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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 4, APRIL 2012
Fast Optimal Resource Allocation is Possible for Multiuser OFDM-Based Cognitive Radio Networks with Heterogeneous Services Mengyao Ge and Shaowei Wang, Member, IEEE
Abstract—In this paper we study the resource allocation in multiuser orthogonal frequency division multiplexing (OFDM)based cognitive radio (CR) networks, where secondary users (SUs) have flexible traffic demands, including heterogeneous realtime (RT) and non-real-time (NRT) services. We try to maximize the sum capacity of the NRT users and maintain the minimal rate requirements of the RT users simultaneously. Additionally, the interference introduced to primary users, which is generated by the access of the SUs, should be kept below a predefined threshold, which makes the optimization task more complex. The contribution of this work is two folds. First, we show that the formulated optimization problem has a special structure which can be exploited to implement a fast barrier method to obtain the optimal solution with a reasonable complexity. Second, we propose an effective measurement criterion to normalize OFDM subchannels’ achievable rates, based on which we develop simple but efficient heuristic algorithm for subchannel assignment and power distribution. Simulation results show that our proposed resource allocation schemes work quite well for concerned wireless scenarios. The fast barrier method converges very fast and can always work out the optimal solution, while the heuristic algorithm produces solution close to the optimal with much lower complexity. Index Terms—Cognitive radio, OFDM, optimization, resource allocation.
I. I NTRODUCTION
D
ESPITE the emerging crisis of spectrum shortage in wireless communication, it is inconsistent that radio spectrum is far from fully utilized due to conventional regulatory policies [1]. Cognitive Radio (CR) is envisioned as a promising paradigm for improving the usage efficiency of radio spectrum [2] [3]. Although radio spectrum has been assigned to primary users (PUs) in a licensed network, CR technology allows secondary users (SUs) in a CR network to sense spectrum environment and dynamically adjust their transmission parameters to access the licensed spectrum in an opportunistic manner, as long as the interference to the PUs is kept below a preset threshold, such as interference temperature level [4]. Since the physical layer of a CR network should be very flexible to meet the requirements of opportunistic access, Manuscript received June 28, 2011; revised October 30, 2011; accepted January 4, 2012. The associate editor coordinating the review of this paper and approving it for publication was C.-C. Chong. M. Ge and S. Wang (corresponding author) are with the School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China (e-mail: [email protected], [email protected]). This work was partially supported by the Jiangsu Science Foundation (BK2011051) and the Fundamental Research Funds for the Central Universities (1095021029, 1118021011). Digital Object Identifier 10.1109/TWC.2012.021512.111233
it necessitates multicarrier methods to operate in CR networks [5] [6]. Owning to the inherent significant advantages of flexibly allocating radio resource [7], Orthogonal Frequency Division Multiplexing (OFDM) is deemed as a promising air interface for CR systems. As one of the most important issues in OFDM systems, adaptive resource allocation (RA) have been studied extensively during the past two decades. A comprehensive survey can be found in [8] and references therein. RA algorithms in conventional OFDM systems can be classified into two categories: margin adaptive and rate adaptive. The optimization objective of the former is generally to minimize the required power under given rate requirements of users, while algorithms for rate adaptive usually try to maximize an OFDM system’s throughput under transmission power limitation. In [9], subchannel, bit and power allocation schemes in OFDM systems are developed. The proposed algorithm shows significant advantages compared with other methods. A margin adaptive algorithm is discussed in [10] for real-time services in a multiuser OFDM system. In [11], sum capacity of an OFDM system is maximized by allocating each subchannel to the user with the highest channel gain over this channel and distributing power among subchannels in a water-filling manner. However, fairness is ignored because the user with poor channel condition may be unable to receive any data. In [12], a max-min algorithm is proposed to maximize the capacity of the user with the worst channel condition. All users can achieve an equal rate with the proposed RA scheme and it provides a good fairness. But for most wireless systems, users may have various rate requirements to subscribe different kinds of services, the scheme proposed in [12] can not meet such requirements. In [13], proportional rate constraints are introduced and an optimal power allocation algorithm is developed to maximize the sum capacity of the OFDM system. In [14], both delay-constrained and non-delay-constrained traffic demands are considered for a heterogeneous multiuser OFDM system. The objective of [14] is to maximize the sum rate of the non-delay-constrained users while maintaining the delay-constrained users’ rate requirements. An iterative algorithm in dual domain is proposed to find the optimal solution of power distribution. In [15], user’s utility is defined as a concave function and a low complexity algorithm is developed, which can work out the optimal solution. However, it is impractical to assume that a user can occupy a non-integer number of OFDM subcarriers. RA for OFDM-based CR networks has been attracted more
c 2012 IEEE 1536-1276/12$31.00
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
and more attentions recently. An overview of the state-ofthe-art research results can be found in [16]. For single SU case, optimal and suboptimal power loading algorithms are presented in [17], where downlink sum capacity is maximized with the constraint that the interference introduced to the PUs is within a tolerable range. In [18], RA for OFDMbased CR network is formulated as a multidimensional knapsack problem, also for single SU case. A greedy heuristic algorithm is proposed, which can produce solution close to the optimal. However, the computational cost would be very high if generalizing it to multiple SUs setting. In [19], an efficient algorithm is derived by converting transmission power and interference constraints into a normalized capacity, which exhibits a tradeoff between capacity and complexity. But there is a capacity gap between the solution produced by the proposed algorithm and the optimal [19]. In [20], an optimal power distribution algorithm is developed, whose complexity is linear related to the number of subchannels. However, only single SU is considered in [19, 20], which limits their applications in practical wireless networks. RA for multiuser OFDM-based CR system is investigated in [21], where a simple near-optimal algorithm is proposed to jointly allocate power and subchannels. In [22], an optimal strategy based on cooperative Game Theory is developed to allocate radio resource among the SUs with given bit-errorrates (BERs) while guaranteeing the required quality of service of the PUs. RA for non-real-time applications is investigated in [23], where SUs’ rates are maintained in proportion to target ones. In [24], a low complexity algorithm is developed to maximize the sum capacity of a CR system with proportional rate constraints. The proposed algorithm exhibits a tradeoff between capacity and fairness by jointly considering channel gain and the interference to the PUs. In [25], the interference to the PUs is assumed to be yielded in two mechanisms: CR out-of-band emissions and imperfect spectrum sensing. Suboptimal RA algorithms are also developed. In this paper, we investigate a general spectrum-sharing case for a CR system: The SUs can access the regulated portion of licensed spectrum as long as the interference to the PUs is kept below their tolerable thresholds. The CR system adopts OFDM modulation and operates in a centralized manner, that is, an access point serves all SUs in the CR network, just as a conventional base station does. We do not assume that the licensed system serving the PUs also operates in an OFDM manner. Heterogeneous services, for both real-time (RT) SUs and non-real-time (NRT) SUs, exist in the CR network. We aim to maximize the downlink sum capacity of the NRT SUs while guaranteeing the minimal rate requirements of the RT SUs. Different from conventional OFDM system model [14], to prohibit the performance degradation of the PUs, the interference introduced to the PUs must be considered carefully. We analyze the formulated optimization problem intensively and develop a fast barrier method to work out the optimal solution by exploiting its structure. Furthermore, we propose an efficient subchannel allocation and power distribution scheme to achieve a tradeoff between capacity and complexity by jointly considering the signal-to-noise ratios (SNRs) of OFDM subchannels and their interference levels.
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The rest of this paper is organized as follows. In Section II, we illustrate system model and formulate the optimization task. In Section III, we proposed a heuristic subchannel allocation scheme, followed by a fast barrier method developed in Section IV. In Section V, an approximate optimal power distribution algorithm is presented. Simulation results are given in Section VI, as well as discussions. Conclusion is drawn in Section VII. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION Consider a CR system with K SUs, denoted by K = {1, 2, . . . , K}, coexisting with L PUs in a licensed system. We assume that perfect channel state information is available at the transceivers of the SUs and the PUs. The total bandwidth W is divided into N OFDM subchannels in the CR system, denoted by N = {1, 2, . . . , N }. The bandwidth of the nth subchannel spans from f0 + (n − 1)W/N to f0 + nW/N , where f0 is the starting frequency and W/N is the bandwidth of each OFDM subchannel. There are K1 RT SUs, each of them has minimal rate requirement Rk,min , k = 1, 2, . . . , K1 . The lth PU’s nominal band ranges from fl to fl + Bl , where fl and Bl are the lth PU’s starting frequency and bandwidth, respectively. The interference to the lth PU introduced by SUs’ access on the nth subchannel with unit transmission power is fl +Bl −f0 −(n−1/2)W/N SP = gn,l φ(f )df, (1) In,l fl −f0 −(n−1/2)W/N
where gn,l represents the power gain from an SU’s transmitter to the lth PU’s receiver on the nth subchannel. φ(f ) is the power spectral density (PSD) of OFDM signal, and φ(f ) = T ( sinπfπfT T )2 , where T is OFDM symbol duration. Let rk,n denote the transmission rate of the kth SU on the nth subchannel, we have pk,n |ck,n |2 rk,n = log 1 + , (2) Γ(N0 W/N + IkP S ) where pk,n is the power allocated to the kth SU over the nth subchannel, ck,n is the channel gain of the kth SU over the nth subchannel, N0 is the PSD of additive white Gaussian noise and Γ is the SNR gap. For an uncoded MQAM, Γ is related to a given BER requirement and Γ = − ln(5BER)/1.5 as derived in [26]. The interference caused by the PUs’ signals is IkP S , which can be regarded as noise and measured by the receiver of the kth SU. For notation brevity, denote Hk,n = |ck,n |2 as the SNR of the nth subchannel used by the Γ(N0 W/N +IkP S ) kth SU with unit power, the rate of the kth SU is Rk =
N
ρk,n log(1 + pk,n Hk,n ),
(3)
n=1
where ρk,n can be either 1 or 0, informing whether the kth SU occupies the nth subchannel or not. We try to maximize the sum rate of the NRT SUs while guaranteeing the minimal rate requirements for the RT SUs, with total transmission power budget of the CR system and the interference constraints of the PUs. Mathematically, the optimization problem can be described as follows,
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 4, APRIL 2012
OP 1 s.t.
K
max
N
pk,n ,ρk,n k=K +1 n=1 1 N
C1 :
n=1
ρk,n rk,n
ρk,n rk,n ≥ Rk,min , k = 1, . . . , K1
C2 : pk,n ≥ 0, ∀k, n K N C3 : ρk,n pk,n ≤ Pt k=1 n=1
C4 : C5 :
K N k=1 n=1 K k=1
(4)
SP ρk,n pk,n In,l ≤ IlP , l = 1, . . . , L
ρk,n = 1, ∀n
C6 : ρk,n ∈ {0, 1}, ∀k, ∀n, where Pt is the power limit of the access point in the CR system and IlP is the interference power threshold of the lth PU. C1 guarantees the target rate requirements of the RT SUs. C2 and C3 are the transmission power constraints, while C4 is the interference constraints of the PUs. C5 and C6 declare that each subchannel is kept from being shared among the SUs. Both binary variables ρk,n ’s and real variables pk,n ’s are involved in the OP1. The main difficulty of solving the OP1 lies in the integer constraints. An intuitive way to tackle them is the time-sharing method, which relaxes the integer variables into continuous ones. Define ρk,n ∈ [0, 1] as the faction of the nth subchannel allocated to kth SU, temporarily permitting that each OFDM subchannel can be shared by multiple SUs, and sk,n = pk,n ρk,n , the OP1 can be converted into
s.t.
K
N
s Hk,n ρk,n log(1 + k,n ρk,n ) pk,n ,ρk,n k=K +1 n=1 1 N s Hk,n ρk,n log(1 + k,n C1 : ρk,n ) ≥ Rk,min , k = 1, . . . , K1 n=1
OP 2
max
C2 : sk,n ≥ 0, ∀k, n K N C3 : sk,n ≤ Pt k=1 n=1
C4 : C5 :
K N k=1 n=1 K k=1
SP sk,n In,l ≤ IlP , l = 1, . . . , L
ρk,n = 1, ∀n
C6 : ρk,n ∈ [0, 1], ∀k, ∀n. (5) The sk,n can be also characterized as the actual power consumption of the kth SU on the nth subchannel in a time frame interval. It is easy to prove that the OP2 is a convex problem [27] which can be solved by standard techniques. The solution to the OP2 contains sharing factor ρk,n ’s which are not binary variables. Rounding is necessary to obtain a feasible solution to the original problem OP1. Intuitively, larger ρk,n implies the kth SU obtains higher rate over the nth subchannel. It is straightforward to allocate the nth subchannel to the k ∗ th SU who has the largest ρk,n , that is, 1 for k ∗ = argmax ρk,n k ρk∗ ,n = . (6) 0 otherwise
However, such a rounding method can result in unreasonable subchannel allocation assignments when K is large because the difference between the largest ρk,n and the second largest one is negligible in many cases. Moreover, power distribution among subchannels has to be re-allocated after rounding procedure in order to maximize the sum capacity of the CR system. Furthermore, solving OP2 with standard convex optimization techniques generally has a complexity of O(2KN + N )3 , which is too high for practical applications. So the optimal solution to the OP2 provides an upper bound for that to the OP1 and is generally infeasible for the OP1. The upper bound can only be approached by feasible solutions of the OP1. In this work, we consider a two-stage approach to tackle the OP1. At the first stage, subchannel allocation is implemented with an efficient heuristic method, which removes the integer constraints in the OP1. At the second stage, power distribution across subchannels is carried out to maximize the sum rate of the NRT SUs and also keep the RT SUs’ rate requirements satisfied. III. S UBCHANNEL A LLOCATION S CHEME We propose an efficient subchannel allocation scheme to figure out the binary variables ρk,n ’s in this section, specifying a subchannel allocation assignment. The intuition of our method is following. In an OFDM-based CR system, an OFDM subchannel with high SNR for an SU may also generate more interference to the PUs. That is to say, the water-filling-like method [9] for conventional OFDM systems is no longer suitable for CR scenarios because interference constraint also lays an upper bound of the transmission power for each subchannel. So the SNR of an OFDM subchannel and the interference introduced to the PUs should be jointly considered to measure the capacity of the subchannel. We propose calculating the achievable rate of the nth subchannel employed by the kth SU as follows, M = log(1 + pM rk,n k,n Hk,n ),
(7)
where pM k,n is the maximum possible power for the kth SU on the nth subchannel, pM k,n = min(Pt , min( l∈L
IlP )). SP In,l
(8)
The channel capacity is normalized by (7) in this way. Let Ωk denote the subchannel set employed by the kth SU, and N is the set of subchannels. The outline of our subchannel allocation scheme is described in Table I, which consists of two steps. First, we allocate the RT SUs subchannels to meet their minimal rate requirements. Then we allocate the remaining subchannels to the NRT SUs. The principle of our subchannel allocation algorithm for the RT SUs is that the SU whose current rate is the farthest away from the target one has the priority to get a subchannel among the available ones. This procedure continues until all RT SUs’ rate requirements are satisfied. Preferably, the subchannel with the highest achievable rate associated with an RT SU will be chosen at this step. To simplify analysis and computation, the power of a subchannel is temperately set as pk,n = SP )) in order to always satisfy the min(Pt /N, minl∈L (IlP /In,l
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
TABLE I S UBOPTIMAL S UBCHANNEL A LLOCATION
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iteration to compute the central point, while the Newton step is the inner iteration executed in the centering step. The complexity of the barrier method is generally O(N 3 ), where N is the number of subchannels in the OP3. Since there are always thousands of subchannels in OFDM systems, the computation cost is unacceptable because RA should be processed online. In this section, we propose a fast barrier method to work out the optimal solution of the OP3 by exploiting its special structure. The high computational load is dramatically reduced. For a given number of PUs, the complexity of our proposed fast barrier method is approximately linear to the number of subchannels.
1.Initialization: 2.Nt = N , Ωk = ∅,∀k 3.Set the RT SUs’ rates to zero: Rti = 0, for i = 1, 2, ..., K1 4.For the RT SUs: 5.While Nt = ∅ and Rtk < Rk,min for any 1 ≤ k ≤ K1 6. Find k ∗ satisfies Rtk∗ − Rk∗ ,min ≤ Rtk − Rk,min 7. For k ∗ , find n∗ satisfies rkM∗ ,n∗ ≥ rkM∗ ,n , ∀n ∈ Nt 8. Update Rtk∗ = Rtk∗ + log(1 + pk∗ ,n∗ Hk∗ ,n∗ ) 9. Update Ωk∗ = Ωk∗ ∪ n∗ ,Nt = Nt \ n∗ 10.Endwhile 11.For the NRT SUs: 12.For i = 1 to length(Nt ) M 13. For n∗ = Nti , find k ∗ satisfies rkM∗ ,n∗ ≥ rk,n ∗ 14. Update Ωk∗ = Ωk∗ ∪ n∗ 15.Endfor
A. The Fast Barrier Method power and the interference constraints. By doing so, we need not to consider the RT SUs when allocating subchannels to the NRT SUs. Each of the remaining subchannels is allocated to the NRT SU who has the highest achievable rate over this channel to roughly maximize the sum capacity of the CR system, as shown in Table I.
When subchannel assignment is given, the binary variables ρk,n ’s are fixed to 1 or 0, and the C5 and C6 constraints in the OP1 vanish. We need to figure out power distribution among subchannels to maximize the sum capacity while keeping all constraints satisfied. Recall that Ωk is the subchannel set of the kth SU, the optimal power allocation is to solve the following optimization problem,
s.t.
max
K
pk,n k=K +1 n∈Ω 1 k
C1 : C2 : C3 :
n∈Ωk
k=1 n∈Ωk K k=1 n∈Ωk
−
−
K 1
log(
k=1 K
n∈Ωk
k=1 n∈Ωk L l=1
log(1 + pk,n Hk,n ) − Rk,min )
log pk,n − log(Pt −
log(IlP −
K k=1 n∈Ωk
log(1 + pk,n Hk,n )
pk,n ≤ Pt
K k=1 n∈Ωk
pk,n )
SP pk,n In,l ),
(10) where x = (p1 , p2 , . . . , pN ). Notice that the subscript k can be omitted now as it has been determined for a given subchannel allocation assignment. Denote f (x) =
K
k=K1 +1 n∈Ωk
log(1 + pk,n Hk,n ),
(11)
the optimal solution to the OP3 can be approximated by solving the following unconstrained minimization problem,
rk,n ≥ Rk,min , 1, . . . , K1
K
φ(x) =
−
IV. O PTIMAL P OWER A LLOCATION : FAST BARRIER M ETHOD
OP 3
The first step of the barrier method is to reformulate the OP3 into a set of unconstrained optimization problems, making all inequality constraints implicit in an objective function. The logarithmic barrier function of the OP3 is following [27],
min Ψt (x) = −tf (x) + φ(x). (9)
SP pk,n In,l ≤ IlP , l = 1, . . . , L
C4 : pk,n ≥ 0, ∀k, n. Obviously, (9) defines a convex optimization problem because the objective function and the constraints are all concave or convex. Generally, barrier method [27] is treated as a standard technique to solve convex optimization problems with inequality constraints. In the barrier method, the original problem is converted into a sequence of unconstrained minimization problems by introducing a logarithmic barrier function with a given parameter t. Particularly, each unconstrained minimization problem determined by the parameter t can be solved by Newton method, and the solution to this problem is called a central point in the central path related to the original problem. As t increases, the central point will be more and more accurate to approximate the optimal solution of the original problem. The barrier method consists of two stages: centering step and Newton step. The former is the outer
(12)
Since function Ψt (x) is convex and twice continuously differentiable, (12) has a unique optimal solution. In each centering step of the barrier method, we adopt Newton method to compute the central point for a given parameter t. The Newton step at x, denoted by Δxnt , is given by 2 Ψt (x)Δxnt = − Ψt (x),
(13)
where 2 Ψt (x) and Ψt (x) are the Hessian and the gradient of Ψt (x), respectively. The outline of the barrier method is summarized in Table II. and n are the tolerances of the barrier method and the Newton step, respectively. α and β are two constants used in the backtracking line search in the Newton step, where α ∈ (0, 0.5) and β ∈ (0, 1), respectively. The step size of the backtracking line search is s with s > 0. t and μ are the parameters which are associated with a trade-off between the number of the outer and the inner iterations. If computing the Δxnt in (13) directly by matrix inversion, the computation complexity is O(N 3 ). It is too high to apply as discussed above. We analyze the problem (12) and develop
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TABLE II T HE BARRIER M ETHOD 1.Initialization for Barrier method 2. Find strictly feasible point x, t := t(0) > 0, tolerance > 0, μ > 1 3.Outer Loop for Barrier method 4. Centering step: Compute x∗ (t) derived by (12) via Newton method 5. Initialization for Newton method 6. Starting point x, tolerance n > 0, α ∈ (0, 1/2), β ∈ (0, 1) 7. Inner Loop for Newton method 8. Compute Δxnt and λ2 := − Ψt (x)Δxnt ; 9. Quit if λ2 /2 ≤ n 10. Backtracking line search on Ψt (x), s := 1 11. While Ψt (x + sΔxnt ) > Ψt (x) − αsλ2 12. s := βs 13. Endwhile 14. Update:x := x + sΔxnt 15. Update:x∗ (t) = x. 16. Stopping criterion:(N + K1 + L + 1)/t < . 17. Increase:t := μt.
a fast algorithm to calculate the Newton step with lower complexity by exploiting the special structure of (12). Denote K pk,n , f0 = Pt − k=1 n∈Ωk fk = rk,n − Rk,min , n∈Ωk
gl = IlP −
K k=1 n∈Ωk
k = 1, . . . , K1 ,
(14)
SP pk,n In,l , l = 1, . . . , L,
k = 1, . . . , K1 k = K1 + 1, . . . , K
The Hessian of Ψt (x) follows ⎡ D1 ⎢ 2 D2 ∂ Ψt (x) ⎢ =⎢ .. 2 ∂pk,n ⎣ .
+
k=1
=D+
fk fk2
M m=1
(15)
.
(16)
⎤ ⎥ ⎥ f0 f0T ⎥+ f02 ⎦
fkT
+
L gl glT gl2 l=1
Fi FiT ,
(17) where D = diag(D1 , D2 , . . . , DN ) ∈ RN ×N and M = K1 + L + 1 with Dn =
1 +χ k
p2k,n
In the initialization step, it requires a strictly feasible starting point for the barrier method. A preparatory procedure is necessary to compute and prove that the feasible points exist or not. As discussed in [27], finding a feasible solution is equivalent to solve a minimization problem by introducing a crucial indicator parameter z. The optimization problem for the warm start procedure can be formulated as follows, OP 4
ez C1 : rk,n + z ≥ Rk,min , k = 1, . . . , K1 min
z,pk,n
C3 :
DN K 1
B. Warm Start Procedure for the Barrier Method
n∈Ωk
where χk yields χk =
The proof is presented in Appendix. Recall that M = K1 + L+1 and generally M N in practical wireless systems. The computational complexity is significantly reduced compared with matrix inversion which needs O(N 3 ) computations.
C2 :
SP L χ H ∂Ψt (x) 1 + 1 + In,l , = − 1 + pk k,n − pk,n ∂pk,n f0 l=1 gl k,n Hk,n
1 fk t
Theorem 1: The equation (13) can be solved with the complexity of O(N M 2 ).
s.t.
the gradient of Ψt (x) is given by
Since it is easy to prove that the matrix D and all Fi FiT ’s are positive definite, it follows that 2 Ψt (x) is also positive definite. Hence, the Karush-Kuhn-Tucker (KKT) matrix 2 Ψt (x) on the left in (13) is invertible.
2 Hk,n
2.
(1 + pk,n Hk,n )
Fi ’s are all vectors with N elements, ⎧ f 0 ⎪ ⎨ f0 , i = 1 fk k = 1, . . . , K1 , i = k + 1 . Fi (x) = fk ⎪ ⎩ gl l = 1, . . . , L, i = K + l + 1 1 gl
(18)
(19)
K k=1 n∈Ωk K k=1 n∈Ωk
pk,n ≤ Pt SP pk,n In,l ≤ IlP + z, l = 1, . . . , L
C4 : pk,n ≥ 0, ∀k, n, (20) where z can be interpreted as a bound on the maximum infeasibility of the inequality C1 and C3 and our goal is to drive the maximum infeasibility below zero. Note that the OP4 is also a convex problem whose structure is as same as the OP3. Since it is easy to find pk,n ’s to satisfy C2 and C4 (i.e. pk,n = Pt /N ), we can choose a feasible z to satisfy C1 and C3. So the feasible solution to the OP4 always exists. Due to the special structure, we can also apply the fast barrier method developed in section IV-A to solve the OP4. By solving the OP4, a strictly feasible point may be computed, or there are no feasible points. If the optimal solution to the OP4 satisfies z ≤ 0, the strictly feasible power allocation pk,n ’s can be used as the starting point of the barrier method to solve the OP3. Otherwise, no feasible point exists for the OP3 and we regard such a case as system outage. C. On the complexity The fast barrier algorithm of solving the OP3 consumes M decomposition, while each decomposition yields an additional equation. We need to solve M + 1 matrix system in the first step of the reverse substitution and the computational complexity for each system is measured by O(N ). After M reverse substitution steps, the total computational cost for the optimal power allocation is O(N M 2 ), which is much lower than O(N 3 ) if using matrix inversion directly.
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
Since we can also apply the proposed fast algorithm to solve the OP4, the computational complexity of the warm start procedure is equal to that of the OP3 because of the same structure. Therefor, we conclude that the complexity of the optimal power allocation is O(N M 2 ). Notice that the number of the PUs is always much smaller than that of the subchannels in wireless systems, that is, M N , the complexity is reduced dramatically. D. On Solving the OP2 Recall that the OP2 is the relaxation form of OP1. The solution to OP2 provides an upper bound of OP1. We will show that OP2 can also be efficiently solved by exploiting its structure with the same analysis method in section IV-A. First, the optimal solution of the OP2 can be approximated by the following equality constrained minimization problem, min Ψt (x) = −tf (x) + φ(x) K ρk,n = 1, ∀n, s.t.
(21)
k=1
where φ(x) is the barrier function, which makes all inequality constraints implicit, and f (x) is the objective function of the OP2. Ψt (x) is convex and twice continuously differentiable, and Newton method can be used to compute the central point with a given parameter t. The Newton step Δxnt at x and the associated dual variable v are given by 2 Ψt (x) d − Ψt (x) Δxnt = , (22) dT 0v 0n ν where zero matrix 0n ∈ RN ×N and vector 0v ∈ RN ×1 . d is a 2KN × N matrix in which d2m,n = 1, m = (k − 1)N + n, ∀k, n, and other elements are zeroes. 2 Ψt (x) and Ψt (x) are the Hessian and the gradient of Ψt (x), respectively. The (2KN + N ) × (2KN + N ) matrix in the left of the equation (22) can be decomposed into 2 M Ψt (x) d Q d Gi GTi , (23) = + dT 0n dT 0n i=1
where Q = (Q1,1 , Q1,2 , . . . , QK,N ) with 1 0 s2k,n Qk,n = + 1 0 + (1−ρ1k,n )2 ρ2k,n ⎡ ⎤ 2 2 χk ⎣
∂ f (x) ∂s2k,n ∂ 2 f (x) ∂sk,n ∂ρk,n
∂ f (x) ∂sk,n ∂ρk,n ∂ 2 f (x) ∂ρ2k,n
A. The Approximation In [19], an index function is defined to measure the cost of allocating each bit, introducing an efficient bit loading algorithm. Similarly, we define a normalized cost function indicating the cost of allocating certain rate on each subchannel, erk,n − 1 . (25) Fc (rk,n ) = M erk,n − 1 The channel SNR and the interference to the PUs are jointly considered in the cost function, reflecting the potential rate of each subchannel. With the cost function, we can convert the transmission power constraint C2 and the interference constraints C3 in the OP3 into a normalized capacity. Given a subchannel assignment, we can transform the OP3 into the following form, OP 5
max
s.t.
C1 :
K
rk,n k=K +1 n∈Ω k 1 n∈Ωk
V. A N E FFICIENT A PPROXIMATION AND A LGORITHM FOR P OWER D ISTRIBUTION In this section, we propose a heuristic power allocation algorithm to approximate the optimal solution with low complexity. The heuristic method can produce solution close to the optimal with lower complexity than the fast barrier method.
(26)
C2 : rk,n ≥ 0, ∀k, n K Fc (rk,n ) ≤ C. C3 : k=1 n∈Ωk
The parameter C in the OP5 is a constant determined by the transmission power and the interference thresholds, informing the maximum sum capacity of all subchannels. The key role of this transformation is that the constraints C2 and C3 in the OP3 are unified into a single constraint C3 in the OP5. The OP5 can be regarded as an approximation to the OP3. Notice that we do not know the value of C in advance and it can be only worked out when all rk,n ’s are known. We will show that it is not necessary to know C when solving the OP5. B. The Algorithm Without consideration of C2 in the OP5, the Lagrangian of the OP5 is given by L(rk,n , λ0 , μk ) = −
K
k=K1 +1 n∈Ωk K K1
+
(24)
and Gi = [ Fi 0v ]T . Obviously, our proposed fast barrier method is also applicable for calculating Newton step efficiently for the OP2. So the computational cost of solving the OP2 can be reduced to O(KN M 2 ), which is much lower than O((2KN + N )3 ) if using standard techniques.
rk,n
rk,n ≥ Rk,min , k = 1, . . . , K1
+λ0 (
⎦,
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k=1 n∈Ωk
k=1
rk,n
Fc (rk,n ) − C)
μk (Rk,min −
n∈Ωk
(27)
rk,n ),
where λ0 and μk (k = 1, . . . , K1 ) are the Lagrange multipliers with λ0 ≥ 0 and μk ≥ 0. Applying KKT conditions [27], the ∗ optimal solution rk,n ’s satisfy the following equations when concerning C2 in the OP5, ∗ ≥0 = 0, rk,n ∂L , (28) ∗ ∗ ∂rk,n =0 > 0, rk,n λ0 (
K
k=1 n∈Ωk
μk (Rk,min −
n∈Ωk
∗ Fc (rk,n ) − C) = 0,
∗ rk,n ) = 0, k = 1, . . . , K1 .
(29) (30)
With simple transformations of (28), we get the optimal solution for the RT SUs, rM k,n −1)
∗ rk,n = [log( μk (e λ0
)]+ , k = 1, . . . , K1 ,
(31)
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TABLE III P OWER D ISTRIBUTION FOR THE RT SU S 1.Initialization: 2.For the RT SUs, denote Ω∗k = Ωk , k = 1, . . . , K1 3.Power Distribution among the RT SUs 4.For k = 1 to K1 5. While true 6. Calculate rk,n by (35) n ∈ Ω∗k and sort rk,n ’s in ascending order 7. If rk,1∗ ≥ 0 8. break 9. Else 10. rk,1∗ = 0, Ω∗k = Ω∗k /1∗ 11. Endif 12. Endwhile 13.Endfor 14.Power loading: pk,n = (erk,n − 1)/Hk,n , n ∈ Ωk , k = 1, . . . , K1
where [x]+ := max(x, 0). And the optimal solutions for the NRT SUs are M rk,n
∗ rk,n = [log( e
λ0
−1
)]+ , k = K1 + 1, . . . , K.
(32)
For the kth SU, denote Ω∗k as the subchannels set in which ∗ ∗ rk,n > 0, ∀n ∈ Ω∗k . Sort the rk,n s in ascending order. Then ∗ for m, n ∈ Ωk , we have ∗ rk,m
−
∗ rk,n
k = Denote wm,n is given by
=
pM Hk,m log( pk,m ), M H k,n k,n
pM k,m Hk,m , pM k,n Hk,n
Rk = Nk rk,1∗ +
n∈Ω∗ k
k = 1, . . . , K.
(33)
Nk = |Ω∗k |, the rate of the kth user
k log(wn,1 k = 1, . . . , K, ∗ ),
(34)
where rk,1∗ represents the minimal element in Ω∗k . Substitute (33) and (34) into (30) for the case of μk = 0, we can obtain the closed-form expression of rk,n for the RT SUs, k k rk,n = N1k (Rk,min − log(wn,1 ∗ )) + log(wn,1∗ ). n∈Ω∗ k
(35) The detail of power allocation for the RT SUs is illustrated in Table III. Denote the power allocated to the kth SU and the interference to the lth PU introduced by the kth SU as Pk and Ikl , respectively, we can get the following equations for each SU based on (2), erk,n − 1 Pk = ∀k Hk,n n∈Ωk SP (36) (erk,n − 1)In,l Ikl = ∀k, l = 1, . . . , L. Hk,n n∈Ω k
Consider the power limitation and the interference constraints, for the K − K1 NRT SUs, we have K k=K1 +1 K k=K1 +1
Pk = Ikl =
where Pr =
K1 k=1
K
erk,n −1 Hk,n
≤ Pt − Pr ,
k=K1 +1 n∈Ωk K (erk,n −1)I SP Hk,n
k=K1 +1 n∈Ωk
Pk and Ilr =
K1 k=1
n,l
(37) ≤ IlP − Ilr ,
Ikl are the power consump-
tion and interference introduced to the lth PU by all RT SUs, respectively.
Substituting (32) into (37), the minimum value of λ0 is determined by solving the following inequalities, M + K (erk,n −1)/λ0 −1 ≤ Pt − Pr , Hk,n k=K1 +1 n∈Ωk (38) + rM K SP In,l (e k,n −1)/λ0 −1 P r ≤ I − I . l l Hk,n k=K1 +1 n∈Ωk
From (32), we know the minimal λ0 can maximize the sum capacity of the NRT SUs. With the value of λ0 obtained from (38), we get the optimal rate allocation for the NRT SUs by (32), which is equivalent to achieve the power distribution among the NRT SUs. The complexity of solving the OP5 can be counted roughly as follows. In the procedure of power loading for the RT SUs, the complexity is O(|Ωk |) for each RT SU, where |Ωk | is the number of subchannels allocated to the kth SU. On the other hand, the rate allocation for the NRT SUs can be obtained in O(L|Nn |), where |Nn | denotes the number of subchannels allocated to the NRT SUs. To sum up, the complexity of the power allocation scheme is bounded by O(LN ). VI. S IMULATION R ESULTS We evaluate our proposed algorithms with a series of experiments. Consider a multiuser OFDM-based CR system, where all users randomly located in a 3 × 3km area, and each receiver uniformly distributed in a circle within 0.5km from its transmitter. The path loss exponent is 4, the variance of shadowing effect is 10dB and the amplitude of multipath fading is Rayleigh. We assume that each PU’s bandwidth is randomly generated by uniform distribution and the maximum value is 2W/3L. The noise power is 10−13 W and the interference thresholds of all PUs are set to 5 × 10−12 W. Consider the following algorithms: Time-sharing (TS) method defined by the OP2, subchannel assignment proposed in Section III followed by the optimal power allocation method proposed in Section IV (Scheme 1), subchannel assignment in Section III followed by the approximation power allocation method proposed in Section V (Scheme 2), maximum SNR priority (MSP) subchannel allocation and minimum interference priority (MIP) subchannel allocation. For the MSP, we always allocate a subchannel to an SU who acquires the highest SNR over this channel, while the MIP scheme allocates a subchannel to an SU that generates the minimum interference to PUs. Both of the MSP and the MIP adopt the optimal power allocation strategy, marked as MSP-OP and MIP-OP in Fig.1. As discussed in Section II, the solution of the TS method is an upper bound of the original problem OP1. Fig.1 shows the average number of bits per symbol of the NRT SUs as a function of transmission power limit. The number of PUs and SUs are 2 and 4, respectively, including 2 RT SUs. There are 64 subchannels in the OFDM-based CR system and the rate requirements of RT SUs are uniformly set to 20bits/symbol. We can observe that the sum capacity of the NRT SUs increases as the transmission power limit increases. Both of our proposed two RA schemes outperform the MSP-OP and the MIP-OP obviously. The average capacity gap between the Scheme 1 and the TS is less than 6%, while the Scheme 2 can achieve more than 92% of the upper bound.
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
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600
500
Average number of bits per symbol of NRT SUs
Average number of bits per symbol of NRT SUs
1500
TS Scheme 1 Scheme 2 MSP−OP MIP−OP
400
300
200
100
−1
10
0
10 Power limit of SUs
1
10
2
2
4
6
8 10 Number of NRT SUs
12
14
16
Fig. 3. Average number of bits per OFDM symbol of NRT SUs as a function of the number of NRT SUs. Rk,min =10bits/symbol, N = 128, K1 = 4, L = 2. 1000
900 TS Scheme 1 Scheme 2 MSP−OP MIP−OP
Average number of bits per symbol of NRT SUs
Average number of bits per symbol of NRT SUs
500
10
Fig. 1. Average number of bits per OFDM symbol of NRT SUs as a function of the transmission power. Rk,min =20bits/symbol, N = 64, K = 4, K1 = 2, L = 2.
700
1000
0
0 −2 10
800
TS Scheme 1 Scheme 2 MSP−OP MIP−OP
600 500 400 300 200
TS Scheme 1 Scheme 2 MSP−OP MIP−OP
900 800 700 600 500 400 300 200 100
100 0
0
50
100 150 Number of subchannels
200
250
Fig. 2. Average number of bits per OFDM symbol of NRT SUs as a function of the number of subchannels. Rk,min =20bits/symbol, Pt = 1W ,K = 4, K1 = 2, L = 2.
Since the MSP and the MIP schemes do not consider the power limit and the interference jointly, they can only achieve about 40% and 30% capacity of the upper bound. Fig.2 illustrates the average number of bits per symbol of NRT SUs as a function of the number of subchannels. The number of SUs is 4 including 2 RT SUs. The transmission power limit is set to 1W and the minimal rate requirement for each RT SU is 20bits/symbol. As the increasing of the number of subchannels, the sum rate of the NRT SUs increases and the rate difference per subchannel between our proposed two schemes and the optimal becomes smaller. The reason is that the SUs can benefit from channel diversity in wireless environment. Our two schemes (Scheme 1 and Scheme 2) significantly outperform the MSP-OP and the MIP-OP. Specially, when N = 256, the difference between the Scheme 1 (Scheme 2) and the upper bound decreases to 3% (4%), while the MSPOP and the MIP-OP just achieve about 50% and 30% capacity
2
4
6
8 10 Number of RT SUs
12
14
16
Fig. 4. Average number of bits per OFDM symbol of NRT SUs as a function of the number of RT SUs. Rk,min =10bits/symbol, N = 128, K − K1 = 4, L = 2.
of the TS. This observation suggests that the performance loss due to the heuristic subchannel allocation method is neglectful when N is sufficiently large, which is suitable for application because there are always thousands of subchannels. We also demonstrate the multiuser diversity effect for the mentioned algorithms in Fig.3. The number of the RT SUs is K1 = 4, and the number of the NRT SUs varies from 1 to 16. Fig.3 shows the average number of bits per symbol as a function of the number of the NRT SUs, Rk,min = 10bits/symbol and Pt = 1W. The number of subchannels is 128. As can be seen in Fig.3, the sum capacity of the CR system increases with the increasing of the number of the NRT SUs. The difference between the TS and the Scheme 1 is roughly staying at 6%, so is the Scheme 2, both of which outperform the MSPOP and the MIP-OP. We can explain this phenomenon as a result of multiuser diversity. When there are more SUs, there are more chances for a subchannel to be allocated to a user with higher channel gain.
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Fast barrier method for optimal RA based on TS Newton iterations
200 150 100 50 0
10
20
30
40 50 60 Random instance(a)
70
80
90
100
Fast barrier method for optimal PA
into a universal subchannel capacity, and produces solution close to the optimal with much lower complexity. Moreover, we also suggest a simple but efficient heuristic method for subchannel allocation. Simulation results verify the effectiveness and the efficiency of the proposed schemes. Our proposal can be naturally extended to the weighted sum rates case, partially dealing with the issue of fairness. However, fairness among all users [28] is not fully considered in this paper and should be investigated in our future work.
Newton iterations
80
A PPENDIX
60
Proof of Theorem 1 Proof : Rewrite the KKT system (13) as
40 20 0
Λ0 Δx = F0 , 10
20
30
40 50 60 Random instance(b)
70
80
90
100
(39)
2
where Λ0 = Ψt and F0 = − Ψt . According to the analysis in section IV-A, we have
Fig. 5. Number of Newton iterations required for convergence for 100 channel realizations. N = 64, K = 4, K1 = 2, L = 2. (a) Fast barrier method for RA based on time-sharing; (b) Fast barrier method for the optimal power allocation with given subchannel allocation assignments.
Λ0 = D +
M
Fi FiT ,
(40)
i=1
which can be decomposed into M equations, Fig.4 depicts the average number of bits per symbol as a function of the number of the RT SUs. The minimal rate requirement Rk,min of each RT SU is 10bit/symbol, and the number of the RT SUs varies from 2 to 16. The number of the NRT SUs and subchannels is 4 and 128, respectively. The transmission power limit is Pt = 1W . Fig.4 shows that the sum capacity of the NRT SUs decreases as the number of the RT SUs increases. It can be explained intuitively. When there are more RT SUs trying to access the CR network, more subchannels or power are consumed by the RT SUs to meet their rate requirements. Moreover, radio resource will be more frequently exhausted when the number of the RT SUs becomes larger, which intensifies the capacity loss of the NRT SUs. The average capacity gap between our proposed two schemes and the TS is about 6%. Again, our proposed algorithms perform much better than the MSP-OP and the MIP-OP. Finally, we investigate the convergence of the fast barrier method. As discussed in section IV, the computational load of the barrier method mainly lies in the number of Newton iterations. From Fig.5(a) and (b), we observe that the average number of Newton iterations required for a guaranteed duality gap of 10−3 is 110 and 40 for the OP2 and the OP3, respectively. Furthermore, the number of iterations varies in a narrow range. We conservatively conclude the proposed fast barrier method is effective and efficient. VII. C ONCLUSION In this paper, we show that for a multiuser OFDM CR system with different rate requirements to support heterogeneous services, the optimal power allocation can be implemented efficiently by exploiting the structure of the problem. We proposed alternative power allocation schemes: Fast barrier method and approximate method. The fast barrier method can work out the optimal solution quickly by updating Newton step in an ingenious way. The approximate method normalizes the transmission power limitation and the interference constraints
T , i = 0, . . . , M − 1, Λi = Λi+1 + Fi+1 Fi+1
where Λi = D +
M j=i+1
(41)
Fj FjT . Note that all Λi ’s are nonsin-
gular. By exploiting the structure of Λi ’s, we give an M -step procedure to compute Δx. Step 1 Use (41) to decompose Λ0 , Λ0 = Λ1 + F1 F1T . F1 υ11 1 1 Then Δx = υ11 − 1+F 1 υ , where Λ1 υ1 = F0 and 1 υ2 2 1 Λ1 υ2 = F1 . So we can figure out the Δx if obtaining the two new variables υ11 and υ21 , instead of solving (39) directly. Step 2 Decompose Λ1 with Λ1 = Λ2 + F2 F2T . Then the two variables introduced in the step 1 can be F2 υi2 2 2 updated by υi1 = υi2 − 1+F 2 υ , i = 1, 2, where Λ2 υi = 2 υ3 3 Fi−1 , i = 1, 2, 3. For the mth step, T Step m Decompose Λm−1 = Λm + Fm Fm . We can update the m variables introduced in the Step F T υim υ m , i = 1, . . . , m, where m−1 by υim−1 = υim − 1+Fm m m υm+1 m+1 Λm υim = Fi−1 , i = 1, . . . , m + 1. Continue the procedure to the M th step, and there are M +1 matrix systems ΛM υiM = Fi−1 ,i = 1, . . . , M + 1. Form the derivation process, we can find that the m variables υim−1 , i = 1, . . . , m in the (m − 1)th Step can be obtained by the m + 1 variables υim , i = 1, . . . , m + 1 in the mth Step. Hence, if we can figure out the M + 1 variables υiM , i = 1, . . . , M + 1, the Δx in (39) will be indirectly obtained. Obviously, a reverse derivation of the M steps is necessary after we solve the M +1 matrix system in the M th Step. The process to solve the matrix equation ΛM υiM = Fi−1 is following. From Section IV-A, we have ΛM = D. Without loss of generality, we unify these equations into ⎡ ⎤ D1 ⎢ ⎥ D2 ⎢ ⎥ (42) ⎢ ⎥υ = g .. ⎣ ⎦ . DN
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
where υ, g ∈ RN ×1 . Since D is a diagonal matrix, we can easily obtained υi = Di−1 gi , i = 1, . . . , N,
(43)
which costs O(N ) to obtain the υ in (42). So it costs O(N M ) to solve the M + 1 variables at the M th Step. Then an M -step reverse iteration is implemented to figure out the Δx, and the total computational complexity can be measured by O(N M 2 ). ACKNOWLEDGEMENT The authors would like to thank the editors and the anonymous reviewers, whose invaluable comments helped improve the presentation of this paper substantially. R EFERENCES [1] F. C. Commission, “Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies,” FCC Report, ET Docket 03-322, Dec. 2003. [2] J. Mitola and G. Q. Maguire, “Cognitive radio: making software radios more personal,” IEEE Person. Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999. [3] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [4] F. C. Commission, “Spectrum policy task force report,” FCC Report, ET Docket 02-135, Nov. 2002. [5] B. Farhang-Boroujeny and R. Kempter, “Multicarrier communication techniques for spectrum sensing and communication in cognitive radios,” IEEE Commun. Mag., vol. 46, no. 4, pp. 80–85, Apr. 2008. [6] H. Mahmoud, T. Yucek, and H. Arslan, “OFDM for cognitive radio: merits and challenges,” IEEE Wireless Commun., vol. 16, no. 2, pp. 6–15, Apr. 2009. [7] T. A. Weiss and F. K. Jondral, “Spectrum pooling: an innovative strategy for the enhancement of spectrum efficiency,” IEEE Commun. Mag., vol. 42, no. 3, pp. 8–14, Mar. 2004. [8] S. Sadr, A. Anpalagan, and K. Raahemifar, “Radio resource allocation algorithms for the downlink of multiuser OFDM communication systems,” IEEE Commun. Surv. & Tut., vol. 11, no. 3, pp. 92–106, Sep. 2009. [9] C. Y. Wong, R. S. Cheng, K. B. Lataief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999. [10] G. Zhang, “Subcarrier and bit allocation for real-time services in multiuser OFDM systems,” in Proc. IEEE ICC’04, vol. 5, pp. 2985– 2989. [11] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171–178, Feb. 2003. [12] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in Proc. 2000 IEEE VTC – Spring, vol. 2, pp. 1085–1089. [13] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2726–2737, Nov. 2005. [14] M. Tao, Y.-C. Liang, and F. Zhang, “Resource allocation for delay differentiated traffic in multiuser OFDM systems,” IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 2190–2201, June 2008. [15] R. Madan, S. P. Boyd, and S. Lall, “Fast algorithms for resource allocation in wireless cellular networks,” IEEE/ACM Trans. Networking, vol. 18, no. 3, pp. 973–984, June 2010.
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[16] R. Zhang, Y.-C. Liang, and S. Cui, “Dynamic resource allocation in cognitive radio networks,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 102–114, May 2010. [17] G. Bansal, M. J. Hossain, and V. K. Bhargava, “Optimal and suboptimal power allocation schemes for OFDM-based cognitive radio systems,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4710–4718, Nov. 2008. [18] Y. Zhang and C. Leung, “Resource allocation in an OFDM-based cognitive radio system,” IEEE Trans. Commun., vol. 57, no. 7, pp. 1928– 1931, July 2009. [19] S. Wang, “Efficient resource allocation algorithm for cognitive OFDM systems,” IEEE Commun. Lett., vol. 14, no. 8, pp. 725–727, Aug. 2010. [20] S. Wang, F. Huang, and Z.-H. Zhou, “Fast power allocation algorithm for cognitive radio networks,” IEEE Commun. Lett., vol. 15, no. 8, pp. 845–847, Aug. 2011. [21] F. Digham, “Joint power and channel allocation for cognitive radios,” in Proc. IEEE WCNC’08, pp. 882–887. [22] A. Attar, M. R. Nakhai, and A. H. Aghvami, “Cognitive radio game for secondary spectrum access problem,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 2121–2131, Apr. 2009. [23] Y. Zhang and C. Leung, “Resource allocation for non-real-time services in OFDM-based cognitive radio systems,” IEEE Commun. Lett., vol. 13, no. 1, pp. 16–18, Jan. 2009. [24] S. Wang, F. Huang, M. Yuan, and S. Du, “Resource allocation for multiuser cognitive OFDM networks with proportional rate constraints,” Int. J. Commun. Sys., DOI:10.1002/dac.1272, 2011 (online). [25] S. M. Almalfouh and G. L. Stuber, “Interference-aware radio resource allocation in OFDMA-based cognitive radio networks,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1699–1713, May 2011. [26] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218– 1230, Oct. 1997. [27] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [28] S. Maharjan, Y. Zhang, C. Yuen, and S. Gjessing, “Distributed spectrum sensing in cognitive radio networks with fairness consideration: Efficiency of correlated equilibrium,” in Proc. IEEE MASS’11, pp. 540– 549. Mengyao Ge received the B.S. degree in communication engineering from Nanjing University, Nanjing, China, in 2011. She is currently pursuing the M.S. degree at the School of Electronic Science & Engineering, Nanjing University, Nanjing, China. She won the Best Bachelor’s Thesis Award from Nanjing University in 2011. Her research interests include wireless communications and convex optimization. Currently, her research focuses on resource allocation in wireless networks.
Shaowei Wang (S’06-M’07) received the B.S., M.S. and Ph.D. degrees in electronic engineering from Wuhan University, Wuhan, China, in 1997, 2003, and 2006, respectively. From 1997 to 2001, he was a research scientist of China Telecom. He joined the School of Electronic Science & Engineering of Nanjing University, Nanjing, China, in 2006, where he has been an Associate Professor since 2008. From 2009 to 2011, He was also a postdoctor at LAMDA group, Nanjing University. His research interests include wireless communications & networks, optimization and machine learning. Currently, his research focuses on resource allocation in wireless networks, cellular networks planning, and traffic engineering in telecommunications.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 4, APRIL 2012
Fast Optimal Resource Allocation is Possible for Multiuser OFDM-Based Cognitive Radio Networks with Heterogeneous Services Mengyao Ge and Shaowei Wang, Member, IEEE
Abstract—In this paper we study the resource allocation in multiuser orthogonal frequency division multiplexing (OFDM)based cognitive radio (CR) networks, where secondary users (SUs) have flexible traffic demands, including heterogeneous realtime (RT) and non-real-time (NRT) services. We try to maximize the sum capacity of the NRT users and maintain the minimal rate requirements of the RT users simultaneously. Additionally, the interference introduced to primary users, which is generated by the access of the SUs, should be kept below a predefined threshold, which makes the optimization task more complex. The contribution of this work is two folds. First, we show that the formulated optimization problem has a special structure which can be exploited to implement a fast barrier method to obtain the optimal solution with a reasonable complexity. Second, we propose an effective measurement criterion to normalize OFDM subchannels’ achievable rates, based on which we develop simple but efficient heuristic algorithm for subchannel assignment and power distribution. Simulation results show that our proposed resource allocation schemes work quite well for concerned wireless scenarios. The fast barrier method converges very fast and can always work out the optimal solution, while the heuristic algorithm produces solution close to the optimal with much lower complexity. Index Terms—Cognitive radio, OFDM, optimization, resource allocation.
I. I NTRODUCTION
D
ESPITE the emerging crisis of spectrum shortage in wireless communication, it is inconsistent that radio spectrum is far from fully utilized due to conventional regulatory policies [1]. Cognitive Radio (CR) is envisioned as a promising paradigm for improving the usage efficiency of radio spectrum [2] [3]. Although radio spectrum has been assigned to primary users (PUs) in a licensed network, CR technology allows secondary users (SUs) in a CR network to sense spectrum environment and dynamically adjust their transmission parameters to access the licensed spectrum in an opportunistic manner, as long as the interference to the PUs is kept below a preset threshold, such as interference temperature level [4]. Since the physical layer of a CR network should be very flexible to meet the requirements of opportunistic access, Manuscript received June 28, 2011; revised October 30, 2011; accepted January 4, 2012. The associate editor coordinating the review of this paper and approving it for publication was C.-C. Chong. M. Ge and S. Wang (corresponding author) are with the School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China (e-mail: [email protected], [email protected]). This work was partially supported by the Jiangsu Science Foundation (BK2011051) and the Fundamental Research Funds for the Central Universities (1095021029, 1118021011). Digital Object Identifier 10.1109/TWC.2012.021512.111233
it necessitates multicarrier methods to operate in CR networks [5] [6]. Owning to the inherent significant advantages of flexibly allocating radio resource [7], Orthogonal Frequency Division Multiplexing (OFDM) is deemed as a promising air interface for CR systems. As one of the most important issues in OFDM systems, adaptive resource allocation (RA) have been studied extensively during the past two decades. A comprehensive survey can be found in [8] and references therein. RA algorithms in conventional OFDM systems can be classified into two categories: margin adaptive and rate adaptive. The optimization objective of the former is generally to minimize the required power under given rate requirements of users, while algorithms for rate adaptive usually try to maximize an OFDM system’s throughput under transmission power limitation. In [9], subchannel, bit and power allocation schemes in OFDM systems are developed. The proposed algorithm shows significant advantages compared with other methods. A margin adaptive algorithm is discussed in [10] for real-time services in a multiuser OFDM system. In [11], sum capacity of an OFDM system is maximized by allocating each subchannel to the user with the highest channel gain over this channel and distributing power among subchannels in a water-filling manner. However, fairness is ignored because the user with poor channel condition may be unable to receive any data. In [12], a max-min algorithm is proposed to maximize the capacity of the user with the worst channel condition. All users can achieve an equal rate with the proposed RA scheme and it provides a good fairness. But for most wireless systems, users may have various rate requirements to subscribe different kinds of services, the scheme proposed in [12] can not meet such requirements. In [13], proportional rate constraints are introduced and an optimal power allocation algorithm is developed to maximize the sum capacity of the OFDM system. In [14], both delay-constrained and non-delay-constrained traffic demands are considered for a heterogeneous multiuser OFDM system. The objective of [14] is to maximize the sum rate of the non-delay-constrained users while maintaining the delay-constrained users’ rate requirements. An iterative algorithm in dual domain is proposed to find the optimal solution of power distribution. In [15], user’s utility is defined as a concave function and a low complexity algorithm is developed, which can work out the optimal solution. However, it is impractical to assume that a user can occupy a non-integer number of OFDM subcarriers. RA for OFDM-based CR networks has been attracted more
c 2012 IEEE 1536-1276/12$31.00
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
and more attentions recently. An overview of the state-ofthe-art research results can be found in [16]. For single SU case, optimal and suboptimal power loading algorithms are presented in [17], where downlink sum capacity is maximized with the constraint that the interference introduced to the PUs is within a tolerable range. In [18], RA for OFDMbased CR network is formulated as a multidimensional knapsack problem, also for single SU case. A greedy heuristic algorithm is proposed, which can produce solution close to the optimal. However, the computational cost would be very high if generalizing it to multiple SUs setting. In [19], an efficient algorithm is derived by converting transmission power and interference constraints into a normalized capacity, which exhibits a tradeoff between capacity and complexity. But there is a capacity gap between the solution produced by the proposed algorithm and the optimal [19]. In [20], an optimal power distribution algorithm is developed, whose complexity is linear related to the number of subchannels. However, only single SU is considered in [19, 20], which limits their applications in practical wireless networks. RA for multiuser OFDM-based CR system is investigated in [21], where a simple near-optimal algorithm is proposed to jointly allocate power and subchannels. In [22], an optimal strategy based on cooperative Game Theory is developed to allocate radio resource among the SUs with given bit-errorrates (BERs) while guaranteeing the required quality of service of the PUs. RA for non-real-time applications is investigated in [23], where SUs’ rates are maintained in proportion to target ones. In [24], a low complexity algorithm is developed to maximize the sum capacity of a CR system with proportional rate constraints. The proposed algorithm exhibits a tradeoff between capacity and fairness by jointly considering channel gain and the interference to the PUs. In [25], the interference to the PUs is assumed to be yielded in two mechanisms: CR out-of-band emissions and imperfect spectrum sensing. Suboptimal RA algorithms are also developed. In this paper, we investigate a general spectrum-sharing case for a CR system: The SUs can access the regulated portion of licensed spectrum as long as the interference to the PUs is kept below their tolerable thresholds. The CR system adopts OFDM modulation and operates in a centralized manner, that is, an access point serves all SUs in the CR network, just as a conventional base station does. We do not assume that the licensed system serving the PUs also operates in an OFDM manner. Heterogeneous services, for both real-time (RT) SUs and non-real-time (NRT) SUs, exist in the CR network. We aim to maximize the downlink sum capacity of the NRT SUs while guaranteeing the minimal rate requirements of the RT SUs. Different from conventional OFDM system model [14], to prohibit the performance degradation of the PUs, the interference introduced to the PUs must be considered carefully. We analyze the formulated optimization problem intensively and develop a fast barrier method to work out the optimal solution by exploiting its structure. Furthermore, we propose an efficient subchannel allocation and power distribution scheme to achieve a tradeoff between capacity and complexity by jointly considering the signal-to-noise ratios (SNRs) of OFDM subchannels and their interference levels.
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The rest of this paper is organized as follows. In Section II, we illustrate system model and formulate the optimization task. In Section III, we proposed a heuristic subchannel allocation scheme, followed by a fast barrier method developed in Section IV. In Section V, an approximate optimal power distribution algorithm is presented. Simulation results are given in Section VI, as well as discussions. Conclusion is drawn in Section VII. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION Consider a CR system with K SUs, denoted by K = {1, 2, . . . , K}, coexisting with L PUs in a licensed system. We assume that perfect channel state information is available at the transceivers of the SUs and the PUs. The total bandwidth W is divided into N OFDM subchannels in the CR system, denoted by N = {1, 2, . . . , N }. The bandwidth of the nth subchannel spans from f0 + (n − 1)W/N to f0 + nW/N , where f0 is the starting frequency and W/N is the bandwidth of each OFDM subchannel. There are K1 RT SUs, each of them has minimal rate requirement Rk,min , k = 1, 2, . . . , K1 . The lth PU’s nominal band ranges from fl to fl + Bl , where fl and Bl are the lth PU’s starting frequency and bandwidth, respectively. The interference to the lth PU introduced by SUs’ access on the nth subchannel with unit transmission power is fl +Bl −f0 −(n−1/2)W/N SP = gn,l φ(f )df, (1) In,l fl −f0 −(n−1/2)W/N
where gn,l represents the power gain from an SU’s transmitter to the lth PU’s receiver on the nth subchannel. φ(f ) is the power spectral density (PSD) of OFDM signal, and φ(f ) = T ( sinπfπfT T )2 , where T is OFDM symbol duration. Let rk,n denote the transmission rate of the kth SU on the nth subchannel, we have pk,n |ck,n |2 rk,n = log 1 + , (2) Γ(N0 W/N + IkP S ) where pk,n is the power allocated to the kth SU over the nth subchannel, ck,n is the channel gain of the kth SU over the nth subchannel, N0 is the PSD of additive white Gaussian noise and Γ is the SNR gap. For an uncoded MQAM, Γ is related to a given BER requirement and Γ = − ln(5BER)/1.5 as derived in [26]. The interference caused by the PUs’ signals is IkP S , which can be regarded as noise and measured by the receiver of the kth SU. For notation brevity, denote Hk,n = |ck,n |2 as the SNR of the nth subchannel used by the Γ(N0 W/N +IkP S ) kth SU with unit power, the rate of the kth SU is Rk =
N
ρk,n log(1 + pk,n Hk,n ),
(3)
n=1
where ρk,n can be either 1 or 0, informing whether the kth SU occupies the nth subchannel or not. We try to maximize the sum rate of the NRT SUs while guaranteeing the minimal rate requirements for the RT SUs, with total transmission power budget of the CR system and the interference constraints of the PUs. Mathematically, the optimization problem can be described as follows,
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OP 1 s.t.
K
max
N
pk,n ,ρk,n k=K +1 n=1 1 N
C1 :
n=1
ρk,n rk,n
ρk,n rk,n ≥ Rk,min , k = 1, . . . , K1
C2 : pk,n ≥ 0, ∀k, n K N C3 : ρk,n pk,n ≤ Pt k=1 n=1
C4 : C5 :
K N k=1 n=1 K k=1
(4)
SP ρk,n pk,n In,l ≤ IlP , l = 1, . . . , L
ρk,n = 1, ∀n
C6 : ρk,n ∈ {0, 1}, ∀k, ∀n, where Pt is the power limit of the access point in the CR system and IlP is the interference power threshold of the lth PU. C1 guarantees the target rate requirements of the RT SUs. C2 and C3 are the transmission power constraints, while C4 is the interference constraints of the PUs. C5 and C6 declare that each subchannel is kept from being shared among the SUs. Both binary variables ρk,n ’s and real variables pk,n ’s are involved in the OP1. The main difficulty of solving the OP1 lies in the integer constraints. An intuitive way to tackle them is the time-sharing method, which relaxes the integer variables into continuous ones. Define ρk,n ∈ [0, 1] as the faction of the nth subchannel allocated to kth SU, temporarily permitting that each OFDM subchannel can be shared by multiple SUs, and sk,n = pk,n ρk,n , the OP1 can be converted into
s.t.
K
N
s Hk,n ρk,n log(1 + k,n ρk,n ) pk,n ,ρk,n k=K +1 n=1 1 N s Hk,n ρk,n log(1 + k,n C1 : ρk,n ) ≥ Rk,min , k = 1, . . . , K1 n=1
OP 2
max
C2 : sk,n ≥ 0, ∀k, n K N C3 : sk,n ≤ Pt k=1 n=1
C4 : C5 :
K N k=1 n=1 K k=1
SP sk,n In,l ≤ IlP , l = 1, . . . , L
ρk,n = 1, ∀n
C6 : ρk,n ∈ [0, 1], ∀k, ∀n. (5) The sk,n can be also characterized as the actual power consumption of the kth SU on the nth subchannel in a time frame interval. It is easy to prove that the OP2 is a convex problem [27] which can be solved by standard techniques. The solution to the OP2 contains sharing factor ρk,n ’s which are not binary variables. Rounding is necessary to obtain a feasible solution to the original problem OP1. Intuitively, larger ρk,n implies the kth SU obtains higher rate over the nth subchannel. It is straightforward to allocate the nth subchannel to the k ∗ th SU who has the largest ρk,n , that is, 1 for k ∗ = argmax ρk,n k ρk∗ ,n = . (6) 0 otherwise
However, such a rounding method can result in unreasonable subchannel allocation assignments when K is large because the difference between the largest ρk,n and the second largest one is negligible in many cases. Moreover, power distribution among subchannels has to be re-allocated after rounding procedure in order to maximize the sum capacity of the CR system. Furthermore, solving OP2 with standard convex optimization techniques generally has a complexity of O(2KN + N )3 , which is too high for practical applications. So the optimal solution to the OP2 provides an upper bound for that to the OP1 and is generally infeasible for the OP1. The upper bound can only be approached by feasible solutions of the OP1. In this work, we consider a two-stage approach to tackle the OP1. At the first stage, subchannel allocation is implemented with an efficient heuristic method, which removes the integer constraints in the OP1. At the second stage, power distribution across subchannels is carried out to maximize the sum rate of the NRT SUs and also keep the RT SUs’ rate requirements satisfied. III. S UBCHANNEL A LLOCATION S CHEME We propose an efficient subchannel allocation scheme to figure out the binary variables ρk,n ’s in this section, specifying a subchannel allocation assignment. The intuition of our method is following. In an OFDM-based CR system, an OFDM subchannel with high SNR for an SU may also generate more interference to the PUs. That is to say, the water-filling-like method [9] for conventional OFDM systems is no longer suitable for CR scenarios because interference constraint also lays an upper bound of the transmission power for each subchannel. So the SNR of an OFDM subchannel and the interference introduced to the PUs should be jointly considered to measure the capacity of the subchannel. We propose calculating the achievable rate of the nth subchannel employed by the kth SU as follows, M = log(1 + pM rk,n k,n Hk,n ),
(7)
where pM k,n is the maximum possible power for the kth SU on the nth subchannel, pM k,n = min(Pt , min( l∈L
IlP )). SP In,l
(8)
The channel capacity is normalized by (7) in this way. Let Ωk denote the subchannel set employed by the kth SU, and N is the set of subchannels. The outline of our subchannel allocation scheme is described in Table I, which consists of two steps. First, we allocate the RT SUs subchannels to meet their minimal rate requirements. Then we allocate the remaining subchannels to the NRT SUs. The principle of our subchannel allocation algorithm for the RT SUs is that the SU whose current rate is the farthest away from the target one has the priority to get a subchannel among the available ones. This procedure continues until all RT SUs’ rate requirements are satisfied. Preferably, the subchannel with the highest achievable rate associated with an RT SU will be chosen at this step. To simplify analysis and computation, the power of a subchannel is temperately set as pk,n = SP )) in order to always satisfy the min(Pt /N, minl∈L (IlP /In,l
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
TABLE I S UBOPTIMAL S UBCHANNEL A LLOCATION
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iteration to compute the central point, while the Newton step is the inner iteration executed in the centering step. The complexity of the barrier method is generally O(N 3 ), where N is the number of subchannels in the OP3. Since there are always thousands of subchannels in OFDM systems, the computation cost is unacceptable because RA should be processed online. In this section, we propose a fast barrier method to work out the optimal solution of the OP3 by exploiting its special structure. The high computational load is dramatically reduced. For a given number of PUs, the complexity of our proposed fast barrier method is approximately linear to the number of subchannels.
1.Initialization: 2.Nt = N , Ωk = ∅,∀k 3.Set the RT SUs’ rates to zero: Rti = 0, for i = 1, 2, ..., K1 4.For the RT SUs: 5.While Nt = ∅ and Rtk < Rk,min for any 1 ≤ k ≤ K1 6. Find k ∗ satisfies Rtk∗ − Rk∗ ,min ≤ Rtk − Rk,min 7. For k ∗ , find n∗ satisfies rkM∗ ,n∗ ≥ rkM∗ ,n , ∀n ∈ Nt 8. Update Rtk∗ = Rtk∗ + log(1 + pk∗ ,n∗ Hk∗ ,n∗ ) 9. Update Ωk∗ = Ωk∗ ∪ n∗ ,Nt = Nt \ n∗ 10.Endwhile 11.For the NRT SUs: 12.For i = 1 to length(Nt ) M 13. For n∗ = Nti , find k ∗ satisfies rkM∗ ,n∗ ≥ rk,n ∗ 14. Update Ωk∗ = Ωk∗ ∪ n∗ 15.Endfor
A. The Fast Barrier Method power and the interference constraints. By doing so, we need not to consider the RT SUs when allocating subchannels to the NRT SUs. Each of the remaining subchannels is allocated to the NRT SU who has the highest achievable rate over this channel to roughly maximize the sum capacity of the CR system, as shown in Table I.
When subchannel assignment is given, the binary variables ρk,n ’s are fixed to 1 or 0, and the C5 and C6 constraints in the OP1 vanish. We need to figure out power distribution among subchannels to maximize the sum capacity while keeping all constraints satisfied. Recall that Ωk is the subchannel set of the kth SU, the optimal power allocation is to solve the following optimization problem,
s.t.
max
K
pk,n k=K +1 n∈Ω 1 k
C1 : C2 : C3 :
n∈Ωk
k=1 n∈Ωk K k=1 n∈Ωk
−
−
K 1
log(
k=1 K
n∈Ωk
k=1 n∈Ωk L l=1
log(1 + pk,n Hk,n ) − Rk,min )
log pk,n − log(Pt −
log(IlP −
K k=1 n∈Ωk
log(1 + pk,n Hk,n )
pk,n ≤ Pt
K k=1 n∈Ωk
pk,n )
SP pk,n In,l ),
(10) where x = (p1 , p2 , . . . , pN ). Notice that the subscript k can be omitted now as it has been determined for a given subchannel allocation assignment. Denote f (x) =
K
k=K1 +1 n∈Ωk
log(1 + pk,n Hk,n ),
(11)
the optimal solution to the OP3 can be approximated by solving the following unconstrained minimization problem,
rk,n ≥ Rk,min , 1, . . . , K1
K
φ(x) =
−
IV. O PTIMAL P OWER A LLOCATION : FAST BARRIER M ETHOD
OP 3
The first step of the barrier method is to reformulate the OP3 into a set of unconstrained optimization problems, making all inequality constraints implicit in an objective function. The logarithmic barrier function of the OP3 is following [27],
min Ψt (x) = −tf (x) + φ(x). (9)
SP pk,n In,l ≤ IlP , l = 1, . . . , L
C4 : pk,n ≥ 0, ∀k, n. Obviously, (9) defines a convex optimization problem because the objective function and the constraints are all concave or convex. Generally, barrier method [27] is treated as a standard technique to solve convex optimization problems with inequality constraints. In the barrier method, the original problem is converted into a sequence of unconstrained minimization problems by introducing a logarithmic barrier function with a given parameter t. Particularly, each unconstrained minimization problem determined by the parameter t can be solved by Newton method, and the solution to this problem is called a central point in the central path related to the original problem. As t increases, the central point will be more and more accurate to approximate the optimal solution of the original problem. The barrier method consists of two stages: centering step and Newton step. The former is the outer
(12)
Since function Ψt (x) is convex and twice continuously differentiable, (12) has a unique optimal solution. In each centering step of the barrier method, we adopt Newton method to compute the central point for a given parameter t. The Newton step at x, denoted by Δxnt , is given by 2 Ψt (x)Δxnt = − Ψt (x),
(13)
where 2 Ψt (x) and Ψt (x) are the Hessian and the gradient of Ψt (x), respectively. The outline of the barrier method is summarized in Table II. and n are the tolerances of the barrier method and the Newton step, respectively. α and β are two constants used in the backtracking line search in the Newton step, where α ∈ (0, 0.5) and β ∈ (0, 1), respectively. The step size of the backtracking line search is s with s > 0. t and μ are the parameters which are associated with a trade-off between the number of the outer and the inner iterations. If computing the Δxnt in (13) directly by matrix inversion, the computation complexity is O(N 3 ). It is too high to apply as discussed above. We analyze the problem (12) and develop
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TABLE II T HE BARRIER M ETHOD 1.Initialization for Barrier method 2. Find strictly feasible point x, t := t(0) > 0, tolerance > 0, μ > 1 3.Outer Loop for Barrier method 4. Centering step: Compute x∗ (t) derived by (12) via Newton method 5. Initialization for Newton method 6. Starting point x, tolerance n > 0, α ∈ (0, 1/2), β ∈ (0, 1) 7. Inner Loop for Newton method 8. Compute Δxnt and λ2 := − Ψt (x)Δxnt ; 9. Quit if λ2 /2 ≤ n 10. Backtracking line search on Ψt (x), s := 1 11. While Ψt (x + sΔxnt ) > Ψt (x) − αsλ2 12. s := βs 13. Endwhile 14. Update:x := x + sΔxnt 15. Update:x∗ (t) = x. 16. Stopping criterion:(N + K1 + L + 1)/t < . 17. Increase:t := μt.
a fast algorithm to calculate the Newton step with lower complexity by exploiting the special structure of (12). Denote K pk,n , f0 = Pt − k=1 n∈Ωk fk = rk,n − Rk,min , n∈Ωk
gl = IlP −
K k=1 n∈Ωk
k = 1, . . . , K1 ,
(14)
SP pk,n In,l , l = 1, . . . , L,
k = 1, . . . , K1 k = K1 + 1, . . . , K
The Hessian of Ψt (x) follows ⎡ D1 ⎢ 2 D2 ∂ Ψt (x) ⎢ =⎢ .. 2 ∂pk,n ⎣ .
+
k=1
=D+
fk fk2
M m=1
(15)
.
(16)
⎤ ⎥ ⎥ f0 f0T ⎥+ f02 ⎦
fkT
+
L gl glT gl2 l=1
Fi FiT ,
(17) where D = diag(D1 , D2 , . . . , DN ) ∈ RN ×N and M = K1 + L + 1 with Dn =
1 +χ k
p2k,n
In the initialization step, it requires a strictly feasible starting point for the barrier method. A preparatory procedure is necessary to compute and prove that the feasible points exist or not. As discussed in [27], finding a feasible solution is equivalent to solve a minimization problem by introducing a crucial indicator parameter z. The optimization problem for the warm start procedure can be formulated as follows, OP 4
ez C1 : rk,n + z ≥ Rk,min , k = 1, . . . , K1 min
z,pk,n
C3 :
DN K 1
B. Warm Start Procedure for the Barrier Method
n∈Ωk
where χk yields χk =
The proof is presented in Appendix. Recall that M = K1 + L+1 and generally M N in practical wireless systems. The computational complexity is significantly reduced compared with matrix inversion which needs O(N 3 ) computations.
C2 :
SP L χ H ∂Ψt (x) 1 + 1 + In,l , = − 1 + pk k,n − pk,n ∂pk,n f0 l=1 gl k,n Hk,n
1 fk t
Theorem 1: The equation (13) can be solved with the complexity of O(N M 2 ).
s.t.
the gradient of Ψt (x) is given by
Since it is easy to prove that the matrix D and all Fi FiT ’s are positive definite, it follows that 2 Ψt (x) is also positive definite. Hence, the Karush-Kuhn-Tucker (KKT) matrix 2 Ψt (x) on the left in (13) is invertible.
2 Hk,n
2.
(1 + pk,n Hk,n )
Fi ’s are all vectors with N elements, ⎧ f 0 ⎪ ⎨ f0 , i = 1 fk k = 1, . . . , K1 , i = k + 1 . Fi (x) = fk ⎪ ⎩ gl l = 1, . . . , L, i = K + l + 1 1 gl
(18)
(19)
K k=1 n∈Ωk K k=1 n∈Ωk
pk,n ≤ Pt SP pk,n In,l ≤ IlP + z, l = 1, . . . , L
C4 : pk,n ≥ 0, ∀k, n, (20) where z can be interpreted as a bound on the maximum infeasibility of the inequality C1 and C3 and our goal is to drive the maximum infeasibility below zero. Note that the OP4 is also a convex problem whose structure is as same as the OP3. Since it is easy to find pk,n ’s to satisfy C2 and C4 (i.e. pk,n = Pt /N ), we can choose a feasible z to satisfy C1 and C3. So the feasible solution to the OP4 always exists. Due to the special structure, we can also apply the fast barrier method developed in section IV-A to solve the OP4. By solving the OP4, a strictly feasible point may be computed, or there are no feasible points. If the optimal solution to the OP4 satisfies z ≤ 0, the strictly feasible power allocation pk,n ’s can be used as the starting point of the barrier method to solve the OP3. Otherwise, no feasible point exists for the OP3 and we regard such a case as system outage. C. On the complexity The fast barrier algorithm of solving the OP3 consumes M decomposition, while each decomposition yields an additional equation. We need to solve M + 1 matrix system in the first step of the reverse substitution and the computational complexity for each system is measured by O(N ). After M reverse substitution steps, the total computational cost for the optimal power allocation is O(N M 2 ), which is much lower than O(N 3 ) if using matrix inversion directly.
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
Since we can also apply the proposed fast algorithm to solve the OP4, the computational complexity of the warm start procedure is equal to that of the OP3 because of the same structure. Therefor, we conclude that the complexity of the optimal power allocation is O(N M 2 ). Notice that the number of the PUs is always much smaller than that of the subchannels in wireless systems, that is, M N , the complexity is reduced dramatically. D. On Solving the OP2 Recall that the OP2 is the relaxation form of OP1. The solution to OP2 provides an upper bound of OP1. We will show that OP2 can also be efficiently solved by exploiting its structure with the same analysis method in section IV-A. First, the optimal solution of the OP2 can be approximated by the following equality constrained minimization problem, min Ψt (x) = −tf (x) + φ(x) K ρk,n = 1, ∀n, s.t.
(21)
k=1
where φ(x) is the barrier function, which makes all inequality constraints implicit, and f (x) is the objective function of the OP2. Ψt (x) is convex and twice continuously differentiable, and Newton method can be used to compute the central point with a given parameter t. The Newton step Δxnt at x and the associated dual variable v are given by 2 Ψt (x) d − Ψt (x) Δxnt = , (22) dT 0v 0n ν where zero matrix 0n ∈ RN ×N and vector 0v ∈ RN ×1 . d is a 2KN × N matrix in which d2m,n = 1, m = (k − 1)N + n, ∀k, n, and other elements are zeroes. 2 Ψt (x) and Ψt (x) are the Hessian and the gradient of Ψt (x), respectively. The (2KN + N ) × (2KN + N ) matrix in the left of the equation (22) can be decomposed into 2 M Ψt (x) d Q d Gi GTi , (23) = + dT 0n dT 0n i=1
where Q = (Q1,1 , Q1,2 , . . . , QK,N ) with 1 0 s2k,n Qk,n = + 1 0 + (1−ρ1k,n )2 ρ2k,n ⎡ ⎤ 2 2 χk ⎣
∂ f (x) ∂s2k,n ∂ 2 f (x) ∂sk,n ∂ρk,n
∂ f (x) ∂sk,n ∂ρk,n ∂ 2 f (x) ∂ρ2k,n
A. The Approximation In [19], an index function is defined to measure the cost of allocating each bit, introducing an efficient bit loading algorithm. Similarly, we define a normalized cost function indicating the cost of allocating certain rate on each subchannel, erk,n − 1 . (25) Fc (rk,n ) = M erk,n − 1 The channel SNR and the interference to the PUs are jointly considered in the cost function, reflecting the potential rate of each subchannel. With the cost function, we can convert the transmission power constraint C2 and the interference constraints C3 in the OP3 into a normalized capacity. Given a subchannel assignment, we can transform the OP3 into the following form, OP 5
max
s.t.
C1 :
K
rk,n k=K +1 n∈Ω k 1 n∈Ωk
V. A N E FFICIENT A PPROXIMATION AND A LGORITHM FOR P OWER D ISTRIBUTION In this section, we propose a heuristic power allocation algorithm to approximate the optimal solution with low complexity. The heuristic method can produce solution close to the optimal with lower complexity than the fast barrier method.
(26)
C2 : rk,n ≥ 0, ∀k, n K Fc (rk,n ) ≤ C. C3 : k=1 n∈Ωk
The parameter C in the OP5 is a constant determined by the transmission power and the interference thresholds, informing the maximum sum capacity of all subchannels. The key role of this transformation is that the constraints C2 and C3 in the OP3 are unified into a single constraint C3 in the OP5. The OP5 can be regarded as an approximation to the OP3. Notice that we do not know the value of C in advance and it can be only worked out when all rk,n ’s are known. We will show that it is not necessary to know C when solving the OP5. B. The Algorithm Without consideration of C2 in the OP5, the Lagrangian of the OP5 is given by L(rk,n , λ0 , μk ) = −
K
k=K1 +1 n∈Ωk K K1
+
(24)
and Gi = [ Fi 0v ]T . Obviously, our proposed fast barrier method is also applicable for calculating Newton step efficiently for the OP2. So the computational cost of solving the OP2 can be reduced to O(KN M 2 ), which is much lower than O((2KN + N )3 ) if using standard techniques.
rk,n
rk,n ≥ Rk,min , k = 1, . . . , K1
+λ0 (
⎦,
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k=1 n∈Ωk
k=1
rk,n
Fc (rk,n ) − C)
μk (Rk,min −
n∈Ωk
(27)
rk,n ),
where λ0 and μk (k = 1, . . . , K1 ) are the Lagrange multipliers with λ0 ≥ 0 and μk ≥ 0. Applying KKT conditions [27], the ∗ optimal solution rk,n ’s satisfy the following equations when concerning C2 in the OP5, ∗ ≥0 = 0, rk,n ∂L , (28) ∗ ∗ ∂rk,n =0 > 0, rk,n λ0 (
K
k=1 n∈Ωk
μk (Rk,min −
n∈Ωk
∗ Fc (rk,n ) − C) = 0,
∗ rk,n ) = 0, k = 1, . . . , K1 .
(29) (30)
With simple transformations of (28), we get the optimal solution for the RT SUs, rM k,n −1)
∗ rk,n = [log( μk (e λ0
)]+ , k = 1, . . . , K1 ,
(31)
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TABLE III P OWER D ISTRIBUTION FOR THE RT SU S 1.Initialization: 2.For the RT SUs, denote Ω∗k = Ωk , k = 1, . . . , K1 3.Power Distribution among the RT SUs 4.For k = 1 to K1 5. While true 6. Calculate rk,n by (35) n ∈ Ω∗k and sort rk,n ’s in ascending order 7. If rk,1∗ ≥ 0 8. break 9. Else 10. rk,1∗ = 0, Ω∗k = Ω∗k /1∗ 11. Endif 12. Endwhile 13.Endfor 14.Power loading: pk,n = (erk,n − 1)/Hk,n , n ∈ Ωk , k = 1, . . . , K1
where [x]+ := max(x, 0). And the optimal solutions for the NRT SUs are M rk,n
∗ rk,n = [log( e
λ0
−1
)]+ , k = K1 + 1, . . . , K.
(32)
For the kth SU, denote Ω∗k as the subchannels set in which ∗ ∗ rk,n > 0, ∀n ∈ Ω∗k . Sort the rk,n s in ascending order. Then ∗ for m, n ∈ Ωk , we have ∗ rk,m
−
∗ rk,n
k = Denote wm,n is given by
=
pM Hk,m log( pk,m ), M H k,n k,n
pM k,m Hk,m , pM k,n Hk,n
Rk = Nk rk,1∗ +
n∈Ω∗ k
k = 1, . . . , K.
(33)
Nk = |Ω∗k |, the rate of the kth user
k log(wn,1 k = 1, . . . , K, ∗ ),
(34)
where rk,1∗ represents the minimal element in Ω∗k . Substitute (33) and (34) into (30) for the case of μk = 0, we can obtain the closed-form expression of rk,n for the RT SUs, k k rk,n = N1k (Rk,min − log(wn,1 ∗ )) + log(wn,1∗ ). n∈Ω∗ k
(35) The detail of power allocation for the RT SUs is illustrated in Table III. Denote the power allocated to the kth SU and the interference to the lth PU introduced by the kth SU as Pk and Ikl , respectively, we can get the following equations for each SU based on (2), erk,n − 1 Pk = ∀k Hk,n n∈Ωk SP (36) (erk,n − 1)In,l Ikl = ∀k, l = 1, . . . , L. Hk,n n∈Ω k
Consider the power limitation and the interference constraints, for the K − K1 NRT SUs, we have K k=K1 +1 K k=K1 +1
Pk = Ikl =
where Pr =
K1 k=1
K
erk,n −1 Hk,n
≤ Pt − Pr ,
k=K1 +1 n∈Ωk K (erk,n −1)I SP Hk,n
k=K1 +1 n∈Ωk
Pk and Ilr =
K1 k=1
n,l
(37) ≤ IlP − Ilr ,
Ikl are the power consump-
tion and interference introduced to the lth PU by all RT SUs, respectively.
Substituting (32) into (37), the minimum value of λ0 is determined by solving the following inequalities, M + K (erk,n −1)/λ0 −1 ≤ Pt − Pr , Hk,n k=K1 +1 n∈Ωk (38) + rM K SP In,l (e k,n −1)/λ0 −1 P r ≤ I − I . l l Hk,n k=K1 +1 n∈Ωk
From (32), we know the minimal λ0 can maximize the sum capacity of the NRT SUs. With the value of λ0 obtained from (38), we get the optimal rate allocation for the NRT SUs by (32), which is equivalent to achieve the power distribution among the NRT SUs. The complexity of solving the OP5 can be counted roughly as follows. In the procedure of power loading for the RT SUs, the complexity is O(|Ωk |) for each RT SU, where |Ωk | is the number of subchannels allocated to the kth SU. On the other hand, the rate allocation for the NRT SUs can be obtained in O(L|Nn |), where |Nn | denotes the number of subchannels allocated to the NRT SUs. To sum up, the complexity of the power allocation scheme is bounded by O(LN ). VI. S IMULATION R ESULTS We evaluate our proposed algorithms with a series of experiments. Consider a multiuser OFDM-based CR system, where all users randomly located in a 3 × 3km area, and each receiver uniformly distributed in a circle within 0.5km from its transmitter. The path loss exponent is 4, the variance of shadowing effect is 10dB and the amplitude of multipath fading is Rayleigh. We assume that each PU’s bandwidth is randomly generated by uniform distribution and the maximum value is 2W/3L. The noise power is 10−13 W and the interference thresholds of all PUs are set to 5 × 10−12 W. Consider the following algorithms: Time-sharing (TS) method defined by the OP2, subchannel assignment proposed in Section III followed by the optimal power allocation method proposed in Section IV (Scheme 1), subchannel assignment in Section III followed by the approximation power allocation method proposed in Section V (Scheme 2), maximum SNR priority (MSP) subchannel allocation and minimum interference priority (MIP) subchannel allocation. For the MSP, we always allocate a subchannel to an SU who acquires the highest SNR over this channel, while the MIP scheme allocates a subchannel to an SU that generates the minimum interference to PUs. Both of the MSP and the MIP adopt the optimal power allocation strategy, marked as MSP-OP and MIP-OP in Fig.1. As discussed in Section II, the solution of the TS method is an upper bound of the original problem OP1. Fig.1 shows the average number of bits per symbol of the NRT SUs as a function of transmission power limit. The number of PUs and SUs are 2 and 4, respectively, including 2 RT SUs. There are 64 subchannels in the OFDM-based CR system and the rate requirements of RT SUs are uniformly set to 20bits/symbol. We can observe that the sum capacity of the NRT SUs increases as the transmission power limit increases. Both of our proposed two RA schemes outperform the MSP-OP and the MIP-OP obviously. The average capacity gap between the Scheme 1 and the TS is less than 6%, while the Scheme 2 can achieve more than 92% of the upper bound.
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
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600
500
Average number of bits per symbol of NRT SUs
Average number of bits per symbol of NRT SUs
1500
TS Scheme 1 Scheme 2 MSP−OP MIP−OP
400
300
200
100
−1
10
0
10 Power limit of SUs
1
10
2
2
4
6
8 10 Number of NRT SUs
12
14
16
Fig. 3. Average number of bits per OFDM symbol of NRT SUs as a function of the number of NRT SUs. Rk,min =10bits/symbol, N = 128, K1 = 4, L = 2. 1000
900 TS Scheme 1 Scheme 2 MSP−OP MIP−OP
Average number of bits per symbol of NRT SUs
Average number of bits per symbol of NRT SUs
500
10
Fig. 1. Average number of bits per OFDM symbol of NRT SUs as a function of the transmission power. Rk,min =20bits/symbol, N = 64, K = 4, K1 = 2, L = 2.
700
1000
0
0 −2 10
800
TS Scheme 1 Scheme 2 MSP−OP MIP−OP
600 500 400 300 200
TS Scheme 1 Scheme 2 MSP−OP MIP−OP
900 800 700 600 500 400 300 200 100
100 0
0
50
100 150 Number of subchannels
200
250
Fig. 2. Average number of bits per OFDM symbol of NRT SUs as a function of the number of subchannels. Rk,min =20bits/symbol, Pt = 1W ,K = 4, K1 = 2, L = 2.
Since the MSP and the MIP schemes do not consider the power limit and the interference jointly, they can only achieve about 40% and 30% capacity of the upper bound. Fig.2 illustrates the average number of bits per symbol of NRT SUs as a function of the number of subchannels. The number of SUs is 4 including 2 RT SUs. The transmission power limit is set to 1W and the minimal rate requirement for each RT SU is 20bits/symbol. As the increasing of the number of subchannels, the sum rate of the NRT SUs increases and the rate difference per subchannel between our proposed two schemes and the optimal becomes smaller. The reason is that the SUs can benefit from channel diversity in wireless environment. Our two schemes (Scheme 1 and Scheme 2) significantly outperform the MSP-OP and the MIP-OP. Specially, when N = 256, the difference between the Scheme 1 (Scheme 2) and the upper bound decreases to 3% (4%), while the MSPOP and the MIP-OP just achieve about 50% and 30% capacity
2
4
6
8 10 Number of RT SUs
12
14
16
Fig. 4. Average number of bits per OFDM symbol of NRT SUs as a function of the number of RT SUs. Rk,min =10bits/symbol, N = 128, K − K1 = 4, L = 2.
of the TS. This observation suggests that the performance loss due to the heuristic subchannel allocation method is neglectful when N is sufficiently large, which is suitable for application because there are always thousands of subchannels. We also demonstrate the multiuser diversity effect for the mentioned algorithms in Fig.3. The number of the RT SUs is K1 = 4, and the number of the NRT SUs varies from 1 to 16. Fig.3 shows the average number of bits per symbol as a function of the number of the NRT SUs, Rk,min = 10bits/symbol and Pt = 1W. The number of subchannels is 128. As can be seen in Fig.3, the sum capacity of the CR system increases with the increasing of the number of the NRT SUs. The difference between the TS and the Scheme 1 is roughly staying at 6%, so is the Scheme 2, both of which outperform the MSPOP and the MIP-OP. We can explain this phenomenon as a result of multiuser diversity. When there are more SUs, there are more chances for a subchannel to be allocated to a user with higher channel gain.
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Fast barrier method for optimal RA based on TS Newton iterations
200 150 100 50 0
10
20
30
40 50 60 Random instance(a)
70
80
90
100
Fast barrier method for optimal PA
into a universal subchannel capacity, and produces solution close to the optimal with much lower complexity. Moreover, we also suggest a simple but efficient heuristic method for subchannel allocation. Simulation results verify the effectiveness and the efficiency of the proposed schemes. Our proposal can be naturally extended to the weighted sum rates case, partially dealing with the issue of fairness. However, fairness among all users [28] is not fully considered in this paper and should be investigated in our future work.
Newton iterations
80
A PPENDIX
60
Proof of Theorem 1 Proof : Rewrite the KKT system (13) as
40 20 0
Λ0 Δx = F0 , 10
20
30
40 50 60 Random instance(b)
70
80
90
100
(39)
2
where Λ0 = Ψt and F0 = − Ψt . According to the analysis in section IV-A, we have
Fig. 5. Number of Newton iterations required for convergence for 100 channel realizations. N = 64, K = 4, K1 = 2, L = 2. (a) Fast barrier method for RA based on time-sharing; (b) Fast barrier method for the optimal power allocation with given subchannel allocation assignments.
Λ0 = D +
M
Fi FiT ,
(40)
i=1
which can be decomposed into M equations, Fig.4 depicts the average number of bits per symbol as a function of the number of the RT SUs. The minimal rate requirement Rk,min of each RT SU is 10bit/symbol, and the number of the RT SUs varies from 2 to 16. The number of the NRT SUs and subchannels is 4 and 128, respectively. The transmission power limit is Pt = 1W . Fig.4 shows that the sum capacity of the NRT SUs decreases as the number of the RT SUs increases. It can be explained intuitively. When there are more RT SUs trying to access the CR network, more subchannels or power are consumed by the RT SUs to meet their rate requirements. Moreover, radio resource will be more frequently exhausted when the number of the RT SUs becomes larger, which intensifies the capacity loss of the NRT SUs. The average capacity gap between our proposed two schemes and the TS is about 6%. Again, our proposed algorithms perform much better than the MSP-OP and the MIP-OP. Finally, we investigate the convergence of the fast barrier method. As discussed in section IV, the computational load of the barrier method mainly lies in the number of Newton iterations. From Fig.5(a) and (b), we observe that the average number of Newton iterations required for a guaranteed duality gap of 10−3 is 110 and 40 for the OP2 and the OP3, respectively. Furthermore, the number of iterations varies in a narrow range. We conservatively conclude the proposed fast barrier method is effective and efficient. VII. C ONCLUSION In this paper, we show that for a multiuser OFDM CR system with different rate requirements to support heterogeneous services, the optimal power allocation can be implemented efficiently by exploiting the structure of the problem. We proposed alternative power allocation schemes: Fast barrier method and approximate method. The fast barrier method can work out the optimal solution quickly by updating Newton step in an ingenious way. The approximate method normalizes the transmission power limitation and the interference constraints
T , i = 0, . . . , M − 1, Λi = Λi+1 + Fi+1 Fi+1
where Λi = D +
M j=i+1
(41)
Fj FjT . Note that all Λi ’s are nonsin-
gular. By exploiting the structure of Λi ’s, we give an M -step procedure to compute Δx. Step 1 Use (41) to decompose Λ0 , Λ0 = Λ1 + F1 F1T . F1 υ11 1 1 Then Δx = υ11 − 1+F 1 υ , where Λ1 υ1 = F0 and 1 υ2 2 1 Λ1 υ2 = F1 . So we can figure out the Δx if obtaining the two new variables υ11 and υ21 , instead of solving (39) directly. Step 2 Decompose Λ1 with Λ1 = Λ2 + F2 F2T . Then the two variables introduced in the step 1 can be F2 υi2 2 2 updated by υi1 = υi2 − 1+F 2 υ , i = 1, 2, where Λ2 υi = 2 υ3 3 Fi−1 , i = 1, 2, 3. For the mth step, T Step m Decompose Λm−1 = Λm + Fm Fm . We can update the m variables introduced in the Step F T υim υ m , i = 1, . . . , m, where m−1 by υim−1 = υim − 1+Fm m m υm+1 m+1 Λm υim = Fi−1 , i = 1, . . . , m + 1. Continue the procedure to the M th step, and there are M +1 matrix systems ΛM υiM = Fi−1 ,i = 1, . . . , M + 1. Form the derivation process, we can find that the m variables υim−1 , i = 1, . . . , m in the (m − 1)th Step can be obtained by the m + 1 variables υim , i = 1, . . . , m + 1 in the mth Step. Hence, if we can figure out the M + 1 variables υiM , i = 1, . . . , M + 1, the Δx in (39) will be indirectly obtained. Obviously, a reverse derivation of the M steps is necessary after we solve the M +1 matrix system in the M th Step. The process to solve the matrix equation ΛM υiM = Fi−1 is following. From Section IV-A, we have ΛM = D. Without loss of generality, we unify these equations into ⎡ ⎤ D1 ⎢ ⎥ D2 ⎢ ⎥ (42) ⎢ ⎥υ = g .. ⎣ ⎦ . DN
GE and WANG: FAST OPTIMAL RESOURCE ALLOCATION IS POSSIBLE FOR MULTIUSER OFDM-BASED COGNITIVE RADIO NETWORKS . . .
where υ, g ∈ RN ×1 . Since D is a diagonal matrix, we can easily obtained υi = Di−1 gi , i = 1, . . . , N,
(43)
which costs O(N ) to obtain the υ in (42). So it costs O(N M ) to solve the M + 1 variables at the M th Step. Then an M -step reverse iteration is implemented to figure out the Δx, and the total computational complexity can be measured by O(N M 2 ). ACKNOWLEDGEMENT The authors would like to thank the editors and the anonymous reviewers, whose invaluable comments helped improve the presentation of this paper substantially. R EFERENCES [1] F. C. Commission, “Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies,” FCC Report, ET Docket 03-322, Dec. 2003. [2] J. Mitola and G. Q. Maguire, “Cognitive radio: making software radios more personal,” IEEE Person. Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999. [3] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [4] F. C. Commission, “Spectrum policy task force report,” FCC Report, ET Docket 02-135, Nov. 2002. [5] B. Farhang-Boroujeny and R. Kempter, “Multicarrier communication techniques for spectrum sensing and communication in cognitive radios,” IEEE Commun. Mag., vol. 46, no. 4, pp. 80–85, Apr. 2008. [6] H. Mahmoud, T. Yucek, and H. Arslan, “OFDM for cognitive radio: merits and challenges,” IEEE Wireless Commun., vol. 16, no. 2, pp. 6–15, Apr. 2009. [7] T. A. Weiss and F. K. Jondral, “Spectrum pooling: an innovative strategy for the enhancement of spectrum efficiency,” IEEE Commun. Mag., vol. 42, no. 3, pp. 8–14, Mar. 2004. [8] S. Sadr, A. Anpalagan, and K. Raahemifar, “Radio resource allocation algorithms for the downlink of multiuser OFDM communication systems,” IEEE Commun. Surv. & Tut., vol. 11, no. 3, pp. 92–106, Sep. 2009. [9] C. Y. Wong, R. S. Cheng, K. B. Lataief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999. [10] G. Zhang, “Subcarrier and bit allocation for real-time services in multiuser OFDM systems,” in Proc. IEEE ICC’04, vol. 5, pp. 2985– 2989. [11] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171–178, Feb. 2003. [12] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in Proc. 2000 IEEE VTC – Spring, vol. 2, pp. 1085–1089. [13] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2726–2737, Nov. 2005. [14] M. Tao, Y.-C. Liang, and F. Zhang, “Resource allocation for delay differentiated traffic in multiuser OFDM systems,” IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 2190–2201, June 2008. [15] R. Madan, S. P. Boyd, and S. Lall, “Fast algorithms for resource allocation in wireless cellular networks,” IEEE/ACM Trans. Networking, vol. 18, no. 3, pp. 973–984, June 2010.
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[16] R. Zhang, Y.-C. Liang, and S. Cui, “Dynamic resource allocation in cognitive radio networks,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 102–114, May 2010. [17] G. Bansal, M. J. Hossain, and V. K. Bhargava, “Optimal and suboptimal power allocation schemes for OFDM-based cognitive radio systems,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4710–4718, Nov. 2008. [18] Y. Zhang and C. Leung, “Resource allocation in an OFDM-based cognitive radio system,” IEEE Trans. Commun., vol. 57, no. 7, pp. 1928– 1931, July 2009. [19] S. Wang, “Efficient resource allocation algorithm for cognitive OFDM systems,” IEEE Commun. Lett., vol. 14, no. 8, pp. 725–727, Aug. 2010. [20] S. Wang, F. Huang, and Z.-H. Zhou, “Fast power allocation algorithm for cognitive radio networks,” IEEE Commun. Lett., vol. 15, no. 8, pp. 845–847, Aug. 2011. [21] F. Digham, “Joint power and channel allocation for cognitive radios,” in Proc. IEEE WCNC’08, pp. 882–887. [22] A. Attar, M. R. Nakhai, and A. H. Aghvami, “Cognitive radio game for secondary spectrum access problem,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 2121–2131, Apr. 2009. [23] Y. Zhang and C. Leung, “Resource allocation for non-real-time services in OFDM-based cognitive radio systems,” IEEE Commun. Lett., vol. 13, no. 1, pp. 16–18, Jan. 2009. [24] S. Wang, F. Huang, M. Yuan, and S. Du, “Resource allocation for multiuser cognitive OFDM networks with proportional rate constraints,” Int. J. Commun. Sys., DOI:10.1002/dac.1272, 2011 (online). [25] S. M. Almalfouh and G. L. Stuber, “Interference-aware radio resource allocation in OFDMA-based cognitive radio networks,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1699–1713, May 2011. [26] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218– 1230, Oct. 1997. [27] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [28] S. Maharjan, Y. Zhang, C. Yuen, and S. Gjessing, “Distributed spectrum sensing in cognitive radio networks with fairness consideration: Efficiency of correlated equilibrium,” in Proc. IEEE MASS’11, pp. 540– 549. Mengyao Ge received the B.S. degree in communication engineering from Nanjing University, Nanjing, China, in 2011. She is currently pursuing the M.S. degree at the School of Electronic Science & Engineering, Nanjing University, Nanjing, China. She won the Best Bachelor’s Thesis Award from Nanjing University in 2011. Her research interests include wireless communications and convex optimization. Currently, her research focuses on resource allocation in wireless networks.
Shaowei Wang (S’06-M’07) received the B.S., M.S. and Ph.D. degrees in electronic engineering from Wuhan University, Wuhan, China, in 1997, 2003, and 2006, respectively. From 1997 to 2001, he was a research scientist of China Telecom. He joined the School of Electronic Science & Engineering of Nanjing University, Nanjing, China, in 2006, where he has been an Associate Professor since 2008. From 2009 to 2011, He was also a postdoctor at LAMDA group, Nanjing University. His research interests include wireless communications & networks, optimization and machine learning. Currently, his research focuses on resource allocation in wireless networks, cellular networks planning, and traffic engineering in telecommunications.
