Optimal Multi-user Space Time Scheduling for Wireless Communications Vincent K.N. Lau and Youjian Liu and Tai Ann Chen
[email protected] Bell Labs, Lucent Technologies
Abstract The multi-user MIMO space time scheduling problem is the prime focus of the paper. Optimizing the link level performance of the MIMO system does not always imply achieving scheduling level optimization. Therefore, the design optimization across the link layer and the scheduling layer is very important to fully exploit the temporal and spatial dimensions of the communication channel. In this paper, we shall address the design of the optimal space time scheduler for the multi-user MIMO system based on an information theoretical approach. We shall assume partial feedback channel, namely a scaler rate-feedback channel is available to the transmitter respectively but the full channel matrix is unknown to the transmitter. With the partial feedback, we found that the optimal resource allocation strategy is water-filling in both the temporal domain and spatial domain. The optimal scheduler should allocate all the power to at most nR users at any particular instant. The optimal scheduler performance depends on the total number of user (K), the number of receive antenna at the base station (nR ), and the number of transmit antenna at the mobile (nT ). The scheduling gain increases with nR due to the distributed MIMO configuration formed between mobile users and base station. With single power feedback channel, the scheduling gain reduces as nT increases, illustrating that transmit diversity hurts the scheduling performance in the case of scalar feedback. Finally, scheduling performance improves as K increases due to the multi-user selection diversity.
1
Introduction
Given a fixed power budget and bandwidth, it is possible to make use of multiple transmit and receive antenna to increase the bit rate of the link. This transforms the channel into a Multiple-Input Multiple-Output (MIMO) system where additional spatial channels are created. It is shown [1] that the channel capacity of MIMO system is proportional to min nT , nR where nT is the number of transmit antenna and nR is the number of receive antenna. Various space-time coding techniques are introduced in [2] to utilized the spatial channel. However, these efforts are all focused on optimizing the link level performance. On the other hand, packet scheduling is an important
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component in the communication systems in addition to the physical layer. In fact, achieving link-level optimization does not always imply achieving system-level optimization, especially for bursty traffic. It is found that selection diversity among different users is the key to the performance gain in scheduling [3]. Intuitively, to have a large scheduling gain, we should have physical links with high fluctuations in channel conditions so that scheduling decisions could ride on the top. Scheduling optimization for SISO multiuser system is thoroughly investigated in [4]. On the other hand, multi-user MIMO optimal scheduling is still an open problem. In this paper, we formulate the optimal space time scheduler for a general multi-user MIMO system based on the information theoretical approach. Instead of feeding back the channel matrix, partial feedback is assumed from the receiver to the transmitter to perform rate and power adaptation. Reverse link is the focus in the paper.
2
Multi-user MIMO System Model
We consider a communication system with K ≥ nR mobile users, each having nT transmit antenna and a base station with nR receive antenna as illustrated in figure 1(a). A widely accepted channel model, namely the quasi-static fading channel model, is assumed where the channel realization is constant over a frame of coded symbols but independent between different frames of coded symbol. We assume that a scalar rate feedback information is available to the transmitter for rate adaptation so as to achieve the ergodic capacity. The rate information is computed at the receiver based on the instantaneous channel matrix realization. In addition to the rate feedback, each user is provided with additional partial feedback information, namely the scalar power feedback for power adaptation. Each of the users is assumed to have an infinite buffer depth and have to compete for resource before uplink transmission of payload. This is to ensure that mobile clients always have packets to transmit whenever a transmission slot is assigned. The resource allocation is centralized at the base station through the scheduling layer. In general, more than one users could transmit at the same instant and signals between users could be separated through successive interference cancellation[5] at the base station. We shall focus on the average system throughput (average bit per channel
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R∗ = max Eh˜ ˜ p
nT
K
rk
(2)
k=1
so that r = [r1 , ..., rK ] resides in the capacity region and the average transmitted power satisfies the average power constraint Eh˜ (pk )] ≤ Pk ∀k ∈ [1, K] where pk is the nT × nT input covariance matrix and Pk is the average power for user k.
user-1
h1 Base Station
To tackle this problem, we have to first reduce the parameter space of the optimization. We found that the capacity hk0 achieving input covariance matrix, in the presence of scalar power feedback, is given by pk = ρk Dk with Dk a constant diagonal matrix with tr(Dk ) = 1 and ρk is a function of the scalar feedback (gf b,k ) chosen to satisfy the average power constraint E[ρk ] = Pk . Furthermore, the diagonal elements of Dk could be interpreted as transmit power allocation user-K 0 among the nT transmit antenna of a client user. We also found that the case of asymmetric power allocation across Figure 1: MIMO system model the nT transmit antenna is equivalent to uniform power allocation across the nT transmit antenna with a transformed N K 1 use per user), ρ¯ = limN →∞ KN n=1 k=1 rk (n), as the channel matrix given by: √ scheduling performance where rk (n) is the bit rate allocated εk,1 0 0 √ √ to user-k at time instance n. = √nT hk D1/2 εk,2 0 hk = nT hk 0 k √ 0 0 εk,nT nR
3
General Formulation of MIMO Capacity Region
Assume that all the users have identical and independent distributed channel matrices. Let Xk be the transmitted vector (nT × 1) symbol from user k and Y be the received ˜ = vector (nR × 1). Given a set of channel matrices, h {h1 , h2 , .., hK } where hk is a realization of the nR × nT ˜ is well channel matrix of user k, the capacity region C[h] known to be: ˜ ∀S ⊂ [1, , , K] r(S) ≤ I(Y, (Xi )i∈S |(Xi )i∈S c , h)
(1)
maximizwith some input distribution α(x1 )α(x2 )...α(xK ) ing the mutual entropy. Note that r(S) denotes i∈S ri . For quasi-static fading channels, the channel matrix realization remains constant for the whole coding block and therefore, the channel is regarded as AWGN within the coding block. The capacity achieving distribution for xk is shown [1] to be circular symmetric complex Gaussian distribution (α(xk ) → CN (0, pk )) with pk denoting the covariance E|h˜ [xk x∗k |gf b,k ]. Note that x∗ denotes the Her˜ denotes the power feedback mitian transpose and gf b,k (h) channel available to the k-th user. The general resource allocation problem of the multiuser MIMO system is now formulated as:
As a result, the optimization parameter space is reduced from a matrix pk to a scalar ρk . On the other hand, the capacity region is expressed as: 1 ∗ ˜ ˜ ˜ ˜ S (hS ) (hS ) I(Y, (Xi )i∈S |(Xi )i∈S c , h) = log2 I + p N0 (3) The optimization problem of this type is referred as maxdet problem where no analytical solution has been found so far. In order to gain insight into the nature of the optimal resource allocation strategy, we shall derive a reasonably tight upper bound for the capacity region. This is given by: 1 ∗˜ ˜ ˜ r(S) ≤ log2 InT |S| + pS hS hS N0 S|S| T β t ≤ log2 1 + ρk vk,t 2 ∀S ⊂ [1, K](4) N 0 t=1 k=S1
where T = min[nR , |S|nT ] is the rank of w, with βt the non˜∗ h ˜ zero eigenvalues of w = h ˜S is the corresponding S S and v nT |S| × nT |S| eigenvector matrix which is partitioned as follows. v1,1 . . . v1,T .. .. .. v ˜S = . . . v|S|,1
. . . v|S|,T
˜ we and vk,t 2 is the norm of the vector vk,t . Problem 1. Given any channel matrices realization, h, ˜ , for each have to find the optimal power control policy, p Introducing Lagrange multipliers (λ1 , ..., λK ), the optiuser so that the aggregate ergodic capacities, R∗ , is maxi- mal scheduling policy is to find the optimal power control mized. policy, {ρ1 , ..., ρK }, so that J ∗ is optimized.
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J = max E ρ1 ,..,ρK
K
[rk − λk ρk ] = E
k=1
max (rk − λk ρk )
ρ1 ,..,ρK
Optimal Sch, Nt=1,Nr=4,K=4 6
k
Average system capacity (bits per ch use per user)
∗
(5) where λk is chosen to satisfy the average power constraint, E[ρk ] = Pk . Combinatorial optimization could be used to solve the above optimization problem. The optimization results is summarized in the following theorem. ˜ let A Theorem 1. Given a channel matrix realization, h, be the set of user indices with non-zero power allocation. We have the cardinality of the set, |A| ≤ T . The algorithm to obtain the set A is given by: Step 1:
Let A = ∅ and K = [1, K]. Set m = 1.
Step 2:
For t
=
{k1 , k2 , .., km :
≥
vk2 ,t 2 λk2
≥ ... ≥
vkm ,t 2 λkm
Optimal Sch Nt=1,Nr=1,K=4
4
3
2
1
−5
0
5
10
SNR (dB)
(a) Scheduling performance of the MIMO vs SISO system with single power feedback. nT = 1, nR = 1, 2, 4 and K = 4.
= ≥
∀k ∈ K − [k1 , .., km ]}.
Optimal Scheduler Nt=2,Nr=2,K=16
14
Step 3: If A1 (m) = A2 (m) = ... = AT (m), then go to step 4. Else m → m + 1 and go to step 2. Set ρ∗ = (0, 0, ..., 0). Set A = ∅. ∂J Step 4b: If (i) argmax ∂ρ ∗ = k ∀j ∈ A1 (m) − A j ρ ∂J and (ii) ∂ρk > 0, then
Step 4a:
12
Average Aggregate Capacity
vk ,t 2 λk
[1, T ], form the sets At (m)
vk1 ,t 2 λk1
Optimal Sch Nt=1,Nr=2,K=4 5
Optimal Scheduler, Nt=2,Nr=2,K=8 Optimal Scheduler Nt=2,Nr=2,K=16
10 Random Scheduler Nt=2,Nr=2 8
6
4
ρ∗
A = A ∪ {k}
2
(6)
−10
∗
(ρ∗1 , ..., ρ∗K )
ρ∗j
and ρ = where = 0 ∀j ∈ / A and ρ∗j > 0 is given by the solution of the equations below for j ∈ A. ∂J =0 ∂ρj ρ∗ and
∂J ∂ρj
T
−6
−4
−2
0 SNR(dB)
2
4
6
8
(b) MIMO scheduling gain vs K.
Figure 2: Numerical Results for scalar feedback. ∀j ∈ A
(7)
3.1
is given by:
∂J = ∂ρj t=1
−8
βt vj,t 2 /nT K N0 + βt k=1 vk,t 2 ρk /nT
− λj (8)
Repeat step 4b. If there is no such k, procedure ends with A obtained above and the optimal power allocation is given by ρ∗ in (7). If A = ∅, the optimal power allocation is ρk = 0 ∀k ∈ [1, K]. The result is that at any particular instant, the optimal resource allocation is to pick at most nR users and the power allocation is thus water-filling in temporal and spatial domain.
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Numerical Results and Discussions
Figure 2(a) shows the scheduling gain of MIMO system (nT = 1, nR = 2, 4) versus the SISO system (nT = nR = 1) at various average SNR for single scalar power feedback channel. The reasons for the gain could be attributed to the multi-user selection diversity and the distributed MIMO configurations formed by the multi-user and the base station. Observe that there is significant gain in capacity and SNR of the MIMO system versus the SISO system (increases with increasing nR ). This corresponds to the distributed 4×4 configuration as illustrated in figure 2(a). The effect of multi-user selection diversity is illustrated in figure 2(b) where the gain increases versus the total number of users in the system. The performance gain with respect to nT is illustrated in figure 3(a) and (b). It is should that although transmit diversity could enhance the link level performance, it is harmful to the system performance because it suppresses the channel fluctuation as perceived by the link
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SISO system, the optimal resource allocation strategy is to allocate power to at most one user, the optimal strategy for the general MIMO system is to allocation power to at most min[nR , KnT ] users. This is a generalization of the SISO results. Furthermore, we illustrate that while transmit diversity is a good attribute with respect to the link level performance, it is detrimental to the scheduling performance in the MISO system and the MIMO system for single power feedback channel.
nT=8,K=1 6 nT=4, K=1
5.5 5
nT=1,K=1
Average link capacity
4.5 4 3.5 3 2.5 2
References
1.5 1 4
6
8
10 12 14 Average SNR per user
16
18
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[1] E. Telatar, “Capacity of Multi-antenna Gaussian Channels,” European Trans on Communs., vol. 39, pp. 585– 595, Nov 1999.
(a) Link level performance vs nT .
[2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Spacetime codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. on Information Theory., vol. 45, pp. 744–765, March 1998.
Optimal Scheduling Nt=1,Nr=2,K=4 Optimal Scheduling Nt=2,Nr=2,K=4
3
Average per user ergodic capacity
Optimal Scheduling, Nt=4,Nr=2,K=4 2.5 Random Scheduler Nt=1,Nr=2
[3] K. Lau and Y. Kwok, “CHARISMA: a novel channeladaptive TDMA-based multiple access control protocol for integrated wireless voice and data services ,” Proceedings of the IEEE Wireless Communications and Networking Confernce, 2000, pp. 507 – 511, Sept. 2000.
2 Random Sch Nt=2,Nr=2 Random Sch Nt=4,Nr=2
1.5
1
0.5
−10
−5
0 5 Average SNR (dB) per user
[4] D. Tse and S. V. Hanly, “Multiaccess Fading Channels - Part I: Polymatroid Structure, Optimal Resource Allocation and Throughput Capacities,” IEEE Trans. on Information Theory., vol. 44, pp. 2797–2815, Nov 1998.
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(b) System level performance vs nT .
[5] R. Ahlswede, “Multi-way communication channels,” Proc. 2nd Int. Symp. Information Theory., pp. 23–52, Figure 3: Role of transmit diversity in system with scalar 1971. power feedback. and therefore, degrades the multi-user selection diversity. Note that the scheduling performance in figure 3(b) is compared against random scheduler. Observe that the random scheduler improves as nT increases. This is because for random scheduler, there is no multi-user selection diversity anyway. Increasing nT shall improve the link performance by transforming the fading statistics into AGWN statistics and therefore, improve the overall performance. This shows that maximizing the link diversity order may not be the right thing to do when our target is system level optimization.
4
Conclusion
In this paper, we have investigated the optimal scheduling strategies for the reverse link of multi-user MIMO system. We constraint our investigation to partial feedback channel only. With the partial feedback channel, water filling of the transmit power on the temporal domain and the spatial domain could be performed. We show that while in
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