Int. J. Communications, Network and System Sciences, 2010, 3, 612-619
doi:10.4236/ijcns.2010.37082 Published Online July 2010 (http://www.SciRP.org/journal/ijcns/).
Techniques of Transmitting Beamforming to Control the
Generated Weights
Imen Sfaihi1,2, Noureddine Hamdi2, Ammar Bouallegue2
1The National school of engineering of Tunis (ENIT), El-Manar University, Tunis, Tunisia
2The communication System Laboratory (SysCom Lab) in ENIT, Tunis, Tunisia
Received April 16, 2010; revised May 25, 2010; accepted June 29, 2010
Abstract
In this paper, we consider the limited feedback Transmitting Beamforming for (multiple in single out) MISO
systems. In conventional techniques, all vectors of a large codebook (CB), used for the feedback of the quan-
tized channel state information (CSI), are broadcasted to all users, in a guard period which is followed by
data burst periods. Instead of transmitting a large number of codevectors, we thought to divide the CB into
several sub-codebooks (SC) and the broadcast would be based on the switch between them. Accordingly, a
good performance can be provided while minimizing the required feedback channel capacity applying some
proposed techniques such as “the switched Sub Codebook (SSC)” and “the Fairness SSC (FSSC)”. To mini-
mize the quantization error, we propose two other techniques. The first is based on making Transmit SSC
vectors controlled by a rotation weight (RW) to obtain almost a zero correlation between the SSC vectors
used for the selected spatial channels. The second is based on introducing “the Schmidt algorithm” to con-
struct an orthonormal weights using the generated weights. These two proposed techniques increase the
probability of the selection of the worst case user on his best codevector to make zero the angle between his
couple codevector and channel response. To analyze and validate the performance of these proposed tech-
niques, simulation results are presented.
Keywords: MISO, Beamforming, Limited Feedback, CSI, SSC, FSSC, RW-FSSC, SSC-Schmidt Algorithm
1. Introduction
In the last few years, Multiuser MISO systems also named
as (Space Division Multiple Access) SDMA based on
Limited Feedback Transmitting Opportunistic Beamform-
ing (OBF) have been a lot of interests in recent research
studies. The goal is to provide high system spectral effi-
ciency [1] while reducing the complexity. Due to the con-
straint of narrowband of the feedback channel, transmitting
OBF on the broadcast channel with limited feedback has
been widely studied in the literature as [2-4] and references
therein.
Moreover, transmitting OBF is provided in the literature
as a more practical design that ameliorates the performance
of SDMA [5] and [6]. Each user selects the correspondent
beamformer from the Beamforming CB. Therefore, we
found many techniques to formulate the Beamforming CB
vectors, for example: random orthonormal beamforming
CB as proposed in [7-9] or transmitting beamforming
based on grassmannian line packing as described in [10]
and [11].
This SDMA design can be combined with multiuser
scheduling [5] and [6]. Therefore BS uses an algorithm to
select the best pair index of user-CB vector that increases
the system capacity such as: Max-rate [4] or sub-optimal
algorithms as proposed in [12] where a selection is based
on the best pair of user-beam vectors. In [13], an algorithm
is proposed, known as semi-orthogonal user selection
scheme. This algorithm is based on upper bounded tech-
niques where the value of the SINR and the value of the
error quantization are compared to predefined thresholds
which are defined in [2-4].
Up to the moment, the generated CB vectors are trans-
mitted to all users and then select the best pair CB vec-
tor-user. After investigating these studies, we thought to
reduce the complexity by introducing new techniques
based on a score that measure frequency of access by using
vectors in SC. Moreover, the second formulation of OBF
CB vectors named Grassmanian method is the most prac-
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I. SFAIHI ET AL.
613
tical and near to the reality but the components of the OBF
CB vectors are with non zero correlation. Consequently,
the correlation between OBF CB vectors and the selected
user channel is not null. Then, the interference level would
be increased and then the system throughput decreases.
Therefore, we thought to minimize the probability to make
errors.
In this paper, we propose four new techniques of trans-
mitting OBF at which the first technique is based on SSC
applied to the Max-Rate scheduler with limited feedback
system. The components of CB composed by N = 2B vec-
tors would be divided into nD SC. In [2-4], the N compo-
nents of the CB vectors are transmitted to all users at each
time slot. This number is reduced in this proposal to N/nD
components of the CB that would be transmitted to all us-
ers. Then, we minimize the complexity and respect the
bandwidth of the feedback channel.
Besides, for the goal of providing fairness among users,
we propose the second technique that is based on fairness
SSC (FSSC). This proposal is the continuity of the SSC
technique when we investigate and improve the SSC by
introducing the proportional fair principle (PF) to switch
the specific SC.
Moreover, we intend to meet the performance of the
FSSC and to reduce the interference level at user receivers.
Whereby, the correlation between OBF CB vectors is
compared to predefined thresholds to make the specific
rotation that minimize this correlation. Then, we put the
threshold to a given value and at each transmission the
value of correlation is controlled and compared to this
threshold.
After this, we thought to use from the beginning an or-
thonormal weights in transmitting OBF in order to reduce
the interference level at user receivers. Whereby, the cor-
relation between OBF CB vectors is converge to zeros by
applying the Schmidt algorithm and construct the new or-
thonormal OBF weights that give a zeros correlation be-
tween the generated OBF CB vectors.
The remainder of this paper is organized as follows. Sec-
tion 2 describes the system and channel models. In Section
3, we present an overview of the design of different CB
proposed in the literature. In section 4, we present an over-
view of the CSI quantization. In section 5, we describe the
different steps of the proposed techniques of transmitting
OBF: SSC, FSSC, RW-FSSC and SSC-Schmidt algorithm.
In section 6, we present the Max-Rate scheduler. In section
7, we analyze the system capacity of the proposed tech-
niques and give the closed form of capacity. In section 8,
we present a selection of simulation results.
2. System Model
We consider a MISO system with Mt antennas at the base
station (BS) and K users when each is equipped with one
receive antenna. Each user has her own rate βk. It is as-
sumed that slow power control is employed to equally
share the total transmitted power Pt on all transmit an-
tennas at the BS. Users symbols are loaded on transmit
antennas using Beamforming, i.e. the BS assigns a
Beamforming vector to each of up to Mt selected active
users.
The Beamforming vectors

1
t
M
ii
W are obtained using
a generated orthogonal unitary beamforming vectors as
defined in [7-9] or using grassmannian line packing
codebook as described in [2-4]. To solve the problem of
limited resources allocated to the feedback channel, users
estimate their CSI and feedback them in a quantized
form on B bits to the BS through an uplink limited ca-
pacity feedback (LCFB) channel. We denote by N = 2B
the number of CB vectors which is defined by


12
,,..., N CBW WW. At each time slot, the number of
components that would be used in optimal side of the
transmission is equal to the number of transmit antennas
Mt.
It is assumed that transmit signals experience path loss,
log-normal shadow fading, and multi-path fading. The
CSI is measured by the vectors which represent the short
term fading CSI on all branches from the BS to the kth
user assumed to be constant during a time slot. Accord-
ing to the slow power control, 1) each entry of the vector
hk is an independent and identically distributed complex
Gaussian random variable CN(0; 1) representing the short
term fading; 2) the CSI experiences flat fading during
each time slot, and varies independently over time slots.
We denote by hk(t) the channel vectors, Wi(t)
the
1
t
M
1

t
MCB, s(t) the transmitted symbol, nk(t)
the additive white Gaussian noise (AWGN) vector with
distribution CN(0; N0/2) for each element, and yk(t) the
received signal. Then, the received signal for the consid-
ered multi-user MISO system in the time slot t is repre-
sented by
1
t
M
1
()()() ()()
t
M
H t
kkii
i
t
P
yth tWtstnt
M

k
(1)
According to Equation (1), the received signal for user
k when using the Wi CB vector can be:
,
1.. ,
()
t
HH tt
kiki ikjjk
jMji
tt
PP
ythWshWs n
MM



 (2)
Hence, the corresponding expression of the signal to
interference plus noise ratio SINR for the kth user and the
ith CB vector is expressed as follows:
2
,2
1.. ,
t
H
ki
ki
Ht
kj
jMji t
hw
SINR
M
hw P




(3)
To evaluate the sum capacity, we need the statistical
distribution of the .
,ki
SINR

I. SFAIHI ET AL.
614
2.1. The Statistical Distribution of the Signal
to Interference Plus Noise Ratio
To simplify the Equation (3), we let 2
H
iki
ahw; i =
1, …, Mt. The SINR on the ith CB vector for user k given
in the Equation (3) can be rewritten as
,
1.. ,
t
i
ki
t
j
jMji t
a
SINR
M
aP




(4)
Consequently,
,1
11
t
i
ki M
i
t
mm
mmi t
a
SINR
M
aa
P





(5)
Then,
,
i
ki
t
ii
t
a
SINR
M
bc P


(6)
where and .
1
1
i
im
m
ba



1
t
M
im
mi
ca


Although, the random variables ai are of independent
2

distribution with two degrees of freedom. Note that
the with different k are independent. Then the
cumulative distribution function (CDF) of the largest
SINR for user k denoted can be calculated in
terms of the joint probability density function (PDF) of
the largest one of Mt i.i.d
,ki
SINR
,ki
SINR
2

random variables, denoted
by

,, ,,
iii
bac

f
xyz, as [14]




,
1
00 0
,, ,,




 
tt
t
ki
iii
MiwM
xzw
P i
SINR
bac
Fx 
f
wyz dydzdw
(7)
After taking derivative with respect to x, the PDF of
is given by
ik
SINR ,

 
,
/1
00
,, ,,









t
ki
iii
Miwi
SINR
t
bac t
Fx
M


f
wxzwz dzdw
P
(8)
It was further shown in [14] that the joint PDF

,, ,,
iii
bac

f
xyz is available in closed form, after some
modifications as given by
 
 
 
  




2
,,
1
0
1
1
,, 12! 1!
1
1
0;1 ;.
iii
t
t
i
t
bac t
t
wyz
Mi jMi
t
j
t
wi y
M
fxyzM
iiMi
eUwiy
Mi zjy Uzjy
j
ywiyzMiy
where U(.) denoted the Heaviside unit step function.
Remark:
If the largest SINR of the user k is the first component
of the CB vectors then i = 1 and the correspondent CDF
is as given in [15]
 
,11 ,
00 ,




 t
t
k
M
xz
P
SINRa b
F
xfyzdydz (10)
And

,11 ,
00 ,


 
 
 

 


k
tt
SINRa b
tt
MM
fxzfx zz
PP
dz
(11)
It was further shown in [15] that the joint PDF is avail-
able in closed form, as given by


  
1,
12
0
,2!
11
t
t
yz t
ab
t
MjM
t
j
M
fyze
M
Mzjy Uzjy
j







 



(12)
After replacing this expression in (24), the PDF of the
largest SINR of the user k () can be written as
1,k
SINR







,1
1
00
21
1
1!!
1
1
t
k
tt
t
M
tt t
SINR
jtt
MM
xz xP
t
t
t
t
MM M
f
xz
MjjP
M
jx zjxe
P
M
Ujxzjxdz
P
 

 




 

 







(13)
3. Conventional Codebook Design
,




 







 


 




 

(9)
According to the considered system model, we can give
a description of the CB vectors design. At each time slot,
Mt symbols of users are multiplied by Mt random or-
thonormal vectors wi 1

t
M for i = 1, ..., M
t. Where
wi’s are generated according to an isotropic distribution
[9] and [10] or are computed according to formulation
described in [7]. This random orthonormal vectors are
used to define a random CB. This CB vectors are gener-
ated with zero correlation which is near to the reality
since it require that the chanal is known at both transmit-
ter and receiver.
Moreover, we found in some prior work such as in [2]
and [3] the term of CB vectors with none zero correlation
such as grassmannian CB vectors defined in [8]. Relying
to previous approaches a good beamforming is specifi-
cally using a grassmannian. Therefore, in this work, we
can use this technique to generate CB vectors with none
zero correlation.

I. SFAIHI ET AL.
615
4. CSI Quantization
Due to constraint of the bandwidth of feedback channel,
each user just feeds back its maximum SINR quantized in
B bits and the index of the corresponding codebook vec-
tor. In [2] and [3], the random vector quantization (RVQ)
method is applied for quantizing the CSI. The CB vec-
tors

1
t
M
ii
W are obtained using a generated orthogonal
unitary beamforming vectors as defined in [7-9] or using
grassmannian line packing codebook as described in
[2-4,12]. At each time slot, each user identified by k se-
lect his 'best' vector from the CB. For that, a quantization
CB vector is selected as:

1
arg max


Mt
jj
H
s
k
W
ih
j
W (14)
The quantization error is expressed as:

 
2
sin ,
s
i
WH  

(15)
In the literature, it is often assumed for simplicity that
the feedback is without errors. Then, the quantization
error δ should be converges to zero.
5. The Proposed Techniques of
Transmitting OBF
5.1. Design of SSC
For transmitting OBF, the CB vectors are used randomly
such as in [2-4]. At each time slot, all of N components
of CB are transmitted to all users. This is clearer in the
previous step of quantization of CSI. Therefore, we pro-
pose an idea that based on dividing the N components of
CB vectors into nD SC to minimize the number of com-
ponents to be transmitted to all users at each time slot.
Then, we define how to switch to a given SC at each
time slot to increase the system capacity and give equal
opportunity among users to the channel accesses. The
switching is based by investigating user scores based on
the historic use of each SC at a number of previous time
slots. This would provide fairness among users.
The switching step is defined as follow:
1) The initialized matrix ((, )
D
Scorezeros Kn) is
updated at each time slot and would be used in the fol-
lowing time slot:
(,) (,)1
ss ss
ScorekiScoreki (16)
where ks is the index of each user among the Mt served
users and is is the index of each CB vector among the Mt
selected CB vectors of the SSC satisfying Equation (14).
2) At each time slot, we search the index of the user
that has the worst capacity to give fairness among users


*min
k
kC (17)
3) After, we search the index of the switched SC vec-
tor that satisfies the following expression



*
max ,
i
iScorek* i (18)
4) Finally, we update the matrix of Scores and com-
pute the system capacity.
The most obvious benefit is to take consideration of
the historic use of the CB vectors that will be represented
by a score based on the user access frequency.
5.2. Design of FSSC
5.2.1. Fairness Criteria
In SSC, the idea to give fairness is derived and the se-
lected score is that has the user with the worst capacity



min
kK C . If we apply this, we can assume that the
number of users to have the worst capacity can be large.
Then, the selected user is chosen randomly and the same
user can be chosen many times. Therefore, in our pro-
posal design, the most obvious goal is to give fairness
among users. And accordingly, we thought to an idea in
basis on max-min schedulers introduced in previous
work such as proportional fairness in [7] when all users
have the equal chance to be served.
Now, we can suppose that each user has her own rate
βk and we propose in this technique to select the index of
the SC that has the minimum of the user’s capacity’s
divided by the proportional componentk

when k

is
expressed in the following sub section in (20). Then, we
can assume that to select user with using
should be given most chance to users who can be served
at the following time slot. Accordingly, the probability to
choose the same user many times is minimized and this
probability converges to zero. Moreover and according
to the most aim of our proposal, we can talk about the
index of Jain for fairness defined in [16] and expressed
as follows

/
kk


mi
kK

nC

2
1
2
1
()
()
K
k
k
K
k
k
Ct
j
K
Ct








(19)
where is the system capacity of kth user and K is
the total number of users. We are going to present and to
discuss this term in the simulation results to validate our
scheme and their results.
()
k
Ct
5.2.2. FSSC Algorithm
The fairness switching algorithm is described as follows:
1) Initialization:

I. SFAIHI ET AL.
616
a) Let βk the rate of user k.
b) Let the proportional components k as

1
kK
k
i
i




 (20)
c) The score matrix is defined in the previous subsec-
tion.
2) The Score matrix is updated at each time slot and
should be used in the following time slot (the same in
(16)).
3) At each time slot, we search the index of the user
that has the worst of the capacity divided by αk to give
fairness among users
*min k
kK k
C
k












(21)
4) After, we search the index of the fairness switched
SC vector that satisfies the following expression



*
max ,
i
iScorek* i
(22)
5) Finally, we update the matrix of Scores and com-
pute the system capacity.
The most obvious benefit is the use of an access con-
trol based on PF constraint to provide the fairness access
channel among users.
5.3. Design of RW-FSSC
In the literature, it is often assumed for simplicity that the
feedback is without errors. Then, the quantization error
expressed in (15) should converge to zero i-e the prob-
ability to make errors should be converge to zero. But in
reality, the use of grassmanian method to generate the
OBF CB vectors when the components of OBF CB vec-
tors are with not zero correlation should be make errors.
Consequently, the correlation between OBF CB vectors
using to select the user channel is not null. Therefore, we
thought to control this value, compared with the value of
threshold and make the rotation weight (RW).
The RW is consisting of the applying a rotation on the
best codevector to make zero the angle between the cou-
ple codevector and channel response. Then, the RW in-
creases the probability of the selection of the worst case
user and if this user is selected he would be assigned the
best possible symbol rate.
The RW is consisting of three cases at which the value
of the angle between the couple codevector and channel
response defined as θ is compared with a specific values
of thresholds that described as follows:
1) Let θmax the threshold: it is the maximum angle be-
tween the couple codevector and channel response and
Wr the OBF codebook vectors after rotation.
2) Let

,
i
WH  
3) if max
4

  Then ri
WW

4) else if max
max 4

  Then
'max
exp 2

 


ri
WW j

5) else if max
max
4


  Then
'max
exp 2

 


ri
WW j

6) else if max

  Then SNR = 0
5.4. Design of the Method that Introduce the
Schmidt Algorithm
5.4.1. Schmidt Algorithm
Theorem: (Process of Gram-Schmidt)
Let {a1, ..., aN} a family of vectors linearly independ-
ent.
Then it exists a family of orthonormal vectors {q1, ...,
qN} when for all i = {1, ..., N}, we have
V ect {a1, ..., ai} = V ect {q1, ..., qi}.
According to the process of Gram-Schmidt and to the
SSC technique, we thought to introduce the Schmidt al-
gorithm to construct a new orthonormal OBF CB vectors
using the generated OBF CB vectors using one of the
conventional CB designs and that used in the following
step to select the user channel.
5.4.2. Steps of this Proposal Technique
On the first hand, we use one of the conventional CB
design such as random orthonormal CB that have an iso-
tropic distribution or the grassmanian method.
1) We denote by

1
N
ii
W

the generated OBF CB vec-
tors
2) We apply the Schmidt algorithm and we obtain the
new orthonormal OBF CB vectors W1
And after, we use this new orthonormal OBF CB vec-
tors W1 to quantize the CSI and use the SSC transmit
technique.
6. Max-Rate Scheduling
To maximize the system capacity, the technique of
scheduling to share resources among active users is stu-
died and applied. In this section, we describe the main
idea of the Max-rate scheduling to select users. Accord-
ingly, the selected Mt users to be served at each time slot
experiences peak level signal to interference plus noise
ratio (SINR) expressed in (3). This can be expressed as:

,
arg max


s
s
k
kSIN
ki
R (23)
where is is the index of the CB vectors selecting with the
proposed technique of transmitting beamforming at the

I. SFAIHI ET AL.
617
R
k
correspondent time slot. We use the Max-rate scheduling
because it gives the optimal performance and the aim is
to investigate the resources with the most efficiently.
7. Capacity Analysis
According to the step of [14], the exact sum capacity
expression for such scheme under consideration can be
written as

2
0log1()



tSIN
CMxfxdx
(24)
Based on the mode of assignment of one of the pro-
posed techniques, we can assume that the largest SINR of
different users are i.i.d and the correspondent PDF is
given by
,1 ,1
1
()() ()
k
K
SINR SINRSINR
f
xKFxfx

 (25)
where and are the CDF and the PDF of
the largest SINR for a particular user, given in (11) and
(24) respectively. Then, the capacity can be written as
,1k
SINR
F,1k
SINR
f

,1 ,1
1
2
0log 1()()


kk
K
tSINRSINR
CMxKFx fxdx (26)
8. Simulation Results
In this section, the performances of the Max-rate sched-
uler with limited feedback are evaluated using the SSC,
FSSC, the RW-FSSC and SSC-Schmidt algorithm to
control the generated CB (for grassmannian CB vectors)
[8] in terms of system capacity. The number of active
users K used for these simulations varies from 1 to 30;
the number of SC nD is equal to 2, the time moving win-
dow T is of 200 and the feedback bits B is of 3.
In Figure 1, we compare the sum capacity perform-
ances of the “SSC” and the RBF techniques (random
Max-rate scheme with LF such as in [2-5]).
Figure 2 illustrates the performance of the FSSC de-
sign in terms of system capacity. These results are com-
pared to the SSC technique.
Figure 3 plots the index of jain for fairness of these
two techniques versus the number of active users K when
the value of SNR = 20 dB. This figure shows the fairness
degree of the capacity in FSSC and SSC techniques. We
can conclude that FSSC and SSC provide respectively a
quasi optimal (~1) and near optimal fair degree. This is
explained by the number of users that have the same
worst capacity can increase with the number of users.
Figure 4 shows the performance of the RW-FSSC in
terms of system capacity. These results are compared to
the FSSC technique. Figure 5 shows the simulation re-
sults of the system capacity of the SSC applying the
Schmidt algorithm and SSC versus the number of active
users.
In Figures 1, 2, 4 and 5, the simulation results of the
510 15 20 253035 40 45 50
0
5
10
15
Active users with n
D
=2, M
t
=2 and B=3
Normalized System Capacity(bits/sec/Hz)
SNR=10dB
SNR=5dB
S NR=0dB
RBF Scheme
SSC Scheme
Figure 1. System capacity of the SSC and OBF techniques
vs. number of active users.
51015 202530
0
2
4
6
8
10
12
14
16
18
20
Active users with M
t
=2, B=3 and n
D
=2
Normalized Capacity System(bits/sec/Hz)
SSC scheme
FSSC scheme
SNR=20dB
SNR=1 0dB
SNR=0dB
Figure 2. System capacity of the FSSC technique vs. num-
ber of active users.
51015 20 2530
0
0.2
0.4
0.6
0.8
1
Active users with M
t
=2, B=3 and n
D
=2
index of jain for fairness
Fairness switched Sub-Codebook
Switched Sub-codebook
Figure 3. The index of jain for fairness.

I. SFAIHI ET AL.
618
24681012 14 16 1820
0
5
10
15
20
Active users
Normalized System Throughput(bits/sec/Hz)
SNR=20dB
SNR=10dB
SNR=0dB
Scheme with RW
Scheme without RW
Figure 4. System capacity of RW-FSSC vs. number of ac-
tive users with variation of the average SNR.
510 1520 25 30
0
2
4
6
8
10
12
14
16
18
20
Active users
Normalized System Throughput(bits/sec/Hz)
Switched only
Switched + Schmidt algorithm
SNR=30dB
SNR=10dB
Figure 5. System capacity of SSC-Schmidt algorithm vs.
number of active users with variation of the average SNR.
system capacity of the proposed techniques are plotted
with different values of average SNR. Since, we can see
that the system capacity applying the proposed tech-
niques is nearly independent of the number of active us-
ers K. As can be seen from these figures, the difference
between the curves of the transmit techniques is very
small. In addition to that, the results of the proposed
techniques are in good concordance and the system ca-
pacity grows as the average SNR increases.
9. Conclusions
The system capacity of limited feedback using OBF CB
vectors and applying one of the new proposed techniques
for Max-rate technique has been analyzed in this paper to
deal with the performance of the coherent transmitting
OBF. The transmit antenna are assigned to different up
to Mt users at each time slot to increase system capacity.
According to the simulation results, we can conclude
that the FSSC technique for Max-rate give fairness
among users for a lot of number of users and a good
performance. Moreover, we can conclude that the tech-
niques to control the generated weights applied the FSSC
for Max-rate minimize the probability to make error and
give fairness among a number of users and a good per-
formance while reducing the complexity of generating
the OBF CB vectors as SSC technique. Accordingly, we
can see that the system capacity can be improved using
our proposed techniques.
1
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