A Practical Approach for Weighted Rate Sum Maximization ... - Supelec

A Practical Approach for Weighted Rate Sum Maximization in MIMO-OFDM Broadcast Channels Mari Kobayashi

Giuseppe Caire

SUPELEC, Gif-sur-Yvette, France Email: [email protected]

University of Sourthern California Los Angeles CA, 90089 USA Email: [email protected]

Abstract— We study the maximization of weighted sum rate in multi-input multi-output OFDM broadcast channels based on MMSE linear beamforming. The problem at hand is a wellknown non-convex problem. Nevertheless, motivated by Yu’s “zero duality gap” result in a multicarrier system, we apply dual decomposition to reduce the original problem to N independent subproblems where N denotes the number of subcarriers. For each subproblem, we propose a simple waterfilling approach that aims at iteratively solving the KKT condition. Numerical examples show that the proposed algorithm achieves near-optimal performance under a typical downlink scenario.

I. M OTIVATION We consider the downlink of a wireless system where the transmitter equipped with M antennas serves K receivers with a single antenna each. Assuming a frequency selective fading channel, we apply orthogonal frequency division multiplexing (OFDM) with N subcarriers to convert the frequency selective channel into N parallel frequency-flat channels. The corresponding system is modeled as a discrete Gaussian multiple input multiple output OFDM broadcast channel (MIMO-OFDM BC), given by yk,n = hH (1) k,n xn + nk,n M

where xn ∈ C denotes the transmit signal vector on subM carrier n, hk,n ∈ C denotes the channel complex vector of user k on subcarrier n and {nk,n } is an independent identically distributed (i.i.d.) sequence of AWGN P ∈ NC (0, 1). The input N 2 is subject to a total power constraint N1 n=1 E[||xn || ] ≤ P , where P denotes the system SNR. The dual uplink channel is the MIMO-OFDM multiple access channel (MAC) where K single-antenna transmitters communicate with a receiver equipped with M antennas. The received signal on subcarrier n is given by rn =

K X

hk,n sk,n + wn

(2)

k=1

where sk,n denotes the symbol of user k on subcarrier n, wn is AWGN with ∼ NC (0, I), and the same total power constraint 1 PK PN 2 k=1 n=1 E[|sk,n | ] ≤ P is imposed. N Assuming perfect channel state information at the transmitter and receivers, the standard results on the uplink-downlink duality [1], [2] state that the capacity regions of the MIMOOFDM BC (1) and of the MIMO-OFDM MAC (2) coincide. Namely, under the same total power constraint the same set

of user rates is achievable by applying a combination of linear beamforming and “dirty-paper coding” (DPC) in downlink and a combination of linear filtering and successive interference cancellation in uplink. In this paper, we study a practical approach for the weighted sum rate maximization (WSRM) in MIMO-OFDM BC. The WSRM problem arises in adaptive resource allocation policies that allocate resource as a function of time-varying channel state information and QoS parameters (weights). In a previous work [3], we proposed an efficient iterative waterfilling algorithm that solves the WSRM building upon the capacity achieving DPC. However, a practical implementation of DPC scheme is complex and remains a challenge. Albeit suboptimal, its simplicity and near-capacity performance [4] makes linear beamforming one of the promising options for next generation high-speed downlink systems. Here, we focus in particular on minimum mean square (MMSE) linear beamforming. Unfortunately, the problem at hand is a challenging one. The difficulties are twofold : 1) under linear MMSE beamforming the WSRM is non-convex due to cross-talk between users , 2) the total power constraint over the users and subcarriers is a “coupling constraint” [5]. To overcome the difficulty 1), in [6] the authors proved that a non-convex optimization problem in multicarrier systems has “zero duality gap” as a number of subcarriers goes to infinity regardless of the convexity of the original problem. This appealing result has motivated a number of contributions in multicarrier systems [7], [8]. This paper is no exception. Assuming a large number of subcarriers, we propose an efficient algorithm via Lagrangian dual decomposition [9]. Dual decomposition allows us to break a coupling constraint (difficulty 2). The original problem reduces to N independent subproblems, each of which corresponds to the WSRM for a given subcarrier. Unfortunately, we have to resort to a heuristic method to solve the subproblems because the zero duality gap result does not imply the convexity of each subproblem. Nevertheless, we propose a simple waterfilling approach that aims at iteratively solving the KKT conditions by including the interference term in the water level inspired by the recent work [10]. Perhaps surprisingly, numerical examples show that the proposed algorithm performs extremely closed to the optimal DPC under a typical downlink setting with M ≤ K, high SNR and for large N . Moreover, the convergence of the proposed algorithm (in terms of number of total iterations) is almost

insensitive to N and K. Interestingly, we also observe that the WSR function is concave in the total power constraint even for a very small N . This might justify the application of dual methods to the non-convex WSRM in MIMO-OFDM BC under the total power constraint for any arbitrary N .

For an easier formulation, we consider the dual MIMOOFDM MAC (2) and let pk,n denote the uplink power of user k on subcarrier n. Notice that from the uplink-downlink duality [1], there exists a set of downlink powers {qk,n } under the P same total power constraint k,n qk,n = N P that achieves the same set of rates. The uplink-downlink transformation {pk,n } → {qk,n } is provided in [2] and shall not be repeated here. At the output of MMSE receive filter the uplink SINR −1 of user k on carrier n is given by pk,n hH k,n Rk,n hk,n where we define the covariance of noise and interference for user k on subcarrier n such that X Rk,n = I + hj,n hH j,n pj,n j6=k

The WSRM under MMSE receive filter is then defined as maximize subject to

1 N

wk

k=1

N X

n=1

fn (pn ) =

L({pn }, {Cn }, µ) ! N N X X = fn (pn ) − µ Cn − N P =

n=1 N X

n=1

[fn (pn ) − µ(Cn − P )]

(5)

n=1

where the last equality follows from a separable structure of the function L. We let Ln = fn (pn ) − µ(Cn − P ) denote the objective function of subproblem n. Let us define the master problem as min g(µ) µ≥0

where µ is a dual variable and

(3)

k

where the weighted sum rate is expressed in bps/Hz. Obviously, due to the SINR expression involving other users’ powers pj,n for j 6= k inside the covariance Rk,n , the objective function is neither convex or concave although the constraint is an affine set. Nevertheless, motivated by Yu’s “zero duality gap” result in multicarrier systems, we will solve the original problem (3) by its Lagrangian dual, namely we will apply Lagrangian dual decomposition in the following. III. P ROPOSED

  −1 wk log 1 + pk,n hH k,n Rk,n hk,n

We form the Lagrangian ofPthe new problem (4) with respect to the coupling constraint n Cn ≤ N P

  −1 R h log 1 + pk,n hH k,n k,n k,n

1 XX pk,n ≤ P N n

K X

k=1

II. P ROBLEM S TATEMENT

K X

where we define the objective function on carrier n ignoring the constant 1/N as a function of a power vector pn = (p1,n , . . . , pK,n )

ALGORITHM

A. Dual decomposition

g(µ) =

n=1

max Ln (pn , Cn , µ)

(6)

pn ,Cn

where each subproblem is subject to a local constraint P k pk,n ≤ Cn . We recall that the dual function g(µ) is the pointwise maximum of a family of affine functions of µ, hence it is a convex function although the primal problem is not convex. In summary, the n-th subproblem consists of maximizing Ln (pn , Cn , µ) subject to a local constraint P k pk,n ≤ Cn , while the master problem minimizes g(µ) so as to satisfy the total power constraint. B. Practical methods Subproblems For each subcarrier, we solve the following problem for a fixed µ given by maximize

In order to apply dual decomposition, we introduce N auxiliary variables {Cn } 1 . Using these new variables, the original problem can be expressed by

N X

subject to

K X

k=1 K X

  −1 wk log 1 + pk,n hH R h k,n k,n k,n − µCn pk,n ≤ Cn

(7)

k=1

maximize subject to

N X

n=1 K X

k=1 N X

fn (pn )

(4)

pk,n ≤ Cn , ∀n Cn ≤ N P

n=1 1 As shown in [3] dual decomposition over users and subcarriers by introducing N K auxiliary variables reduces to the same solution.

Each subproblem is a non-convex optimization problem. Nevertheless, we look for an efficient algorithm based on a recent work [10] in order to find local optimal solutions. The Lagrangian corresponding to the n-th subproblem (7) is given by Ln (pn , Cn , βn )

=

K X

k=1

  −1 wk log 1 + pk,n hH k,n Rk,n hk,n

−µCn − βn

X k

pk,n − Cn

!

where βn is a Lagrangian variable associated to the constraint P k pk,n ≤ Cn . The KKT conditions, which are necessary for (7), can be obtained by letting ∂Ln /∂pk,n = 0 and ∂Ln /∂Cn = 0. ∂Ln ∂pk,n

=

−1 wk hH k,n Rk,n hk,n −1 1 + pk,n hH k,n Rk,n hk,n −1 2 X wi pi,n |hH i,n Ri,n hk,n | − −1 1 + pi,n hH i,n Ri,n hi,n i6=k

− βn = 0, ∀k

∂Ln = −µ + βn = 0 ∂Cn By combining the both KKT conditions we obtain the condition for any pk,n > 0 −1 wk hH k,n Rk,n hk,n −1 1 + pk,n hH k,n Rk,n hk,n

=

−1 2 X wi pi,n |hH i,n Ri,n hk,n | i6=k

−1 1 + pi,n hH i,n Ri,n hi,n

+µ

(8) Solving (8) does not seem straightforward. However, Yu made a progress to solve the WSRM in the presence of cross-talk for multiuser digital subscriber line applications [10]. The key feature of his algorithm is that it incorporates the interference term in the waterfilling process. Following this idea, we let the interference that user k causes to others on subcarrier n denote Ik,n , given by Ik,n =

−1 2 X wi pi,n |hH i,n Ri,n hk,n | i6=k

−1 1 + pi,n hH i,n Ri,n hi,n

(9)

−1 which is non-negative. We also let αk,n = hH k,n Rk,n hk,n . Notice that Ik,n is a function of p1,n , . . . , pK,n while αk,n depends on pj,n for j 6= k. By treating both αk,n and Ik,n fixed, we can derive a simple expression for pk,n given by   wk 1 pk,n = − (10) µ + Ik,n αk,n +

In order to solve (8) for a fixed water level µ, we have to find both power and interference terms iteratively. Namely, we find {pk,n } by treating {Ik,n } fixed and then update {Ik,n }, and repeat these procedures until both terms converge. Remark that the user powers should be updated sequentially one user after another rather than simultaneously. This is because the subproblem (7) is not subject to a total power constraint provided that Cn is an optimization variable. Master problem The water level µ, common for all users and subcarriers, should be determined to satisfy the total power constraint. By letting Kn (µ) denote the set of users with positive powers on subcarrier n for a level µ, the corresponding total power is given by X X X X wk 1 P(µ) = − (11) µ + Ik,n αk,n n n k∈Kn (µ)

k∈Kn (µ)

which is clearly a monotonically decreasing function of µ for fixed {Ik,n } and {αk,n }. Therefore, the water level should be adjusted as follows : increase µ if P(µ) > Ptot , decrease µ if P(µ) < Ptot . Since the search is one-dimensional, we

can apply simply a bisection search. The overall algorithm to find the local optimal solutions is summarized as follows. Iterative waterfilling algorithm for WSRM 1) Initialize pk,n = 0 (Ik,n = 0) for all k, n 2) Water level set : at iteration l, let µ = (µmin + µmax )/2 3) Inner iteration : repeat until the solution of (7) converges. • Compute Ik,n according to (9) N • For fixed {Ik,n }, update {pk,n }n=1 by (10) over k = 1, . . . , K until converges • Repeat until {Ik,n } converges P P 4) Water interval update : If n k pk,n > Ptot then set µmin = µ, else set µmax = µ 5) Repeat until a desired accuracy on |µmax − µmin | is reached C. Convergence In this section, we discuss the convergence of the proposed algorithm. First, we consider the convergence of power terms {pk,n } for a fixed set of {Ik,n } and a water level µ. For simplicity, we focus on the equal weight case. By dropping the subcarrier index n, consider a new optimization problem given by   K X X − maximize log det I + pj hj hH pk Ik − µC j j=1

subject to

K X

k

pk ≤ C

k=1

where the optimization variables are {pk } and C and we let fnew denote the objective function. Clearly fnew is concave in {pk } thus the new problem is convex. It is easy to verify that the new problem yields the same KKT condition (8) of the inner problem (7) for each n and for fixed {Ik,n }. In other words, the solution of the inner problem can be equivalently found as the solution of the new problem. Notice that for Ik = 0, ∀k the new problem reduces to the inner problem of [5] which is guaranteed to converge by updating sequentially the user powers by using (10). Since for the general case where Ik > 0 the concavity of fnew is preserved, the new problem also converges to the optimal solution with the same power update for a fixed set of {Ik }. Next, we remark that the alternate update of {pk,n } and {Ik,n } corresponds to solving iteratively a fixed point solution of the KKT condition. Therefore, if it converges it yields at least a local optimum of (7). In fact, the proposed algorithm is observed to always converge to a fixed point that satisfies the KKT condition. Finally, the master problem is convex, hence if each of subproblems converge to its local optimum, the original problem also converges to at least a local optimum while satisfying the total power constraint up to a desired accuracy. Under a typical downlink setting, with M ≤ K, moderate to high SNR and the channel matrix of rank M , at the convergence there are at most M users allocated positive power at each subcarrier.

IV. N UMERICAL EXAMPLES This section provides some numerical examples to illustrate the behavior of the proposed algorithm. Unless otherwise mentioned, we consider that the inputs of {hk,n } are i.i.d. with ∼ NC (0, 1) over antennas, users, and frequencies. Concavity of the objective function In order to verify if Yu’s time-sharing condition, i.e. sufficient condition for zero duality gap is fulfilled, we plot the optimal weighted sum rate value as a function of the total power for a fixed channel realization. We P recall that the time-sharing condition holds if the optimal n fn (p∗n ) is concave in the total power constraint Ptot = P N . Fig.1 shows the optimized objective value for M = K = 2 and N = 1, 2, 4. We consider equal weights and following fixed channels :     1 2 1 0 H1 = H3 = , H2 = H4 = 2 1 1 0 where we let Hn = [h1,n , h2,n ] denote the channel matrix of subcarrier n. For N = 4, the proposed algorithm yields the following power allocation : for Ptot = 1, p1 = p3 = (0.5, 0) and p2 = p4 = (0, 0) , for Ptot = 2, p1 = p3 = (0.458, 0.458) and p2 = p4 = (0.084, 0), and for Ptot = 10 p1 = p3 = (1.725, 1.725) and p2 = p4 = (1.55, 0). Notice that as Ptot increases, the algorithm tends to allocate positive power at most M users at each subcarrier. For the subcarrier 2 and 4 whose matrix has a rank one, the algorithm allocates only one user in order not to be interference-limited. Surprisingly, these plots clearly shows that the objective is concave in Ptot even for such a small N . Although not shown here, our numerical examples under different channel realizations (M, K and weights) show the concavity of the objective function regardless of the number of subcarriers N , even for N = 1. This justifies at least the use of dual method to solve the non-convex WSRM under MMSE beamforming in MIMO-OFDM BC. Two-user ergodic rate region Fig. 2 shows a two-user ergodic capacity region with N = 8 and M = 2. We compare the ergodic capacity region achieved by DPC and MMSE beamforming for different values of SNR P = 5, 10, 20 dB. It is interesting to notice that as the SNR increases, the dominant face enlarges and the DPC region gets closed to a pentagon. This means that each user achieves the power of P/2 in average. For a two-user case, the suboptimality of MMSE beamforming increases for larger SNR especially under balanced weights. Sum rate vs. SNR A poor performance of the previous two user example leads us to a following question. Does multiuser diversity under M ≪ K can recover the sum rate loss with respect to DPC ? In order to examine the question, Fig. 3 plots the sum rate performance as a function of SNR for M = 4, N = 64 and K = 4, 10. The proposed algorithm is able to close the gap with respect to optimal DPC. We explain this fact in terms of its ability to find a subset of M users whose channel vectors are almost orthogonal and their channel magnitude is not small. It is known that for K ≫ M such a subset exists with high probability [4].

Average convergence vs iterations Figs. 4, 5, 6 show the averaged objective value vs. the total number of iterations for two strategies : DPC implemented by the waterfilling [3] and MMSE linear beamforming with the proposed algorithm. The four classes of weights w2 /w1 = 8, w3 /w1 = 16, w4 /w1 = 32 are considered and the total power is set to be Ptot = 10. The averaged objective value is normalized with respect to the one achieved by DPC. From these figures, we remark that the proposed MMSE linear beamforming achieves nearoptimal performance and that the required number of iterations is insensitive to K and N . V. C ONCLUSIONS We studied the weighted sum rate maximization problem in MIMO-OFDM BCs by resticting ourselves to linear MMSE beamforming. The problem at hand is a well-known nonconvex problem involving cross-talk between users and thus calls for a heuristic method. We proposed a simple iterative waterfilling approach together with dual decomposition. Although we are unable to prove the convergence of our proposed algorithm to at least a local optimum in full generality, numerical results show its near optimal performance especially under a typical downlink setting with M ≤ K, high SNR and large N , both under equal and unequal weights. It should be remarked that the optimal transmit strategy generally reduces to a non-trivial mixture of FDMA and SDMA, i.e. a set of users that are allocated positive power might be different from one subcarrier to another. Finally, we observed that the objective function for a given channel realization is concave in the total power constraint even for a very small N . This might justify the application of dual decomposition to the nonconvex WSRM in MIMO-OFDM BC for an arbitrary N . ACKNOWLEDGMENT The work of Giuseppe Caire has been partially supported by NSF Grant CCF 0635326.

R EFERENCES [1] N. Jindal, S. Vishwanath, and A. J. Goldsmith, “Duality, achieving rates, and sum-rate capacity of gaussian MIMO broadcast channels,” IEEE Trans. on Inform. Theory, , no. 10, October 2003. [2] P. Viswanath and D. N. C. Tse, “Sum capacity of the vector gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. on Inform. Theory, vol. 49, pp. 1912 – 1921, August 2003. [3] M. Kobayashi and G. Caire, “Iterative Water-Filling for Weighted SumRate Maximization in MIMO-OFDM Broadcast Channels,” Proceeding of ICASSP’2007, Honolulu, HI, 2007. [4] T. Yoo and A. Goldsmith, “On the optimality of Multiantenna Broadcast Scheduling using Zero-Forcing Beamforming,” IEEE J. Select. Areas Commun., vol. 24, no. 3, pp. 528–541, 2006. [5] W. Yu, “Sum-Capacity Computation for the Gaussian Vector Broadcast Channel via Dual Decomposition,” IEEE Trans. on Inform. Theory, vol. 52, February 2006. [6] W. Yu and R. Lui, “Dual Methods for Non-Convex Spectrum Optimization of Multi-carrier Systems,” IEEE Trans. on Commun., 2005. [7] K. Seong, M. Mohseni, and J.M. Cioffi, “Optimal Resource Allocation for OFDMA Downlink Systems,” IEEE ISIT, 2006. [8] C. Chen, K. Seong, R. Zhang, and J.M. Cioffi, “Optimized Transmission for Upstream Vectored DSL Systems Using Zero-Forcing Generalized Decision-Feedback Equalizers,” Globecom, 2006. [9] D. P. Bertsekas, Nonlinear Programming, Belmont, MA : Athena Scientific, 1999. [10] W. Yu, “Multiuser Water-filling in the Presence of Crosstalk,” Information Theory and Applications Workshop, San Diego, CA, Jan-Feb 2007.

12

Optimized objective value

10

normalized average objective

N=1 N=2 N=4

11

9 8 7 6 5 4 3 2

M=K=2 equal weight 1

2

3

4

5

6

7

8

9

1

0.5 K=10 M=4 N=64 0 0

10

P

dual DPC MMSE

1.5

25

tot

Fig. 1.

Fig. 4.

Optimized objective value vs. a total power constraint

50 total iterations

75

100

Convergence for K = 10, M = 4, N = 64

Ergodic capacity region 7

DPC MMSE normalized average objective

R2 [bps.Hz]

6 SNR=20dB

5 4

SNR=10dB

3 2 1

SNR=5dB

dual DPC MMSE

1.5

1

0.5 K=20 M=4 N=64

M=K=2, N=8 0 0

1

2

3 4 5 R1 [bps/Hz]

6

0 0

7

Fig. 5. Fig. 2.

25

50 total iterations

75

100

Convergence for K = 20, M = 4, N = 64

Two-user ergodic capacity region with M = 2, N = 8

25

normalized average objective

sum rate [bps/Hz]

20

DPC MMSE K=10

15

10 K=4 5

dual DPC MMSE 1.5

1

0.5 K=20 M=4 N=128

M=4, N=64 0 0

−5

0

5

10 SNR [dB]

15

20 Fig. 6.

Fig. 3.

Sum rate vs. SNR with M = 4, N = 64, K = 4, 10

25

50 total iterations

75

Convergence for K = 20, M = 4, N = 128

100