Email: {masoud,hassibi}@systems.caltech.edu. Abstract-Since having full channel state information in the transmitter is not reasonable in many applications and ...
On the Capacity of MIMO Broadcast Channel with Partial Side Information Masoud Sharif and Babak Hassibi Department of Electrical Engineering California Institute of Technology Pasadena, CA 91 125 Email: {masoud,hassibi}@systems.caltech.edu Abstract-Since having full channel state information in the therefore the sum rate capacity does not scale with the number transmitter is not reasonable in many applications and lack of of transmit antennas for high signal to noise ratios (SNRs). channel knonledge does not lead to linear growth of the sum In many applications, however, it is not reasonable to rate capacity as the number transmit antennas increases, it is therefore of interest to investigate transmission schemes that assume that all the channel coefficients to every user can be employ only partial CSI. I n this paper, we propose a scheme that made available to the transmitter. This is especially true if the constructs M random beams and that transmits information to number of transmit antennas M andlor the number of users n the users with the highest signal-to-noire-plus-inferfe~"ceratios is large (or if the users are mobile and are moving rapidly). (SINRs), which can be made available to the transmitter with Since perfect channel state information may be impractical, yet very little feedback. For fixed M and n increasing, the sum-rate capacity of our scheme scales as Mloglogn, which is precisely no channel state information is useless, it is very important to the same scaling obtained uith perfect channel information. We devise and study transmission schemes that require only partial furthermore show that linear increase in capacity can be obtained channel state information at the transmitter. This is the main provided that M does not not grow faster than O(log n). We also goal of the current paper. study the fairness of our scheduling scheme and show that, when The scheme we propose is one that constructs M random M is large enough, the system becomes interferencedominated orthonormal beams and transmits to users with the highest and the probabilily of transmitting to any user converges to i, irrespective of its path-loss. I n fact, using M = a l o g n transmit signal-to-noise-plus-interference ratios (SINRs). In this sense antennas emerges as a desirable operating point, both in terms of it is in the same spirit as the work of [4] where the transmission providing linear increase in capacity as well as in guaranteeing of one random beam is also proposed. However, our scheme fairness.. differs in several key respects. First, we send multiple beams (in fact, M of them) whereas [4] sends only a single beam. 1. INTRODUCTION Second, whereas the main concern in [4] is to improve the Multiple-antenna communications systems have generated proportional fairness of the system (by giving different users a great deal of interest since they are capable of considerably more of a chance to be the best user) our scheme aims increasing the capacity of a point to point wireless link. There at capturing as much of the broadcast channel capacity as has also been recent interest in the role of multiple antenna possible. Fairness is achieved in our system as a convenient systems in a multi-user network environment, and especially by-product. in broadcast and multi-access scenarios. For multiple-input Based on asymptotic analysis, we show that, for fixed M multiple-output (MIMO) broadcast channels the capacity reand n increasing, our proposed scheme achieves a sum-rate gion has been studied in [I], 121, [3] and it has been shown capacity of Mloglogn. Happily, this is the same as the that the sum rate capacity i s achieved by dirt):paper coding. sum-rate capacity when perfect channel state information is While the above results suggest that capacity increases available and so, asymptotically, our scheme does not suffer . linearly in the number of transmit antennas, they all rely on a loss. One may ask how large may M grow to guarantee the assumption that the channel is known'perfectly at the a linear increase in capacity? We show that the answer is transmitter. One may speculate whether, as in the point-toM = O(1ogn). point case, it is possible to get the same gains without having In schemes (such a s ours) that exploit multi-user diversity channel knowledge at the transmitter. Unfortunately, it can there is often tension between increasing capacity and fairness. be proved that, if no channel knowledge is available at the The reason being that the strongest users may dominate the transmitter no matter whether the receivers have full CSI or network. Fortunately, we show that in our scheme, provided not, the Gaussian MIMO broadcast channel is degraded and the number of transmit antennas is large enough, the system becomes interference dominated and so, although close users This work was supported in part by the Sational Science Foundation receive strong signal they also receive strong interference. under grant no. CCR-0133818, by the omre of Naval Research under giant no. 500014-02-1-0578. and by Calteeh's Lee Center for Advanced Therefore it can be shown that, for large enough hi, the probability of any user having the highest SINR converges to Networking.
0-7803-8104-1/03/%17.00 02003 IEEE
958
i.
interference as follows,
A more careful study of this issue reveals that the choice of M = a log n transmit antennas is a desirable operating point, both in terms of providing linear increase in capacity as well as in guaranteeing fairness. This paper is organized as follows. Section 2 introduces our notation and our scheduling algorithm. The asymptotic analysis of the sum rate throughput of our scheme is done in Section 3 when M is fixed and N = 1, Le., single antenna receivers. In Section 4, it is shown that the linear increase in the throughput is retained as long as M is growing not faster than logn. Section 5 deals with the case heterogeneous users and investigates the fairness of our scheduling. Finally Sections 6 and 7 present simulation results and conclude the paper, respectively.
for m = 1,. . ., M. Suppose now each user feeds back its maximum SINR, i.e. max SIN&,,, when the maximum is l 1 receive antennas is omitted in this paper for the sake of brevity [ 6 ] .Then at each time instance, the m'th vector is multiplied by the m'th transmit symbol ,s so that the transmitted signal is,
IS FIXED
In this section we obtain lower and upper bounds for the sum rate capacity when M is fixed, N = 1 and n is going to infinity. Using M random beams and sending to the users with the highest SINRs, we can bound the sum rate throughput R,
where this is an upper bound since we ignored the probability that user i be the maximum SlNR user twice (if this is the case, the transmitter has to choose another user with SlNR less than the maximum SINR which decreases the capacity). On the other hand, in [ 6 ] , the following lower bound for the sum rate throughput is proved
R 2 M (1 - {Pr {SINKJ 5 1))")
x
M
~ ( t=)
+msm,
t=1,
...,T.
(2)
m=l
We also assume that the average transmit power per antenna is one, equivalently, E { ( s # } = 1, and henceforth the total transmit power is E{S'S} = M . After T channel uses, we independently choose another set of orthonormal vectors {&}. and so on. From now on, we drop the time index from S and Y, and therefore, the received signal at the i'th receiver is, M
Y,=
Hi+,s,
+ Wi,
i = l , ...,n.
(3)
*=
We further assume the receiver knows H ias well as + j ' s . Therefore, the i'th receiver (i = 1,.. . ,n) can compute the following M signal to interference and noise ratio (SINR) by assuming that s, is the desired signal and the other si's are
As we shall show later, the lower and upper b u n d s for the sum rate capacity become tight for sufficiently large n and when lim JL = 0. In this case, conditioning ,+m'og n on max SINR;,, 2 1 in Eq. (6) can be replaced by l
