programming (SDP) relaxation method to solve the robust beam- forming problem. We show that the semidefinite relaxation is tight. (i.e., giving a rank-one ...
ROBUST SINR-CONSTRAINED MISO DOWNLINK BEAMFORMING: WHEN IS SEMIDEFINITE PROGRAMMING RELAXATION TIGHT? Enbin Song
Qingjiang Shi
Maziar Sanjabi, Ruoyu Sun, Zhi-Quan Luo∗
Shanghai Jiao Tong University Sichuan University Dept. of EE College of Mathematics Chengdu, Sichuan 610064, China Shanghai 200240, China ABSTRACT We consider the robust beamforming problem under imperfect channel state information (CSI) subject to SINR constraints in a downlink multiuser MISO system. One popular approach to solve this nonconvex optimization problem is via semidefinite relaxation (SDR). In this paper, we prove that the SDR method is tight when the channel uncertainty bound is small or when the base station is equipped with two antennas. Index Terms— Robust beamforming, SINR, channel uncertainty, SDR 1. INTRODUCTION A popular measure of quality of service (QoS) in a multi-user communication system is the signal-to-interference-plus-noise ratio (SINR) [1]. In a MISO (multi-input single-output) system, it is known that if the SINR is used as the QoS constraints and if the perfect channel state information is available at the transmitter (CSIT), the problem of designing downlink transmit beamformers can be solved by using convex optimization [1, 2, 3]. In practice, however, due to the finite length of training signal and/or limited feedback bandwidth we cannot obtain the perfect CSIT. This is unfortunate, because we cannot simply ignore the CSIT imperfection in the design of downlink beamformers, for otherwise the resulting solution might violate the SINR constraints for some users. As a remedy, various researchers [6, 7, 8, 9, 10] have proposed robust beamforming formulations by taking the CSIT imperfection into consideration. Unfortunately the resulting robust downlink beamforming problem is nonconvex and has infinitely many constraints. It is difficult to solve in practice. To overcome the computational difficulties associated with the robust downlink beamforming, researchers have proposed various optimization methods to approximate the problem by restricting the non-convex feasible region to a smaller but convex set [6, 7]. Due to convex restriction, these methods can not necessarily reach the global optimal solution although they do guarantee the satisfaction of the QoS constraints. In this paper, we consider a natural semidefinite programming (SDP) relaxation method to solve the robust beamforming problem. We show that the semidefinite relaxation is tight (i.e., giving a rank-one solution) under norm-constrained CSIT errors [6, 7, 8], provided that the size of channel errors is sufficiently small.
Notations: Throughout this paper, (·)H denotes the Hermitian conjugate of a matrix or a vector, and · denotes the spectral norm of a matrix or the Euclidean norm of a vector. In addition, {a} denotes the real part of a. 2. PROBLEM FORMULATION Consider a K-user downlink MISO system where the base station (BS) is equipped with M antennas. The BS wants to transmit K independent data streams s1 , s2 , . . . , sK to K users and each sk is complex Gaussian with unit variance. The BS uses linear beamforming vector gk ∈ C M , k = 1, 2, . . . , K, to send the stream sk .The received signal at receiver k is sˆk = hH k
K
si gi + wk ,
k = 1, 2, . . . , K,
i=1
where, hk denotes the channel vector from BS to the k-th user, {wk }’s are zero-mean, independent white complex Gaussian noises, each with a variance of σk2 . Therefore, the SINR of the k-th user is SINRk = i=k
2 |hH k gk | . H |hk gi |2 + σk2
Because of the finite length of training signal and/or limited feedback bandwidth, the perfect CSIT are not available in practice. The imperfect CSIT can dramatically impact the system performance and the users’ quality of service. To maintain the desired QoS for all users, we consider robust transmit beamforming problem under imperfect channel knowledge. In particular, let us assume that the channel vector hk lies within a ball with radius k around ˜ k , i.e., the estimated channel vector h ˜ k + δk | ||δk || ≤ k hk ∈ U k = h ˜ k is the channel estimate available for all k = 1, 2, . . . , K, where h at the BS and δk ∈ C M is the channel estimation error whose norm is assumed to be bounded by k . In addition, we suppose that the desired QoS constraints are given in the form of SINR constraints. Hence, the robust worst case QoS-constrained beamforming problem can be formulated as below: min gk ’s
∗ This
K
gk 2
k=1
s.t.
work was supported in part by the Army Research Office, grant number W911NF-09-1-0279, and in part by the National Science Foundation, grant number CMMI-0726336. This work was completed during a visit of the first two authors to the University of Minnesota.
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
University of Minnesota Dept. of ECE Minneapolis, MN 55455, USA
i=k
2 |hH k gk | ≥ γk , ∀ hk ∈ Uk , H 2 |hk gi | + σk2
(PR )
k = 1, 2, . . . , K.
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ICASSP 2011
where γk > 0 is the SINR requirement for receiver k.
To this end, we first study the relationship between the robust beamforming problem and its non-robust version:
3. SDP RELAXATION min
Using the S-Procedure [4, 5] from control theory, we can rewrite the problem (PR ) as K
min
λk ,gk ,Gk
gk ’s
K
s.t. Tr (Gk )
gk 2
k=1
i=k
˜k Xk h H ˜ ˜ hk Xk hk − σk2 − λk 2k
Gk = gk gkH , 1 Gk γk
where Xk =
−
(1)
0
k = 1, 2, . . . , K.
i=k
Gi . The nonlinear constraint Gk =
Obviously, if the above problem is not feasible, then the robust version cannot be feasible either. Hence, a reasonable and necessary assumption is that the problem (PN R ) is feasible.
˜ = g ˜2 . . . g ˜K be the optimal solu˜1 g Lemma 1 Let G tion of the problem (PN R ), and for k = 1, 2, . . . , K define
gk gkH is equivalent to: Gk 0 and rank(Gk ) = 1. Because of the rank constraint the above problem is nonconvex and difficult to solve. If we drop this rank constraint, we obtain the following convex SDP problem which can be efficiently solved. min
K
1 Gk γk
˜k Xk h 2 2 ˜H ˜ h k Xk hk − σk − λk k
i=k
Yk ,yk ,αk
(P )
Δ
Ψ=
0
k = 1, 2, . . . , K. −
g ˜j g ˜jH ,
(2)
˜ k 2 − A ˜k ˜ k + A ˜ kh ˜ kh γk σk2 A ˜ k A
,
(3)
and
Proof: Based on the results of [1], all the constraints of the problem
(PN R ) should be active at the optimal solution, i.e., ˜ = g ˜2 . . . g ˜K satisfies ˜1 g G
αk σk2
˜H |h ˜k |2 k g = γk , ˜H 2 |hk g ˜i | + σk2
k=1 Δ
s.t. μk = αk 2k − Tr (Yk ) ≥ 0 Yk yk Δ Uk = 0 H yk αk 1 Δ ˜ H ˜H ˜ ˜H Zk = I − Yk + yk h k + hk yk + αk hk hk γk ˜H ˜ ˜ H 0, ˜ H + Yi + yi h i + hi yi + αi hi hi
= (1 , · · · , K ) 0 ≤ k < ηk , k = 1, 2, . . . , K T
Then, for any ∈ Ψ, the problem (PR ) must have a feasible point.
Gi .
The dual of this SDP problem is given by K
Tr (Gk )
Gk 0,
j=k
ηk =
k=1
where Xk =
Δ ˜k = g ˜k g ˜kH − γk A
Δ
s.t. λk ≥ 0, Xk + λk I H ˜H h k Xk
max
k = 1, 2, . . . , K.
k=1
s.t. λk ≥ 0, Xk + λk I H ˜H h k Xk
λk ,Gk
(PN R )
2 ˜H |h k gk | ≥ γk , H 2 ˜ |hk gi | + σk2
k = 1, 2, . . . , K
i=k
(D )
which is equivalent to 2 ˜H ˜ ˜ h k Ak hk = γk σk ,
k = 1, 2, . . . , K.
(4)
Using the definition of ηk (c.f. (3)), we can easily see that for any k ∈ [0, ηk ),
i=k
k = 1, 2, . . . , K.
˜ k k > 0, ˜ k 2k − 2A ˜ kh γk σk2 − A
It can be checked that the feasible region of (D ) has a nonempty interior. If, in addition, the primal SDP (P ) also has a nonempty interior, then the strong duality holds and the dual SDP (D ) is equivalent to (P ). Also, notice that the channel uncertainty bounds = (1 , ..., K )T can be regarded as a vector of parameters for problem (P ) and its dual (D ).
k = 1, 2, . . . , K.
(5)
For each k = 1, 2, . . . , K, we let tk =
γk σk2
γk σk2 ˜ k k ˜ ˜ kh − Ak 2k − 2A
(6)
and 4. MAIN RESULTS
t = max {tk } .
In what follows, we will give a region of uncertainty radius, and show that for k ’s lying in that region, the solution of the problem (P ) must be of rank-one, which means that the SDP-relaxation (P ) is tight in this case.
Combining the fact that t ≥ tk with equality (6) and using the inequality in (5), we can show
1≤k≤K
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˜ k k + (t − 1)γk2 σk2 ≥ 0. ˜ kh ˜ k 2k − 2tA − tA
(7)
Now for any δk ∈ Ek , we have ⎛ ⎞ H ˜ k + δk − γk σk2 ˜ k + δk ⎝t˜ gk g ˜kH − tγk g ˜j g ˜jH ⎠ h h
For any ∈ Ω(¯ ), let V () denote the optimal value of (P ) (or its dual (D )). Then, it is obvious that V () ≤ V (¯ ) .
j=k
2 ˜H ˜ ˜ ˜H ˜ k δk + 2t h ˜ = tδkH A k Ak δk + thk Ak hk − γk σk 2 ˜H ˜ k δk + 2t h ˜ = tδkH A k Ak δk + (t − 1) γk σk 2 ˜H ˜ k δk − 2th ˜ ≥ −tδkH A k Ak δk + (t − 1) γk σk ˜ k k + (t − 1)γk2 σk2 ≥ 0, ˜ k 2k − 2tA ˜ kh ≥ −tA
where the second equality is a direct consequence of (4), and the first inequality is due to the Cauchy-Schwartz inequality and the matrix √ norm inequality. follows from (7). Hence, √ √ The last inequality √ ˜ = tG t˜ g1 t˜ g2 . . . t˜ gK is a feasible solution for problem (PR ). This proves Lemma 1. We will also need the following lemma whose proof is omitted due to space limitation. Lemma 2 Suppose that the strong duality holds for problem (P ) and its dual problem (D ) and that γk ≥ 1, k = 1, 2, . . . , K. Let {λk }, {Gk }, {Yk }, {yk }, {αk } be any primal and dual optimal solutions. Then, the optimal solution Gk of problem (P ) and Zk of problem (D ) satisfy
Let {λk }, {Gk }, {Yk }, {yk }, and {αk } be any primal and dual optimal solutions satisfying complementary slackness for the SDP problem (P ) and its dual (D ). As we proved in Lemma 2, we have rank(Gk ) ≥ 1. To establish rank(Gk ) ≤ 1, we use the complementarity slackness condition Tr(Gk Zk ) = 0 and prove rank(Zk ) = M − 1. We claim that each αk must be strictly positive, for otherwise we could increase its value without compromising dual feasibility, which would then contradict the optimality of {αk }. As αk > 0, it follows that for any k = 1, · · · , K: 1 H H H ˜ ˜H ˜ ˜ + h y + α = Y − y y Yk + yk h + h h k k k k k k k k k αk √ ˜ √ ˜ H 1 1 0. (9) √ yk + αk h √ yk + αk h k k αk αk In addition, we have Tr (Yk ) ≤ αk 2k ≤ ≤
1 ≤ rank (Gk ) ≤ M − 1,
1 ≤ rank (Zk ) ≤ M − 1.
Hence, Tr
1 Yk γk
Lemma 2 implies that if the base station is equipped with two antennas (M = 2), then the optimal solutions of (P ) must be rank one. In other words, the SDP relaxation is tight. Theorem 1 Suppose that for some choice of uncertainty bounds T ¯1 ¯2 · · · ¯K ¯ = > 0, the problem (P¯) is strictly feasible. Let V (¯ ) denote the optimal value of (P¯) and let
T Δ Ω (¯ ) = = 1 2 . . . K k ≤ ¯k , γk σk2 and k < , k = 1, 2, . . . , K . V (¯ ) Then, for any vector of uncertainty bounds ∈ Ω(¯ ) the problem (P ) is feasible, and each optimal solution {Gk } must be rank one, i.e., rank(Gk ) = 1 for all k. Proof : Let 0 ≤ ≤ ¯. First we consider the SDP problem (P ) and its dual (D ). Notice that Yk yk Δ Uk = ykH αk ⎡ ⎤ 2 min γ4k , γ4k × 2M h˜k h˜H IM 0 ⎦ k k = ⎣ γk 0 ˜ h ˜H 4h k
k
is a strictly feasible point of problem (D ). Also, it can be checked that the strict feasibility of (P¯) implies the same for (P ) since ≤ ¯. Hence, Slater condition holds for both (P ) and its dual (D ). Consequently, the strong duality [4] holds and the optimal values of (P ) and its dual (D ) are attained.
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(8)
V () 2 k σk2
V (¯ ) 2 V (¯ ) γk σk2 k < = γk . 2 2 σk σk V (¯ )
< 1 which further implies I−
1 Yk 0. γk
(10)
Now we can compute rank(Zk ): rank (Zk ) = rank I − −
1 γk
Yk −
1 yk ykH αk
√ ˜ √ ˜ H 1 1 √ yk + αk h √ yk + αk h k k αk αk H H H ˜ ˜ ˜ ˜ Yi + yi hi + hi yi + αi hi hi +
1 γk
i=k
1 1 Yk + yk ykH γk γk αk ˜ H ˜H ˜ ˜H + Yi + yi h i + hi yi + αi hi hi
= rank
I−
i=k
1 − γk
√ ˜ 1 √ yk + αk h k αk
√ ˜ 1 √ yk + αk h k αk
H
≥ M − 1, where the last inequality holds true because √ ˜ 1 √ ˜ H 1 √ √ y + αk h y + αk h is rank one and k k αk k αk k H y y ˜ H ˜H ˜ ˜ H is Yi + yi h I − γ1k Yk + γkk αkk + i + hi yi + αi hi hi i=k
full rank, due to (9) and (10).
5. SIMULATION RESULTS We now present simulation results to corroborate the findings of Theorem 1 and to demonstrate the effectiveness of the SDR method. In the simulations, the base station has three antennas (i.e., M = 3) and serves three users (i.e., K = 3); the coefficients of the channel estimates are i.i.d complex Gaussian random variables with zero mean and unit variance; the channel uncertainty bounds k ’s are all 0.1; the noise variances of all users are set equally to be 0.1. To demonstrate the tightness of SDR method we compute the rank one test (ROT) ratio defined as M λi (Gk ) ROT = max i=2 k λ1 (Gk ) where λi (Gk ) denotes the i-th largest eigenvalue of Gk . The simulation result is shown in Fig.1 where each data point represents the maximum ROT over 2000 Monte Carlo simulations with the 2000 data sets generated independently for each SINR target. From this figure, we can see that the rank one test ratio is very close to zero, suggesting that the solution of the SDR method is indeed always rank one.
power versus the target SINR level, where each data point is averaged over 2000 Monte Carlo simulations. The 2000 sets of channel vectors used to generate Figure 2 are identical to those used to obtain Figure 1. Note that, to guarantee the feasibility of the problem, we compare two methods only when the convex restriction method is feasible. From the figure, it can be observed that the SDR method gives more power-efficient solution than the convex restriction method (which is abbreviated as ‘cvxRes’ in the figure). 6. CONCLUSIONS In this paper we have considered the robust beamforming design problem (PR ) for a MISO downlink channel. Our work has focused on the existence of a rank-one solution for the corresponding SDP relaxation (P ). It is shown that the problem (PR ) can be solved by the SDR method to global optimality as long as the size of channel estimation errors are sufficiently small, or when the BS is equipped with two antennas. The numerical simulations show that the SDR method gives better results in comparison with the existing robust beamforming methods. More work is needed to understand the relation between the robust beamforming problem (PR ) and its SDP relaxation (P ).
−8
2.5
x 10
7. REFERENCES
Rank one test ratio
2
1.5
1
0.5
0
6
8
10
12
14
16
γ (dB)
Fig. 1. Rank one test ratio Vs. the target SINR.
22 SDR cvxRes
Avergage transmission power (dB)
20 18 16 14 12 10 8 6
6
8
10
12
14
16
γ (dB)
Fig. 2. Average transmission power Vs. the target SINR. We also compare the SDR method with the convex restriction method proposed in [6]. Figure 2 presents the average transmission
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[1] M. Bengtsson and B. Ottersten, “Optimal and suboptimal transmit beamforming,” Chapter 18 in Handbook of Antennas in Wireless Communications, L. C. Godara, Ed., CRC Press, Aug. 2001. [2] H. Boche and M. Schubert, “A general duality theory for uplink and downlink beamforming,” in Proc. IEEE VTC-Fall, Vancouver, British Columbia, Canada, Sep. 2002. [3] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson, and B. Ottersten, “Convex optimization-based beamforming,” IEEE Signal Process. Mag., pp. 62-75, May 2010. [4] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University Press, 2004. [5] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [6] N. Vuˇ ci´ c and H. Boche, “Robust QoS-constrained optimization of downlink multiuser MISO systems,” IEEE Trans. Signal Process., vol. 57, no. 2, pp. 714-725, Feb. 2009. [7] M. B. Shenouda and T. N. Davidson, “Convex conic formulations of robust downlink precoder designs with quality of service constraints,” IEEE J. Sel. Topics in Signal Process., vol. 1, pp. 714-724, Dec. 2007. [8] G. Zheng, K.-K. Wong, and T.-S. Ng, “Robust linear MIMO in the downlink: A worst-case optimization with ellipsoidal uncertainty regions,” EURASIP J. on Advances in Signal Process., vol. 2008, pp. 1-15, June 2008. [9] I. Wajid, Y. C. Eldar, and A. B. Gershman, “Robust downlink beamforming using covariance channel state information,” Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP’09), Taipei, Taiwan, Apr. 2009, pp. 2285-2288. [10] V. Sharma, I. Wajid, A. Gershman, H. Chen and S. Lambotharan, “Robust downlink beamforming using positive semidefinite covaraince constraints,” Int. ITG Workshop on Smart Antennas, WSA, Darmstadt, Germany, Feb. 2008.
