INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XI, November 2025
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Resource Allocation Optimization for Orthogonal Frequency Division
Multiplexing Access using Modified Utility Function
Kon Kim
*
Faculty of Communication, Kim Chaek University of Technology, Pyongyang 999093, Democratic
People’s Republic of Korea
*Corresponding author
DOI: https://doi.org/10.51583/IJLTEMAS.2025.1411000028
Received: 10 November 2025; Accepted: 20 November 2025; Published: 03 December 2025
ABSTRACT
In this paper, we propose an optimal method for resource allocation in OFDMA (Orthogonal Frequency
Division Multiplexing Access) system. The utility function is used to balance the efficiency and fairness of
wireless resource allocation. We analyze the mathematical property of resource allocation by modified utility
function. We develop optimal algorithm for dynamic subcarrier assignment and adaptive power allocation
based on modified utility function, and demonstrate the convergence properties of optimization. Simulation
results show that the proposed method provides the effective tradeoff between fairness and efficiency in radio
resource allocation.
Keywords: OFDMA, Utility Function, Fairness and efficiency, Adaptive power allocation and subcarrier
assignment.
INTRODUCTION
OFDMA is a kind of multiple access technology. It has overcome the disadvantage of OFDM-TDMA which
allocates all resource to a single user at certain time. The principle of OFDMA is that it can vary the resource
allocation according to some conditions containing path condition [1-10].
Impulse response of a general time-varying multi-path channel can be represented as
I
i
ii
tth
1
)()(),(
(1)
And transfer function is
dethtfH
fj2
),(),(
(2)
It is assumed that the channel fading rate is slow enough so that the frequency response does not change during
an OFDM symbol. The condition of channel can be based on signal nose ratio function.
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),(
),(
),(
2
tfN
tfH
tf
(3)
Using adaptive modulation, the transmitter can send higher data rates over the subcarriers with better channel
conditions to improve throughput and simultaneously ensure an acceptable BER in all subcarriers.
And the achievable throughput can be expressed as
)),(),(1(log
)
),(
),(),(
1(log),(
2
2
2
tftfp
tfN
tfHtfp
tfc
(4)
System consisted of M users is shown in Fig.1.
Fig. 1. System of M users
And the m
th
user’s throughput is
)),(),(1(log
)
)(
),(),(
1(log),(
2
2
2
tftfp
fN
tfHtfp
tfc
m
m
m
m
(5)
And the rate of m
th
user is
m
m
D
m
D
mm
dftftfp
dftfctr
)],(),(1[log
),()(
2
(6)
And then the target is to maximize the entire rate sum of users in some essays. That is
M
m
m
fpD
r
1
)(,
max
(7)
To solve this problem a lot of approaches have implemented.[1],[4]
But the target is emphasized on maximizing the entire rate sum, so this problem has disadvantage of unfairness
among users. To overcome this shortcoming weighted-sum method is adopted.
M
m
mm
fpD
rw
1
)(,
max
(8)
But this method is not as good as utility function method.
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The approaches used to solve the problem of OFDMA resource allocation formulated as rate sum or weighted
sum are shown in Refs. [1]-[10].
Table 1 shows the approaches used in the resource allocation.
Table 1. Approaches used in resource allocation
Approach
Terminate at
Complexity
Lagrangian method
None
)(MKO
SQP
Iteration
)(
3
KO
Simulated Annealing
Condition Satisfaction
)(
333
LKMO
Genetic Algorithm
Generation
)265.1(
21.0
MKO
K
Branch and bound method
None
MKL
2
As can be seen from Table 1, every approach has own character and advantage and disadvantage. First the
accuracy of approach simulated annealing and genetic algorithm cannot get the exact results. And the
complexity is lowest in Lagrangian function.
On the other hand, the formulation of problem is most adjectival in utility function method. So we proposed
three problems in this paper.
First is to analyze the mathematical property of resource allocation by modified utility function,Second is to
develop optimal algorithm for the resource allocation in OFDMA system,Third is to simulate the proposed
method and verify the efficiency of the system.
METHODS
Modified utility function
Utility functions are used to quantify the benefit of usage of certain resources. And the target function using
Utility function is
M
)(
1
max
)(,
mm
fpD
rU
M
(9)
For the global optimality, Eq. (9) must be a concave function.
If all
)(
mm
rU
are concave functions, then the objective function(9) is also a concave function. But in reality,
all
)(
mm
rU
are not concave functions.
So utility function is modified
)(rU
into
)(
~
rU
as follows.
crt
crt
0
0
)(
)(
~
rr
rr
rU
rU
(10)
Then
)(
~
rU
is concave in
),0[
. All functions concave in
),[
crt
r
can be modified into
)(
~
rU
.
As the objective function is concave, there is a unique global maximum solution to the optimization problems.
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Continuous rate optimization
Let
m
and
K
be
m
th user’s SNR and the number of available subcarriers, respectively.
m
can be
described by
),,,(
,2,1, Kmmmm
(11)
where
km,
is
m
th user’s SNR at
k
th subcarrier.
A transmission power
p
k
at kth subcarrier can be expressed as follows.
),,(
,,2,1 kMkkk
pppp
(12)
where p
m,k
is a transmission power at kth subcarrier allocated to the mth user.
Define D
i
as the frequency set assigned to user i. Then
nm
DD
,
nmnm ,, M
where
denotes an empty set.
The transmission throughput of mth user at kth subcarrier can be expressed as
fpr
kmkmkm
)1log(
,,,
. (13)
where
f
is the bandwidth of the subcarrier.
Then optimization problem can be regarded as
M
m
K
k
kmkmkmm
p
prU
M
U
1 1
,,,
))((
1
maxmax
(14)
subject to
M
m
K
k
km
Pp
1 1
,
.
where
P
is the maximum transmission power of the transmitter.
Using the Lagrangian method, the above optimization problem with the power constraint becomes to
maximize
M
m
M
m
K
k
km
K
k
kmkmkmm
pPprU
M
pL
1 1 1
,
1
,,,
)())((
1
),(
. (15)
Then the dual problem for (15) is defined as
)(min
0
*
g
. (16)
where
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M
m
km
K
k
kmkmkmm
pPp
pprU
M
PpL
1
,
1
,,,
))((
1
max),(max)(
(17)
As the subcarrier assignment is independent of the power allocation,
K
k
M
m
kmkmkmkmm
p
pprU
M
P
k
1 1
,,,,
))((
1
max)(
. (18)
As a subcarrier is not shared by two or more users and the channel gain is not independent of subcarriers but
distributed ideally
))))(((max(
1
max)(
,,,,
0,
kmkmkmkmm
pM
pprU
M
KP
km
. (19)
In (19)
)))(((max
,,,,
,
kmkmkmkmm
p
pprU
km
satisfies the following equation for the optimal power allocation.
kmm
km
p
,,0
*
,
1
)(
1
)(
(20)
Where
)(
)(
*
,
'
,0
kmm
m
rU
. (21)
By (19) and (20), (16) can be described as follows.
)],([min*
0
mk
KgPg
(22)
where
)],([max),(
,,
kmkm
M
mk
gg
(23)
)())(()(),(
*
,,
*
,,
*
,
'
,,
kmkmkmkmkmmkmkm
pprrUg
(24)
Let I be the number of the function evaluations to converge.
If
]
max
,
min
[
*
,
1
)618.0log(
)/(ln
maxmin
I
(25)
where
is an allowable error.
If
is small enough to satisfy the power constraint, i.e
K
km
Pp
*
,
,
max
*
,
'*
,
*
,
'
min
)(max
2ln
1
max
)(min
2ln
kmm
m
K
km
m
kmm
M
rU
P
K
P
rUK
. (26)
Then
K
km
kmm
rU
P
,
*
*
,
'
1
2ln
)(
(27)
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K
km
kmm
m
rU
,
*
*
,
'
1
2ln
)(
min
(28)
K
km
m
kmm
m
rU
,
*
*
,
'
1
max
2ln
)(min
(29)
K
km
m
kmm
m
rU
K
,
*
*
,
'
1
max
2ln
)(min
(30)
K
km
m
kmm
m
P
rUK
,
*
,
'
*
1
max
)(min
2ln
(31)
Also
K
km
kmm
rU
P
,
*
*
,
'
1
2ln
)(
(32)
K
km
kmm
m
rU
,
*
*
,
'
1
2ln
)(
max
(33)
)(max
2ln
*
,
'
*
kmm
m
rU
K
(34)
)(max
2ln
*
,
'*
kmm
m
rU
P
K
(35)
Then, from (31) and (25), we found (26) is satisfied.
After getting
*
, the optimal multiplier
*
determines the user selection and power allocation per subcarrier
k:
)}(
~
))(
~
()({maxarg
*
,
*
,
*
,,
*
,
'*
kmkmkmkmkmm
m
k
pprrUm
(36)
)(1)(
~
**
,
*
, kkmkm
mmpp
(37)
where
false is
trueis
,0
,1
)(1
x
x
x
. (38)
Note, however, that it is possible that the sum of the candidate power allocation vector
M
m
K
k
km
pP
1 1
*
,
*
)(
does not satisfy the total power constraint, since the constraint is not enforced explicitly. Hence, our final
power allocation values should be multiplied by a constant
)(/
*
PP
which is then plugged back into the
objective in (14) to arrive at our computed primal optimal value
M K
))1(log(
ˆ
*
,,2
*
kmkmm
pUf
. (39)
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In continuous modulation the resource allocation algorithm is as follows.
Step 1:
km,
,
])))(
~
))(
~
(((max[minarg
,,,,
0
*
K
M
k
kmkmkmkmm
m
pprUP
Step 2:
km,
,
kmm
km
p
,
*
,0
**
,
1
)(
1
)(
Step 3:
)}(
~
))(
~
()({maxarg
*
,
*
,
*
,,
*
,
'*
kmkmkmkmkmm
m
k
pprrUm
Step 4:
)(1)(
~
**
,
*
, kkmkm
mmpp
The complexity of this algorithm is
)(IMKO
,
)(IO
,
)(IO
, and
)(KO
in each step, respectively.
If we let
*
f
(a
0
) and
*
g
(
0
) be the optimal values of the primal and dual problems given in (14) and (23),
and let
0
ˆ
*
f
be the computed feasible primal value, the relative duality gap can be bounded as
*
**
*
**
ˆ
ˆ
0
f
fg
f
fg
(40)
The left inequality follows directly from the non-negativity of
*
f
and the weak duality theorem and the right
inequality follow from
**
ˆ
ff
.
The following equation is derived by dividing the numerator of (40) by any feasible solution to the primal
problem.
K
))(
ˆ
())))(
~
1((log(
ˆ
**
,
*
,2
**
**
PPpUfg
km
km
m
kk
K
)))
)(
ˆ
)(
~
1((log(
*
,
*
,2
**
P
P
pU
km
km
m
kk
(41)
))(
ˆ
()))
)(
ˆ
)(
~
1
)(
~
1
((log(
**
*,
*
,
,
*
,
2
*
*
*
PP
P
P
p
p
U
km
km
km
km
m
k
k
k
K
(42)
One can notice that if
)(
*
PP
, i.e. if our dual optimal powers satisfy the power constraint tightly, the
duality gap upper bound is zero, thus the dual optimal and primal optimal solutions are equal and we have
solved our problem exactly.
However, the existence of
*
such that
)(
*
PP
cannot be guaranteed in general, since
)(
ˆ
*
P
is a
(possibly) discontinuous function of
, and the discontinuity may actually happen at
*
such that the
total power does not meet the constraint tightly.
Fortunately, the height of the discontinuity (if it exists) is quite small, and actually diminishes quickly as K
increases, and thus the duality gap also diminishes quickly.
This behavior can also be explained analytically by using a generic bound for the duality gap of separable
integer programming problems, which when applied to our problem results in
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)))(((max2
,,
**
kmkmm
k
PrUfg
MmK,
(43)
which can be interpreted as twice the maximum weighted conditional expected rate over all users and
subcarriers when all the power is allocated to it.
From (43), we can find that the duality gap bound does not scale with K.
If we include the bandwidth term
KB/
into the per-subcarrier rate, it can be seen that the duality gap
diminishes as
K
.
Discrete rate optimization
In the discrete rate case, the data rate of the kth subcarrier for the mth user can be given by the following
staircase function.
LkmkmLL
kmkm
kmkm
kmkm
d
km
pr
pr
pr
pr
,,11
2,,11
1,,00
,,,
,
,
,
)(
(44)
where
Ll
l
r
,
},...,1{ LL
are the L available discrete information rates in increasing order, and
l
is the SNR
boundary chosen in such a way that the information rate
l
r
is supportable subject to an instantaneous BER
constraint.
Then, (14) can be described as follows.
M
m
K
k
km
Pp
1 1
,
M
m
K
k
kmkm
d
kmm
p
d
prU
M
U
1 1
,,,
))((
1
maxmax
(45)
And the dual problem becomes
))))(((max(
1
max)(
,,,,
,
kmkmkm
d
kmm
pM
d
pprU
M
KP
km
. (46)
For
km
p
,
, we have L power allocation functions to choose from.
km
l
km
l
l
R
,
1
,
,
,
10 Ll
kmlkmmkmkmkm
d
kmkmm
prrUpprrU
,,
'
,,,,,
'
)()()(
km
l
lkmm
rrU
,
,
'
)(
,
l
km
Rp
,
(47)
km
L
km
d
km
p
,
1
,
0
,
,,
~
(48)
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Here we should find a function that
km
l
lkmm
rrU
,
,
'
)(
is maximized.
km
l
d
km
km
p
,
,
*
,
~
(49)
where
km
l
lkmm
l
l
rrU
km
,
,
'
)(maxarg
*
,
. (50)
For
10, Lll
km
l
lkmm
km
l
l
kmm
rrUrrU
km
km
,
,
'
,
,
'
)()(
*
,
*
,
(51)
l
l
l
l
kmkmm
l
l
l
l
km
km
km
km
rr
rU
rr
*
,
*
,
*
,
*
,
,,
'
)(
,
*
,km
ll
,
*
,km
ll
(52)
l
l
l
l
ll
kmkmm
l
l
l
l
ll
km
km
km
km
km
km
rr
rU
rr
*
,
*
,
*
,
*
,
*
,
*
,
min
)(
max
,,
'
(53)
1
1
1
1
,,
'
*
,
,
)(
:10
ll
ll
ll
ll
kmkmm
km
rrrr
rU
Lll
(54)
After getting
*
in the same way with the continuous modulation,
*
determines the optimal subcarrier, rate,
and power allocation:
}))({maxarg
,
**
,
'*
*
,
*
,
km
l
l
kmm
m
k
km
km
rrUm
(55)
)(1
**
,
*
,
k
l
km
mmrR
km
(56)
)(1
*
,
*
,
*
,
k
km
l
km
mmp
km
(57)
The algorithm for discontinuous modulation is as following.
Step 1:
km,
,
])))(
~
))(
~
(((max[minarg
,,,,
0
*
K
M
k
kmkmkm
d
kmm
m
pprUP
Step 2:
km,
,
1
1
1
1
,,
'
*
,
,
)(
ll
ll
ll
ll
kmkmm
km
rrrr
rU
ll
:L
.
Step 3:
}))({maxarg
,
**
,
'*
*
,
*
,
km
l
l
kmm
m
k
km
km
rrUm
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Step 4:
)(1
*
,
*
,
*
,
k
km
l
km
mmp
km
,
)(1
**
,
*
,
k
l
km
mmrR
km
The complexity of each algorithm is
)(IMKO
,
))log(( LMKO
,
)(MKO
and
)(KO
, respectively.
Simulation Results
Simulation parameters
Table 2 shows the simulation parameters.
Table 2. Simulation parameters
System parameters
Value
Bandwidth (
B
)
1.25MHz
Number of sub-carriers (
fft
K
)
128
Number of used sub-carriers (
K
)
76
Sampling Frequency (
s
F
)
1.92MHz
Doppler Frequency (
d
F
)
200Hz
Cyclic prefix length(
cp
L
)
6 samples
Bite Error Rate (BER)
10
6
Maximum Modulation Level (
L
)
3
Modulation mode
QPSK,16-QAM,64-QAM
Simulation by MATLAB
Channel simulation
The Rayleigh channel is used to simulate the envelope of an individual multipath component.
The parameters of the Rayleigh channel are as follow.
SampTime = 1/1920000;
FreqD = 200;
gainVector = [0 -1.5 -3 -4.5 -6 -7.5 -9];
delayVector = 1.0e-4 * [0 0.01 0.02 0.03 0.04 0.05 0.06];
snr=[0,1,2,3,4,5,6,7,8,9,10];
The Rayleigh channel object is generated by
rayleighchan commander in Matlab.
rayChanObj = rayleighchan(SampTime, FreqD, delayVector, gainVector);
modulation
The modulation object is
generated by following command.
modObj = modem.qamkmod(2^l);
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modObj.InputType = 'Bit';
where l is modulation level and its default value is 2.
Then number of bit per frame is specified and modulation process is as follow.
msg = randi([0 1],bitsPerFrame,1);
modSignal = modulate(modObj, msg);
The search of
*
The search of
*
is as follow.
x = fminbnd(fun,x1,x2);
RESULTS AND DISCUSSION
First let’s confirm the multiuser diversity effect in OFDMA system.
Fig. 2. Multiuser diversity
As can be seen on Figure 2, more increased the number of user, more increased channel capacity in proportion
to logarithmic value of users.
Second, the fairness of proposed method was confirmed in Fig. 3.
Fig. 3. Fairness among users
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As can be seen on Figure 3, the rate sum method cannot provide fairness of resource allocation but utility
function provides fairness of resource allocation.
Third, the efficiency of proposed method is shown in Fig. 4.
Fig. 4. Efficiency of proposed system
As can be seen on Figure 4, when the jointed DSA and APA are used simultaneously for optimization by utility
function, the efficiency and fairness of resource allocation are provided.
CONCLUSION
We proposed an optimal method for resource allocation in OFDMA (Orthogonal Frequency Division
Multiplexing Access) system. We have derived optimal resource allocation algorithms for continuous and
discrete rate maximization by modified utility function, and demonstrated the convergence properties of
optimization. Simulation results show that the proposed scheme provides the effective tradeoff between
fairness and efficiency in radio resource allocation.
Resolved scientific and technical contents in this paper are as follows:
First, we proposed
the resource allocation algorithm
of OFDMA system based on utility function.
Second, we simulated proposed approach with practical parameters using MATLAB and confirm the
advantages of the system proposed in this paper.
We will investigate the optimization under more realistic conditions and develop low complexity approaches
for the OFDM wireless network.
ACKNOWLEDGEMENTS
This work was supported by Kim Chaek University of Technology, Democratic People’s Republic of Korea.
The supports are gratefully acknowledged. The authors express their gratitude to the editors and the reviewers
for their helpful suggestions for improvement of this paper.
CRediT authorship contribution statement
Kon Kim: Investigation, Methodology, Project administration, Software, Writing-original draft.
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XI, November 2025
www.ijltemas.in Page 296
Funding
This research did not receive the external funding.
Data availability
All data that support the findings of this work are included within this article.
Declarations
Ethics approval
The authors approve to observe the ethics standard of this journal.
Conflict of interest
The authors declare that they have no conflict of interest.
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