa rae requiremen ρ d +ρ u, bps/hz Fig. 4: CDFs of he maximum suppored DL daa rae requiremen in he fullduplex and hybrid relay schemes, shown for differen UL daa rae requiremens. Fig. 5: CDFs of he maximum suppored sum daa rae requiremen in he full-duplex and hybrid relay schemes under differen fixed UL/DL daa rae raios. hybrid relay scheme. In paricular, wih he highes daa rae requiremens, he full-duplex scheme is feasible wih lower SI cancellaion performance han he hybrid relay scheme, while he opposie is rue for he lowes considered daa rae requiremens. In he laer case, he hybrid relay scheme benefis from he fac ha i only needs o ransmi o he s or o he N, unlike he full-duplex scheme which mus ransmi everyhing a he same ime. This resuls in less sringen SI cancellaion requiremens. However, wih he higher daa rae requiremens, his benefi is overshadowed by he need o perform ime-division duplexing. Anoher perspecive ino he feasibiliy is he highes suppored DL daa rae requiremen. The corresponding CDFs are ploed in Fig. 4, which again show he approximaed boundaries given by 65 and 69, alongside wih he exac feasibiliy boundaries obained from 64 and 67. Also now, he approximaed feasibiliy boundaries are essenially similar o he exac boundaries, furher confirming heir accuracy under he sudied condiions. I can also be concluded ha he fullduplex scheme can suppor a higher DL daa rae requiremen wih all he considered UL daa rae requiremens. However, i should be noed ha here is more uncerainy regarding he maximum suppored DL daa rae requiremen in he fullduplex scheme, since he slope of he CDF is lower han in he hybrid relay scheme. Hence, even hough he full-duplex scheme suppors a higher median DL daa rae requiremen, here is a higher probabiliy ha i canno fulfill ha for differen randomly chosen posiions in he nework. This indicaes ha here is a rade-off beween he maximum performance and robusness when comparing he full-duplex and hybrid relay schemes. The maximum suppored sum daa rae requiremen is hen analyzed in Fig. 5. There, he CDFs of he feasibiliy boundary are shown under a scenario where he raio beween he UL and DL daa raes is fixed, ha is, ρ u /ρ d = c for some consan c. In his case, he CDFs are only shown for he approximaed equaions in order o make he figure more readable. In general, he full-duplex scheme suppors also a higher median sum daa rae requiremen, alhough he uncerainy in he suppored daa rae requiremen is again somewha higher han in he hybrid relay scheme. I can also be observed from Fig. 5 ha he hybrid relay scheme suppors higher sum daa rae requiremens wih lower daa rae raios. This sems from he sysem parameers having been chosen o suppor a higher DL daa rae r > o reflec he daa raffic disribuion of a pracical nework [5], which resuls in he hybrid relay scheme benefiing from a DL-oriened daa rae disribuion. On he oher hand, he fullduplex scheme seems o be beer suied for a more even disribuion of he DL and UL daa rae requiremens, which is eviden from Fig. 5 when invesigaing he median values of he highes suppored sum-rae requiremens. This is due o he more symmeric naure of he full-duplex scheme since i has less opions for dividing he resources beween UL and DL. Hence, unlike he hybrid relay scheme, which has he benefi of a duplexing parameer, he full-duplex scheme requires a more even daa rae disribuion o suppor he highes sum-raes. Finally, Fig. 6 shows he probabiliy of feasibiliy wih respec o he number of s in he full-duplex and hybrid relay schemes, assuming D = U. The probabiliies have been obained by evaluaing he approximaed feasibiliy boundaries in 65 and 68 for differen numbers of s. Firsly, i can be observed from Fig. 6 ha he full-duplex scheme can in general fulfill he QoS requiremens for a larger number of randomly posiioned s, especially when he is capable of efficien SI cancellaion. Wih he lower SI cancellaion performances, he hybrid relay scheme is more evenly mached wih he full-duplex scheme, being again he more robus opion in erms of fulfilling he QoS requiremens. Namely, while he full-duplex scheme can in general suppor a larger number of s, he slope of he probabiliy curve is seeper wih he hybrid relay scheme, indicaing ha he laer is he more predicable opion when here is a moderae number of s in he cell. This somewha resembles he behaviour of he maximum suppored DL daa rae requiremens in Fig. 4. Neverheless, wih sufficienly high SI cancellaion performance, he full-duplex scheme is clearly he superior opion wih regard o he number of suppored s.

16 15 Probabiliy of feasibiliy FD, FD, FD, RL, RL, RL, = -110 d = -10 d = -130 d = -110 d = -10 d = -130 d Number of s D=U Fig. 6: The probabiliy of feasibiliy wih respec o he number of s U = D in he full-duplex and hybrid relay schemes, shown for differen SI cancellaion performances. Cumulaive probabiliy FD, ransmi power FD, ransmi power FD, N ransmi power RL, ransmi power RL, ransmi power RL, N ransmi power HD, ransmi power HD, ransmi power HD, N ransmi power Transmi power dm Fig. 7: CDFs of he ransmi powers of he individual paries wih he defaul sysem parameers.. Transmi Powers To hen invesigae he ransmi power efficiency of he differen communicaion schemes, he CDFs of he ransmi powers of he, each individual, and he N are firs shown in Fig. 7 using he defaul sysem parameers. I can be observed ha he full-duplex scheme obains he lowes ransmi powers for all he communicaing paries. However, he downside of he full-duplex scheme is is inabiliy o fulfill he QoS requiremens in some cases, evidenced by he fac ha he CDFs saurae o a value below 1. These cases represen he siuaions where he feasibiliy condiions in 64 are no fulfilled, and herefore he highes value of he CDF is in fac he probabiliy of feasibiliy of he corresponding sysem, illusraed also in Fig. 6 wih respec o he number of s. This deducion is furher confirmed by Fig. 3, which shows ha he SI cancellaion requiremen is indeed more han 10 d in some cases when ρ d = 8 and ρ u =. From he perspecive of he overall ransmi power consumpion, he hybrid relay scheme is hen he nex bes opion, while he half-duplex scheme ouperforms he hybrid relay scheme in erms of minimizing he ransmi powers. The reason for his sems from he fac ha in he half-duplex scheme he ransmissions occur in he same ime slo where he DL daa is backhauled. Due o he higher DL daa rae requiremens, his resuls in a somewha longer ime slo for he ransmissions, allowing for a lower ransmi power. Noe ha his occurs due o he opimal duplexing parameer being chosen by minimizing he oal ransmi power. A differen oucome would be obained if a -ransmi-powerminimizing duplexing parameer was used. Wha is more, he hybrid relay scheme also suffers from he inabiliy o fulfill he QoS requiremens under some circumsances, similar o he full-duplex scheme. To observe he effec of he SI cancellaion capabiliy of he, Fig. 8 shows hen he CDFs of he oal ransmi power of he whole radio access sysem for differen values of. Again, he ransmi power usage of he full-duplex scheme is significanly lower han ha of he oher schemes, regardless of he SI cancellaion performance. However, wih he lower values of, he probabiliy of fulfilling he QoS requiremens wih he full-duplex scheme drops raher low. This is also eviden from Fig. 3, where i can clearly be seen ha he SI cancellaion requiremen is beyond 110 d wih a large probabiliy when ρ d = 8 and ρ u =. Hence, he lower probabiliy of feasibiliy is he cos of he low ransmi power consumpion. The hybrid relay scheme also ouperforms he half-duplex scheme when is 10 d or beer, while i performs very poorly wih he lowes considered SI cancellaion performance. This is explained by he CDF of he SI cancellaion requiremen shown in Fig. 3, which indicaes ha he SI cancellaion requiremen of he hybrid relay scheme is in he majoriy of he cases more han 110 d. Sill, even wih = 10 d, he probabiliy of he hybrid relay scheme having o use more power for he ransmissions han he half-duplex scheme is raher high, suggesing ha i requires relaively high SI cancellaion performance in he in order o be a viable opion. To invesigae he effec of he cell size on he differen schemes, Fig. 9 shows he CDFs of he oal ransmi power for differen cell radii. Again, wih all considered cell sizes, he full-duplex scheme is he mos power-efficien opion, while he hybrid relay scheme and he half-duplex scheme are quie closely mached. Especially wih he larger cell sizes, heir median ransmi power usages are nearly he same. However, he hybrid relay scheme again suffers from he fac ha i canno fulfill he QoS requiremens for some posiions and hus, regardless of he higher median power, he half-duplex scheme migh be he more favorable opion of hese wo. On a more general noe, he cell size has a raher significan impac on he required ransmi power, as can be expeced. For insance, he oal median ransmi power of he fullduplex scheme is increased by almos 0 d when he cell radius is increased from 5 m o 75 m. oreover, wih he highes considered cell radius of 75 m, he full-duplex and he hybrid relay schemes canno fulfill he QoS requiremens for a significan porion of he posiions. Hence, i can be concluded ha especially he schemes uilizing inband fullduplex communicaions are bes suied for relaively small cells.

17 Cumulaive probabiliy FD, = -110 d FD, = -10 d FD, = - d RL, = -110 d RL, = -10 d RL, = - d Toal ransmi power dm Fig. 8: CDFs of he oal used ransmi power of each scheme, shown for differen values of SI cancellaion. HD Cumulaive probabiliy FD, cell radius: 5 m FD, cell radius: 50 m FD, cell radius: 75 m RL, cell radius: 5 m RL, cell radius: 50 m RL, cell radius: 75 m HD, cell radius: 5 m HD, cell radius: 50 m HD, cell radius: 75 m Toal ransmi power dm Fig. 9: CDFs of he oal used ransmi power of each scheme, shown for differen cell radii. The disance beween he and he N is reained a 3 imes he cell radius. VI. CONCLUSION In his paper, we invesigaed a self-backhauling inband fullduplex access wih large anenna arrays, which can use he same ime-frequency resource for serving he mobile users as well as for backhauling, hereby significanly reducing he cos of deploymen in ulra-dense neworks. Three differen communicaion schemes for he access were analyzed: a purely full-duplex scheme, a purely half-duplex scheme, and a hybrid scheme where he access acs as a one-direcional full-duplex relay. Especially, we derived he opimal ransmi powers for he differen communicaion schemes in closed form when a QoS requiremen for each mobile user is given. In his work, QoS was defined as a minimum achievable daa rae. In addiion, we showed ha he QoS requiremens canno always be achieved when using a full-duplex-capable access, expressing his feasibiliy condiion also in closed form. Evaluaing hen he ransmi powers and feasibiliy condiions wih realisic sysem parameer values, i was observed ha having a purely full-duplex access provides usually he lowes ransmi powers for all communicaing paries. However, he downside of he purely full-duplex scheme is is inabiliy o fulfill he QoS requiremens under some circumsances, characerized by he closed-form feasibiliy condiions. 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Conf., Dec. 013, pp Dani Korpi S was born in Ilmajoki, Finland, in He received his.sc. and D.Sc. degrees boh wih disincion in communicaions engineering and elecrical engineering from Tampere Universiy of Technology, Finland, in 014 and 017, respecively. He is currenly a posdocoral researcher in he Laboraory of Elecronics and Communicaions Engineering a he same universiy. So far, he has auhored or co-auhored 37 refereed aricles and one book chaper. His curren research ineress include inband full-duplex radios, modeling and compensaion of analog circui impairmens in radio ransceivers, and 5G New Radio sysems. Taneli Riihonen S received he D.Sc. degree in elecrical engineering wih disincion from Aalo Universiy, Finland, in Augus 014. He is currenly an Assisan Professor a he Laboraory of Elecronics and Communicaions Engineering, Tampere Universiy of Technology, Finland. He held various research posiions a Helsinki Universiy of Technology and Aalo Universiy School of Elecrical Engineering from Sepember 005 hrough December 017. He was a Visiing Associae Research Scienis and an Adjunc Assisan Professor a Columbia Universiy, USA, from November 014 hrough December 015. He has been nominaed eleven imes as an Exemplary/Top Reviewer of various IEEE journals and is serving as an Edior for IEEE COUNICATIONS LETTERS since Ocober 014 and for IEEE WIRELESS COUNICATIONS LETTERS since ay 017. He received he Finnish echnical secor s award for he bes docoral disseraion of he year and he EURASIP es PhD Thesis Award 017. His research aciviy is focused on physical-layer OFDA, mulianenna, relaying and full-duplex wireless echniques wih curren ineres in he evoluion of beyond 5G sysems. Ashuosh Sabharwal S S 06 F 14 received he.tech. degree from Indian Insiue of Technology IIT Delhi, New Delhi, India, in 1993, and he.s. and Ph.D. degrees from The Ohio Sae Universiy, Columbus, OH, USA, in 1995 and 1999, respecively. He is currenly a Professor wih he Deparmen of Elecrical and Compuer Engineering, Rice Universiy, Houson, TX, USA. His research ineress include informaion heory, communicaion algorihms, and he experimen-driven design of wireless neworks. He was a recipien of he 1998 Presidenial Disseraion Fellowship Award and 017 Jack Neubauer emorial Award. ikko Valkama S S 15 was born in Pirkkala, Finland, on November 7, He received he.sc. and Ph.D. Degrees boh wih honors in elecrical engineering EE from Tampere Universiy of Technology TUT, Finland, in 000 and 001, respecively. In 00, he received he es Ph.D. Thesis -award by he Finnish Academy of Science and Leers for his disseraion eniled Advanced I/Q signal processing for wideband receivers: odels and algorihms. In 003, he was working as a visiing pos-doc research fellow wih he Communicaions Sysems and Signal Processing Insiue a SDSU, San Diego, CA. Currenly, he is a Full Professor and Laboraory Head a he Laboraory of Elecronics and Communicaions Engineering a TUT, Finland. His general research ineress include radio communicaions, communicaions signal processing, esimaion and deecion echniques, signal processing algorihms for flexible radios, cogniive radio, full-duplex radio, radio localizaion, and 5G mobile radio neworks.