A New Bi-Allocation Scheme for Uplink OFDMA
Wireless System Using Compound Symbols for the
Detection Process
PABLO ADASME(1), ISMAEL SOTO(1) , ROLANDO CARRASCO(2)
(1)
Industrial Engineering Department
Engineering Faculty
University of Santiago of Chile
Av. Ecuador 3769, Santiago
CHILE
isoto@lauca.usach.cl
(2)
School of Electrical, Electronic and
Computer Engineering.
University of Newcastle upon Tyne
Merz Court Newcastle upon Tyne
(NE1 7RU)
UNITED KINGDOM
Abstract: - In this paper, we propose a new heuristic for a Bi-Allocation uplink problem when using OFDMA multiuser
wireless systems. This is done by assuming a new detection scheme of “two” or fewer incoming signals from different users
to each receiving detector subcarrier of the base station which is based on product trellises. A gain of 13.56% in (bits/s/Hz)
is achieved in capacity bandwith channel by the proposed heuristic compared with non-polynomial solvers. On the other
hand, under the new detection scheme, there’s a gain of 1.5 dB’s achieved for a BER of less than 10-2 when sending double
the number of bits over a MIMO Rayleigh fading channel.
Key-Words: Optimization, Wireless Networks, Adaptive Modulation, OFDMA, Trellis Structure.
1. Introduction
Recently, there has been a growing interest in wireless
multiuser systems, such as WLAN and mobile
communication systems. OFDM is considered a promising
solution due to its ability to combat multipath fading
problems. For multiuser access, in OFDMA each user
communicates with the base station over a set of dedicated
subchannels. This way, OFDMA is also referred to as
Multiuser-OFDM [1], and for these reasons, it is being
considered as a Multiple access modulation method for 4th
generation wireless networks [2]. OFDMA is an extension of
Orthogonal Frequency Division Multiplexing (OFDM), which
is currently the modulation of choice for high speed data
access systems such as IEEE 802.11a/g wireless LAN [3] and
IEEE 802.16a fixed wireless broadband access [4] systems.
In a typical wireless transmission environment, the transfer
function is different for each user. As a consequence, some
subchannels might be in deep fade for one user while they
might be fine for others. Thus, in case of a fixed wireless or a
slowly fading channel, the subchannel allocation should be
adapted to the channel characteristics and adaptive
modulation should be applied on each subchannel. If the
channel is known to the transmitter and the receiver, it can be
shown that OFDMA clearly outperforms other multiuser
techniques [8]. For single-user OFDM, several algorithms for
adaptive modulation based on the classical waterpouring
theorem [9] have been developed [10]. They are referred to as
bit loading algorithms and determine the number of bits and
the transmitting power for each subchannel. Hence, in
OFDMA, first a subcarrier allocation algorithm assigns the
subchannels to the users and then a bit loading algorithm
determines the constellation size and transmit power for each
subchannel.
The subcarrier allocation problem has been studied under
various premises. Wong et al. [11], [12] presented an
algorithm which is based on Lagrange optimization and
which minimizes the total transmit power under bit rate
constraints. This algorithm nearly reaches the optimal
solution, but due to its complexity and its slow convergence it
is computationally very expensive. Later, the same authors
presented a strongly simplified faster algorithm [13]. Another
step towards a fast implementation was made by Yin and Liu
[14] who partitioned the task into two steps. Nevertheless,
their algorithm still contains a highly complex assignment
problem whose solution is shown for only two users.
Regarding uplink or downlink the math formulation is indeed
complex to solve. Since there are two decision variables,
some authors try to separate the problem into two
subproblems. For instance, the approach in [5] for the
downlink problem, which is very similar to the uplink math
formulation, was to decouple the solution into an initial
resource allocation step, wherein the number of subcarriers
and amount of power for each user is determined; and
subsequently into a subcarrier assignment step, wherein each
user is assigned the corresponding subcarriers. In [6] on the
other hand, the approach was to first determine the subcarrier
allocation, then the power allocation.
In the next section, the system model is introduced and a
new Bi-Allocation scheme is sketched for the uplink OFDMA
problem. The Bi-Allocation scheme referes to a new method
of detection at the receiver. That is, in each subcarrier, “two”
or fewer incoming signals are detected from “two” users
simultaneously using different modulations such as M-PSK,
thus taking advantage of the subcarrier allocation. This
approach is similar to a MIMO system in which there are two
transmitting antennas and one receiving antenna, but differs
in that the transmitted data is different and in that it also uses
different transmission modulations since we are sending
symbols from different users. A proposed heuristic solution
for the Bi-Allocation scheme is developed in order to achieve
a high performance in bandwith channel capacity and to
diminish CPU time.
The paper is organized as follows: Section 2 provides the
system description, details the mathematic formulation for the
uplink problem and explains the new detection scheme
supporting this new idea. Section 3 explains the proposed
solution for the emerging Bi-Allocation problem. Section 4
shows the results obtained in simulations which are
compared using the optimization solver. Also, results for the
new detection scheme are shown in one case using 4-PSK and
QPSK adaptive modulation. Section 5 gives main conclusions
of this work.
experiences independent fading. The channel gain of user
in subcarrier
k
n is denoted as g k ,n , with additive white
ni  N 0 B / N
gaussian noise (AWGN)
where
N0
represents the noise power spectral density, B is the
bandwith and N is the total number of subcarriers in the
base station. The corresponding subchannel signal-to-noise
ratio (SNR) is thus denoted as hk ,n  g k2,n / ni2 and the n-th
subcarrier's received SNR is  k ,n  p k ,n  hk ,n . The slowly
time-varying assumption is very important since it is also
assumed that the base station is able to estimate the channel
perfectly. These channel estimates are then used as input to
the resource allocation algorithms. In order to satisfy the
BER constraints, the effective SNR has to be adjusted
accordingly. The BER of an M-PSK modulation as a
function of received SNR  k , n and number of bits rk ,n can
be approximated to within 1.5 dB of error for rk ,n  2 and
2. System Description
First, the Bi-allocation system model for the uplink
problem is described and later the new detection scheme
emerges as an effort to maximize the spectral efficiency since
each subcarrier is being used for “two” incoming signals
simultaneously, in contrast to the classic OFDMA which
receives one signal from one subchannel. With the new
scheme, the detector will be able to detect compound symbols
at the same time. This scheme allows use of M-PSK adaptive
modulation in order to increase data rate over the channel.
Nothing in the literature reports on the assignation of “two”
users to each one of the subcarriers of the base station, so in
that sense, this constitutes a novel approach for the uplink
problem. The algorithm proposed for solving the problem
using this novel approach corresponds to a simple way of
first, assigning each subcarriers to a “Bi-tuples” of users and
later, allocating the amount of power energy among the
subchannels for the signals tranmitted by each user.
2.1 Math Formulation for the Bi-Allocation
Model
The term Bi-allocation is used in the sense that each
subcarrier is being assigned to a Bi-tuple of sending users.
Figure 1 shows the Bi-allocation system model and the
emerging uplink problem since in this case, at the receiver
base station, each of the n  N subcarriers must be
allocated to several users k  K . In addition, in each
sending node, the power that each user has, must be
distributed in an efficient way. For example, user 1 is
sending information to subcarrier 1 and also to subcarrier n ,
so the total power of user 1 must be assigned in these
subchannels in such a way that capacity of each subchannel
is maximized. Each of the user's bits are then modulated into
k M-level PSK symbols, which are subsequently combined
using the IFFT into an OFDMA symbol. This is then
transmitted through a slowly time-varying, frequencyselective Rayleigh channel with a bandwidth B . It is
assumed that the subcarrier allocation is known to all users
through a control channel. It is also assumed that each user
BER  10 3 as [20]:
  6 
BERMPSK  k ,n   0.05 exp  rk ,n k ,n 
 2  1
(1)
And solving for rk , n , we have:
  
rk ,n  log 2 1  k ,n   log 2 1  pk ,n H k ,n 
 

(2)
Where    ln 20  BER  / 6 is a constant SNR gap, and
H k , n  hk , n /  is the efective subchannel SNR.
Then, the maximization problem is formulated as follows:
Max f c k , n , p k , n  
subject to :
B
N
K
N
c
k 1 n 1
k ,n
log 2 1  p k , n H k , n 
C1 : c k , n  0,1  k , n
(3)
C 2 : c k ,n  p k ,n  0  k , n
K
C3 :
c
k 1
N
C4 :

n 1
k ,n
p k , n  Pk  k
N
C5 :
c
n 1
 2 n
k ,n
 Nk  k
Where ck ,n is the subcarrier allocation variable, such that if
ck ,n  1 , it means that subcarrier n is assigned to user k .
k for subcarrier n is p k ,n and
the amount of energy that user k has is Pk . The input
The power assigned to user
variable N k corresponds to the maximum number of
subcarriers that can be used by user k . This parameter
depends on what quality of service is being required by a
particular user. Constraints C1 and C2 in (3) ensure the
correct assignation, for instance if c k ,n  0 , then p k ,n must
also be zero since that subchannel wouldn’t exist for the
subcarrier allocation problem. C3 imposes the restriction that
each subcarrier can be assigned to at most “two” users. C4 is
the power constraint, meaning that the power assigned to
each subchannel going out from each user must not exceed
the total power that the user has and C5 is a quality of
service constraint meaning that each user is not allowed to
use more than N k number of subcarriers.
2.2 New Detection Scheme
For the detection scheme, the main idea, is to detect
symbols at the receiver in a new trellis structure which is
assumed to be a product of trellises of “two”-sent
codewords. Supposing that two sending users are considered,
then if a 4-PSK modulation is used for the first sending user
and a QPSK modulation is used for the second sending user,
then the number of possible states in the detection process
for each codeword will be 4  4  16 states in each stage of
the new product trellis. The required condition is that the two
sending users have to rotate the phases of their
constellations, guaranteeing the uniqueness of the new
formed symbols on the product trellis. As an example let’s
consider these two codewords coming from each sending
user
respectively,
and
C Node _ A  a, d , c, b
is well known that the derivative of logarithm function is a
decreasing function of x , which intuitively leads us to think
that the contribution in the objective function, among p k ,n
variables doesn’t have to be concentrated in the highest value
of H k ,n , rather than that, it is better to distribute it among all
subcarriers that a user is allowed to use. This suggests that
we must generate a simple algorithm. As it is possible to
observe, the problem consists of a decision matrix ck ,n
indicating the assignation of subcarrier n to user k and a
decision matrix p k ,n indicating the amount of power
allocated to user k for transmitting to subcarrier n . Both
matrices depend on the H k , n coefficients, which describe the
quality of a particular subchannel, the better the H k , n is , the
more attractive the subchannel is to be chosen. So the
problem can be expressed as a matrix decision function
satisfying C1, C2, C3, C4 and C5 from (3) as:
 p1,1 H1,1

 p2,1 H 2,1
H  PH   

 
p H
 k ,1 k ,1
p1, 2 H1, 2   p1,n H1,n 


  p 2 ,n H 2 ,n 

 
 


 
 

  pk ,n H k ,n 
(5)
C Node _ B   f , e, h, g . This way, user 1 can only send a ,
In order to satisfy the constraints, we have to choose at most
two H k ,n from each column and at most N k subcarriers
b , c and d symbols and user 2 can only send e , f , g and
h symbols, so the constellations from each one of them are
from each row, this way maximizing the objective function
of (3).
as shown in Figure 2. That is, changing their phases by 45
degrees. The symbols sent by user A and user B are shown in
Table Nº 1 and the 16 possible states in the product trellis are
formed by summing the first symbol of user A with the four
symbols of user B, summing the second symbol of user A
with the four symbols of user B, and so forth. (See Table Nº
1). Note that all these new symbols are unique and form a
new trellis constellation of size 16. The detection process
applied on these new trellis structure corresponds to a Viterbi
Detection process in which symbols are recovered by the
following extended metric:
The algorithm is as follows:
 ri, j ,n  g i,n xi,n  g j ,k x j ,n
2
 i j
(4)
Initialize:
ck ,n  0, pk ,n  0, k  K n  N
Read:
N k , Pk , k  K
for i  1 to K {
H k ,n 
t
Where x k , n are the signals sent by user
H k ,n
N
H
n 1
k using subcarrier
 Pk ,  n  N }
k ,n
Sort H k ,n  k  K ,  n  N in a in
n . The g k ,n are fading rayleigh coefficents and ri , j ,n are the
Descending order.
composed symbols to be detected which correponds to the
signals transmitted from user i and user j respectively.
Save the position of each H k ,n .
for i  1 to K  N
3. Proposed Algorithm
Thanks to some properties of the logarithm function it is
posible to say that it is a concave increasing function. Also, it
 Max _ carriers  posi . pos _ k   N k and 

if 
 Max _ usuarios  posi . pos _ n  2

c k ,n  1
while  i  K and ck ,n  1{
p k ,n 
H k ,n
N
H
n 1
Calculate:
 Pk }
k ,n
B K N
 ck ,n log 2 1  p k ,n H k ,n 
N k 1 n 1
The first step of the algorithm consists of distributing the
total power of user k between all components of row k of
H as a fraction of the total power of user k . In the
second step, each H k ,n has to be sorted in a descending
matrix
order. This way, it is possible to choose the better
subchannels of matrix H . The third step consists of
constructing a solution to the problem without violating the
restrictions of equation (3). After that, the power is
allocated, but now only in the chosen subchannels of the
previous step. Finally the objective function is calculated.
4. Simulation Results
Simulation results are obtained first, for the Bi-Allocation
uplink problem in order to determine the performance of the
capacity Bandwith channel and second for the determination
of the performance related to CPU time between nonpolynomial solvers and the proposed heuristic for a “2-1”
scheme.
After that, results for the new detection scheme of “2-1”
are shown in order to determine the BER performance for a
case of study using 4-PSK and QPSK adaptive modulation
over an AWGN channel and over a Rayleigh MIMO channel
with two antennas at the transmiter and also two antennas at
the receiver.
algorithm. Figure 4 shows results obtained for the maximum
channel capacity achieved by the solver software and those
results obtained by the proposed heuristic for the Biallocation model.
For the results obtained with the solver and Heuristic when
assigning 64 subcarriers under the 2-1 assignation scheme,
the solver uses 478 seconds to solve a problem for 10 users,
1800 seconds for 30 users and 5386 seconds for 50 users
which is very high compared to the proposed heuristic. On the
other hand, the heuristic takes 1.7 seconds to solve the
problem for 10 users, 6.46 seconds for 30 users and 11.53
seconds for 50 users. Moreover, related to the subchannel
capacities, the heuristic is also better than the solver local
optimum, since for 10 users the solver reaches 1.83
(bits/s/Hz), for 30 users reaches 5.879 (bits/s/Hz) and for 50
users reaches 7.89 (bits/s/Hz). The heuristic achieves 2.17
(bits/s/Hz) for 10 users, 6.48 (bits/s/Hz) for 30 users and 8.96
(bits/s/Hz) for 50 users. In the same way, it is possible to say
that the heuristic is better than the solver local optimum in
13.56% for 50 users.
4.2 Results for the Detection Scheme
For the simulation of the new detection scheme under
study, a particular 2,1,3 convolutional encoder is used for a
case of study under the scheme of “2-1”. The equations used
by the convolutional encoder are the following:
 l1  cl  cl 2 ,
function of independent Rayleigh values g k ,n  x 2  y 2 .
Where x and y are assumed to be normally distributed


N   0, 2  1 . The total power was assumed to be
randomly between 5 and 105 W in each user, the total
bandwidth as 1 MHz, and total subcarriers as 64. The SNR
gap is    ln 20  BER  / 6   ln 20  10 3  / 6  0.65 and is
included in the value of each H k ,n of matrix H . The
number of users is varied from 10 to 50 with increments of
10. Results are obtained on one hand, with the optimization
solver and, on the other hand, with the proposed algorithm in
order to compare them. Java programming language was used
to develop the Bi-Allocation algorithm and was used for
experiments over the new detection scheme. In both cases,
taking the average over fifty random experiments. Figure 3
shows results in a logarithmic scale for the average CPU time
in seconds between solver software and the proposed
(6)
There have been two situations simulated. The first one, for a
conventional system, which is formed with one sending user
and one receiving node, and the second situation, which
corresponds to a system with two sending nodes to one
receiving node respectively. The metrics used for the viterbi
detection process in both situations are shown below in
equations (7) .
MR
4.1 Results for the Bi-Allocation Model
The gain of the fading coefficients g k ,n , is obtained as a
 l2   cl  cl 1  cl 2
MT
 r   g
t
i 1
j
t
MR
i 1
MT
 r   g
t
i 1
j
t
i 1
2
t
i
j ,i ,1 t ,1
x
(7)
t
j ,i
MT
xti   g tj ,i xti
2
i 1
Where g i , j corresponds to the channel Rayleigh fading
coefficients, it is assumed that the receiver has complete
channel state information. These results were obtained for a
million bits from each one of the sending users. The amount
of bits per packet used was 100. In this case, we assume 100
symbols sent in each packet, since we are modulating two
bits per symbol. It can be observed, from Figure 5, that in an
AWGN channel, the probability error rate is lower for the
conventional system with one sending node than for the
system with two sending nodes with a gain of 2 dB’s for a
BER less than 10-3. Related to the fading channel, it is
observed that both systems have a few differences, but it is
clear that the 2-1 scheme system is better for a BER of less
than 10-2, and the 2-1 scheme system achieves a gain of 1.5
dB’s for a Rayleigh fading channel. This can be seen as a
good result since double the amount of bit information is
being detected simultaneously.
5. Conclusions
In this paper, we proposed a new heuristic for a BiAllocation uplink problem using OFDMA multiuser wireless
systems under a new “2-1” detection scheme of “two”
sending users to one receiving user, a detection scheme which
is based on product trellises.
Simulations were tested for 10, 20, 30, 40 and 50 users in
the uplink system, and the results obtained related to the
subchannel capacities showed that the heuristic is better than
the solver local optimum, in all the experiments. As an
example, for 50 users using the heuristic, an improvement of
13.56% of capacity in (bits/s/Hz) over the solver was
achieved.
On the other hand, for the new detection scheme, in the
case of study using 4-PSK adaptive modulation and sending
double the amount of bits over a MIMO channel, there was a
loss of 2 dB’s in the case of an AWGN channel for a Bit
Error Rate of less than 10-3, but there was a gain of 1.5 dB’s
achieved in the case of the Rayleigh channel for a Bit Error
Rate of less than 10-2.
Acknowledgment
The authors would like to thank CONICYT/PBCT
(Project ACT-11/2004) Chile for their financial
support.
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1,0
User B
0,1
 1,0
0,1
User A
 2 2


 2 , 2 



2 2
1 

,

2
2 


2
2
1 

,


2
2



2 2
1 

,

2 2 


2
2
1 

,

2
2 

 2
2
  , 
 2

2


 2 2 


 2 , 2 1



2
2 


 2 ,  2  1



2 2 


 2 , 2  1


 2
2 


 2 , 2  1



2 2


 2 , 2 



2 2
 1 

,

2 2 


2
2
 1 

,


2
2



2 2
 1 

,

2 2 


2
2
 1 

,


2
2


 2
2


 2 , 2 


 2 2 


 2 , 2 1



2
2 


 2 ,  2  1



2 2 


 2 , 2  1


 2
2 


 2 ,  2  1


Table Nº 1: Compound Symbols at the Detector of the Receiver.
10
Capacity (Bits/s/Hz)
n1
h1,1
User 1

h2,1
+
SubCarrier 1
SubCarrier 2
User 2
SubCarrier 3
Base Station
h1, n
nn

hK , n
+
SubCarrier n
8
6
Local Optimum
Solver
4
Local Optimum
Heuristic
2
0
User
K
0
10
20
30
40
50
60
Number of Users
Figure 1: Bi-Allocation System Model.
a
Figure 4: Capacity for the Uplink wireless System.
e
b
1,00E+00
h
f
1,00E-01
1,00E-02
BER
d
c
Conventional Rayleigh
g
1,00E-03
2-1 scheme Gaussian
2-1 scheme Rayleigh
1,00E-04
Conventional Gaussian
1,00E-05
1,00E-06
Figure 2: Constellations for 4-PSK and QPSK Modulations.(2-1)
1,00E-07
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
dB
Average Time (Sec)
1,E+04
Figure 5: Case Study one using 4-PSK and QPSK adaptive
Modulation.
1,E+03
Solver Time
1,E+02
Heuristic Time
1,E+01
1,E+00
0
10
20
30
40
50
60
Number of Users
Figure 3: Comparison for the Average CPU time.