Fair scheduling and orthogonal linear precoding/decoding. in broadcast MIMO systems

Fair scheduling and orthogonal linear precoding/decoding in broadcast MIMO systems R Bosisio, G Primolevo, O Simeone and U Spagnolini Dip di Elettronica e Informazione, Politecnico di Milano Pzza L da Vinci, 3 I 133 Milano (Italy) Abstract In the downlink of a multi user MIMO system over a fading channel, the base station can assign the available spatial streams to different users by capitalizing on multiuser diversity or enforcing fairness constraints Assuming linear precoding (beamforming) at the base station, the problem amounts to the joint design of precoding matrices and channel aware scheduling, according to the cross layer paradigm In this paper we formulate the joint optimization of scheduling and linear precoding within a physical layer oriented framework, where the performance metric is the transmission rate Moreover, we constraint the precoding scheme to ensure interference free reception of spatial streams (Orthogonal Space Division Multiple Access, OSDMA) Fairness constraints are either inspired by the max min or the proportional fair criterion Performance of different schemes is evaluated by numerical simulations allowing to assess the tradeoffs between sum rate (multi user diversity) and fairness Keywords: channel aware scheduling, MIMO, downlink, precoding, fairness I INTRODUCTION In the downlink of a multi user system, the deployment of an antenna array at the base station (or access point) allows the simultaneous transmission to multiple users with controlled interference In particular if the base station is equipped with N T transmitting antennas and is provided with the channel state information of different users (ie, through a feedback link), up to N T single antenna terminals can be served simultaneously by N T independently encoded substreams More generally, for a multi user MIMO system, each terminal has an antenna array with, say, N R N T antennas, and thus spatial multiplexing can assign up to N R data streams to each user out of the total amount of N T available spatial channels [1] It is then the task of a scheduling algorithm to select within each time slot, say the tth, the subset of K(t) N T users to be served and the number of spatial streams d k (t) N R to be granted to each terminal Conventional design of the transmission strategy at the physical layer (ie, coding, beamforming and power allocation) assumes that the scheduling step is performed separately by higher layers according to quality of service requirements and system parameters The optimal transmission strategy, from an information theoretic point of view, employs joint encoding of the signal streams (Dirty paper coding) and linear spatial filtering [] Suboptimal and more practical algorithms that perform separate encoding of different streams have been proposed Specifically, a linear precoding/ decoding algorithm that for large signal to noise ratio maximizes the sum rate This is an invited paper to the special session on Cross layer scheduling for MIMO systems which is jointly organized by the IST NEWCOM and IST ACE projects under the assumption of independent encoding has been introduced in [3] (OSDMA) More recently, it has been recognized that remarkable performance benefits can be obtained by joint optimization of the transmission strategy at the physical layer and scheduling, according to the cross layer paradigm [] In this work, we tackle this joint optimization problem within a physical layer oriented paradigm as in references [1] [5] [] As a first step toward a fully cross layer approach, the scheduling algorithm is thus designed by considering physical layer (information theoretic) quantities, namely the transmission rate The aim can be either the maximization of the overall system performance (ie, the sum rate) with no concern about fair sharing of system resources among different users, or the enforcement of fairness constraints This framework, referred to as channel aware scheduling, does not take into account higher layer parameters such as transmission delays and buffer length Therefore, it is well suited for delayinsensitive applications where call admission algorithms run at higher layers guarantee a stable behavior of the users queues The state of the art on the subject has focused on independent encoding of different spatial streams for its practicality as compared to the Dirty paper approach In particular, channel aware scheduling aimed at maximizing the sum rate with straightforward linear processing at the base station (each spatial channel is directly assigned to a transmitting antenna) and linear interfaces at the receivers has been considered in [1] (for zero forcing equalizers) and [5] (for MMSE equalizers) More recently, reference [7] introduced fairness constraints in the design of channel aware scheduling for zero forcing precoding at the base station Here we focus on channel aware scheduling with and without fairness constraints for OSDMA Optimal design of OSDMA and channel aware scheduling with the aim of maximizing the sum rate, ie, exploiting multi user diversity, was studied in [] Here we gear the optimization toward the goal of granting widely used (longterm) fairness constraints such as proportional fairness and max min fairness [9] The three approaches are compared by simulation according to the framework derived from portfolio theory [], following to the proposal of [] This investigation allows to assess the trade offs between sum rate (multi user diversity) and fairness The paper is organized as follows The signal model and the main assumptions are presented in Sec II Joint precoding and channel aware scheduling that maximizes the sum rate is discussed in Sec III, whereas Sec IV introduces fairness constraints Finally, in Sec V numerical results are presented in order to evaluate the performance of different schemes in

x 1 x K M 1 M K H 1 H K n 1 n K y 1 y K H B 1 H B K ~ y1 y~ K equalization decoding equalization decoding Fig 1 Block diagram of a broadcast channel with linear interfaces at the transmitter (base station) and receivers (users) terms of trade off between sum rate and fairness II SIGNAL MODEL A broadcast channel with linear interfaces at the transmitter and receivers is depicted in fig 1 Let K be the set of K available users The base station (or access point) is equipped with an antenna array of N T elements, whereas each user has N R antennas The subset of K(t) users that are served by the base station within the tth time slot is denoted as K(t) K and its elements are indexed by k =1,,, K(t) The scheduler allocates d k (t) N R spatial channels to the kth user so that all the available N T spatial channels are used: X K(t) d k(t) =N T (1) k=1 The signal intended for the kth user, collected in the d k (t) 1 vector x k (t) is linearly precoded by the N T d k (t) matrix M k (t) Following the conventional notation for block fading channels (see, eg, []) and referring to fig 1, the signal received by the kth user across its N R receiving antennas within the tth time slot can be written as the N R 1 vector y k (t) y k (t) =H k (t)m k (t)x k (t)+ X i=k H k (t)m i (t)x i (t)+n k (t) i K(t) () where H k (t) is the N R N T channel matrix of the kth user and n k (t) is the zero mean additive Gaussian noise with E[n k (t)n k (t) H ]=σ ni NR The channel matrix H k (t) is assumed to be zero mean circularly symmetric complex Gaussian with independent identically distributed entries and variance ρ k The received signal y k (t) lies in a N R dimensional linear space However, only d k (t) N R spatial channels are assigned to the kth user Therefore, the useful part of the received signal spans a d k (t) dimensional subspace that we refer to as receiving subspace In order to account for this, at the receiver, the N R 1 received signal y k (t) is pre filtered by the d k (t) N R matrix B k (t) H ỹ k (t) = B k (t) H y k (t) = H k (t)m k (t)x k (t)+ + X i=k H k (t)m i (t)x i (t)+ñ k (t), (3) i K(t) wherewehavedefined the d k (t) N T equivalent channel H k (t) =B k (t) H H k (t) and ñ k (t) =B k (t) H n k (t) In order to simplify the analysis and without limiting the generality of the approach, we assume B k (t) H B k (t) =I dk (t), () so that E[ñ k (t)ñ k (t) H ]=σ ni dk (t) The range space of B k (t) corresponds to the receiving subspace for the kth terminal As a last step, equalization and detection is performed on ỹ k (t) Joint optimization of channel aware scheduling and transmission strategy amounts to i) the selection of the subset of K(t) users K(t), the number of spatial channels d k (t) and the corresponding receiving subspaces B(t) ={B k (t)} K(t) k=1 (channel aware scheduling); ii) the design of precoding matrices M(t)={M k (t)} K k=1 (precoding) In the following, we solve this problem by considering different optimization criteria As stated in the Introduction, it is assumed that separate encoding is performed on each spatial stream Moreover, the instantaneous power constraint tr(m im H i ) P (5) is enforced In Sec III, the considered optimization criterion is the maximization of the sum rate On the other hand, Sec IV introduces long term fairness constraints, namely proportional fairness and max min fairness Notice that throughout this work, we assume that the channel matrices H k (t) are known to the transmitter and receivers, eg, by transmission of pilot symbols and feedback of the channel state information from the receivers to the base station III MAXIMIZATION OF THE SUM RATE In this Section, we brie y review the results in [] related to the joint design of scheduling and precoding aimed at maximizing the sum rate in each time slot (therefore, the argument t is herein dropped for simplicity of notation) In this case, the problem can be formulated as follows: {B, M} = argmax C i(b, M) () B,M st (5), (1) and (), where C i (B, M) is the link capacity for the ith user [] C i (B, M) =log I di +R 1 i ( H i M i M H i H H i ) (7) with R i = σ ni di + X k=i H i M k M H k H H i () k K Notice that as a result of the optimization problem () the ith user belongs to the set of active users K if d i > or equivalently (M i, B i ) are not empty matrices Solution of the optimization problem () is not known, even for the case of given scheduling, ie, for given sets K and B Following [3], in order to simplify the problem, in [] it was proposed to set the additional zero interference (OSDMA) constraint among the users H i M j = if i = j, (9)

which does not affect optimality for large signal to noise ratios As a direct consequence of (9), the precoding matrix M k has the following form M k = V k Q k, () where V k is a N T d k orthonormal matrix matrix selected so as to ensure (9) and Q k is a d k d k matrix that performs beamforming and power allocation on the interference free single user MIMO channels created by OSDMA Therefore, it is easy to conclude the optimization of matrices V ={V k } K k=1 and {Q k } K k=1 can be carried out separately In particular, once the set V and B are given, matrix Q k can be easily computed so as to satisfy () from the known results regarding singlelink MIMO systems [] To be specific, Q k can be factorized as Q k = U k D k, (11) where U k is a d k d k orthonormal matrix obtained from the right eigenvectors of the equivalent channel matrix H k and D k is a diagonal matrix that defines the power allocation over the d k spatial channels assigned to the kth user Power allocation amounts to a simple multi user filling as discussed in [3] We are then left with the problem of jointly designing V and scheduling B If the condition N T KN R is satisfied, there is no need for the scheduling step since all the KN R deployable spatial channels can be allocated In this case, optimality () with the additional constraint (9) is guaranteed by setting the number of spatial channels for each user to d k = N R and B k = I NR Computation of V k then amounts to the evaluation of the null space of the compound channel matrices of all the users j = k We refer to [3] for details In the practical case where the number of transmitting antennas is not large enough to guarantee the allocation of all the deployable spatial channels (ie, N T < KN R ), joint optimization of the sets V and B isnecessarytosolve() with the constraint (9) In [3] the problem was simplified by carrying out a separate optimization of the two sets Here, to get a feasible problem, we propose to simplify the optimization by considering the first term Taylor expansion for capacity C i (B, M) 1/σ n B H i H im i The resulting joint optimization problem reads {B, V} = argmax N i(b, V) () B,V st (1), () and (9), where N i (B, V) = B H i H iv i The objective function () is amenable to the efficient greedy numerical optimization proposed in [], to which we refer for further details IV LONG TERM FAIRNESS CONSTRAINTS The joint design of channel aware scheduling and precoding proposed in the previous Section aims at maximizing the same rate Therefore, whenever different users have unbalanced channel conditions, the discussed solution tends to favour the users with the more advantageous propagation conditions, leading to an unfair sharing of resources In this Section, two fairness criteria are taken into account, that attempt to strike a balance between the long term rates T k of different users, defined as T k (B(t), M(t)) = lim inf 1 X t t C k(b(t), M(t)) (13) t In Sec IV A, system design under the max min fairness criterion is considered whereas Sec IV B the proportional fair criterion is discussed A Max min fairness The simplest way to enforce a (long term) fairness constraint among the users is to require the long term rates of all users to be equal: T k = T j for every j and k This condition can be stated analytically as the following optimization problem [9] {B(t), M(t)} = max min B(t),M(t) i T i (B(t), M(t)) st (5), (1), () and (9) (1) and is known as max min fairness A joint channel aware scheduling and precoding that solves this problem is not known However, here we consider a simple algorithm that approximately enforces the istantanous equal rate condition as follows For each time slot t, perform a random scheduling that selects a subset K(t) of K(t) =K users, where KN R = N T (assuming N T to be an integer multiple of N R ) Therefore, all the N T available spatial channels can be assigned by setting d k (t) =d k = N R for k =1,, K and consequently B k (t) =I NR as explained in Sec III Moreover, the factors V k and U k of the precoding matrices M k (recall () (11)) are then obtained according to Sec III On the other hand, the power allocation matrices D k are computed so as to ensure the equality among the instantaneous rates C i (B(t), M(t)) of the selected users This last step amounts to solve the following optimization problem (define D(t) ={D k (t)} K k=1 ) D(t) =max min X NR log (1 + D ijλ ij D(t) i j=1 σ ), (15) n where D ij is the jth diagonal element of D i (t) and λ ij the jth eigenvalue of the equivalent single user channel matrix H k V k This is a convex problem that can be solved by standard techniques [11] The performance of this algorithm will be evaluated in Sec V through numerical simulations B Proportional fairness The proportional fair criterion enforces a weaker requirement on the long term rates of all users as compared to the max min criterion The idea is that each user should be granted a transmission rate that is fair as compared to its long term channel conditions More precisely, the transmission rate toward each terminal, say T k, should be such that the ratio T k /T j for j = k equals the ratio between the corresponding long term (or ergodic []) single users capacities

E[log I di +P/K H i H H i ] The problem can be stated as following optimization: {B(t), M(t)} = argmax log T i(b(t), M(t))() B(t),M(t) st (5), (1), () and (9) Similarly to [], a scheduling procedure that is expected to (approximately) converge to the solution of () works as follows For each time slot t apply the algorithm described in Sec III that maximizes the sum rate on normalized channel matrices with α i (t) = H N i (t) = H i(t) p αi (t) (17) µ1 1tc ˆα i (t)+ 1 t c N i (B(t 1), V(t 1)), (1) where parameter t c rules the memory of the algorithm Parameter α i (t) measures the channel power that each user has been allowed to use within a window of t c time slots The rationale of the algorithm is that if a given user has been ignored by the scheduling procedure in the considered time window, the channel scaling (17) will force the transmitter to allocate resources to it The convergence properties of the algorithm were studied by simulation in [], where it is shown that the scaling (17) (1) actually guarantees that the proportional fair criterion is satisfied by the long term rates T k V NUMERICAL RESULTS In this Section, the downlink of a MIMO system with N T = transmitting antenna at the base station and N R =antennas at each terminal is simulated Where not stated otherwise, the variance ρ k of the channel entries for each user is selected as ρ k =1/(k ) for k =,, K 1 Therefore, the propagation conditions experienced by different users are such that the channel norms can be ordered for decreasing values so that they differ by 3dB At first, the long term rates T k are plotted versus the signal to noise ratio, defined as SNR = P/σ n, for K =users in fig As expected, the algorithm the maximizes that sumrate presented in Sec III (max sum rate) yields the most unfair sharing of transmission rates (upper figure) whereas the approximate max min algorithm discussed in Sec IV A provides similar long term rates for all the users (lower figure) Exact equal rates can be achieved by devising an alternative max min approach (based on an utility function) that takes into account channel conditions while performing scheduling as proposed in [13] On the other hand, the proportional fair algorithm () represents a trade off between the two solutions (middle figure) Dashed lines denote the fraction of the long term sum rate T T = KX T i (19) as distributed among the users according to the ratio between the single user capacities From the discussion in Sec IV B, 1 1 1 5 15 SNR [db] 5 3 Fig Long term rates T k versus SNR for the three schemes: max sum rate (upper), (middle), max min (lower figure) (N T =,N R =, K =) this proves (via simulation) that the proposed algorithm is able to enforce the proportional fairness conditions The performance of the three schemes is further investigated below Following to the proposal of [], the three schemes for channel aware joint precoding and scheduling discussed in the previous Sections are compared by numerical simulation within the framework derived from portfolio theory [] In particular, in order to assess both the overall performance and the fairness properties of the system, the algorithms are evaluated in terms of both the long term sum rate T (19) and the standard deviation σ, where σ = 1 K KX T T T T T T (T i T K ) () The latter quantity measures the average (over the users) dispersion of the long term rates with respect to the mean T/ K Fig 3 shows the sum rate T versus the dispersion σ for the three schemes as a function of the signal to noise ratio SNR The max sum rate algorithm yields for each SNR the larger sum rate but at the expense of a larger dispersion () In other words, the increase in sum rate entails an increasingly unfair sharing of the system resources On the other hand, the max min algorithm is able to drastically reduce the dispersion σ (exact max minfairnesswouldimplyσ =for all SNR), thus enforcing a fairness constraint on the long term rates

5 19 1 17 σ 3 SNR = 5dB 1 SNR = db T 15 1 13 5 15 5 3 35 T 3 5 7 9 K ( Fig 3 Sum rate T versus standard deviation σ () for different SNR (arrows denote the direction of increasing SNR from db to 3dB) (N T =, N R =, K =) However, this redistribution of resources causes an equally dramatic decrease of the sum rate Halfway between max sumrate and max min fairness lies the proportional fair criterion As discussed in Sec IV B, this alleviates the fairness constraint imposed by the max min principle by allowing a controlled dispersion σ according to the long term channel conditions of each user In fact, fig 3 shows that the proportional fair algorithm allows to increase the sum rate as compared to maxmin at the expense of an increase of the dispersion of the long term rates The performance of the three schemes is then evaluated in fig by varying the number of users K available in the system for a fixed SNR =15dB In this case, the channel variance is assumed to be the same for all users, ρ k =1, in such a way to better evaluate the impact of multi user diversity [] From fig, the max sum rate algorithm is able to exploit the multiuser diversity by scheduling in each time slot the users with the best channel conditions This effect is apparent in the increase of the sum rate T for increasing number of users K However, this performance benefit is attained at the expense of an unfair allocation of resources, as explained above Fairness constraints such as max min and proportional fair prevent the base station from being able to fully exploit multi user diversity In fact, they force the scheduling algorithm to serve a given user irrespective of its channel state in order to fulfill the fairness criteria This behavior is confirmed by fig VI CONCLUDING REMARKS Joint optimization of linear precoding and channel aware scheduling has been considered within a physical layeroriented framework where the performance metric is the transmission rate Both the maximization of the sum rate, which exploits multi user diversity, and fairness constraints, namely max min and proportional fairness, have been taken into account with the additional constraint on the precoding scheme to ensure zero inter user interference The performance Fig Sum rate T versus the number of users K for SNR = 15dB (N T =,N R =) analysis of different schemes has been carried out by numerical simulations, showing the trade off between multi user diversity and fairness criteria This physical layer oriented approach is a first step toward a fully cross layer framework in which physical parameters (eg, linear precoding) and scheduling are jointly optimized by using higher layer performance criteria, such as buffer length and queuing delay REFERENCES [1] R W Heath Jr, M Airy, A J Paulraj, Multiuser diversity for MIMO wireless systems with linear receivers, Proc Asilomar Conf Signals, Systems and Computers, pp 119 1199, Nov 1 [] A Goldsmith, S A Jafar, N Jindal and S Vishwanath, "Capacity limits of MIMO channels," IEEE J Select Areas Commun, vol 1, no 5, pp 7, June 3 [3] Q H Spencer, A Lee Swindlehurst and M Haardt, "Zero forcing methods for downlink spatial multiplexing in multiuser MIMO channels", IEEE Trans Signal Processing, vol 5, no, pp 1 71, Feb [] R A Berry and E M Yeh, "Fundamental Performance Limits for Wireless Fading Channels," IEEE Signal Processing Magazine, special issue on "Signal Processing for Networking," vol 1, no 5, pp 59, September [5] O Shin and K Bok, Antenna assisted round robin scheduling for MIMO cellular systems, IEEE Comm Letters, vol 7, no 3, pp 9 111, March 3 [] D Bartolomé, "Fairness analysis of wireless beamforming schedulers," PhD dissertation, Universitata Politecnica de Catalunya (UPC), Nov [7] DBartolomé,AIPérez Neira, "A unified fairness framework in multiantenna multi user channels," Proc IEEE International Conference on Electronics, Circuits ans Systems, [] GPrimolevo,OSimeone,USpagnolini, Channelawarescheduling for broadcast MIMO systems with orthogonal linear precoding and fairness constraints, Proc IEEE International Conference on Communications, May17, Korea, 5 [9] A Banchs, "User fair queuing: fair allocation of bandwidth for users", Proc of IEEE INFOCOM, June [] H Markowitz, "Foundation of portfolio theory," The Journal of Finance, vol, no, pp 9 77, June 1991 [11] S Boyd and L Vanderberghe, Convex optimization, Cambridge University Press, [] P Viswanath, D N C Tse and R Laroia, "Opportunistic Beamforming UsingDumbAntennas,"IEEE Trans Inform Theory, vol,no,pp 77 9, June [13] G Song and Y Li, "Cross layer optimization for OFDM wireless networks Part I: Theoretical framework", IEEE Trans Wireless Comm, Vol, no,pp 1, March 5