Due to the rapid growth of wireless technologies, increasing
demands have been made for spectrum usage, and the
conventional approaches to spectrum managements have been
challenged by the new insights. Studies have shown that most
of the radio frequency spectrum is largely underutilized [1-3].
In this context, Cognitive Radio (CR) with its unique ability to
change the transmission or reception parameters, to
communicate efficiently avoiding interference with licensed or
unlicensed users, has been developed as an efficient
technology to utilize the spectrum [4]. CR system provides
access for a group of unlicensed users (i.e. SUs) to the
frequency bands which are originally allocated to licensed
users (i.e. PUs), so that no harmful interference to the PUs is
caused.
Underlying sensing, spectrum shaping, flexibility in
dynamically allocating the radio resources, and adaptiveness
are essential requirements of CR systems. Given this fact,
OFDMA has been reported in the literature to be the best
candidate for cognitive radio systems [5-6].
Many resource allocation algorithms for OFDM-based
cognitive radio have been proposed during these years,
considering different conditions. An optimal power loading
algorithm for single user OFDM-based cognitive radio has
been proposed in [7], which aims to maximize transmission
capacity while the interference introduced to PU (licensed
user) remains within a tolerable range. In [8], an efficient joint
subcarrier and power allocation algorithm for multi-user
OFDM-based CR system has been presented, considering the
peak power constraint. A suboptimal subcarrier and power
allocation algorithm has been offered in [9] which maximizes
the sum throughput of SUs without causing adverse
interference to Pus.
With rapid development of wireless communication and
new emerging technologies, throughput guarantee becomes
more important. Services that new generations of networks are
presenting such as video conferencing, internet gaming, and
online TV clarify the importance of a guaranteed throughput
[10]. Besides, one of the challenges in resource management
of wireless networks is power consumption. So the idea of a
network, which is capable of allocating the network resources
such as power and subcarriers to provide a guaranteed
throughput for its users with minimum power consumption,
becomes so interesting. In this paper, a resource allocation
algorithm for a multi user OFDM-based CR system is
presented which minimizes the power consumption while
providing a guaranteed throughput. Considering this fact that,
in CR scenario the interference introduced to PUs should be
below a certain threshold. The dual Lagrange method is used
to find the optimum solution for the problem. For the sake of
comparison, the proposed algorithm is compared with
Uniform Resource Allocation Algorithm (URAA) and Sum
Throughput Maximization Algorithm (STMA) [9].
This paper is organized as follows: In Sec.2 system model is presented. Sec.3 formulates the problem and describes the problem solution and proposed algorithm. Simulation results are presented in Sec.4, and finally Sec.5 concludes the paper.
II. SYSTEM MODEL
In this paper, the resource allocation problem in the downlink of a multiuser OFDM-based cognitive radio system in an underlay manner is considered. The model of the system is shown in Fig.1.
This article is granted by Iran Telecommunication Research Center (ITRC).
Figure 1. The downlink of OFDM-based Cognitive Radio system
The system has M Pus, K SUs, and N subcarriers. Besides, this model includes a primary base station (P-BS), which provides services to the PUs, and a secondary base station (S-BS), which is responsible for allocating power and subcarriers to the SUs, sensing the spectrum, and finding spectrum opportunities. A channel is said to be a spectrum opportunity if the interference to the PU receivers caused by S-BS transmission is acceptable.
Due to spectrum sharing between SUs and PUs, the signal transmitted to SUs imposes interference on the PUs (as shown in Fig.1 with red lines). CR systems are typically designed in a way that the interference introduced to PUs remains below a certain threshold. Based on [11], the power spectral density of
thn subcarrier’s signal which is allocated to
thk SU can be
expressed as:
sin( ) ( ) (1)s
kn kn s
s
fTf p T
fT
πφ
π
=
Where, knp
is the allocated power to
thk SU on the thn
subcarrier. sT is the symbol duration. The interference power
introduced to PU by this CR subcarrier is [9]
2 2 2sin( ) ( ) ( ) ( ) ( )
/ (2)
sn kn s
sf F f F
kn n kn n n
fTI n H n f df p H n T df
fT
p I I p I I p
πφ
π∈ ∈
= =
= ≤ → ≤ =
Where, ( )nH is the channel gain between S-BS and the PU
for the thn subcarrier. F denotes the frequency band licensed
to the PU. I is the PU’s maximum tolerable interference
power. nI denotes the interference factor for th
n subcarrier.
Using nI , the interference constraint can be expressed as the
subcarrier power limit,nP . It is considered that the each
channel gain is known and a perfect spectrum sensing is
performed. Also the transmission rate for the th
k SU on the
thn subcarrier, knR , is given by Shannon capacity formula as
follows:
2
2 2( , ) log 1 (3)
kn knkn kn kn
h pR p h f
σ
= ∆ +
Where, 2
σ is the variance of the received noise on each
subcarrier including the thermal noise and the interference
received from the primary system.
III. PROBLEM FORMULATION AND PROPOSED ALGORITHM
Our objective is to minimize the power consumption while satisfying guaranteed throughputs for SUs and keeping interference introduced to each of PUs below a certain threshold. The problem can be formulated as follows:
, 1
2
1
min (4)
. .1) log (1 ) , 1,2,..., (4 )
2)0 1,2,..., (4 )
3) (4 )
kn kk
K
knp
k n
N
k kn kn k
n
kn n k
l i
p
s t R p D k K a
p p k K and n b
l i c
γ
Ω= ∈Ω
=
= + ≥ ∈ −
≤ ≤ ∈ ∈Ω −
Ω Ω ≠ Φ ≠ −
Where
k is the set of subcarriers allocated to the
thk SU. D
is the set of guaranteed (promised) throughputs. (4-a) is due to rate guarantee constraints. (4-b) results from the interference constraints on the PUs and (4-c) implies that one subcarrier cannot be allocated to more than one SU. The Lagrangian of the above problem can be expressed as follows:
2
1 1
1 1
( , , , ) log (1 )
... ( ) (5)
k k
k k
K K
kn k kn kn k
k n k n
K K
kn kn kn n kn
k n k n
p p D
p p p
λ γ
α β
= ∈Ω = ∈Ω
= ∈Ω = ∈Ω
= − + −
− − −
L p
Where, 1 2 , ,..., Kλ λ λ= ,
11 21 , ,..., KNα α α= and
11 21 , ,..., KNβ β β= are Lagrangian multiplier vectors or dual
variables and 11 12 , ,..., KNp p p=p is the vector of primal
variables. The dual function g (, , ) is defined as the infimum of the Lagrangian over p; that is:
( , , ) inf ( , , , ) (6)g =p
L p Based on lower band property [12], the dual function, is a
lower bound on the optimal value, ∗
p , of the problem that is:
, , ,1
min max ( , , ) (7)kn k
k
K
knp
k n
p gΩ
= ∈Ω
≥
When attempting to solve the primal problem, the best lower
bound of its optimal value is considered. From lower band
property [12], it is natural that the following optimization
problem, called the dual problem, is then examined:
701
, ,max ( , , ) (8)
. . , , 0
g
s t ≥
The difference between the original problem and the dual problem is called the duality gap. The dual problem answer will converge to the primal problem solution when the number of subcarrier goes to infinity [13]. In applied systems, number of subcarriers is big enough. . The Karush-Kuhn-Tucker (KKT) conditions are given as follows:
( , , , )1 0 (9 )
ln 2(1 )
1(9 )
ln 2(1 )
kn k kn kn k knkn kn
kn kn kn
kkn
kn kn kn
L pa
p p
p b
λ α β λ γα β
γ
λ
α β γ
∂= − − + = −
∂ +
= − −
− +
2
1
( , , , )log (1 ) 0 (10)
Nkn k kn kn
k k kn kn k
nk
L pp D
λ α βλ λ γ
λ=
∂× = × + − =
∂
( , , , )0 (11)kn k kn kn
kn kn kn
kn
L pp
λ α βα α
α
∂× = × =
∂
( , , , )( ) 0 (12)kn k kn kn
kn n kn
kn
kn
L pp pβ
λ α ββ
β
×
∂= × − =
∂
From (9-b) and (11), it can be concluded that:
1(13)
(1 ) ln 2
kkn
kn kn
pλ
β γ
+
= −
+
Where max( ,0)x x+
=. Now, from (12) and (13), we can
conclude that the power allocation for prblem1 is:
1min , (14)
ln 2
kkn n
kn
p pλ
γ
+ = −
It is possible to decompose the Lagrange dual function of (5)
into N independent optimization problems, each for one
subcarrier, as follows:
2
1 1
1 1
( , , , ) log (1 )
( ) (15)
K K
kn k kn kn kn k kn kn
k k
K K
kn kn kn n kn
k k
L p p p
p p p
λ α β λ γ
α β
= =
= =
= − +
− − −
Where ( , , , )kn k kn kn
p λ α βL can be expressed by
( , , , )kn k kn kn
L p λ α β , as shown in (19):
1 1
( , , , ) ( , , , ) (16)N K
kn k kn kn kn k kn kn k k
n k
p L p Dλ α β λ α β λ
= =
= + L
And per-subcarrier optimization problem is:
2
1 1 1 1
min ( , , , )
min log (1 ) ( ) (17)
kn
kn
kn k kn knp
K K K K
kn k kn kn kn kn kn n knp
k k k k
L p or
p p p p p
λ α β
λ γ α β
= = = =
− + − − −
Using (11) and (12), equation (17) is reduced to:
2
1 1
min ( , , , ) min log (1 ) (18)kn kn
K K
kn k kn kn kn k kn knp p
k k
L p p pλ α β λ γ
= =
= − +
Due to OFDMA orthogonality constraint, each of the
subcarriers should be allocated to only one of K SUs. For
each of N subcarriers, the user that minimizes
2log (1 )kn k kn kn
p pλ γ− + is chosen, and subcarriers
allocation is performed as follows.
2arg min log (1 ) (19)
kn k kn knk
p p nλ γ− + ∀
The optimum value of should be found to reach the best
lower bound of the dual problem. This can be efficiently done
by iterative updating using a subgradient-based method or a
bisection method until the convergence of . In [9], a sub-
gradient method has been used, which suffers from slow
convergence. In our study, bisection method, which has a
faster convergence, is adopted to find the optimum value of
. According to (10), the optimal value of kλ is obtained so that
the achieved rate of SUs meets the guaranteed rate (
2
1
log (1 )N
kn kn k
n
p Dγ
=
+ = ). Otherwise, kλ converges to
maxλ and the problem does not have any solution. Our
devised algorithm, which is presented in table1, has two
phases. In phase 1, both power and subcarriers are allocated.
In phase 2, only power is allocated to subcarriers, based on the
subcarrier allocations obtained from phase 1.
TABLE I. THE PROPOSED ALGORITHM
The proposed resource allocation algorithm with throughput guarantee
1) Phase one: primitive power and subcarrier allocation
1-1: Initializing:
702
max 1max 2max max
min 1min 2min min
, ,..., , ,..., where issufficiently larg
, ,..., 0,0,..., 0
K
K
λ λ λ δ δ δ
λ λ λ
= =
= =
1-2: While sum( max min− ) > epsilon
a) max min
2
+
=
b) power will be allocated based on (14) c) subcarriers will be allocated based on (19)
d) rate is, ( ) ( )rate k log 1 p 2 kn knn k
= +
∈
e) updating :
if kminrate(k)kD =>
elseif kmaxrate(k)kD =<
if end.
1-3: While end.
2) Phase two: final power allocation 2-1: While (abs(D-rate)) > epsilon1
a) power will be allocated based on (14)
b) subcarriers allocation will be the same as phase1
c) recalculate the rate with powers achieved from part a
d) updating :
if kminrate(k)kD =>
elseif kmaxrate(k)kD =<
if end.
2-2: While end
End of algoritim.
The reason for applying a two-phase approach is as follows: according to (14) and (19), power and subcarrier
allocation is affected by kλ . When in each iteration
kλ
changes, it may lead to a different subcarrier allocation. In
bisection method, the searching space of kλ is divided by 2 in
each iteration. When kλ is so close to its optimum value, this
may cause a fluctuation between two specific subcarrier
allocations. This fluctuation will not let kλ reach to its optimal
value and provide the promised rate for GUs. Therefore, after achieving the desired accuracy, which is imposed to the algorithm by epsilon in phase 1, the subcarrier allocation is saved, and in phase 2, only power allocation is performed to satisfy the guaranteed rates.
IV. NUMERICAL RESULTS
An OFDM-based cognitive radio system with N 256
subcarriers and K 4 SUs is considered in our simulation.
The users’ power limits are generated from uniform
distribution [9]:
10 30
256 256~ ,p Un
The values of epsilon, epsilon1 are 0.0001, and 0.1
respectively. The channel gain is assumed to be Rayleigh
fading with an average power gain of 1dB. Since the channel
fading gains for different realizations can be different, an
average sum throughput of 10000 independent simulation runs is considered.
Our devised algorithm is compared with two resource
allocation algorithms, the Sum Throughput Maximization
Algorithm (STMA) from [9] and Uniform Resource
Allocation Algorithm (URAA) with the throughput guarantee.
To make a fair comparison, the available power for these two
algorithms is the same as consumed power of our devised
algorithm (named as Power Minimization Algorithm, PMA).
The STMA maximizes the sum throughput of the system
considering maximum tolerable interference introduced to PUs
from SUs, but throughput is not guaranteed for any of SUs.
The URAA tries to allocate resources among SUs so that the
guaranteed throughput is provided. Interference threshold for
PUs is also considered in this algorithm. In URAA, the total
available power is distributed among all subcarriers equally. If
the allocated power to each subcarrier is more than the power
threshold related to the interference constraint, then it is replaced with the value of the power threshold. Subcarriers are
allocated to the first SU until the guaranteed rate is achieved.
If free subcarriers are still available, this procedure is repeated
for the second SU, and so is for the third and finally, the forth
SU. In the next step, remaining subcarriers are equally
distributed among all SUs. Our devised algorithm, PMA, as
discussed before, minimizes the power consumption while
provide the guaranteed throughputs for all SUs.
Figure 2. Throughput of each user versus different values of guaranteed throughput for PMA
703
Figure 3. Throughput of each user versus different values of guaranteed
throughput for STMA
Figure 4. Throughput of each user versus different values of guaranteed
throughput for URAA
The throughput of each user for guaranteed throughput
equal to 10, 20, and 30bps/Hz is presented in Fig.2, Fig.3, and
Fig.4, for PMA, STMA, and URAA respectively. By
comparing these Figs it can be concluded that:
1- PMA, provide the guaranteed throughputs for all of
SUs as it was expected.
2- STMA, provide the highest sum throughput but it
does not provide the guaranteed throughputs. To be
more precise, while guaranteed throughput is
10bps/Hz only second SU achieves the guaranteed
throughput. For guaranteed throughput equal to
20ps/Hz, only third and forth SUs can meet the
throughput, and for 30bps/Hz, the first 3 SUs can
meet the guaranteed throughput.
3- URAA by consuming the same amount of power as
two previous algorithms can only provide the
guaranteed throughput for first SU. In this algorithm
all of the power is used for first and second SUs, so
that no power is allocated to the third and fourth SUs.
For better conclusion, the sum throughput of these three
algorithms is presented in table 2. From this table it can be
seen that:
1- Sum throughput of PMA is so close to sum
throughput of STMA, while PMA provide a
guaranteed throughput for all of SUs while STMA
does not guarantee the throughput.
2- With the same amount of water, URAA, which is a
common method of resource allocation (because it
distributes resources equally among all users); sum
throughput of URAA is not comparable with that of
PMA.
V. CONCLUSION
In this paper, we have developed an efficient resource
allocation algorithm that provides the guaranteed throughputs
for the SUs of CR with minimum power consumption while
the per-subcarrier interference introduced to PUs has been
limited to a certain threshold. Numerical results have shown
that the sum throughput of the proposed algorithm is close to
that of the STMA as the upper bound, and is much higher than
that of the URAA as the lower bound of the problem.
TABLE II. SUM THROUGHPUT OF ALGORITHMS
Guaranteed
Throughput
Algorithm Name
10 bps/Hz
20 bps/Hz
30 bps/Hz
Sum Throughput Maximization Algorithm
40.17
81.02
120.83
Power Minimization Algorithm
40
80
120
Uniform Resource Allocation Algorithm
12.80
28.44
51.75
ACKNOWLEDGMENT
This research was partly supported by Iran
Telecommunication Research Center (ITRC).
REFERENCES
[1] S. M. Mishra et al., “A real time cognitive radio testbed for physical and
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[2] Federal Communications Commission, “Spectrum Policy Task Force,” Rep. ET Docket no. 02-135, Nov. 2002.
[4] J. Mitola III and G. Q. Maguire, Jr. , “Cognitive radio: making software radios more personal,” IEEE Personal Communications., vol. 6, no. 4, pp. 13– 18, Aug. 1999.
[5] H. Arsalan, Cognitive Radio, Software Defined Radio and Adaptive Wireless Systems. Springer
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[6] T. Weis s and F. K. Jondral, “Spectrum pooling: an innovative strategy for the enhancement of spectrum efficiency” IEEE Commun. Mag., vol. 43, no. 3, pp. S8-S14, Mar. 2004.
[7] G. Bansal, Md. J. Hossain, and V. K. Bhargava, “Optimal and Suboptimal Power Allocation Schemes for OFDM-based Cognitive Radio Systems” in IEEE Trans. On Wireless commun. vol. 7, No. 11, pp. 4710-4718, Nov 2008.
[8] Chen. H .C, and Wang. L.C, “Joint Subcarrier and Power Allocation in Multiuser OFDM-based Cognitive Radio Systems” IEEE ICC 2010.
[9] Zhihua Tang, Guo Wei and Youtuan Zhu, “Weighted sum rate maximization for OFDM-based cognitive radio systems” in
[11] T. Weiss, J. Hillenbrand, A. Krohn, and F. K. Jondral, “Mutual interference in OFDM-based spectrum pooling systems,” in Proc. IEEE Vehicular Technol. Conf. (VTC’04), vol. 4, pp. 1873-1877, May 2004.
[12] Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press, Cambridge.
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