A Rate-Splitting Strategy for Max-Min Fair Multigroup Multicasting

Transcription

1 A Rate-Splittin Stratey for Max-Min Fair Multiroup Multicastin Hamdi Joudeh and Bruno Clercx Department of Electrical and Electronic Enineerin, Imperial Collee London, United Kindom School of Electrical Enineerin, Korea University, Seoul, Korea {hamdi.joudeh0, Abstract We consider the problem of transmit beamformin to multiple cochannel multicast roups. The conventional approach is to beamform a desinated data stream to each roup, while treatin potential inter-roup interference as noise at the receivers. In overloaded systems where the number of transmit antennas is insufficient to perform interference nullin, we show that inter-roup interference dominates at hih SNRs, leadin to a saturatin max-min fair performance. We propose a rather unconventional approach to cope with this issue based on the concept of Rate-Splittin (. In particular, part of the interference is broadcasted to all roups such that it is decoded and canceled before the desinated beams are decoded. We show that the stratey achieves sinificant performance ains over the conventional multiroup multicast beamformin stratey. Index Terms Broadcastin, multicastin, downlin beamformin, derees of freedom, W approach. I. INTRODUCTION Since the wor of Sidiropoulos et al. [], beamformin for physical-layer multicastin has received considerable research attention. In the most basic setup, the Base Station (BS transmits a common data stream to all receivers. This was later eneralized to multiple cochannel multicast roups, also nown as multiroup multicastin []. The main problems considered in the multicastin literature are those of classical multiuser beamformin, namely the Quality of Service (QoS constrained power minimization problem and the power constrained Max-Min Fair (MMF problem. Such problems were shown to be NP-hard, and the solutions advised in [], [] are based on Semidefinite Relaxation (SDR and Gaussian randomization techniques. Alternative solutions based on convex approximation methods were later proposed, exhibitin marinally improved performances under certain setups, and more importantly, lower complexities [3], [4]. The multiroup multicastin problem was also extended to incorporate perantenna power constraints [5] and lare-scale arrays [6]. In addition to the QoS and MMF problems, the sum-rate maximization problem was considered in [7]. The common transmission stratey adopted in multiroup multicastin is based on extendin the multiuser beamformin paradim, i.e. each messae is first encoded into an independent data stream then transmitted throuh linear precodin (or beamformin. However, the multicast nature of each This wor has been partially supported by the UK EPSRC under rant number EP/N053/ /6/$3.00 c 06 IEEE stream results in different, and more difficult, desin problems compared to their multiuser counterparts. In the beamformin stratey, each receiver decodes its desired stream while treatin all interferin streams as noise. Hence, inter-roup interference is inevitable under an insufficient number of BS antennas. Althouh rarely hihlihted or treated in the multiroup multicastin literature, such interference can be hihly detrimental. We propose a beamformin stratey based on the concept of Rate-Splittin (, where the messae intended to each roup is split into a common part and a desinated part. All common parts are paced into one super common messae, broadcasted to all users in the system. Desinated parts on the other hand are transmitted in the conventional beamformin manner. While the concept of is not particularly new (it appears in the interference channel literature, it has only been applied recently to multiuser beamformin, where it was shown to enhance the performance under residual interference arisin from imperfect Channel State Information (CSI at the BS [8], [9]. We show that brins sinificant performance ains to multiroup multicastin, particularly in inter-roup interference limited scenarios. While the focus is on the MMF problem in this paper, can be extended to the QoS problem. The rest of the paper is oranized as follows. Section II presents the system model. The limitations of the conventional transmission stratey are analysed in Section III. In Section IV, the stratey is introduced and the performance ains over the conventional stratey are derived. The precoders are optimized usin a Weihted Minimum Mean Square Error (W alorithm in Section V. Simulation results are presented in Section VI, and Section VII concludes the paper. II. SYSTEM MODEL Consider a transmitter equipped with N antennas communicatin with K sinle-antenna receivers rouped into the M multicast roups {G,..., G M }, where K {,..., K} and M {,..., M}. We assume that m M G m = K, and G m G j =, for all m, j M and m j. Let x C N denote the sinal vector transmitted by the BS in a iven channel use, which is subject to an averae power constraint E { x H x } P. Denotin the correspondin sinal received by the th user as y, the input-output relationship writes as y = h H x + n, where h C N is the narrow-band channel vector from the BS to the th user, and n CN (0, σn, is the Additive White Gaussian Noise (AWGN at the receiver.

2 We assume, without loss of enerality, that σn,,..., σn,k = σn, from which the transmit SNR is iven by P/σn. Moreover, the transmitter perfectly nows all K channel vectors, and each receiver nows its own channel vector. In multiroup multicast transmission, the BS communicates the messaes W,..., W M to G,..., G M respectively. Consider a conventional linear precodin (beamformin transmission model. Messaes are first encoded into independent data streams, where the vector of coded data symbols in a iven channel use writes as s p [s,..., s M ] T C M. We assume that E { } s p s H p = I, where power allocation is considered part of the beamformin. Data streams are then mapped to the transmit antennas throuh a linear precodin matrix P p [p,..., p M ], where p m C N is the mth roup s precodin vector. The resultin transmit sinal is x = p m s m ( where the power constraint reduces to M p m P. The th user s averae receive power (over multiple channel uses in which the channel is fixed writes as S {}}{{ }}{ T = h H p µ( + h H p m + σn. ( m µ( where µ : K M maps a user-index to the correspondin roup-index, i.e. µ( = m such that G m. In the followin, µ( is referred to as µ for brevity where the arument of the function is clear from the context. In (, S and I denote the desired receive power and the interference plus noise power, respectively. Hence, the Sinal to Interference plus Noise Ratio (SINR experienced by the th user is defined as γ S I. Under Gaussian sinallin, the th achievable user-rate is iven by R = lo ( + γ. In multiroup multicastin, users belonin to the same roup decode the same data stream. Therefore, to uarantee that all users in the mth roup are able to recover W m successfully, the correspondin code-rate should not exceed the roup-rate defined as r m min i Gm R i. I III. MAX-MIN FAIRNESS AND INTER-GROUP INTERFERENCE In the liht of the conventional multi-stream beamformin model, the MMF problem is formulated as max min min R i P p m M i G m R(P : (3 s.t. p m P where the inner minimization in (3 accounts for the multicast nature within each roup, while the outer minimization accounts for the fairness across roups. It is common practice to formulate the above problem in terms of the SINRs [], [4] [6]. Since each roup receives a sinle stream, and due to the Rate-SINR monotonic relationship, the two formulations are equivalent. The rate formulation is preferred in this wor in order to compare the performance to the scheme. A. Inter-Group Interference and Derees of Freedom An optimum MMF desin achieves balanced roup rates, requirin a simultaneous increase in powers allocated to all streams as P increases. In scenarios where the number of transmit antennas in insufficient to place each beam in the null space of all its unintended roups, inter-roup interference is expected to limit the MMF performance. To characterize this, we resort to hih SNR analysis throuh the Derees of Freedom (DoF. This reime is of particular interest as the effect of noise can be nelected, and inter-roup interference is the main limitin factor. The DoF can be rouhly interpreted as the number of interference-free streams that can be simultaneously communicated in a sinle channel use. To facilitate the definition of the DoF, we first define a precodin scheme {P p (P } P as a family of feasible precoders with one precodin matrix for each power level. The correspondin achievable user-rates write as {R (P,..., R K (P } P, and the th user-dof is defined as D lim P R (P lo (P. It follows that the mth roup-dof, denoted by d m, satisfies 0 d m D i for all i G m. The correspondin symmetric- DoF is iven by d = min m M d m. For a iven setup, the optimum MMF precodin scheme is denoted by { P p(p }. The correspondin MMF-DoF is P iven by d R(P = lim P, which is the maximum lo (P symmetric-dof. Since each user is equipped with a sinle antenna, then D,..., D K, and d for any precodin scheme. Hence, when d = is achievable, it is also optimum. It should be noted that althouh a rate-optimal precoder is also optimum in a DoF sense, the converse is usually untrue. B. MMF-DoF of Multi-Stream Beamformin In the DoF analysis, we mae the followin assumptions. Assumption. The channel vectors h,..., h K are independently drawn from a set of continuous distribution functions. Hence, for any N K sub matrix in which the K sub column vectors constitute any subset of the K channel vectors, it holds with probability one that the ran is min{n, K sub }. Assumption. We assume equal size roups. i.e. G,..., G M = G, where G = K/M is a positive inteer. Next, the MMF-DoF of the conventional multi-stream transmission scheme is characterized. Proposition. Under Assumptions and, the optimum MMF-DoF achieved by solvin (3 is iven by { R(P lim P lo (P =, N N min (4 0, N < N min where N min = + K G. To show this, let us define H m as the matrix with columns constitutin channel vectors of all users in G m, and H m = [H,..., H m, H m+,..., H M ] as the complementary set

3 of channel vectors. By Assumptions and, null ( Hm has a dimension of max{n + G K, 0} for all m M. Hence, N N min is sufficient to place each beamformin vector in the null space of all roups it is not intended to, i.e. p m null ( Hm for all m M. Each roup sees no inter-roup interference, and a DoF of per roup is achievable. Such DoF is optimum as it cannot be surpassed. On the contrary, when N < N min, this is not possible, and inter-roup interference limits the MMF-DoF to 0 as shown in the Appendix. We refer to this case as an overloaded system. Finally, we conclude this section by hihlihtin the impact of a collapsin DoF on the rate performance. When d = 0, the MMF rate stops rowin as SNR rows lare, reachin a saturated performance. Althouh the DoF analysis is carried out as SNR oes to infinity, its results are hihly visible in finite SNR reimes as we see in the simulation results. IV. RATE-SPLITTING FOR MULTIGROUP MULTICASTING The saturatin performance can be avoided by sinle-stream multiroup transmission. In particular, the M messaes are paced into one super messae, encoded into a sinle data stream. This is broadcasted such that it is decoded by all roups, hence retrievin their correspondin messaes. Since this interference-free transmission achieves a total DoF of, each roup is uaranteed a non-saturatin performance with a DoF of /M. However, relyin solely on this stratey jeopardizes partial ains potentially achieved by multi-stream beamformin. A simple example is the low-snr reime, where interference is overwhelmed by noise, and beamformin each messae to its correspondin roup is a preferred stratey. Hence, we introduce the followin unifyin stratey. A. The Rate-Splittin Stratey Each roup-messae is split into a common part and a roup-desinated part, e.. W m = {W m0, W m }, with W m0 and W m bein the common and desinated parts respectively. All common parts are paced into one super common messae W c {W 0,..., W M0 }, encoded into the stream s c, and then precoded usin p c C N. On the other hand, the desinated messaes are encoded into s,..., s M and precoded in the conventional multi-stream manner described in Section II. The transmit sinal writes as a superposition of the common stream and the desinated streams such that x = p c s c + p m s m. (5 The power constraint writes as p c + M p m P. The common stream can be interpreted as the part of the interference that is decoded (hence eliminated by all roups, while interference from desinated streams is treated as noise. The th user s averae received power now writes as T c, = h H p c + T (6 To be more precise, this corresponds to a rate scalin as o ( lo (P, which either stops rowin or rows extremely slow with P compared to the interference free scenario, reachin a flat or almost-flat performance. where S c, = h H p c denotes the common stream s receive power and I c, = T is the interference plus noise power experienced by the common stream. By treatin all desinated streams as noise, the SINR of the common stream at the th receiver is iven by γ c, S c, I c,. Hence, transmittin W c at a rate of R c, = lo (+γ c, uarantees successful decodin by the th receiver. To uarantee that W c is successfully recovered by all receivers, the rate of the common data stream should not exceed the common-rate defined as R c = min K R c,. After decodin the common stream, the receiver removes it from y usin Successive Interference Cancellation (SIC. This is followed by decodin the desinated stream in the presence of the remainin interference and noise, achievin the rate R defined in Section II. The common rate writes as a sum of M portions: R c = M C m, where C m is associated with W m0. It follows that the mth roup-rate is defined as R,m C m + min i Gm R i, consistin of a common-rate portion plus a desinated-rate. It is easy to see that the roup-rates reduce to the conventional roup-rates defined in Section III when W c = 0. B. MMF With Rate-Splittin The MMF problem is formulated in terms of as follows ( max min C m + min R i c,p m M i G m s.t. R c, M C m, K R (P : C m 0, m M (7 p c + p m P where c [C,..., C M ] T is the vector of common-rate portions. The first set of constraints in (7 accounts for the lobal multicast nature of the common stream and uarantees that it can be decoded by all users. The second set of constraints uarantees that no user is allocated a neative common-rate portion. Solvin (7 yields the optimum precodin matrix, in addition to the splittin ratio for each roup-messae. Next, the DoF performance of the scheme is characterized. Proposition. The MMF-DoF achieved by solvin the problem in (7 is lower-bounded by { R (P lim P lo (P, N N min N < N (8 min M, where the lower-bound is tiht for N N min. This follows directly from Proposition, and the fact that the sinle-stream multiroup solution described at the beinnin of this section is feasible for problem (7. V. PRECODER OPTIMIZATION In the scheme, each roup-rate writes as a sum of two rate components. Hence, the MMF solutions in [], [4] do not apply here, as the performance metric of each user cannot be expressed as a sinle SINR. Alternatively, we resort to the W approach [0], [], which is particularly effective

4 in dealin with problems incorporatin non-convex coupled sum-rate expressions, includin problems [8], [9]. A. Rate-W Relationship We start by definin the MSEs. The th user s estimate of s c, denoted by ŝ c,, is obtained by applyin the equalizer c, to the receive sinal such that ŝ c, = c, y. After removin the common stream usin SIC, the equalizer is applied to the remainin sinal to obtain an estimate of ŝ iven by ŝ = (y h H p cs c. The common and private MSEs at the output of the th receiver, defined as ε c, E{ ŝ c, s c } and ε E{ ŝ s } respectively, write as: ε c, = c, T c, R { c, h H p c } + ε = T R { h H p µ } +. (9a (9b The s are defined as ε c, min c, ε c, = T c, I c, and ε min ε = T I, where the correspondin optimum equalizers are the well-nown weihts written as c, = p H c h T c, and = p H h T. The s are related to the SINRs such that γ c, =, from which ( /ε c, the achievable rates write as R c, and γ = ( /ε = lo (ε c, and R = lo (ε. Next we introduce the main buildin blocs of the solution, the aumented WMSEs defined for the th user as: ξ c, u c, ε c, lo (u c, and ξ u ε lo (u (0 where u c,, u > 0 are the correspondin weihts. In the followin, ξ c, and ξ are referred to as the WMSEs for brevity. The Rate-W relationship is established by optimizin (0 over the equalizers and weihts such that: ξ c, min u c,, c, ξ c, = R c, ξ (a min u, ξ = R (b where the optimum equalizers are iven by: c, = c, and =, and the optimum weihts are iven by: u c, = u c, ( ε c, and u = u (, ε obtained by checin the first order optimality conditions. By closely examinin each WMSE, it can be seen that it is convex in each variable while fixin the other two. B. WMSE Reformulation and Alorithm Motivated by the relationship in (, an equivalent WMSE reformulation of problem (7 writes as R (P : max r,r,c,p,,u s.t. r C m + r m r, m M ξ i r m, i G m, m M ξ c, M C m, K C m 0, m M p c + p m P ( where r and r {r,..., r M } are auxiliary variables, u {u c,, u K} is is the set of weihts, and { c,, K} is the set of equalizers. The equivalence between ( and (7 is established by observin that the WMSEs are decoupled in their equalizers and weihts. Hence, optimum and u are obtained by minimizin each WMSE separately as shown in (, yieldin the solution. The equivalence follows by substitutin ( into (. The WMSE problem in ( is solved usin an Alternatin Optimization (AO alorithm, which exploits its bloc-wise convexity. In a iven iteration of the alorithm, and u are firstly updated usin the optimum solution of (. Next, the set of precoders P alonside all auxiliary variables R in ( are updated by solvin (P, formulated by fixin and u in (. This is a convex problem which can be efficiently solved usin interior-point methods []. The steps of the AO procedure are summarized in Alorithm. Alorithm Alternatin Optimization : Initialize: n 0, P, r (n 0 : repeat 3: n( n + ( 4: c,,, 5: 6: (r (n ( ( c, uc,, u u c, 7: until r (n, r, c, P ar r (n < ɛ, u R (P, K, K Each iteration of Alorithm increases the objective function, which is bounded above for a iven power constraint, until converence. The lobal optimality of the limit point cannot be uaranteed due to non-convexity. However, the stationarity (KKT optimality of the solution can be arued based on the ideas in [3], avoided here due to space limitations. VI. SIMULATION RESULTS We consider i.i.d channels with entries drawn from CN (0,, and all results are averaed over 00 channel realizations. We compare the MMF rates for: conventional beamformin (No, Sinle-Stream (SS multiroup transmission described at the beinnin of Section IV, and 3 the stratey. The results for No and SS are obtained usin the SDR method in [], []. We plot the SDR upper-bounds (no randomization, hence presentin optimistic performances for No and SS. On the other hand, the results and obtained by Alorithm, and represent the actual performances. The MMF rates for a system with N = transmit antennas and M = roups with G = users each are presented in Fi.. As predicted from the DoF result in Proposition, No exhibits a saturatin performance. Both SS and achieve non-saturatin rates with DoFs of / and an improved rate performance for which comes from the desinated beams. The ains of over No are very pronounced. The results for a system with N = 4 transmit antennas and M = 3 roups with G = 3 users per roup are shown in Fi.. The benefits of usin desinated beams over SS transmission at low SNRs are clearer in this scenario, as the performance

5 Rate (bps/hz Fi.. Rate (bps/hz No (SDR UB SS (SDR UB SNR (db Fi.. N = antennas, K = 4 users, M = equal roups. No (SDR UB SS (SDR UB SNR (db N = 4 antennas, K = 9 users, M = 3 equal roups. of the latter is constrained by the worst out of 9 users. While SS achieves a DoF of /3, seems to surpass this DoF, which is evident from its slope at hih SNR. This suests that the trivial achievable lower-bound in Proposition is in fact loose, and can achieve even hiher MMF-DoF. VII. CONCLUSION In this paper, we proposed a multi-roup multicast beamformin stratey. We showed throuh DoF analysis that the proposed stratey outperforms the conventional beamformin stratey in overloaded scenarios, i.e. when the number of transmit antennas is insufficient to cope with inter-roup interference. An AO alorithm based on the W method was used to obtain the precoders. The effectiveness of the proposed alorithm and the sinificant ains associated with the stratey were demonstrated throuh simulations. Simulations also revealed that the trivial MMF-DoF lower-bound is in fact loose, which calls for a riorous characterization of the optimum MMF-DoF achieved throuh. APPENDIX Proof of d = 0 for N < N min : First, we write a precodin vector as p m = q m p m, where q m is the power and p m is the unit-norm beamformin direction. For a iven precodin scheme characterized by one precoder for each power level, the mth power scales as q m = O(P am with a m, further assumed to be non-neative as MMF necessitates non-vanishin powers allocated to all roups. Let I m M be the index set of roups that interfere with the mth roup, dependin on the precoder desin. From the DoF definitions in Section III-A, it can be shown that d m ( a m max j I m a j +. (3 The (. + can be omitted as when it is active, the MMF-DoF is limited to zero which is also achieved when all a m are equal. It is sufficient to show that the MMF-DoF is upper-bounded by 0 for N = N min = (M G, as decreasin the number of antennas does not increase the DoF. For this case, p m can be placed in the null space of at most M roups, i.e. each beam interferes with at least one roup. It follows that m M I m = M. We assume that each beam interferes with exactly one roup, as the contrary does not increase the DoF. It follows that at least two roups see non-zero interference. Let m be the index of the roup receivin the dominant interference, i.e. max{a m } m M I m, and m be the index of the roup receivin interference from m, i.e. a m I m. It follows from (3 that d m a m a m and d m a m a m. Since the symmetric-dof is upper-bounded by the averae of any number of roup DoFs, we write d dm +dm 0, which holds for any possible precoder desin. REFERENCES [] N. Sidiropoulos, T. Davidson, and Z.-Q. Luo, Transmit beamformin for physical-layer multicastin, IEEE Trans. Sinal Process., vol. 54, no. 6, pp. 39 5, Jun [] E. Karipidis, N. Sidiropoulos, and Z.-Q. Luo, Quality of service and max-min fair transmit beamformin to multiple cochannel multicast roups, IEEE Trans. Sinal Process., vol. 56, no. 3, pp , Mar [3] N. Bornhorst and M. Pesavento, An iterative convex approximation approach for transmit beamformin in multi-roup multicastin, in Proc. IEEE SPAWC, Jun. 0, pp [4] A. Schad and M. Pesavento, Max-min fair transmit beamformin for multi-roup multicastin, in Proc. Int. ITG WSA, Mar. 0, pp [5] D. Christopoulos, S. Chatzinotas, and B. Ottersten, Weihted fair multicast multiroup beamformin under per-antenna power constraints, IEEE Trans. Sinal Process., vol. 6, no. 9, pp , Oct. 04. [6] D. Christopoulos, S. Chatzinotas, and B. Ottersten, Multicast multiroup beamformin for per-antenna power constrained lare-scale arrays, in Proc. IEEE SPAWC, Jun. 05, pp [7] D. Christopoulos, S. Chatzinotas, and B. Ottersten, Sum rate maximizin multiroup multicast beamformin under per-antenna power constraints, in Proc. IEEE GLOBECOM, Dec 04, pp [8] H. Joudeh and B. Clercx, Robust transmission in downlin multiuser MISO systems: A rate-splittin approach, arxiv preprint arxiv: , 06. [9] H. Joudeh and B. Clercx, Sum-rate maximization for linearly precoded downlin multiuser MISO systems with partial CSIT: A rate-splittin approach, arxiv preprint arxiv: , 06. [0] S. Christensen, R. Aarwal, E. Carvalho, and J. Cioffi, Weihted sumrate maximization usin weihted for MIMO-BC beamformin desin, IEEE Trans. Wireless Commun., vol. 7, no., pp , Dec [] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, An iteratively weihted approach to distributed sum-utility maximization for a MIMO interferin broadcast channel, IEEE Trans. Sinal Process., vol. 59, no. 9, pp , Sept 0. [] S. P. Boyd and L. Vandenberhe, Convex Optimization. Cambride university press, 004. [3] M. Razaviyayn, M. Hon, and Z.-Q. Luo, A unified converence analysis of bloc successive minimization methods for nonsmooth optimization, SIAM Journal on Optimization, vol. 3, no., pp. 6 53, 03.