Wireless Sensor Network, 2009, 3, 189-195
doi:10.4236/wsn.2009.13025 ctober 2009 (http://www.SciRP.org/journal/wsn/).
Published Online O
Generation of Multiple Weights in the Opportunistic
Beamforming Systems
Guangyue LU1,2, Lei ZHANG2, Houquan YU1, Chao SHAO2
1Electronics and Information College, Yangtze University, Jingzhou, China
2Department of Telecommunications Engineering, Xi’an Institute of Posts and Telecommunications, Xi’an, China
Received April 18, 2009; revised April 29, 2009; accepted May 31, 2009
Abstract
A new scheme to generate multiple weights used in opportunistic beamforming (OBF) system is proposed to
deal with the performance degradation due to the fewer active users in the OBF system. In the proposed
scheme, only two mini-slots are employed to create effective channels, while more channel candidates can be
obtained via linearly combining the two effective channels obtained during the two mini-slots, thus increas-
ing the multiuser diversity and the system throughputs. The simulation results verify the effectiveness of the
proposed scheme.
Keywords: Opportunistic Beamforming (OBF), Multiuser Diversity, System Throughputs, Scheduling
1. Introduction
With the development of the wireless communication,
increasing the spectrum efficiency and data rates is be-
coming the major task, especially in the downlink case.
Multiple-Input-Multiple-Output (MIMO) technique [1]
can improve the spectrum efficiency with no need of
more bandwidth by employing multiple antennas at both
transmitter and receiver. Therefore MIMO technique is
becoming one of the most promising techniques in the
future communication systems (e.g., LTE, B3G), and
coherent beamforming [2] and dirty paper coding [3] are
two ways to improving the spectrum efficiency. How-
ever the full channel information for all users at the
transmitter is required to realize the coherent beamform-
ing and dirty paper coding, which is not realistic with the
increasing of the number of the users and antennas be-
cause of the waste of the systems resource to feedback
the channel information from the receivers to the trans-
mitter.
In wireless communication system, many users are
communicating with the base station, and the system
throughput can be improved by suitably scheduling
(through, e.g., maximum throughput (MAX) scheduling
algorithm or proportional fairness (PF) scheduling algo-
rithm) the user with large channel gains to transmit its
packets, which is known as the multiuser diversity
(MUD) [4]. In contrast to the channel equalization used
in the traditional communication systems to combat the
effect of the multipath fading channel on the data trans-
mission, it is the channel fluctuations that is the source of
the MUD and the MUD will be enlarged with the in-
crease of the dynamic range of the channel fading. The
larger the dynamic range of the channel fluctuations, the
higher peak of the channels and the larger the multiuser
diversity gain. Hence to achieve large MUD requires the
large channel dynamic range and the suitable scheduling
scheme.
However, the MUD gain will be limited by the small
dynamic range of the channel fluctuations due to the
availability of light-of-sight (LOS) path and little scatting
in the environment and the slowly channel fading com-
pared to the delay constraint of the services. Thus those
users with small channel gain and fluctuations may not
be scheduled to transmit their packets and their QoS can
not be met.
In [5], random fading is induced purposely in multi-
ple-input-single-output (MISO) systems when the envi-
ronment has little scatting and/or the fading is slow to
increase the MUD gain of the system by multiplying the
transmit data with different weighting factors at each
transmitting antenna. When the weighting factors are
phase-conjugate with the independent channels from
the user to the transmitting antennas, this user is in its

G. Y. LU ET AL. 190
beamforming configuration state and its channel peak
values occur. When the number of the users in the sys-
tems is large enough, the probability that at lease one
user is in its beamforming configuration state is large and
the throughput of the system can approach that of the
coherent beamforming with only partial channel infor-
mation (i.e., the overall SNR) feedback. And the scheme
in [5] is interpreted as the opportunistic beamforming
(OBF).
However, one of the limits of the OBF is the require-
ment of large number of users in the system simultane-
ously and the system throughput will be degraded when
the number of the users in the system is not too large.
When fewer users are active in the system, the MISO
system in [5,6] is extended to MIMO in [7], that is, mul-
tiple antennas are also employed at the receivers, which
equivalently increase the number of visual active users
and, thus, the system throughput. However, the feedback
and the costs of each user will be inevitably increased
with the increase of the number of the users and the em-
ployed receiving antennas. The weighting factors used at
the transmitting antennas in [5] are totally random
among different time slots. However, since the base sta-
tion possesses all the users’ channels information at cur-
rent time slot and the previous time slots, the weighting
factors can be generated in an pseudo-random manner,
that is, the former weighting factors that create beam-
forming configuration state for one user can be used, in
some way, to generate the current former weighting fac-
tors only if the coherent time of the channels is large
enough [8,9].
Since the random weighting factors strongly affect the
channel states, multiple weighting vectors at several
mini-slots in one time slot [10] are used to create multi-
ple induced channels, and the one with larger channel
gain is selected and the corresponding weighting vector
is used as the current weighting vector. The OBF with
multiple weighting factors (MW-OBF) can improve the
throughput of OBF-CDMA systems. Since several
mini-slot are used to ‘train’ the best weighting factors,
some mini-slots and power resources are wasted in
MW-OBF.
In [11], two multiple weight OBF schemes tailored for
fast fading and slow fading scenarios respectively are
investigated and the tight upper bounds of the data rates
for both schemes are derived. It is claimed that the faster
the fading is, the less weight vectors are desired; and the
more users there are, the less weight vectors are desired.
To overcome the problem of limited multiuser diversity
in a small population, [12] devises a codebook-based
OBF (COBF) technique, where the employed unitary
matrix changes with time slot to induce larger and faster
channel fluctuations in the static channel and to provide
further selection diversity to the conventional OBF tech-
nique. Compared with [10], the COBF technique reduces
the required number of mini-time slots, and, since it is
the size of codebook, not the number of mini-time slots,
that determines the amount of supplementary selection
diversity, the system throughput can be increased with-
out limitation from the number of mini-time slots. How-
ever, the receiver should estimate all of channels from it
to the transmitters.
In this paper, a new scheme to generate multiple
weights used in OBF is proposed to deal with the per-
formance degradation due to the less number of users in
the OBF system. In the proposed scheme, only the
equivalent channels at two mini-slots are required to be
estimated, as in the normal OBF. The paper is outlined as
follow: after the introduction of conventional OBF and
MW-OBF in Section 2, the proposed scheme with only
two mini-slots to create more channel candidates via
linearly combining the two effective channels at the re-
ceiver is developed and analyzed in Section 3. Section 4
gives the numerical results to verify the effectiveness of
the proposed scheme from different aspects. The paper is
concluded in Section 5.
2. Conventional OBF and MW-OBF
Assume there are N transmitting antennas at the base
station and one receiving antenna at each user side, the
channel gain vector for the k-th user is
, where hnk(t) (n=1,…,N) is the
channel gain from the n-th antennas to the k-th user at
time t. And the transmitting signal
()
ktH
1
[( ),...,( )]T
kNk
hth t
()
x
t
()
T
tt

α
is multiplied
with the weight vector , where () ()Ve t
1
() N tC

V, diagonal matrix 1( ),...,t( (iagα)td 
())
Nt
 denotes the power allocation on each transmit-
ting antenna, and is random
phase vector applied to the signal, θn(t) are the inde-
pendent random variables uniformly distributed over [0,
2π). In order to preserving the total power,
1()
,..., ]
jt T 
()
N
jt
e

( )[te

e
1()
N
n
nt

1

, where random variable ()t n

varies
from 0 to 1. Then the received signal for the k-th user is,
()
1
()()() ()()
n
N
jt
knnk
n
k
y
ttehtxt




zt
k
()()() ()()
T
kk
tt txtt

eαHz
() ()()
def
k
H
txt t
z (1)
where ()() ()()
T
kk
H
ttt

 
eHt() ()
k
ttVH is the
equivalent channel (i.e., overall channel) for user k, and
be the independent and identically distributed
AWGN.
()
ktz

G. Y. LU ET AL. 191
nk
From (1), when are phase-conjugate with
, that is,
()
ktH
()t()t

e( ())
n
angle ht

 (n=1,…,N),
()
k
H
t
 are the coherent sum of hnk(t), and user k is in its
beamforming configuration state. Thus large channel
gain for user k is obtainable.
In a heavy load system (i.e., the number of active user
are large enough), by varying the weights V(t), there is a
large possibility that some users are in or nearly in their
beamforming configuration states. Using the propor-
tional fair (PF) scheduling algorithm [5], the users with
their overall channel SNR near to the peaks are possibly
scheduled and the system throughput is approaching to
that of the coherent beamforming system.
However, in order to obtain the high throughput by the
opportunistic beamforming, a large number of users must
exist in each cell. In particular, as the number of transmit
antennas of the base station increases, the number of
required users grows rapidly. In [9], the conventional
OBF is generalized by allowing multiple random
weighting vectors at each time slot.
In the multiple weights OBF (MW-OBF) systems,
there exist Q mini-slots in each time slot. During each
mini-slots, respectively, Q known signals multiplied by
Q randomly selected independent weighting vectors
() are transmitted. Then, during the q-th
mini-slot, the overall channel gain is
()
qtV1,...,qQ
,()()() qkq k
H
tt
VHt1,...,q, (2) Q
Each user measures its overall channel gain, ,()
qk
H
t
,
and feeds it back to the base station, then the base station
determines the optimum weighting vector, wopt(t), for
data transmission and the selected user, k*,


*
,
1,.., 1,...,
,()argmaxmax ()
opt
qk
qQk K
kq tRt

 (3)
()
() ()
opt
opt
qt
wt wt (4)
where is the transmitted rate for user k if the
q-th weight vector is used.
,() qk
Rt
3. New Scheme to Generate the Multiple
Weights
By allowing multiple random weighting vectors at each
time slot, the throughput of the MW-OBF scheme is
considerably improves compared to the conventional
OBF since the employing the weights-selective diversity.
However the using of several mini-slots will waste sev-
eral radio resources and, thus, lower the spectrum effi-
ciency.
In this section, a novel multiple weights generation
method is developed by using only two mini-slots at each
time slot. This novel scheme is illustrated with N=2.
Similar to the MW-OBF, two independent random
vectors, and
1()tV()()
Ttt

eα2()tV() ()
Ttt


e, are
used at two mini-slots to create two equivalent channels,
where


12
,diag

 α,

12
,diag


Te
β,
, . And the two equivalent channels
are, respectively,
1
(,e

2)T
e

12
(, )
T
ee

T

e
(1)
,1
()()()()()()
T
eq kkk
H
ttt tt


VHeαHt
(2)
,2
()()()() ()()
T
eq kkk
H
ttttt


VHe βHt
)
k
At the receiver, after the estimation of the two equiva-
lent channels, linearly combining them as (the time vari-
able t is omitted for simplicity in the following),
(1)( 2 )
,, ,12 eq keq keq kkk
HHbH b 
 
VHV H
(
TTTT
kk
bb

 eαHeβHeαeβ
H
(5)
where


12
,
kkk
hhH, the complex value b is the system
parameter to be designed as followed.
Denoting
ˆ() TT bb

γeαe
β
(6)
then ,ˆ()
eq kk
Hb
γ
H
can be viewed as the OBF channel
using weighting factors . As in the conventional
OBF, to preserve the total transmit power, should
be normalized as
ˆ()b
γ
ˆ()b
γ
ˆˆ ()() ()bbb γγγ (7)
Since is the function of parameter b, selecting
different b can resulting in different multiple weighting
vectors using only two mini-slots. Then the newly gener-
ated channel is the linear combination of
and . Suppose that parameter b is selected from a
set with W elements, then W new channel can be gener-
ated.
ˆ()b
γ
)
,k
,eqk
H
(1)
, eq k
H

(2
eq
H

In order not to increase the number of multiple opera-
tions, suppose b is selected from the following set,


1,1,,jj

, with four elements (i.e., W=4). Then six
weight vectors can be generated using only two mini-
slots, thus improving the spectrum efficiency. Compar-
ing with the original MW-OBF, the proposed scheme
needs to estimate the equivalent channels at the two
mini-time slots; however, this is easier than the quantized
codebook scheme in [11] where channel gains from all
users to each antenna must be estimated.
In the proposed scheme, users need feedback its
maximum channel gain and the selected parameter b.
Then transmitter schedules the users and calculating the
current weights, using (6) and (7) based on the b, V1(t)
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G. Y. LU ET AL. 192
and V2(t).
4. Numerical Results
In this section, we present an extensive set of simulations
to verify the effectiveness of the proposed scheme from
different aspects. Firstly, since the achievable MUD gain
in the system is determined by the dynamic range of the
overall channel, which can be described by the probabil-
ity density function (PDF) of the channels, our simula-
tions depict the PDFs for different schemes. Then, if
channels fade very slowly compared to the delay con-
straint of the application so that transmissions cannot
wait until the channel reaches its peak, its QoS cannot be
met. Therefore, the channel fluctuation speed, which can
be described by the correlation function (CF) of the
overall channel, is simulated and given for different
schemes. Finally the average throughput of the system
for different schemes is simulated for comparison, using
both maximum throughput (MAX) scheduling scheme
and the PF scheduling scheme.
In the following simulations, we consider two transmit
antennas at the base station under the Rician channel
with different Rician factor

and average SNR=0dB.
We also suppose the availability of an error-free feed-
back channel from each user to the base station and the
data rate achieved in each time slot is given by the
Shannon limit.
4.1. The PDFs and CFs of the Overall Channels
To compare the performance of increasing the dynamic
range of the equivalent channels, the PDFs of the chan-
nels are plotted in Figure 1 for Rician channel (with
10


) using different schemes, that is, none-OBF,
OBF, normal MW-OBF and the proposed scheme. The
width of the PDF plot shows the dynamic range of the
overall channel. From Figure 1, we can see that the dy-
namic range of the equivalent channels after OBF and
the proposed scheme is much greater than that of the
none-OBF, which ensures the larger obtainable MUD
gain after OBF and the new MW-OBF scheme. Also
comparing the proposed scheme with the normal MW-
OBF, OBF and none-OBF, the probabilities that the
overall channels have large amplitude are in descending
order, which means that the proposed scheme has larger
probability to approach high amplitude and, hence, the
larger MUD gain.
If the maximum throughput scheduling scheme is em-
ployed at the transmitter, the user with the largest chan-
nel gain at a time slot will be scheduled to transmit data
and the distribution of the peaks of the overall channels
will be related to the system throughput directly. Hence,
Figure 2 gives the PDFs of the channels’ peak for
none-OBF, OBF, normal MW-OBF and the proposed
scheme, and 10 active users are in the system in the
simulation. The four vertical bars, from left to right, in-
dicate the mean values for the four schemes, respectively.
The proposed scheme obtains the largest mean values
and dynamic range among the four schemes.
Since the fluctuating speed within the time scale of
interest is another source of the MUD gain, here we use
the normalized correlation function (CF) of the overall
channel as the indicator of the fluctuating speed, which is
illustrated in Figure 3. And the Rician channels with
30


are employed in this simulation. For the same
00.5 11.5 22.5 3
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Channel amplitude
Density
None OBF
OBF
Normal MW-OBF
proposed scheme
Figure 1. Channels PDFs for Rician channel.

G. Y. LU ET AL. 193
00.5 11.5 22.5 3
0
0. 00 5
0. 0 1
0. 01 5
0. 0 2
0. 02 5
0. 0 3
0. 03 5
Channel amplitude
Densi ty
None OBF
OBF
Normal MW-OBF
proposed scheme
Figure 2. PDFs of channels’ peak for Rician channel.
-100 -80 -60-40 -20 020406080 100
0. 84
0. 86
0. 88
0. 9
0. 92
0. 94
0. 96
0. 98
1
Time lag
correlation coefficient
None OBF
OB F
Normal MW-OBF
proposed scheme
Figure 3. The normalized correlation function of the overall channel.
time lag, the larger the correlation coefficient is, the sma-
ller the fluctuation speed is. So the proposed scheme has
less correlation for same time lag, especially for small
time lag compared to none-OBF and normal MW-OBF.
Since the channels are generated via linearly combining
the two equivalent channels, there is correlation among
the channels generated in the proposed scheme. So com-
paring the proposed scheme with OBF, the correlation of
the proposed scheme is larger than that of OBF. For ex-
ample, when the time lag equals 1, the correlation coef-
ficient of none-OBF, OBF, normal MW-OBF and the
proposed scheme are 0.984, 0.98, 0.86 and 0.925, re-
spectively.
From the above simulations, the resulting channels in
the proposed scheme have larger dynamic range, larger
probability to have high amplitudes, and larger fluctuat-
ing rate. We, therefore, can expect that the proposed
scheme can obtain larger MUD gain, which will be illus-
trated in the following simulations.
4.2. The System Average Throughput for
Different Schemes
The simulating parameters are same as those in [10]. The
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G. Y. LU ET AL. 194
05 1015 2025 30 3540 45 50
0.8
1
1.2
1.4
1.6
1.8
2
users number
average throughput,bps/Hz
None OBF
OB F
Normal MW-OBF
proposed scheme
Figure 4. Average throughput using the PF scheme.
051015 20 25 30 3540 4550
0. 8
1
1. 2
1. 4
1. 6
1. 8
2
2. 2
2. 4
2. 6
users numbe
r
average throughput,bps/Hz
None OBF
OB F
Normal MW-OBF
proposed scheme
Figure 5. Average throughput using MAX scheduling scheme.
Rician channel with . Six mini-slots are used to
generate six equivalent channels in MW-OBF, whereas
two mini-slots are used in the proposed scheme to create
two overall channels, and four additional channels are
generated via linearly combining the available two over-
all channels.
10  
Figures 4 and 5 illustrate the average throughput of the
system using PF and MAX scheme for different schemes,
respectively. The results show that, in both scheduling
schemes, the average throughput are improved greatly,
especially when the system with small number of users
in MW-OBF and the proposed scheme. Meantime, the
proposed scheme has larger throughput than MW-OBF.
4.3. Throughput Variation with the Rician Factors
Finally we study the performance variation of different
schemes with the Rician factor , that is, to investigate
the influence of the light of sight (LOS) on the system
throughput. From the Figure 6, it can be seen that with
the increase of


, the throughput for all scheme de-
grades because the throughput rely on the peak values of
the instant overall channel. When the factor in-
creases, the channel fluctuations are reduced and the
peak values of the instant overall channel are reduced, too.
Compared with normal OBF, the proposed scheme can be
improved the throughput, for example, for


=10,

G. Y. LU ET AL. 195
05 10 15
1. 15
1. 2
1. 25
1. 3
1. 35
1. 4
1. 45
1. 5
1. 55
1. 6
Rician facto
r
average throughput,bps/Hz
None OBF
OB F
Normal MW-OBF
proposed scheme
Figure 6. Throughput versus Rician factor of the channel.
more than 10% throughput enhancement can be ob-
tained.
5. Conclusions
A new simple scheme to generate multiple weights used
in opportunistic beamforming (OBF) system is proposed
in this paper to deal with the performance degradation
due to the fewer users in the OBF system. Only two
mini-slots are employed to create effective channels,
while more channel candidates can be obtained via line-
arly combining the two effective channels at the receiver
side, thus increasing the multiuser diversity and the sys-
tem throughputs. The simulation results show that the
throughput can be improved using the proposed scheme.
6. Acknowledgements
This work is supported by Program for New Century
Excellent Talents in University (NCET-08-0891), the
Natural Science Foundation of China under the grant No.
60602053, and the Natural Science Foundation of
Shaanxi Province under the grant No. 2007F02.
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