Minimum Rates Scheduling for MIMO OFDM Broadcast Channels

IEEE Ninth International Syposiu on Spread Spectru Techniques and Applications Miniu Rates Scheduling for MIMO OFDM Broadcast Channels Gerhard Wunder and Thoas Michel Fraunhofer Geran-Sino Mobile Counications Lab, Heinrich-Hertz-Institut Einstein-Ufer 37, D-587 Berlin {wunder,ichel}@hhi.fhg.de Abstract In this paper we study the ultiple input ultiple output (MIMO orthogonal freqency division ultiplexing (OFDM Gaussian Broadcast Channel (BC. Several fundaental probles are considered: The axiization of a weighted su of rates and the dual iniization of su power subject to rate requireents. Further we study the cobined proble of weighted rate su axiization under iniu rate requireents. To show the connections aong the, all probles are ebedded into an enhanced convex set of rate-power tuples (R,P. This not only gives insights into the structure of the MIMO OFDM BC capacity region, but also otivates algoriths exploiting these properties. I. INTRODUCTION It was not until recently that the capacity region of the MIMO Gaussian BC was copletely solved. In the line of work [] [3] it was shown that the MIMO BC capacity region in fact equals the MIMO BC dirty-paper-coding region and thus is equivalent to the capacity region of a dual MIMO ultiple access channel (MAC with heritian transposed channels. These results are of fundaental interest, since they allow to carry out any analysis and optiization of the MIMO BC in the dual MAC and to carry over the results to the downlink via the corresponding duality relations []. Based on the established duality, soe iportant resource allocation results can be derived: It is known that the boundary (i.e. the efficient points of the capacity region can be paraeterized by a weighted rate suaxiization over the capacity region. Further, an algorith ore efficient than plain convex optiization for weighted su rate axiization in the MIMO MAC based on a siplified gradient approach exists []. Substituting the weights in the weighted rate su-optiization by the corresponding queue-length yields the stability optial transission policy [5], which can be carried over to the MIMO BC via duality. For the special case of throughput axiization under a power constraint an efficient algorith based on an enhanced waterfilling principle was presented in []. However, iportant questions reain open. So the inverse proble of iniizing the su power while achieving given required rates was faced not until recently [7], [8] and weighted rate su-optiization with iniu rates to be guaranteed was not considered up to now. In [9] the case of * The authors are supported in part by the Bundesinisteriu für Bildung und Forschung (BMBF under grant FK BU 35 supporting su rate requireents with iniu power was solved. For the special case of parallel Gaussian channels with a single antenna, both probles were studied recently using the notion of arginal utility functions fro [] in [], []. In this paper, we study the probles of weighted rate su-axiization under a power constraint and su power iniization under given rate requireents extending the work fro [3]. First, it is shown that finding the iniu su power can be forulated directly as a convex progra. Then, even though the analytic structure of the proble in the single antenna case can not be applied to the MIMO case, we exploit ideas derived in [], [] to find the iniu su power. Further, the structural insights allow to face the weighted rate su-axiization under given iniu rates. Finally, we coent on the use of subgradient ethods for this proble. The reainder of the paper is organized as follows. Section II introduces the syste odel. In Section III the probles are stated and basic properties are derived. Subsequently, in Section IV and V the su power iniization and the iniu rates proble are studied, respectively. We conclude with Section VI. II. SYSTEM MODEL Consider a frequency selective MIMO OFDM Gaussian BC with K subcarriers, a base station equipped with U antennas and M obiles each owning antennas. In the following the set of users will be denoted as M = {,..., M}. Assue a block fading process with independent and identical fading fro block to block and let the fixed channel realization of user Mon subcarrier k during a tie block be H H,k C V U where [H H,k ] i,j is the coplex channel coefficient between antenna j of the base station and antenna i. Then the signal received by user on subcarrier k is given by y,k = H H,kx k + n,k k =,..., K. ( with x k C U and y,k C V being the transitted and received signals, respectively. Here n,k C V denotes the white Gaussian receiver noise with E{n,k n H,k } = I V with I n being the identity atrix of diension n. Now consider the following ultiple access channel r k = H,k s,k + z k k =,..., K ( = -783-978-//$. IEEE 5

with r k C U, s,k C V and (noralized additive white Gaussian noise E{z k z H k } = I U. This syste is coonly called the dual MAC to the BC in ( and it is known that their capacity regions under a coon su power constraint P coincide : Stacking the channel atrices into a single atrix H =[H,,..., H M,K ] T, we have fro [3] C BC (H, P C MAC (H H, P (3 where the capacity region of the Gaussian MIMO OFDM MAC is given by C MAC (H, P = P,k tr(q,k P Q,k with R(I := I R and f H,Q (I := { R : R (I f H,Q (I, I M ( K log det I + H,k Q,k H H,k I k= }. ( being a rank function. Here, A eans that A is a positive seidefinite atrix, tr( denotes the trace operator and Q,k = E{s,k s H,k } is the covariance atrix of the th user s signal on subcarrier k stacked to the atrix Q = [Q,,..., Q M,K ] T for brevity. Any rate tuple R C BC (H, P achievable in the MIMO OFDM BC is also achievable in the dual MAC and vice versa. Further, the transforation laws relating the covariance atrices achieving the sae rate tuple in MAC and BC are known []. Note that the characterization of the dual MIMO OFDM MAC in ( involves the polyatroidal structure of eleentary capacity regions - i.e. capacity regions for a fixed set of covariance atrices - which can be exploited in ultiple ways. Thus we can focus on the dual MAC in the following and the subscripts MAC and BC will be oitted. III. PROBLEM STATEMENT AND BASIC PROPERTIES The fundaental proble of axiizing a weighted su of rates yields a characterization of the boundary of C(H, P and was studied in [3] for the MIMO OFDM BC: Proble : ax μ T R subj. to R C(H, P Here μ =[μ,..., μ M ] T is the vector of individual weights. Proble can be turned into a convex proble with trace constraints via the following well known Lea. Lea : The solution to Proble is equivalent to solving ax μ T R (7 S πn (H, P (5 ( where the set S πn (H, P denotes the region achievable by one specific decoding order { S πn (H,P= R : R πn( P,k tr(q,k P Q,k } f H,Q ({π n (,..., π n (} f H,Q ({π n (,..., π n ( }, (8 where π n Π is a perutation sorting the μ in decreasing order: μ πn( μ πn(... μ πn(m. (9 The proof exploits the polyatroidal structure of eleentary capacity regions. We want to consider two different iportant resource allocation strategies, which contrast with the pure weighted rate suaxiization. Although this strategy is known to be stabilityoptial in the sense of non-evanescence of queues [5], it ight penalize users with a low queue but strict Quality of Service deands. Thus, we first consider the proble of finding the optial transit strategy and therefore the iniu transit power for achieving a set of required rates R =[ R,..., R M ] T. This proble can be forulated as: Proble : in P ( subj. to R C(H,P Geoetrically speaking, we search for the sallest possible capacity region still containing the desired rate vector R. If the su power is liited to a fixed budget P in each fading state and stability issues can not be neglected copletely, both perspectives can be cobined. This leads to the proble of weighted rate su-axiization under given iniu rates R (the inequality refers to coponent-wise greater or equal: Proble 3: ax μ T R ( R C(H, P To solve the presented probles, we ebed the in a higher diensional space, leading to a forulation siilar to that in [], []. First we need the following definition. Definition : Let G(H be the set that contains all feasible rate-power tuples for given channel realizations H: G(H :={(R,P:R C(H,P}. ( Obviously the boundary of G(H are the points we are interested in, since they doinate all points in the interior. This eans, for all (R,P in the interior of G(H at least one coponent of R can be increased while all others reain fixed without leaving the set. In analogy to the single antenna case [] we can state the following lea. 5

Lea : The set G(H is a convex set and duality holds also for the expanded set. The proof follows easily fro the concavity of f H, (I over the set of positive seidefinite atrices Q. Note that the lea can be also iediately seen if the objective function and the affine trace constraint is ade explicite by using Lea. The boundary of the set G(H for rando channels H is depicted in Figure. The convexity is obvious. Note, that each horizontal slice represents the capacity region for a specific su power P. P (linear 5 5 3 5 3 5 R R Fig.. Boundary of the region G(H for a exeplary MIMO BC with M =users, U = V = V =antennas and rando channel realizations. IV. MINIMUM SUM POWER In this section we study Proble. There is a siple approach to solve this proble. Following the polyatroid structure of the MIMO OFDM MAC capacity region the proble can be written explicitely as in G(H,μ P μt R subj. to f H,Q (I R(I, I M. (3 Note that the nuber of constraints is M since for each subset of users (in fact each face of the polyatroid a constraint has to be et. This proble is a convex porgra and can be solved using standard convex optiization tools such as the YALMIP package [5]. Since the nuber of constraints grows exponentially with M, the proble becoes very coplex even for oderate nubers of users. In the following, a different approach is chosen. Equivalently to (3, the subsequent proble can be considered: in G(H,μ P μt R ( Unfortunately, the notion of arginal utility functions introduced in [] can not be applied to the MIMO (OFDM BC. However, we will have a look at the objective function in (. To this end we define [ ] R (μ, H = arg ax G(H μt R P (5 to be the th rate-coponent of the optiizing set ( R, P and propose Algorith to solve the iniu su power proble. Algorith MIMO-OFDM Su Power Miniization ( initialize μ ( = while desired accuracy not reached do for =to M do ( find upper bound μ (i+ on μ (i+ such that R ([μ (i (3 find μ (i+ end for end while R ([μ (i,..., μ(i+,..., μ (i M ], H > R by bisection such that,..., μ(i+,..., μ (i M ], H = R Note that the desired rate vector R ight not be a vertex of the optial polyatroid but ay lie on the su capacity plane (iagine the red circle in Figure 3 lying on the front plane. Then after convergence of the Algorith (which is indicated by very sall variations of the weights, a linear syste of M! equations has to be solved deterining the tiesharing factors {α i } M! i= with i α i =corresponding to the M! vertices (encoding orders of the plane. R = α R π +... + α M! R πm! ( For this special case where only a su rate constraint is active, further a fast iterative algorith is presented in [9] based on an iterative water-filling procedure. Interestingly, the optial Lagrangian ultipliers ˆμ reveal the necessary Dirty-Paper Precoding order for the MIMO OFDM BC: Since ˆμ constitutes the noral vector of a tangent hyperplane to the rate vector R, the optial encoding order (and reverse decoding order in the dual MAC π is given by the ordering of the Lagrangian factors such that: ˆμ π(... ˆμ π(m. (7 This is in analogy to the weighted rate su axiization. The convergence properties of Algorith is discussed in the following section. V. MINIMUM RATES The developed ethodology can be odified to guarantee iniu rates in the weighted su rate axiization. In analogy to ( and ( Proble 3 can be ebedded in the 5

enhanced set G(H. A reforulation is given by: in λp (μ + μ T R G(H λ,μ P P (8 Equation (8 reveals the intiate connection to Proble. However, the su power is liited in this case and the initial weights μ constitute an additional value to the Lagrangian factors μ. In analogy to (5 define [ ] ˆR (μ, H = arg ax (μ + μ T R λp (9 G(H,P P and, siilarly, we can use Algorith for Proble 3. Obviously, if the algorith settles down and delivers soe weight vector we have found a solution to the proble. Note, however, that (apart fro the two user case and the single antenna case [], [] convergence of the algorith is not fully clear yet. In fact we need to establish the following: Let μ (, μ ( be two weight vectors differing only in the nth coponent, i.e. μ ( n >μ ( n,μ ( = μ ( for all n. Suppose that the optiizing rates are related by ˆR (μ (, H ˆR (μ (, H n ( and ˆR n (μ (, H ˆR n (μ (, H. ( Then, assuing feasibility convergence follows fro a siple onotonicity arguent. At first, it is clear that by the convexity and by the fact that the user-wise extree point are achieved on the coordinate axes that for μ ( n > μ ( n, and μ ( = μ (, n, the rate R n ust increase indicated by condition (. More involved is condition (. Here, observe that (while fixing the other coponents the functions R n (R, n, are onotonously decreasing and convex. Now, suppose for the oent differentiability. Since any point on the boundary of the region is the solution to ax μ T R we have fro the optiality conditions R n / R = μ /μ n. Hence, if we increase μ n while fixing the other weights we increase the slope in the regarded direction at the osculation point of a tangent hyperplane with noral vector μ. However, although quite intuitive, the trajectory on the boundary ay not be such that the rates decrease. A rigorous proof has not yet been found. The optial Dirty-Paper-Encoding order is again a byproduct of the optiizing Lagrangian factors μ : In contrast to Proble, now the ordering of the su of weights μ and Lagrangian factors μ constitutes the optial encoding order π: μ π( + μ π(... μ π(m + μ π(m. ( Note, that this fact was already observed for the OFDM case in []. Regarding coputational coplexity, step (3 of both algoriths although seeing siple contains a considerable coputational challenge. In case of Algorith this proble Algorith MIMO-OFDM Miniu Rates Algorith ( initialize μ ( = while desired accuracy not reached do for =to M do if ˆR (μ (i, H < R then ( find upper bound μ (i+ on μ (i+ such that end if end for end while ˆR ([μ (i (3 find μ (i+ ˆR ([μ (i,..., μ(i+,..., μ (i M ], H > R ( by bisection such that,..., μ(i+,..., μ (i M ], H = R (3 can be solved by standard interior point ethods while for Algorith the algorith fro [] can be used. However, both Algorith and consist of a sequence of convex optiization probles bringing along a high coputational deand. Recently, an interesting subgradient ethod was proposed in [8]. We can substantially odify this approach for the proble at hand []. Define the function g (μ := ax R C BC(H, P μ T R + = μ ( R R for μ i. It is clear that g (μ ax R CBC(H, P μ T R for any such μ i. Hence, we can iniize g (μ over all possible μ. To do so the ellipsoid ethod can be invoked. Let R be the solution for μ: R =arg ax R C BC(H, P μ T R + = μ ( R R Then a subgradient is easily found by observing that g (ˆμ g ( μ + = ( (ˆμ μ R R. Thus we can iniize the Lagrangian dual function. This approach will be intensively studied in []. Let us illustrate the presented algoriths. Figures illustrates the convergence behavior of the Miniu Rates Algorith for a rando channel with K =subcarrier, U = antennas at the base station and V = V = V 3 =antennas at each obile. The required rates are set to R =[] T bps/hz. Figure shows that after a oderate nuber of iterations the algorith converges. Figure 3 illustrates the polyatroid with the rate tuple R = [ 5. ] T bps/hz corresponding to the optial covariance atrices. This vertex is achieved by the encoding order 3. VI. CONCLUSIONS We provided an analysis of resource allocation schees for the MIMO OFDM Broadcast Channel. Copleentary 53

.8.7. + *.5..3.. User User User 3 3 5 7 8 9 Iterations R 3 3 R R R 5 3 User User User 3 3 5 7 8 9 Iteration Fig.. Convergence of weights plus Lagrangian ultipliers μ + μ and achieved rates R in Algorith. to the weighted rate su-axiization, we considered the probles of iniizing the su power to achieve a certain set of rates and to axiize the weighted rate su for a fixed power budget and iniu required rates. For both probles we presented algoriths. To this end we odified concepts known fro the single antenna case. Further the optial Dirty- Paper Encoding order was derived and shown to consist of the decreasing ordering of Lagrangian ultipliers μ and in the first case and the decreasing ordering of the su of Lagrangian ultipliers and weights (μ + μ in the iniu rates case. REFERENCES [] S. Vishwanath, N. Jindal, and A.J. Goldsith, On the capacity of ultiple input ultiple output broadcast channels, in Proc. IEEE Int. Conf. on Counications (ICC, New York, April. [] H. Weingarten, Y. Steinberg, and S. Shaai (Shitz, Capacity region of the degraded MIMO broadcast channel, in Fourth ETH-Technion Meeting on Inforation Theory and Counications, Zürich, Switzerland, Feb 5-7. [3] H. Weingarten, Y. Steinberg, and S. Shaai (Shitz, The capacity region of the Gaussian MIMO broadcast channel, in Proc. Conf. on Inforation Sciences and Systes (CISS, Princeton, NJ, Mar. [] H. Viswanathan, S. Venkatesan, and H. Huang, Downlink capacity evaluation of cellular networks with known interference cancellation, IEEE Journal on Selected Areas in Co., vol., no. 5, pp. 8 8, 3. Fig. 3. Polyatroid corresponding to the set of converged covariance atrices fro Figure. The optial rate tuple with encoding order 3 fulfilling the rate constraints R =[]bps/hz is indicated by a red circle. [5] H. Boche and Marcin Wiczanowski, Stability-Optial Transission Policy for Multiple Antenna Multiple Access Channel in the Geoetric View, EURASIP Signal Processing Journal, Special Issue on Advances in Signal Processing-assisted Cross-layer Designs,, to appear. [] N. Jindal, W. Rhee, S. Vishwanath, S.A. Jafar, and A. Goldsith, Su power iterative water-filling for ulti-antenna Gaussian broadcast channels, IEEE Trans. Infor. Theory, vol. 5, no., pp. 57 58, Apr 5. [7] C.-H. F. Fung, W. Yu, and T.J. Li, Multi-antenna downlink precoding with individual rate constraints: power iniization and user ordering, in International Conference on Counication Systes (ICCS, Singapore,, pp. 5 9. [8] J. Lee and N. Jindal, Syetric capacity of MIMO downlink channels, in Proc. IEEE Int. Syp. Inforation Theory (ISIT, Seattle, USA,. [9] T. Michel and G. Wunder, Su Rate Iterative Water-Filling for Gaussian MIMO Broadcast Channels, in Proc. Intern. Syp. On Wireless Personal Multiedia Counications (WPMC, San Diego,. [] D. Tse, Optial power allocation over parallel Gaussian broadcast channels, unpublished, available at http://www.eecs.berkeley.edu/ dtse/broadcast.pdf, 998. [] T. Michel and G. Wunder, Miniu rates scheduling for OFDM broadcast channels, in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc. (ICASSP, Toulouse, May. [] G. Wunder and T. Michel, Optial resource allocation for OFDM ultiuser systes, subitted for publication,, available at: ftp://ftp.hhi.de/ichel/wundermicheltransp.pdf. [3] T. Michel and G. Wunder, Optial and low coplexity suboptial transission schees for MIMO-OFDM broadcast channels, in Proc. IEEE Int. Conf. on Counications (ICC, Seoul, May 5. [] D.N.C. Tse and S.V. Hanly, Multiaccess fading channels - part I: Polyatroid structure, optial resource allocation and throughput capacities, IEEE Trans. Infor. Theory, vol., no. 7, pp. 79 85, Nov 998. [5] J. Löfberg, YALMIP : A toolbox for odeling and optiization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan,, Available fro http://control.ee.ethz.ch/ joloef/yalip.php. [] T. Michel and G. Wunder, Miniu Power and Miniu Rates - Resource Allocation in MIMO-OFDM Broadcast Channels, in preparation,. 5