Communications and Network, 2013, 5, 204-210
http://dx.doi.org/10.4236/cn.2013.53B2039 Published Online September 2013 (http://www.scirp.org/journal/cn)
A Resource Allocation Algorithm of Physical-Layer
Security for OFDMA System under Non-ideal Condition
Xiao-min Ran, You-quan Mo, Yu-lei Chen
National Digital Switching System Engineering & Technological Research Center, Zhengzhou, China
Received June, 2013
ABSTRACT
In this paper, a resource allocation scheme based on physical layer security under non-ideal condition for OFDMA sys-
tem is introduced. Firstly, the program uses the information security constru cting an OFDMA system Wiretap Channel
Model under non-id eal condition. Based on this model, artificial noise is generated for secure communications combat-
ting passive multiple eavesdroppers. In order to maximize the average secrecy outage capacity without channel state
information of eavesdroppers, we use dual decomposition method to implement subcarriers and power allocation in
joint optimization. Simulation results show that the average secrecy outage capacity can achieve 7.81 bit/s/Hz while
secrecy outage probability is 0.05 with 50 dB mtransmitpower and 64 sub-carrier for 8 authorized users.
Keywords: OFDMA System; Physical Layer Security; Secrecy Outage Capacity; Resource Allocation
1. Introduction
With many good characteristics, OFDM (Orthogonal
Frequency Division Multiplex ing) not only has the resis-
tance to multipath fading, but also the allocation of re-
sources can be flexible depending on the constraints. It
has been chosen as an important candidate technology by
Wireless communication standards such as 3GPP LTE,
IEEE802.16WiMAX and IEEE802.22 WRAN.
In recent years, there are many researches about physi-
cal layer security of OFDM system, unlike traditional
channel encryption method, the new method takes ad-
vantages of the channel characteristics differences be-
tween the communicating parties, it achieves the secure
transmission in wireless signal while ensure the commu-
nication quality of a legitimate user by increasing the
difficulty of the eavesdropper to intercept or restore the
signal and measure the security of the system by Confi-
dential capacity[1,2]. Csiszar[3]proposes a differential
encoding method to lower the probability of interception
in OFDM system; Renna[4,5] analyze the Confidential
capacity of OFDM system with different receiver. Jor-
swieck[6] and Li[7] discuss how to deal with the alloca-
tion of power to maximize the confidential rate in sin-
gle-user and multi-user OFDM system.
Existing literatures about physical resource allocation
in multi-user OFDMA(Orthogonal Frequency Division
Multiple Access) system are always based on the ideal
condition that authorized user`s channel quality is better
than the eavesdropper`s, and lack of attention on the
physical resource allocation methods to obtain higher
safety in bad communication environment. In this paper,
an algorithm about OFDMA physical resource allocation
for non-ideal environment is proposed. First, we con-
struct an OFDMA network security model. Then, the
concept of system average confiden tial capacity is putted
forward as an index to measure the safety of system
when the state of wiretap channel is unknown. Finally,
the joint optimization distribution of subcarrier and
power is realized by dual decomposition to maximize the
average confidential interrup t capacity.
2. OFDMA System Network Model
Assume that there are
K
authorized users,
E
N eaves-
droppers in OFDM network. Each eavesdropper shares
the received message (equally, one eavesdropper with
E
N
N
antenna), as shown in Figure 1. Because there are
many eavesdroppers in the system, In order to guarantee
that the system can secure communication, the sender
need to use multi-antenna, and the number of antenna
AE
. Assume that each authorized user uses single
antenna when receiving. And there are
N
F
N subcarriers
in OFDMA system, consider that all the carrier channels
are slow fading channels, in a short time the channel im-
pulse response remains unchanged. The signal that the


,K1,kth user and eavesdropper received on the


,1,
F
ith Ncarrier can be written as
,,,kiki kii
yn

hx (1)
C
opyright © 2013 SciRes. CN

X.-M. RAN ET AL. 205
Figure 1. OFDMA network model.
,,,
E
iEiki
i 
y
Gx e (2)
where is the signal vector sent by author-
ized user k on carrier i,
1
,A
N
ki C
x
N
M
C

is a matrix.
,
NM
1
A
N
ki is the CSI vector of the sender and the au-
thorized user k on the i-thcarrier, ,
C h
E
A
N
N
C 
Ei is the
CSI matrix of the sender and eavesdropper on the i-th
carrier, ,ki， ,
G
h
E
icontains the influence of path loss
and the multipath fading. i is the AWGN (Additive
White Gaussian Noise) of legitimate user k on the i-th
carrier, andit submitsGaussian distribution with mean of
0 and variance of 0, i is the AWGN vector
of eavesdropper on i-th carrier, and it obeys Gaussian
distribution with mean of 0 and variance of . Nor-
malize all the noise variance at the receiver, set 0
Gn
1
E
N
C Ne
0
N
N1

.
Assume that the sender can fully know the CSI(Channel
State Information) of each authorized user, namely ,ki
is already known. Because of the passive interception of
the eavesdropper, the sender can't learn the CSI of the
eavesdrop per, name l y
h
,
E
i
G is unknown, where

1, ,
F
iN, and .

1,kK
Giving that the sender can't acquire the accurate num-
ber and position of eavesd ro p pers，so let us image a poor
situation, set . In order to achieve safe
transmission while the sender and eavesdropper are very
near, we can use artificial noise[8] to guarantee security,
namely, let the sending signal ,ki contains the two
parts of useful signa l and noise signal
1
EA
NN
,ki
ux
,ki
v
,,, ,kiki kikiki
uxb Wv
,
(3)
where, ,ki is an independent and identically distributed
complex Gaussian random variable with the variance of
v
2
i

, ,ki is the orthogonal basis on the null space of
,ki, namely ,
and ,,,
W
h,,ki kiki
WW I0
kiki ki

hWv . ,ki
is a unit matrix. In order to maxi-
mize the signal-to-noise ratio of authorized user k, we
can use the method of specific maximum beam forming,
so the received signal can be written as
I

11
AA
N N
,,,,kikiki kii
yuhb n
(4)
,,,,,,,
E
iEikikiEikiki
u
Assume that the total transmitting power of the k-th
user on the i-th carrier is , the power of useful signal
is ,ki, so ,ki
P
p


2
,, 1 ki ki
Pp A Ni

 . ,ki

is defined as
the power ratio of useful signal, then there will be
,,kiki ki
p, P


(6)


,,
21
1
ki ki
iA
P
N



 (7)
According to the CSI between the sender and the au-
thorized user, we can achieve the channel capacity of
user k on carrier i

2
,2 ,,
log 1
kiki ki
Cph
(8)
while the channel capacity of eavesdropper on carrier i
can be indicated as
,
,2
2
log 1
ki i i
Ei
iii
p
C



 
ψψ
IΦΦ (9)
where, ,,iEiki

ψGb, ,,iEiki

ΦGW . Thus, (9) can be
converted to:


,
,22
1
,
2,
log
1
log11
ki i i
Ei
iii
ki Aiii i
ki
p
C
N















ψψ
IΦΦ
ψΦΦ ψ
(10)
So the secrecy capacity of user k achieved on carrier i
can be shown as



,,,
2
2,,
1
,
2,
log1
1
log 11
s
kiki Ei
ki ki
ki Aiiii
ki
CCC
ph
N






 
















ψΦΦ ψ
(11)
3. The Resource Allocation Algorithm of
Physical Layer Security in Non-ideal
Environment
The concept of average confidential outage capacity is
the general average security bits from the sender to all
the authorized users per secondindicate, it can be de-
scribed as
,,,,, ,
11
Pr |
F
N
Kss
outki ikikkiEiki
ki
CRRCC 






 h (12)
Among them, ,ki

is the indication of sub-carrier’s
allocation, when user k employ in sub-carrier i , ,=1
ki

,
otherwise ,=0
ki

. In order to make the security of the
system reaching a maximum, in the conditions of total
i

y
GbGWv e (5)

X.-M. RAN ET AL.
206
power constraint, the optimization problem of making
the confidential outage capacity maximization is estab-
lished. Assuming that th e authorized users can meet their
needs of confidential outage probability, so the mathe-
matical problems are as follows

,,,,,,
,,
max, ,
ki ki kioutki ki ki
pCp  
Subject,
,,,,
Pr|,, ,
s
ikkiEikik
RCC ki 



h

,, ,
11
,,
1
,
,0,
1,0,1,, ,
01,,.
F
N
K
ki kitki
ki
K
ki ki
k
ki
PPandPk i
andk i
ki










,,
(13)
The essence of the problem is to complete the maxi-
mization of system performance through rational alloca-
tion of each user’s carrier and power, using channel state
information and the added artificial noise which have
been achieved. (13) Indicates that, constraint conditions
that exist in the form of probability. In order to simplify
the problem, the constraint cond itions can be transformed
into objective function. Then it can be solved through
dual decompositi on method.
3.1. The Transformation of Optimization
Problem
In order to make the design of resource allocation algo-
rithm not be effected by the CSI of eavesdropper’
schannel, the following lemma can be given firstly.
Lemma：For a given security outage probability k

,
the security rate of authorized user k in carrier i is
equivalent to the following form

 
2
2,,,
1
,,
2,
log 1
1
log 11
ki kiki
s
ik ki Azk
ki
Ph
RNF























(14)
Among them,

1
z
k
F

 is the inverse function of
 
1
1
1
A
nn
0
1
E
A
N
N
n
z
k
z
N
Fz z






C
, the proof is given in Appen-
dix. The lemma uses the interference characteristics of
multiple antenna, computes the security outage probabil-
ity by tapping channel information of the probability
distribution functio n to express the security rate.
When is fixed, in order to make the security out-
,ki
P
age capacity achieve the maximum, let ,
,
0
s
ik
ki
R



. Then
the optimum solution of ,ki

is:
 
 
 
21
,,
2
,, 21
,,
,21
,,
1
11
11
Akikizk
ki ki
ki kiAzk
ki
ki kiAzk
NPhF
Ph PhN F
PhNF











 







(15)
With the increase of transmission power, 2
,,ki ki
Ph
will continue growing, but

 
1
1
Azk
NF


 maintain a
fixed value of constant. So (15) can be predigested as




 
1
,11
111
111
Azk
ki
Azk Azk
NF
NF NF








 (16)
Combining (16) with (14)

,2,,2,
log1 log1
s
ikki kiki
RP






 (17)
where




 
2
,
,1
1
,1
1
1
11
ki
ki
Azk
Azk
ki
Azk
h
NF
NF
NF






 




，
So when the SNR is high, the SINR of eavesdropper is
a fixed value, and is never influenced by the transmitted
power .
t
Taking (17) into (13), the optimization problem is still
a NP-hard problem. Using the method of literature[10] to
relax ,ki
P

, the problem can be converted to a convex
optimization problems. Literature[11]indicates that when
the system’s sub-carriers are enough, the error caused by
relaxation will be close to zero. In order to simply (13)
further, the constraint conditions of k

can narrow the
range only an equality. Let


,ki ,
0, 1  ,,kiki ki
PP
,


,
then (13) can be converted to the below

,, ,,
,11
max 1F
ki ki
N
K
s
kkii
Pki
R






k


Subject,
,,
11 ,and0,,,
F
N
K
ki tki
ki
PPPki


 

,,
1
1,0,1 ,,,
K
ki ki
k
andk i 


 (18)
wher e . Because
,,,
,, /
|kiki ki
ss
ikik PP
RR
 


,,
,|kiki
s
ik
P
P
R is a
concave function to , and
,ki
P
,,
s
ki ik
R 
 is the perspective
function of ,,
,|kiki
s
ik
P
P
R
,
, according to literature[12],
,
s
ki ik
R 
 is also aconcave function of , then(18) is a
,ki
P

convex optimization problem. For the optimization vari-

X.-M. RAN ET AL. 207
ables of above-mentioned problem are coupling, the dual
decomposition method of convex optimization theory
can be used to solve the problem.
3.2. The Optimization Problem Solved by Dual
Method
3.2.1. Dual Transformation and Decomposition
Dual decomposition method [13] can decompose the
original complex optimization problem into main prob-
lem and sub-problem which can be solved easily. We can
get the final results of the original problem by solving th e
main problem and the sub-problem. At first, structure the
Lagrangiandual of (18).


12, ,
11
1,
11
2,
11
,, ,1
1
F
F
F
N
K
s
kki
ki
NK
ik
ik
N
K
Tk
ki
ik
i
i
L
R
PP
 



















ρP
(19)
where is the sum of allocation instruction of each
carrier, P is the sum of allocation power of each carrier.
ρ
111121
,,,F
N
 




, 2

is the Lagrange multiplier
of each constraint condition. The dual problem is to solve
the original problem through tight upper bound. There-
fore, the dual problem of (18) can be expressed as

12 12
, ,
min max,,,L





ρpρp. Solving the dual problem is di-
vided into two layers, a main problem to be solved at a
higher level; K independent sub-problems to be solved at
a lower level, each sub-problem correspond to a user.
Therefore, the sub - pr oblem k can be described as


,, ,, 12,
,1
max 1FF
ki ki
NN
s
kkiiki
Pii
RP

 





1
ki

Subject


,,
0,1 0,
ki ki
and Pi  

2
T
P

(21)
The main problem can be described as

12 121 2
,11
1
min ,
Subject 0,and0.
F
i
N
K
ki
ki
i
G
i

 



 


(22)
where is the optimal value of the objective func-
tion in sub-problem, which can be obtained by solving-
sub-problem.

k
G
3.2.2. The Solving of Main Problem and
Sub-problems
When the Lagrange coefficient in (21) is a fixed value,
K-T conditions can deduce the optimal power allocation
of user k on carrier i.

 
1
,,,, 2
2,
1
1
ln 2
Fzk
k
kiki kiki
ki
NF
PP h













 





(23)
According the above formula， the optimal power
allocation can be observed as water injection in a multi-
horizontal surface, different users have different water
lines

2
1
ln 2
k


. To obtain the allocating power of each user’s
sub-carrier, we should find the partial derivative of ,ki

using 21. According to the K-T conditions
,
,1 ,
,
00 1
01
ki
kki iki
ki
GA







 (24)
where






,
,,
2,, 2,
,,
1
log 1log 1ln 21
ki k
ki ki
ki kiki
ki ki
A
P
PP








 



it can be seen from the above formula that when
,
0
ki 1


, there is ,1ki i A


,ki
, in order to be able to get
the integer value of

, ,ki

can be defined as
,1
,
1,
0,
ki i
ki
A
otherwise





 (25)
For each of the sub-carriers there is no more than one
nonzero ,kn

. The above formula is equivalent to allo-
cate sub-carrier i to the user of the sub-carrier which has
the maximum ,ki
A
. The optimal power allocation of the
carrier can be written as
,, ,
*,
1, max0
0,
ki kjkj
j
ki
AAandA
otherwise








(26)
After the solving of the sub-problem, we need to solve
the 1i

, 2

in main problem. From (25) it can be de-
duced that 1i

in the optimal solution can take any one
value between the largest and the second largest ,ki
A
of
sub-carrier i. The value of 2

can be obtained by sub-
gradientiterative algorithm
 
22 ,
11
1F
N
K
tki
ki
tttPP 






 

 



  (27)
where t is the number of iterations,


t 
is the iterative
step. As long as iteration step meets certain conditions,
you can guarantee iterations to converge to the optimal
solution. Therefore, performing loop iteration on solving
process of the main problem until all parameters conver-
gence, we will get the optimal solution of the original
problem.

X.-M. RAN ET AL.
208
3.3. Process of Resource Allocation Algorithms
and Complexity Analysis
The algorithm processes can be expressed as Table 1.
The calculation amount of the proposed algorithm is
mainly focused on the dual decomposition algorithm.
The total computational complexity can be approximated
as , which is greatly less compared to the com-
putational complexity
(OKN))(
N
OK of the exhaustive search.
In the same time, the sender does not participate in solv-
ing each sub-problem, only according 1i

, 2

to con-
trol the resource allocation of each user, so the calcula-
tion complexity for the sende r is reduce d greatly .
4. Simulation and Analysis
4.1. Simulation Conditions
It is assumed that the carrier number of OFDMA network
, the number of authorized users
64
F
N8K

, the
secrecy outage probability which is required by each user
0.05,
kk

. It is assumed that the coverage of trans-
mission signal is 1 km, the distance between the eaves-
dropper and the sender should be closer than that be-
tween authorized user and the sender, path loss uses the
modified Hata path loss model, shadowing takes log-
normal shadow fading. Small scale fading takes Cost 231
Table 1. Resource allocation algorithm.
It is known that sender can get all the channel state
information of each authorized user, the total
transmit poweris .
,ki
h
t
P
First, initialization: 1i

, 2

will be initialized to
random positive number.
Second, the iterative process:
Step1: calculate the allocating power value for each
carrier by (23), and the negative value to be zero.
Step2: ,ki
A
is calculated by (24), execution on any
sub-carrier i: carrier i is allocated
to the users with the largest
*,
argmax ki
k
k
model [14], the noise power spectral density takes-120
dBm. And set the iteration step , where a is a
constant. Perform simulation of the proposed algorithm
through Monte Carlo method, take the average of the 200
times Channel implementation result as the final simula-
tion results.

/ta  t
Comparing the performance of the two basic OFDMA
resource allocation method, method 1 considers to allo-
cate carrier evenly to all authorized users, in this case,
each user will get 8 subcarriers, each user gets the same
power 8
T
P, we use the method in article[6] to com-
plete the carrier power allocation within each user.
Method 2 considers to allocate subcarrier to user with the
largest channel gain, then carriers power is allocated e-
qually, the power obtained by each carrier is 64
T
P.
4.2. Comparison of Secrecy Outage Capacity of
Different Allocation Methods
Take the number of eavesdroppers , compare
achievable throughput of eavesdropper and secrecy out-
age capacity of each allocation method in different trans-
mit power. As shown in Figure 2, it can be clearly seen
from the figure. The average secrecy outage capacity of
this allocation method system is far higher than that of
the other two allocation method systems in the same
transmit power, and with the increasing of transmit
power, the rise of secrecy outage capacity of this alloca-
tion method is obviously higher than that of the other two
allocation methods. In addition, it can also be seen that
when carrier allocation is equivalent, secrecy outage ca-
pacity of the system is the lowest. This shows that the
optimal allocation of the carrier is more important than
the optimal allocation of power and therefore has a
greater impact on the security of the system. It can be
found from the figure, the number of transmit antennas
will influence the size of secrecy outage capacity. Com-
2
E
N
A
,ki
A
. If more than one
user has the same ,ki
A
value, we randomly choose
one of them to make the optional user’s serial number
as ; judge
*
k,ki
A
is positive or not , if all ,ki
A
are
negative, the carrier i will not be assigned.
Step3: according to (27) update 2

, let ,ki
A

be a
second largest value in ,ki
A
, then 1i

can be
.

*
,,
/2
ki ki
AA

Step4: If 2

don’t convergence, continue step1, 2, 3;
Otherwise, the algorithm terminates.
8
A
N

8
A
N

8
A
N

8
A
N

6
A
N

6
A
N

6
A
N

6
A
N

6, 8
A
N
Figure 2. The comparison of Secrecy outage capacity of
different allocation methods.

X.-M. RAN ET AL. 209
pared to , when 6
A
N8
A
N

, secrecy outage capac-
ity improves. This is because when the number of emis-
sion antennas increases, the overall signal-to-noise ratio
of the system will also increase. For the eavesdropper,
proportion of artificial noise will also increase. Therefore,
secrecy outage capacity of the system will increase. It
can be seen from the figure that when the eavesdropper
through put is the same, the artificial noise has a great
influence on the eavesdropper. Although the total trans-
mit power is increasing, the average throughput of the
eavesdropper is always maintained at a very low pos
4.3. The Influence of the Number of
Eavesdroppers on Secrecy outage capacity
Define the number of transmit antennas 8
A
N

. As
shown in Figure 3, with increasing number of eaves-
droppers, secrecy outage capacity is sustained declining.
This is due to when the number of the eavesdroppers
increase, the suppression of increase in throughput will
need more power to produce artificial noise, in case that
the total power is constant, the share of the power of the
useful signal is bound to reduce, thereby the average se-
crecy outage capacity will decline. It can also be seen
from the figure that although the increasing number of
the eavesdroppers will decrease the secrecy outage ca-
pacity of the system, as long as AE , the secrecy
outage capacity of this method does not tend to 0, for two
other allocation methods when , the secrecy out-
age capacity of the system has dropped to a very low
level, which further illustrates that the method has certain
advantages in protecting the security of the system.
N
7
E
N
N
4.4. The Influence of Outage Probability on
Secrecy outage capacity
Take the transmission power , the number
of transmitting antennas 44dBm
t
P
8
A
N

, and the number of the
48 dBm
T
P
48 dBm
T
P
48 dBm
T
P
44dBm
T
P
44dBm
T
P
44dBm
T
P
Figure 3. The influence of the number of eavesdroppers on
Secrecy outage capacity.
eavesdroppers 2
E
N

, we analyze the influence of
outage probability on the secrecy outage capacity. As
shown in Figure 4, with the increasing of selected outage
probability, the average secrecy outage capacity which
system can achieve will slightly upgrade. This shows that
when the user is able to endure high outage probability,
the secrecy outage capacity which system can achieve
will increase, but as artificial noise shares more power,
the power of useful signal is relatively small, so the level
of increase is not large. It can also be seen from the fig-
ure that when the number of authorized users 16K

,
the secrecy outage capacity which system can achieve is
higher than that when the number of authorized users
8K

. This illustrates that the method can take advan-
tage of the diversity of multi-user to achieve the effect of
diversity, but for the other two allocation methods, this
effect is not very obvious.
5. Conclusions
In this paper, a resource allocation algorithm for OF-
DMA physical layer security under a non-ideal environ-
mentis proposed to solve the problem of secure transmis-
sion of OFDM system. Firstly, we build the security
model of OFDMA system network based on artificial
noise, in the presence of multiple eavesdroppers. Then,
average secrecy outage capacity of the system is defined
as the optimal goal without eavesdropper channel state.
Finally, to further simplify the optimization problem, the
joint optimal allocation of subcarrier and power is real-
ized via dual decomposition method. Simulation results
show that, total transmit power of the system is 50 dBm,
64 subcarriers are chosen to provide services for eight
authorized users, when the secrecy outage capacity of
each user is 0.05, the average secrecy outage capacity
can be up to 7.81 bit/s/Hz.
16K
8K
8K
8K
16K
16K
Figure 4. The influence of outage probability on secrecy-
outage capacity.

X.-M. RAN ET AL.
210
REFERENCES
[1] A. D. Wyner, “The Wire-tap Channel,” The Bell System
Technical Journal, Vol. 54, No. 8, 1975, pp. 1355-1387.
doi:10.1002/j.1538-7305.1975.tb02040.x
[2] I. Csiszar and J. Koner, “Broadcast Channels with Confi-
dential Messages,” IEEE Transactions on Information
Theory, Vol. 24, No. 3, 1978, pp. 339-348.
doi:10.1109/TIT.1978.1055892
[3] Z. Li and X. G. Xia, “A Distributed Differentially En-
coded OFDM Scheme for Asynchronous Cooperative
Systems with Low Probability of Interception,” IEEE
Transactions on Wireless Communication, Vol. 8, No. 7,
2009, pp. 3372-3379. doi:10.1109/TWC.2009.080365
[4] F. Renna, N. Laurenti and H. V. Poor, “Physical Layer
Secrecy for OFDM Systems,” in Proceedings of the IEEE
European Wireless Conference, Lucca, Italy, 2010, pp.
782-789.
[5] F. Renna, N. Laurenti and H. V. Poor, “High SNR Se-
crecy Rates with OFDM Signaling over Fading Chan-
nels,” in Proceedings of the IEEE 21thInternational
Symposium on Personal Indoor and Mobile Radio Com-
munications,Istanbul, Turkey, 2010, pp. 2692-2697.
[6] E. Jorswieck and A. Wolf, “Resource Allocation for the
Wire-tap Multi-carrier Broadcast Channel,” in Proceed-
ings of International Workshop on Multiple Access
Communications, Petersburg, Russia, 2008.
[7] Z. Li, R. Yates and W. Trappe, “Secrecy Capacity ofIn-
dependent Parallel Channels,” in Proceedings of Allerton
Conference on Communications, 2006, pp. 841-848.
[8] X. Zhou and M. R. McKay, “Secure Transmission with
Artificial Noiseover Fading Channels: AchievableRate
and Optimal Power Allocation,” IEEE Transactions on
Vehicular Technology, 2010, pp. 3831-3842.
doi:10.1109/TVT.2010.2059057
[9] M. Bloch, J. Barros, M. R. S. Rodrigues and S. W.
McLaughlin, “Wireless Information Theoretic Security,”
IEEE Transactions on Information Theory, Vol. 54, No. 6,
2008, pp. 2515-2534. doi:10.1109/TIT.2008.921908
[10] D. W. K. Ng and R. Schober, Cross-layer Scheduling for
Ofdma Amplify and Forward Relay Networks,” IEEE
Transactions on Vehicular Technology, Vol. 59, No. 3,
2010, pp. 1443-1458. doi:10.1109/TVT.2009.2039814
[11] W. Yu and R. Liu, “Dual Methods for Non-convex Spec-
trum Optimization of Multicarrier Systems,” IEEE Trans-
actions on Communications, Vol. 54, No. 7, 2006, pp.
1310- 1322. doi:10.1109/TCOMM.2006.877962
[12] S. Boyd and L. Vandenberghe, Convex Optimization,
Cambridge University Press, 2004.
doi:10.1017/CBO9780511804441
[13] D. P. Palomar and M. Chiang, “Chiang Tutorial on De-
composition Methods for Network Utility Maximization,”
IEEE Journal on Selected Areas in Communications, Vol.
24, No. 8, 2006, pp. 1439-1451.
doi:10.1109/JSAC.2006.879350
[14] 3GPP TR25.996 V9.0.0, Spatial Channel Model for Mul-
tiple Input Multiple Output Simulations [S]. 3GPP, 2009:
[15] H. Gao, P. J. Smith and M. V. Clark, “Theoretical Reli-
ability of MMSE Linear Diversity Combining in Rayleigh
Fading Additive Interference Channels,” IEEE Transac-
tion Communications, 1998, Vol. 46, pp. 666-672.