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A Scheduling Algorithm Combined with Zero-forcing Beamforming for a Multiuser MIMO Wireless System Jinsu Kim1, Sungwoo Park1, Jae Hong Lee1, Joonho Lee2, and Hanwook Jung2 1

2

School of Electrical Engineering, Seoul National University Shillim-dong, Gwanak-gu, Seoul 151-742, Korea Tel: +82-2-880-8430 E-mail: {jskim97, painamb0}@snu.ac.kr

Convergence Lab. KT 17 Woomyneon-dong, Seocho-gu, Seoul 137-792, Korea Tel: +82-2-526-6092 E-mail: [email protected]

Abstract– In this paper, we introduce a scheduling algorithm over a MIMO broadcast channel. Given a set of users, the scheduler selects more than one user and transmits independent data to them simultaneously by using zero-forcing beamforming. Taking computational complexity into account, a greedy method finding the best and most orthogonal channel vectors is proposed. Additionally, considering fairness among asymmetric users, we also propose an asymptotically fair scheduling algorithm.

information-theoretic result motivates the recent intense researches on the effective packet scheduling schemes [4], [5]. In parallel with works on scheduling, recently, there has been a large amount of interest in the area of multiuser MIMO systems, specially, the capacity of MIMO broadcast channels (BC) where a single transmitter with multiple antennas sends independent information to multiple receivers simultaneously [6], [7], [8]. Built on Costa’s work [9], recent development in information theory have indicated that in an interferencelimited fading environment, so-called dirty paper coding achieves the sum capacity of a MIMO BC when the transmitter has full knowledge of the channel state information to all the receivers. The idea of dirty paper coding is based on successive interference cancellation at the transmit side by using the knowledge of the other users’ signal. In addition, transmit beamforming, which is also referred to as space-division multiple access (SDMA), have also drawn a lot of attention as suboptimal technique due to the difficulty of practical implementation of dirty paper coding [10]. The research results on MIMO BC have shown that the sum capacity, which is the maximum achievable throughput by using dirty paper coding or transmit beamforming, is much higher than that of TDMA scheduling [11]. In other words, it is possible to boost the sum capacity when the base station transmits to more than one user simultaneously instead of assigning all the antennas to a single user at one time slot. In this reason, the packet scheduling algorithm for the downlink of a MIMO system must choose dynamically the best set of users at each time slot, contrary to the SISO case where the optimum strategy is to schedule only the single best user at each time. In this paper, we introduce the scheduling algorithm that has both low complexity and resource-fairness as well as increases the system throughput in multiuser MIMO system. We focus on the downlink where the zero-forcing beamforming is used at the base station. The remainder of this paper is organized as follows. In Section II, the system model of MIMO BC is described and zero-forcing beamforming is reviewed briefly. In Section III, a scheduling algorithm combined with zeroforcing beamforming is proposed, and its complexity and fairness are considered. Section IV provides some simulation results and discussions for this scheme. Finally, conclusions are

I. INTRODUCTION Multiple-input multiple output (MIMO) technique is one of key technologies to increase the spectral efficiency by using the same frequency bandwidth through all transmit antennas. Recent research in information theory has shown that dramatic capacity gains can be achieved by the use of multiple antennas at both transmit and receive sides in a point-to-point communication system [1], [2]. In a multiuser system, however, achieving link-level optimization does not always imply system-level optimization, especially for bursty packet data traffic. In addition to the physical (or link) layer, the multiple access control (MAC) layer plays a critical role in determining the multiuser system performance. Traditionally, the design of the PHY and MAC layers is done separately. This approach is effective for wire systems in which the state of physical layer is essentially timeinvariant. However, for wireless systems, the physical layer performance is time varying due to fading effects. This motivates the cross-layer approach that MAC layers effectively adapt to changing channel conditions. One potential technique of the cross-layer approach is the packet scheduling exploiting multiuser diversity. In [3], Knopp and Humblet pointed out a form of diversity inherent in a wireless system with multiple users, provided by independent timevarying channels across the different users. The diversity benefit is exploited by tracking the channel fluctuations of the users and scheduling transmissions to the user with the best channel condition. Since the fading, which is a fundamental characteristic of the wireless channel, is exploited rather than mitigated in this strategy, the system throughput can be enhanced. This This work was supported in part by the National Research Laboratory (NRL) Program, the Brain Korea 21 Project, and KT Corp., Korea.

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drawn in Section V. II. SYSTEM MODEL AND ZERO-FORCING BEAMFORNING We consider the downlink channel of a multiuser MIMO system with M antennas at the base station and K users with a single antenna at each receiver. The transmitter sends independent information to multiple receivers simultaneously. The system model is given by y = Hx + n (1) M ×1 K ×1 K ×1 K ×M where x ∈ ,y ∈ ,n ∈ and H ∈ are the transmitted signal vector, the received signal vector, the noise signal vector, and the channel matrix whose ith row is the ith channel vector hi ∈ 1× M , respectively. In case that the number of users K is less than or equal to the number of transmit antennas M and the transmitter has the perfect channel knowledge, the zero-forcing beamforming can achieve the sum rate adjacent to maximum sum capacity obtained by using a dirty-paper approach. Zero-forcing beamforming consists of inverting the channel matrix at the transmitter in order to create orthogonal channels between the transmitter and the receivers without receivers’ cooperation. Thus, there is no interference among users. The beamforming matrix is given by

B =H

†

(HH )

† −1

D

(2)

where D = diag ( d1 ,… , d K ) is a diagonal matrix which keeps the transmit power unchanged after beamforming. The kth diagonal elements d k is defined as

( HH † )− 1  .   k , k

dk = 1

(3)

Let u ∈ M ×1 be the information signal vector whose element uk is the original data for each user k . The actual transmitted signal vector through transmit antennas after beamforming is given by x = Bu . (4) Then, the received signal vector is represented as y = HBu + n = Du + n . (5) Since D is a diagonal matrix, a multiuser MIMO downlink channel is decomposed into parallel single-user MIMO channels. The maximum achievable sum rate is given by K

(

WF Rsum = ∑ log ξ dk  k =1

2

)

(6) +

where ξ is the solution of the water-filling equation K

ξ − 1 dk 2  = P ∑  + k =1

(7)

and P is the total transmit power.

The goal of this approach is to select proper M users among K users and to transmit data to selected users simultaneously by using zero-forcing beamforming. An optimum policy in the sense of maximization of sum capacity is so-called max-sumrate scheduling which selects the user set that maximizes the sum rate by using exhaustive search. However, it is a combinatorial optimization problem so that the associated complexity is too high. In this section, we propose a scheduling algorithm which has low complexity by using a greedy method.

A. Greedy algorithm to find “the best and most orthogonal” channels To maximize the sum rate, the scheduler has to select a 2 user set that provides large values of d k ’s. The following 2 theorem relates how to increase the values of d k ’s, which motivates the greedy method that can be applied to max-sumrate scheduling. Theorem 1: Let H ∈ M × M be the channel matrix corre2 sponding to the selected user set. Then d k is given by 2 1 2 (8) dk = = h k ⋅ Null (H k ) 1 − ( HH † )    k , k H k is defined where h k is the channel vector of k th user, T h kT−1 h kT+1 h MT  and a denotes as H k = h1T the vector norm of a . Null (A) denotes the orthonormal basis corresponding to the null space of A and can be obtained by using the singular value decomposition as follows VD†  A = UDV † = U Drow 0   row  , Null (A) = V0 . (9) †  V0  −1 Proof: The ( k , k ) th element of ( HH † ) can be expressed as Minork , k ( HH † ) ( HH † )− 1  = . (10)   k , k det HH † where Minork , k ( HH ) is the expressed using H k as †

(

( k , k ) th

(

)

(11)

†

The determinant of HH is given by det ( HH † ) = det Hˆ Hˆ †

(

k

k

(

)

)

= det H k H k† − H k h k† (h k h k† ) h k H k† ⋅ det (h k h k† ) −1

T

)

(

(

2

)

The second equality is obtained by using

(

)

(13)

)

= h k ⋅ Null (H k ) ⋅ det Dk 2 .

212

(12)

where Hˆ k is defined as Hˆ k = h k H k  ,which is a permutational version of H . Using the singular value decomposition of H k , det ( HH † ) can be derived as det ( HH † ) = h k I M × M −VDkVD† k h k† ⋅ det Dk 2 = h kV0 kV0†k h k† ⋅ det Dk 2

The zero-forcing beamforming scheme in the previous section has the constraint that the number of users is less than or equal to the number of transmit antennas. But in a practical system, it is hard to satisfy this constraint. Thus, there is a need to be combined with scheduling in a multiuser MIMO system.

minor and can be

Minork , k ( HH † ) = det H k H k † .

(

III. SCHEDULING

)

V †  I M × M = VkVk † = VDk V0 k   D†k  = VDkVD† k +V0 kV0†k .(14) V0 k  2 † 2 Since det H k H k = det Dk , d k is expressed as (8) by using (10) and (13). The proof is completed.

(

)

(

)

†

Since Null (H k ) represents a unit vector that is orthogonal to all the channel vector except the k th channel vector h k , h k ⋅ Null (H k ) denotes the amplitude of Hermitian in2 ner product of h k and Null (H k )† . Therefore, d k has the higher value as the norm of h k increases and the direction of h k is closer to that of Null (H k )† , in other words, h k is more orthogonal to all the other channel vectors. This result can be understood intuitively. The high norm of h k indicates the channel of a good quality. Additionally, the orthogonality of channel vectors represents how correlated the channel vectors of users are. If the channel vectors of users are highly correlated, that is, the orthogonality of channel vectors is low, 2 then d k s have low values even though the each user has a channel of a good quality. Considering these two factors – channel quality and orthogonality, the scheduler has to choose the users whose channels are both the best and most orthogonal. Instead of an exhaustive search which is an optimum solution, a greedy algorithm using Theorem 1 can achieve nearoptimum results. First, the scheduler selects the user with the best channel. Next, it selects the user who has the largest vector component perpendicular to the first user’s channel vector. It iterates finding the vector which has the largest component perpendicular to all the already selected users’ channel vectors until the number of selected users is equal to M. We call this algorithm greedy max-sum-rate scheduling. The details of the algorithm are given below. Greedy max-sum-rate scheduling algorithm: 1. Initialize G = {1, 2,… , K } and S = {

}.

2. Select i1 = arg max h j and let Ω = hi1  . j∈G

Update S ← S ∪ {i1} and G ← G − S .

3. Find Ω ⊥ by using SVD of Ω . VD†  Ω = UDV † = U [D1 0 ]  †1  , Ω ⊥ = Null ( Ω ) =V0 V0 

4. Select i = arg max h j Ω ⊥ . j∈G Ω  Update Ω ←   , S ← S ∪ {i1} and G ← G − S . hi  5. Repeat step 3,4 until S = M . The complexity of the above algorithm is given by M  K − i + 1 M ( 2 K − M + 1) ∑  = 1  2 i =1 

(15)

K which is much less than that of exhaustive search –   . M  B. Fairness When the channels experienced by various users have the symmetrical statistics, the greedy max-sum-rate scheduling algorithm in Section III-A guarantees the asymptotic fairness as well as maximizes the sum capacity of the system and the throughput of individual users. However, in practice, the channel statistics of users are not symmetrical due to many different conditions such as distance from the basestation, moving speed, and antenna correlation, etc. In this case, greedy maxsum-rate scheduling still maximizes the system capacity but tends to assign a larger fraction of time to users with higher mean capacity, possibly causing starvation of users with low mean capacity. This may result in too unfair throughputs among the users. Slight modification of the greedy max-sum-rate scheduling can achieve fairness as well as high scheduling gain. A simple way to guarantee fairness is to equalize the norms of channel vectors before applying to the algorithm. The scheduler finds the most orthonormal vectors among those normalized by their instantaneous norms, hj hˆ j = , j = 1,… , K (16) hj

which have unit norms for all users. At step 2 in Section III-A, the scheduler selects the first user randomly with equal probability. We call this modified scheme instantaneously fair greedy max-sum-rate scheduling. This strategy equalizes the probabilities that each user will be selected so that the scheduler can provide fairness among users at any instantaneous time. However, it does not consider the instantaneous channel quality but only the orthogonality of channels. We consider another noble strategy dependent on both channel quality and orthogonality. We assume asymmetric Rayleigh fading over each transmit-receive antenna pair. We also assume that the total transmit power and the variance of noise is normalized to one. Then, the channel vector of user j can be modeled as h j = ρ j h wj = ρ j ( hwj1 ,… , hwjM ) (17)

where ρ j is the average signal-to-noise ratio (SNR) at each receiver and h wj ∈ 1× M is a vector whose entries hwjk are complex Gaussian random variables with zero mean and unit variance. Since the channel coefficient hwjk is a random variable, the associated channel vector h j and the norm h j are also random2variables. has a Chi-square distribution of 2 M deSince h wj gree, the statistics (pdf and mean) of h j can be derived as follows. − h2 / ρ j

fh

213

j

2 ⋅ h 2 M −1e (h) = ρ Mj −1Γ ( M )

(18)

ρ j Γ ( M + 1/ 2 )

IV. SIMULATION RESULTS

(19)

Γ(M )

These channel vector norms, h j ’s, are used as the metric of instantaneous channel quality at the greedy max-sum-rate scheduling. Although these are random quantities, the probability that the j th user will be selected tends to increase as the mean E  h j  increases, which is related to the user’s average SNR ρ j . This fact causes increment of sum rate but results in unfairness at the same time. On the contrary, the instantaneously fair greedy max-sum-rate scheduling uses hˆ j ’s as the metric of channel quality. Since hˆ j ’s are deterministic and equal to one at all time, there is no gain exploiting the instantaneous channel quality, while each user has equal chance to be selected. A noble strategy that takes advantage of the best of both above algorithms uses the metric of channel quality as follows hj hj = , j = 1,… , K (20)

Consider the asymptotic case that window length tc is extended to infinity in order to have an insight into how this scheme guarantee fairness. In this case, the time average µ h j can be regarded as the ensemble mean of the random variable h j and thus h j can be rewritten as hj hj = , j = 1,… , K (22) E  h j  The pdf of h j are given by j

2 ( Γ( M + 1/ 2) )

( Γ( M ) )

2 M +1

2

2M

h

2 M −1

e

 Γ ( M +1/ 2)  2 −  h  Γ(M ) 

(23)

We note the interesting fact that the statistics of h j are not related to a user’s average SNR ρ j and are all the same for various users. This means that h j ’s for all users are independent and identically distributed (i.i.d.) random variables. Thus, the each user has equal probability to be selected. In addition, the random characteristic of h j enables the scheduler to exploit the channel quality by selecting the user whose instantaneous channel quality is high relative to its own mean. In this reason, we call this scheme asymptotically fair greedy max-sum-rate scheduling.

214

ZF-BF MSR by exh. ZF-BF G-MSR TDMA MR

20

15

10

5

0 -10

-5

0

5

10

15

20

25

30

SNR(dB)

Fig. 1. Sum rate of ZF-BF with scheduling and TDMA scheduling versus SNR for a symmetric 10 user system with 4 transmit antennas.

j

where µ h is an estimate of the time average for h j j which can be estimated by keeping track of h j in a past window. In the strict sense, the h j , h j , and µ h in equation j (20) are the variables related to the time slot index t and are merely abbreviated. So µ h , to be exact, µ h ( t ) can be j j obtained using a low pass filter with a time constant of tc time slots as  1 1 µ h (t + 1) =  1 −  µ h (t ) + h j (t ) . (21) j j t t  c  c

f h (h) =

25

40

ZF-BF MSR ZF-BF RR TDMA MR TDMA RR

35 30

Sum rate (nats/s/Hz)

µh

Fig. 1 shows the sum rate of zero-forcing beamforming(ZFBF) with max-sum-rate scheduling (MSR) over MIMO BC and TDMA max-rate scheduling (MR) as a function of signal-tonoise ratio in a symmetric user case where all the statistics of

Sum rate (nats/s/Hz)

E  h j  =

25 20 15 10 5 0

0

2

4

6

8

10

Number of Tx antennas, M

Fig. 2. Sum rate of ZF-BF with scheduling and TDMA scheduling versus the number of transmit antennas, M , for a symmetric 30 user system where each user’s average SNR is 20dB.

users in the system are identical. It is assumed that the transmitter has four transmit antennas and the number of users is set to 10. We can see that the sum rate of ZF-BF with scheduling is much higher that that of TDMA scheduling. We can also see that the sum rate of greedy max-sum-rate (G-MSR) is very close to that of exhaustive search. In Fig. 2, we plot the sum rate versus the number of transmit antennas M , along with those of round-robin scheduling (RR) for the sake of comparison. We assumed a 30 symmetric user case. As expected, the sum rate of ZF-BF grows almost linearly with M while the sum rate of TDMA scheduling grows similarly to logarithmic fashion. In addition, we can show that the addition of transmit antennas yields an increase of scheduling gain over round-robin to a ZF-BF system, but a decrease to a TDMA scheduling system. This effect can be explained by channel hardening effect [12]. Fig. 3 and Fig. 4 show the sum rates of proposed scheduling algorithms as a function of the number of users, K in asymmetric environment where the statistics of users are not

identical. The average SNRs of users are assumed to be uniformly distributed over [ 0 dB, 20 dB] and the number of tra nsmit antennas is two for Fig. 3 and four for Fig. 4, respectively. For the sake of comparison, the sum rate of round-robin scheduling is plotted. We can see that the greedy max-sum-rate

downlink of multiple antenna multiuser wireless system, which is so-called MIMO broadcast channel. We showed that it is no longer optimum to transmit data to only one user at a time in MIMO broadcast channel. We also proposed a novel scheduling scheme that has both low computational complexity and 40 30 20 10 0

0

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Number of users, K

Fig. 3. Sum rate of ZF-BF with G-MSR, AFG-MSR, IFG-MSR, RR scheduling for M = 2 in asymmetric case.

5

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User index

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User index

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Round robin

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Instantaneously fair greedy max-sum-rate

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Asymptotically fair greedy max-sum-rate

15 10 5 0

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25

30

Fig. 5. Percentage of fraction of time allocated to each user with the same environment as in Fig. 4. The number of users is 30.

G-MSR AFG-MSR IFG-MSR RR

16

Sum rate (nats/s/Hz)

50

7

6

20

Greedy max-sum-rate

60

Fraction of time (%)

8

Fraction of time (%)

Sum rate (nats/s/Hz)

9

Fraction of time (%)

G-MSR AFG-MSR IFG-MSR RR

Fraction of time (%)

10

good performance close to the optimum max-sum-rate scheduling in throughput sense. In addition, the proposed scheme can be modified to guarantee fairness among users. Simulation results demonstrate the proposed scheme provides a good trade-off between throughput and fairness in practical system.

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REFERENCES

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I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecommun., vol. 10, no. 6, pp. 586-595, Nov./Dec. 1999. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, no. 3, pp. 311-335, Mar. 1998. [3] R. Knopp and P. Humblet, “Information capacity and power control in single cell mulituser communications,” in Proc. IEEE Int. Computer Conf. (ICC’95), Seattle, WA, June 1995, pp. 331-335. [4] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayane, and A. Viterbi, “CDMA/HDR: A bandwidth efficient high speed wireless data service for nomadic users,” IEEE Commun. Mag., vol. 38, pp. 70-78, July 2000. [5] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforing using dumb antennas,” IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1277-1294, June 2002. [6] G. Caire and S. Shamai, “On achievable rates in a multi-antenna broadcast downlink,” in Proc. 38th Annual Allerton Conf. Communications, Control, Computing, Oct. 2000, pp. 1183-1193. [7] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sum-rate capacity of MIMO broadcast channels,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2658-2668, Oct. 2003. [8] P. Viswanath and D. N. C. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. Inform. Theory, vol. 49, no. 8, pp. 1912-1921, Aug. 2003. [9] M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. IT29, no. 3, pp. 439-441, May 1983. [10] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Tran. Signal Processing, vol. 52, no. 2, pp. 461-471, Feb. 2004. [11] N. Jindal, A. Goldsmith, “Dirty paper coding vs. TDMA for MIMO broadcast channels,” submitted to IEEE Trans. Inform. Theory. [12] B. M. Hochwald, T. L. Marzetta, and V. Tarohk, “Multiple-antenna channel hardening and its implications for rate feedback and scheduling,” IEEE Trans. Inform. Theory, vol. 50, no. 9, pp. 1893-1909, Sep. 2004.

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Number of users, K

Fig. 4. Sum rate of ZF-BF with G-MSR, AFG-MSR, IFG-MSR, RR scheduling for M = 4 in asymmetric case.

scheduling has the best performance over others. It is because it does not consider a trade-off relationship between throughput and fairness at all. We can see that the asymmetrically fair greedy max-sum-rate (AFG-MSR) scheduling outperforms the instantaneously fair greedy max-sum-rate (IFG-MSR) scheduling as expected. To compare fairness, we plot the percentage fraction of time allocated to each of 30 users for various scheduling algorithms in Fig. 5. The simulation environment is the same as in Fig. 4. The round-robin scheduling is perfect resource-fair due to inherent characteristic. We can see that the greedy maxsum-rate scheduling suffers from starvation of some users due to unfairness, while both the instantaneously fair greedy maxsum-rate and the asymptotically fair greedy max-sum-rate scheduling have fairness similar to that of round-robin scheduling. Thus, Fig. 4 and Fig. 5 demonstrate the asymptotically fair greedy max-sum-rate scheduling exhibits good performance in both the throughput and the fairness senses. V. CONCLUSIONS In this paper, we investigated scheduling algorithms for the

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