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Efficient Resource Allocation Algorithm with Rate
Requirement Consideration in Multicarrier-Based
Cognitive Radio Networks
Ling Zhuang, Lu Liu, Kai Shao, Guangyu Wang, and Kai Wang Chongqing Key Laboratory of Mobile Communication
Chongqing University of Posts and Telecommunications, Chongqing, 400065, China
Email: {zhuangling, shaokai}@cqupt.edu.cn; {liulucqupt, wkcqupt}@163.com; [email protected]
Abstract—In this paper, we present an efficient downlink
resource allocation algorithm with rate requirement
consideration in multicarrier-based cognitive radio (CR)
networks. The algorithm is divided into two steps aiming to
maximize the downlink capacity of the network under the total
power, the interference introduced to the primary users (PUs)
and the transmission rate of the secondary users (SUs)
constraints. The transmission rate requirement is considered by
ensuring the allocated number of subcarriers satisfies the
required number of subcarriers for every SU within the
subcarrier assignment step and turning the minimum
transmission rate threshold constraints into the minimum power
constraint on each subcarrier within the power allocation step.
Simulation results demonstrate that the proposed algorithm
provides a good guarantee of transmission rate requirement and
prove that filter bank multicarrier (FBMC) has higher efficiency
than orthogonal frequency division multiplexing (OFDM) in CR
networks.
Index Terms—Cognitive radio, multicarrier, rate requirement,
resource allocation
I. INTRODUCTION
Spectrum scarcity crisis exists for many wireless
applications, but investigations show that many licensed
frequency bands are far underutilized [1]. Cognitive
Radio (CR), a technique first introduced by Mitola [2], is
considered as a promising method to solve the spectrum
efficiency problem, in which unlicensed secondary users
(SUs) are allowed to use the licensed frequency bands
without causing a harmful interference to the licensed
primary users (PUs).
The multicarrier technique is a hot topic in 5G
communication systems [3] and has been suggested as a
promising candidate for CR systems due to its flexibility
in allocating resources among different SUs. Orthogonal
frequency division multiplexing (OFDM) is the common
transmission technique in current CR systems.
Nevertheless, with the large side lobes of its filter
frequency response, the OFDM-based CR system suffers
Manuscript received October 10, 2014; revised January 29, 2015. This work was supported by Sino-Finland Cooperation Project (No.
1018) and Chongqing Municipal Education Commission Project (No. KJ1400437).
Corresponding author email: [email protected].
from considerable amount of interference to the PUs [4].
Moreover, inserting the cyclic prefix (CP) in each OFDM
symbol decreases the system total capacity. As a
candidate for the physical layer in the future 5G
communications systems, filter bank multicarrier (FBMC)
can overcome the spectral leakage problem without any
CP extension by decreasing the side lobes of each
subcarrier, which leads to small interference to the PUs
and high efficiency [5].
In multicarrier-based CR systems, the resource
allocation problem is an attractive research issue and
many algorithms have been proposed with different
degree of success. In [6], the author focused on
maximizing the downlink capacity of the CR system
while keeping the interference induced to the PU bands
below the prespecified thresholds, but the total power
constraint was not considered and the optimal solution
obtained by the Lagrange method had a high
computational complexity. In order to reduce the
complexity, some suboptimal algorithms were proposed
to maximize the total system capacity under both total
power and interference introduced to the PUs constraints
[7]-[9], but the complexity still remained high. In [7], two
kinds of subcarriers were considered, the underlay
subcarriers and the overlay subcarriers. The equal amount
of power was allocated on each underlay subcarrier and
the overlay subcarriers were allocated power according to
a ladder profile. The proposed algorithm in [8] started
with an initial power allocation step, where an initial
power level was allocated to the subcarriers according to
four different criteria and then the subcarrier allocation
problem was formulated as a multiple-choice knapsack
problem. In [9], based on the structure of solutions that
were obtained through the Lagrangian duality, a
distributed algorithm was proposed, which only required
some limited cooperation among primary and secondary
networks while offering optimum performance. In [10],
an algorithm called PI algorithm was presented, where
the subcarriers were assigned based on the best channel
gain criterion, the interference constraints were converted
into the maximum power constraint on each subcarrier
and the optimization problem was solved using the
concept of the conventional water-filling with lower
complexity. However, the minimum transmission rate
Journal of Communications Vol. 10, No. 1, January 2015
16©2015 Engineering and Technology Publishing
doi:10.12720/jcm.10.1.16-23
constraint for each SU was not taken into consideration in
the algorithms proposed in [6]-[10].
In this paper, an efficient resource allocation algorithm
with rate requirement consideration in downlink
multicarrier-based CR systems is presented to maximize
the downlink capacity of the CR system while keeping
the total power and the interference induced to the PU
bands below the prespecified thresholds with the
consideration of the transmission rate of the SUs
constraints as well. The algorithm is divided into two
phases and the minimum transmission rate limit is
considered in both phases. The subcarriers to SUs
assignment is performed first according to the required
number of subcarriers for every SU using the proposed
RS algorithm. Then the power is allocated to the different
subcarriers using the proposed PIR algorithm, where the
minimum transmission rate constraints are transformed
into the minimum power constraint on each subcarrier
and the optimization problem is solved efficiently with
the modified “cap-limited” water-filling to reduce the
computational complexity. The efficiency of the proposed
algorithm will be investigated in both OFDM and FBMC-
based CR systems. The rest of this paper is organized as
follow. Section II presents the system model and
formulates the problem. The subcarrier and power
allocation algorithms are developed in Section III.
Simulation results are provided to evaluate the
performance of the proposed algorithm in Section IV.
Finally, Section V concludes the paper.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System Model
Consider the downlink of a multiuser multicarrier-
based CR network. The CR system’s frequency spectrum
is equally divided into K subcarriers each being assigned
a f bandwidth, where L PUs are in the licensed bands
and the others are the unlicensed bands. All subcarriers
are shared by M SUs, which means the SUs can use both
the unlicensed bands and the licensed PU bands.
Assume that each subcarrier goes under frequency flat
fading gains and the instantaneous fading gains are
perfectly known at the CR system. The transmission rate
of the kth subcarrier kR , the transmit power emitted by
the kth subcarrier kP , and the channel gain kh are related
via the Shannon capacity formula and is given by
2
2 2log 1
k k
k
k
P hR
(1)
where 2
k is the sum of the additive white Gaussian noise
(AWGN) variance and the interference introduced by PUs
on the kth subcarrier, which is assumed to be equal on
each subcarrier. In turn, given a transmission rate kR , the
required transmit power kP can be expressed as follows:
2
2
2 1kR
k
k
k
Ph
(2)
B. Modeling of Interference to Primary Users
The interference of the lth PU band introduced by the
kth subcarrier with unit transmission power l
kIF can be
written as [4]
22
2
( )k l
k l
d B
l l
k k k
d B
IF g f df
(3)
where kd is the spectral distance between the kth
subcarrier and the center frequency of the lth PU band, lB
is the bandwidth occupied by the lth PU, l
kg is the
channel gain between the kth subcarrier and the lth PU
band, ( )k f is the power spectrum density (PSD) of the
kth subcarrier and can be written as
2
( ) ( )k kf H f (4)
where ( )kH f is the frequency response of ( )kh n , which
denotes the shaping filter on the kth subcarrier. The
expression of ( )kH f depends on the used multicarrier
technique.
If an OFDM-based CR system is assumed, the
prototype filter ( )h n can be chosen as the rectangular
window with the length T K C in number of samples,
where K is the number of subcarriers and C is the length
of the CP in number of samples. Hence we get
1
2
1
( ) 2 cos 2T
kr
kH f T T r r f
K
(5)
If an FBMC-based CR system is assumed, the
prototype filter can be chosen as ( )h n with 0,1,...,n W ,
where W NK and N is the length of each polyphase
components (overlapping factor). Assuming that (0)h is
zero, and ( )h n have even symmetry around ( 2)h W [11],
we get
2 1
1
( ) 2 cos 22 2
W
kr
W W kH f h h r r f
K
(6)
C. Problem Formulation
The optimization objective is to maximize the total
capacity of the CR system under the total transmit power, the
interference introduced to the PUs and the transmission
rate of SUs constraints. The constrained optimization
problem can be formulated as follows:
,
2
. ,
, 2 21 1
1: max log 1k m
M Kk m k m
k mP
m k k
P hOP a
(7)
subject to
,m 0,1 , ,ka k m , (8)
Journal of Communications Vol. 10, No. 1, January 2015
17©2015 Engineering and Technology Publishing
,1
1, 1,2,...,M
k mm
a k K
(9)
, ,1 1
M K
k m k m Tm k
a P P
(10)
, 0, ,k mP k m (11)
, ,1 1
, 1,2,...,M K
l l
k m k m k thm k
a P IF I l L
(12)
2
, ,
, 2 21
log 1 , 1,2,...,K
k m k m m
k m thk k
P ha R m M
(13)
where ,k ma denotes the subcarrier allocation indicator, i.e.
, 1k ma only if the kth subcarrier is allocated to the mth
SU. Inequality (9) ensures that each subcarrier can be
allocated to one SU at most. ,k mP is the transmit power on
the kth subcarrier, ,k mh is the channel gain from the base
station to the mth SU on the kth subcarrier, PT is the total
SUs power budget, l
thI is the interference threshold of the
lth PU, and m
thR is minimum transmission rate limit of the
mth SU.
III. DOWNLINK SUBCARRIER AND POWER ALLOCATION
The optimal solution for the combinatorial
optimization problem OP1 has a high computational
complexity which grows exponentially with the input size.
The problem is solved in two steps to reduce the
computational complexity, where in the first step, the
subcarriers are assigned to the SUs and then the power is
allocated to these subcarriers in the second step. Once the
subcarriers assignment is complete, the multiuser
multicarrier system can be viewed virtually as a single
user multicarrier system which makes the optimization
problem computationally simpler.
A. Subcarrier Assignment
As proved in [12], the CR system can obtain the
maximum transmission rate in downlink if the subcarriers
are assigned to the SU with the best channel gain on that
subcarrier, however, it will lead to the situation that some
SUs cannot reach to the required transmission rate. In this
section, we present a subcarrier assignment algorithm, in
which the subcarriers are assigned by jointly considering
the channel gain and the subcarrier requirement of each SU.
22
2log 1m
m th m mK R P h
is the estimated
number of subcarriers required for the m th SU, where
m TP P K is the average power allocated to each
subcarrier of the mth SU, ,1
K
m k mkh h K
is the average
channel gain on each subcarrier for the mth SU, and 2 2
k is the AWGN variance. In the Requirement-
based Subcarrier (RS) assignment algorithm, referred to as
RS algorithm, the subcarriers are assigned successively
based on the best channel gain criterion in [12] and if the
number of subcarriers allocated to the mth SU mK satisfies
the subcarrier requirement of that SU mK , the subcarriers
assignment for it will stop. When the subcarrier
requirement is met for every SU, the remaining subcarriers
are allocated according to the previous criterion. The
implementation of the RS algorithm is described in Table I.
B. Power Allocation
By subcarrier assignment, the values of the subcarrier
allocation indicator ,k ma are determined and hence for
notation simplicity, single user notation can be used. The
channel gain on the kth subcarrier kh is determined as
follows:
, ,1
M
k k m k mm
h a h
(14)
It can be assumed that each subcarrier belongs to the
closest PU band and only introduces interference to it
according to the fact mentioned in [4] that most of the
interference of the PU bands is caused by cognitive
transmission on the subcarriers which are in the PU bands
as well as directly adjacent to the PU bands. And hence
the optimization problem OP1 in (7) can be reformulated
as follows:
2
2 21
2 : max log 1k
Kk k
Pk k
P hOP
(15)
subject to
1
K
k Tk
P P
, (16)
0, 1,2,...,kP k K , (17)
, 1,2,...,l
l l
k k thk K
P IF I l L
, (18)
2
2 2log 1 , 1,2,...,
m
k k m
thk K k
P hR m M
, (19)
where lK denotes the set of the subcarriers belong to the
lth PU band and mK denotes the set of the subcarriers
assigned to the m th SU.
TABLE I: RS ALGORITHM
1. Initialize , 0, ,k ma k m
2. for 1k to 1
M
mmK
do
*
,arg max k mm
m h , *,1
k ma , * * 1
m mK K
if * *m mK K
let *,0, 1,2,...,
k mh k K
end if
end for
3. for 1
1M
mmk K
to K do
*
,arg max k mm
m h , *,1
k ma
end for
Journal of Communications Vol. 10, No. 1, January 2015
18©2015 Engineering and Technology Publishing
Theorem 1: The solution to the optimization problem
OP2 is given by
2
*
2
1 m kk l
l k k
PIF h
(20)
where max(0, )x x and , , 1,2, ,l l L , ,m
1,2,...,m M are the Lagrange multipliers for (16),
(18), (19), respectively. Three Lagrange multiplies can be
obtained based on interior point method with a complexity 3( )O K [13].
Proof: See Appendix.
The optimal solution is computationally complex so
that it is unsuitable for practical wireless communication
systems, and hence a low complexity algorithm will be
proposed. If the total power and the transmission rate
constraints are ignored, applying the Lagrange multiplier
method, the optimal solution can be written as
2
2
1IF kk IF l
l k k
PIF h
(21)
where , 1,2,...,IF
l l L is the Lagrange multiplier for
(18) and can be calculated by substituting (21) into
l
IF l l
k k thk K
P IF I
to get
22
l
lIF
ll l
th k k kk K
K
I IF h
(22)
Consider the optimization problem whose optimization
goal is to minimize the total transmit power of the CR
system subject to the transmission rate of the SUs
constraints. According to (2), it can be formulated as
2
21
2 13 : min
TRk
TRk
RK k
R kk
OPh
(23)
subject to
0, 1,2,...,TR
kR k K (24)
, 1,2,...,m
TR m
k thk K
R R m M
(25)
Applying the Lagrange multiplier method, the optimal
solution can be written as
2
2ln lnTR TR k
k m
k
Rh
(26)
where , 1,2,...,TR
m m M is the Lagrange multiplier for
(25) and can be calculated by substituting (26) into
m
TR m
k thk KR R
to get
22ln
exp m
m
th k kk KTR
m
m
R h
K
(27)
Therefore, TR
kP can be obtained by substituting (26) into
(2).
In order to solve the optimization problem OP2, the
Power Interference Rate (PIR) constrained algorithm,
referred to as PIR algorithm, is proposed. We can start by
assuming that the maximum power which can be
allocated to a given subcarrier Max
kP is determined
according to the interference constraints only by using
(21) and (22) for every subcarrier , 1,2,...,lk K l L .
By such an assumption, we can guarantee that the
interference introduced to the PU bands will not exceed
the prespecified thresholds. Moreover, assume that the
initial minimum power which can be allocated to a given
subcarrier Min
kP is determined according to the
transmission rate constraints only by using (2), (26) and
(27) for every subcarrier , 1,2,...,mk K m M . The
results that violate the maximum power Max
kP are
determined with the upper bound Max
kP and the minimum
transmission rate threshold of each SU is reduced by
subtracting the transmission rate that SU have obtained so
far. Then use (2), (26) and (27) to update the minimum
power Min
kP of the subcarriers that did not violate the
maximum power Max
kP . This procedure is repeated until
the minimum power Min
kP does not violate the maximum
power Max
kP on any of the subcarriers in the new iteration.
By such an assumption, we can guarantee that the
transmission rate of the SUs will not lower than the
prespecified thresholds.
Once the maximum power Max
kP and the minimum
power Min
kP are determined, the total power constraint TP
is tested. If 1
K Min
T kkP P
, it is obvious that there does
not exist any solution to OP2, the problem is studied
except for the minimum power constraints [14].
Moreover, if 1
K Max
k TkP P
, then the solution is equal to
the tolerated maximum power allocated to each subcarrier,
i.e. * Max
k kP P . In most of the cases,
1 1
K KMin Max
k T kk kP P P
, the following problem
should be solved:
2
2 21
4 : max log 1WFk
WFKk k
P k k
P hOP
(28)
subject to
1
KWF
k Tk
P P
(29)
, 1,2,...,Min WF Max
k k kP P P k K (30)
The problem can be solved efficiently using the
modified “cap-limited” water-filling. To begin with, the
total power budget PT is reduced by subtracting the
Journal of Communications Vol. 10, No. 1, January 2015
19©2015 Engineering and Technology Publishing
summation of the minimum power Min
kP , the maximum
power Max
kP is reduced by subtracting the minimum
power Min
kP and the height of the container’s bottom
22
k k kd h is increased by adding the minimum
power Min
kP . Then we execute the “cap-limited” water-
filling [14] and obtain the result CWF
kP . The final solution
WF
kP of the problem OP4 is the result of the “cap-
limited” water-filling CWF
kP plus the minimum power
Min
kP . The implementation of the PIR algorithm is
described in Table II.
TABLE II: PIR ALGORITHM
1. Initialize l lO K ,
m m m mU V W K , m
m thQ R , TS P
2. Find the Max
kP as follows:
1) 1,2,...,l L , sort 22 ,l
k k k k lT IF h k O in
decreasing order with i being the sorted index
2) l
sum kk OT T
, IF l
l l th sumO I T , 1n
3) while 1
( )
IF
l i nT do
( )sum sum i nT T T , \ ( )l lO O i n ,
IF l
l l th sumO I T , 1n n
end while
4) Set 221Max IF l
k l k k kP IF h
, Max
k kP P
3. Find the Min
kP as follows:
1) 1,2,...,m M , sort 22ln ,k k k mH h k W
in decreasing order with j being the sorted index
2) m
sum kk WH H
, expTR
m m sum mQ H W ,
1n
3) while (n)ln TR
m jH do
( )sum sum j nH H H , \ ( )m mW W j n ,
expTR
m m sum mQ H W , 1n n
end while
4) Set lnMin TR
k m kR H
, 222 1kRMin
k k kP h ,
mk V
5) Repeat
if Min
k kP P
let Min
k kP P , 2 2
2log 1 Min
m m k k kQ Q P h ,
\m mV V k , m mW V , and go to step 1)
end if
until ,Min
k k mP P k U
4. if 1
K
kkP S
let *
k kP P and stop the algorithm
end if
5. Let 1
K Min
kkS S P
, Min
k k kP P P ,
2 22 2 Min
k k k k kh h P , execute the “cap-limited” water-
filling and set WF CWF Min
k k kP P P , * WF
k kP P
The computational complexity of step 2 in the proposed
PIR algorithm is 1
( log ) ( log )L
l llO K K O K K
. Step
3 has a complexity of 1
( log )M
m m m mmO K K K
( log max( ) )mO K K K , where m mK is the
number of the iterations. Step 5 of the algorithm executes
the modified “cap-limited” water-filling which has a
complexity of ( log )O K K K , where K is the
number of the iterations. Thus, the overall complexity of
the algorithm is lower than ( log )O K K K , where
0,5 is estimated via simulation. Comparing to the
computational complexity of the optimal solution, 3( )O K , the proposed PIR algorithm has much lower
computational complexity, especially for large number of
subcarriers.
IV. PERFORMANCE EVALUATION
A. Simulation Configuration
Simulations are performed with MATLAB 7.8.0 to
evaluate the performance of the proposed algorithm.
Consider a CR system of 2L PUs that each has six
subcarriers, 3M SUs and 32N subcarriers. The
bandwidth of subcarriers f is selected as 0.3125 MHz
and the AWGN variance 2
k is 610 . The channel gains h
and g are outcomes of independent, identically distributed
Rayleigh distributed random variables with mean of 1. The
OFDM and FBMC-based CR systems are evaluated. The
OFDM system is assumed to have a 6.67% of its symbol
time as the CP. For FBMC system, the prototype filter
coefficients are assumed to be equal to PHYDYAS
coefficients with overlapping factor 4N [15].
Furthermore, 1 2
th th thI I I and 1 2 3
th th th thR R R R are
assumed.
The efficiency of the proposed algorithm, which
consists of RS algorithm and PIR algorithm, is compared
with the PI algorithm proposed in [10] without the step that
make the allocated power satisfy approximately the
interference constraints. The PI algorithm maximizes the
downlink capacity of the CR system under only two
constraints, the total power and interference induced to
the PUs constraints. In the subcarrier assignment step, it
always assigns the subcarriers based on the best channel
gain criterion in order to obtain the highest
transmission rate. In the power allocation step, it turns the
interference constraints into the maximum power
constraint on each subcarrier and uses the concept of the
conventional water-filling to solve the optimization
problem.
B. Simulation Results
The capacity of the CR system using proposed and PI
algorithms for different interference thresholds with
0.5TP W and 120thR bit/s/Hz is plotted in Fig. 1. It
Journal of Communications Vol. 10, No. 1, January 2015
20©2015 Engineering and Technology Publishing
can be observed that the capacity increases as the
interference threshold increases. The proposed algorithm is
slightly inferior to the PI algorithm. This is because in the
subcarrier assignment step, the subcarriers are allocated
based on the requirement using the proposed algorithm, in
which the subcarriers are not always assigned to the SU
with the best channel gain, unlike the PI algorithm,
which only takes the best channel gain into account to
obtain the maximum capacity. The appearance of steep
drop is due to the guarantee that each SU can reach to the
required transmission rate in the power allocation step using
the proposed algorithm which is not the case in the PI
algorithm. The proposed algorithm transforms the
minimum transmission rate constraints into the minimum
power constraint on each subcarrier and then solves the
optimization problem with the modified “cap-limited”
water-filling, so that it may allocate the power to the
subcarriers with poor channel gain, which leads to the
drop in capacity.
In Fig. 1, the capacity achieved using the proposed
algorithm is not as ideal as that achieved using the PI
algorithm. The reason is that in the proposed algorithm,
the subcarriers may be assigned to the SUs with poor
channel gain owing to the subcarrier requirement of each
SU within the subcarrier assignment step and the power is
probably allocated to the subcarriers with poor channel
gain on account of the minimum power constraint on
each subcarrier within the power allocation step. Fig. 1
also shows that the capacity of FBMC-based CR system
is higher than that of OFDM-based CR system because
the side lobes of the subcarrier’s PSD in FBMC system
are smaller than that in OFDM one which introduces less
interference to the PU bands. Another reason is the
inserted CP in OFDM-based CR system which can
reduce the system capacity. The significant advantage in
the capacity of FBMC-based CR system over the OFDM-
based one recommends the FBMC as a candidate for the
physical layer in the future CR systems.
By sacrificing a little system capacity, the proposed
algorithm has a good guarantee of transmission rate
requirement as shown in Fig. 2, which plots the
transmission rate of a specific SU versus the required
transmission rate of the SU with 0.5TP W and 10thI
mW for different algorithms. It can be noted that the
practical transmission rate of the SU varies with the
transmission rate requirement and always strictly satisfies
it in the proposed algorithm while it remains constant
regardless of the transmission rate requirement in the PI
algorithm. This is because in the proposed algorithm, with
the change of the required transmission rate, both the
assignment of the subcarriers to SUs in the first step and
the allocation of the power to subcarriers in the second
step will vary, which leads to a different transmission rate.
Fig. 3 illustrates the transmission rate per SU versus
SU index with 0.5TP W, 10thI mW and 120thR
bit/s/Hz using different algorithms. We can observe that all
three SUs can satisfy its transmission rate requirement in
the proposed algorithm. Nevertheless, in the PI algorithm,
only SU 2 can achieve the required transmission rate since
there are not any efficient transmission rate requirement
assurance schemes in it. More subcarriers and power are
allocated to SU 2 in virtue of its better channel gains,
which results in that the transmission rate of SU 2 is
much higher than the required transmission rate and the
other two SUs cannot reach to it.
Fig. 1. Capacity of secondary users versus interference threshold for OFDM and FBMC-based CR systems.
Fig. 2. Transmission rate versus required transmission rate of a specific secondary user for OFDM and FBMC-based CR systems.
Fig. 3. Transmission rate per secondary user versus secondary user index for OFDM and FBMC-based CR systems.
Journal of Communications Vol. 10, No. 1, January 2015
21©2015 Engineering and Technology Publishing
Fig. 4 shows the instantaneous transmission rate of a
specific SU over time for different algorithms with 0.5TP
W, 10thI mW and 120thR bit/s/Hz. In both
algorithms, the instantaneous channel gains are exactly the
same. It can be noted that the proposed algorithm keeps the
instantaneous transmission rate above 120thR bit/s/Hz
all the time within this observation period while the PI
algorithm cannot. This demonstrates that the proposed
algorithm can achieve a better guarantee of transmission
rate requirement than the PI algorithm.
Fig. 4. Instantaneous transmission rate of a specific secondary user for OFDM and FBMC-based CR systems.
V. CONCLUSION
In this paper, an efficient resource allocation algorithm
with rate requirement consideration for downlink in
multicarrier-based CR networks is proposed. The
proposed algorithm maximizes the total downlink capacity
of the multicarrier-based CR networks while respecting
the available power budget and guaranteeing that no
excessive interference is introduced to the PU bands and
each SU can achieve the required transmission rate. The
minimum transmission rate limit is considered in both
subcarrier assignment and power allocation steps. In the
subcarrier assignment step, the subcarriers are allocated to
the SUs based on the required number of subcarriers for
every SU using the proposed RS algorithm. In the power
allocation step, the proposed PIR algorithm is performed,
where the minimum transmission rate constraints are
converted into the minimum power constraint on each
subcarrier and the modified “cap-limited” water-filling is
executed to solve the optimization problem efficiently.
Simulation results prove that the proposed algorithm has a
significant advantage in the strict guarantee of
transmission rate requirement over the PI algorithm with a
little expense of the system capacity. They also
demonstrate that the capacity of FBMC-based CR system
is more than OFDM-based one and recommend FBMC as
a candidate for transmission in CR networks. In our future
work, we plan to extend the applications of 5G in CR
networks.
APPENDIX PROOF OF THEOREM 1
The problem OP2 can be considered as a convex
optimization problem, which can be solved using the
Lagrange multiplier method [13] and the Lagrange
function can be written as
2
2 21 1
log 1K K
k k
k Tk kk
P hL P P
1 1
2
2 21
log 1
0, 0, 0, 0,
l
m
K Ll l
k k l k k thk l k K
Mk km
m thm k K k
k l m
P P IF I
P hR
(31)
where , , 1,2,...,k k K , , 1,2,...,l l L and
, 1,2,...,m m M are the Lagrange multipliers.
According to the Karush-Kuhn-Tucker (KKT)
conditions, we have
* 0, 1,2,...,kP k K
0, 1,2,...,k k K
* 0, 1,2,...,k kP k K (32)
* 22 *
10lm
k l k
k k k k
LIF
P h P
Eliminate k from (32) and rewrite the KKT
conditions as follows:
* 0, 1,2,...,kP k K
22 *
1, 1,2,...,l m
l k
k k k
IF k Kh P
(33)
*
22 *
10, 1,2,...,l m
l k k
k k k
IF P k Kh P
.
If 2 21l
l k m k kIF h , the second condition
in (33) holds if * 0kP , which implies
2
*
2
1 m kk l
l k k
PIF h
(34)
Moreover, if 2 21l
l k m k kIF h , we assume
* 0kP , which implies 2 21l
l k m k kIF h
22 *1 m k k kh P and violates the third condition
in (33). Thus, the only possible solution in this case is * 0kP .
Therefore, the optimal solution can be written as
follows:
Journal of Communications Vol. 10, No. 1, January 2015
22©2015 Engineering and Technology Publishing
22
2 2
*
2
2
11, if
10, if
m klm kl kl
l k kk
k
m kl
l k
k
hIF
IF hP
hIF
(35)
which is equal to
2
*
2
1 m kk l
l k k
PIF h
(36)
ACKNOWLEDGMENT
This work was supported by Sino-Finland Cooperation
Project (No. 1018) and Chongqing Municipal Education
Commission Project (No. KJ1400437).
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Ling Zhuang is an associate professor since
2010 in the College of Communication and
Information Engineering, Chongqing
University of Posts and Telecommunications
(CUPT), China. She received her Ph.D.
degree in 2009 in Control Theory and Control
Engineering from the Chongqing University,
her M.S. in 2003 in Communication and
Information System and her 4-year B.S.
degree in 2000 in Communication Engineering from CUPT. Her main
research interests are cognitive radio communications and networking,
multicarrier modulation technique for mobile communication, multi-rate
digital signal processing, and filter bank theory.
Lu Liu was born in Hunan Province, China,
in 1990. She received the B.S. degree from
CUPT in 2011. Currently she is working on
her M.S. degree in Chongqing University of
Posts and Telecommunications. Her research
interests are cognitive radio communications
and networking, multicarrier modulation
techniques for mobile communication, and
filter bank theory.
Kai Shao is an Associate Professor since
2012 in the College of Communication and
Information Engineering, CUPT. His main
research interests are filter bank theory,
multicarrier modulation technique for mobile
communication, and related topics.
Guangyu Wang is a Professor in the College
of Communication and Information
Engineering, CUPT. He received a Ph.D.
degree in 1999 in Electrical Engineering from
the Kiel University, Germany, an MSc degree
in 1988 in Telecommunication Engineering
from Beijing University of Posts and
Telecommunications, China, and a 4-year BS
degree in 1985 from CUPT. His main research
interests include high-speed signal processing and multi-rate filter bank
theory.
Kai Wang was born in China, in 1989. He
received his B.S. degree from Hubei
University of Arts and Science, China in 2012.
Currently he is working on his M.S. degree in
CUPT. His research interests are multicarrier
modulation techniques for mobile
communication and multi-rate digital signal
processing.
Journal of Communications Vol. 10, No. 1, January 2015
23©2015 Engineering and Technology Publishing