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370 IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 8, AUGUST 2003
Multiuser Diversity for a Dirty Paper ApproachZhenyu Tu, Student
Member, IEEE, and Rick S. Blum, Senior Member, IEEE
AbstractMultiuser diversity has attracted significant
attention
recently. In this letter, we propose a greedy algorithm based on
QRdecomposition of the channel and dirty paper precoding to
exploitthe multiuser diversity of the Gaussian vector broadcast
channel.Simulations show the approach provides performance which is
ex-tremely close to a well-known upper bound on the sum rate.
Fur-ther, exploiting multiuser diversity can provide large gain
over ap-proaches ignoring this resource.
Index TermsBroadcast channel, dirty paper, multiple
inputmultiple output (MIMO), sum rate.
I. INTRODUCTION
I
N RECENT years, using antenna arrays to form multipleinput
multiple output (MIMO) systems has shown promise
for achieving high capacities over wireless links [1], [2].
TheMIMO enhanced broadcast channel is of great interest lately
asobserved by the large number of papers at [3]. Finding simpleways
to achieve performance close to capacity is still of greatinterest
for the MIMO enhanced broadcast channel. Recent re-sults in [5]
demonstrate that the Sato bound [4] is achievable,but do not tell
how to achieve this bound. In [6], a simple ap-proach based on QR
decomposition is proposed to optimize thesum rate for channel
matrices with full row rank. It is referred aszero-forcing
dirty-paper (ZF-DP) coding. For a channel matrixto have full row
rank, the system must have as many transmitantennas as users which
is quite restrictive. In this paper, we ex-tend ZF-DP to a scenario
in which the channel does not have
full row rank. This is accomplished by addressing the
orderingand selection of active users to exploit the multiuser
diversity.The resulting approach is a simple greedy algorithm
referred toas greedy ZF-DP. Simulation results show it provides
perfor-mance very close to the Sato bound for some widely
acceptedchannel models and provides significant gain over
ZF-DP.
II. SYSTEM MODEL
We assume antennas are deployed at the transmitter (base)and
there are geographically dispersed receivers (mobiles),each of
which has one receive antenna. The base is transmit-ting signals
intended for all users (mobiles). In typical cases,
, which means there are more users than transmit antennas.
Since the propagation environment of each user is generally
dif-ferent, the possibility of multiuser diversity [7] exists. The
white
Manuscript received January 22, 2003. The associate editor
coordinating thereview of this letter and approving it for
publication was Dr. J. Ritcey. Thiswork was supported by the Air
Force Research Laboratory under AgreementF49620-01-1-0372 and
F49620-03-1-0214, and by the National Science Foun-dation under
Grant CCR-0112501.
The authors are with the Electrical Engineering and Computer
Sci-ence Department, Lehigh University, Bethlehem, PA 18015 USA
(e-mail:[email protected]).
Digital Object Identifier 10.1109/LCOMM.2003.815652
Gaussian noise observed at each antenna is assumed to be
inde-
pendent from the noise observed at other antennas. The channelis
assumed to be memoryless and quasi-static and known to boththe
transmitter and receivers. The broadcast channel is charac-terized
by
(1)
where is a complex matrix whose th entry is thepath gain from
the th transmit antenna to the th receiver, andis an vector whose
th entry represents the received signalat the th receiver (the
receiver for user where ).We denote as the th row of which
corresponds to thechannel between the base station and the th
receiver. In (1),
is the circularly symmetric complex Gaussian noisevector with
identity covariance matrix and is the
transmitted vector signal with a power constraint , i.e,, where
is the signal intended for user .
A rate vector is said to be achievable if there exists a
sequence of codes with
, where is the probability of error [8]. In thispaper, we
attempt to find a simple way to achieve a sum rate
close to the sum capacity.
III. PROBLEM FORMATION
Let be the covariance matrix for user .
Since the transmitter knows , we can apply thedirty paper
results in [9] and [10]. For any fixed permutation of
denoted by , the achievable ratewith dirty paper precoding is
given in [11] as
(2)
The dirty paper rate region is the union of all such rate
vectors over all covariance matrices suchthat and over all
permutations. This region appears to be the capacity
region of the Gaussian broadcast channel based on the
resultspresented at a recent meeting [3]. However, simple methods
forachieving an arbitrary point in the region are still
unavailable.The difficulty lies in handling the interference
between users.The conventional approach to address the interference
problem,called zero-forcing (ZF), is discussed and compared
withZF-DP coding in [6]. It is shown that ZF-DP coding is
superiorto ZF scheme in the sense the former can provide no less
sumrate than the latter does for any channel realization.
1089-7798/03$17.00 2003 IEEE
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372 IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 8, AUGUST 2003
Fig. 1. Ergodic sumrate comparison foruncorrelated channel
without full rowrank. r = 1 8 ; t = 3 .
Fig. 2. Ergodic sum rate comparison for correlated channel
without full rowrank. r = 1 8 t = 3 , = = 3 and L = 2 0 .
Since the approach employed is not optimized, is a lowerbound
for . Hence (13) also holds for . Q.E.D.
If we apply ZF-DP and randomly select and orderrows (we call
this ZF-DP with random ordering) of , the re-
sulting random variables are statistically independent and(see
[6] and therein). In this case does not
depend on at all, so we can not exploit the multiuser
diversityof the system if is fixed. Therefore, even with a large
numberusers in the system, the sum rate will not benefit.
In Fig. 1, we plot the ergodic sum rate versus SNR. Since
thenoise is normalized, the SNR is actually the power constraint
.As we can see, when applying the greedy ZF-DP, the sum rateis
pretty close to the Sato bound. We can also see a significant( 4
dB) gain of the greedy scheduling over ZF-DP with
randomordering.
We also test the greedy ZF-DP for correlated fading
channels.Here we adopt the channel model used in [7]. Suppose there
are
distinct paths from a mobile to the base station. The channelis
uncorrelated at the mobile side and correlated at the base. The
paths have equally spaced angles of departure from to, where is
the angle spread at the base. We also assume
the base has a linear array of antennas with half carrier
wave-length spacing. Then the correlated channel can be
representedby
(14)
where and are the array response vectors at the re-ceiver and
transmit sides respectively. is a complexmatrix whose th column is
, and is a complexmatrix whose th row is . Based on the assumptions
wehave made, can be modeled as a matrix with i.i.d. complexGaussian
entries. The array response vector at the transmitter is
, where is the angle ofdeparture of th path.
We plot the ergodic sum rate vs SNR for the correlatedchannel in
Fig. 2. As we can see, for the correlated channel, the
greedy ZF-DP is still close to the Sato bound and
outperformsZF-DP with random ordering significantly.
VI. CONCLUSION
In this letter, we proposed a greedy algorithm to exploit
themultiuser diversity for the vector broadcast Gaussian channelto
increase the systems instantaneous sum rate. Our algorithmexhibits
good performance for some accepted channel models.We expect each
user will get approximately equal rate if the sta-tistical
description of each user is identical and the schedulingalgorithm
is employed for a time much longer than that overwhich the channel
changes. If it is not the case, some additionaltime sharing or
scheduling algorithm can be employed to pro-
vide fairness.
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