Lnköpng Unversty Post Prnt Cooperatve Beamformng for the MISO Interference Channel Johannes Lndblom and Eleftheros Karpds N.B.: When ctng ths work, cte the orgnal artcle. 9 IEEE. Personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths work n other works must be obtaned from the IEEE: Johannes Lndblom and Eleftheros Karpds, Cooperatve Beamformng for the MISO Interference Channel,, Proceedngs of the 6th European Wreless Conference (EW'). Postprnt avalable at: Lnköpng Unversty Electronc Press http://urn.kb.se/resolve?urn=urn:nbn:se:lu:dva-796
Cooperatve Beamformng for the MISO Interference Channel (Invted Paper) Johannes Lndblom and Eleftheros Karpds Communcaton Systems Dvson, Department of Electrcal Engneerng (ISY) Lnköpng Unversty, SE-8 8 Lnköpng, Sweden. {lndblom,karpds}@sy.lu.se Abstract A dstrbuted beamformng algorthm s proposed for the two-user multple-nput sngle-output (MISO) nterference channel (IFC). The algorthm s teratve and uses as barganng value the nterference that each transmtter generates towards the recever of the other user. It enables cooperaton among the transmtters n order to ncrease both users rates by lowerng the overall nterference. In every teraton, as long as both rates keep on ncreasng, the transmtters mutually decrease the generated nterference. They choose ther beamformng vectors dstrbutvely, solvng the constraned optmzaton problem of mzng the useful sgnal power for a gven level of generated nterference. The algorthm s equally applcable when the transmtters have ether nstantaneous or statstcal channel state nformaton (CSI). The dfference s that the core optmzaton problem s solved n closed-form for nstantaneous CSI, whereas for statstcal CSI an effcent soluton s found numercally va semdefnte programmng. The outcome of the proposed algorthm s approxmately Pareto-optmal. Extensve numercal llustratons are provded, comparng the proposed soluton to the Nash equlbrum, zero-forcng, Nash barganng, and mum sum-rate operatng ponts. I. INTRODUCTION The stuaton when two wreless lnks operate n the same spectrum, and create mutual nterference to one another, s well modeled by the nterference channel (IFC). Assocated wth any IFC there s an achevable rate regon, consstng of all pars of transmsson rates R (for lnk ) and R (for lnk ) that can be acheved, subject to constrants on the power used by the transmtters. The Pareto boundary of the rate regon s the part of the outer boundary consstng of rate ponts, where ncreasng R necessarly requres decreasng R and vce versa. It s generally desrable to operate at rate ponts that le on the Pareto boundary, such as the mum-sum-rate () pont and the Nash barganng soluton () []. In ths paper, we consder the two-user multple-nput sngle-output (MISO) IFC, where the transmtters (TX and TX ) have multple antennas and the recevers (RX and RX ) have a sngle antenna each. By usng beamformng the transmtters are able to steer power n arbtrary drectons. On one extreme, when the transmtters do not cooperate, t s natural to act selfshly and use the mum-rato (MR) Ths work has been supported n part by the Swedsh Foundaton of Strategc Research (SSF). Ths work has been performed n the framework of the European research project SAPHYRE, whch s partly funded by the European Unon under ts FP7 ICT Objectve. - The Network of the Future. beamformng vector, whch mzes the useful sgnal power wthout takng nto account the nterference generated towards the other recever. Then, the outcome s the so-called n gametheoretc studes Nash equlbrum (), at whch none of the users can ncrease ts rate by unlaterally changng ts transmt strategy []. On the other extreme, the transmtters can be enforced by regulaton to act altrustcally and use the zero-forcng () beamformng vector, whch mzes the useful sgnal power wthout generatng any nterference towards the other recever. When both transmtters use the altrustc strategy, we wll refer to the correspondng rate par as the pont. In general, both the and the ponts le far nsde the Pareto boundary. Pareto-optmal (PO) operatng ponts can only be acheved by combnatons of the two aforementoned extreme strateges [] [4]. Ths s because the mzaton of the useful sgnal power and the mnmzaton of the generated nterference are conflctng objectves. It s evdent that the transmtters need to cooperate and agree to mutually decrease the generated nterference, n order to acheve larger rates than the. Heren, we propose a smple and self-enforcng algorthm for the dstrbuted desgn of the beamformng vectors, wth mnmum requred channel knowledge. We assume that the transmtters are synchronzed and that there are feedback channels from all the recevers to all the transmtters. Each transmtter has CSI only of the drect lnk to ts ntended recever and the crosstalk lnk to the other recever. The feedback channels are ntally used to provde channel state nformaton (CSI) and n the sequel nformaton about the algorthm evoluton. The proposed algorthm s teratve and uses as barganng value the level of generated nterference. It s natural to ntalze the algorthm wth the MR transmt strategy. Then, n every teraton, each transmtter wll decrease the upper bound on the generated nterference and dstrbutvely compute a new beamformng vector solvng the optmzaton problem of mzng the useful power, gven the nterference bound and the power constrant. Inevtably, n every teraton the optmal value ( useful sgnal) of each system decreases, snce the feasblty set of the optmzaton s constrcted. But, n return, the experenced nterference decreases too. The teratons contnue as long as the sgnal-to-nterferenceplus-nose rato (SINR), hence the rate, of both users benefts from them. The algorthm can be nterpreted as a walk from
selfsh towards altrustc choces of beamformng vectors. The barganng outcome s approxmately PO. When the sgnal-to-nose rato (SNR) s hgh and the spatal correlaton among the drect and crosstalk channels s strong, the pont corresponds to larger rates, than the,.e. t s closer to the Pareto boundary. In such a case, the algorthm wll converge to a soluton faster f t s ntalzed wth the transmt strategy. Then, the nterference levels need to be ncreased n every teraton, to expand the feasblty set, and the walk s from altrustc towards selfsh transmt strateges. The proposed algorthm can be equally used when the transmtters have ether nstantaneous CSI (.e., perfect knowledge of the channel vectors) or statstcal CSI (.e., knowledge of the channel dstrbutons). In the former case, the achevable regon s comprsed by nstantaneous rates, whereas n the latter by ergodc rates. We propose a generc formulaton of the beamformng problem as constraned optmzaton, whch s common for both CSI cases. For nstantaneous CSI, we solve the optmzaton n closed-form, usng the parameterzaton n []. For statstcal CSI, we fnd an effcent numercal soluton va semdefnte programmng (SDP), as descrbed n [4]. In ths paragraph, we summarze some known barganng algorthms and cooperatve beamformng solutons. In [6], the authors presented a barganng algorthm, smlar n sprt to the one we propose heren. Ther algorthm requres nstantaneous CSI, starts wth the MR strategy, and explots the parametrzaton n []. In every teraton, a porton of the strategy s added to the prevously computed strategy untl a stoppng crteron s met. Ths algorthm converges to an operatng pont whch s better than the. Compared to [6], the man contrbutons of our paper are the use of the generated nterference level as barganng value and the fact that the proposed algorthm works for both nstantaneous and statstcal CSI. In [7], the authors presented a soluton to the cooperatve beamformng problem n the case of nstantaneous CSI, usng the noton of vrtual SINR. The aforementoned algorthms were also extended n the context of multcell MIMO channels n [8]. We compare the outcome of our algorthm to the,,, and ponts, for dfferent cases of CSI, SNR values and spatal correlaton levels. Also, we provde exemplary llustratons of the barganng trajectory and dscuss the complexty of the algorthm. Notaton: Tr{ }, rank{ }, R{ }, and N{ } denote the trace, rank, range, and nullspace, respectvely, of a matrx. Π Z Z(Z H Z) Z H s the orthogonal projecton onto the column space of Z, whle Π Z I Π Z s the orthogonal projecton onto the orthogonal complement of the column space of Z and I s the dentty matrx. E{ } s the expectaton operator. A. System Model II. PRELIMINARIES We assume that transmsson conssts of scalar codng followed by beamformng and that all propagaton channels are Ths s optmal n the case of nstantaneous CSI, but not necessarly for statstcal CSI, see []. frequency-flat. The matched-fltered symbol-sampled complex baseband data receved by RX s modeled as y = h H w s +h H j w js j +e, j =,,j {,}, () where s CN(,) and w C n are the transmtted symbol and the beamformng vector, respectvely, employed by TX. Also, e CN(,σ ) models the recever nose. The (conjugated) channel vector between TX and RX j s modeled as h j CN(,Q j ). We denote r j rank{q j }. In the case of nstantaneous CSI, TX accurately knows the channel realzatons h and h j, whereas for statstcal CSI t only knows the channel covarance matrces Q and Q j. The transmsson power s bounded due to regulatory and hardware constrants, such as battery and amplfers. Wthout loss of generalty, we set ths bound to. Hence, the set of feasble beamformng vectors s W {w C n w }. () Note that the set W s convex. In what follows, a specfc choce of w W s denoted as a transmt strategy of TX. B. Instantaneous CSI When the transmtters perfectly know the channel vectors and the recevers treat nterference as nose, the achevable nstantaneous rate (n bts/channel use) for lnk s [] R (w,w j ) = log ( + hh w h H j w j +σ ). () It s evdent that the rate on each lnk depends on the choce of both beamformng vectors. We defne the power that RX receves from TX j as p j (w j ) h H jw j = w H j h j h H jw j. (4) Then, we can wrte () as ) p (w ) R (w,w j ) = log (+ p j (w j )+σ, () whch s monotonously ncreasng wth the useful sgnal power p (w ) for fxed receved nterference power p j (w j ) and monotonously decreasng wth p j (w j ) for fxed p (w ). The man goal of the barganng algorthm we ntroduce n Secton III s to agree on a PO soluton. Hence, we restrct our attenton to the beamformng vectors whch are canddates to acheve PO ponts. From [], we know that the PO beamformng vectors use full power and that they are lnear combnatons of the MR and strateges (λ ) = λ w MR λ w MR w PO for λ [,], where w MR = h h and +( λ )w +( λ )w w (6) = Π h j h. (7) Π h j h The outcome when both transmtters use ther MR strateges s the. When both use ther we refer to the pont. Whenever an expresson s vald for both systems, t s denoted once wth respect to system and the ndex j = refers to the other system.
C. Statstcal CSI When the transmtters only have statstcal knowledge of the channels, t s natural to desgn the achevable the beamformng vectors wth respect to the ergodc rates, whch are obtaned by averagng over the channel realzatons. From [], we have where R (w,w j ) E {log ( = p (w ) ln f (x) e σ /x + hh w h H j w j +σ )} f (p (w )) f (p j (w j )), p (w ) p j (w j ) e t σ /x t (8) dt. (9) In (8), p j (w j ) denotes the average power that RX receves from TX j p j (w j ) = E { w H j h j h H jw j } = w H j Q j w j. () Note that the fnal terms n both (4) and () are convex homogeneous quadratcs. The dfference s that the parameter (channel) matrx n (4) s rank- by defnton, whereas n () t can have any rank. The ergodc rate (8) has the same behavor as the nstantaneous rate (),.e., t s monotonously ncreasng (decreasng) wth p (w ) (p j (w j )) for fxed p (w ) (p j (w j )) []. Also, for ponts on the Pareto boundary we know that w R{Q,Q j } []. The MR strategy w MR s the domnant egenvector of Q []. When R{Q } R{Q j }, the strategy w s the domnant egenvector of Π N{Qj}Q Π N{Qj} and when R{Q } R{Q j }, e.g., when Q j s full-rank, then w = []. D. Important Operatng Ponts In the followng, we ntroduce some operatng ponts, whch are mportant n the sense that they le on the outer boundary of the rate regon; see, e.g., [] and references theren. Sngle-user (SU): The ponts acheved when one transmtter employs ts MR strategy whle the other refrans from transmsson. Maxmum sum-rate (): The pont where the sum of the rates s mum. Graphcally, t s the pont where a lne of slope touches the Pareto boundary of the rate regon. Nash barganng soluton (): The outcome of a Nash barganng s a pont ( R, R ) such that ( R R )( R R ) s mzed for some threat pont (R,R ) and R R. It s natural to use the as the threat pont, snce t s the only reasonable outcome f the systems are not able to agree on a soluton. The s only defned on convex utlty regons, but we wll call the soluton to the correspondng optmzaton problem the. We delberately use the same symbols, as n the case of nstantaneous CSI, to denote the rate and the power (R and p, respectvely) n order to facltate n the sequel a unform treatment of both CSI scenaros. III. COOPERATIVE BEAMFORMING ALGORITHM In ths secton, we elaborate the proposed barganng algorthm that enables the transmtters to dstrbutvely desgn ther beamformng vectors. We assume that there exsts a feedback lnk from every recever to every transmtter. The recevers use these lnks to feedback CSI. Each transmtter has CSI only on the lnks t s affectng. The transmtters are assumed synchronzed, but no nformaton (CSI or user data) s exchanged between them. In the algorthm, we use as barganng value an upper bound on the nterference generated by system to system j. Ths bound, denoted c j, s adjusted n every teraton. Durng the barganng, the recevers feed back a one-bt message that tells the transmtters whether the teraton was successful or not,.e. whether the rates ncreased or not. We denote l the teraton counter, whch also acts as a quanttatve measure of the overhead (total number of bts per RX-TX feedback lnk) and the computatonal complexty (total number of optmzaton problems that need to be solved). A flowchart of the algorthm s llustrated n Fg.. The frst step of the algorthm s the decson whether the ntalzaton pont wll be the or the pont. For ths reason, the transmtters send two plots usng ther MR and beamformng vectors. The recevers measure the SINR for each transmsson and feed back one-bt of nformaton tellng the transmtters whch strategy yelds hgher SINR, hence rate. If R (w,wj ) R (w MR,wj MR ) for both systems, the algorthm s ntalzed wth the pont, snce t s closer than the to the Pareto boundary. Hence, the algorthm wll requre fewer teratons to converge to a soluton. If only one system acheves hgher rate wth the strateges, there s no ncentve for the other to accept the pont as ntal pont. The algorthm s then ntalzed wth the pont. Then, the algorthm sets the stepsze for updatng c j. As wth any teratve algorthm, the best output s obtaned for an nfntesmal stepsze. However, ths s not practcal, so we consder nstead a fxed stepsze 4. We assume that TX samples the nterval [,p j (w MR )] unformly n N + ponts, to allow up to N teratons. Ths gves the step δ j = ±p j (w MR )/N. The sgn of δ j depends on the ntal pont. If the algorthm s ntalzed wth the, δ j wll be negatve (decreasng nterference). Otherwse, δ j wll be postve (ncreasng nterference). At teraton l, TX updates the nterference level as c l j = cl j +δ j and solves the problem p (w ) () w W s.t. p j (w ) c l j. () The optmal soluton of problem () () s the beamformng vector whch mzes the useful power gven that the generated nterference s c l j. As long as cl j s chosen n the range [,p j (w MR )], there always exsts a feasble soluton to () () [4]. The lower and upper end on the nterference level correspond to the and MR strateges, respectvely. 4 Also, an adaptve stepsze can easly be ncorporated to the algorthm.
Transmt two plots usng w MR and w are canddates for achevng PO ponts. Any other beamformng vector wll be a waste of power. Usng (6) we get No l = w = wmr c j = pj(wmr ) δ j = p j(w MR )/N R (w,w ) R (w MR,w MR ) R (w,w ) R(wMR,wMR ) w l l l+ c l j = cl j +δ j soluton of () () Transmt plot usng w l Yes l = w = w c j = δ j = p j(w MR )/N s.t. λ [,] h H wpo (λ ) () h H jw PO (λ ) = c j. (6) Note that the optmzaton () (6) s now only wth respect to the real scalar λ. Furthermore, the power constrant s obsolete, snce the PO beamformng vectors use full power. That s, the nequalty constrant n () s met wth equalty. Instead, we have a constrant on the range of the weghtng factor λ. Fnally, t s straghtforward to see that the objectve functon () s monotonously ncreasng wth λ. Thus, we can equvalently rewrte () (6) as Yes R (w l,wl ) R(wl,w l ) R (w l,wl ) R(wl,w l ) s.t. λ [,] λ (7) h H jw PO (λ ) = c j. (8) No Use w l To smplfy notaton, we defne α ( h H jh / h j ) and β Π h j h / h. (9) Fg.. Flowchart descrbng the proposed algorthm The values (9) are only calculated once per channel realzaton. For c j > we wrte (8) as Furthermore, the bound wll be tght at the optmum; hence, the nequalty n () can be equvalently replaced wth equalty. We propose a soluton to the optmzaton problem () () n Sectons III-A and III-B for the case of nstantaneous and statstcal CSI, respectvely. TX uses w l to transmt a plot. RX measures R (w l,wl j ) and f t s no smaller than R (w l,wj l ), t feeds back a one-bt message tellng the transmtters to contnue updatng the nterference level. As soon as the rate decreases for at least one of the recevers, the algorthm termnates and the transmtters wll use the beamformng vectors from the prevous teraton. We clam that the algorthm s self-enforced. Suppose that, n one of the steps, TX chooses to cheat by not decreasng the nterference level. Then, the rate of system j wll decrease and RX j wll feedback a negatve bt. Accordng to the last step of the algorthm, the transmtters are expected to choose the beamformng vectors from the prevous teraton. If TX does not, RX j wll notce and report t to TX j. Then, TX j wll leave the barganng and employ ts MR beamformng vector nstead. That s, f one system tres to cheat, then the cooperaton s canceled and the operaton falls back to the (the so-called threat pont, n the context of Nash barganng). A. Instantaneous CSI By nsertng the expresson (4) n () (), wth nequalty changed to equalty, we get the problem w hh w () W s.t. h H j w = c j. (4) Snce the objectve of the algorthm s to fnd a PO pont, the transmtters are only wllng to use beamformng vectors that λ α λ +( λ ) +λ ( λ )β = c j λ (α /c j +β )+λ ( β ) =. When c j =, the strategy s the optmal soluton (.e, λ = ). Now, we wrte (7) (8) as λ () s.t. λ (α /c j +β )+λ ( β ) =, () λ. () The soluton to () () s the largest of the two solutons to () that satsfes (). B. Statstcal CSI By nsertng () n () () we get w wh C n Q w () s.t. w H Q j w c j, (4) w H w. () Problem () () s a quadratcally constraned quadratc program (QCQP). The feasblty set determned by (4) () s convex. However, the optmzaton s non-convex owng to the form of the objectve functon. However, t can stll be solved optmally and effcently usng semdefnte relaxaton. Ths s because semdefnte relaxaton s tght for QCQP problems of the form n () (4), as shown n [9]. We brefly elaborate the procedure, smlar to the way we dd n [4]. We change the optmzaton varables to W w w H. Note that W = w w H W ર and rank{w } =. (6)
Usng (6) and the property that Tr{YZ} = Tr{ZY} for matrces Y, Z of compatble dmensons, the average power term n (4) can be wrtten as w H Q j w = Tr { w H } { Q j w = Tr Qj w w H } = Tr{Q j W }. (7) Due to (6) and (7), we equvalently recast () (4) as W C n n Tr{Q W } (8) s.t. Tr{Q j W } c j, (9) Tr{W }, () W ર, () rank{w } =. () The objectve functon (8), the constrants (9) and () are lnear. The cone of postve semdefnte matrces () s convex. But the rank constrant () s non-convex. Droppng t, the remanng problem (8) () s a semdefnte programmng (SDP) problem, whch can be solved effcently. Due to the absence of (), the SDP problem wll not necessarly return rank- optmal matrces. We experenced through extensve smulatons that t actually does yeld rank- matrces. IV. NUMERICAL ILLUSTRATIONS In ths secton, we present extensve smulaton results to evaluate the performance of the algorthm we propose. We focus on the case of statstcal CSI, but also provde some results for nstantaneous CSI. In Secton IV-A, we explan how we generate CSI (.e., channel covarance matrces or channel vectors) for smulaton purposes. In Sectons IV-B and IV-C, we compare the outcome of the algorthm to the,,, and. Furthermore, n Secton IV-D, we show exemplary barganng trajectores. Fnally, n Secton IV-E, we dscuss the overhead and the complexty assocated wth our algorthm. Throughout the smulatons, we assume that the transmtters use n = antennas. We allow our algorthm run up to N = teratons. The results reported n Fgs. 7 and, are averages over Monte-Carlo (MC) runs. Fgs. 7 llustrate the sum of the transmsson rates,.e., R + R. Fgs. 8, 9,, and show examples of achevable rate regons,.e., for a sngle CSI realzaton. Fgs. and depct the average number of teratons needed tll the algorthm termnates, dependng on whether the algorthm s ntalzed wth the or ether of the and, respectvely. A. Generatng the Channels We generate the drect and the crosstalk channels n two dfferent ways, to model the scenaros of weak or strong spatal correlaton. Specfcally, n the case of nstantaneous CSI and weak correlaton, we generate the channel vectors h and h j drawng ndependent samples from CN(, I). For the scenaro of strong correlaton, we use the formula h j = μ h + μ h j, () whereh and h j are drawn fromcn(,i), andμ [,]. A value of μ close to refers to the case of strong nterference. In the case of statstcal CSI, we construct the covarance matrces, of rank r, randomly as r Q = q k q H k, (4) k= where q k CN(,I). For the scenaro of weak correlaton, we generate the covarance matrces Q and Q j ndependently accordng to (4). For the scenaro of strong correlaton, we construct the matrces such that the angle between the egenvectors of the drect matrx and the egenvectors of the crosstalk matrx s small. Assumng that r r j, we frst generate Q as n (4). Then, we construct the vectors {q j,k } k that defne Q j as { qj,k = μ q,k + μ q j,k, k r q j,k = q j,k, k > r () where q j,k CN(,I) and μ [,]. If r > r j, the matrces are constructed the other way around. B. Statstcal CSI In ths secton, we provde results for statstcal CSI, both for weak and strong spatal correlaton. Also, we dstngush among the cases of havng full-rank and low-rank covarance matrces. In the low-rank scenaro, the covarance matrces of the drect-channels have rank r = r =, and covarance matrces of the cross-talk channels have rank r = r = 4. For strong correlaton, we use μ =.8. In Fgs. and, we study the full-rank scenaro. Frst, we note that the sum rate s equal to snce full-rank crosstalk matrces correspond to w =. Second, we see that the sum rates for the proposed algorthm, the, and the saturate for hgh SNR. The reason s that when the SNR s hgh, nterference s the man lmtng factor. Snce the nterference cannot become zero, except for the SU-ponts, there should be a lmtaton. Thrd, snce there s no nterference at the SU ponts, the correspondng rates wll grow unbounded wth SNR and the wll be found at a SU pont for hgh SNR. In Fgs. and 6, we llustrate the low-rank scenaro. Here, all ponts but the converge to the same sum rate at hgh SNR. The dfference s that, for strong correlaton they converge at hgher SNR value than for weak correlaton. Also, we see that the rates grow almost lnearly wth the SNR. In general, there exsts a non-trval zero-forcng pont for the case of low-rank matrces. Usng ths, the nose s the only lmtaton. When the nose decreases, the rates ncrease. At low SNR, the starts growng later for strong correlaton than for weak correlaton. Furthermore, we evdence that weak correlaton (Fgs. and ) gves hgher rates for the proposed algorthm, the, and the, than strong correlaton (Fgs. and 6). As a general remark, low SNR means operaton n the nose-lmted regme and all the rates but the are almost the same. Concludng, we see that the performance of the proposed algorthm s slghtly below the and close to the,
except for full-rank matrces and hgh SNR. Most mportant, the algorthm performs consstently much better than the, whch would be the outcome f there was no cooperaton. C. Instantaneous CSI In Fgs. 4 and 7 we report the results for weak and strong correlaton (μ =.9), respectvely. We note that the curves behave smlarly to the ones n Fgs. and 6. The reason for ths s that the case of nstantaneous CSI can be regarded as a specfc nstance of the low-rank statstcal CSI when all covarance matrces are rank-. D. Barganng Trajectory In ths secton, we gve examples of the barganng trajectory,.e., the rate ponts (marked wth stars) reached at every teraton of the proposed algorthm. Here, the mum number of teratons used s N =. Fgs. 8 and llustrate the trajectores for statstcal CSI wth full-rank covarance matrces and SNR equal to and db, respectvely. The Pareto boundary s calculated usng the technque proposed n [4]. Fgs. 9 and llustrate the trajectores for nstantaneous CSI and SNR equal to and db, respectvely. The Pareto boundary s calculated usng the technque proposed n []. For statstcal CSI and full-rank matrces, the algorthm s always ntalzed wth the, snce a non-trval pont does not exst. For nstantaneous CSI and low SNR t s ntalzed wth the pont, whle for hgh SNR wth the pont. We note that the fnal outcome of the barganng algorthm s close to the Pareto boundary, but does not necessarly le on t. On one hand, the fnal outcome depends on the stepsze of the algorthm. On the other hand, the algorthm termnates when ether of the rates stops ncreasng,.e., when the tangent of the trajectory stops beng postve. More on, the outcome s close to, but generally far from. We note that the does not mply that both systems have ncreased ther rates compared to, whle the proposed algorthm and guarantee that both systems get at least ther rates. Fnally, n all fgures we show what the barganng trajectory would look lke f the algorthm went the entre way from one extreme pont ( or ) to the other wth small steps. Note that for statstcal CSI and full-rank matrces the pont corresponds to the orgn of the rate regon. E. Overhead and Complexty In ths secton, we study the number of teratons requred for the algorthm to termnate. Specfcally, we study the dependence on the SNR and the startng pont. In Fg., the algorthm s ntalzed wth the and we llustrate the three CSI scenaros (statstcal CSI, full- and low-rank, and nstantaneous CSI). For each CSI scenaro, the sold lne corresponds to weak spatal correlaton and the dashed lne corresponds to strong spatal correlaton. At low SNR, we see that the algorthm needs fewer teratons for strong correlaton than for weak correlaton. As the SNR ncreases, the number of teratons approaches the threshold N. Ths s because at hgh SNR the pont s closer to the Pareto boundary and the transmtters use all avalable teratons to reach t. When the systems are nterested n mnmzng the amount of overhead and complexty, they start at the pont that requres the smallest number of teratons. That s, the algorthm starts at the fr (w,wj ) R (w MR,wj MR ) for both systems. We llustrate ths n Fg.. Compared to Fg., we see that the methods perform equally for low SNR, snce ths s the nose-lmted regme. When the SNR s medum, the number of teratons s ncreased because both extreme ponts ( or ) are away from the Pareto boundary. For hgh SNR, the number of teratons for nstantaneous CSI and statstcal CSI wth low-rank matrces goes to one. Ths s because ths s the nterference-lmted regme and s optmal. For statstcal CSI wth full-rank matrces, a non-trval strategy does not exst. Therefore, the algorthm wll always start at, as a comparson of Fgs. and reveals. V. CONCLUSIONS We consdered the dstrbuted desgn of beamformng vectors for the MISO IFC. We proposed a cooperatve algorthm that acheves an operatng pont whch s almost Pareto optmal. The fnal soluton s n all cases better than the, whch would be the outcome f there was no cooperaton. The novel element of the proposed algorthm s the use of the generated nterference level as barganng value. The algorthm s equally applcable to the case of nstantaneous and statstcal CSI. We valdated the mert of our algorthm va extensve numercal llustratons. REFERENCES [] E. G. Larsson, E. Jorsweck, J. Lndblom, and R. Mochaourab, Game theory and the flat fadng Gaussan nterference channel, IEEE Sgnal Processng Magazne, vol. 6, no., pp. 8 7, Sep. 9. [] E. A. Jorsweck, E. G. Larsson, and D. Danev, Complete characterzaton of the Pareto boundary for the MISO nterference channel, IEEE Trans. Sgnal Process., vol. 6, no., pp. 9 96, Oct. 8. [] J. Lndblom, E. Karpds, and E. G. 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R + R [bts/channel use] Stat. CSI (full-rank), μ =, n =, N =, MC R + R [bts/channel use] Stat. CSI (full-rank), μ =.8, n =, N =, MC - - - - R + R [bts/channel use] Fg.. Sum rate; stat. CSI (full-rank), μ = Stat. CSI (low-rank), μ =, n =, N =, MC R + R [bts/channel use] Fg.. Sum rate; stat. CSI (full-rank), μ =.8 Stat. CSI (low-rank), μ =.8, n =, N =, MC - - - - Fg.. Sum rate; stat. CSI (low-rank), μ = Fg. 6. Sum rate; stat. CSI (low-rank), μ =.8 R + R [bts/channel use] Inst. CSI, μ =, n =, N =, MC R + R [bts/channel use] Inst. CSI, μ =.9, n =, N =, MC - - - - Fg. 4. Sum rate; nst. CSI, μ = Fg. 7. Sum rate; nst. CSI, μ =.9
R [bts/channel use] 4 Statstcal CSI; SNR = db, n =, μ =. Pareto boundary Barganng trajectory Barganng steps R [bts/channel use] 7 6 4 Statstcal CSI; SNR = db, n =, μ =. Pareto boundary Barganng trajectory Barganng steps R [bts/channel use] 4 4 R [bts/channel use] 6 7 Fg. 8. Barganng trajectory; stat. CSI, SNR = db Fg.. Barganng trajectory; stat. CSI, SNR = db Instantaneous CSI; SNR = db, n =, μ =.8 Instantaneous CSI; SNR = db, n =, μ =.8 R [bts/channel use] Pareto boundary Barganng trajectory Barganng steps R [bts/channel use] 6 4 Pareto boundary Barganng trajectory Barganng steps R [bts/channel use] 4 R [bts/channel use] 6 Fg. 9. Barganng trajectory; nst. CSI, SNR = db Fg.. Barganng trajectory; nst. CSI, SNR = db Average number of teratons Stat. CSI (full-rank), μ = Stat. CSI (full-rank), μ =.8 Stat. CSI (low-rank), μ = Stat. CSI (low-rank), μ =.8 Inst. CSI, μ = Inst. CSI, μ =.9 Average number of teratons Stat. CSI (full-rank), μ = Stat. CSI (full-rank), μ =.8 Stat. CSI (low-rank), μ = Stat. CSI (low-rank), μ =.8 Inst. CSI, μ = Inst. CSI, μ =.9 - - - - Fg.. Average number of teratons; algorthm ntalzaton: Fg.. Average number of teratons; algorthm ntalzaton: or