2512
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
On the Balance of Multiuser Diversity and
Spatial Multiplexing Gain in Random Beamforming
Jörg Wagner, Student Member, IEEE, Ying-Chang Liang, Senior Member, IEEE,
and Rui Zhang, Member, IEEE
Abstract—This paper is concerned with the problem of exploiting spatial multiplexing gain through opportunistic beamforming
with multiple random beams. The base station transmits multiple
pilot sequences using orthogonal beams, and each user feeds the
channel gain of each beam back. Then, the base station selects the
subset of users and beams for which the sum-rate is maximized.
We aim to exploit spatial multiplexing gain in a controlled
fashion, i.e., the number of beams used for data transmission
is chosen depending on the available multiuser diversity in the
system. Two schemes are proposed: dynamic Orthonormal Random Beamforming with Systematic Beam Selection (ORBF/SBS),
for which the number of beams used for data transmission is
dynamically changed from one fading block to another so that
the sum-rate is maximized; and static ORBF/SBS, which selects
a fixed number of beams for a given signal-to-noise ratio and
user number. The static scheme achieves the sum-rate of the
dynamic one up to a small gap with much reduced computational
and feedback complexity. We derive an approximate expression
for the sum-rate achieved by the dynamic scheme, and provide
insight into the optimal data stream number in the static scheme
by means of asymptotical results. Computer simulations are
provided to evaluate the performance of the proposed schemes.
Index Terms—Broadcast channel, downlink, MIMO, opportunistic beamforming, random beamforming.
I. I NTRODUCTION
W
IRELESS downlink transmission with multiple transmit antennas at the base station (BS) and possibly
multiple receive antennas at each mobile station (MS) – also
known as the fading MIMO broadcast channel (MIMO-BC)
– has motivated a great deal of valuable scholarly work
for assessing its fundamental performance limits. From an
information-theoretic point of view, the sum-capacity of the
Gaussian MIMO-BC can be achieved by “dirty-paper coding
(DPC)” when channel state information (CSI) of each mobile user is perfectly known at both the transmitter and the
receiver [1]–[2]. Consider a fading Gaussian MIMO-BC with
N transmit antennas and single receive antenna at each of
K mobile users, the asymptotic sum-capacity is known to
scale as N log log K when K becomes infinitely large [3].
Some insights can be drawn from this asymptotic capacity.
Manuscript received March 27, 2006; revised August 20, 2006 and November 1, 2006; accepted November 26, 2006. The associate editor coordinating
the review of this paper and approving it for publication was J. Andrews.
J. Wagner is with ETH Zurich, Sternwartstrasse 9, CH-8092 Zurich,
Switzerland. The work was done when Jörg Wagner visited the Institute for
Infocomm Research, Singapore (e-mail: [email protected]).
Y.-C. Liang and R. Zhang are with the Institute for Infocomm Research, 21
Heng Mui Keng Terrace, 119613 Singapore (e-mail: ycliang, rzhang @i2r.astar.edu.sg).
Digital Object Identifier 10.1109/TWC.2008.060111.
On the one hand, the BS should transmit at one time to
multiple users in order to achieve the multiplicative N factor,
which can be regarded as the fundamental spatial multiplexing
gain of MIMO-BC. On the other hand, the log log K factor
demonstrates another inherent capacity gain in fading MIMOBC, namely, the multiuser diversity, which can be attained by
selecting a subset of users that have good channel conditions
for transmission at one time. Therefore, a reasonably sound
transmission scheme for fading MIMO-BC should be able to
capture both the spatial multiplexing gain and the multiuser
diversity in order to approach its fundamental throughput limit.
To achieve this, the availability of CSI at the BS becomes
crucial. On the one hand, suboptimal schemes such as zeroforcing beamforming (see, e.g., [4] [5]) and zero-forcing DPC
precoding (see, e.g, [6] [7]) give relatively close performace
of the optimal DPC, when precise CSI is known at the
transmitter. Nevertheless, these schemes become too costly to
implement in systems for which the transmitter can acquire
CSI only via a feedback channel from the receiver. On the
other hand, completely unknown CSI at the transmitter leads
to neither spatial multiplexing gain nor the multiuser diversity
in the achievable throughput [8]. Therefore, the exploration
of partial CSI through a limited feedback channel is highly
valuable for realistic MIMO-BC.
Opportunistic Beamforming (OBF) was introduced by
Viswanath et al. in [9] as a powerful method to sustain the
multiuser diversity of the fading MISO-BC 1 with only partial
CSI feedback to the transmitter. The basic idea of OBF is
to randomly vary phase and amplitude of the beamforming
weight at each transmit antenna in order to artificially accelerate and increase the fluctuation of each user’s individual
channel. In order to enable OBF, it is necessary to divide
the transmission time into time slots dedicated either to pilot
transmission (pilot mode) or data transmission (data mode). In
pilot mode, each user measures the resultant channel gain that
is represented as the absolute value of the inner-product of
its own channel vector and the random beamforming vector
and then feeds it back to the BS. Thereupon, the user with
the largest channel gain is scheduled for transmission in data
mode. When there is a large number of users, one user can be
selected for transmission only if its channel condition is among
the best of all users and at the same time its channel matches
well to the pre-selected beamforming vector. Therefore, OBF
1 A simple extension to the MIMO case is to treat the individual receive
antennas as independent users, a slightly more elaborate extension might also
make use of receive antenna beamforming.
c 2008 IEEE
1536-1276/08$25.00 WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
captures the multiuser diversity to a certain extent and also
avoids feedback of the complete CSI.
In the meanwhile, OBF has experienced several extensions,
including e.g., multiple receive antennas [10]–[11], multiple
random beams in pilot and/or data mode [12]–[13], and
exploitation of channel time-correlations [14], among others.
In [12], the OBF scheme is modified to make use of a single
beam in data mode transmission, but multiple beams in pilot
mode. Each user now feeds back the largest channel gain
among all beams in pilot mode as well as the associated
beam index. We thus refer to this scheme as Opportunistic
Beamforming with Multiple Weighting Vectors (OBF/MWV).
As it compensates for the limitation in OBF by providing
several beams for each user to choose from, OBF/MWV
attains an improved multiuser diversity as compared to OBF
particularly when the number of users is small. However,
as multiple beam trainings increase the pilot overhead, the
scheme becomes less efficient than OBF if the user number
is large.
One common drawback in both OBF and OBF/MWV lies in
their incapability of capturing the channel spatial multiplexing
gain with only a single beam transmission in data mode.
To overcome this, in [15], a major extension of OBF was
proposed with simultaneous transmissions through multiple
beams in both pilot and data mode. The idea is to generate
not a single, but N random beams in pilot mode and to
schedule the N best users for data transmission accordingly.
The scheduling criterion is the signal-to-interference-plusnoise ratio (SINR) instead of the channel gain. This scheme –
referred to as Orthonormal Random Beamforming (ORBF) –
allows for a significant gain in the sum-rate in both low and
medium signal-to-noise ratio (SNR) regimes. However, ORBF
performs poorly under high SNR when the interference signals
between beams dominate over the receiver additive noise [16].
To summarize, a practical transmission scheme should be
able to balance well the spatial multiplexing gain and the
multiuser diversity and in the meanwhile require low-rate
feedback from the receiver in approaching the sum-capacity of
fading MIMO-BC. To achieve this end, this paper introduces
a generic class of random beamforming schemes that incorporate previous schemes [9] [12] [15] as special cases and
within this class proposes the optimal scheme. Following the
approach in previous random beamforming schemes, three key
parameters are defined in the following. Firstly, the number of
beams advertised by the BS in pilot mode, B(≤ N ); secondly,
a feedback vector uk for each user k = 1, 2, . . . , K; and
thirdly, the number of beams actually used in data mode,
B0 (≤ B). In the most general and optimal case, each user
feeds back uk that consists of channel gains pertaining to each
single random beam in pilot mode. Based on this information,
the BS can select a subset of B0 users, which reside in
spatially (almost) orthogonal beams for the transmission. We
call this scheme Orthonormal Random Beamforming with
Systematic Beam Selection (ORBF/SBS).
In order to find the optimal subset of users and beams, the
BS has to perform a search over all possible combinations
of users and beams. In the optimal case this means to search
through all sets of sizes 1 ≤ B0 ≤ B. The scheme performing this search over sets of variable size is called dynamic
2513
ORBF/SBS, as it chooses the number of beams used for data
mode dynamically. However, it will turn out, that given the
number of pilot beams as well as the average SNR and the user
number, the optimal data stream number varies slightly only
over random beam and channel realizations. Accordingly, a
large portion of the optimal sum-rate can actually be achieved
by searching over the user combinations of a fixed size B0
only. This scheme, which is of lower computational complexity, will be referred to as static ORBF/SBS. Intuitively stated,
both ORBF/SBS schemes schedule users in a way such that
multiuser diversity and spatial multiplexing gain are balanced
optimally.
Previously proposed schemes in general require both less
feedback and lower computational complexity, and approach
the performance of the optimal scheme in few special cases,
such as extremely small/large user numbers. In most (realistic)
scenarios, however, there is a significant gap to the optimal
sum-rate, as they do not efficiently exploit the degrees of
freedom available in the generic scheme. We will therefore
discuss possible ways to reduce feedback (and thereby also
computational complexity) in practical cases, while sustaining
close to optimal performance.
Recently, there has been a different approach to generalize
existing random beamforming techniques denoted by Opportunistic Space Division Multiple Access with Beam Selection
(OSDMA/BS) [17]. In this paper, the authors suggest to
generate M sets of B orthonormal random beams each in
pilot mode. The MSs are then required to feed back their best
SINR in each of these sets as well as the respective beam
indices. Compared to our scheme, which is restricted to a
maximum of N pilot sequences, the length of the pilot mode
periods is multiplied by a factor of M . Interestingly, it will
turn out that we can achieve the (up to a small gap) same
enhancement in multiuser diversity even in ORBF/SBS at the
expense of less extended pilot mode periods only. On the other
hand OSDMA/BS requires less feedback than ORBF/SBS.
This is, since the MSs quantize the precisely computed SINR
in OSDMA/BS, which can be done more efficiently than the
quantization of the channel gains fed back in ORBF/SBS for
the SINR evaluation at the BS.
In [18] a scheme closely related to OSDMA/BS called
limited feedback OSDMA/BS has been proposed. This scheme
passes completely on broadcasting random beamforming vectors. Instead the MSs feed back quantized versions of their
respective channel vectors, which allows for a centralized
beam selection at the base station and thus overcomes the
drawback of numerous training iterations. In this method
both the amount of required feedback and the computational
complexity are increased again. However, the authors use an
efficient quantization method and a low complexity beam
selection algorithm to reduce this overhead again. Although
this scheme cannot be classified as a classical opportunistic
beamforming scheme according to our definition below (feedback is not restricted to gain or SINR values) its flavor is
somewhat similar to the scheme proposed in this paper.
The remainder of the paper is structured as follows: In
Section II we describe the signal model considered in this
paper. The ORBF/SBS scheme is introduced in Section III. An
approximate expression for the sum-rate achieved by dynamic
2514
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
ORBF/SBS is derived in Section IV. In Section V we discuss
the fundamental dependencies of the optimal number of data
streams on other system parameters. The performance of both
ORBF/SBS schemes is discussed and compared to existing
schemes in Section VI. We consider ways to reduce the
amount of feedback as well as the computational complexity in
Section VII. In Section VIII we discuss fairness issues, when
random beamforming is applied to heterogeneous channels.
Finally, Section IX concludes this paper with some remarks.
II. S IGNAL M ODEL
We consider a scenario, where the BS is equipped with
N antennas and the K MSs are equipped with one antenna
each. Thus, there are K multiple-input single-output (MISO)
channels from the BS to the MS. These channels are assumed
to be flat fading and also to remain constant over several pilot
and data mode periods. The received signal at the k-th MS
during pilot mode in time slot n is given by
yk [n] =
B
hk φi si [n] + nk [n],
(1)
i=1
yk =
In this section, we introduce a class of random beamforming
schemes as defined below.
Definition: The class of random beamforming schemes
without any additional preprocessing at the transmitter is
defined to comprise all schemes that proceed according to the
subsequent protocol:
• In pilot mode the BS broadcasts pilot sequences through
B ≤ N orthonormal random beams to the MSs.
• Each MS measures the information required by the
scheme and feeds it back to the BS in a vector uk either
containing the channel gains (|hk φ1 |, . . . , |hk φB |) or
a sufficient statistic derived from these gain values to
enable a specific (suboptimal) beam selection method.
• In data mode, the BS schedules – based on the available
information – the optimal subset of B0 ≤ B users for
transmission through a subset of the pilot beams.
• Thereby, the fixed power budget is distributed equally
over the assigned data streams.
A scheme within the class is then completely identified by
the number of pilot beams B generated by the BS,
• the number of beams B0 actually used for data transmission,
• the information fed back from the MS in u.
In such a scheme the sum-rate is determined by the SINRs
of the scheduled users. Accordingly, the informations necessary for the BS to determine the optimal – i.e., sum-rate
maximizing – subsets of users and beams are all the individual
channel gain values |hk φb |2 values pertaining to each beam
for each user. Once these values are available at the BS, the
SINRs for arbitrary combinations of beams and users can be
computed according to
•
where hk is the (1 × N ) channel vector for the k-th user containing independent zero mean circularly symmetric complex
Gaussian (ZM-CSCG) random variables with unit variance,
and {φi }B
i=1 are the B (N × 1) realizations of orthonormal
random vectors containing the beamforming weights obtained
from an isotropic distribution2 [19]. si [n] is the n-th symbol
of a sequence of pilot symbols transmitted over the i-th pilot
beam. The B simultaneously transmitted symbol sequences
are orthonormal (Walsh-Hadamard sequences, e.g.), known
to the MSs, and used to estimate the channel gains |hk φi |,
i = 1, . . . , B, necessary to compute
users’ SINRs at the BS.
The power budget ρ = trace E[ssH ] , where s = [s1 . . . sB ]T
and E[·] denotes expectation, is equally distributed over all
beams, i.e., the average signal power at each receiver is given
by ρ/B. Finally, nk [n] respresents the additive white (with
respect to both k and n) Gaussian noise at the k-th receiver
modeled as ZM-CSCG with unit variance.
During data mode the received signal at the k-th MS is
given by
B0
III. O RTHONORMAL R ANDOM B EAMFORMING W ITH
S YSTEMATIC B EAM S ELECTION
(d)
hk φi si + nk ,
(2)
i=1
2
SINRki ,bi =
j=1,j=i
(3)
2 ,
hki φbj Rki ,bi = log2 (1 + SINRki ,bi ).
(4)
If we allow the BS to search over user combinations of
different sizes, we obtain the optimal scheme within the
class under consideration. This scheme is termed dynamic
ORBF/SBS, and chooses beams and users according to
(dyn)
2 From a purely information theoretic point of view using deterministic
beamforming weights yields the same sum-rate. The randomness is introduced
to ensure fairness in slow fading environments.
+
where ki is the i-th selected user and bi its assigned beam.
Then, the user’s rate in bits per complex dimension is
{(b1
(d)
0
where we dropped the time index and B0 ≤ B. {φi }B
i=1 ⊆
B
{φi }i=1 are the B0 beams chosen for data transmission, and
the si ’s now contain the actual data symbols. Power again is
uniformly distributed, but normalized according to the number
of simultaneously scheduled users, B0 . Thus, the average
signal power received at the MSs is ρ/B0 . Note that we
assume a homogeneous channel, i.e., equal E[hk ], ∀k, if
not stated differently and explicitly.
B0
ρ
|hki φbi |
B0
=
(dyn)
, k1
(dyn)
(dyn)
), . . . , (bB0 , kB0 )}
B0
argmax
{(b1 ,k1 ),...,(bB0 ,kB0 )}∈∪B
B
0 =1
BB0 ×KB0
Rki ,bi ,(5)
i=1
where BB0 and KB0 are the sets of all possible combinations
of B0 beams and users, respectively. The term “dynamic”
refers to the dynamically chosen number of data streams.
Fixing the number of data streams to
Bd
B0 = argmax E
max
Rki ,bi
1≤Bd ≤B
{(b1 ,k1 ),...,(bBd ,kBd )}∈BBd ×KBd
i=1
(6)
WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
in advance yields the selection criterion for the static
ORBF/SBS scheme:
(stat) (stat)
(stat) (stat)
{(b1 , k1 ), . . . , (bB0 , kB0 )}
=
argmax
{(b1 ,k1 ),...,(bB0 ,kB0 )}∈BB0 ×KB0
B0
Rki ,bi , (7)
i=1
Here, the optimal static B0 depends on ρ, K and B. The
BS needs to store this B0 for each set of values of these
parameters in a look-up table. Then, the search space can be
reduced significantly compared to the dynamic case.
Finally, Table I compares the associated parameters used in
the different random beamforming schemes discussed so far.
It points out that both ORBF/SBS schemes basically can be
seen as generalizations of the previous schemes.
2515
B. R(b1 ,...,bBd ) – Sum-Rate of Best Users in Bd Beams
In order to sum up the rates in the single beams, we need
to switch to the characteristic function domain. As the PDF
of Rbi cannot be transformed in closed form, we approximate
it by the PDF of a Gamma distribution. This distribution is
particularly suitable for our purposes for the following reasons:
• Skewness and tail in both distributions are similar over
the whole parameter range of interest.
• The sum of two Gamma distributed random variables is
Gamma distributed again, what makes it easy to derive
the PDF of the sum-rates.
• The Gamma distribution is uniquely determined by its
mean and variance, what allows for an easy mapping
from the parameters of the original distribution to the
parameters of the Gamma distribution.
• Excellent match with simulation results will justify the
approximation.
We map the parameters of (13) to the two parameters of
the Gamma distribution such that the first two moments (we
denote the variance of the random variable X by Var[X]) of
both distributions 5 coincide, i.e.,
IV. S UM -R ATE ACHIEVED BY DYNAMIC ORBF/SBS
In this section we derive an approximate expression for the
average sum-rate achieved by dynamic ORBF/SBS. For this
purpose, we formally decompose the maximization problem
into three steps4 (see (8), (9) on next page), where the approximation is from the fact, that we neglect the small probability
2
that a user is the optimal one for several beams. Originating
(E[Rbi ])
Var[Rbi ]
α∗ =
and θ∗ =
,
(14)
in the statistics of the SINRbi , we derive the distribution
Var[Rbi ]
E[Rbi ]
of Rbi by the rule of variable transformation. We will then
approximate this distribution by a Gamma distribution [20] where mean and variance of fRbi (r) are given by
which will allow us to switch to the characteristic function
K
K
Bd k
Bd k
domain. Finally, we will be able to compute an approximate E[Rb ] = 1
EBd −Bd k+1
(−1)k−1 exp
i
k
ln(2)
ρ
ρ
expression for the average sum-rate achieved by the dynamic
k=1
(15)
scheme.
and
K A. Rbi – The Best User’s Rate in a Beam
Bd k
K
2
k−1
Var[Rbi ] =
exp
(−1)
The SINR for an arbitrary user ki when using beam bi given
k
(ln(2))2
ρ
k=1
that the set of beams {b1 , . . . , bBd } ∈ BBd is used is given by
∞
Bd k
−(Bd −1)k−1
2
x
ln(x)x
exp
−
×
|hki φbi |
ρ
1
SINRki ,bi =
(10)
.
Bd
Bd
2
hki φbj 2
+
×
dx
−
E[R
]
.
(16)
bi
j=1,j=i
ρ
The probability distribution function (PDF) of SINRki ,bi
has been derived in [15] and is given by
exp − Bρd s Bd
(1 + s) . (11)
fSINRki ,bi (s) =
Bd − 1 +
(1 + s)Bd
ρ
The according cumulative distribution function (CDF) is
found by integration and given by
exp − Bρd s
.
(12)
FSINRki ,bi (s) = 1 −
(1 + s)Bd −1
We change variables according to Rki ,bi = g(SINRki ,bi ) =
log2 (1 + SINRki ,bi ) and obtain the CDF of the best user’s rate
in a beam as
K
FRbi (r) = FRki ,bi (r)
K
B
Bd
· 2−(Bd −1)r exp − 2r
= 1 − exp
. (13)
ρ
ρ
3Q
denotes the number of values the quantity to be fed back can take on.
works as maxa,b f (a, b)
=
maxa maxb|a f (a, b)
=
maxb maxa|b f (a, b) (can be generalized for arbitrary numbers of
arguments).
4 This
The respective intermediate steps in the evaluation of (15)
and (16) can be found in Appendix A.
In order to simplify the further analysis we will round the
first parameter to the closest integer and adjust the second
parameter to sustain the mean, i.e., α = α∗ + 1/2 and θ =
E[Rbi ]/α∗ . This does not hurt a lot, since the first parameter
is large usually, i.e., α∗ 1.
Fig. 1 shows both original and approximated curves in the
case that Bd = 3, K = 16 and ρ = 15 dB. Although this is
only a sample plot, the approximation is very close over the
whole range of system parameters.
The PDF of the achieved sum-rate given the use of Bd
beams now can be found by means of characteristic functions.
The characteristic function of the sum-rate R(b1 ,...,bBd ) =
Bd
i=1 Rbi , where the Rbi are independently and identically
distributed (i.i.d), is given by
Bd 1 αBd
φR(b1 ,...,bB ) (ω) = φRbi (ω)
=
, (17)
d
1 + ıωθ
5 We
use the PDF fX (x) =
distribution.
xα−1 exp(−x/θ)
Γ(α)θ α
for describing the Gamma
2516
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
TABLE I
C HARACTERIZATION OF R ANDOM B EAMFORMING S CHEMES
dynamic ORBF/SBS
static ORBF/SBS
B
1≤B≤N
1≤B≤N
B0
1 ≤ B0 ≤ B (variable)
1 ≤ B0 ≤ B (fixed)
ORBF
N
N
OBF/MWV
OBF
1≤B≤N
1
1
1
R(dyn)
=
≈
max
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
uk
(|hk φ1 |, . . . , |hk φB |)
(|hk φ1 |, . . . , |hk φB |)
(maxi=1,...,N
N
ρ
+
|h φ |2
N k i
, imax )
|hk φ j |2
j=1,j=i
(maxi=1,...,B |hk φi |, imax )
(|hk φ|)
⎧
⎪
⎪
⎪
⎪
⎪
Bd
⎨
max
1≤Bd ≤B ⎪
⎪{b1 ,...,bBd }∈BBd
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎧
⎪
⎪
⎪
⎨
R(b1 ,...,bB
d
feedback bits
≤ B log Q
≤ B log Q
3
log Q + log N log Q + log B
log Q
⎫
⎪
⎫⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎬⎪
)
max
Rki ,bi
⎪
⎪
{k1 ,...,kBd }∈KBd
⎪
⎪
⎪⎪
i=1 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
Rbi
⎪
⎪
⎪
⎪
⎭
RBd
⎧
⎪
⎪
⎪ Bd
⎨
(8)
⎞ ⎫⎫
⎪
⎪
⎪
⎪
⎪⎪
⎜
⎟⎬
⎬
⎜
⎟
max
max
,
log2 ⎜1 + max SINRki ,bi ⎟
1≤Bd ≤B ⎪
1≤ki ≤K
{b1 ,...,bBd }∈BBd ⎪
⎝
⎠⎪
⎪
⎪
⎪
⎪⎪
i=1
⎪
⎪
⎪
⎪
⎩
⎩
⎭⎭
⎛
(9)
SINRbi
0.7
original
approximated
0.6
0.5
PDF
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
rate of best user in a beam (bps/Hz)
Fig. 1. Approximation of fRb (r) by the PDF of the Gamma distribution
i
for B0 = 3, K = 16 and ρ = 15 dB.
where
√ φRbi (ω) is the characteristic function of fRbi (ω) and
ı = −1. It follows that
R(b1 ,...,bBd ) ∼ Γ(αBd , θ).
(18)
We will use the CDF of R(b1 ,...,bBd ) in the further analysis. It
is found by integration and given by
r
FR(b1 ,...,bB ) (r) = P (αBd , ),
(19)
d
θ
where P (·, ·) is the normalized Gamma function 6 .
C. Average Sum-Rates
We finally are interested in the distributions of the sumrates achieved by using Bd data beams and by dynamic
6 The
γ(a,x)
normalized Gamma function is defined as P (a, x) = Γ(a) , where
!
!
γ(a, x) = 0x ta−1 exp(−t)dt and Γ(a) = 0∞ ta−1 exp(−t)dt.
ORBF/SBS.
The first sum-rate is obtained by optimizing
over all BBd combinations of beams, the latter one by
further optimizing over the total number of used beams
1 ≤ Bd ≤ B. Rewriting (8) this is equivalent to computing
the distributions of RBd = max{b1 ,...,bBd }∈BBd R(b1 ,...,bBd ) and
R(dyn) = max1≤Bd ≤B RBd .
Since the rates under different combinations of beams are
statistically dependent in general, the CDFs of RBd and R(dyn)
cannot be written as a product of marginal CDFs. However,
as the dependence is rather weak and furthermore vanishes
asymptotically for large user numbers, we use that expression
as an approximation (actually an upper bound as the correlation between the rates under different beam combinations is
always positive). The same argument holds for the rates when
different numbers of beams are used.
Thus, we can write the CDFs of RBd as (20) (see next page)
and
FR(dyn) (r)
= FR1 ,...,RB (r, . . . , r)
B
"
≈
FRBd (r)
Bd =1
≈
B "
Bd =1
(BB )
d
FR(b1 ,...,bB ) (r)
.
d
(21)
The approximations for the average sum-rates of the schemes
are finally given by:
(BBd ) B Bd
E[RBd ] = θ ·
(−1)l−1
l
l=1
l
αB
−1
αB
−1 min
d
d
i=1 ki , 1 − 1 !
. (22)
···
l
#l
l1+ i=1 ki · i=1 ki !
k1 =0
kl =0
E[R(dyn) ] is given in (23) (see next page).
WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
FRBd (r) = FR(b1,1 ,...,bB
d ,1
E[R
(dyn)
)
,...,R⎛
⎜
⎝b
1,
⎟
B ,...,b
B ⎠
Bd ,
Bd
Bd
( )
(B1 ) (B2 )
] =
⎞
···
l1 =1 l2 =0
i=1
( )
(B
B ) α−1
lB =0 k11 =0
⎡⎛
(r, . . . , r) ≈
(BBd )
"
···
α−1
k1l1 =0
···
(BB )
d
FR
(r) = FR(b1 ,...,bB ) (r)
d
(b1,i ,...,bBd ,i )
αB−1
kB1 =0
···
kBlB =0
sum−rate capacity (bps/Hz)
(23)
V. F UNDAMENTAL D EPENDENCIES
simulated
analytical
30 dB
16
14
In this section we analyze the dependence of the optimal
data stream number in static ORBF/SBS on the user number,
the SNR, and the pilot beam number. Furthermore, we provide
asymptotical results in analytical form.
20 dB
12
10
10 dB
A. User Number
8
The more users are requesting data, the more of them
we can expect to reside in (close to) orthogonal beams.
Accordingly, it is intuitive that B0 increases with the user
number. Indeed, the following lemma holds:
6
0 dB
4
2
0
(20)
αB−1
⎞ ⎤
(BB )
lBd
B
lBd −1
d
B
(−1)
k
,
1
−
1
!
min
iBd =1 iBd
Bd =1
⎢⎜ " lBd
⎟
⎥
⎣⎝
⎠
⎦.
#lBd
lBd
B
#
1+
k
i
k
!
B
l
Bd =1
iB =1
Bd
Bd =1
iBd =1 iBd
d
( Bd =1 lBd )
· i=1 ki !
18
2517
20
40
60
user number
80
100
120
Lemma 1: For fixed ρ,
lim B0 = B,
Fig. 2.
Comparison of simulated and analytical sum-rate achieved by
dynamic ORBF/SBS under SNR of 0, 10, 20 and 30 dB.
The respective intermediate steps again can be found in
Appendix B.
In Fig. 2 we plot both the simulated and the analytical sumrate according to (23) (see next page) for various SNR values
and numbers of users. As expected the analytical expression
tends to yield slightly too optimistic values. This is as we
used the product of marginal CDFs instead of the correct joint
CDFs for RBd and R(dyn) , what slightly overrated the actually
available diversity. The analytical curves are not smooth and
not monotonically increasing. This is due to the rounding of
the first parameter of the Gamma distribution. Nevertheless,
it can be seen that (23) is a tight approximation of the actual
sum-rate, which is very robust with respect to variations in
the system parameters.
Note that (23) can also be interpreted as the sum-rate
achieved by OSDMA/BS with Bd data and BBd pilot beams.
Indeed, we have approximated the sum-rate of ORBF/SBS by
the sum-rate of the respective OSDMA/BS scheme. Fig. 2 thus
also illustrates that ORBF/SBS approaches the performance
of ORBF/OSDMA while requiring a significantly smaller
extension of the pilot mode period. On the other hand – as
explained in the introduction – OSDMA/BS is less costly with
respect to the amount of feedback necessary.
K→∞
(24)
i.e., all beams advertised to the MSs in pilot mode should be
used for data transmission in data mode if the user number is
sufficiently large.
Proof: We upper-bound the sum-rate of a scheme that
makes use of Bd (optimally chosen) out of B beams, RBd ,
by Bd times the rate of the strongest out of K users in
beamforming configuration in absence of any interference
with full power budget ρ assigned, R(BF) . The channel gain
Xk hk 2 of each user is χ2 distributed with 2N degrees
of freedom. In [15] (Example 1) it has been shown that
*
+
Pr max Xk ≤ log K + N log log K + O(log log log K)
1≤i≤K
1
= 1−O
.
(25)
log K
Borrowing a technique from [3], the expected rate can
be upper-bounded as (26)–(27) (see next page), where we
used the fact that E[R(BF )] ≤ O(N log K) which is the
single user MIMO capacity with full CSI at the transmitter,
N transmit and K receive antennas [15]. Thus,
E[R(BF) ]
log log K + O(log log log K)])
≤ lim
= 1,
K→∞ log log K
K→∞
log log K
(28)
lim
2518
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
E[R(BF) ] ≤
log (1 + ρ [log K + N log log K + O(log log log K)])
*
+
·Pr max Xk ≤ log K + N log log K + O(log log log K)
1≤i≤K
+
*
+O(N log K) · Pr max Xk > log K + N log log K + O(log log log K)
(26)
log (1 + ρ log K) + O(log log log K),
(27)
1≤i≤K
=
i.e., only one data stream should be used under sufficiently
low SNR.
4
Data streams B
0
3
2
1
−20
−15
−10
−5
0
5
10
15
20
25
30
Average SNR ρ [dB]
35
40
2
4
8
16
32
64
1024
512
256
128
Number of users K
Fig. 3. The optimal B0 in static ORBF/SBS as a function of K and ρ for
N = B = 4.
and
E[RBd ]
K→∞ Bd log log K
Bd E[R(BF) ]
K→∞ Bd log log K
E[R(BF) ]
≤ 1. (29)
= lim
K→∞ log log K
On the other hand, we know from Theorem 1 in [15] that
limK→∞ E[RB ]/(B log log K) = 1 if all B beams are used
for data transmission. Finally, we conclude that for Bd < B
lim
E [RBd ]
K→∞ E [RB ]
lim
=
≤
lim
Bd
lim
B K→∞
E[RBd ]
Bd log log K
E[RB ]
B log log K
E[RBd ]
Bd limK→∞ Bd log log K
·
=
B limK→∞ E[RB ]
B log log K
Bd
< 1.
≤
B
Consequently, B data beams are optimal, if K → ∞.
(30)
Fig. 3 confirms the above discussion (for small and large ρ
the plot needs to be extended to arrive at B0 = B).
B. SNR
Next, we consider the dependence of the number of data
streams on the average SNR. We state the following lemma:
Proof: We consider the ratio of the sum-rates when using
Bd > 1 beams and one beam, respectively, i.e., RBd /R1 , when
the SNR goes to zero. In the first case, we denote the gains of
the data beams for the scheduled users by α1 = |hk1 φb1 |2 >
. . . > αBd = |hkBd φbBd |2 , and the largest individual channel
2
gain by β = maxx,y {|hkx φb
y | }. Clearly, β ≥ α1 > α2 >
. . . > αBd > 0. Finally, ii = j=i |hki φbj |2 denotes the gain
of the interfering data beam of the i-th chosen user. Now, by
denoting the mean of the αi by α, we can write according to
(3) and (4)
Bd
αi
i=1 log 1 + Bd +ii
RBd
ρ
lim
= lim
ρ→0 R1
ρ→0
log (1 + βρ)
Bd
αi ρ
log
1
+
i=1
Bd
≤ lim
ρ→0
log (1 + βρ)
Bd log 1 + αρ
Bd
(32)
≤ lim
ρ→0
log (1 + βρ)
α
ρ − o(ρ)
α1
α
= <
< 1, (33)
= lim Bd Bd
ρ→0
βρ − o(ρ)
β
β
where we used Jensen’s inequality to obtain (32). o(·) is
the Landau symbol7 . As RBd < R1 holds for any channel
realization, it finally follows that E[RBd ] < E[R1 ].
The result thus goes along with the waterfilling strategy
in point-to point MIMO channels, which suggests to close
all subchannels but the strongest one at low SNR. Although
waterfilling relates to interference free channels, it allows
to intuitively understand the lemma above, which considers
interference channels: using only the best channel allows
for interference free operation. By the waterfilling argument
this is better than dividing the transmit power over multiple
interference free channels at low SNR. Finally, this in turn is
better than dividing the transmit power over multiple interfering channels. Indeed, this reasoning constitutes an alternative
proof to the one we gave above.
In the high SNR region, the optimal B0 can be well
understood intuitively, again. In such a scenario, a multi-beam
system is interference limited, i.e., the sum-rate capacity remains bounded for arbitrarily high signal power. Accordingly,
we have the following lemma:
Lemma 3: For fixed K,
lim B0 = 1,
Lemma 2: For fixed K,
lim B0 = 1,
ρ→0
ρ→∞
(31)
7 f (φ)
= o(φ) means limφ→φ0
f (φ)
φ
= 0 for the considered limit.
(34)
WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
10
i.e., only one data stream should be used under sufficiently
high SNR.
lim
ρ→∞
RBd
= lim
ρ→∞
R1
Bd
log
1+
i=1
αi
+ii
Bd
ρ
log (1 + βρ)
= 0,
(35)
where we used the same notation as in the previous proof.
With the same argument as in the proof of Lemma 2 we
conclude E[RBd ] < E[R1 ] and have shown that the use of
one data stream is optimal.
Again, we find our results confirmed in Fig. 3 (for large K
the plot needs to be extended in order to arrive at B0 = 1).
From Fig. 3 we also see that for all reasonable pairs (ρ, K)
more than one, but less than all pilot beams should be used in
data mode. This strongly motivates our ORBF/SBS scheme.
C. Number of Pilot Beams
So far we did not consider how many pilot beams should
be generated at the BS in pilot mode. Each beam has to
be advertised by the BS and its channel gain measured by
the MSs in a pilot slot of duration tpilot . The duration of
the pilot mode period is proportional to the number of pilot
beams. This is as – depending on whether the pilot beams
are transmitted simultaneously or subsequently – the pilot
sequences themselves have to be extended over time to allow
for further orthogonal sequences or additional pilot slots must
be added for each additional beam.
In order to account for the pilot sequences, which are
not used for data transmission, we introduce the notion of
throughput in a similar way as in [12] in this section. Firstly,
we need to define the latency time tlatency as the average length
of the period a user has to wait before being scheduled again
after having lost the channel. This means that each user is
served in slots of duration tlatency /(K/B0 ) comprising both
pilot and data mode. We now define the throughput TB0 (B) as
the average number of transmitted data bits per latency period
when B pilot beams and B0 data beams are used. Within each
such period it is necessary to switch to pilot mode K/B0 times
in order to fulfill the given delay constraint. As a single pilot
slot lasts Btpilot seconds we write the throughput as
TB0 (B) =
tlatency −
K
Btpilot
B0
tlatency
t
· RB0 (B) =
B
1−
τ · RB0 (B),
B0
(36)
pilot
where τ = tlatency
/K and RB0 (B) is the sum-rate for B0 data
beams and B pilot beams.
Clearly, if the length of the pilot slots is negligible compared
to the duration of the overall transmission period tlatency /K the
throughput coincides with achievable sum-rate. In this case
it is optimal to generate as many orthogonal pilot beams as
possible, i.e., as many as transmit antennas are available. This
maximizes the number of possible subsets of beams that can
be used afterwards in data mode and therefore also diversity
at negligible costs. On the other hand, if τ is significant the
B0=2
B0=3
9
throughput (bps/Hz)
Proof: We consider the ratio of the sum-rates when using
one and Bd > 1 beams, i.e., RBd /R1 , when the SNR goes to
infinity. As all ii > 0 it follows that
9.5
2519
8.5
8
7.5
7
6.5
τ=0, 2.5, 5, 10 %
6
5.5
5
3
4
5
6
number of pilot beams B
7
8
Fig. 4. Throughput vs. number of pilot beams for different τ , when using
two and three data streams, respectively. For SNR=15dB, and N = B = 8.
diversity gain obtained by generating a further pilot beam
might become less than the loss in terms of the wasted
transmission time.
With this in mind the following result for a fixed B0 is
obvious:
,
N, τ → 0,
(37)
argmax TB0 (B) =
B0 , τ 0.
B
Furthermore, in general the optimal number of used data
streams increases for a fixed B, when τ increases. This is
as the BS has to switch to pilot mode less frequently, if
it can serve more users at a time. Note that for the ORBF
scheme throughput maximization coincides with sum-rate
maximization, since TB (B) = (1 − τ ) RB (B). On the other
hand, OBF/MWV, which yields TB (1) = (1 − Bτ ) RB (B),
makes use of the pilot beams most inefficiently in terms of
data mode duration among all schemes, what results in poor
performance for large τ .
In Fig. 4 we plot the throughput when using different numbers of pilot beams and two or three data streams, respectively.
If τ increases the optimal number of pilot beams decreases.
This is as the gain in the sum-rate generated by the additional
beam combination opportunities becomes less than the loss
due to the shorter data mode at some point.
VI. P ERFORMANCE C OMPARISONS
In this section we compare the achievable sum-rates of both
ORBF/SBS schemes to the previous schemes. The according
plots are found in Fig. 5. In our simulations, we assume N =
4. The number of pilot beams is B = 4 for ORBF/SBS and
OBF/MWV. The pilot overhead is assumed to be negligible,
i.e., τ = 0.
We start with a comparison of the sum-rates achievable by
the two ORBF/SBS schemes. The dynamic scheme – due to its
optimality – is clearly superior to the static scheme. However,
the performance gap between both schemes is surprisingly
small over the complete range of SNR values and user
numbers. The reason for the only moderate performance loss
2520
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
9
OBF
OBF/MWV
ORBF
stat. ORBF/SBS
dyn. ORBF/SBS
OBF
OBF/MWV
ORBF
stat. ORBF/SBS
dyn. ORBF/SBS
8
sum−rate capacity (bps/Hz)
sum−rate capacity (bps/Hz)
4
3
7
6
2
5
0 dB
1
0
10
20
30
40
50
60
70
80
90
10 dB
100
4
0
10
20
30
user number
50
60
70
80
90
100
user number
13
12
40
16
20 dB
30 dB
14
sum−rate capacity (bps/Hz)
sum−rate capacity (bps/Hz)
11
12
10
9
10
8
7
6
OBF
OBF/MWV
ORBF
stat. ORBF/SBS
dyn. ORBF/SBS
5
4
3
0
10
20
30
40
50
60
70
80
90
8
6
OBF
OBF/MWV
ORBF
stat. ORBF/SBS
dyn. ORBF/SBS
4
100
2
0
user number
10
20
30
40
50
60
70
80
90
100
user number
Fig. 5. Comparison of random beamforming schemes for ρ = 0, 10, 20 and 30 dB. N = B = 4 for ORBF/SBS, OBF/MWV, and ORBF, and N = 4, B = 1
for OBF.
can be seen by examining Fig. 6, which shows the corresponding probability mass functions (PMF) for the number of
data streams dynamically chosen by the dynamic ORBF/SBS
scheme. One can see that the scheme does not make use
of the whole spectrum of data stream numbers effectively.
Rather, it focuses on one or two different stream numbers
mainly. Furthermore, one of these stream numbers is usually
dominant. Accordingly, the loss in sum-rate when fixing the
beam number is small.
The largest gaps between the static and the dynamic scheme
will arise, when K and ρ are such that the B0 in static
ORBF/SBS changes from one integer to another. In this case,
the data stream number PMF of the dynamic scheme under
the same setup will not have a dominant peak, but two almost
equally likely beam numbers. This effect can be identified
most obviously through the curve simulated for ρ = 30dB in
the region around K = 18 users, where the optimal static data
stream number changes from one to two (results in a sharp
bend in the curve). Also one should refer to the respective
PMF in Fig. 6. In the simulated scenarios it turns out that
the static ORBF/SBS scheme always guarantees at least 93%
of the sum-rate achieved by the static scheme, while the
computational complexity is reduced.
Finally, we also consider the performance of the existing
schemes mentioned earlier, i.e., OBF, OBF/MWV, and ORBF.
Three key observations are listed as follows:
•
•
•
In contrast to static ORBF/SBS none of the other schemes
approaches the performance of the optimal dynamic
ORBF/SBS schemes over the complete range of K and
ρ.
For very high SNR (30 dB) or very low SNR (0 dB) and
small numbers of users OBF/MWV coincides with static
ORBF/SBS. This is consistent with the Lemmas 2 and 3
and can be exploited to reduce the amount of feedback
(one channel gain value, one beam index).
For extremely (unrealistically) high numbers of users,
ORBF converges to static ORBF/SBS (which again converges to dynamic ORBF/SBS). This is consistent with
Lemma 1.
Based on the above three points, we see that existing
schemes approach the optimal performance in few unrealistic scenarios. However, in real world applications, i.e., for
reasonable user numbers and average SNR values, ORBF,
OBF/MWV, and OBF can not approach the optimal sum-rate.
WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
VII. R EDUCING C OMPLEXITY
0.9
In this section, we focus on feasibility issues, when
ORBF/SBS is applied in practical applications. Particularly,
we consider the computational complexity of the search algorithm at the BS, which finds the scheduled beam and user
subsets, and the feedback complexity. Compared to previous
schemes, we have increased both, if we implement the optimal
dynamic ORBF/SBS scheme literally. However, it turns out,
that both the search complexity at the BS, and the amount
of feedback, can be reduced efficiently in static ORBF/SBS
without major loss in performance.
0.8
30 dB
20 dB
10 dB
0 dB
0.7
PMF
0.6
0.5
0.4
0.3
0.2
0.1
A. Feedback Complexity
0
1
2
3
4
number of data streams
Fig. 6. Probability mass function of number of data streams chosen by the
dynamic ORBF/SBS when K = 18, N = B = 4, and SNR=0,10,20,30 dB.
13
12
sum−rate capacity (bps/Hz)
Requiring feedback of all the channel gains of all B pilot
beams sounds very demanding at first sight. Before we consider techniques to reduce the feedback amount efficiently, we
point out that even the feedback of B channel gains is far less
than the required amount in DPC or coherent beamforming. In
the latter cases, we feed back all channel coefficients, which
are complex instead of real numbers. Furthermore, these must
be made available at BS with high accuracy. In ORBF/SBS
the channel gains can be quantized more roughly, as they
are not used for precise beamforming weight derivation or
interference pre-subtraction, but only for SINR computations.
The robustness of random beamforming to quantization of
feedback is shown in [21], where the authors proof that 1bit feedback suffices to sustain the log log K scaling law in a
single data beam scheme.
In static ORBF/SBS the amount of feedback can be further
reduced without suffering from performance loss significantly.
The basic idea is to ignore all beams with moderate channel
gain at the MS, as these – with high probability – are too weak
to be used as data beam, and too strong to be scheduled as
interfering beam. Thus, the users need to feed back the index
and channel gain of the strongest and few indices and channel
gains of the weakest beams only. This technique efficiently
exploits that a user usually will not be scheduled together
with beams that impose major interference to it with high
probability.
The number of weak beams that should be considered in
general depends on K. The a-priori-probability that a user
will be scheduled in a beam where |hk φi | < thu or interfered
in a beam where |hk φi | > thl for some threshold values thu
and thl , decreases, when K increases. Thus, we can reduce
the feedback most efficiently for large user numbers, which is
exactly the scenario, where the amount of feedback is most
crucial.
If B is not too large and B0 = 2, it is even possible to
feedback a single SINR (each MS computes its SINR when
scheduled in its strongest and interfered in its weakest beam)
and two beam indices for each user while sustaining close to
optimal performance. In Fig. 7, it can be seen that even for
realistic user numbers the performance of static ORBF/SBS
can be approached. For 30 users the gap is smaller than 3%
already.
In general, the amount of feedback might also vary from
user to user depending on its concrete channel gain realizations. E.g., a user without a single strong beam or with an
2521
dyn. ORBF/SUS
static ORBF/SUS
st. ORBF/SUS w. red. feedback
11
10
9
8
7
6
0
10
20
30
40
user number
50
60
70
Fig. 7. Approaching optimal performance with reduced feedback, for B0 =
2, N = B = 4 at SNR=20 dB.
insufficient number of weak beams does not need to feed back
anything at all, since the apriori probability of being scheduled
is negligible.
B. Computation Complexity
A naive approach to perform the optimization in (5) and
(7) is to search over all possible user and beam combinations.
Assuming K B this algorithm is of complexity O(K +
K 2 +...+K B ) = O(K B ), when applied with the dynamic and
of complexity O(K B0 ) when applied with the static scheme.
However, in a practical application the search can be
performed as follows:
• Compute only one SINR for each user assuming that
it is scheduled in its strongest beam for each particular
beam combination and add it to a list of candidates for
its strongest beam.
• For each beam choose the user with highest SINR from
the corresponding candidate list.
If the candidate lists for all beams are non-empty, this
algorithm yields the optimal user and beam combinations. As
2522
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
the probability of an empty candidate list goes to zero rapidly
for increasing user numbers, the algorithm almost surely yields
the optimal result for moderate user numbers already. This fact
has already been used in [15] and also in (9) to analyse the
respective rate distributions.
Both dynamic and static scheme are of complexity O(K)
under the above algorithm. More precisely,
the complexity in
the static scheme scales like K BB0 , while the one of the
B
dynamic scheme scales like K B
B0 =1 B0 . Thus, the runB
( B )
B =1 B0
0
time of static ORBF/SBS is
times shorter than
(BB0 )
the one of dynamic ORBF/SBS.
Note that by decreasing the feedback complexity according
to the previous subsection, automatically also the computational complexity is further reduced.
Finally, we mention an efficient method to determine the
data beam number B0 in static ORBF/SBS, which is hard to
compute analytically. In the previous section, we have seen
that the dynamic scheme makes use of few different data
stream numbers only. Thereby, the peak in the data stream
number PMF coincides with B0 in static ORBF/SBS. Thus,
an easy way to find B0 for the static scheme, is to run the
dynamic scheme over some cycles, and then to decide for B0
according to the peak in the empirical PMF.
Fig. 8. Average SNR (normalized by K
k=1 E[SNRk ]) and fraction of time
being scheduled for the individual users, when PFS is applied to dynamic
ORBF/SBS in a scenario, where K = 12, average SNR = 20 dB, tc = 500
and N = B = 6.
apply PFS to ORBF/SBS according to
(dyn)
{(b1
=
VIII. FAIRNESS
So far we have considered a homogeneous channel, i.e., the
scenario where all users have equal average SNR. However,
this assumption is only reasonable, as long as all users are
roughly in the same distance from the BS and shadowing effects can be neglected. In a more general scenario,
users’ signals experience different path losses. In [9] the
authors proposed a scheduling algorithm called proportional
fair scheduling (PFS) for their scheme. It schedules users not
directly based on their current rate, but rather based on the
ratio of current rate to the average throughput it received
in the past. The latter is computed for user k according to
(38) (see next page), where the parameter tc is related to the
window length back in the past that is taken into account for
the computation. It has been shown that in OBF asymptotically
(when K → ∞), all users are scheduled equally often,
while always being close to their beamforming configuration.
This also holds for the ORBF scheme, where users are only
scheduled when they are in beamforming configuration with
respect to their data beam and nulled by all interfering beams.
According to the previous discussion all beams should be used
in the ORBF/SBS, when K → ∞, what reduces it to ORBF.
Therefore, asymptotically PFS has the same ideal properties
combined with ORBF/SBS, as when applied to the original
OBF scheme. While a small user number (around 30) suffices,
to approach the asymptotic behavior in OBF, in schemes
deploying multiple beams simultaneously the required user
number is huge. Thus, fairness is not guaranteed necessarily,
if the user number is only moderate.
Thus, the question arises, whether users are still scheduled
equally often under the PFS algorithm, when applied with
ORBF/SBS in practical applications. In simulations, where we
(dyn)
, k1
(dyn)
(dyn)
), . . . , (bB0 , kB0 )}
argmax
{(b1 ,k1 ),...,(bB0 ,kB0 )}∈∪B
B0 =1 BB0 ×KB0
B0
Rk
i ,bi
i=1
Tki
(39)
it turns out that this – although not ideally – is fulfilled reasonably. This can be seen from Fig. 8, where we
show the differK
ently chosen average SNRs (normalized by k=1 E[SNRk ])
and the fractions of time being scheduled as obtained from
simulations for twelve individual users in a system. The
average SNR (over all users) is 20 dB. It can be seen that
the algorithm is still reasonably fair. User 5, who has an
extremely poor channel, is even served most often. In general
there is no rule, such as ”stronger/weaker users are scheduled
more often/seldom than weaker/stronger ones“, while it can
be observed that users with similar average channel gain
are scheduled similarly often (consider, e.g., users 4, 7, and
10). The simulation results are obtained through 100000
iterations, where tc = 500. tc is chosen large enough such
that the average computation is reasonably precise. The Tk
are initialized according to their stationary distribution.
IX. C ONCLUSIONS
In this paper we have proposed a generalization of several
existing random beamforming schemes. With this generalization, two methods are studied: dynamic ORBF/SBS and static
ORBF/SBS. The dynamic ORBF/SBS achieves the ultimate
limit on the sum-rate achievable by the random beamforming
methods that are based on channel gain feedback and do not
make use of additional preprocessing at the transmitter. The
static ORBF/SBS, on the other hand, approximates the optimal
scheme with lower computational complexity. The proposed
schemes enable a significant performance improvement in
terms of sum-rate as compared to the previously introduced
schemes in most practical scenarios, i.e., for realistic user
numbers and in reasonable SNR regimes. This is achieved by
WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
⎧ ⎨ 1−
Tk (t + 1) =
⎩ 1−
=
=
=
1
tc
· Tk (t) +
1
tc Rk (t),
· Tk (t),
if user k is scheduled,
if user k is not scheduled,
K
Bd
Bd r
−(Bd −1)r
1 − FRbi (r)dr =
1 − 1 − exp
exp − 2
· dr
·2
ρ
ρ
0
0
K
∞
Bd
1
Bd
1 − 1 − exp
· dx
· x−(Bd −1) exp − x
ln(2)x
ρ
ρ
1
∞
K Bd k
1 K
Bd k
x · dx
x−(Bd −1)k−1 exp −
(−1)k−1 exp
·
k
ln(2)
ρ
ρ
1
k=1
K Bd k
1 K
Bd k
(−1)k−1 exp
EBd −Bd k+1
,
k
ln(2)
ρ
ρ
E[Rbi ] =
1
tc ∞
2523
(38)
∞
(40)
(41)
(42)
k=1
∞
1 − FR2b (a)da − E[Rbi ]2
Var[Rbi ] = E[Rb2i ] − E[Rbi ]2 =
i
0
K
∞
√
Bd
Bd √
=
1 − 1 − exp
· da − E[Rbi ]2
· 2−(Bd −1) a exp − 2 a
ρ
ρ
0
K ∞
Bd
2 ln(x)
Bd
−(Bd −1)
1 − 1 − exp
=
exp − x
· dx − E[Rbi ]2
·x
(ln(2))2 x
ρ
ρ
1
∞
K Bd k
K
2
Bd k
k−1
−(Bd −1)k−1
=
x
· dx − E[Rbi ]2 ,
exp
ln(x)x
exp
−
(−1)
·
k
(ln(2))2
ρ
ρ
1
(43)
(44)
(45)
(46)
k=1
(BBd ) ⎞
γ αBd , θr
⎠ dr
⎝1 −
(1 − FRBd (r))dr ≈
Γ(αBd )
0
0
B ∞
γ(αBd , r) (Bd )
θ·
1−
dr
Γ(αBd )
0
⎛
B ⎞
(BBd ) 1 − exp(−r) αBd −1 rk (Bd )
∞
[(αB
−
1)!]
d
⎜
⎟
k=0
k!
⎜1 −
⎟ dr
θ·
B
⎝
⎠
(
Bd )
0
[(αBd − 1)!]
⎞
⎛
l
(BBd ) B ∞
αB
−1
αB
−1
d
d
k
r i=1 i ⎟
⎜
Bd
···
(−1)l exp(−lr)
θ·
⎠ dr
⎝1 −
#l
l
0
i=1 ki !
l=0
k1 =0
kl =0
E[RBd ] =
=
=
=
∞
∞
⎛
(BBd ) B ∞
αB
αB
d −1
d −1
l
1
Bd
= θ·
···
exp(−lr)r i=1 ki · dr
(−1)l−1
#l
l
i=1 ki ! 0
l=1
k1 =0
kl =0
l
(BBd ) B αB
αB
d −1
d −1
ki , 1 − 1 !
min
i=1
Bd
···
= θ·
(−1)l−1
l
#l
l
l1+ i=1 ki · i=1 ki !
l=1
k1 =0
kl =0
(47)
(48)
(49)
(50)
(51)
(52)
2524
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
E[R(dyn) ] =
⎞(BB ) ⎞
⎛ d
r
B
∞
∞⎜
γ αBd , θB
"
⎟
d
⎟ dr
⎜
⎠
⎝
(1 − FR(dyn) (r)) dr ≈
⎠
⎝1 −
Γ(αBd )
0
0
⎛
(53)
Bd =1
⎛
(BBd ) B αB
αB
d −1
d −1
∞⎜
r
l
Bd
⎜1 −
)
···
(−1) exp(−l
⎝
θBd
l
0
B =1 l=0
k =0
k =0
=
B
"
d
=
l
1
li=1 ki ⎞
⎟
θBd
⎟ dr
#l
k! ⎠
r
i=1
(54)
i
⎛
∞
0
(B
(B1 )
B)
α−1
α−1
αB−1
αB−1
⎜
·
·
·
·
·
·
·
·
·
·
·
·
1
−
⎝
l1 =0
B
"
Bd =1
=
(B2 )
(B1 ) lB =0 k11 =0
(BB )
d
(−1)lBd
lBd
#lBd
iBd =1
···
l1 =1 l2 =0
kiBd !
(B
B ) α−1
exp −
B
Bd =1
···
lB =0 k11 =0
⎡⎛
k1l1 =0
α−1
···
k1l =0
kB1 =0
lBd
r
θBd
αB−1
···
kB1 =0
kBlB =0
r
θBd
B
B
d =1
lBd
iB =1
d
⎞
ki Bd
⎟
⎠ dr
(55)
αB−1
kBl =0
1
B
⎞
⎤
(BB )
B
lBd
l
−1
Bd
d
B
B
k
B
∞
(−1)
i
d
Bd =1
iB =1
r
r
d
⎢⎜ " lBd
⎟
⎥
exp −
lBd
dr⎦
⎣⎝
⎠
#lBd
θ
θ
B
B
0
d
d
Bd =1
Bd =1
iB =1 kiBd !
(56)
d
( )
( ) B
1
=
B
2
l1 =1 l2 =0
⎡⎛
( ) α−1
B
B
···
···
lB =0 k11 =0
α−1
k1l1 =0
···
αB−1
kB1 =0
···
αB−1
kBlB =0
⎤
⎞ (BB )
lBd
B
lBd −1
d
B
(−1)
k
,
1
−
1
!
min
"
iBd =1 iBd
Bd =1
lBd
⎢⎜
⎥
⎟
⎣⎝
⎦.
⎠
#lBd
lBd
B
B
#l
1+ B =1 i =1 kiB
Bd =1
d
Bd
d
iBd =1 kiBd !
( Bd =1 lBd )
· i=1 ki !
efficiently balancing spatial multiplexing gain and multiuser
diversity. Both schemes in general require an increased amount
of feedback as compared to previously proposed schemes.
However, this is still far smaller as compared to precise
channel coefficient feedback. We also indicated the potential
of further feedback reduction in ORBF/SBS, while still sustaining close to optimal performance. Finally, our computer
simulations have shown, that fairness is reasonably guaranteed
by applying the proportional fair scheduler to the ORBF/SBS
scheme.
A PPENDIX A
E VALUATION OF E[Rbi ] AND VAR [Rbi ]
We evaluate mean and variance of Rbi starting from the
following integral including the CDF of Rbi ,√(40)–(46), where
we used the substitutions x = 2r resp. x = 2 a , and expressed
the first result
! ∞ in terms of the exponential integral function
Ek (x) = 1 x−k exp(−kx) dx.
A PPENDIX B
E VALUATION OF E[RBd ] AND E[R(dyn) ]
We evaluate the average rates E[RBd ] and E[R(dyn) ] using
Gamma approximations of the respective distributions (see
(47)–(57)).
(57)
R EFERENCES
[1] G. Caire and S. Shamai, “On the achievable throughput for a multiantenna Gaussian broadcast channel,” IEEE Trans. Inform. Theory,
vol. 49, no. 7, pp. 1691–1706, July 2003.
[2] W. Yu and J. M. Cioffi, “Sum capacity of Gaussian vector broadcast
channels,” IEEE Trans. Inform. Theory, vol. 50, no. 9, pp. 1857–1892,
Sep. 2004.
[3] M. Sharif and B. Hassibi, “Scaling laws of sum-rate using time-sharing,
DPC, and beamforming for MIMO broadcast channels,” in Proc. IEEE
Int. Symposium Inform. Theory, June/July 2004, p. 175.
[4] C. B. Peel, B. Hochwald, and A. L. Swindlehurst, “A vector perturbation
technique for near capacity multi-antenna multi-user communication –
Part I: Channel inversion and regularization,” IEEE Trans. Commun.,
vol. 53, no. 1, pp. 195–202, Jan. 2005.
[5] T. Yoo and A. Goldsmith, “On the optimality of multi-antenna broadcast
scheduling using zero-forcing beamforming,” IEEE J. Select. Areas
Commun., Special Issue on 4G Wireless Systems, vol. 24, no. 3, pp.
528–541, Mar. 2006.
[6] Z. Tu and R. Blum, “Multiuser diversity for a dirty paper approach,”
IEEE Commun. Lett., vol. 7, no. 8, pp. 370–372, Aug. 2006.
[7] M. A. Maddah-Ali, M. Ansari, and A. K. Khandani, “An efficient
signaling scheme for MIMO broadcast systems: design and performance
evaluation,” IEEE Trans. Inform. Theory, to be published.
[8] S. A. Jafar and A. Goldsmith, “On the capacity region of the vector
fading broadcast channel with no CSIT,” in Proc. IEEE ICC, June
2004, vol. 1, pp. 468–472.
[9] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using
dumb antennas,” IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1277–
1294, June 2002.
[10] J. Chung, C. S. Hwang, K. Kim, and Y. K. Kim, “A random beamforming technique in MIMO systems exploiting multiuser diversity,” IEEE
J. Select. Areas Commun., vol. 21, no. 5, pp. 848–855, June 2003.
WAGNER et al.: ON THE BALANCE OF MULTIUSER DIVERSITY AND SPATIAL MULTIPLEXING GAIN IN RANDOM BEAMFORMING
[11] Y.-C. Liang and R. Zhang, “Multiuser MIMO systems with random
transmit beamforming,” Int. J. Wireless Inform. Networks, Special Issue
on MIMO, vol. 12, no. 4, pp. 235–247, Dec. 2005.
[12] I. M. Kim, S. C. Hong, S. S. Ghassemzadeh, and T. V., “Opportunistic
beamforming based on multiple weighting vectors,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2683–2687, Nov. 2005.
[13] K. Zhang and Z. Niu, “Random beamforming with multi-beam selection
for MIMO broadcast channels,” in Proc. IEEE ICC, June 2006, vol. 9,
pp. 4191–4195.
[14] M. Kountouris and D. Gesbert, “Memory-based opportunistic multi-user
beamforming,” in Proc. IEEE Int. Symposium Inform. Theory, Sep.
2005, pp. 1426–1430.
[15] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels
with partial side information,” IEEE Trans. Inform. Theory, vol. 51,
no. 2, pp. 506–522, Feb. 2005.
[16] R. Zhang, Y.-C. Liang, and J. M. Cioffi, “Throughput comparison of
wireless downlink transmission schemes with multiple antennas,” in
Proc. IEEE ICC, May 2005, vol. 4, pp. 2700–2704.
[17] W. Choi, A. Forenza, J. G. Andrews, and R. W. Heath Jr., “Opportunistic
space division multiple access with beam selection,” IEEE Trans.
Commun., to be published.
[18] K. Huang, J. G. Andrews, and R. W. Heath Jr., “Joint beamforming
and scheduling for sdma systems with limited feedback,” IEEE Trans.
Commun., to be published.
[19] B. Hassibi and T. L. Marzetto, “Multiple-antennas and isotropically
random unitary inputs: The received signal density in closed form,”
IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1473–1484, June 2002.
[20] A. Papoulis, Probability, Random Variables, and Stochastic Processes.
New York: McGraw-Hill, 1984.
[21] S. Sanayei and A. Nosratinia, “Exploiting multiuser diversity with only
1-bit feedback,” in Proc. IEEE Wireless Commun. Networking Conf.
(WCNC), Mar. 2005, vol. 2, pp. 978–983.
Jörg Wagner (S’06) received the Master’s degree in
Information Technology and Electrical Engineering
from ETH Zurich in October 2006. He is currently
working towards his PhD at the same university.
He was a visiting student in the Institute for Infocomm Research, Singapore, from September 2005
to February 2006. His research interests lie in the
broad fields of physical layer design and applied
information theory.
2525
Ying-Chang Liang (SM’00) received the PhD degree in Electrical Engineering in 1993. He is now
a Senior Scientist in the Institute for Infocomm Research (I2R), Singapore. He also holds adjunct associate professorship positions in Nanyang Technological University (NTU) and the National University
of Singapore, Singapore (NUS). From Dec. 2002 to
Dec. 2003, Dr. Liang was a visiting scholar with
the Department of Electrical Engineering, Stanford
University. He has been teaching graduate courses
in NUS since 2004. In I2R, he has been leading
the research activities in cognitive radio and standardization activities in
IEEE 802.22 wireless regional networks (WRAN) for which his team has
made fundamental contributions in physical layer, MAC layer, as well as
channel sensing solutions. His research interests include cognitive radio,
reconfigurable signal processing for broadband communications, and spacetime wireless communications and information theory, for which he has
published over 140 international journal and conference papers.
Dr. Liang received the Best Paper Awards from IEEE VTC-Fall’1999 and
IEEE PIMRC’2005. He served as an Associate Editor for the IEEE T RANS ACTIONS ON W IRELESS C OMMUNICATIONS from 2002 to 2005, and guesteditor for the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS ,
S PECIAL I SSUE ON C OGNITIVE R ADIO : T HEORY AND A PPLICATIONS. He
was Publication Chair for the 2001 IEEE Workshop on Statistical Signal
Processing, TPC Co-Chair for IEEE ICCS’2006, and Co-Chair, Thematic Program on Random matrix theory and its applications in statistics and wireless
communications, Institute for Mathematical Sciences, National University of
Singapore, 2006. Dr. Liang is a Senior Member of IEEE.
Rui Zhang (S’00-M’07) received the B.S. and
M.S. degrees in electrical and computer engineering
from National University of Singapore in 2000 and
2001, respectively, and the Ph.D. degree in electrical
engineering from Stanford University, Stanford, CA,
in 2007. Since 2007, he has been a research fellow
with the Institute for Infocomm Research (I2R), Singapore. His main research interests include digital
transmission and coding, statistical signal processing, multi-antenna systems, and wireless networks.