Communications and Network, 2013, 5, 303-307
http://dx.doi.org/10.4236/cn.2013.53B2056 Published Online September 2013 (http://www.scirp.org/journal/cn)
Resource Allocation for OFDMA-MIMO Relay Systems
with Proportional Fairness Constraints
Cuiru Zhao, Youming Li, Bin Chen, Zhao Wang, Jiongtao Wang
Institute of Communication Technology, Ningbo University, Ningbo 315211, China
Received July, 2013
ABSTRACT
In this paper, we study resource allocation problem in orthogonal frequency division multiple access multiple-input
multiple-output (OFDMA-MIMO) relay systems and formulate the optimal instantaneous resource allocation problem
including subcarrier assignment, relay selection and power allocation to maximize system capacity. Based on the as-
sumption that the availability of perfect channel state information (CSI) is known at the resource allocation controller,
we propose a new resource allocation algorithm which can guarantee proportional fairness among users. In the proposed
algorithm, a two-step suboptimal method is taken into account. Firstly, we assume equal power allocation for each user
to linearize the problem and propose the subcarrier assignment and relay selection scheme based on equivalent channel
gain. Secondly, we derive the closed-form expressions for power allocation through relaxing the proportional fairness
constraints. Numerical simulations show that the proposed algorithm performs well in terms of satisfying proportional
fairness among users in strict sense and achieving improvement on system total capacity.
Keywords: Relay Network; OFDMA; MIMO; Proportional Fairness; Resource Allocation
1. Introduction
The Orthogonal Frequency Division Multiple Access
(OFDMA) is regarded as a leading candidate for the
fourth generation (4G) wireless communication system
because of its high spectral efficiency, flexible resource
allocation and inherent robustness against frequency-
selective fading. Furthermore, Multiple-Input Multiple-
Output (MIMO) system has been extensively studied in
recent years. Since the multiple antenna technology pro-
vides extra spatial degrees of freedom, an OFDMA-
MIMO system can improve transmission reliability and
capacity without the need of increasing power or band-
width. On the other hand, due to its potential benefits of
enlarging the coverage of communication systems, in-
creasing the capacity and enhancing the link reliability,
cooperative relaying has attracted significant attention of
many researchers.
Recently, there have been many research efforts on
improving the system throughput by resource allocation
in OFDMA-MIMO relay system. Resource allocations of
Amplify-and-Forward (AF) and Decode-and-Forward
(DF) MIMO-OFDM relay systems are proposed in [1,2]
respectively. But, both [1] and [2] only focus on a single
user scenario.
A number of results have been published on resource
allocation for multi-user MIMO-OFDMA relay systems
[3-5]. The optimal instantaneous resource allocation
problem, including path selection, power allocation and
sub-channel scheduling, is formulated in [3]. Optimal
and suboptimal resource allocation algorithms for
weighted sum rate maximization are presented in [4]. In
[5], heterogeneous data rate requirements for delay sensi-
tive and non-delay sensitive users are taken into consid-
eration; the authors derive a distributed iterative resource
allocation and scheduling algorithm with closed-form
power and subcarrier allocation via employing dual de-
composition.
However, most existing research resource allocation in
OFDMA-MIMO relay systems focus on maximizing the
system capacity, fairness among multiple users has not
received much attention. In most practical communica-
tion system, the different business types of users have
different rate, are endowed with different resource allo-
cation priority. Therefore, to study the resource alloca-
tion problem with proportional fairness constraint is es-
sential for OFDMA-MIMO relay systems.
In this paper, we investigate subcarrier assignment,
relay selection and power allocation problem with pro-
portional fairness constraint for OFDMA-MIMO relay
assisted cellular system, and formulate the problem as a
joint optimization problem. Since the problem is a
NP-hard combination optimization problem with non-
linear constraints, we use a two-step suboptimal method
to solve it. Firstly, we assume equal power allocation for

C. R. ZHAO ET AL.
304
each user to linearize the problem and propose the sub-
carrier assignment and relay selection scheme based on
equivalent channel gain. Secondly, we derive the closed-
form expressions for power allocation through relaxing
the proportional fairness constraints. Simulation results
show that the proposed algorithm performs well in terms
of satisfying proportional date rate constrains in strict
sense and achieving improvement on system total rate.
The rest of this paper is organized as follows. The
OFDMA-MIMO relay assisted cellular communication
system model and optimization problem formulation is
described in Section Ⅱ. The proposed resource alloca-
tion solution is presented in Section Ⅲ. In Section Ⅳ,
simulations results are given and discussed. Finally,
conclusions are drawn in section Ⅴ.
2. System Model
We consider a OFDMA-MIMO relay assisted single cel-
lular communication system as shown in Figure 1. In the
adopted system, the overall bandwidth is B which is di-
vided into N orthogonal subcarriers. A base station (BS)
is located at the center, while K relay stations (RS) at the
inner boundary, operating in decode-and-forward (DF)
mode. M mobile stations (MS) are separated between the
inner and the outer boundaries. The BS and the RSs are
equipped with ,
B
SRS
NN antennas, respectively. While,
each MS is equipped with only one antenna. In order to
guarantee all users can receive signals from the base sta-
tion, the system operates in a time division duplex (TDD)
mode, where transmission takes place in two time slots.
In the first time slot, each RS receives and decodes the
signal from BS, then re-enc odes and forwards the signal
to the relevant MSs during the second time slot. For sim-
plicity, we assume that the subcarriers used in the first
time slot and the second time slot are the same. It is fur-
ther assumed that each subcarrier can be assigned to only
link in each slot [6].
Let ,,, ,
,
B
Skn kmn
HH represent the channel gain matrix
of the link from BS to relay k on the subcarrier n and the
link from relay k to user m on the subcarrier n, respec-
tively. And the corresponding transmission power is
,,, ,
,
B
Skn kmn
pp.
,, ,,,,
1,1 1,21,
,, ,,,,
2,12,22,
,,
,, ,,,,
,1 ,2,
,, ,,,,
, ,1,11,21,
BS
BS
RSRSRSBS
RS
BS k nBSk nBS k n
N
BS k nBSk nBS k n
N
BS k n
BS k nBSk nBS k n
NN NN
kmn kmnkmn
kmn N
hh h
hh h
H
hh h
Hhh h














The singular value decomposition of ,,
B
Skn
H and
,,kmn
H are given by

'
,,,, ,,,,
1
N
H
ii i
BS knBS k nBSknBS kn
i
Hu 


,,,,,, ,,
H
kmnkmnkmn kmn
Hu  
where '
,, (1, 2,...)
i
BS kniN
  and ,,kmn

are the singular
value of the matrix,,
B
Skn
H, ,,kmn H, respectively.


',
B
SRS
NNN is the rank of the matrix
,, ,,
H
B
Skn BSkn
HH.
The instantaneous rate between BS and relay k on the
subcarrier n can be written:

,,2,, ,,
log 1
2
B
SknBSkn BSkn
B
Rpg
N

Similarly, the instantaneous rate between relay k and
user m on the subcarrier n in the second time slot is given
by

,,2,, ,,
log 1
2
kmnkmnkmn
B
Rpg
N

The achievable rate of user m assisted by the relay k on
the subcarrier n is the minimum rate of ,,
B
Skn
R and
,,kmn
R[7]

,,,,, ,
min ,
n
B
SkmBSkn kmn
RRR (1)
where


2
1
,,
,,
0/
BS k n
BS k n
gNB N


 and

2
,,
,,
0/
kmn
kmn
gNB N



are the main characteristic sub-channel gain of the link
from BS to relay k on the subcarrier n and the link from
relay k to user m on the subcarrier n, respectively.

is the
SNR gap related to BER, ln(5)/1.5BER

 and 0 N
denotes the power spectral density of the Gaussian white
noise.
Define


,, 0,1 kmn
  as the joint user selection, path
selection and subcarrier allocation indicator. ,, 1
kmn


if and only if the subcarrier n is assigned to the user m
and relayed by relay k. Thus the achievable data rate of
the user m is ,, ,,
11
KN n
mkmnBSkm
kn
RR 

 . Therefore, the
adaptive optimal resource allocation problem subject to
total transmit power constraint and guaranteeing propor-
tional fairness among users is expressed as follows
MS
BS
2
RS
Figure 1. Single cellular OFDMA-MIMO relay system model.

C. R. ZHAO ET AL.
305
,, ,,
111
,,
,,
11
,,
11
,, ,
11
12 12
max
:
1: {0,1}
2: 1
3:
4:
5 :::... ::: ... :
KMN n
kmnBSkm
kmn
kmn
KM
kmn
km
KN
BS k nT
kn
MN
kmn kT
mn
M
M
R
subject to
A
A
ApP
ApP
ARRR


















(2)
where 1
A
and 2
A
emphasize that every subcarrier
can be allocated at most one link in each slot. 3
A
and
4
A
are the total transmit power constraints for BS and
RS, respectively. 5
A
denotes the proportional data rate
constraint.
3. Proposed Resource Allocation Algorithm
It is difficult to solve the optimization problem in (2),
because it includes both integer and continuous variables.
In this section, we use a two-step suboptimal algorithm
to reduce the computational complexity. In step one, the
subcarrier allocation and relay selection under equal
power allocation are discussed. In step two, the problem
of power allocation is solved.
3.1. Subcarrier Allocation and Relay Selection
Scheme
In order to approaches the maximum rate in (1), the fol-
lowing equation should be satisfied
,,,,, ,, ,
B
Skn BSknkmnkmn
pg pg
So we obtain
,, ,,
,,
,,
B
SknBSkn
kmn kmn
pg
pg  (3)
and the equivalent channel gain of link [8],
,,, ,
,,
,,, ,
BSkn kmn
n
BS k m
B
Skn kmn
gg
ggg 
Thus (1) can be rewritten as


,,2,, ,,
2,,,,
log 1
2
log 1
2
n
B
SkmBSkn BSkn
kmnkmn
B
Rpg
N
Bpg
N


(4)
The subcarrier allocation and relay selection algorithm
is described as follows
A) Determine the minimum number of subcarriers as-
signed for each user
1
/,1
M
mm m
m
NN mM 





 



B) Implement the subcarrier allocation and relay se-
lection for each user
a) Initialization
{1,2 ,,},{1,2 ,,} ,
{1,2 ,,},0 ,
KN
T
Mm
K
N
P
MRp N






b) While 0
m
N

1) Find m satisfying

arg min/
M
mm
mR


2) For the found m, find n* and * ksatisfying

**
**
,,
,
(, )argmax
NK
n
B
Skm
nk
nk g


3) Let
 **
*
,, 1, /
NN
kmn n
  , update *
*
,,
n
B
Skm
R
according to (3).
C) The remaining subcarrier allocation
a) While 0
N


, for 1n to N and n doesn’t be
used, find the user * m and relay * k satisfying

**
**
,,
,
(,) argmax
MK
n
B
Skm
mk
mk g


b) Let 
**
,, 1, /
NN
kmnn
  and update **
,,
n
B
Sk m
R
according to (3).
3.2. Proposed Power Allocation Scheme
After the subcarrier allocation and relay selection, the
next is to assign the available power on subcarriers. The
optimization problem can be reformed as
2,,,,
1
,,
11
1212
maxlog (1)
2
:
1:
2::: ...:::... :
k
K
BS knBSkn
knC
KN
BS k nT
kn
K
K
BpH
N
subject to
BpP
BRRR NNN







(5)
Formulate Lagrangian problem as following
00
0
0
0
,,
2,,,,
11
,,
11
2,,,,
1
2,,,,
(,,)
log (1)
2
()
[log(1 )
2
log (1)]
2
k
k
BS knk
KN
BS knBSkn
kn
KN
BS kn
kn
K
kBSknBSkn
knC
kk
k
BS k nBS kn
nC
k
Lp
BpH
N
pP
BpH
N
NBpH
NN


















C. R. ZHAO ET AL.
306
Then differentiate

,, ,,
B
Skn k
Lp

 with respect to



0
,,
,,
,,
,, ,,
,,
,, ,,
,,
12ln2
12ln2
0
BS k nk
BS kn
BS k n
BS k nBS kn
kBS kn
kkBS knBS k n
Lp
p
BH
pH N
NBH
NpH N









(6)
From (6) we have
'
''
,, ,,
,, ,,
,, ,,
11
BS k nBS kn
B
Skn BSkn
BS k nBS k n
HH
pH pH


(7)
Then equation (7) can be reformed as
,, ,,1
1
11
BS k nBS kn
pp
HH
BS,k, BS,k,
+- (8)
The rate of link k
2,,1 2,,
,,1
1
log log
2k
kkBSk BSkn
nC
BS k
B
RNp H
NH



 






According to the last constraint in (5), we can obtain:
00 0
0
2,,1
,,1
2,,1
,,1
1
log
1
log
kBSk k
BS k
kBSk k
BS k
Np w
H
Np w
H



 













(9)
where 2,,
log
k
kBSkn
nC
wH

, 0 k is a reference relay.
From (9), we can derive
0
,,1, ,1
B
Skk BSkk
papb
(10)
where
00
0
0
,,1 ,,1
1
2,
kkk k
kk
NwN w
NN k
kk
B
Sk BSk
a
ab
HH


Therefore,

0
1
,,1
1
K
Tkkk
k
BS kK
kk
k
PNbe
p
Na





(11)
According to (8) and (11), we have
0
,,, ,1
,,1 ,,
,, ,,
,,
,,
11
BS knkBS kk
B
Sk BSkn
BS k nBS k n
kmn kmn
papb
HH
pg
pg







(12)
4. Numerical Results and Analysis
4.1. Simulation Parameters
In this section, we consider a single cellular OFDMA-
MIMO relay communication system with a BS located in
the center, RSs equally distributed at the inner boundary
and MSs separated between the inner and the outer
boundaries. Both BS and RS are equipped with multiple
antennas, each MS is equipped with one antenna. We
adopt the channel for simulation consists of six-path
Rayleigh fading and the maximum doppler shift is 30Hz.
The total bandwidth is set to be 1MHz. Assume that
Gaussian white noise single PSD is 8 10  and BER is
3
10

.
4.2. Results and Analysis
In our simulation, a heuristic algorithm, PAARS+EquPo
and a static algorithm, Static+ EquPo are compared with
the proposed algorithm. PAARS+EquPo means the
subcarriers allocation and relay selection according to the
proposed scheme, equal power allocation for each
subcarrier, which is adopted in [6]. Static+ EquPo means
each user is assigned fixed subcarriers and the power
equally allocated to subcarriers. In the following, the
performance of proposed algorithm is evaluated from
two aspects: user fairness and system capacity.
To evaluate user fairness, we first define the normal-
ized capacity
The normalized capacity ＝ m
M
m
im
R
M
R


Figure 2 and Figure 3 depict the normalized capacity
of each user with different subcarrier numbers, respec-
tively. It is shown that, the normalized capacity of pro-
posed algorithm is close to the original normalized fair-
ness constraint. PAARS+EquPo algorithm can achieve
better fairness among users when the number of subcar-
riers is large. However, Static+ EquPo algorithm can’t
obtain proportional fairness among users.
Figure 4 shows the performance of system capacity
with respect to the number of users. As shown in this
figure, the capacity increases as the number of users in
the system increases. This is due to multiuser diversity
gain, which is more prominent in systems with larger
number of users. Furthermore, the capacity using
PAARS + EquPo algorithm is higher than that adopting
the proposed algorithm, because in the proposed algo-
rithm, we consider proportional fairness among users in
subcarrier allocation, relay selection scheme, and power
allocation scheme. The proposed algorithm outperforms
Static+EquPo algorithm, because the proposed algorithm
is an adaptive resource allocation algorithm which can
adapt to different channel information.

C. R. ZHAO ET AL.
307
12345678910 1112
0
0. 02
0. 04
0. 06
0. 08
0. 1
0. 12
0. 14
User i ndex
Normalized capacity
PF
PSARS+EquPo
Propos ed
Stat ic+E qupo
Figure 2. Normalized capacity versus user index after 50
iterations, the number of subcarriers is 1024.
1 234 5 6 78910 11 12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
User index
Normalized capacity
PF
PSARS+EquPo
Propos ed
St at i c + EquP o
Figure 3. Normalized capacity versus user index after 50
iterations, the number of subcarriers is 2048.
234567891011 12
4.1
4.12
4.14
4.16
4.18
4.2
4.22
4.24
4.26
4.28
Number of us ers
System capacity(Mbit/s/Hz)
Proposed al gori t hm
PSARA+EquPo
St at i c + E quP o
Figure 4. System capacity versus number of users after 50
iterations.
5. Conclusions
In this paper, we propose a new resource allocation algo-
rithm for OFDMA-MIMO relay assisted single cellular
communication system, which consider proportional
fairness among users. The performance of the proposed
algorithm is compared with other algorithms and simula-
tion results demonstrate that he proposed algorithm per-
forms well in terms of satisfying proportional fairness
among users in strict sense and achieving improvement
on system total capacity.
6. Acknowledgements
This work was supported in part by the National Science
Foundation of China (61071119), the Ningbo Natural
Science Foundation (2012A610017), and the Innovation
Team of Ningbo (2011B81002).
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