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Multiuser MIMO Downlink Beamforming Based on Group Maximum SINR Filtering Yu-Han Yang, Shih-Chun Lin and Hsuan-Jung Su Graduate Institute of Communication Engineering Department of Electrical Engineering National Taiwan University, Taipei, Taiwan Email: {r94942109, sclin2}@ntu.edu.tw, [email protected] Abstract—In this paper we aim to solve the multiuser multiinput multi-output (MIMO) downlink beamforming problem. The transmitter is a multi-antenna base-station broadcasting to users. Each user has multiple antennas at the receiver. A solution of the joint transmit-receive beamforming and power allocation under signal-to-interference-plus-noise-ratio (SINR) constraints in this system is proposed. The beamforming filter is a group maximum SINR filter which exploits intra-group cooperation. Taking advantage of the uplink-downlink duality property, the proposed algorithm iteratively computes the transmit-receive beamforming filters and the power allocation matrix. Simulation results verify the superiority of the proposed algorithm over previous works using per stream maximum SINR method.
I. I NTRODUCTION In this paper, a general case of the downlink channel is considered. One transmitter with multiple antennas is communicating with multiple independent users, each having multiple antennas. Each user is assigned a target signal-to-interferenceplus-noise (SINR) ratio. To suppress interference between users, transmit beamforming is used to point signals to the respective users. Each user also applies receive beamforming to further suppress interference. Power allocation at the transmitter is performed to optimize the performance. Joint beamforming and power allocation for the downlink channel have been studied previously. In [1], [2], [3], a precoding method is employed to block diagonalize the overall channel so that each user does not see the signals transmitted to other users. Thus the equivalent channel between any user and the base station is interference-free. However, such a zero-forcing approach is not optimal, because it thoroughly eliminates the co-channel interference (CCI) and takes no consideration of the noise. Hence the performance can be improved if the balance between CCI suppression and noise enhancement can be found. In [4], Schubert et al. consider the situation where a multiantenna base station is transmitting independent data streams to users each having single antenna and an individual SINR constriant. Khachan et al. [5] generalize the scheme in [4] to allow several transmission beams to be grouped to serve one user. However, in [5], in addition to the interferences among users, there are intra-group interferences between transmission This work was supported by the National Science Council, Taiwan, R.O.C., under grant NSC 95-2219-E-002-005.
beams. In order to tackle the intra-group interferences, we apply a group maximum SINR filter [6] to collect all the desired signal energy in the beam group of each user. The proposed scheme can be applied together with joint coding, and the group maximum SINR filtering can exploit intra-group cooperation and improve the performance. In this paper, we extend the work in [4] to the multiantenna setting based on group maximum SINR filtering which exhibits better performance compared to [5]. New optimization methods to iteratively calculate the beamforming filters and the power allocation matrix are proposed. Two optimization criteria are investigated. One is minimizing total transmitted power while satisfying a set of SINR targets. The other is maximizing the achieved SINR to target ratio under a total power constraint. For simplicity, we consider group power allocation which restricts equal power on beams in each group. Under such a restriction, our algorithm can find the optimal solution with the uplink-downlink duality [7]. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model Consider the downlink scenario with K users, where a base station is equipped with M antennas. The upper part of Fig. 1 shows the overall system block diagram for user k, Lk data streams. who has Nk receive antennas and receives K Thus the K users have a total of N = k=1 Nk receive K antennas receiving a total of L = k=1 Lk data streams. In a specific symbol time, symbols of the data streams intended for user k are denoted by a vector xk = [xk1 , xk2 , ..., xkLk ]T . Symbols of the L data streams are concatenated in a vector x = [xT1 , ..., xTK ]T . The precoder Uk ∈ C M ×Lk processes user k’s data streams before they are transmitted over the M antennas. These individual precoders together form the global transmitter beamforming matrix UM ×L = [U1 , U2 , ..., UK ]. The power allocation matrix for user k is a diagonal matrix Pk = diag{pk1 , pk2 , ..., pkLk },
(1)
and the global power allocation matrix, P = diag{P1 , P2 , ..., PK },
(2)
is a block diagonal matrix of dimension L × L. The channel between the transmitter and user k is represented by the Nk ×
978-1-4244-2075-9/08/$25.00 ©2008 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
Downlink
U1 Ξ
xK
P1 Ξ
x1
PK
UK
nk
H kH
+
+
xˆ k
VkH
where γk and SINRk are the SINR target and the achieved SINR for user k, respectively. We refer to SINRk /γk as the SINR to target ratio for user k. If the minimum SINR to target ratio in equation (5) can be made greater than or equal to one, then the second optimization problem is to find the minimum power required such that the SINR targets can be all satisfied. That is, min
Virtual Uplink
x1
V1
Q1
VK
QK
Fig. 1.
nkUL
H1
U
+
xˆ k
H k
HK
MIMO downlink system model for user k and its virtual uplink.
M matrix HH k , the Hermitian of Hk . The resulting N × M global channel matrix is HH , with H = [H1 , H2 , ..., HK ]. User k receives a length Nk vector K Pk xk +HH Uj Pj xj +nk (3) yk = HH k Uk k j=k,j=1
where nk represents the additive white Gaussian noise (AWGN) at the user’s receive antennas with variance σ 2 . That 2 is, E[nk nH k ] = σ INk , where E[·] denotes the expectation operator. The second term in equation (3) is the inter-group interference for user k. To estimate its Lk symbols xk , user k processes yk with its Lk × Nk receive beamforming matrix VkH . The resulting estimated signal vector is H x ˆk = VkH HH k Ux + Vk nk
= VkH HH k Uk
K
Pk xk + VkH HH k
Uj
(4a) Pj xj
j=k,j=1
+ VkH nk
(4b)
The global receive filter VH is a block diagonal matrix of dimension L × N , V = diag{[V1 , V2 , ..., VK ]} owing to the non-cooperative nature between users.
k
SINRk subj. to γk
k=1 j=1
SINRk ≥1 γk
(6)
MATRICES SELECTION
A. Maximum SINR Beamforming Lk SINRDL For given U and P, kj is maximized by using j=1
the maximum SINR filter as the receive beamformer Vk , where SINRDL kj is the SINR of the jth stream of user k in the downlink,. This optimization problem can be proved to be equivalent to a generalized eigenvalue problem [6]. That is, DL DL RDL s,k vkj = λkj Rn,k vkj
Based on the system model described in Section II-A, we consider two problems in this paper as follows. We assume that the base station has perfect knowledge of the channel matrix H, and receiver k knows its Hk perfectly. The first optimization problem is : with a total power constraint Pmax , maximize min SINRk /γk over all beamformers U, V, and k power allocation matrix P. That is, max min
k
We briefly review the uplink-downlink duality which plays an important role in our algorithm design. According to [7], [8], [9], [10], a virtual uplink system for our downlink system can always be found. We plot the virtual uplink for user k in the lower part of Fig. 1, where Qk is the corresponding power allocation matrix in the virtual uplink defined similarly as Pk . To be more specific, with fixed beamforming filters U, V, SINR targets γ1 , ..., γK , and a common sum power constraint Pmax for both the downlink and the virtual uplink, the optimal P and Q will make the downlink and its virtual uplink have the same SINR to target ratio. We will restricts the filters V and U as the maximum SINR array processors proposed in [6]. Then with the aids of the uplink-downlink duality, the optimization problems in Section II-B can be solved efficiently with iterative algorithms. That is, for each iteration we will use power allocations P and Q to find V and U; with new filters V and U, we can find new P and Q, and so on. When the SINR to target ratios of the downlink and virtual uplink equal to each other, the optimization problems are solved. We will first show how to calculate the beamforming filters and then show how to use these filters to find new power allocations P and Q in the next section.
B. Problem Formulation
U,V,P
pkj subj. to min
III. I TERATIVE ALGORITHMS AND BEAMFORMING
Ξ
Ξ
Ξ
xK
U,V,P
Lk K
Lk K k=1 j=1
pkj = Pmax ,
(5)
with λDL kj =
H RDL vkj s,k vkj
H RDL v vkj n,k kj
= SINRDL kj ,
(7)
(8)
H H where Vk = [vk1 , . . . , vkLk ], RDL s,k = Hk Uk Pk Uk Hk DL denotes the signal covariance matrix, and Rn,k = H 2 Hk Uj Pj UH j Hk + σ INk denotes the interference and j=k
noise covariance matrix. The receive beamforming filter Uk for the virtual uplink system can be derived similarly. As a result, the receive
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
beamforming filter designed for the downlink can be carried over to the transmit beamforming filter for uplink, and vice versa. IV. P OWER A LLOCATION We now consider finding the power allocation strategies for problems corresponding to (5) and (6), respectively, given fixed beamforming filters U, V and a set of SINR targets γ1 , ..., γK . The two problems are: Lk K SINRDL k s.t. pkj = Pmax , max min P k γk j=1
(9)
k=1
and
DL
The expression of SINRk has the same structure as the case considered in [4] with Nk = 1, ∀k. Therefore, we can apply the result in [4, Lemma 1], which states that since DL each SINRk is strictly monotonically increasing in pk and monotonically decreasing in pj for j = k, the maximizer of the optimization problem (9) satisfies DL
SINRk γk
= C DL , 1 ≤ k ≤ K.
(15)
That is, all users have the same SINR to target ratio C DL , termed balanced level. Otherwise, the users with higher SINR to target ratios can give some of their power to the user with the lowest ratio to help it, which contradicts the optimality.
A. Sum Power Constraint SINRDL k pkj = Pmax s.t. min ≥ 1, k = 1, ..., K. As in [11], with (15) and (9), equation (14) can be rewritten min k γk k=1 j=1 as 1 (10) (16) p DL = DΨp + Dσ, C For simplicity, we consider group power allocation. That is, power is allocated to users, and the allocated power for a user where is evenly distributed over all streams of that user: L2K γK L21 γ1 , (17) D = diag , ..., VH HH U1 2 VH HH UK 2 (11) pk1 = pk2 = ... = pkLk , 1 ≤ k ≤ K 1 1 K K F F Lk K
Let the power allocated to user k be pk . Consequently, the diagonal power allocation matrix Pk for user k can be written as a scaled identity matrix, i.e., pk IL . (12) Pk = Lk k We also define a vector p = [p1 , . . . , pK ]T to replace matrix P in the optimization problems. We will first show an important property when using maximum SINR filters in problems (9) and (10). That is, all users will have the same SINR to target ratios. This property will simplify the problems (9) and (10) to eigensystems. First, the vector norms of the beamforming filters vkj , j = 1...Lk , can be adjusted such that • VkH RDL n,k Vk is a scaled identity matrix [6]. • trace VkH Vk = Lk . Once the above two conditions are satisfied, the average SINR of each data stream of user k in the downlink scenario can be expressed as
H DL Lk trace V R V k k s,k 1 DL . (13)
SINRk = SINRDL kj = Lk j=1 H DL trace V R V k
Substituting Pk = SINR is DL SINRk
pk Lk ILk
n,k
into equation (13), the average
H H 2 V H Uk k k F = p j VH HH Uj 2 + σ 2 k k Lj Lk F pk L2k
k
(14)
j=k
the Frobenius norm, which is defined as where ·F
denotes
MF = trace MMH .
and [Ψ]ij =
2
ViH HH i Uj F Li Lj
0
j = i j = i.
(18)
Here [Ψ]ij denotes the ijth element of the K × K matrix Ψ, and the K × 1 vector σ = σ 2 1. Multiplying both sides of (16) by 1T , we have 1 1 1 = 1T DΨp + 1T Dσ. (19) DL C Pmax Pmax p Defining an extended power vector p ˜ = and an 1 extended coupling matrix DΨ Dσ Υ= , (20) 1 1 T T Pmax 1 DΨ Pmax 1 Dσ (16) and (19) together form an eigensystem 1 ˜ with [˜ p]K+1 = 1 (21) Υ˜ p = DL p C It can be observed that C DL is a reciprocal eigenvalue of the nonnegative extended coupling matrix Υ. However, only positive C DL is meaningful and its corresponding eigenvector p ˜ should also comprise nonnegative components. The existence of positive eigenvalue and eigenvector is guaranteed by the Perron-Frobenius theory [12], which states that for any nonnegative real matrix B ≥ 0 with spectral radius ρ(B), there exists a vector y ≥ 0 and λmax (B) = ρ(B), such that By = λmax (B)y. Therefore, we can conclude that the maximal eigenvalue λmax (Υ) and the associated eigenvector, corresponding to the reciprocal of C DL and the extended power vector p ˜ , respectively, are always nonnegative. To obtain the optimal power vector p, we can scale the extended power vector p ˜ , such that its last element equals to one.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
TABLE I G ROUP P OWER A LLOCATION WITH S UM P OWER C ONSTRAINT
TABLE II G ROUP P OWER M INIMIZATION
Initialization: U = I, V = I Iteration: 1: Downlink Receive Maximum SINR Beamforming for k = 1 : K DL Vk = eig(RDL s,k , Rn,k ) 2: Virtual Uplink Power Allocation with Sum Power Constraint q q 1 Λ = C UL 1 1 3: Virtual Uplink Receive Maximum SINR Beamforming for k = 1 : K UL Uk = eig(RUL s,k , Rn,k ) 4: Downlink Power Allocation with Sum Power Constraint p p 1 Υ = C DL 1 1 5: Repeat steps 1-4 until convergence, i.e., C DL = C UL .
Initialization: U = I, V = I. If λmax (DΨ) < 1, then Iteration: 1: Downlink Receive Maximum SINR Beamforming for k = 1 : K DL Vk = eig(RDL s,k , Rn,k ) 2: Virtual Uplink Power Minimization Allocation q = (I − DΨT )−1 Dσ 3: Virtual Uplink Receive Maximum SINR Beamforming for k = 1 : K UL Uk = eig(RUL s,k , Rn,k ) 4: Downlink Power Minimization Allocation p = (I − DΨ)−1 Dσ 5: Repeat steps 1-4 until convergence, i.e., C DL = C UL = 1.
V. S IMULATION R ESULTS The optimal q for the virtual uplink can also be obtained through similar derivation as in the downlink system. Using the uplink-downlink duality described in Section III, it is guaranteed that C DL = C UL . We summarize the optimization algorithm in Table I, which iteratively calculates the optimal beamforming filter and power allocation vector between the downlink and the uplink. Here eig denotes the generalized eigenvalue solver which outputs the eigenvectors. B. Power Minimization As discussed above, the minimizer of (10) satisfies DL
SINRk = γk , 1 ≤ k ≤ K.
(22)
Substituting (22) into (14), we have
and similarly,
(I − DΨ)p = Dσ,
(23)
(I − DΨT )q = Dσ.
(24)
Thus the resulting power allocation vectors are
and
p = (I − DΨ)−1 Dσ
(25)
q = (I − DΨT )−1 Dσ.
(26)
As in [4][5], there is a feasibility issue for the starting point U = I, V = I. It is proved in [13] that there exists a positive solution p if λmax (DΨ) < 1 is satisfied. Similarly, if λmax (DΨT ) < 1, then the existence of a positive solution q can be guaranteed. In [13], it is shown that λmax (DΨ) = λmax (DΨT ). In other words, the set of SINR constraints are achievable if λmax (DΨ) < 1 is fulfilled for both the downlink and the uplink. Therefore, if the feasibility condition is not satisfied, the SINR constraints should be relaxed. The proposed algorithm is summarized in Table II, which first checks whether λmax (DΨ) < 1 holds to ensure the algorithm does not diverge, and then iteratively finds out the optimal solution minimizing the required power.
In this section we provide some numerical results to illustrate the advantages of the proposed algorithms. The channel matrix HH is assumed flat Rayleigh fading with i.i.d. complex Gaussian elements with zero mean and unit variance. The noise is AWGN. The transmitter is assumed to have perfect knowledge of the channel matrix HH , and each user knows its own channel as well as the transmit beamforming filter U. Without loss of generality, assume the same SINR constraint γ is required to be satisfied by all users, i.e., γk = γ for all k. In Figure 2, we plot minimum total power Pmin versus SINR constraint γ. Simulation parameters are K = 2 users, M = 8 transmit antennas, and each user is equipped with Nk = 4 receive antennas. The proposed algorithm in Table II is compared with the method proposed in [5]. As shown in the figure, the proposed group power allocation performs better than the method in [5] at high SINR. However, at low SINR it requires more power. This is because group power allocation suffers for the fact that it cannot adjust the power within the group as the method in [5]. At low SINR, the interference is larger and the method in [5] can adjust the power within a group to better deal with the interference. Figure 3 gives the simulation results of the balanced level C DL versus total power Pmax , where C DL is defined as in equation (15) for the proposed algorithm in Table I, and on the per stream achievable SINR for the method in [5]. Simulation parameters are: K = 4 users, M = 8 transmit antennas, and each user has 2 receive antennas. It can be seen that the proposed group power allocation can achieve higher balanced level than the method in [5] at positive SINR region. VI. D ISCUSSION For the proposed methods, the SINR constraints are defined as the per-user average SINR constraints, and the balanced levels C DL are the associated balanced average SINR levels. This per-user group SINR definition is reasonable since all data streams for a certain user are combined by the maximum SINR filtering for intra-group cooperation. For the method in
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
K=4 M=8 N=[2 2 2 2]
K=2 M=8 N=[4 4] 12
45 Proposed Group Power Allocation Khachan
Proposed Group Power Allocation Khachan
10
Balanced level C
Minimum total power (dBm)
40
35
30
8
6
4
2
25
0 20 −6
−4
−2
0 2 SINR constraint γ (dB)
4
6
Fig. 2. Comparison with [5] when all SINR targets must be satisfied. K=2, M=8, N=[4 4].
[5] which does not exploit intra-group cooperation, the same SINR constraint is to be satisfied by all data streams for a user. The average SINR constraint will thus equal to the per stream SINR constraint. Although the average SINR definition seems intuitively less restrictive than the per stream SINR constraints, the restriction of even power distribution within a group in our method is a stricter constraint compared to [5]. Other than these two differences, there is a main distinction making the proposed algorithm outperform [5]. Better transmit and receive beamformers are used, and data streams within a group can cooperate to resolve the transmitted signals. The proposed algorithm can exploit cooperation and consequently give gain over the method in [5], in which the data streams within a group are processed independently. Equivalently, the so-called multi-antenna user in [5] is a collection of several singleantenna users. In the proposed scheme, the assumption of even power distribution within a group is made to simplify the optimization. This even power allocation is suboptimal. Further improvement can be found in [14]. VII. C ONCLUSION A solution to the joint transmit-receive beamforming and power allocation under SINR constraints in the downlink of multiuser MIMO systems was proposed. The beamforming filter is a group maximum SINR filter which exploits the intragroup cooperation. Two optimization problems were considered. One is to maximize the minimum SINR to target ratio under a total power constraint; the other is to minimize the total transmission power while satisfying a set of SINR targets. The proposed group power allocation generalized the one in [4] to allow users to have multiple receive antennas. Based on the uplink-downlink duality, we formulated the dual problem in the virtual uplink, and iteratively solved the optimal beamforming filters and power allocation matrix. Simulation
Fig. 3. 2 2 2].
0
5 10 Total transmitted power Pmax (dB)
15
Comparison with [5] with sum power constraint. K=4, M=8, N=[2
results verified the superiority of the proposed algorithms over the per stream maximum SINR methods in [5]. R EFERENCES [1] A. Bourdoux and N. Khaled, “Joint Tx-Rx optimisation for MIMOSDMA based on a null-space constraint,” in Proc. of IEEE Vehicular Technology Conference (VTC-02 Fall), Sep. 2002, pp. 171–174. [2] L.-U. Choi and R. D. Murch, “A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 2–24, Jan. 2004. [3] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Processing, vol. 52, no. 2, pp. 461–471, Feb. 2004. [4] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 18–28, Jan. 2004. [5] A. Khachan, A. Tenenbaum, and R. S. Adve, “Linear processing for the downlink in multiuser MIMO systems with multiple data streams,” in IEEE International Conf. on Communications, Jun. 2006. [6] H.-J. Su and E. Geraniotis, “Maximum signal-to-noise array processing for space-time coded systems,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1419–1422, Sep. 2002. [7] D. Tse and P. Viswanath, “Downlink-uplink duality and effective bandwidths,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2002. [8] H. Boche and M. Schubert, “Optimal multi-user interference balancing using transmit beamforming,” in Wireless Personal Communications (WPC), 2003. [9] M. Schubert and H. Boche, “A unifying theory for uplink and downlink multi-user beamforming,” in Proc. IEEE Intern. Zurich Seminar, Jul. 2002. [10] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of Gaussian multiple-access and broadcast channels,” IEEE Trans. Inform. Theory, vol. 50, no. 5, pp. 768–783, May 2004. [11] W. Yang and G. Xu, “Optimal downlink power assignment for smart antenna systems,” in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc. (ICASSP), May 1998. [12] R. Horn and C. Johnson, Matrix Analysis. Cambridge University Press, 1990. [13] H. Boche and M. Schubert, “A general duality theory for uplink and downlink beamforming,” in Proc. Of IEEE Vehicular Technology Conference (VTC-02 Fall), 2002. [14] Y.-H. Yang, “Multiuser MIMO downlink beamforming with SINR constraints,” Master’s thesis, Graduate Institute of Communication Engineering, National Taiwan University, 2007.
Multiuser MIMO Downlink Beamforming Based on Group Maximum SINR Filtering Yu-Han Yang, Shih-Chun Lin and Hsuan-Jung Su Graduate Institute of Communication Engineering Department of Electrical Engineering National Taiwan University, Taipei, Taiwan Email: {r94942109, sclin2}@ntu.edu.tw, [email protected] Abstract—In this paper we aim to solve the multiuser multiinput multi-output (MIMO) downlink beamforming problem. The transmitter is a multi-antenna base-station broadcasting to users. Each user has multiple antennas at the receiver. A solution of the joint transmit-receive beamforming and power allocation under signal-to-interference-plus-noise-ratio (SINR) constraints in this system is proposed. The beamforming filter is a group maximum SINR filter which exploits intra-group cooperation. Taking advantage of the uplink-downlink duality property, the proposed algorithm iteratively computes the transmit-receive beamforming filters and the power allocation matrix. Simulation results verify the superiority of the proposed algorithm over previous works using per stream maximum SINR method.
I. I NTRODUCTION In this paper, a general case of the downlink channel is considered. One transmitter with multiple antennas is communicating with multiple independent users, each having multiple antennas. Each user is assigned a target signal-to-interferenceplus-noise (SINR) ratio. To suppress interference between users, transmit beamforming is used to point signals to the respective users. Each user also applies receive beamforming to further suppress interference. Power allocation at the transmitter is performed to optimize the performance. Joint beamforming and power allocation for the downlink channel have been studied previously. In [1], [2], [3], a precoding method is employed to block diagonalize the overall channel so that each user does not see the signals transmitted to other users. Thus the equivalent channel between any user and the base station is interference-free. However, such a zero-forcing approach is not optimal, because it thoroughly eliminates the co-channel interference (CCI) and takes no consideration of the noise. Hence the performance can be improved if the balance between CCI suppression and noise enhancement can be found. In [4], Schubert et al. consider the situation where a multiantenna base station is transmitting independent data streams to users each having single antenna and an individual SINR constriant. Khachan et al. [5] generalize the scheme in [4] to allow several transmission beams to be grouped to serve one user. However, in [5], in addition to the interferences among users, there are intra-group interferences between transmission This work was supported by the National Science Council, Taiwan, R.O.C., under grant NSC 95-2219-E-002-005.
beams. In order to tackle the intra-group interferences, we apply a group maximum SINR filter [6] to collect all the desired signal energy in the beam group of each user. The proposed scheme can be applied together with joint coding, and the group maximum SINR filtering can exploit intra-group cooperation and improve the performance. In this paper, we extend the work in [4] to the multiantenna setting based on group maximum SINR filtering which exhibits better performance compared to [5]. New optimization methods to iteratively calculate the beamforming filters and the power allocation matrix are proposed. Two optimization criteria are investigated. One is minimizing total transmitted power while satisfying a set of SINR targets. The other is maximizing the achieved SINR to target ratio under a total power constraint. For simplicity, we consider group power allocation which restricts equal power on beams in each group. Under such a restriction, our algorithm can find the optimal solution with the uplink-downlink duality [7]. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model Consider the downlink scenario with K users, where a base station is equipped with M antennas. The upper part of Fig. 1 shows the overall system block diagram for user k, Lk data streams. who has Nk receive antennas and receives K Thus the K users have a total of N = k=1 Nk receive K antennas receiving a total of L = k=1 Lk data streams. In a specific symbol time, symbols of the data streams intended for user k are denoted by a vector xk = [xk1 , xk2 , ..., xkLk ]T . Symbols of the L data streams are concatenated in a vector x = [xT1 , ..., xTK ]T . The precoder Uk ∈ C M ×Lk processes user k’s data streams before they are transmitted over the M antennas. These individual precoders together form the global transmitter beamforming matrix UM ×L = [U1 , U2 , ..., UK ]. The power allocation matrix for user k is a diagonal matrix Pk = diag{pk1 , pk2 , ..., pkLk },
(1)
and the global power allocation matrix, P = diag{P1 , P2 , ..., PK },
(2)
is a block diagonal matrix of dimension L × L. The channel between the transmitter and user k is represented by the Nk ×
978-1-4244-2075-9/08/$25.00 ©2008 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
Downlink
U1 Ξ
xK
P1 Ξ
x1
PK
UK
nk
H kH
+
+
xˆ k
VkH
where γk and SINRk are the SINR target and the achieved SINR for user k, respectively. We refer to SINRk /γk as the SINR to target ratio for user k. If the minimum SINR to target ratio in equation (5) can be made greater than or equal to one, then the second optimization problem is to find the minimum power required such that the SINR targets can be all satisfied. That is, min
Virtual Uplink
x1
V1
Q1
VK
QK
Fig. 1.
nkUL
H1
U
+
xˆ k
H k
HK
MIMO downlink system model for user k and its virtual uplink.
M matrix HH k , the Hermitian of Hk . The resulting N × M global channel matrix is HH , with H = [H1 , H2 , ..., HK ]. User k receives a length Nk vector K Pk xk +HH Uj Pj xj +nk (3) yk = HH k Uk k j=k,j=1
where nk represents the additive white Gaussian noise (AWGN) at the user’s receive antennas with variance σ 2 . That 2 is, E[nk nH k ] = σ INk , where E[·] denotes the expectation operator. The second term in equation (3) is the inter-group interference for user k. To estimate its Lk symbols xk , user k processes yk with its Lk × Nk receive beamforming matrix VkH . The resulting estimated signal vector is H x ˆk = VkH HH k Ux + Vk nk
= VkH HH k Uk
K
Pk xk + VkH HH k
Uj
(4a) Pj xj
j=k,j=1
+ VkH nk
(4b)
The global receive filter VH is a block diagonal matrix of dimension L × N , V = diag{[V1 , V2 , ..., VK ]} owing to the non-cooperative nature between users.
k
SINRk subj. to γk
k=1 j=1
SINRk ≥1 γk
(6)
MATRICES SELECTION
A. Maximum SINR Beamforming Lk SINRDL For given U and P, kj is maximized by using j=1
the maximum SINR filter as the receive beamformer Vk , where SINRDL kj is the SINR of the jth stream of user k in the downlink,. This optimization problem can be proved to be equivalent to a generalized eigenvalue problem [6]. That is, DL DL RDL s,k vkj = λkj Rn,k vkj
Based on the system model described in Section II-A, we consider two problems in this paper as follows. We assume that the base station has perfect knowledge of the channel matrix H, and receiver k knows its Hk perfectly. The first optimization problem is : with a total power constraint Pmax , maximize min SINRk /γk over all beamformers U, V, and k power allocation matrix P. That is, max min
k
We briefly review the uplink-downlink duality which plays an important role in our algorithm design. According to [7], [8], [9], [10], a virtual uplink system for our downlink system can always be found. We plot the virtual uplink for user k in the lower part of Fig. 1, where Qk is the corresponding power allocation matrix in the virtual uplink defined similarly as Pk . To be more specific, with fixed beamforming filters U, V, SINR targets γ1 , ..., γK , and a common sum power constraint Pmax for both the downlink and the virtual uplink, the optimal P and Q will make the downlink and its virtual uplink have the same SINR to target ratio. We will restricts the filters V and U as the maximum SINR array processors proposed in [6]. Then with the aids of the uplink-downlink duality, the optimization problems in Section II-B can be solved efficiently with iterative algorithms. That is, for each iteration we will use power allocations P and Q to find V and U; with new filters V and U, we can find new P and Q, and so on. When the SINR to target ratios of the downlink and virtual uplink equal to each other, the optimization problems are solved. We will first show how to calculate the beamforming filters and then show how to use these filters to find new power allocations P and Q in the next section.
B. Problem Formulation
U,V,P
pkj subj. to min
III. I TERATIVE ALGORITHMS AND BEAMFORMING
Ξ
Ξ
Ξ
xK
U,V,P
Lk K
Lk K k=1 j=1
pkj = Pmax ,
(5)
with λDL kj =
H RDL vkj s,k vkj
H RDL v vkj n,k kj
= SINRDL kj ,
(7)
(8)
H H where Vk = [vk1 , . . . , vkLk ], RDL s,k = Hk Uk Pk Uk Hk DL denotes the signal covariance matrix, and Rn,k = H 2 Hk Uj Pj UH j Hk + σ INk denotes the interference and j=k
noise covariance matrix. The receive beamforming filter Uk for the virtual uplink system can be derived similarly. As a result, the receive
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
beamforming filter designed for the downlink can be carried over to the transmit beamforming filter for uplink, and vice versa. IV. P OWER A LLOCATION We now consider finding the power allocation strategies for problems corresponding to (5) and (6), respectively, given fixed beamforming filters U, V and a set of SINR targets γ1 , ..., γK . The two problems are: Lk K SINRDL k s.t. pkj = Pmax , max min P k γk j=1
(9)
k=1
and
DL
The expression of SINRk has the same structure as the case considered in [4] with Nk = 1, ∀k. Therefore, we can apply the result in [4, Lemma 1], which states that since DL each SINRk is strictly monotonically increasing in pk and monotonically decreasing in pj for j = k, the maximizer of the optimization problem (9) satisfies DL
SINRk γk
= C DL , 1 ≤ k ≤ K.
(15)
That is, all users have the same SINR to target ratio C DL , termed balanced level. Otherwise, the users with higher SINR to target ratios can give some of their power to the user with the lowest ratio to help it, which contradicts the optimality.
A. Sum Power Constraint SINRDL k pkj = Pmax s.t. min ≥ 1, k = 1, ..., K. As in [11], with (15) and (9), equation (14) can be rewritten min k γk k=1 j=1 as 1 (10) (16) p DL = DΨp + Dσ, C For simplicity, we consider group power allocation. That is, power is allocated to users, and the allocated power for a user where is evenly distributed over all streams of that user: L2K γK L21 γ1 , (17) D = diag , ..., VH HH U1 2 VH HH UK 2 (11) pk1 = pk2 = ... = pkLk , 1 ≤ k ≤ K 1 1 K K F F Lk K
Let the power allocated to user k be pk . Consequently, the diagonal power allocation matrix Pk for user k can be written as a scaled identity matrix, i.e., pk IL . (12) Pk = Lk k We also define a vector p = [p1 , . . . , pK ]T to replace matrix P in the optimization problems. We will first show an important property when using maximum SINR filters in problems (9) and (10). That is, all users will have the same SINR to target ratios. This property will simplify the problems (9) and (10) to eigensystems. First, the vector norms of the beamforming filters vkj , j = 1...Lk , can be adjusted such that • VkH RDL n,k Vk is a scaled identity matrix [6]. • trace VkH Vk = Lk . Once the above two conditions are satisfied, the average SINR of each data stream of user k in the downlink scenario can be expressed as
H DL Lk trace V R V k k s,k 1 DL . (13)
SINRk = SINRDL kj = Lk j=1 H DL trace V R V k
Substituting Pk = SINR is DL SINRk
pk Lk ILk
n,k
into equation (13), the average
H H 2 V H Uk k k F = p j VH HH Uj 2 + σ 2 k k Lj Lk F pk L2k
k
(14)
j=k
the Frobenius norm, which is defined as where ·F
denotes
MF = trace MMH .
and [Ψ]ij =
2
ViH HH i Uj F Li Lj
0
j = i j = i.
(18)
Here [Ψ]ij denotes the ijth element of the K × K matrix Ψ, and the K × 1 vector σ = σ 2 1. Multiplying both sides of (16) by 1T , we have 1 1 1 = 1T DΨp + 1T Dσ. (19) DL C Pmax Pmax p Defining an extended power vector p ˜ = and an 1 extended coupling matrix DΨ Dσ Υ= , (20) 1 1 T T Pmax 1 DΨ Pmax 1 Dσ (16) and (19) together form an eigensystem 1 ˜ with [˜ p]K+1 = 1 (21) Υ˜ p = DL p C It can be observed that C DL is a reciprocal eigenvalue of the nonnegative extended coupling matrix Υ. However, only positive C DL is meaningful and its corresponding eigenvector p ˜ should also comprise nonnegative components. The existence of positive eigenvalue and eigenvector is guaranteed by the Perron-Frobenius theory [12], which states that for any nonnegative real matrix B ≥ 0 with spectral radius ρ(B), there exists a vector y ≥ 0 and λmax (B) = ρ(B), such that By = λmax (B)y. Therefore, we can conclude that the maximal eigenvalue λmax (Υ) and the associated eigenvector, corresponding to the reciprocal of C DL and the extended power vector p ˜ , respectively, are always nonnegative. To obtain the optimal power vector p, we can scale the extended power vector p ˜ , such that its last element equals to one.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
TABLE I G ROUP P OWER A LLOCATION WITH S UM P OWER C ONSTRAINT
TABLE II G ROUP P OWER M INIMIZATION
Initialization: U = I, V = I Iteration: 1: Downlink Receive Maximum SINR Beamforming for k = 1 : K DL Vk = eig(RDL s,k , Rn,k ) 2: Virtual Uplink Power Allocation with Sum Power Constraint q q 1 Λ = C UL 1 1 3: Virtual Uplink Receive Maximum SINR Beamforming for k = 1 : K UL Uk = eig(RUL s,k , Rn,k ) 4: Downlink Power Allocation with Sum Power Constraint p p 1 Υ = C DL 1 1 5: Repeat steps 1-4 until convergence, i.e., C DL = C UL .
Initialization: U = I, V = I. If λmax (DΨ) < 1, then Iteration: 1: Downlink Receive Maximum SINR Beamforming for k = 1 : K DL Vk = eig(RDL s,k , Rn,k ) 2: Virtual Uplink Power Minimization Allocation q = (I − DΨT )−1 Dσ 3: Virtual Uplink Receive Maximum SINR Beamforming for k = 1 : K UL Uk = eig(RUL s,k , Rn,k ) 4: Downlink Power Minimization Allocation p = (I − DΨ)−1 Dσ 5: Repeat steps 1-4 until convergence, i.e., C DL = C UL = 1.
V. S IMULATION R ESULTS The optimal q for the virtual uplink can also be obtained through similar derivation as in the downlink system. Using the uplink-downlink duality described in Section III, it is guaranteed that C DL = C UL . We summarize the optimization algorithm in Table I, which iteratively calculates the optimal beamforming filter and power allocation vector between the downlink and the uplink. Here eig denotes the generalized eigenvalue solver which outputs the eigenvectors. B. Power Minimization As discussed above, the minimizer of (10) satisfies DL
SINRk = γk , 1 ≤ k ≤ K.
(22)
Substituting (22) into (14), we have
and similarly,
(I − DΨ)p = Dσ,
(23)
(I − DΨT )q = Dσ.
(24)
Thus the resulting power allocation vectors are
and
p = (I − DΨ)−1 Dσ
(25)
q = (I − DΨT )−1 Dσ.
(26)
As in [4][5], there is a feasibility issue for the starting point U = I, V = I. It is proved in [13] that there exists a positive solution p if λmax (DΨ) < 1 is satisfied. Similarly, if λmax (DΨT ) < 1, then the existence of a positive solution q can be guaranteed. In [13], it is shown that λmax (DΨ) = λmax (DΨT ). In other words, the set of SINR constraints are achievable if λmax (DΨ) < 1 is fulfilled for both the downlink and the uplink. Therefore, if the feasibility condition is not satisfied, the SINR constraints should be relaxed. The proposed algorithm is summarized in Table II, which first checks whether λmax (DΨ) < 1 holds to ensure the algorithm does not diverge, and then iteratively finds out the optimal solution minimizing the required power.
In this section we provide some numerical results to illustrate the advantages of the proposed algorithms. The channel matrix HH is assumed flat Rayleigh fading with i.i.d. complex Gaussian elements with zero mean and unit variance. The noise is AWGN. The transmitter is assumed to have perfect knowledge of the channel matrix HH , and each user knows its own channel as well as the transmit beamforming filter U. Without loss of generality, assume the same SINR constraint γ is required to be satisfied by all users, i.e., γk = γ for all k. In Figure 2, we plot minimum total power Pmin versus SINR constraint γ. Simulation parameters are K = 2 users, M = 8 transmit antennas, and each user is equipped with Nk = 4 receive antennas. The proposed algorithm in Table II is compared with the method proposed in [5]. As shown in the figure, the proposed group power allocation performs better than the method in [5] at high SINR. However, at low SINR it requires more power. This is because group power allocation suffers for the fact that it cannot adjust the power within the group as the method in [5]. At low SINR, the interference is larger and the method in [5] can adjust the power within a group to better deal with the interference. Figure 3 gives the simulation results of the balanced level C DL versus total power Pmax , where C DL is defined as in equation (15) for the proposed algorithm in Table I, and on the per stream achievable SINR for the method in [5]. Simulation parameters are: K = 4 users, M = 8 transmit antennas, and each user has 2 receive antennas. It can be seen that the proposed group power allocation can achieve higher balanced level than the method in [5] at positive SINR region. VI. D ISCUSSION For the proposed methods, the SINR constraints are defined as the per-user average SINR constraints, and the balanced levels C DL are the associated balanced average SINR levels. This per-user group SINR definition is reasonable since all data streams for a certain user are combined by the maximum SINR filtering for intra-group cooperation. For the method in
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
K=4 M=8 N=[2 2 2 2]
K=2 M=8 N=[4 4] 12
45 Proposed Group Power Allocation Khachan
Proposed Group Power Allocation Khachan
10
Balanced level C
Minimum total power (dBm)
40
35
30
8
6
4
2
25
0 20 −6
−4
−2
0 2 SINR constraint γ (dB)
4
6
Fig. 2. Comparison with [5] when all SINR targets must be satisfied. K=2, M=8, N=[4 4].
[5] which does not exploit intra-group cooperation, the same SINR constraint is to be satisfied by all data streams for a user. The average SINR constraint will thus equal to the per stream SINR constraint. Although the average SINR definition seems intuitively less restrictive than the per stream SINR constraints, the restriction of even power distribution within a group in our method is a stricter constraint compared to [5]. Other than these two differences, there is a main distinction making the proposed algorithm outperform [5]. Better transmit and receive beamformers are used, and data streams within a group can cooperate to resolve the transmitted signals. The proposed algorithm can exploit cooperation and consequently give gain over the method in [5], in which the data streams within a group are processed independently. Equivalently, the so-called multi-antenna user in [5] is a collection of several singleantenna users. In the proposed scheme, the assumption of even power distribution within a group is made to simplify the optimization. This even power allocation is suboptimal. Further improvement can be found in [14]. VII. C ONCLUSION A solution to the joint transmit-receive beamforming and power allocation under SINR constraints in the downlink of multiuser MIMO systems was proposed. The beamforming filter is a group maximum SINR filter which exploits the intragroup cooperation. Two optimization problems were considered. One is to maximize the minimum SINR to target ratio under a total power constraint; the other is to minimize the total transmission power while satisfying a set of SINR targets. The proposed group power allocation generalized the one in [4] to allow users to have multiple receive antennas. Based on the uplink-downlink duality, we formulated the dual problem in the virtual uplink, and iteratively solved the optimal beamforming filters and power allocation matrix. Simulation
Fig. 3. 2 2 2].
0
5 10 Total transmitted power Pmax (dB)
15
Comparison with [5] with sum power constraint. K=4, M=8, N=[2
results verified the superiority of the proposed algorithms over the per stream maximum SINR methods in [5]. R EFERENCES [1] A. Bourdoux and N. Khaled, “Joint Tx-Rx optimisation for MIMOSDMA based on a null-space constraint,” in Proc. of IEEE Vehicular Technology Conference (VTC-02 Fall), Sep. 2002, pp. 171–174. [2] L.-U. Choi and R. D. Murch, “A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 2–24, Jan. 2004. [3] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Processing, vol. 52, no. 2, pp. 461–471, Feb. 2004. [4] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 18–28, Jan. 2004. [5] A. Khachan, A. Tenenbaum, and R. S. Adve, “Linear processing for the downlink in multiuser MIMO systems with multiple data streams,” in IEEE International Conf. on Communications, Jun. 2006. [6] H.-J. Su and E. Geraniotis, “Maximum signal-to-noise array processing for space-time coded systems,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1419–1422, Sep. 2002. [7] D. Tse and P. Viswanath, “Downlink-uplink duality and effective bandwidths,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2002. [8] H. Boche and M. Schubert, “Optimal multi-user interference balancing using transmit beamforming,” in Wireless Personal Communications (WPC), 2003. [9] M. Schubert and H. Boche, “A unifying theory for uplink and downlink multi-user beamforming,” in Proc. IEEE Intern. Zurich Seminar, Jul. 2002. [10] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of Gaussian multiple-access and broadcast channels,” IEEE Trans. Inform. Theory, vol. 50, no. 5, pp. 768–783, May 2004. [11] W. Yang and G. Xu, “Optimal downlink power assignment for smart antenna systems,” in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc. (ICASSP), May 1998. [12] R. Horn and C. Johnson, Matrix Analysis. Cambridge University Press, 1990. [13] H. Boche and M. Schubert, “A general duality theory for uplink and downlink beamforming,” in Proc. Of IEEE Vehicular Technology Conference (VTC-02 Fall), 2002. [14] Y.-H. Yang, “Multiuser MIMO downlink beamforming with SINR constraints,” Master’s thesis, Graduate Institute of Communication Engineering, National Taiwan University, 2007.
