ITERATIVE WATERFILLING FOR WEIGHTED RATE SUM MAXIMIZATION IN MIMO-MAC

ITERATIVE WATERFILLING FOR WEIGHTED RATE SUM MAXIMIZATION IN MIMO-MAC Mari Kobayashi and Giuseppe Caire Centre Tecnològic de Telecomunicacions de Catalunya, Barcelona, Spain University of Southern California, Los Angeles, CA, USA E-mail: mari.obayashi@cttc.es, caire@usc.edu ABSTRACT We consider the weighted sum rate maximization in Gaussian MIMO multiple access channel (MAC) under individual power constraints. This problem arises in the stabilitywise optimal scheduling policy that allocates the resource as a function of the buffer queue states and the channel matrices in each time slot. The straightforward generalization of Yu et al. s well-nown iterative waterfilling algorithm for the sum rate maximization is non-trivial because the problem cannot reduce to decoupled single-user waterfilling-type solutions with arbitrary weights. Therefore, we propose a simple alternative treating multiple antennas at each transmitter as virtual single-antenna transmitters, which enables a iterative waterfilling-type algorithm. For a special case such as a OFDM-MAC, the proposed algorithm converges to the optimal solution faster than a steepest ascent algorithm and maes the convergence speed independent of the number of subcarriers.. MOTIVATION We consider the queued uplin system where K transmitters equipped with N antennas communicate with a M-antenna receiver. The received signal is given by y = H x + n () = where each component of H C M N is Gaussian with zero-mean and unit variance and i.i.d. over, n CN(, I M ) is AWGN. We assume that the channel matrices vary according to an ergodic bloc-fading model and remain constant over T seconds (such interval of T seconds will be referred to as a slot ). We assume that each transmitter has a short-term individual power constraint, i.e. for each channel realization the covariance matrix of the transmit vector x C N satisfies tr(s ) P, S (2) This wor was partially funded by the Generalitat de Catalunya under grant SGR25-69. where S denotes a positive definite matrix. We let P =(P,...,P K ) denote the required power vector. Furthermore, we assume that each user generates random traffic with stationary ergodic arrival process A (t): at each slot t, a pacet of A (t) bits enters the transmission queue of user, with an average arrival rate of λ = T EA (t) bit/sec. The arrived bits are locally stored into K queues, each associated to one user. The buffer size of user in slot t is denoted by q (t) in bit. With perfect channel state information at transmitter (CSIT), a centralized resource allocation policy observes the buffer queue states {q (t)} and the channel matrices H(t) = {H (t)} at the beginning of each slot t in order to stabilize the queue buffers. Under the assumption of random pacet arrival, the stabilization of the buffers is the single most important criterion for fairness. It is proved in, 2, 3, 4 that for any arrival rate inside the stability region, there exists an adaptive policy that stabilizes the all buffers without a priori nowledge of the arrival rates. Such adaptive policy (referred to as max-stability policy ) is given as follows, 2, 3, 4: in each slot, allocate the covariances in order to maximize the weighted rate sum K = q R subject to R C(H; P), where {θ } is a set of arbitrary positive coefficients and C(H; P) denotes the capacity region of the MIMO-MAC under individual power constraints P for a given channel H, given by C(H; P) = {R R K : (3) tr(s ) P,S, R log I + H S H H K {,...,K}} K K where the union is over all possible sets of the input covariances satisfying the constraints in (2). It is also proved in, 2, 4, 3 that under mild conditions on the joint ergodicity and stationarity of the arrival and channel processes, the stability region achieved by the max-stability policy coincides

with the ergodic capacity region, given by C(P) = {R R K : (4) tr(s ) P,S, " X R E log I + X # H S H H, K {,...,K}} K K The above stability results motivate the study of efficient algorithms for the weighted sum rate maximization problem under individual power constraints: max w R, subject to R C(H; P) (5) for an arbitrary set of non-negative weight coefficients {w }. This optimization problem has been studied in literature for some special cases. In 5, Tse and Hanly solved the problem in a single-antenna MAC under both the short-term and long-term user power constraints. In 6, Yu et al. proposed an iterative waterfilling algorithm for the sum rate maximization in the MIMO-MAC. In 4, the authors studied the problem from the stability point of view and proved that the solution of (5) is found in the set of successively decodable rate points with the stability-optimal decoding order. This paper aims to generalize Yu et al. s well-nown iterative waterfilling 6 for the sum rate maximization to the case of arbitrary weights. The problem is non-trivial and in particular cannot reduce to decoupled single-user waterfilling solutions, because the input covariance cannot be generally diagonalized. Hence, we propose a simple alternative treating multiple antennas at each transmitter as virtual single-antenna transmitters, which enables a iterative waterfilling-type algorithm based on a one-dimensional line search. It should be remared that the proposed algorithm finds the optimal solution for some special MIMO-MAC cases such as frequency-selective MAC where an individual channel matrix is diagonalized in the frequency domain by using OFDM. In the OFDM-MAC case, the proposed algorithm is found to converge faster than a steepest ascent algorithm and mae the convergence speed independent of the number of subcarriers. 2. PROBLEM STATEMENT Under the capacity-achieving strategy in the MIMO-MAC based on successive interference cancellation (SIC) decoding, the original weighted rate sum maximization problem in (5) reduces to max {S },tr(s ) P, = I M + j= H π j S πj H H π j w log I M + (6) j= H π j S πj H H π j where π =(π,...,π K ) denotes a permutation of {,...,K} (such that user π K is decoded first and π is decoded last). The ey result of 4 is that the solution of the original problem in (5) is always found in the set of successively decodable rate points and that the solution of (6) is given by the decoding order π that sorts the weights in non-increasing order w π w π2 w πk With this decoding order, the resulting maximization problem is convex (the objective is concave and tr(s ) P S is a convex set). Since the decoding order is fixed solely by the weights, without loss of generality we can consider π =, i.e., users are decoded in the order K (first), K,..., (last). Then, we rewrite the objective function in the more convenient form. max f({s }) =max j j log I M + H i S i H H i (7) j= i= subject to : tr(s ) P, =,...,K (8) S, =,...,K (9) where we let = w w + and we define w K+ =. To consider the KKT conditions 7 which are sufficient and necessary condition for optimality, we form the Lagrangian function as follows. L({S j }, {µ j }, {Φ j }) = f({s }) µ j (tr(s j ) P j )+ tr(s j Φ j ) j= j= where {Φ j } are the matrix dual variables associated with the positive definiteness constraints. By differentiating the Lagrangian function with respect to S for any, we obtain the KKT conditions given by ( ) j µ I N = j H H I M + H i S i H H i H + Φ () j= i= tr(s )=P, tr(φ S )=, Φ, S,µ, Although the KKT conditions can be solved by any standard convex optimization tool, we propose an iterative waterfilling algorithm that taes advantage of the problem structure in the following section. 3. ITERATIVE WATER-FILLING ALGORITHM In the analogy with the approach in 8, we can consider the covariance of noise plus interference for user when user K to j +are decoded and subtracted such that Σ,j = I M + j l=,l H l S l H H l

for j. By imposing a positive semidefinite covariance, the KKT equality condition can be rewritten as µ I N = j H H ( H S H H ) + Σ,j H () j= which is a function of S only when treating Σ,,...,Σ,K fixed. Based on Theorem 6 stating that a set of covariances can achieve the maximum sum rate if and only if each covariance is the solution of a single-user waterfilling while treating other users signals as noise, we propose the following iterative waterfilling algorithm for the weighted rate sum maximization. Iterative waterfilling algorithm for weighted sum rate maximization Initialize S () = N N for =,...,K At each iteration n, For =,...,K S (n) = arg max U :tr(u )=P j= j log H U H H + Σ(n),j (2) where Σ (n),j is a function of S(n),...,S(n), S(n ) +,...,S(n ) j End repeat By noticing that the maximization in (2) is a convex optimization problem which guarantees a unique solution and a strict increase of the objective, it can be easily shown that the above algorithm converges to the global optimum of (7) and the covariances {S } converge to the optimal set of the input covariances, similarly to the proof of Theorem 2 6. Some remars are in order : ) It is clear that the above algorithm is a generalization to arbitrary weights of the iterative waterfilling 6 for the sum rate maximization. In fact, if we let w =for all, which yields =for =,...,K and K =,wefind exactly the algorithm of 6. 2) The solution of (2) cannot generally reduce to a waterfillingtype solution because the input covariance U cannot be diagonalized, contrary to the equal weight case where the input covariance is diagonalized via singular-value decomposition 6. 3) For a special MIMO-MAC case, we have frequency selective MAC where an individual channel matrix is diagonalized in the frequency domain by using OFDM with cyclic prefix. As a result, the channel matrices as well as the input covariances become all diagonal and the decoupled KKT conditions can be solved by a waterfilling-type approach based on a one-dimensional line search (specified later). Since an efficient method to solve (2) seems non-trivial, we propose a simple alternative that consists of treating N antennas at each user terminal as N virtual single-antenna users. These N virtual users belonging to a subset share the weight w and should satisfy the total power constraint P. Under this setting, let x,m denote the transmit signal of virtual user m in subset and let h,m C M (the m- th column of H ) denote the channel of the corresponding virtual user. Notice that the power constraint is given by N m= p,m P for each subset =,...,K where we let E x,m 2 =p,m. We assume that the subsets are decoded in a decreasing weight order K,..., and inside each subset virtual users are decoded in the order N,...,. With this decoding order, we obtain the rate of virtual user m in subset given by I M + N j= l= h j,lh H j,l p j,l + m l= h,lh H,l p,l log I M + N j= l= h j,lh H j,l p j,l + m l= h,lh H,l p,l It is not difficult to see that the new optimization problem can be stated as N max log = I M + h j,l h H j,l p j,l j= l= N subject to : p,l P, =,...,K l= which coincides with the original problem in (7) by letting the input covariances diagonal, i.e. S j = diag(p j,,...,p j,n ) for j =,...,K. This means that treating N transmit antennas at each user terminal as N virtual single-antenna users is equivalent to restricting the input covariances to be diagonal. Under this setting, the solution of (2) can be readily obtained by solving the decoupled KKT conditions µ = j= j α,j,m +p,m α,j,m for m =,...,N (3) where we define α,j,m as the equivalent channel gain of user at antenna m when users K to j +are subtracted, given by jx NX α,j,m = h H,m @I M + h j,l h H j,l p j,l h,m h H,m p A,m h,m i= l= for =,...,K, j, and m =,...,N. We can interpret µ > as a water level that must be the same for all m, while satisfying the power constraint of user, i.e. m p,m = P. The solution of (3) cannot be given in a closed-form. However, using the fact that the RHS of (3) denoted by g,m (p,m ) is a monotonically decreasing

function of p,m, a one-dimensional line search (e.g. a bisection method) can be applied to find the water level efficiently (see 8 for more details). Fig. illustrates the function g,m (p,m ) for three channels of a given user. It is obvious that the proposed algorithm can be directly applied to the OFDM-MAC case. In this case, the individual channels are diagonal H = diag(h,,...,h,n ) with N parallel no-interfered channels and the equivalent channel gain α,j,m is given by α,j,m = h,m 2 + j i=,i p i,m h i,m 2 Since the optimal input covariances must be diagonal, the proposed algorithm finds the optimal solution. 4. NUMERICAL EXAMPLES In this section, we provide some numerical examples to show the behavior of our proposed algorithm. Throughout the examples, we let the user power constraint P = for all and let M = N. Fig.2 shows a two-user capacity region C(H; P) for M =2when we have H = 2, H 2 =.5 2 and P = P 2 =. The boundary of the region was plotted by varying w and w 2. By cross mars, we show the convergence behavior of our proposed algorithm for different weights w /w 2 =/5,, 5. It can be observed that the algorithm converges to the optimal set of the covariances only after three iterations for each case. We also show the capacity region (pentagon) corresponding to the optimal covariance sets. Under an equal weight, the following set maximizes the sum rate. S =, S 2 = Since the columns of these covariances are orthogonal, the vertex points corresponding to each decoding order collapse into one point 6 yielding R = R 2 =5.72 bit/sec. Under the weight w /w 2 =/5, the algorithm finds the following set S =, S 2 = 2.37 7.63 which yields the rates R =4.72,R 2 =5.67 bit/sec. Under the weight w /w 2 =5, the optimal set is given by 5.92 S =, S 4.8 2 = which yields the rates R =6.98,R 2 =3.6 bit/sec. Figs 3 and 4 show the evolution of the objective value as a function of the number of total iterations (including user iterations) for our proposed waterfilling algorithm and the gradient steepest ascent algorithm similar to the one proposed in 9. In the gradient algorithm, we update the input covariance for =,...,K at each iteration n such that S (n+) = t S (n) +( t )P v (n) H v(n) where v (n) denotes the principal eigen vector of the gradient f (S (n+),...,s (n+), S(n),...,S(n) K ) corresponding to the first term of RHS of () and t denotes a step size that can be found by a one-dimensional line search. The fading is randomly generated with i.i.d. components and averaged over a large number of realizations. We consider four classes of weights such that w 4 /w =32,w 3 /w = 6,w 2 /w =8. In Fig.3 we consider 4 4 MIMO channel with K = 2 users, while in Fig. 4 we have diagonal OFDM channels with M = 8, 6, 32 subcarriers and K =8users. The objective value is normalized so that the final value obtained by the gradient algorithm should be. We remar that the complexity of the gradient algorithm is higher due to the eigen decomposition requiring a complexity of roughly O(N 3 ). Despite its low complexity, the proposed algorithm is suboptimal in a general MIMO channel and fails to converge to the optimum by restricting the input covariances to diagonal ones. On the other hand, in diagonal OFDM channels our proposed algorithm converges to the optimum faster than the gradient algorithm and moreover maes the convergence speed independent of the number of subcarriers. This is because the gradient algorithm updates the power of one subcarrier at each user iteration whereas our waterfilling algorithm optimizes the power of all subcarrier simultaneously. Notice that the both algorithms have a comparable complexity in the OFDM-MAC case. 5. CONCLUSION In this paper, we considered the weighted sum rate maximization in the MIMO-MAC under individual user power constraints. The generalization of Yu et al. s well-nown iterative waterfilling algorithm to arbitrary weights is a nontrivial because the input covariance of the decoupled maximization for each user in (2) cannot be diagonalized and this prevents a waterfilling-type solution. An efficient method for solving (2) remains as a future investigation. Nevertheless, we proposed a iterative waterfilling-type algorithm based on a one-dimensional line search by treating the multiple antennas at a user terminal as multiple virtual users with a single antenna each. Although the proposed algorithm fails to converge to the optimal solution in a general MIMO channel, it converges to the optimum faster than a gradient steepest ascent algorithm and maes the convergence speed independent of the number of subcarriers in the OFDM-MAC case.

6. REFERENCES M. J. Neely, E. Modiano, and C. E. Rohrs, Power Allocation and Routing in Multibeam Satellites With Time-Varying Channels, IEEE/ACM Transaction on Networing, vol., pp. 38 52, February 23. 2 E. M. Yeh and A. S. Cohen, Information Theory, Queueing, and Resource Allocation in Multi-user Fading Communications, Proceedings of the 24 CISS, Princeton, NJ, March 24. 3 L. Georgiadis, M. Neely, and L. Tassiulas, Resource Allocation and Cross Layer Control in Wireless Networs, To appear in Foundations and Trends in Communications and Information Theory. 4 H. Boche and M. Wiczanowsi, Stability Optimal Transmission Policy for the Multiple Antenna Multiple Access Channel in the Geometric View, To appear in EURASIP Signal Processing Journal, Special Issue on Advances in Signal Processing-assisted Cross-layer Designs, 26. 5 D. Tse and S. Hanly, Multi-Access Fading Channels: Part I: Polymatroid Structure, Optimal Resource Allocation and Throughput Capacities, IEEE Trans. on Inform. Theory, vol. 44, no. 7, November 998. 6 W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, Iterative Water- Filling for Gaussian Vector Multiple-Access Channels, IEEE Trans. on Inform. Theory, vol. 5, January 24. 7 S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, 24. 8 M. Kobayashi and G. Caire, An Iterative Waterfilling Algorithm for Maximum Weighted Sum-Rate of Gaussian MIMO- BC, To appear in IEEE J. Select. Areas Commun., Special Issue on Nonlinear Optimization, 26. 9 H. Viswanathan, S. Venatesan, and H. Huang, Downlin Capacity Evaluation of Cellular Networ with Known- Interference Cancellation, IEEE J. Select. Areas Commun., vol. 2, no. 5, June 23. rate 2 bit/sec normalized objective value 6 5 w /w 2 =/5 w 2 /w = 4 w /w 2 =5 3 R +R 2 maximized 2 R +5R 2 maximized 5R +R 2 maximized SNR per user = db 2 3 4 5 6 7 8 rate bit/sec.2.8.6.4 Fig. 2. Two-user capacity region M=N=K=4 M=N=4 K=2 gradient waterfilling.2 Unequal weight SNR=dB/user 5 5 2 25 3 35 4 number of total iterations Fig. 3. Convergence of the gradient and waterfilling algorithm M = N =4.2 g, () g,2 () g,3 () K j δ,j,m 2 g,m (p,m )= Σ j= +p,m δ,j,m 2 µ p,3 p, p,2 power normalized objective value.8.6.4 M=8,6, 32 M=8 M=6 M=32 gradient waterfilling.2 Unequal weight, K=8 SNR=dB/user 2 3 4 5 6 7 8 number of total iterations Fig.. The water level µ satisfying the user power constraint Fig. 4. Convergence of the gradient and waterfilling algorithm K =8