A Scheduling Algorithm with One-Bit Feedback for Multi-User MIMO System

Jianguo Liu, Qinghua Ma, Min Lin, Luxi Yang, Member, IEEESchool of Information Science and Engineering

Southeast University Nanjing, Jiangsu, 210096, China

hohailiu@163.com, dspmqh@gmail.com, linmin63@sina.com, lxyang@seu.edu.cn

Abstract— In this paper, we present a scheduling algorithm to exploit multi-user diversity with one bit feedback information for multi-user MIMO system. Instead, the users in a cell only need feed back one-bit information about the comparison between the channel quality indicator and the defined threshold thC , and then BS allocates the channel resource to one user experiencing favorable channel conditions. Specially, we solve this problem achieving the optimum threshold for multiple antennas systems in presence of transmit correlation. Simulation results show that the proposed scheme achieves better performance in terms of the average system spectral efficiency than round robin scheduling scheme.

Keywords- MIMO, feedback, multi-user diversity, scheduling algorithm

I. INTRODUCTION

The scheduling scheme based on multi-user diversity for multi-user multiple-input multiple-output (MIMO) system has been addressed in various works to maximize average system throughput, e.g., [1]-[3]. General scheduling algorithms have been proposed based on knowledge of channel station information at transmitter (CSIT) and/or channel quality indicator (CQI). As shown in [2], the more feedback loads of each user, the higher system throughput improvement. However, in frequency-division duplex (FDD) systems where the downlink and uplink are not reciprocal, CSIT and/or CQI must be conveyed to the base station (BS) from all users through a feedback channel. This is not impractical for all users to feed back too much information to BS over finite-rate feedback control channels in FDD mode.

Consequently, literature [4] has developed a scheduling algorithm that exploits multi-user diversity with one-bit feedback information for multi-user single-input single-output (SISO) system; however it has not discussed the scheduling problem for multiple antenna system. Therefore, in this paper we proposed a scheduling algorithm that exploits multi-user diversity with one-bit feedback information for multi-user MIMO system. Herein, we separate the users to two sets with

one-bit feedback information about the comparison between the CQI for each user and the defined threshold thC , then BS will allocate the channel resource to one user belonging to the set experiencing favorable channel conditions, with an appropriate threshold thC , some degree of multi-user diversity benefit can be achieved. As the threshold thC is important for system performance, we solve this problem achieving the optimum threshold for multiple antennas systems in presence of transmit correlation.

II. SYSTEM MODEL

We consider a narrowband MIMO system consisting of one

tN -antenna BS and K user, each with rN antennas,

where r tN N≥ . The channel for each user is assumed to be Rayleigh flat fading with uncorrelated receive antennas and with transmit correlation modeled as

1/ 2k kw t=H H R , 1 k K≤ ≤ (1)

Where kwH is a matrix of independent identically distributed

(i.i.d.) complex Gaussian random variable with zero mean and unit variance for the thk user, 1/ 2 *1/ 2

t t t=R R R is the spatial covariance matrix at transmitter and is further assumed to have an exponential form for all users

( )

1*

* 1*

11

1

t

t

tt

N

N

t

NN

ρ ρρ ρ

ρ ρ

−

−

=R (2)

Therefore the received vector at the thk user is,

1/ 2k ksw t

t

EN

= +y H R s n (3)

Where ky is the 1rN × received signal vector, sE is total This work was supported in part by NSFC (60496310, 60672093), NSFJS (BK2005061) and Huawei Technologies Co., Ltd.

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transmit energy, s is 1tN × transmit signal vector with elements picks from a unit energy constellation and n is the 1rN × circular complex additive with Gaussian noise

vector with variance 2rn Nσ I .

The CSIT is assumed to be perfectly known at the receivers of all users. And suppose the zero-forcing (ZF) receiver is utilized, the ZF receiver decompose the MIMO link into tNparallel streams, the post signal to noise ratio (SNR) on the thi ( )1, , tN decoded stream for the thk user can be shown to be:

( ) 12 *

k si

k kt n

ii

E

Nγ

σ−

=H H

(4)

As shown in [5], tN parallel streams are i.i.d with

1r tN N− + order diversity, and the probability distribution function (PDF) of the post SNR for each stream can be expressed as

( ) ( )

2 2

2 2 2 2

!

ki t n i

r ts

NN NE k

k t n i i t n ii

s r t s

N e NfE N N E

γ σ σ

σ σ γ σ σγ− −

=−

(5)

Where 1, ,k K= , 1, , ti N= , and 2iσ is the thi diagonal

entry of 1t−R .

III. SCHEDULING WITH ONE BIT FEEDBACK

For the TDMA systems, only one user can receive or transmit, and this user is selected by the scheduler during one time slot. In the multi-user MIMO system, when user k is scheduled by the BS, the instantaneous channel capacity

kC can be defined as the tN sub-channel capacity sum:

,1

tN

k k ii

C C=

= , 1 k K≤ ≤ (6)

Where ,k iC denotes the thi sub-channel capacity for user k ,and be expressed as

( ), 2log 1 kk i iC γ= + (7)

Where kiγ is the post SNR on the thi decoded stream of

the thk user in (4).

As shown in [3], in order to maximize the system throughput, the system resource should be assigned to the user with maximal instantaneous channel capacity, the benefit of multi-user diversify can be exploited through the above algorithm,

however, in this case, the scheduler requests CSIT and/or CQI from each user. In order to make a trade-off between the feedback load and system available throughput, we let each user feed back one-bit information to the BS by defining a specific threshold thC as follows:

th1, > 0, otherwise

kk

C CI = (8)

Accordingly for each time slot, each user will measure the downlink channel on the assumption that that the perfect CSIT and the number of users in the system can be obtained for each user, and then reports kI to the BS, the BS separates all users to two sets according with received one-bit feedback information, one set denotes the qualified set { }1 | 1, 1kSet k I k K= = ≤ ≤ with the feedback bit “1” and another set denotes the unqualified set { }0 | 0, 1kSet k I k K= = ≤ ≤ with the feedback bit “0”. Then the BS will allocate the channel resource to one user belonging to qualified set 1Set or randomly schedule one user

from the unqualified set 0Set when the qualified set 1Set is empty. This can be expressed as:

( ) ( )( )

1 1th

0 1

, Set ,

, Set

rand Set if is not emptyk K C

rand Set if is empty= (9)

As shown above, the value of threshold thC is pivotal for system performance, if the throughput threshold thC is too small, there is a higher probability that the scheduler will have to select one user with an unfavorable channel condition, otherwise that will lead to a higher probability to select a random user, and the average system throughput will decrease due to the scheduling outage.

IV. CHOICE OF THRESHOLD

In the TDMA system described, the capacity can be described using the maximal average system spectral efficiency (MASSE), so we use the MASSE herein as the objective function to perform an optimization on the threshold thC for the proposed scheduling algorithm, and MASSE can be given by

( ) { }th, kMASSE K C E C= (10)

Where k is the selected user for the next time slot in (9).

By analysis of the derivative of (10), we can see that this function has an absolute maximal point, which means it possible to find a threshold which gives optimal MASSE. In order to obtain optimal threshold thC , the PDF of

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instantaneous channel capacity kC must be taken into account.

Firstly we shall determine the PDF of ,k iC which can be derived from (5)

( ) ( ) ( )

( )( )

( )( )

,

2 2,

,,

, ,2 1

2 12 22 2 2 12 ln2

!

Ck k ii

Ck it n i

r tk ik i s

k kk i i i k i

NN NCC E

t n it n i

s r t s

f C f J C

NN eE N N E

γ

σ σ

γ γ

σ σσ σ

= +

+−−

= →

+=

−

(11)

Where J denotes the Jacobian of the transformation, defined by the function

( ) ( )( )

, 1,

,

2 ln 2k i

ki Ck

i k ik i

J CC

γγ +∂

→ = =∂

(12)

Because kiγ ( )1,2, , ti N= are statistically i.i.d. random

variables for user k ( )1 k K≤ < , ,k iC ( )1,2, , ti N= are

also i.i.d. random variables for user k ( )1 k K≤ < , which is

characterized by its PDF ( )1 ,1kf f C= . Suppose the random

variable 2kC is defined as 2

,1 ,2k k kC C C= + , and then we can

obtain the PDF of 2kC according to the PDF of ,k iC in the

form[6]

( ) ( ) ( )2 22 1 10k kf c f c x f x dx

+∞= − (13)

So the PDF of the variable 3 2,3k k kC C C= + can be derived

from the PDF of ,3kC in (11) and the PDF of 2kC in (13) as:

( ) ( ) ( )3 33 2 10k kf c f c x f x dx

+∞= − (14)

Finally, equivalently we can obtain the desired PDF of random variable 1

,t t

t

N Nk k k NC C C−= + through recursive

method, which is characterized by ( )c kf C .

Suppose that the precise threshold thC is given and the information kI is transmitted to the BS with the error-free, low-delay feedback link. In this case, according the scheduling algorithm in section IV, the objective function in (10) can be derived as

( ) ( ) ( )( )( ) ( )

th th th

th th

, 0, ,

1 0, ,

Q U

Q Q

MASSE K C f C K R C

f C K R C

= +

− (15)

Where ( )th, ,Qf k C K is the probability that there are k users

in qualified set 1Set for a given threshold thC and defined as

( ) ( )( ) ( )( )th, , 1th th

k k

Q c cc c

Kf k C K f x dx f x dx

k+∞ +∞

= − (16)

Then it is easily shown that UR denotes the expected

dynamic data rate contributions from the unqualified set 0Set ,

and QR denotes the expected dynamic data rate contributions

from the unqualified set 1Set . They are defined as

( ) ( ) ( )

( )

( ) ( ) ( )( )

th

th th

th

th

th

th 0

0

th

k

k

C

cU k C C C

c

cCQ k C C

cC

x f x dxR C E C

f y dy

x f x dxR C E C

f y dy

<

+∞

+∞>

⋅= =

⋅= =

(17)

Consequently, given the system model and the number of users in system, the problem of finding the MASSE in (15) is now an optimization problem as

( ) ( ){ }th arg max ,C

C K MASSE K C= , subject to 0C > (18)

V. SIMULATION RESULTS

For the simplicity, the simulations are performed based on the assumption that there are 2 antennas for the BS and 4 antennas for each user, and the channels are realizations of a Rayleigh flat fading channel model, which is described in section II. As shown in TABEL I, we obtain some optimal threshold values through computer simulation approach based on different configures, where the received signal to noise ratio (SNR) is defined as 2/s nSNR E σ= .

2 4 6 8 10 12 14 16 18 2010.5

11

11.5

12

12.5

13

Number of Users

Ave

rage

S

pect

ral E

ffic

ency

(bps

/Hz)

Round Robin Scheme

Optimal SchemeProposed Scheme(Simulation)

Proposed Scheme(Theoretic)

Figure 1. The maximal average system spectral efficiency for different scheduling scheme with 0ρ = and 15SNR dB=

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Fig.1 illustrates the performance of MASSE with different number of users for different scheduling schemes including the proposed scheme, the optimal scheme where all user feedback their channel capacity to the BS, and the Round-Robin scheme [7]. As shown in Fig.1, the proposed scheme further outperforms the Round Robin scheme because of exploiting multi-user diversity. Moreover, we observe though there is a little of gap between the proposed scheme and the optimal scheme, the proposed scheme achieves high MASSE which is close to the theoretic results. It proves that the developed scheduling scheme is effective.

2 4 6 8 10 12 14 16 18 209.5

10

10.5

11

11.5

12

12.5

Number of Users

Ave

rage

Spe

ctra

l Eff

icie

ncy(

bps/

Hz)

Round Robin Scheme

Optimal Scheme

Proposed Scheme(Simulation)Proposed Scheme(Theoretic)

Figure 2. The maximal average system spectral efficiency for different scheduling scheme with 0.5ρ = and 15SNR dB=

In Fig.1, we showed that the proposed scheduling algorithm significantly enhances the MASSE of the multi-user MIMO system in absence of transmit correlation. Fig.2 demonstrates the MASSE performance of the system for different schemes as a function of the transmit correlation coefficient ρ using correlation model (2), it is shown that the proposed scheme still achieve good performance as in Fig.1, however, the gain shrinks with the increasing of spatial correlation.

VI. CONCLUSIONS

The proposed scheduling algorithm exploits multi-user diversity with one bit feedback information for time slotted multi-user MIMO systems. Instead, we solve this problem achieving the optimum threshold for multiple antennas systems in presence of transmit correlation. Simulation results show that the proposed scheme is most promising scheduling algorithm, which may give low feedback, high MASSE.

However, this paper focuses on the ideal channel model with zero-delay, error-free feedback link, in particular, the feedback link is inherently noise and the users can not obtain the precise channel, further research will consider a realistic model for the feedback channel (delay, errors to be included), and we will develop a robust scheduling algorithm to improve system performance.

REFERENCES

[1]. P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inform. Theory, vol. 48, pp.1277-1294, June, 2002.

[2]. D. Gesbert and M. S. Alouini, “How much feedback is multi-user diversity really worth?” In IEEE Int. Conf. on Commun.,Paris, France, pp.234-238, June 2004.

[3]. G. Corral-Briones, and A. A. Dowhuszko, etc, “Downlink multiuser scheduling algorithms with HSDPA closed-loop feedback information,” IEEE VTC (Vehicular Technology Conference) Spring, Stockholm, Sweden. May. 2005.

[4]. Y. Xue, T. Kaise, “Exploiting multiuser diversity with imperfect one-bit channel state feedback,” IEEE Trans. on Vehicular Technology, vol.56, no.1, pp.183-193, Jan. 2007.

[5]. D. Gore, R. W. Heath, A. Paulraj, “On performance of the zero forcing receiver in presence of transmit,” ISIT 2002, Lausanne, Switzerland, pp.159-159, July, 2002.

[6]. D. Kannan, An introduction to stochastic processes. New York: Elsevier North Hollond, Inc, 1979.

[7]. O. S. Shin and K. B. Lee, “Antenna-Assisted Round Robin Scheduling for MIMO Cellular Systems,” IEEE Commun. Lett.,vol. 7, no. 3, pp. 109-111, Mar. 2003.

TABLE I. THE OOPTIMAL THRESHOLD THROUGH COMPUTER SIMULATIONS FOR THE PROPOSED SCHEDULING ALGORITHM

K 2 4 6 8 10 12 14 16 18 20

( )15 , 0thC SNR dB ρ= = 10.7170 11.2181 11.4773 11.6490 11.7756 11.8750 11.9565 12.0252 12.0845 12.1364

( )15 , 0.5thC SNR dB ρ= = 9.9155 10.4108 10.6682 10.8385 10.9641 11.0629 11.1438 11.2121 11.2710 11.3226

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