21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications

Fair and Efficient Resource Allocation Algorithm for Uplink Multicarrier Based Cognitive Networks Musbah Shaat and Faouzi Bader Centre Tecnol`ogic de Telecomunicacions de Catalunya (CTTC) Parc Mediterrani de la Tecnolog´ıa, Av. Carl Friedrich Gauss 7, 08860 , Castelldefels-Barcelona, Spain. Phone: +34 93 6452911, Fax: +34 93 6452900, Email:{musbah.shaat,faouzi.bader}@cttc.es

Abstract— This paper presents an efficient and fair uplink resource allocation algorithm in multicarrier based cognitive radio systems. The proposed resource allocation is divided into two steps. The subcarriers to user assignment is first performed and then the power is allocated to the different subcarriers. The subcarriers are allocated based on the channel quality, amount of interference imposed to the primary bands, instantaneous rate achieved by every user and the increment in the total data rate. The fairness among users is considered within the subcarrier allocation by reducing the probability of having users whose instantaneous rates are below a given minimum rate. The power is distributed among subcarriers so that the total data rate is maximized while the interference introduced to the primary system is guaranteed to be under the prescribed interference temperature limit. Simulation results confirm the efficiency of the proposed algorithm.

I. I NTRODUCTION Cognitive radio (CR) has been proposed to improve the spectrum utilization by using the vacant channels without causing unacceptable interference to the licensed system. Multicarrier communications has been suggested as a candidate for the CR systems [1]. The problem of uplink resource allocation has been already studied in classical multicarrier systems [2], [3] and references therein. The algorithms used in classical (non-cognitive) multicarrier systems are not efficient for the CR ones due to the existence of the interference constraints. In [4], a single carrier CR system is analyzed for both downlink and uplink scenarios. The results can be used -with some modifications- to find the optimal power allocation in multicarrier based CR system when the subcarrier channel allocation is already known. There is few existed research on the subcarrier and power allocation in uplink multicarrier based CR systems. In [5], the author proposed an algorithm for jointly allocating channels and powers among different users under the individual users power constraints. The problem is relaxed to get a convex version and then the solution is quantized to yield a binary channel allocation. Afterwards, the solution is modified to consider the interference constraints to the licensed system. Wang et al. proposed in [6] an algorithm to allocate resources This work was partially supported by the European ICT-2008-211887 project PHYDYAS, COST Action IC0902 and Generalitat de Catalunya under grant 2009-SGR-940.

978-1-4244-8015-9/10/$26.00 ©2010 IEEE

1210

CR base station (CBS)

Transmission Interference

Secondary User (SU1)

(SU3) (PU2) Primary User (PU1)

Fig. 1.

(SU2)

Uplink Cognitive Radio Network

in the uplink OFDMA based CR systems under the persubcarrier power constraints. The subcarriers are allocated initially to the users with the best channel quality and then adjusted according to the different user’s waterfilling levels. The algorithm has high computational complexity and limited performance. The instantaneous fairness among users was not taken into consideration for both of the algorithms proposed in [5] and [6]. In this paper, we propose an efficient resource allocation algorithm in multicarrier based CR systems. The objective is to maximize the total CR data rate while limiting the interference introduced to the primary system. The fairness among users is considered within the subcarrier allocation by reducing the probability of having users whose instantaneous rate are below a given value. The proposed resource allocation is divided into two steps. The subcarriers to user assignment is first performed and then the power is allocated to the different subcarriers. The subcarrier to user assignment scheme proposed in [3] for non-cognitive systems is developed in order to be suitable for the cognitive ones. The subcarriers are allocated based on the channel quality, amount of interference imposed to the primary bands, instantaneous rate achieved by every user and the increment in the total data rate. For the power allocation step, an efficient power allocation algorithm is proposed to distribute the powers among the subcarriers under the per-user total power and interference constraints. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION In this paper, the uplink scenario will be considered. As shown in Fig. 1, the CR system coexists with the primary system in the same geographical location. The unlicensed users [referred to as secondary users (SUs)] is able to opportunistically access the available spectrum holes without

B1

Active PU1 band 1 2

Fig. 2.

NonActive band

……….

B2

BL

Active PU2 band

PUL band

Active Frerquency

N

Δf

Frequency distribution of the active and non-active primary bands

causing harmful interference to the licensed users [referred to as primary users (PU)]. The CR system’s frequency spectrum is divided into N subcarriers each having a Δf bandwidth. The side by side frequency distribution of the PU bands and the CR bands will be assumed (see Fig. 2). The frequency bands B1 , B2 ,· · · ,BL has been occupied by the PUs (active PU bands) while the other bands represent the inactive PU bands (CR bands). It is assumed that the CR system can use the inactive PU bands provided that the total interference l , where introduced to the lth PU band does not exceed Ith l l Ith = Tth Bl denotes the maximum interference power that can l is the interference temperature be tolerated by the P Ul and Tth limit for P Ul . Assume that Φi is the power spectrum density (PSD) of the ith subcarrier. The expression of the PSD depends on the used multicarrier technique. If an OFDM based CR is assumed, the PSD of the ith subcarrier can be written as [7] 2 sin πf Ts Φi (f ) = Pi Ts (1) πf Ts where Pi is the total transmit power emitted by the ith subcarrier and Ts is the symbol duration. The mutual interference introduced by the ith subcarrier to lth PU, Iil (di , Pi ) , is the integration of the PSD of the ith subcarrier across the lth PU band, Bl , and can be expressed as [7] di +B l /2

Iil

(di , Pi ) =

l 2 gi Φi (f ) df = Pi Ωli

(2)

where di is the spectral distance between the ith subcarrier and the lth PU band. gil denotes the channel gain between the ith subcarrier and the lth PU band while Ωli denotes the interference factor of the ith subcarrier to the lth PU band. The interference power introduced by the lth PU signal into the band of the ith subcarrier is [7] Jil =

l 2 jw y i ψl e dw

where hi,m is the ith subcarrier fading gain from the CBS to L

2 2 Jil where σAW the mth SU. σi2 = σAW GN + GN is the l=1

mean variance of the additive white Gaussian noise (AWGN) and Jil is the interference introduced by the lth PU’s band into the ith subcarrier and can be evaluated using (3). The noise variance is assumed to be constant for all the users and subcarriers. The fairness among the SUs is guaranteed by assuming that every SU has the minimum instantaneous rate Rmin . Our objective is to maximize the total capacity of the CR system subject to the instantaneous interference introduced to the primary system, the per-user transmit power constraints and the per-user minimum rate constraints. Therefore, the optimization problem can be formulated as follows P1 :

di −Bl /2

di +Δf /2

can be obtained by the CBS/SUs via, e.g., estimating the received signal power from each primary terminal when it transmits, under the assumptions of pre-knowledge on the primary transmit power levels and the channel reciprocity [4]. Based on this information, the CBS assigns the subcarrier and power to each SU through a reliable low-rate signaling channel. Let ai,m to be the subcarrier allocation indicator, i.e. ai,m = 1 if and only if the ith subcarrier is allocated to mth user. It is assumed that each subcarrier can be used for transmission to at most one user at any given time. The transmission rate of the ith subcarrier, Ri , with the transmit power Pi,m can be evaluated using the Shannon capacity formula and is given by 2 Pi,m |hi,m | Ri (Pi,m , hi,m ) = Δf log2 1 + (4) σi2

(3)

di −Δf /2

where ψl ejw is the power spectrum density of the lth P U signal and yil is the channel gain between the ith subcarrier and lth PU signal. It will be assumed that all the instantaneous fading gains are perfectly known at the cognitive base station (CBS) and there is no synchronization between the PUs and SUs. The channel gains between the SUs and the CBS can be obtained practically by the classical channel estimation techniques while the channel gains between the CBS/SUs and the PUs

1211

s.t.

max

M

N

Pi,m ,ai,m m=1 i=1 N M

m=1 i=1 N

ai,m Ri (Pi,m , hi,m )

l ai,m Pi,m Ωli,m ≤ Ith

ai,m Pi,m ≤ Pm ,

i=1

Pi,m ≥ 0, ai,m ∈ {0, 1} , M

ai,m ≤ 1,

m=1

R (m) ≥ Rmin ,

∀l ∈ {1, · · · , L}

∀m ∀i, m ∀i, m ∀i ∀m

(5) where N denotes the total number of subcarriers while M denotes the number of SUs. L is the number of active PU bands l is the interference threshold prescribed by the lth PU. and Ith Pm is the mth SU total power budget. R (m) is the mth SU N

ai,m Ri (Pi,m , hi,m ). instantaneous rate, i.e. R (m) = i=1

The optimization problem P 1 is a NP-hard optimization problem in which achieving the optimal solution needs unacceptable computational complexity. In order to reduce the computational complexity, the resource allocation problem can be solved in two steps. In the first step, the subcarriers are assigned to the users and then the power is allocated for these

subcarriers in the second step. Depending on the values of l , Pm and the channel gains, the minimum rate Rmin may Ith not be satisfied for all the users. Therefore, the probability of having users whose rates are below the minimum rate should be reduced [3]. The outage probability can be defined as Poutage = P r{Mlow ≥ 1}

(6)

where Mlow is the number of SUs whose instantaneous rate below Rmin . In the sequel, the proposed subcarrier to user assignment scheme with low outage probability is introduced and then an efficient power allocation algorithm is presented. III. P ROPOSED S UBCARRIER A LLOCATION A LGORITHM WITH FAIRNESS C ONSIDERATION The optimal downlink subcarriers to users allocation scheme in multicarrier systems is achieved by allocating each subcarrier to the user with the maximum signal to noise ratio (SNR). This scheme of subcarrier allocation is not efficient in uplink case due to the per-user power constraints. Moreover, the interference introduced to the primary system by each SU should be considered in CR context which makes the schemes used in classical multicarrier systems not efficient. In this section, a heuristic subcarrier allocation algorithm is presented. The proposed algorithm tries to assign the subcarriers to the different SUs considering not only their channel quality and per-user power constraints but also the interference that will be induced to the PU bands. Moreover, the proposed scheme tries to minimize the probability of having users with instantaneous rates below the minimum rate. The scheme initially assumes that the interference introduced to the lth PU band is divided uniformly among the different CR subcarriers which means that every subcarrier is allowed to introduce the samel amount of interference to the I PU band, i.e. IUl nif orm = Nth . Considering L interference constraints and by using (2), the maximum power that can be allocated to the ith subcarrier when it is allocated to the mth SU is IU1 nif orm IULnif orm U ni = min{ , · · · , } (7) Pi,m Ω1i,m ΩL i,m Let C to be the set of unassigned subcarriers and U to be the set that contains the indices of the users whose rates below the minimum. Define the sets Am and Bm to include all the subcarriers already allocated to the mth SU according to U ni , and average power respectively. maximum power, i.e Pi,m The average power means that the left power for the mth user

U ni Pi,m ) is divided equally among the subcarriers (Pm − i∈Am

in the set Bm . The algorithm begins by the allocation of the subcarriers that are located next to the PU bands, i.e. subcarriers that have more interference to the PU bands, and moving towards the distant ones. The assigning procedures of a particular subcarrier i∗ ∈ C are as follows 1) if U is empty, let Z = {1, · · · , M } else Z = U 2) ∀m ∈ Z,

Evaluate PT est =

Pm −

r∈Am

U ni Pr,m

|Bm |+1

1212

ni if PT est ≥ PiU∗ ,m let A m = Am ∪ {i∗ } and B m = Bm else let B m = Bm ∪ {i∗ } and A m = Am . |X | means the cardinality of the set X . 3) Compute the amount of increment Δm in the data rate when the subcarrier {i∗ } is assigned to mth SU, i.e, new old Δm = Rm − Rm

where new Rm

=

P

m−

r∈A m |B m |

U ni Pr,m

Ri U ni

Ri Pi,m , hi,m

i∈B m

, hi,m +

i∈A m

old Rm =

n∈Bm i∈Am

Ri

Pm −

r∈Am

|Bm |

U ni Ri Pi,m , hi,m

U ni Pr,m

, hi,m +

and Ri (Pi,m , hi,m ) is evaluated using (4). 4) Find m∗ satisfying m∗ = arg maxm (Δm ), set ai∗ ,m∗ = 1, and update the sets Am∗ = A m∗ and Bm∗ = B m∗ . If R (m∗ ) ≥ Rmin , remove m∗ form the set U. 5) Remove the subcarrier i∗ form the set C and repeat the above procedures until the set C is empty. The first step determines whether all the users are having the minimum rate or not. In case that there are users whose rates below the minimum, the subcarrier allocation will be confined within these users. If the minimum rate constraints are satisfied for all the users, i.e. U is empty, the algorithm will be performed considering that the subcarrier can be allocated to any one of the SUs. The second step considers the limitation that will be introduced to any subcarrier assignment due the interference constraints. Generally speaking, the subcarriers that will introduce high interference to the PU bands will have a low transmitting power which will reduce the total data rate even that they have a good channel quality. Hence, the algorithm initially assign the subcarrier to the group Bm and evaluate the average power. If the average power exceeds the U ni , then the subcarrier should be moved to maximum power Pi,m the group Am . Therefore, the second step helps in classifying the subcarriers according to their interference to the PU bands. Afterwards, the increments of the individual data rates due to the allocation of a particular subcarrier to different SUs are evaluated and the subcarrier is allocated to the SU with maximum data rate increment. The scheme is repeated until the allocation of all subcarriers. Note that the final set of allocated subcarriers to mth SU is Nm = Am ∪ Bm . IV. P ROPOSED P OWER A LLOCATION A LGORITHM By the subcarrier to users assignment step, the subcarriers are allocated to the different users with the consideration of the minimum rates constraints. Therefore, the values of the

subcarrier indicators, i.e. ai,m , are already known and the power allocation problem can be formulated as follows P2 : s.t.

max

N

Pi,m i=1 N

i=1

(Int)

αl∗

Ri (Pi,m , hi,m )

l Pi,m Ωli,m ≤ Ith

Pi,m ≤ Pm

i∈Nm

Pi,m ≥ 0

∀l ∈ {1, · · · , L}

(8)

∀m

∀i

N ∗

∗ l l Ri Pi,m , hi,m + L α P Ω − I l i,m i,m th l=1 i=1 i=1 M N

∗ ∗ βm Pi,m − Pm − Pi,m μi + N

m=1

i=1

i∈Nm

(9)

where αl , μi , and βm are the non-negative Lagrange multipliers. Solving for the optimal solution, we can get ⎤+ ⎡ ∗ Pi,m

⎢ =⎢ ⎣

1 L l=1

αl Ωli,m

M

+

− βm

σi2 |hi,m

⎥ ⎥ 2⎦ |

(10)

m=1

+

where [x] = max (0, x). The same expression can be found in [4] when the single carrier system is assumed as a multicarrier system with one user and different subcarriers. Solving for (M + L) Lagrangian multipliers is computational complex. These multipliers can be found numerically using ellipsoid or interior point method with a complexity O N 3 [8]. The high computational complexity makes the optimal solution unsuitable for practical application and hence a low complexity suboptimal algorithm is proposed. Ignoring the per-user power constraints and assuming that l∗ , the following we have only one interference constraint, i.e. Ith problem can be formulated P3 : s.t.

N

max

(Int)

Pi,m N

i=1

(Int) Ri Pi,m , hi,m

(Int)

∗

∗

l Pi,m Ωli,m ≤ Ith

i=1 (Int)

Pi,m

≥0

= ∗

l + Ith

where m in Pi,m , hi,m and Ωi,m refers to the user who’s already got the subcarrier i, i.e. ωi,m = 1. Nm denotes the set of subcarriers allocated to the mth SU. Remark that having too much power comparing to the interference constraints will lead to an interference-only optimization problem while having high interference constraints in relative with the total power budgets will lead to a non-cognitive, i.e. classical, resource allocation problem. The problem P 2 is a convex optimization problem. The Lagrangian can be written as G=−

(Int)

is the Lagrange multiplier and can be evaluated where αl∗ from the following relation

(11)

∀i

where (Int) stands for optimization under the interference constraint only. Following the analysis given in [9], we get + 1 σi2 (Int) ∗ Pi,m (l ) = − (12) 2 (Int) ∗ |hi,m | αl∗ Ωli,m

1213

|N | N Ω l∗ σ 2

i,m i i=1

(13)

|hi,m |2

Problem P 2 can be solved by modifying the downlink power allocation algorithm (PI-Algorithm) presented in [10]. We can M ax start by assuming that the maximum power Pi,m that can be allocated to each subcarrier is determined according to the different interference constraints using (12)-(13) as follows (Int)

(Int)

(Int)

M ax Pi,m = min{Pi,m (1) , Pi,m (2) , · · · , Pi,m (L)} (14)

Afterwards, the per-user power constraints are tested to check

(Int) whether Pi,m ≤ Pm , ∀m holds or not. If the relation i∈Nm

∗ M ax holds, then the solution is found where Pi,m = Pi,m . Otherwise, the available power Pm for each SU should be distributed among the subcarriers in Nm giving that the power M ax . allocated to each subcarrier is lower than or equal to Pi,m Hence, the following problem should be solved for every SU W.F

P 4 : max Ri Pi,m , hi,m W.F Pi,m i∈Nm

W.F (15) s.t. Pi,m ≤ Pm ; i∈Nm

W.F M ax 0 ≤ Pi,m ≤ Pi,m

The problem P 4 is called ”cap-limited” waterfilling [11], [12]. The problem can be solved efficiently using the concept of the conventional waterfilling. Given the initial waterfilling M ax solution, the channels that violate the maximum power Pi,m M ax are determined and upper bounded with Pi,m . The total power budget of the user is reduced by subtracting the power assigned so far. At the next step, the algorithm proceeds to successive waterfilling over the subcarriers that did not violate M ax in the last step. These procedures the maximum power Pi,m W.F are repeated until the allocated power Pi,m doesn’t violate the M ax maximum power Pi,m in any of the subcarriers in the new W.F iteration. The solution Pi,m of the problem P 4 is satisfying the per-user power constraints of the problem P 2 with equality l . Since which is not the case for the interference constraints Ith W.F M ax ≤ Pi,m , some of the powers allocated it’s assumed that Pi,m to subcarriers will not reach the maximum allowable values which will make the interference introduced to the PU bands l . In order to take the advantage of below the thresholds Ith the allowable interference, some power can be taken from one subcarrier and given to another hoping to increase the total system capacity. Therefore, the values of the maximum M ax should be power that can be allocated to each subcarrier Pi,m updated depending on the residual interference. The residual interference can be determined as follows l l = Ith − Iresidual

N

W.F l Pi,m Ωi,m

(16)

i=1

Assuming that Sm ⊂ Nm is the set of the subcarriers that W.F M ax reach its maximum, i.e. Pi,m = Pi,m , ∀i ∈ Sm , then,

TABLE I COMPLEXITY COMPARISON Algorithm Complexity Optimal O N 3 M N 2 Wang [6] ∈ O N M , O N 3M Fadel [5] ≥ O (N log N ) + O (N M ) Proposed ≤ O (N (log N + 1)) + O (N M )

Subcarriers allocated to User 1 Subcarriers allocated to User 2

Power

Set {S}

Set {S}

Pmax

PU band

Updated Pmax

CR allocates zero power in these subcarriers

Subcarriers

Initial Pi Updated Pi Power

User 1 power allocation

Pmax

Power Set {S1}

Set {S2}

Updated Pmax

Updated Pmax

PU band

User 2 power allocation Pmax

PU band

Subcarriers

Subcarriers

Fig. 3. An Example of the SUs allocated power using proposed power allocation algorithm.

Algorithm 1 Power Allocation Algorithm l 1) Initialize N = {1, 2, · · · , N }, ILef t = 0 and S = ∅. σi2 l 2) ∀l ∈ {1, · · · , L}, Sort Hi = Ω , i ∈ N in decreasing |hi,m |2 i,m M ax order with k being the sorted index. Find the Pi as follows: l

(Int) = |N | / Ith + Hsum , n = 1. a) Hsum = i∈Nl Hi , αl

b)

(Int)

while αl

−1 > Hk(n) do

(Int)

Hsum = Hsum − H k(n) , N = N \ {k (n)}, αl l +H |N | / Ith sum , n = n + 1 end while + (Int)

c) Set Pi,m

(l) =

1

(Int) l αl Ωi,m

(Int)

−

σi2

|hi,m |

(Int)

=

[5] has a complexity of O (N M ) with the assumption of sorted channel gains matrices. Therefore, including the sorting complexity of the different matrices as well as the iterative nature of the algorithm, the complexity will be more than O (N log N ) + O (N M ). Moreover, the algorithm proposed 2 by Wang et al. in [6] 3has a complexity larger than O N M and lower than O N M . Recall that our proposed algorithm to solve problem P 1 is divided into two steps: the subcarriers to users allocation step and the power allocation step. Each subcarrier in the first step requires no more than M function evaluations to be assigned to one user. Hence, the computational complexity of the proposed subcarrier to user allocation algorithm is O (N M ). In the power allocation algorithm, Step 2 has a computational complexity of O (N log N ). Steps 5 and 7 of the algorithm execute the ”cap-limited” waterfilling for every SU with a complexity M

O (|Nm | log |Nm | + Nm ) ≤ O (N log N + N ) [12]. of m=1

2

(Int)

M ax = min{P 3) Evaluate Pi,m i,m (1) , Pi,m (2) , · · · , Pi,m (L)}

M ax ≤ P ; ∀m 4) if i∈Nm Pi,m m ∗ M ax and stop the algorithm. Let Pi,m = Pi,m end if 5) ∀m, Perform the ”cap-limited” waterfilling on the set of subcarriers Nm under the per-user constraint Pm and the maximum power that M ax and find the set S can be allocated to each subcarrier Pi,m m ⊂ Nm M ax . where Pi,m iW.F = Pi,m l l − = Ith 6) Let S = {S1 ∪ S2 · · · ∪ Sm }, evaluate Iresidual

N W.F l l l P Ωi,m , set N = S, Ith = Iresidual +

i=1 i,m W.F l i∈S Pi,m Ωi,m and apply again only steps (2 − 3) to update M ax . Pi,m 7) ∀m, Perform the ”cap-limited” waterfilling on the set of subcarriers Nm under the per-user constraint Pm and the maximum power that M ax and set P ∗ W.F can be allocated to each subcarrier Pi,m i,m = Pi,m .

M ax Pi,m , ∀i ∈ Sm can be updated by applying the equations (12)-(14) on the subcarriers in the set S = {S1 ∪ S2 · · · ∪ Sm } with the following interference constraints l l W.F l = Iresidual + Pi,m Ωi,m (17) Ith i∈S M ax After determining the updated values of Pi,m , the ”caplimited” waterfilling is performed again for every SU to find ∗ W.F = Pi,m . A graphical description of the the final solution Pi,m proposed power allocation algorithm is given in Fig. 3 while the implementation procedures is described in Algorithm 1.

V. C OMPLEXITY A NALYSIS AND S IMULATIONS The exhaustive enumeration scheme needs to iterate M N times all the cases, and its complexity of to exhaust O N 3 M N is hard to afford. The algorithm proposed in

1214

Step 6 has a complexity of O (|S| log |S|) ≤ O (N log N ). Hence, The complexity of the power allocation algorithm is lower than O (N log N + N ). Thus, the overall computational complexity of the proposed resource allocation algorithm is O (N log N + N )+O (N M ). Tab. I summarizes the complexity of the different algorithms. The simulations are performed under the scenario given in Fig.1. An OFDM system of M = 10 SUs and N = 128 subcarriers is assumed. The values of Ts , Δf , and σi2 are assumed to be 4μ seconds, 0.3125 MHz and 10−6 respectively. The channel gains h and g are outcomes of independent Rayleigh distributed random variables with mean equal to 1. The per-user power budget is set to be Pm = 1mWatt. Two interference constraints belonging to two active PU bands , i.e. L = 2, are assumed with B 1 = B 2 = 10 MHz (see Fig. 2). The minimum rate for each user is set to be 20 Mbits/s, i.e. Rmin = 80 bits per OFDM symbol. All the results have been averaged over 1000 iterations. We refer to the algorithms proposed in [5] and [6] by Fadel and Wang respectively. The interference constraints are converted into per-subcarrier power constraints in Fadel and Wang algorithms by using (7). Moreover, Classical+Pr will refer to the method in which the subcarriers are allocated according to the scheme used in non-cognitive OFDM [2] while the power is allocated using our proposed power allocation algorithm. Fig. 4 plots the average capacity vs. the interference thresh1 2 = Ith . It can be observed that as the interference olds with Ith thresholds increase, the average sum rate increases since each SU is allowed to have more flexibility in allocating more power on its subcarriers. Remark that the algorithms Wang, Fadel and Classical+Pr aren’t considering any fairness among users. The performance of the proposed algorithm without considering the fairness among the users outperforms the reference algo-

9

200 180 Instantaneous Rate (bits/symbol)

8

Capacity (Bit/Hz/sec)

7 6 5 Proposed without Fairness Wang Classical+Pr Fadel Proposed with Fairness

4 3

Fig. 4.

−35

−30 −25 −20 −15 −10 Interfernce threshold Ith1=Ith2(dBm)

−5

Achieved capacity vs allowed interference thresholds.

80 60

Propsed without Fairness Proposed with Fairness

−2

10

−3

10

Proposed without Fairness Classical+Pr Wang Fadel Proposed with Fairness

−4

−5

−35

−30 −25 −20 −15 −10 Interference thresholds Ith1=Ith2 (dBm)

−5

20

Fig. 6.

−1

10

0

40 60 Sample index

80

100

Instantaneous rates over time.

the users according to their channel quality as well as the interference that they may introduce to the primary system. In the second step, the per-user power budget is distributed among the subcarriers so that the total system capacity is maximized without causing excessive interference to the primary system. The fairness among users is considered within the subcarrier allocation by reducing the probability of having users whose instantaneous rate are below a given minimum rate. Without applying the fairness constraints, the proposed algorithm enhances the sum rate of the system and outperforms the exciting ones in the literature in which the fairness among users are not considered. The proposed algorithm achieves superior outage performance when the fairness among users is considered. Developing a distributed resource allocation algorithm will be the guideline of our future research.

10

Outage Probability

100

0

0

0

Fig. 5.

120

20

10

10 −40

140

40

2 1 −40

160

0

Outage probability vs allowed interference thresholds.

rithms. Moreover, its worth noting that the performance of the proposed algorithm without fairness is considered as an upper bound for the case when fairness is taken into account. From this fact, numerical results reveal that the proposed algorithm with fairness consideration achieves a very good performance. The gap between the different algorithms decreases with the interference thresholds as the CR system becomes more closer to the classical (non-cognitive) system. Fig. 5 plots the outage probability of the different algorithms. The outage probability of proposed algorithm with fairness is much lower than that of the reference algorithms. Moreover, the outage probability decreases with the increase of the interference constraints because the algorithm becomes more able to give the minimum instantaneous rate for the different users. Fig. 6 plots the instantaneous data rate for a given user over time for the proposed algorithm with and 1 2 = Ith = −20 dBm. It without fairness consideration when Ith can be noted that the proposed algorithm with fairness keeps the instantaneous rate above Rmin = 80 bits/symbol. VI. C ONCLUSION In this paper, we proposed an efficient resource allocation algorithm for uplink in multicarrier based CR networks with fairness consideration. The allocation process is separated into two steps. In the first step, the subcarriers are allocated to

1215

R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Select. Areas Commun., pp. 201–220, Feb. 2005. [2] K. Kim, Y. Han, and S.-L. Kim, “Joint subcarrier and power allocation in uplink OFDMA systems,” IEEE Commun. Lett., vol. 9, no. 6, pp. 526–528, Jun 2005. [3] L. Gao and S. Cui, “Efficient subcarrier, power, and rate allocation with fairness consideration for OFDMA uplink,” IEEE Trans. Wireless Communications, vol. 7, no. 5, pp. 1507–1511, May 2008. [4] R. Zhang, S. Cui, and Y.-C. Liang, “On ergodic sum capacity of fading cognitive multiple-access and broadcast channels,” IEEE Trans. Inf. Theor., vol. 55, no. 11, pp. 5161–5178, 2009. [5] F. Digham, “Joint power and channel allocation for cognitive radios,” in IEEE WCNC’08, April 2008, pp. 882–887. [6] W. Wang, W. Wang, Q. Lu, and T. Peng, “An uplink resource allocation scheme for OFDMA-based cognitive radio networks,” Int. J. Commun. Syst., vol. 22, no. 5, pp. 603–623, 2009. [7] T. Weiss and J. Hillenbrand, “Mutual interference in OFDM-based spectrum pooling systems,” in VTC-Spring, vol. 4, May 2004. [8] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [9] G. Bansal, M. J. Hossain, and V. K. Bhargava, “Optimal and suboptimal power allocation schemes for OFDM-based cognitive radio systems,” IEEE Trans. Wireless Communications, vol. 7, no. 11, pp. 4710–4718, November 2008. [10] M. Shaat and F. Bader, “Low complexity power loading scheme in cognitive radio networks: FBMC capability,” in IEEE PIMRC, TokyoJapan, Sept. 2009. [11] N. Papandreou and T. Antonakopoulos, “Bit and power allocation in constrained multicarrier systems: The single-user case,” EURASIP Journal on Advances in Signal Processing, vol. 2008, Article ID 643081, 2008. [12] C. Zhao, M. Zou, and K. Kwak, “Mutual interference considered power allocation in OFDM-based cognitive networks: The single SU case,” Computer Communications, vol. 32, no. 18, pp. 1965 – 1974, 2009.

Fair and Efficient Resource Allocation Algorithm for Uplink Multicarrier Based Cognitive Networks Musbah Shaat and Faouzi Bader Centre Tecnol`ogic de Telecomunicacions de Catalunya (CTTC) Parc Mediterrani de la Tecnolog´ıa, Av. Carl Friedrich Gauss 7, 08860 , Castelldefels-Barcelona, Spain. Phone: +34 93 6452911, Fax: +34 93 6452900, Email:{musbah.shaat,faouzi.bader}@cttc.es

Abstract— This paper presents an efficient and fair uplink resource allocation algorithm in multicarrier based cognitive radio systems. The proposed resource allocation is divided into two steps. The subcarriers to user assignment is first performed and then the power is allocated to the different subcarriers. The subcarriers are allocated based on the channel quality, amount of interference imposed to the primary bands, instantaneous rate achieved by every user and the increment in the total data rate. The fairness among users is considered within the subcarrier allocation by reducing the probability of having users whose instantaneous rates are below a given minimum rate. The power is distributed among subcarriers so that the total data rate is maximized while the interference introduced to the primary system is guaranteed to be under the prescribed interference temperature limit. Simulation results confirm the efficiency of the proposed algorithm.

I. I NTRODUCTION Cognitive radio (CR) has been proposed to improve the spectrum utilization by using the vacant channels without causing unacceptable interference to the licensed system. Multicarrier communications has been suggested as a candidate for the CR systems [1]. The problem of uplink resource allocation has been already studied in classical multicarrier systems [2], [3] and references therein. The algorithms used in classical (non-cognitive) multicarrier systems are not efficient for the CR ones due to the existence of the interference constraints. In [4], a single carrier CR system is analyzed for both downlink and uplink scenarios. The results can be used -with some modifications- to find the optimal power allocation in multicarrier based CR system when the subcarrier channel allocation is already known. There is few existed research on the subcarrier and power allocation in uplink multicarrier based CR systems. In [5], the author proposed an algorithm for jointly allocating channels and powers among different users under the individual users power constraints. The problem is relaxed to get a convex version and then the solution is quantized to yield a binary channel allocation. Afterwards, the solution is modified to consider the interference constraints to the licensed system. Wang et al. proposed in [6] an algorithm to allocate resources This work was partially supported by the European ICT-2008-211887 project PHYDYAS, COST Action IC0902 and Generalitat de Catalunya under grant 2009-SGR-940.

978-1-4244-8015-9/10/$26.00 ©2010 IEEE

1210

CR base station (CBS)

Transmission Interference

Secondary User (SU1)

(SU3) (PU2) Primary User (PU1)

Fig. 1.

(SU2)

Uplink Cognitive Radio Network

in the uplink OFDMA based CR systems under the persubcarrier power constraints. The subcarriers are allocated initially to the users with the best channel quality and then adjusted according to the different user’s waterfilling levels. The algorithm has high computational complexity and limited performance. The instantaneous fairness among users was not taken into consideration for both of the algorithms proposed in [5] and [6]. In this paper, we propose an efficient resource allocation algorithm in multicarrier based CR systems. The objective is to maximize the total CR data rate while limiting the interference introduced to the primary system. The fairness among users is considered within the subcarrier allocation by reducing the probability of having users whose instantaneous rate are below a given value. The proposed resource allocation is divided into two steps. The subcarriers to user assignment is first performed and then the power is allocated to the different subcarriers. The subcarrier to user assignment scheme proposed in [3] for non-cognitive systems is developed in order to be suitable for the cognitive ones. The subcarriers are allocated based on the channel quality, amount of interference imposed to the primary bands, instantaneous rate achieved by every user and the increment in the total data rate. For the power allocation step, an efficient power allocation algorithm is proposed to distribute the powers among the subcarriers under the per-user total power and interference constraints. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION In this paper, the uplink scenario will be considered. As shown in Fig. 1, the CR system coexists with the primary system in the same geographical location. The unlicensed users [referred to as secondary users (SUs)] is able to opportunistically access the available spectrum holes without

B1

Active PU1 band 1 2

Fig. 2.

NonActive band

……….

B2

BL

Active PU2 band

PUL band

Active Frerquency

N

Δf

Frequency distribution of the active and non-active primary bands

causing harmful interference to the licensed users [referred to as primary users (PU)]. The CR system’s frequency spectrum is divided into N subcarriers each having a Δf bandwidth. The side by side frequency distribution of the PU bands and the CR bands will be assumed (see Fig. 2). The frequency bands B1 , B2 ,· · · ,BL has been occupied by the PUs (active PU bands) while the other bands represent the inactive PU bands (CR bands). It is assumed that the CR system can use the inactive PU bands provided that the total interference l , where introduced to the lth PU band does not exceed Ith l l Ith = Tth Bl denotes the maximum interference power that can l is the interference temperature be tolerated by the P Ul and Tth limit for P Ul . Assume that Φi is the power spectrum density (PSD) of the ith subcarrier. The expression of the PSD depends on the used multicarrier technique. If an OFDM based CR is assumed, the PSD of the ith subcarrier can be written as [7] 2 sin πf Ts Φi (f ) = Pi Ts (1) πf Ts where Pi is the total transmit power emitted by the ith subcarrier and Ts is the symbol duration. The mutual interference introduced by the ith subcarrier to lth PU, Iil (di , Pi ) , is the integration of the PSD of the ith subcarrier across the lth PU band, Bl , and can be expressed as [7] di +B l /2

Iil

(di , Pi ) =

l 2 gi Φi (f ) df = Pi Ωli

(2)

where di is the spectral distance between the ith subcarrier and the lth PU band. gil denotes the channel gain between the ith subcarrier and the lth PU band while Ωli denotes the interference factor of the ith subcarrier to the lth PU band. The interference power introduced by the lth PU signal into the band of the ith subcarrier is [7] Jil =

l 2 jw y i ψl e dw

where hi,m is the ith subcarrier fading gain from the CBS to L

2 2 Jil where σAW the mth SU. σi2 = σAW GN + GN is the l=1

mean variance of the additive white Gaussian noise (AWGN) and Jil is the interference introduced by the lth PU’s band into the ith subcarrier and can be evaluated using (3). The noise variance is assumed to be constant for all the users and subcarriers. The fairness among the SUs is guaranteed by assuming that every SU has the minimum instantaneous rate Rmin . Our objective is to maximize the total capacity of the CR system subject to the instantaneous interference introduced to the primary system, the per-user transmit power constraints and the per-user minimum rate constraints. Therefore, the optimization problem can be formulated as follows P1 :

di −Bl /2

di +Δf /2

can be obtained by the CBS/SUs via, e.g., estimating the received signal power from each primary terminal when it transmits, under the assumptions of pre-knowledge on the primary transmit power levels and the channel reciprocity [4]. Based on this information, the CBS assigns the subcarrier and power to each SU through a reliable low-rate signaling channel. Let ai,m to be the subcarrier allocation indicator, i.e. ai,m = 1 if and only if the ith subcarrier is allocated to mth user. It is assumed that each subcarrier can be used for transmission to at most one user at any given time. The transmission rate of the ith subcarrier, Ri , with the transmit power Pi,m can be evaluated using the Shannon capacity formula and is given by 2 Pi,m |hi,m | Ri (Pi,m , hi,m ) = Δf log2 1 + (4) σi2

(3)

di −Δf /2

where ψl ejw is the power spectrum density of the lth P U signal and yil is the channel gain between the ith subcarrier and lth PU signal. It will be assumed that all the instantaneous fading gains are perfectly known at the cognitive base station (CBS) and there is no synchronization between the PUs and SUs. The channel gains between the SUs and the CBS can be obtained practically by the classical channel estimation techniques while the channel gains between the CBS/SUs and the PUs

1211

s.t.

max

M

N

Pi,m ,ai,m m=1 i=1 N M

m=1 i=1 N

ai,m Ri (Pi,m , hi,m )

l ai,m Pi,m Ωli,m ≤ Ith

ai,m Pi,m ≤ Pm ,

i=1

Pi,m ≥ 0, ai,m ∈ {0, 1} , M

ai,m ≤ 1,

m=1

R (m) ≥ Rmin ,

∀l ∈ {1, · · · , L}

∀m ∀i, m ∀i, m ∀i ∀m

(5) where N denotes the total number of subcarriers while M denotes the number of SUs. L is the number of active PU bands l is the interference threshold prescribed by the lth PU. and Ith Pm is the mth SU total power budget. R (m) is the mth SU N

ai,m Ri (Pi,m , hi,m ). instantaneous rate, i.e. R (m) = i=1

The optimization problem P 1 is a NP-hard optimization problem in which achieving the optimal solution needs unacceptable computational complexity. In order to reduce the computational complexity, the resource allocation problem can be solved in two steps. In the first step, the subcarriers are assigned to the users and then the power is allocated for these

subcarriers in the second step. Depending on the values of l , Pm and the channel gains, the minimum rate Rmin may Ith not be satisfied for all the users. Therefore, the probability of having users whose rates are below the minimum rate should be reduced [3]. The outage probability can be defined as Poutage = P r{Mlow ≥ 1}

(6)

where Mlow is the number of SUs whose instantaneous rate below Rmin . In the sequel, the proposed subcarrier to user assignment scheme with low outage probability is introduced and then an efficient power allocation algorithm is presented. III. P ROPOSED S UBCARRIER A LLOCATION A LGORITHM WITH FAIRNESS C ONSIDERATION The optimal downlink subcarriers to users allocation scheme in multicarrier systems is achieved by allocating each subcarrier to the user with the maximum signal to noise ratio (SNR). This scheme of subcarrier allocation is not efficient in uplink case due to the per-user power constraints. Moreover, the interference introduced to the primary system by each SU should be considered in CR context which makes the schemes used in classical multicarrier systems not efficient. In this section, a heuristic subcarrier allocation algorithm is presented. The proposed algorithm tries to assign the subcarriers to the different SUs considering not only their channel quality and per-user power constraints but also the interference that will be induced to the PU bands. Moreover, the proposed scheme tries to minimize the probability of having users with instantaneous rates below the minimum rate. The scheme initially assumes that the interference introduced to the lth PU band is divided uniformly among the different CR subcarriers which means that every subcarrier is allowed to introduce the samel amount of interference to the I PU band, i.e. IUl nif orm = Nth . Considering L interference constraints and by using (2), the maximum power that can be allocated to the ith subcarrier when it is allocated to the mth SU is IU1 nif orm IULnif orm U ni = min{ , · · · , } (7) Pi,m Ω1i,m ΩL i,m Let C to be the set of unassigned subcarriers and U to be the set that contains the indices of the users whose rates below the minimum. Define the sets Am and Bm to include all the subcarriers already allocated to the mth SU according to U ni , and average power respectively. maximum power, i.e Pi,m The average power means that the left power for the mth user

U ni Pi,m ) is divided equally among the subcarriers (Pm − i∈Am

in the set Bm . The algorithm begins by the allocation of the subcarriers that are located next to the PU bands, i.e. subcarriers that have more interference to the PU bands, and moving towards the distant ones. The assigning procedures of a particular subcarrier i∗ ∈ C are as follows 1) if U is empty, let Z = {1, · · · , M } else Z = U 2) ∀m ∈ Z,

Evaluate PT est =

Pm −

r∈Am

U ni Pr,m

|Bm |+1

1212

ni if PT est ≥ PiU∗ ,m let A m = Am ∪ {i∗ } and B m = Bm else let B m = Bm ∪ {i∗ } and A m = Am . |X | means the cardinality of the set X . 3) Compute the amount of increment Δm in the data rate when the subcarrier {i∗ } is assigned to mth SU, i.e, new old Δm = Rm − Rm

where new Rm

=

P

m−

r∈A m |B m |

U ni Pr,m

Ri U ni

Ri Pi,m , hi,m

i∈B m

, hi,m +

i∈A m

old Rm =

n∈Bm i∈Am

Ri

Pm −

r∈Am

|Bm |

U ni Ri Pi,m , hi,m

U ni Pr,m

, hi,m +

and Ri (Pi,m , hi,m ) is evaluated using (4). 4) Find m∗ satisfying m∗ = arg maxm (Δm ), set ai∗ ,m∗ = 1, and update the sets Am∗ = A m∗ and Bm∗ = B m∗ . If R (m∗ ) ≥ Rmin , remove m∗ form the set U. 5) Remove the subcarrier i∗ form the set C and repeat the above procedures until the set C is empty. The first step determines whether all the users are having the minimum rate or not. In case that there are users whose rates below the minimum, the subcarrier allocation will be confined within these users. If the minimum rate constraints are satisfied for all the users, i.e. U is empty, the algorithm will be performed considering that the subcarrier can be allocated to any one of the SUs. The second step considers the limitation that will be introduced to any subcarrier assignment due the interference constraints. Generally speaking, the subcarriers that will introduce high interference to the PU bands will have a low transmitting power which will reduce the total data rate even that they have a good channel quality. Hence, the algorithm initially assign the subcarrier to the group Bm and evaluate the average power. If the average power exceeds the U ni , then the subcarrier should be moved to maximum power Pi,m the group Am . Therefore, the second step helps in classifying the subcarriers according to their interference to the PU bands. Afterwards, the increments of the individual data rates due to the allocation of a particular subcarrier to different SUs are evaluated and the subcarrier is allocated to the SU with maximum data rate increment. The scheme is repeated until the allocation of all subcarriers. Note that the final set of allocated subcarriers to mth SU is Nm = Am ∪ Bm . IV. P ROPOSED P OWER A LLOCATION A LGORITHM By the subcarrier to users assignment step, the subcarriers are allocated to the different users with the consideration of the minimum rates constraints. Therefore, the values of the

subcarrier indicators, i.e. ai,m , are already known and the power allocation problem can be formulated as follows P2 : s.t.

max

N

Pi,m i=1 N

i=1

(Int)

αl∗

Ri (Pi,m , hi,m )

l Pi,m Ωli,m ≤ Ith

Pi,m ≤ Pm

i∈Nm

Pi,m ≥ 0

∀l ∈ {1, · · · , L}

(8)

∀m

∀i

N ∗

∗ l l Ri Pi,m , hi,m + L α P Ω − I l i,m i,m th l=1 i=1 i=1 M N

∗ ∗ βm Pi,m − Pm − Pi,m μi + N

m=1

i=1

i∈Nm

(9)

where αl , μi , and βm are the non-negative Lagrange multipliers. Solving for the optimal solution, we can get ⎤+ ⎡ ∗ Pi,m

⎢ =⎢ ⎣

1 L l=1

αl Ωli,m

M

+

− βm

σi2 |hi,m

⎥ ⎥ 2⎦ |

(10)

m=1

+

where [x] = max (0, x). The same expression can be found in [4] when the single carrier system is assumed as a multicarrier system with one user and different subcarriers. Solving for (M + L) Lagrangian multipliers is computational complex. These multipliers can be found numerically using ellipsoid or interior point method with a complexity O N 3 [8]. The high computational complexity makes the optimal solution unsuitable for practical application and hence a low complexity suboptimal algorithm is proposed. Ignoring the per-user power constraints and assuming that l∗ , the following we have only one interference constraint, i.e. Ith problem can be formulated P3 : s.t.

N

max

(Int)

Pi,m N

i=1

(Int) Ri Pi,m , hi,m

(Int)

∗

∗

l Pi,m Ωli,m ≤ Ith

i=1 (Int)

Pi,m

≥0

= ∗

l + Ith

where m in Pi,m , hi,m and Ωi,m refers to the user who’s already got the subcarrier i, i.e. ωi,m = 1. Nm denotes the set of subcarriers allocated to the mth SU. Remark that having too much power comparing to the interference constraints will lead to an interference-only optimization problem while having high interference constraints in relative with the total power budgets will lead to a non-cognitive, i.e. classical, resource allocation problem. The problem P 2 is a convex optimization problem. The Lagrangian can be written as G=−

(Int)

is the Lagrange multiplier and can be evaluated where αl∗ from the following relation

(11)

∀i

where (Int) stands for optimization under the interference constraint only. Following the analysis given in [9], we get + 1 σi2 (Int) ∗ Pi,m (l ) = − (12) 2 (Int) ∗ |hi,m | αl∗ Ωli,m

1213

|N | N Ω l∗ σ 2

i,m i i=1

(13)

|hi,m |2

Problem P 2 can be solved by modifying the downlink power allocation algorithm (PI-Algorithm) presented in [10]. We can M ax start by assuming that the maximum power Pi,m that can be allocated to each subcarrier is determined according to the different interference constraints using (12)-(13) as follows (Int)

(Int)

(Int)

M ax Pi,m = min{Pi,m (1) , Pi,m (2) , · · · , Pi,m (L)} (14)

Afterwards, the per-user power constraints are tested to check

(Int) whether Pi,m ≤ Pm , ∀m holds or not. If the relation i∈Nm

∗ M ax holds, then the solution is found where Pi,m = Pi,m . Otherwise, the available power Pm for each SU should be distributed among the subcarriers in Nm giving that the power M ax . allocated to each subcarrier is lower than or equal to Pi,m Hence, the following problem should be solved for every SU W.F

P 4 : max Ri Pi,m , hi,m W.F Pi,m i∈Nm

W.F (15) s.t. Pi,m ≤ Pm ; i∈Nm

W.F M ax 0 ≤ Pi,m ≤ Pi,m

The problem P 4 is called ”cap-limited” waterfilling [11], [12]. The problem can be solved efficiently using the concept of the conventional waterfilling. Given the initial waterfilling M ax solution, the channels that violate the maximum power Pi,m M ax are determined and upper bounded with Pi,m . The total power budget of the user is reduced by subtracting the power assigned so far. At the next step, the algorithm proceeds to successive waterfilling over the subcarriers that did not violate M ax in the last step. These procedures the maximum power Pi,m W.F are repeated until the allocated power Pi,m doesn’t violate the M ax maximum power Pi,m in any of the subcarriers in the new W.F iteration. The solution Pi,m of the problem P 4 is satisfying the per-user power constraints of the problem P 2 with equality l . Since which is not the case for the interference constraints Ith W.F M ax ≤ Pi,m , some of the powers allocated it’s assumed that Pi,m to subcarriers will not reach the maximum allowable values which will make the interference introduced to the PU bands l . In order to take the advantage of below the thresholds Ith the allowable interference, some power can be taken from one subcarrier and given to another hoping to increase the total system capacity. Therefore, the values of the maximum M ax should be power that can be allocated to each subcarrier Pi,m updated depending on the residual interference. The residual interference can be determined as follows l l = Ith − Iresidual

N

W.F l Pi,m Ωi,m

(16)

i=1

Assuming that Sm ⊂ Nm is the set of the subcarriers that W.F M ax reach its maximum, i.e. Pi,m = Pi,m , ∀i ∈ Sm , then,

TABLE I COMPLEXITY COMPARISON Algorithm Complexity Optimal O N 3 M N 2 Wang [6] ∈ O N M , O N 3M Fadel [5] ≥ O (N log N ) + O (N M ) Proposed ≤ O (N (log N + 1)) + O (N M )

Subcarriers allocated to User 1 Subcarriers allocated to User 2

Power

Set {S}

Set {S}

Pmax

PU band

Updated Pmax

CR allocates zero power in these subcarriers

Subcarriers

Initial Pi Updated Pi Power

User 1 power allocation

Pmax

Power Set {S1}

Set {S2}

Updated Pmax

Updated Pmax

PU band

User 2 power allocation Pmax

PU band

Subcarriers

Subcarriers

Fig. 3. An Example of the SUs allocated power using proposed power allocation algorithm.

Algorithm 1 Power Allocation Algorithm l 1) Initialize N = {1, 2, · · · , N }, ILef t = 0 and S = ∅. σi2 l 2) ∀l ∈ {1, · · · , L}, Sort Hi = Ω , i ∈ N in decreasing |hi,m |2 i,m M ax order with k being the sorted index. Find the Pi as follows: l

(Int) = |N | / Ith + Hsum , n = 1. a) Hsum = i∈Nl Hi , αl

b)

(Int)

while αl

−1 > Hk(n) do

(Int)

Hsum = Hsum − H k(n) , N = N \ {k (n)}, αl l +H |N | / Ith sum , n = n + 1 end while + (Int)

c) Set Pi,m

(l) =

1

(Int) l αl Ωi,m

(Int)

−

σi2

|hi,m |

(Int)

=

[5] has a complexity of O (N M ) with the assumption of sorted channel gains matrices. Therefore, including the sorting complexity of the different matrices as well as the iterative nature of the algorithm, the complexity will be more than O (N log N ) + O (N M ). Moreover, the algorithm proposed 2 by Wang et al. in [6] 3has a complexity larger than O N M and lower than O N M . Recall that our proposed algorithm to solve problem P 1 is divided into two steps: the subcarriers to users allocation step and the power allocation step. Each subcarrier in the first step requires no more than M function evaluations to be assigned to one user. Hence, the computational complexity of the proposed subcarrier to user allocation algorithm is O (N M ). In the power allocation algorithm, Step 2 has a computational complexity of O (N log N ). Steps 5 and 7 of the algorithm execute the ”cap-limited” waterfilling for every SU with a complexity M

O (|Nm | log |Nm | + Nm ) ≤ O (N log N + N ) [12]. of m=1

2

(Int)

M ax = min{P 3) Evaluate Pi,m i,m (1) , Pi,m (2) , · · · , Pi,m (L)}

M ax ≤ P ; ∀m 4) if i∈Nm Pi,m m ∗ M ax and stop the algorithm. Let Pi,m = Pi,m end if 5) ∀m, Perform the ”cap-limited” waterfilling on the set of subcarriers Nm under the per-user constraint Pm and the maximum power that M ax and find the set S can be allocated to each subcarrier Pi,m m ⊂ Nm M ax . where Pi,m iW.F = Pi,m l l − = Ith 6) Let S = {S1 ∪ S2 · · · ∪ Sm }, evaluate Iresidual

N W.F l l l P Ωi,m , set N = S, Ith = Iresidual +

i=1 i,m W.F l i∈S Pi,m Ωi,m and apply again only steps (2 − 3) to update M ax . Pi,m 7) ∀m, Perform the ”cap-limited” waterfilling on the set of subcarriers Nm under the per-user constraint Pm and the maximum power that M ax and set P ∗ W.F can be allocated to each subcarrier Pi,m i,m = Pi,m .

M ax Pi,m , ∀i ∈ Sm can be updated by applying the equations (12)-(14) on the subcarriers in the set S = {S1 ∪ S2 · · · ∪ Sm } with the following interference constraints l l W.F l = Iresidual + Pi,m Ωi,m (17) Ith i∈S M ax After determining the updated values of Pi,m , the ”caplimited” waterfilling is performed again for every SU to find ∗ W.F = Pi,m . A graphical description of the the final solution Pi,m proposed power allocation algorithm is given in Fig. 3 while the implementation procedures is described in Algorithm 1.

V. C OMPLEXITY A NALYSIS AND S IMULATIONS The exhaustive enumeration scheme needs to iterate M N times all the cases, and its complexity of to exhaust O N 3 M N is hard to afford. The algorithm proposed in

1214

Step 6 has a complexity of O (|S| log |S|) ≤ O (N log N ). Hence, The complexity of the power allocation algorithm is lower than O (N log N + N ). Thus, the overall computational complexity of the proposed resource allocation algorithm is O (N log N + N )+O (N M ). Tab. I summarizes the complexity of the different algorithms. The simulations are performed under the scenario given in Fig.1. An OFDM system of M = 10 SUs and N = 128 subcarriers is assumed. The values of Ts , Δf , and σi2 are assumed to be 4μ seconds, 0.3125 MHz and 10−6 respectively. The channel gains h and g are outcomes of independent Rayleigh distributed random variables with mean equal to 1. The per-user power budget is set to be Pm = 1mWatt. Two interference constraints belonging to two active PU bands , i.e. L = 2, are assumed with B 1 = B 2 = 10 MHz (see Fig. 2). The minimum rate for each user is set to be 20 Mbits/s, i.e. Rmin = 80 bits per OFDM symbol. All the results have been averaged over 1000 iterations. We refer to the algorithms proposed in [5] and [6] by Fadel and Wang respectively. The interference constraints are converted into per-subcarrier power constraints in Fadel and Wang algorithms by using (7). Moreover, Classical+Pr will refer to the method in which the subcarriers are allocated according to the scheme used in non-cognitive OFDM [2] while the power is allocated using our proposed power allocation algorithm. Fig. 4 plots the average capacity vs. the interference thresh1 2 = Ith . It can be observed that as the interference olds with Ith thresholds increase, the average sum rate increases since each SU is allowed to have more flexibility in allocating more power on its subcarriers. Remark that the algorithms Wang, Fadel and Classical+Pr aren’t considering any fairness among users. The performance of the proposed algorithm without considering the fairness among the users outperforms the reference algo-

9

200 180 Instantaneous Rate (bits/symbol)

8

Capacity (Bit/Hz/sec)

7 6 5 Proposed without Fairness Wang Classical+Pr Fadel Proposed with Fairness

4 3

Fig. 4.

−35

−30 −25 −20 −15 −10 Interfernce threshold Ith1=Ith2(dBm)

−5

Achieved capacity vs allowed interference thresholds.

80 60

Propsed without Fairness Proposed with Fairness

−2

10

−3

10

Proposed without Fairness Classical+Pr Wang Fadel Proposed with Fairness

−4

−5

−35

−30 −25 −20 −15 −10 Interference thresholds Ith1=Ith2 (dBm)

−5

20

Fig. 6.

−1

10

0

40 60 Sample index

80

100

Instantaneous rates over time.

the users according to their channel quality as well as the interference that they may introduce to the primary system. In the second step, the per-user power budget is distributed among the subcarriers so that the total system capacity is maximized without causing excessive interference to the primary system. The fairness among users is considered within the subcarrier allocation by reducing the probability of having users whose instantaneous rate are below a given minimum rate. Without applying the fairness constraints, the proposed algorithm enhances the sum rate of the system and outperforms the exciting ones in the literature in which the fairness among users are not considered. The proposed algorithm achieves superior outage performance when the fairness among users is considered. Developing a distributed resource allocation algorithm will be the guideline of our future research.

10

Outage Probability

100

0

0

0

Fig. 5.

120

20

10

10 −40

140

40

2 1 −40

160

0

Outage probability vs allowed interference thresholds.

rithms. Moreover, its worth noting that the performance of the proposed algorithm without fairness is considered as an upper bound for the case when fairness is taken into account. From this fact, numerical results reveal that the proposed algorithm with fairness consideration achieves a very good performance. The gap between the different algorithms decreases with the interference thresholds as the CR system becomes more closer to the classical (non-cognitive) system. Fig. 5 plots the outage probability of the different algorithms. The outage probability of proposed algorithm with fairness is much lower than that of the reference algorithms. Moreover, the outage probability decreases with the increase of the interference constraints because the algorithm becomes more able to give the minimum instantaneous rate for the different users. Fig. 6 plots the instantaneous data rate for a given user over time for the proposed algorithm with and 1 2 = Ith = −20 dBm. It without fairness consideration when Ith can be noted that the proposed algorithm with fairness keeps the instantaneous rate above Rmin = 80 bits/symbol. VI. C ONCLUSION In this paper, we proposed an efficient resource allocation algorithm for uplink in multicarrier based CR networks with fairness consideration. The allocation process is separated into two steps. In the first step, the subcarriers are allocated to

1215

R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Select. Areas Commun., pp. 201–220, Feb. 2005. [2] K. Kim, Y. Han, and S.-L. Kim, “Joint subcarrier and power allocation in uplink OFDMA systems,” IEEE Commun. Lett., vol. 9, no. 6, pp. 526–528, Jun 2005. [3] L. Gao and S. Cui, “Efficient subcarrier, power, and rate allocation with fairness consideration for OFDMA uplink,” IEEE Trans. Wireless Communications, vol. 7, no. 5, pp. 1507–1511, May 2008. [4] R. Zhang, S. Cui, and Y.-C. Liang, “On ergodic sum capacity of fading cognitive multiple-access and broadcast channels,” IEEE Trans. Inf. Theor., vol. 55, no. 11, pp. 5161–5178, 2009. [5] F. Digham, “Joint power and channel allocation for cognitive radios,” in IEEE WCNC’08, April 2008, pp. 882–887. [6] W. Wang, W. Wang, Q. Lu, and T. Peng, “An uplink resource allocation scheme for OFDMA-based cognitive radio networks,” Int. J. Commun. Syst., vol. 22, no. 5, pp. 603–623, 2009. [7] T. Weiss and J. Hillenbrand, “Mutual interference in OFDM-based spectrum pooling systems,” in VTC-Spring, vol. 4, May 2004. [8] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [9] G. Bansal, M. J. Hossain, and V. K. Bhargava, “Optimal and suboptimal power allocation schemes for OFDM-based cognitive radio systems,” IEEE Trans. Wireless Communications, vol. 7, no. 11, pp. 4710–4718, November 2008. [10] M. Shaat and F. Bader, “Low complexity power loading scheme in cognitive radio networks: FBMC capability,” in IEEE PIMRC, TokyoJapan, Sept. 2009. [11] N. Papandreou and T. Antonakopoulos, “Bit and power allocation in constrained multicarrier systems: The single-user case,” EURASIP Journal on Advances in Signal Processing, vol. 2008, Article ID 643081, 2008. [12] C. Zhao, M. Zou, and K. Kwak, “Mutual interference considered power allocation in OFDM-based cognitive networks: The single SU case,” Computer Communications, vol. 32, no. 18, pp. 1965 – 1974, 2009.