Robust Precoder for Multiuser MISO Downlink with SINR Constraints

R OBUST PRECODER FOR MUL TIUSER MISO DO WNLINK WITH SINR CONSTRAINTS P . Ubaidulla and A. Choc kalingam Department of ECE, Indian Institute of Science, Bangalore 560012, INDIA ABSTRA CT In this paper, we consider linear pr ecoding with SINR con- straints for the downlink of a multiuser MISO (multiple-input single-outp ut) c ommunica tion system in the presence of im- perfect ch annel state informa tion (CSI). The ba se station is equippe d with mu ltiple transmit antenna s a nd e ach u ser ter- minal is equipped with a single receive antenna . W e pro- pose a robust design of linear p recoder which tran smits min- imum p ower to provide th e req uired SINR at the user termi- nals when the true channel state lies in a region of a given size around the channel state a vailable at the transmitter . W e s how that this design problem can be formu lated a s a Second Or der Cone Pr ogram ( SOCP) which ca n be solved efficiently . W e compare the p erform ance of the pro posed design with some of the robust designs reported in the literature. Simulation results show that the proposed ro bust design provid es better perfor mance with reduced complexity . 1. INTR ODUCTION There has be en considerable interest in multiuser multip le- input multiple-outp ut (MIMO) wir eless communicatio ns in view of their potential for transmit diversity and increased channel c apacity [1],[2]. Sin ce it is difficult to provid e mo - bile user termin als with large numb er of antennas du e to space constraints, multiu ser multiple-input single-output (MISO) wi- reless commun ications on the downlink, whe re th e base sta- tion is equipp ed with multiple transmit anten nas and each user terminal is equipp ed with a sing le receive an tenna is of sig- nificant pra ctical interest. I n such multiuser MISO systems, multiuser interfer ence at the receiver is a crucial issue. One way to deal with this interference issue is to use multiuser d e- tection [3] at the receivers, wh ich increases the receiv er com- plexity . As an alter nate way , transmit side p rocessing in the form of pr ecoding is being studied wid ely [2],[4]. Sev eral linear preco ders such a s tra nsmit zero- forcing (ZF) an d m in- imum mean squ are erro r (MMSE) filters, and non -linear pre- coders including T omlinson-Hara shima p recoder (THP) hav e been proposed and widely inv estigated in the literature [5],[6]. Both linear and nonlinear preco ding strategies which meet SINR constraints at in dividual users have also been investi- gated [7],[8]. Non-linear preco ding strategies, though more complex th an the linear strategies, resu lt in impr oved per- forman ce com pared to linear pre-pr ocessing. Transmit side precod ing techniques, linear or n on-linear, can rend er the re - ceiv er side p rocessing at the u ser term inal simpler . Howe ver , transmit side pr ecoding techniqu es requ ire channel state in- formation (CSI) at the transmitter . Sev eral studies on transmit prec oding assume perfect know- ledge of CSI at the transmitter . Howev er, in practice, CSI at the transmitter suffers fro m inaccur acies caused by errors in channel estimation and/or limited, delayed or errone ous feedback . The performan ce of precod ing schemes is sensi- ti ve to such inaccuracies [ 9]. Several p apers in th e literature have prop osed preco der design s, bo th linear and non-linea r , which are robust in th e presence of ch annel estimation err ors [10],[11]. Linear robust preco ding for MISO downlink with SINR constraints with imperfect CSI a t t he transmitter is con- sidered in [ 12], where the rob ust d esign is formulated as Semi Definite Programs (SDP) of different perf ormance an d c om- plexity . I n [13], Payaro et a l., con sider ro bust power alloca- tion for fixed beamforme rs with Mean Squ are Erro r (MSE) constraints and formu late the problem as con vex op timization problem . Robust power control for fixed beam former s with SINR constraints is considered in [14]. In this paper , we consider robust linear pre coding und er SINR constraints for the do wnlink o f a multiuser MISO wire- less com munication system in the presen ce of imperf ect CSI at the transmitter . Th e C SI at the transmitter is assumed to be perturb ed by estimation or qu antization error . The objective of the robust design considered here is to minimize the to- tal tran smit power , while ensur ing the requ ired SINR at e ach user . The r obustness of the d esign co nsists in en suring th e target SINR for all e rrors in th e CSI belo nging to a given un- certainty region. W e show that the design of the robust p re- coder with SINR constraints can be fo rmulated a s a conv ex optimization p rogram . Sp ecifically , we show th at it is possi- ble to formulate this problem as a SOCP wh ich can be solved very ef ficiently . This is in contrast to the formu lations in [12] which are comp utationally demandin g. Our simulation re- sults i llustrate the imp rovement in perfor mance and complex- ity compared to other robust precoders i n the literatu re. The rest of the pap er is organ ized as follows. In sectio n 2, we presen t the system model. The propo sed robust pre- coder design is presented in section 3. Perfo rmance results and comp arisons are p resented in section 4. Conclu sions are presented in section 5. 2. SYSTEM MODEL W e con sider a multiuser MISO system, wher e a base station (BS) comm unicates with N u users on the downlink. The BS employs N t transmit antennas a nd each user is equipp ed with one re ceiv e anten na. Let u denote 1 the N u × 1 data sy mbol vector , where u i , i = 1 , 2 , · · · , N u , de notes the complex val- ued data sym bol meant f or user i . The linear pr ecoding matrix B ∈ C N t × N u acts on this vector u . The output o f the precod - ing operation is den oted by the N t × 1 vector x , whe re x j , j = 1 , 2 , · · · , N t , d enotes th e com plex-valued symbol trans- mitted on the j th transmit antenna. The received signal at user i , denoted by y i , can be written as y i = h i Bu i + n i , (1) where h i is the row vector containing complex channe l gains from the tran smit antennas to the receive antenna of user i , and n i is an i.i.d complex Gaussian rando m variable with zero mean and variance of σ 2 n representin g the no ise at the i th r e- ceiv er . The channel g ains are assumed to be in depend ent zero mean complex Gaussian v ariable and E { h H i h i } = I . 3. ROB UST PRECODER DESIGN WITH IMPERFECT CSI A T THE TRANSMITTE R In this section, we co nsider the design o f ro bust precoder which transmits minim um p ower in ord er to pr ovide th e re- quired SINR at each user when the CSI at the transmitter is imperfect. The SINR at user termina l k is gi ven by SINR k = | h k b k | 2 P N u j =1 ,j 6 = k | h k b j | 2 + σ 2 k , (2) where h k is th e k th r ow o f the matrix H and b j is th e j th column of matrix B . Assuming E { uu H } = I , total transmit power is given by P T = E { x H x } = T r  B H B  (3) = k b k 2 , (4) where b = vec ( B ) . When the transmitter has the perfect knowledge o f CSI, the problem o f designing a precoder which transmits m ini- mum power while ensuring th e req uired SINR at each user can be posed as min B trace  B H B  subject to SINR k ≥ γ k , 1 ≤ k ≤ N u , (5) 1 V ectors a re de noted by boldface lowerca se letters, an d matri ces are de- noted by boldface uppercase letters. [ . ] T , [ . ] H , and [ . ] † denote transpose, Hermitia n, and pseudo-in verse operat ions, respecti vely . [ A ] ij denotes the element on the i th row and j th column of the m atrix A . ve c ( . ) operator stacks the columns of the input matrix into one colu mn-vector . where γ k is the SINR requ ired a t the k th user . The ab ove problem can be solved in different ways [ 8], [15]. The p rob- lem as stated in ( 5) is not conve x. Bu t it is shown in [15] that this p roblem c an b e f ormulated as a SOCP . This conve x for- mulation e nables th e use of efficient nu merical algor ithms to solve the precoder d esign problem. T he SOCP formu lation of (5) is gi ven by [16] min B τ subject to k b k − τ ≤ 0 , (6)   [ h k B σ k ]   − a k h k b k ≤ 0 , 1 ≤ k ≤ N u , where a k = q 1 1+ γ k . Here, we have assumed th at the imag i- nary part of h k b k is zero. Th is is possible because we c an add arbitrary phase rotation to the c olumns of B withou t affecting the SINR. 3.1. Imperfect CSI If the tran smitter’ s kn owledge of CSI is imperfe ct, then the precod er desig ned ba sed on the above fo rmulation in the as- sumption of perfe ct CSI may fail to achieve the req uired SINR. Here, we consider the design of precoder s wh ich will meet the SINR requirem ents of all users even when the CSI at the transmitter is imperfect. 3.1.1. Ch annel Err o r Model In the pr esent context, we co nsider the situation where the transmitter CSI b h k is related to the true channel h k as h k = b h k + e k . (7) In one model, b h k is an imperf ect estimate of the true chan nel h k and e k is a vector of i.i.d comp lex Gaussian ran dom vari- ables. This mo del is app licable when the u ser estimates its own channe l and feeds it back to the tr ansmitter th rough an ideal feed back l ink with n o q uantization, or if there is a delay between the estimation and the actual ch annel in fast varying en viron ments. In ano ther mode l, b h k is a quan tized version o f the ac tual channel h k , and e k represents the quantization error . Th is model is app licable when the u ser which knows the perf ect channel h k quantizes it and feeds back to th e transmitter throug h a digital fe edback link. Equa tion ( 8) can be used to m odel the uncertainty region in this case als o. In the robust precoder design, we consider the set Z k = { h k   h k = b h k + e k , k e k k ≤ δ k } . (8) The uncertainty r e gion Z k is the set o f all channel vectors which lie in a sphere of ra dius δ k around th e estimated chan- nel vector b h k . This characterizatio n of the uncer tainty region is also related to the outage probab ility [13]. 3.1.2. R obust Pr ecod er Design When the CSI at the transmitter is known to be im perfect, a robust precod er is designed to meet the target SINR of all the users. Whe n the imperf ections in the CSI are of the typ es described above, the r obustness re quiremen t of the prec oder can be represented , in terms of the SOCP formulatio n as min B τ subject to k b k − τ ≤ 0 , (9) max h k ∈Z k    [ h k B σ k ]   − a k h k  ≤ 0 , 1 ≤ k ≤ N u . This pr oblem is akin to the Robust Optimization (R O) [1 8], which is o ne of th e meth odolog ies for so lving optimization problem s u nder parameter uncertain ties. The general problem of optimization under parameter un- certainties has the following form: min f 0 ( ζ ) (10) subject to f i ( ζ , d ) ≤ 0 , ∀ d ∈ R , 1 ≤ i ≤ m, where ζ ∈ R n is the vector of decision variables, d ∈ R k are data vectors, f i are the co nstraints, an d R is the uncer- tainty set . Compu tationally tractable appro aches to the p rob- lem (10) for different classes o f constraint func tions f an d uncertainty regions R are reported in the literature [19]. For the case o f robust precoder design (9 ), it is possible to use directly the results re ported in [20] after converting the Sec- ond Order Cone (SOC) co nstraints to equiv alent semi- definite constraints [21 ]. Th is appro ach is adopted in [12]. But the re- sulting ro bust co unterpar t is a Semi Definite Progra m (SDP) which is of higher c omplexity and computatio nally dem and- ing than the SOCP formula tion of the perfec t CSI preco ding problem . Recent results h av e shown that its possible to have ro- bust counterp arts which preserve the stru cture of the nomi- nal p roblem [22]. For the preco der design, this means that the r obust desig n is a SOCP pro blem, which is a sign ificant improvement over the SDP form ulation. In this con text, con- sider the data perturbatio n model d = d 0 + X j ∈ N ∆d j z j , (11) where d 0 is the nominal da ta value, ∆d j are the directions of data pertu rbations, and { z j , 1 ≤ j ≤ N } are the zero mean i.i.d rand om variables. Robust o ptimization aims at fin ding a robust optimal ζ wh ich will meet the following c onstraint: max d ∈R Ω f ( ζ , d ) ≤ 0 , (12) where R Ω =    d 0 + X j ∈ N ∆d j u j     k u k ≤ Ω    . (13) The follo wing linearized v ersion of th e constraint (12) is con- sidered in [22]: max ( v , w ) ∈V Ω f ( ζ , d 0 )+ X j ∈ N { f ( ζ , ∆d ) v j + f ( ζ , − ∆d ) w j } ≤ 0 , (14) where V Ω =  ( v , w ) ∈ R 2 | N | +     k v + w k ≤ Ω  . It is shown in [22] that, for an SOC constraint, ζ is feasible in (12) if ζ is feasible in (14). W e state the following theor em for the specific case of SOCP constraints. Theorem 1 (Bertsimas-Sim [22]) a) Constraint (14) is equivalent to f ( ζ , d 0 ) + Ω k s k ≤ 0 , (15) wher e s j = max  f ( ζ , ∆d j ) , f ( ζ , − ∆d j )  . b) Equatio n (15) can be written as ∃ ( y , t ) ∈ R | N | +1 f ( ζ , d 0 ) ≤ − Ω y (16a) f ( ζ , ∆d ) ≤ t j ∀ j ∈ N (16b) f ( ζ , − ∆ d ) ≤ t j ∀ j ∈ N (16c) k t k ≤ y . (16d) The following transfor mation [12] will en able us to write the precoder design problem in real variables: B =  Re ( B ) Im ( B ) − Im ( B ) Re ( B )  , (17) h k = [ Re ( h k ) Im ( h k )] , (18) b k = [ Re ( b k ) − Im ( b k )] , (19) e k = [ Re ( e k ) Im ( e k )] . (20) The precoder design proble m can b e for mulated in terms o f real variables by replacing th e complex vectors and matrices in ( 9) by th e co rrespond ing real vectors a nd matrices o btained by the transform ations given above. The d ata per turbation model ( 11) fo r the seco nd ord er cone co nstraint of th e p recoder design problem takes the f orm d k = d 0 k + X N u ∆d j k ¯ e k,j , 1 ≤ k ≤ N u , 1 ≤ j ≤ 2 N t . (2 1) where d k = [ h k h k ] T , d 0 = [ b h k b h k ] T , ∆d j k = [ i j i j ] T , and i j is the j th row of 2 N t × 2 N t identity matrix. d is the vector of all data in the pro blem and has the structure gi ven above a s h k appears twice in the constraint in (9). ∆d j k in- dicates how the error in th e j th com ponent of h k affects d . Based on this data perturbatio n mod el, it is obviou s that the channel uncertainty region Z k of the r obust pr ecoder design problem of (9) co rrespond s to the u ncertainty region in (13), with Ω = δ k . 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ Pr{SINR ≤ λ } RPC [14] SDP−1 [12] SDP−2 [12] Proposed Method Fig. 1 . CDF of achieved SINR at the downlink user s. Min i- mum req uired SINR γ 1 = γ 2 = γ 3 = 5 dB. N t = N u = 3 , uncertainty size, δ 1 = δ 2 = δ 3 = 0 . 01 5 . Using the data pertu rbation mod el in (21) and app lying Theorem -1 to (9), we obtain the follo wing S OCP formulation of the propo sed robust precod er design: min B τ (22a) subject to    b k    ≤ τ (22b)    [ h T k B σ k ]    − a k h T k b k ≤ − δ k y k (22c)    [ b i T σ k ]    − a k B i,k ≤ t k,i (22d)    [ b i T σ k ]    + a k B i,k ≤ t k,i (22e) k t k k ≤ y k (22f) 1 ≤ k ≤ N u , 1 ≤ i ≤ 2 N t . (22g) where b i is the i th ro w of B . In this formulation o f the ro bust precod er design, the constrain ts are of the same typ e of the nominal pro blem (11). Hence, the comp utational complexity is of the same order as the nominal problem. The robustness con straint in (14) is a relaxatio n of the constraint in (1 2). By selecting app ropriate value of κ, 0 ≤ κ ≤ 1 , and replacing δ k in (22) by κδ k , it is po ssible to ge t a ro bust pr ecoder which transmits less power while achiev- ing th e req uired SINR con straints. Through extensive sim- ulations, it was found that κ = 1 / 4 provides good balan ce between the achiev ed SINR and the transmit po wer . 4. SIMULA TION RESUL TS In this section, we p resent the perfo rmance of the pr oposed robust preco der design (22) through simulations. W e com- pare this perfo rmance with some oth er ro bust d esigns avail- able in the literature. The componen ts of the estimated ch an- 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 120 140 160 180 200 Target SINR (dB) Transmit Power RPC [14] SDP−1 [12] SDP−2 [12] Proposed Method Fig. 2 . Transmit power versus SINR r equireme nt of th e users. Uncertainty size δ = δ 1 = δ 2 = δ 3 = 0 . 02 , N t = N u = 3 . nel vectors b h k , 1 ≤ k ≤ N u are i.i.d zero mean unit variance proper co mplex Gaussian rand om variables. Th e n oise sam- ples n k , 1 ≤ k ≤ N u are i.i.d pr oper complex Gaussian random v ariables with ze ro mean and unit variance. W e com- pare th e per forman ce of the propo sed d esign with the robust SOCP design (den oted here by SDP-1 ) and the un structured SDP design (denoted by SDP-2) in [12], an d th e robust power control (denoted by RPC) in [14]. First, we co mpare the C DF of the achieved SINR of SDP- 1, SDP-2, and RPC with the CDF of SINR of the proposed robust design. Figure 1 shows the CDF fo r various metho ds. In this experiment, we c onsider a system with a base station having N t = 3 transmit anten nas and N u = 3 single an- tenna recei vers. Th e uncertainty size of CSI at the transmitter is assumed to be same for all users and is δ = 0 . 0 1 5 . The target SINR fo r a ll users is γ = 5 dB. In case of SDP-1 and RPC, it is e viden t th at, most of the time, the u sers get SINR much high er than the target SINR. This implies that th ese al- gorithms result in much hig her transmit power than requir ed. The SDP-2 and the p roposed de sign have a lmost sam e CDF , and is very near to the required SIN R. That is, perform ance wise, the proposed design ac hiev es almost the same p erfor- mance as SDP-2 in [12] but with reduce d complexity . Figure 2 shows the transmit power T r { B H B } for v arious robust designs in order to achieve d ifferent target SINRs. Th is experiment also has the same setting a s above, except for th e target SINR which is varied from 0 dB to 10 d B. T he SDP-1 and RPC metho ds transmits more power compar ed to SDP-2 and the prop osed method. This high er transmit power results in the higher SINRs at the users. Figure 3 s hows the transmit po wer for th e dif ferent robust designs for different values of the size of channel uncertainty . The SINR requirem ent for all u sers is 5 dB. The SDP-1 and RPC methods end up in h igher tra nsmit power co mpared to SDP-2 and the p roposed method. This higher transmit p ower T able 1 . Comparison of Run-Time in Seconds for different precod ing m ethods Method N u , N t =3 N u , N t =4 N u , N t =5 N u , N t =6 Proposed 0.10 0.2 0.44 0.6 SDP-1 [12] 0.2 0.3 0.6 1 SDP-2 [12] 4.5 16 61 121 0 0.05 0.1 0.15 0 10 20 30 40 50 60 70 80 90 100 δ Transmit Power RPC [14] SDP−1 [12] SDP−2 [12] Proposed Method Fig. 3 . T ransmit power versus chann el u ncertainty size, δ . N t = N u = 3 , SINR requiremen t of the users γ 1 = γ 2 = γ 3 = 5 dB. results in the higher SINRs at the users. Also the rang e of δ for which the propo sed method is feasible is lar ger than other methods. T able- 1 shows the compariso n of co mputation time in sec- onds r equired fo r solving the robust p recoder usin g different methods on a 2.66 GHz mach ine u sing the solver SeDuMi [17]. Compu tation time for SDP-2 is the highest. Comp uta- tion tim e for SDP-1 and the propo sed metho d are com para- ble. T he proposed method is ab le to achieve the perf ormance compara ble to SDP-2 at the computatio nal cost of SDP-1. In su mmary , th e proposed robust desig n ach ie ves better perfor mance than the other metho ds compared while being computatio nally less intensiv e. 5. CONCLUSION W e p roposed a de sign of ro bust pr ecoder with SINR con- straints for m ultiuser MISO downlink with imp erfect CSI at the transmitter . W e showed tha t the robust p recoder design problem can be formu lated as a SOCP . The SOCP formula- tion has the advantage that the co mputationa l com plexity o f the ro bust d esign is of the same or der as that of the design with pe rfect CSI. A comparison with o ther r obust precod er designs reported in the literatu re showed that the pro posed ro- bust design per forms better wh ile being com putationally less complex. 6. REFERENCES [1] D. Tse and P . V iswana th, Fundamenta ls of W ir eless Communication , Cambridge Uni versity Press, 2006. [2] H. Bolcskei, D. 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