FUll-duplex (FD) is a promising technique to increase

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1 1 Jont Relay-User Beamformng Desgn n Full-Duplex Two-Way Relay Channel Zhgang Wen, Shua Wang, Xaoqng Lu, and Junwe Zou arxv: v1 cs.it] 17 Oct 2016 Abstract A full-duplex two-way relay channel wth multple antennas s consdered. For ths three-node network, the beamformng desgn needs to suppress self-nterference. Whle a tradtonal way s to apply zero-forcng for self-nterference mtgaton, t may harm the desred sgnals. In ths paper, a desgn whch reserves a fracton of self-nterference s proposed by solvng a qualty-of-servce constraned beamformng desgn problem. Snce the problem s challengng due to the loop self-nterference, a convergence-guaranteed alternatng optmzaton algorthm s proposed to jontly desgn the relay-user beamformers. Numercal results show that the proposed scheme outperforms zero-forcng method, and acheves a transmt power close to the deal case. Index Terms Beamformng desgn, convex optmzaton, fullduplex, self-nterference, two-way relay channel. I. INTRODUCTION FUll-duplex FD s a promsng technque to ncrease the spectral effcency n relay systems. However, the performance of FD relay suffers from the self nterference SI 1]. Although natural solaton and tme-doman cancellaton can be appled for mtgaton of SI, measurements show that resdual SI stll exsts 2]. To ths end, null-space projecton usng multple antennas s proposed n one-way relay systems 3]. Furthermore, snce zero-forcng ZF may harm the desred sgnal, relay beamformng desgn based on mnmum mean square error MMSE s also proposed for one-way relays 4]. Very recently, FD two-way relay channel TWRC receves sgnfcant attentons 5], whle the majorty of lteratures focus on the sngle antenna scenaro 6]. To analyze the performance of mult-antenna FD TWRC, beamformng desgn based on ZF s proposed n 7]. However, t s known that the method n 7] s suboptmal due to the manually added ZF constrant. On the other hand, beamformng desgn whch reserves a fracton of SI can overcome the drawback of 7], and currently has not been dscussed n FD TWRC. Ths paper provdes the frst attempt to desgn such beamformers by solvng the sgnal-tonterference-plus-nose rato SINR qualty-of-servce QoS constraned beamformng desgn problem. Compared wth Copyrght c 2015 IEEE. Personal use of ths materal s permtted. However, permsson to use ths materal for any other purposes must be obtaned from the IEEE by sendng a request to pubs-permssons@eee.org. Ths work was supported by the NSFC Zhgang Wen, Xaoqng Lu and Junwe Zou are wth the Bejng Key Laboratory of Work Safety Intellgent Montorng, School of Electronc Engneerng, Bejng Unversty of Posts and Telecommuncatons, Bejng , P.R.Chna. Shua Wang correspondng author s wth the Department of Electrcal and Electronc Engneerng, The Unversty of Hong Kong, Hong Kong e-mal: swang@eee.hku.hk. Fg. 1. Tx Sgnal Tx SI Rx Self Interference Loop SI Rx User 1 Relay Staton User 2 System model of mult-antenna FD-TWRC. one-way relays, the consdered problem n FD TWRC nvolves a sngle beamformer servng multple users, and the MMSE based method n 4] s not applcable. Furthermore, due to the loop self-nterference 1], the relay transmt power and users SINRs are nonlnear fractonal, leadng to a challengng nonconvex problem. To solve the nonconvex problem, a convergence guaranteed alternatng optmzaton AO algorthm s proposed to dvde t nto four subproblems. In partcular, the subproblems of relay beamformers, user transmtters, and user recevers are solved usng successve convex approxmaton SCA, secondorder cone programmng SOCP, and MMSE crteron, respectvely. Besdes, a low-complexty ZF based desgn s presented as a benchmark. Fnally, numercal results show that the proposed algorthm outperforms exstng algorthms, and acheves a transmt power close to the deal case. II. SYSTEM MODEL AND PROBLEM FORMULATION We consder a FD-TWRC system consstng of a relay staton and two users. The relay, the frst user, and the second user are equpped wth M R,M 1,M 2 transmt antennas and N R,N 1,N 2 receve antennas, respectvely. As shown n Fg. 1, the two users have no drect lnk and wsh to exchange nformaton through the relay. Wth the help of FD, the communcaton only requres one transmsson phase, and all the nodes transmt ther sgnals smultaneously. We assume that the requred processng tme to mplement the FD operaton at relay s gven byτ-symbol duraton, whch s short compared to a tme slot 7]. At the tme nstance n, the symbol x n] wth E x n] ] = 1 s transmtted at the th user, through the correspondng beamformer f C M 1 wth f 2 = P. Meanwhle, the symbol x R n] s transmtted at the relay. Therefore, the receved sgnal rn] C NR 1 at relay can be expressed as rn] = H 1,R f 1 x 1 n]+h 2,R f 2 x 2 n]+h R,R x R n]+nn], 1 Tx SI Rx

2 2 where H,R C NR M represents the channel from to R, and H R,R C NR MR represents the resdual SI channel. The term nn] CN0,σ 2 I s the Gaussan nose at relay. Snce the receved sgnal rn τ] at tme nstant n τ s fltered by receve beamformer w H C 1 N and transmt beamformer v C MR 1, the transmtted sgnal at relay x R n] s of H R, can be obtaned wth channel recprocty. The CSI of H, can be obtaned from feedback of users. However, P1 s dffcult to solve because varablesv,w, f andu are coupled, and the terms of v, w are nonlnear fractonal. To decouple the varables v, w, {f } and {u }, we propose the AO algorthm 8] n the next secton. x R n] = vw H rn τ]. 2 Puttng equaton 1 nto 2, we have x R n] = v w H H R,R v r w H H 1,R f 1 x 1 n rτ τ] r=0 +H 2,R f 2 x 2 n rτ τ]+nn rτ τ]. 3 Observng that the power of x R n] n equaton 3 s fnte and w H H R,R v < 1, we derve the relay output power as: P r = Ex H R n]x Rn] = v H 1 w H H R,R v 2n v lm n 1 w H H R,R v 2 w H H 1,R f w H H 2,R f 2 2 +σ 2 w H w v H v w H H 1,R f w H H 2,R f 2 2 +σ 2 w 2 = 1 w H H R,R v 2, where the equalty n the thrd lne s due to the sum formula of geometrc sequence 1]. Based on 3, the receved sgnal y n] at the th user n the downlnk phase s y n] = H R, x R n]+h, f x n]+z n] = H R, v w H H 1,R f 1 x 1 n τ]+w H H 2,R f 2 x 2 n τ] +H R, v w H H R,R v r w H H 1,R f 1 x 1 n rτ τ] +H R, v r=1 w H H R,R v r w H nn rτ τ] r=0 4 +H 2,R f 2 x 2 n rτ τ] +H, f x n]+z n], 5 where H R, C N MR represents the channel from relay to user, and H, C N M represents the resdual SI channel at user. The term z n] CN0,σ 2 I s Gaussan nose at the th user. Snce the frst term n 5 s the desred sgnal and the resdual s the nterference, by applyng user recever u H C 1 N wth u = 1 to y n] n 5, the SINR γ at the th user can be expressed as equaton 6. To provde relable communcaton for both users at ther requred SINR threshold, we must have γ θ, where θ s the SINR requrement at the th user. On the other hand, havng the QoS requrement satsfed, t s crucal to reduce the total transmt power for cost reducton and envronment benefts. Therefore, we can wrte the SINR QoS constraned beamformng desgn problem as P1. The optmzaton problem P1 s carred out at the relay node. The channel state nformaton CSI ofh,r,h R,R s avalable at relay usng plots. The CSI III. JOINT RELAY-USER BEAMFORMER WITH ALTERNATING OPTIMIZATION A. Subproblem of v Ths subsecton dscusses the optmzaton of v when other varables are fxed. To smplfy the notaton, let g,r = w H H,R f, g, = u H H, f 2, g H R, = uh H R,, and g H R,R = w H H R,R. Then problem P1 reduces to P2 : mn g 1,R +g 2,R +σ 2 v H v v 1 gr,r H v 2 s.t. g 3,R g H R,v 2 +θ g, +σ 2 g H R,Rv 2 θ g 1,R +g 2,R +g 3,R g H R,Rv 2 g H R,v 2 7a +θ σ 2 w 2 g H R,v 2 +θ g, +σ 2, 7b g H R,R v 2 < 1. 7c Problem P2 s challengng because the objectve functon s quadratc fractonal and the constrants are quartc. To tackle t, we ntroduce a slack varable ξ v H v/1 g H R,R v 2, whch s equvalent to 1 g H R,R v 2 v H v ξi ] 0 8 accordng to Schur Complement Lemma 9]. Then the objectve functon 7a becomes g 1,R +g 2,R +σ 2 ξ, whch s lnear n ξ. To further transform 8 nto a lnear matrx nequalty LMI, we can ntroduce another slack varable µ such that 1 gr,r H v 2 µ, and ths nequalty s convex. On the other hand, we deal wth the constrant 7b as follows. Specfcally, rearrange the left hand sde of 7b as g 3,R gr, H v 2 +θ g, +σ 2 gr,r H v 2 = v H g 3,R g R, gr, H +θ g, +σ 2 g R,R gr,r H v. 9 } {{ } Φ Moreover, to deal wth the quartc term on the rght hand sde of 7b, ntroduce slack varable λ gr,r H v 2 gr, H v 2. Snce gr, H v 0, ths newly added nequalty can be recast as an LMI λ gr,r H v ] v H 0, 10 g R,R ρ where ρ s a slack varable wth g H R, v 2 ρ. Wth the above procedure, problem P2 s equvalently

3 3 γ v,w,{f,u } = u H H R,v 2 w H H 3,R f 3 2 u H H R,v 2 w H H R,R v 2 1 w H H R,R v 2 w H H 1,R f w H H 2,R f 2 2 +σ 2 uh H R,v 2 w 2. 1 w H H R,R v 2 + uh H,f 2 +σ 2 6 P1 : mn v,w,{f,u } v H v 1 w H H R,R v 2 w H H 1,R f w H H 2,R f 2 2 +σ 2 w 2 +f1 H f 1 +f2 H f 2 s.t. γ v,w,{f,u } θ, = 1,2, w H H R,R v 2 < 1, u = 1. transformed nto P2 : mn ξ v,ξ,µ,{ρ,λ } s.t. v H Φ v θ g 1,R +g 2,R +g 3,R λ 11a +θ σ 2 w 2 gr,v H 2 +θ g, +σ 2, 11b 1 g H λ R, ρ v 2, gr,r H v ] v H 0, 11c g R,R ρ ] µ v H 0, µ+ g H v ξi R,Rv d Problem P2 s stll nonconvex due to term v H Φ v n 11b and 1/ρ n 11c. However, we can apply SCA to construct a lnear approxmaton for them. In partcular, defne Υ v := v H Φ v, and ρ := 1/ρ. Assumng that the soluton at the n th teraton s gven by v n],{ρ n] } 1, now defne functons Υ n] v = 2Rev n] H Φ v] v n] H Φ v n], n] ρ = 2 ρ n] 1 ρ n] 2ρ, 12 and the followng property can be establshed. Υ n] n] Property 1. The functons and satsfy the followng: n] n] n] Υ v Υ v, ρ ρ ; Υ v n] = Υ v n] n], ρ n] = ρ n] ; and v v ρ ρ Υ n] n] v=v n] ρ=ρ n] = Υ v v = ρ ρ v=v n], ρ=ρ n] Proof: See Appendx A. Wth the result of Property 1, the followng problem s consdered at the n+1 th teraton: s.t. P2 n+1] : mn ξ v,ξ,µ,{ρ,λ } Υ n] θ g 1,R +g 2,R +g 3,R λ +θ σ 2 w 2 gr, H v 2 +θ g, +σ 2, n] gr,v H 2 λ, gr,r H v ] v H 0,, 11d. g R,R ρ 1 The superscrpt n] n ths subsecton refers to the ndex of nner SCA teraton, and the ntal v 0],{ρ 0] } s obtaned from last teraton of AO.. Problem P2 n + 1] s a semdefnte programmng SDP, whch can be optmally solved by CVX, a Matlab-based software for convex optmzaton 10]. Denotng the soluton of P2 n+1] as v,ξ,µ,{ρ,λ } and settng vn+1] = v and ρ n+1] = ρ, we can proceed to solve P2 n + 2]. Accordng to Property 1 and 11, Theorem 1], the teratve algorthm s convergent, and the converged pont would be a local optmal soluton for P2 based on 12]. B. Subproblem of w Ths subsecton dscusses the optmzaton of w when other varables are fxed. Let q,r = H,R f, q, = u H H,f 2, q R, = u H H R,v 2, and q R,R = H R,R v. Then problem P1 reduces to P3 : mn w v 2 wh q 1,R 2 + w H q 2,R 2 +σ 2 w H w 1 w H q R,R 2 s.t. q R, w H q 3,R 2 q R, w H q R,R 2 2 w H q 3,R 2 +θ w H q j,r 2 j=1 +θ σ 2 q R, w 2 +θ q, +σ 2 1 w H q R,R 2, w H q R,R 2 < 1. Problem P3 has the same structure wth P2, and we can fnd the local optmal soluton of w. Due to space lmtaton, the detals are omtted here. C. Subproblem of f Now suppose that v,w and u are gven. Let a H,R = w H H,R, a H, = uh H,, a R, = u H H R,v 2, and a R,R = w H H R,R v 2. Then problem P1 reduces to P4 : mn {f } v 2 2 a H 1 a j,rf j 2 +σ 2 + R,R j=1 s.t. 1 a R,R a R, a H 3,R f 3 2 θ a R, a R,R a H 1,Rf a H 2,Rf f j 2 j=1 +θ 1 a R,R a H,f 2 +σ 2 +θ σ 2 a R,,. The objectve functon of P4 s convex quadratc wth respect to f. On the other hand, by takng the square root of the constrants of P4 on both sdes and replacng a H 3,R f 3

4 4 wth Rea H 3,R f 3 accordng to phase rotaton 13], the constrants of P4 can be equvalently reformulated as 1 a R,R a R, Rea H 3,R f 3 θ a R, a R,R a H 1,R f θ a R, a R,R a H 2,R f θ 1 a R,R a H, f 2 + θ 1 a R,R σ 2 +θ a R, σ 2 2 1/2,. 15 Wth constrant 15, problem P4 can be reformulated as an SOCP and solved by CVX 10]. D. Subproblem of u Suppose thatv,w andf are gven. Snceu s only nvolved n γ, the optmal u can be derved by maxmzng γ n 6. Specfcally, dvdng u H H R,v 2 by the numerator and denomnator of 6, the optmal u s the MMSE recever 14] wth the followng closed form: u = β σ 2 I N +H, f f H H H, 1HR, v, 16 where I N s an N N dentty matrx, β s a normalzaton coeffcent such that u = 1. E. Convergence and Complexty Analyss Based on the solutons for subproblems, we develop the followng teratve algorthm for problem P1. That s, ntalzng the varables as v 0],w 0],{f 0],u 0] } based on Appendx B, and then optmzng v,w,f,u alternatvely accordng to Secton A-D. Snce the proposed method s an nexact AO algorthm subproblems of P2 and P3 are not solved optmally, the convergence s not straghtforward. To ths end, we establsh the followng property. Property 2. The proposed AO algorthm s convergent. {u m 1] Proof: At the m th teraton, gven w m 1],{f m 1] }, and }, we updatev m] usng SCA. Snce the ntal pont s v m 1],{ρ m 1] = h R, v m 1] 2 }, and SCA s guaranteed to produce a monotoncally decreasng sequence 11], the objectve value correspondng to v m],w m 1],{f m 1],u m 1] },u m 1] s no larger than that of v m 1],w m 1],{f m 1] }. After we obtan v m], update w m] by solvng problem P3 wth SCA. Smlarly, the objectve value correspondng to v m],w m],{f m 1],u m 1] } would be no larger than that of v m],w m 1],{f m 1],u m 1] }. Fnally, we update f m] and u m] based on SCOP n Secton C and MMSE n Secton D. Snce the two subproblems are optmally solved, the objectve value correspondng to v m],w m],{f m],u m] } would be no larger than that of v m],w m],{f m 1],u m 1] }. Therefore, the whole AO procedure wll produce a monotoncally decreasng sequence of the objectve values, and the sequence s lower bounded by zero, whch proves the convergence of the AO algorthm. In terms of complexty, the AO algorthm s domnated by the subproblems of v and w. Specfcally, solvng the problem P2 n + 1] requres complexty OM R ]. For p 1 teratons of SCA, the complexty of solvng P2 would be Op 1 M R , and that of solvng P3 would be Op 1 N R Therefore, the total complexty s O p 2 p 1 M R p 1 N R ], where p 2 s the number of teratons for AO. F. Low-Complexty Zero-Forcng Desgn Ths subsecton presents a low-complexty ZF based algorthm as a benchmark, whch s a generalzed verson of 7]. Specfcally, the users apply u H HH, f = 0 to cancel the SI, and the relay apples w H H H R,Rv = 0 to cancel the loop SI. Therefore the problem P1 reduces to P5 : mn v,w,{f,u } vh v w H H 1,R f w H H 2,R f 2 2 +σ 2 w 2 +f H 1 f 1 +f H 2 f 2 s.t. u H H R, v 2 w H H 3,R f 3 2 θ σ 2 u H H R,v 2 w 2 +σ 2,, u H H,f = 0, u = 1,, w H H R,R v = 0. Followng the aforesad AO algorthm, P5 can be dvded nto four subproblems as well, and all the subproblems have closedform optmal solutons. Specfcally, the closed-form solutons of w and v are gven n 7, Secton III-A] and 7, Secton III-B], respectvely. To derve the closed-form soluton of{f }, applyng sngular value decomposton SVD to H H, u, we have H H, u = z Z ]Λ V H, where Z C M M 1. Usng the null space matrx Z, we perform SVD w H H,R Z = A B c C ] H, wherec C M 1 1. Then, the optmal ZF user beamformer s f = α Z c, where θ 3 σ 2 u H 3 HH R,3 v 2 w 2 +σ 2 α = u H 3 H. R,3 v 2 w H H,R Z c 2 The closed-form soluton of {u } can be smlarly derved usng the null space of H, f and range space of H R, v. Snce all the subproblems of P5 have closed-form optmal solutons, wth p 2 teratons of AO, the total complexty s O p 2 MR 3 +M3 1 +M3 2 +N3 R +N3 1 +N3 2 ]. IV. SIMULATION RESULTS In ths secton, smulaton results are provded to verfy the proposed algorthm. The number of transmt antennas s set as M R,M 1,M 2 = 4,2,2 and the number of receve antennas s set as N R,N 1,N 2 = 2,2,2. All the channel entres are generated accordng to CN0, ρ, wth large scale fadng ρ = It s assumed that the resdual SI channels are further multpled by SI coeffcent κ = 0.1 7], and nose power σ 2 = 30dBm. The same SINR QoS targets θ 1 = θ 2 = θ n db are requested by all users. Each pont n the fgures s obtaned by averagng over 100 smulaton runs, wth ndependent channels n each run. The followng schemes are smulated: the proposed FD AO scheme; the deal FD scheme wth no SI solvng P5 wthout ZF constrants; the ZF-based FD AO scheme 7]; the FD baselne scheme n Appendx B; the half-duplex

5 5 Total Transmt Power dbm Total Transmt Power dbm Half duplex baselne scheme Half duplex AO scheme FD baselne scheme ZF based FD AO scheme Proposed FD AO scheme Ideal FD wthout SI SINR QoS θ db a ZF based FD AO scheme Proposed FD AO scheme Number of Iteratons b Fg. 2. Total transmt power for the case of M R,M 1,M 2 = 4,2,2 and N R,N 1,N 2 = 2,2,2 a Versus θ at nose power σ 2 = 30dBm; b Versus number of teratons at θ = 10dB and σ 2 = 30dBm. AO scheme; and the half-duplex baselne scheme. For a far comparson, the SINR target θ for half-duplex schemes s θ = 1+θ 2 1 snce the requred data-rate for half-duplex schemes s 2log1+θ. We frst analyze the total transmt powers of dfferent schemes versus SINR target θ. As shown n Fg. 2a, the proposed FD AO scheme sgnfcantly outperforms the exstng schemes, and approaches the deal case very tghtly over a wde range of SINR target θ. In partcular, t saves 5dB transmt power compared to the ZF-based FD AO scheme, whch demonstrates the advantage of reservng a fracton of SI. Besdes, the FD schemes generally outperform the halfduplex schemes due to the reduced number of tme slots, and are less senstve to θ. To verfy the convergence of the proposed FD AO algorthm, Fg. 2b shows the total transmt power versus number of teratons when θ = 10dB. It can be seen that whle startng from the same ntal pont, the proposed FD AO algorthm shows much better performance than the ZF-based scheme after only 1 teraton. Furthermore, the number of teratons needed for the proposed FD AO algorthm to convergence s smaller than 10, whch ndcates fast convergence and moderate complexty of the proposed algorthm. V. CONCLUSION In ths paper, we proposed a jont desgn of relay-user beamformers n FD-TWRC whch reserves a fracton of SI. The local optmum was obtaned for relay beamformers, and the global optmum was obtaned for user beamformers. Furthermore, the alternatng optmzaton algorthm was proved to be convergent. Smulaton results ndcated that the proposed algorthm outperforms exstng algorthms, and s close to the deal case. APPENDIX A PROOF OF PROPERTY 1 To prove part, consder the followng nequaltes v v n] H 2 ρ gr, 0, ρ Then from 18, we further have ρ n] v H Φ v 2Rev n] H Φ v] v n] H Φ v n], ρ ρ n] ρ n] 2ρ, 19 whch mmedately proves part. Part can be easly verfed by substtutng v n],{ρ n] } nto the defnton of Υ n] n]. To prove part, we frst calculate the followng, dervatves: Υ n] / v = v n] H Φ ] T, Υ / v = v H Φ T 20 n] / ρ = 1 ρ n] 2, / ρ = 1 ρ Then by puttng v = v n],{ρ = ρ n] } nto the above equatons, the proof for part s completed. APPENDIX B THE INITIALIZATION OF v 0],w 0],{f 0],u 0] } To begn wth, set w 0] = 1/N r 1 Nr and u 0] = 1/N 1 N, where 1 N s ann 1 vector wth all the elements beng 1. The ntal v 0],{f 0] } can be obtaned from problem P5. Specfcally, the feasblty condton for f n P5 s gven by w 0] H H 3,R f 3 2 > θ σ 2 w 0] 2, u 0] H H, f = 0. Usng SVD H H, u0] = z Z ]Λ V H and w 0] H H,R Z = A B c C ] H, we set f 0] = 2θ3 σ 2 w 0] 2 w 0] H H,R Z c Z c. On the other hand, usng SVDH H R,R w0] = z R Z R ]Λ R VR H and u 0] 1 H H R,1 + u 0] 2 H H R,2 Z R = A R B R c R C R ] H

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