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and H. Vincent Poor, Fellow, IEEE. Abstract—Cognitive ...... 1570–1580,. Apr. 2005. [18] D. Tse and S. Hanly, “Multiaccess fading channels-part I: Polymatriod.

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Cognitive Multiple Access Channels: Optimal Power Allocation for Weighted Sum Rate Maximization Lan Zhang, Student Member, IEEE, Yan Xin, Member, IEEE, Ying-Chang Liang, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE

Abstract— Cognitive radio is an emerging technology that shows great promise to dramatically improve the spectrum utilization efficiency. In this paper, a cognitive radio model is considered in which the secondary network is allowed to use the radio spectrum concurrently with primary users (PUs) provided that interference from the secondary users (SUs) to the PUs is constrained by certain thresholds. The weighted sum rate maximization problem is studied under interference power constraints and individual transmit power constraints, for a cognitive multiple access channel (C-MAC), in which each SU having a single transmit antenna communicates with the base station having multiple receive antennas. An iterative algorithm is developed to efficiently obtain the optimal solution of the weighted sum rate problem for the C-MAC. It is further shown that the proposed algorithm, although developed for single channel transmission, can be extended to the case of multiple channel transmission. Corroborating numerical examples illustrate the convergence behavior of the algorithm and present comparisons with other existing alternative algorithms. Index Terms—Cognitive radio networks, multiple access channel, power allocation, sum rate maximization

I. I NTRODUCTION Traditional spectrum regulation is based primarily on the command-and-control strategy that assigns users to prescribed frequency bands, thus restricting the potential users to dynamically access allocated radio spectrum. This policy, together with the rapid deployment of various wireless services, leads to increasing scarcity and congestion in the radio spectrum. On the other hand, many frequency bands, particularly television bands, experience considerably low utilization. Being widely considered as a promising solution to this dilemma, cognitive radio (CR) is able to dramatically improve spectrum Manuscript received March 24, 2008; revised September 9, 2008. The editor coordinating the review of this paper and approving it for publication is Syed Jafar. The work was supported by the National University of Singapore (NUS) under Grants R-263-000-314-101, R-263-000-314-112, and R-263-000-478112, by a NUS Research Scholarship, and by the U.S. National Science Foundation under Grants ANI-03-38807 and CNS-06-25637. This work was presented in part at Vehicular Technology Conference (VTC), Singapore, May, 2008. L. Zhang and Y. Xin are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 118622 (email: [email protected], [email protected]). Y.-C. Liang is with the Institute for Infocomm Research, 1 Fusionopolis Way, ♯21-01 Connexis, South Tower, Singapore 138632 (email: [email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA, (email: [email protected]). Publisher Item Identifier

utilization by allowing the secondary (unlicensed) users to opportunistically or concurrently access spectrum allocated to primary (licensed) users [1], [2], [3], [4], [5], [6]. However, such spectrum access by secondary users (SUs) needs to avoid causing detrimental interference to the primary users (PUs). To meet the requirement, in an opportunistic access CR model the SUs are allowed to communicate only over unoccupied holes in the spectrum [7], [8], [9] whereas in the concurrent access CR model, interference constraints for the SUs are typically imposed [10], [11], [12]. In this paper, we consider a concurrent CR model, in which the SUs and PUs may simultaneously transmit over the same spectral band provided that the interference from the SUs to the PUs is constrained by certain thresholds. We study the weighted sum rate maximization problem for the single-input multiple-output (SIMO) cognitive multiple access channel (C-MAC), in which each SU having a single transmit antenna simultaneously communicates with the base station (BS) having multiple receive antennas over the same spectral band. Notably, unlike the conventional sum rate maximization problem under non-CR settings, two sets of constraints are considered: interference power constraints to the PUs, and transmit power constraints on the SUs. Moreover, compared with the setting from an information theoretic perspective [6], [13], the SU considered in this paper does not have a priori knowledge about the message being sent by the PU. Most previous work on the MAC sum rate maximization problem has considered non-CR settings [14], [15], [16], [17]. Specifically, the work [14] developed an iterative water-filling algorithm to optimally solve the multiple-input multipleoutput (MIMO) MAC equally weighted sum rate maximization problem under individual power constraints. Subject to a sum power constraint instead of individual power constraints, the MIMO MAC equally weighted sum rate maximization problem was also investigated in [15], where the sum power constraint was decomposed into individual power constraints such that an iterative water-filling algorithm can be applied. In [16], a weighted sum rate maximization problem for the SIMO MAC subject to a sum power constraint was solved by using a cyclic coordinate ascent algorithm [17]. However, these aforementioned algorithms are not applicable to the weighted sum rate maximization problem under the CR setting, where not only individual transmit power constraints but also interference power constraints are imposed. Very recently, a weighted sum rate maximization problem

2

x1 h1 y

x2

PU1

g1,N ......

h2 ......

...

BS (SU)

g1,1

gK,1

hK xK

gK,N

PUN

Fig. 1. Single-input multiple-output cognitive multiple access channel (SIMO C-MAC) model with K SUs and N PUs.

for the SIMO MAC was investigated under a CR setting in [10]. However, only suboptimal solutions have been derived. The question of how to efficiently obtain the globally optimal solution for the SIMO C-MAC weighted sum rate maximization problem remains unsolved thus far. In this paper we develop an efficient iterative algorithm that obtains the globally optimal solution to this optimization problem. We further show that the algorithm, although developed for single channel transmission, can be generalized to the optimization problem with general multiple linear power constraints, and thus is applicable to the case of multiple channel transmission. The rest of this paper is organized as follows. Section II presents the system model and problem formulation. In Section III considers a single channel transmission scenario. We develop iterative algorithms that solve the SIMO C-MAC weighted sum rate maximization problem in the single PU and multiple PU cases. We further prove the convergence properties of our algorithm. Section IV extends the developed algorithm to the case of multiple channel transmission. Simulation results are presented in Section V. Section VI concludes the paper. The following notation is used in this paper. Uppercase boldface and lowercase boldface letters are used to denote matrices and vectors, respectively, (·)H and (·)T denote the conjugate transpose and transpose operation, respectively, and I M denotes an M × M identity matrix. II. C HANNEL M ODEL

AND

P ROBLEM F ORMULATION

With reference to Fig. 1, we consider a SIMO C-MAC with K SUs, where each SU has a single transmit antenna, and the K SUs communicate simultaneously with the BS having Nr receive antennas. The SUs, as depicted in Fig. 1, share the same spectrum with N PUs. Mathematically, the transmitreceive signal model from the SUs to the BS can be described as1 : y = Hx + z, (1)

random variables (RVs) having zero means and variances σ 2 , i.e., z ∼ CN (0, σ 2 I Nr ). Precisely speaking, due to the effect from thePPU transmission, the distribution of z is ˇ nh ˇ H ), where pˇn is the signal power CN (0, σ 2 IN r + N ˇn h n n=1 p ˇ of nth PU, and hn is the channel response from the nth PU to the BS. Since the noise covariance matrix does not affect the algorithm, for simplicity we consider only the case in which the noise covariance matrix is σ 2 I Nr . We now turn our attention to the SIMO C-MAC weighted sum rate maximization problem, in which the SUs are subject to individual transmit power constraints as well as interference power constraints. Mathematically, the problem can be formulated as follows2 : Problem 1 (C-MAC Weighted Rate Maximization): max pi

K X

wi ri ,

(2)

i=1

subject to: pi ≤ P¯i , i = 1, . . . , K, K X

(j)

gk,j pk ≤ Pt , j = 1, . . . , N,

(3) (4)

k=1

where wi ≥ 0 is the weight of SUi , ri is the rate achieved by SUi , pi is the power allocated to SUi , gk,j is the power gain between SUk to the jth PU (denoted by PUj ), and (j) Pt is the interference threshold for PUj . The interference power received by PUj from all SUs is characterized by T g Tj p with g j := [g1,j , . . . , gK,j ] and p := [p1 , . . . , pK ]T . We assume that as1) the channel matrix H and vectors g j (j = 1, . . . , N ) are fixed during each transmission block and change independently from block to block according to ergodic random processes, and as2) the BS of the SUs has perfect knowledge about the channel matrix H and vectors g j (j = 1, . . . , N ). In practice, acquiring the knowledge of the channel vectors g j (j = 1, . . . , N ) at the BS of the SUs entails a certain degree of cooperation among the PUs, the SUs, and the BS of the SUs, which can be realized via a carefully designed protocol (see [10] for details). Moreover, it is worth noting that the capacity PKregion of the C-MAC can be characterized by maximizing i=1 wi ri [18]. By varying the values of wi , the optimal solution of Problem 1 can reach the entire capacity region of the C-MAC. III. S INGLE C HANNEL T RANSMISSION As is well-known from [18], the capacity of the MAC is achieved by successive interference cancellation (SIC), and the corresponding achievable rate of SUπi can be written as Pi |I Nr + j=1 hπj hH πj p πj | rπi = log , i = 1, . . . , K, (5) Pi−1 H |I Nr + j=1 hπj hπj pπj |

where y denotes the Nr × 1 received signal vector, x is the K × 1 transmit signal vector of SUs with the ith element xi denoting the signal transmitted from the ith SU (denoted by SUi ), H = [h1 , . . . , hK ] denotes the Nr × K channel matrix with hi being the channel response from SUi to the BS, and z is a Gaussian noise vector. The entries of z are assumed to be independent and identically distributed (i.i.d.)

where π = π1 , . . . , πK , a permutation of 1, . . . , K, represents a decoding order (πK is decoded first, πK−1 is decoded second, and π1 is decoded last). From (5), we can see that the interference from SUπj , where j > i, has no influence on the achievable rate of SUπi . Therefore, it is necessary to

1 In this model, we treat the interference from the PUs to the BS of the SUs as part of Gaussian noise.

2 We start with the case of single channel transmission, and will further investigate the case of multiple channel transmission in Section IV.

3

determine the optimal decoding order of the SIC. According to [18], the optimal decoding order π can be obtained by sorting the weights in a non-increasing order as follows:

Problem 2 (Equivalent Problem): K X  min max f (p) − λ qk − Pt , λ≥0

wπ1 ≥ wπ2 ≥ . . . ≥ wπK . Without loss of generality, we assume πk = k in the sequel. Therefore, the objective function (2) can be rewritten as f (p) :=

K X

wi ri =

i=1

K X

k=1

k X ∆k log I Nr + hj hH p , j j

(6)

j=1

where ∆k = wk − wk+1 , and wK+1 = 0. Due to the facts that f (p) is concave and constraints (3) and (4) are linear, Problem 1 is a convex optimization problem that can be solved via standard convex optimization techniques such as interiorpoint algorithms [19]. However, the standard interior-point methods cannot take advantage of the special structure of this problem, and thus is computationally costly even for moderate values of Nr and K. By exploiting the special structure of this problem, we develop efficient algorithms to solve this optimization problem as shown in the following subsections.

p ,q

(8)

k=1

subject to: pi ≤ P¯i , i = 1, . . . , K (9) gi pi ≤ qi , i = 1, . . . , K, (10) where λ is an auxiliary variable. Problem 2 is a minimax problem with the optimization variables λ, p, and q. In the following lemma, we show that Problem 1 and Problem 2 have the same optimal value. Lemma 1: Problem 2 and the transformed problem (7) of Problem 1 have the same optimal solution. Proof: The Lagrangian function of Problem 1 with P respect to the coupled constraint K k=1 qk ≤ Pt is given by K X  L(p, q, λ) = f (p) − λ qk − Pt .

(11)

k=1

Moreover, the dual objective function is g(λ) = max L(p, q, λ), p ,q

(12)

where the constraints of the maximization are pi ≤ P¯i and gi pi ≤ qi . Thus, the dual problem of Problem 1 can be written as follows:

A. A Single PU Case For convenience of description, we first consider the relatively simple scenario in which only one PU is present. The multiple PU scenario will be discussed in Section III-B. In the single PU case, we simply omit the respective subscript and (j) superscript j in gi,j and Pt . 1) Dual decomposition algorithm: Problem 1 consists of two types of constraints: the individual transmit power constraint and the interference power constraint. The major difficulty in solving this problem is that all variables pi are coupled in the interference power constraint, which can be viewed as a weighted sum power constraint. One method that overcomes the difficulty of the coupled linear constraints is to use the dual decomposition algorithm [15], [20], which decomposes the sum power constraint into individual power constraints. By applying the dual decomposition method, Problem 1 can be transformed as follows: max f (p), subject to: pi ≤ P¯i , i = 1, . . . , K K X

λ

(7)

qk ≤ Pt ,

k=1

where qi denotes the interference power from SUi to the PU, which we term interference allocation. Moreover, we define q := [q1 , . . . , qK ]T . To furtherPsimplify the computations, we can incorporate the constraint K k=1 qk ≤ Pt into the objective function by forming the Lagrangian function with respect to this constraint, and transform the problem into its equivalent form:

(13)

By virtue of the convexity of Problem 1, the dual problem (13) and Problem 1 achieve the same optimal solution. Since the dual problem (13) is equivalent to Problem 2, the lemma follows immediately. 2) Multiple constraint decoupling: Problem 2 consists of two layers of optimization. The inner layer optimization involves the maximization of L(p, q, λ) over p and q for a fixed λ as given in (12), whereas the outer layer one involves the minimization of g(λ) in (13) over λ. We start with the inner layer. For a fixed variable λ, Eq. (12) can be considered as a convex maximization problem over p and q. We can rewrite the inner layer optimization as follows: K X  max f (p) − λ qk − Pt ,

p ,q

p ,q

gi pi ≤ qi , i = 1, . . . , K

min g(λ) subject to: λ ≥ 0.

k=1

subject to: pi ≤ P¯i , i = 1, . . . , K gi pi ≤ qi , i = 1, . . . , K,

(14)

where λ is a fixed constant. To determine the optimal power pi for each SU in (14), we apply an iterative algorithm similar to iterative water filling [14]. In each iteration step, we sequentially compute the optimal power allocation pi and interference allocation qi assuming that all pj and qj with j 6= i are fixed. In the iterative process, the objective value will increase continuously until the globally optimal power allocation is achieved due to the convexity of the problem. In the following, we first present the power allocation update rule in each iteration step.

4

The parameter pi is updated in each iteration step according to the solution to the following problem: K X  max f (p) − λ qk − Pt , pi ,qi

(15)

k=1

subject to: pi ≤ P¯i , gi pi ≤ qi . In order to obtain the solution of (15), we decouple this twoconstraint optimization problem (15) into two single-constraint sub-problems as follows: Subproblem 1: K X  max f (p) − λ qk − Pt , subject to: pi ≤ P¯i , pi ,qi

Subproblem 2:

(16)

k=1

K X  max f (p) − λ qk − Pt , subject to: gi pi ≤ qi . (17) pi ,qi

k=1

(1)

(2)

In each iteration step, we assume that pi and pi are the solutions to Subproblems 1 and 2, respectively. Since pi is a (1) (2) scalar, the minimum of pi and pi satisfies both constraints in (15) simultaneously. It is clear that this solution is the globally optimal solution of (15). This implies that only one constraint is active. Thus, in each iteration step, we update pi (1) (2) with the minimum of pi and pi . (1) For Subproblem 1, the solution is pi = P¯i . However, for Subproblem 2, the solution cannot be obtained directly. To solve Subproblem 2, we express the Lagrangian function of Subproblem 2 as K X  L(p, q, ν) = f (p) − λ qk − Pt − ν(gi pi − qi ). (18) k=1

Thus, the Karush-Kuhn-Tucker (KKT) conditions are obtained by applying ∂L(p, q, ν)/∂pi = 0 and ∂L(p, q, ν)/∂qi = 0, which are ∂f (p) − νgi = 0, ∂pi ν = λ, ν(gi pi − qi ) = 0.

(19) (20) (21)

For each i, we have K

∂f (p) X H −1 = ∆j hH hj , j (Qi,j (p) + pi hi hi ) ∂pi j=i

(22)

H k=1,k6=i pk hk hk .

where Qi,j (p) = I Nr + Applying the matrix inversion lemma3 , we can rewrite (22) as

3 The

(23)

j=i

∆j

−1 hH j Qi,j (p)hj −1 1 + pi hH j Qi,j (p)hj

= λgi .

A−1 cdH A−1 , 1 + cH A−1 d

where A is a nonsingular matrix, and c and d are vectors with appropriate dimensions.

(24)

The parameter λ is a constant for all the users, and thus can be viewed as a generalized water level in the water-filling principle [10]. Note that if the power quantities pj with j 6= i are all fixed, then the quantities Qi,j (p) are constants independent of pi . Due to the monotonically decreasing property of the left hand side of (24) in pi , there is a unique p¯i satisfying (24). A simple bisection method can efficiently determine p¯i , (2) which is the solution pi for Subproblem 2. Moreover, Eq. (24) belongs to a well known family of nonlinear equations, secular equations, for which there are highly efficient root finding algorithms [21]. After obtaining the optimal solution (1) (2) ( ( of pi and pi , we have p∗i = min(pi 1), pi 2)) as the optimal power solution for the problem (15) by following the previous discussion. The corresponding optimal interference allocation qi∗ can be obtained by using the slackness condition (21), i.e., qi∗ = gi p∗i . In summary, the problem (14) can be solved by the iterative algorithm. In each iteration step, pi is cyclicly updated by computing the decoupled Subproblems 1 and 2, while keeping pj with j 6= i as constant. The iterative process continues until a fixed point is achieved. The convergence of the proposed algorithm is guaranteed by applying the monotonically increasing property of each step and the convexity of the problem (14). We now examine the outer layer of Problem 2, which finds the optimal λ for (13). Since the Lagrangian function g(λ) is convex over λ, the optimal λ can be obtained via a onedimensional search. However, because g(λ) is not necessarily differentiable, the gradient algorithm cannot be applied. Alternatively, the subgradient based bisection method can be used to find the optimal solution. In each iteration step, the parameter λ is updated according to the subgradient direction. PK Lemma 2: The subgradient of g(λ) is Pt − i=1 qi , where qi is the corresponding optimal interference allocation for a fixed positive λ. ˜ For a given Proof: Let s be the subgradient of g(λ). ′ ˜ ˜ λ ≥ 0 and any feasible value of λ , the subgradient s of g(λ) satisfies the following inequality: (25)

This inequality can be proved as follows: K   X g(λ′ ) = max f (p) − λ′ ( qi − Pt )

p ,q

i=1

= f (p′ ) − λ′

matrix inversion lemma states that (A + cdH )−1 = A−1 −

K X

˜ + s(λ′ − λ). ˜ g(λ′ ) ≥ g(λ)

Pj

K −1 hH ∂f (p) X j Qi,j (p)hj = ∆j . −1 ∂pi 1 + pi h H j Qi,j (p)hj j=i

Substituting (20) and (23) into (19), we have

K X

qi′ − Pt



i=1

≥ f (˜ p) − λ′

K X i=1

q˜i − Pt



(26)

5 K K K  X  X  X ˜ ˜ = f (˜ p)− λ q˜i −Pt + λ q˜i −Pt −λ′ q˜i −Pt i=1

i=1

i=1

K   X ˜ + Pt − ˜ q˜i (λ′ − λ), = g(λ) i=1

˜ p′ and q ′ where s := Pt − i=1 q˜i is the subgradient of g(λ), i ′ ˜ and q˜i are are the optimal variables in (12) for λ = λ , and p ˜ The inequality (26) the optimal variables in (12) for λ = λ. ′ ′ is due to the fact that p and qi are the optimal variables in (12) for λ = λ′ . 2 indicates that the value of λ should increase if PLemma K q > Pt , and vice versa. i i=1 In the following, we outline the iterative algorithm that solves Problem 2. Algorithm 1: (For C-MAC With A Single PU) 1) Initialization: λmin , λmax , p. 2) repeat a) λ = (λmin + λmax )/2, m = 1, b) repeat, i) for i = 1, . . . , K, (2) compute pi according to (24) (2) pi = min(P¯i , pi ), qi = gi · pi , ii) m = m + 1, c) until PitKconverges d) if q > Pt , then λmin = λ, elseif PK k=1 k k=1 qk < Pt , then λmax = λ, 3) until |λmin − λmax | ≤ ǫ, where m is the iteration index of the inner loop, and ǫ > 0 is a constant. The following proposition assures the convergence of Algorithm 1. Proposition 1: Algorithm 1 converges to a solution of Problem 2. Proof: Algorithm 1 includes the inner and outer loops. The inner loop is to compute pi and qi for i = 1, . . . , K. In each iteration step, we update the ith user’s power pi while fixing the other users’ power pj , where j 6= i. Since the objective function (12) is nondecreasing with each iteration, the algorithm converges to a fixed point. Moreover, due to the convexity of (12), the fixed point is the optimal solution. The outer loop is to compute the Lagrangian coefficient λ. Due to the convexity of the Lagrangian dual function [19], [22], there is a unique λ that optimizes (13). Hence, one dimensional bisection search converges to the optimal λ. The proposition follows. Remark 1: If the wi s for all SUs are equal, then the objective function (6) can be expressed in the form of a determinant, and correspondingly the problem is reduced to a determinant maximization problem, which can be solved by using an interior point method, e.g., MAXDET [23]. In Section V, we compare the MAXDET algorithm with our algorithm. Remark 2: The dual decomposition algorithm has a desirable property that its asymptotic convergence rate is independent of K. As the complexity per iteration scales linearly with K, the computational complexity of the inner loop is roughly O(cK log(1/ǫ1 )), where c denotes the number of the inner loop iterations, and ǫ1 denotes the error tolerance for PK

computing pi in (24). Moreover, the bisection search of the outer loop guarantees error tolerance of ǫ2 after O(log(1/ǫ2 )) iterations. Compared with the complexity of the interior point algorithm O(K 3.5 log(1/ǫ)), the complexity of the proposed algorithm is significantly reduced. B. The Multiple PU Case In the preceding subsection, we have studied the weighted sum rate maximization problem with a single interference power constraint. It is worth noting that its dual problem g(λ) is always one dimensional, and can be solved via a highly efficient bisection method. However, this method cannot be applied to the multiple PU scenario. In this subsection, we extend the dual decomposition algorithm to solve the case with multiple PUs. The equivalent problem in this case can be expressed as min max f (p) − λ≥0

p ,q

N X

λj

j=1

K X

(j)

qi,j − Pt



,

i=1

subject to: pi ≤ P¯i ,

(27)

gi,j pi ≤ qi,j , i = 1, . . . , K, j = 1, . . . , N, where λj is the auxiliary variable for the interference power constraint of PUj , and qi,j denotes the interference allocation from SUi to PUj . Since there are N auxiliary variables, the optimal values cannot be obtained via the bisection search. However, the problem can be solved via the subgradient algorithm, i.e., in each iteration step of the outer loop, each auxiliary variable λj is updated according to its subgradient direction. In what follows, we summarize the iterative algorithm that solves problem (27). Algorithm 2: (For C-MAC With Multiple PU) (0)

1) Initialization: λj , j = 1, . . . , N , n = 1, 2) repeat a) initialization: pi , i = 1, . . . , K, m = 1, b) repeat, i) for ∀i, for decoupled subproblem with a single PUj (j = 1, . . . , N ) interference constraint, find the optimal solution p∗i,j according to (24), pi = min(p∗i,1 , . . . , p∗i,N , P¯i ), qi,j = gi,j · pi , ii) m = m + 1, c) until it converges. (n) d) for each j = 1, . . . , N , update λj via a subgradient algorithm as follows: PK (n+1) (n) (j) λj = λj + t( k=1 qk,j − Pt ), e) n = n + 1, (n) PK (j) 3) stop when |λj ( k=1 qk,j − Pt )| < ǫ for j = 1, . . . , N are satisfied simultaneously, where n and m denote the iteration indices of the outer loop and inner loop, respectively, t denotes the step size of the (n) subgradient algorithm, and λj denotes the estimate of λj at the nth iteration step. It has been shown in [24] that, for a fixed step size, the subgradient algorithm converges to the optimal value within a small range, i.e., (n)

lim |λj

n→∞

− λ∗j | < ǫ,

(28)

6

TABLE I P OWER ALLOCATION FOR THE MULTIPLE CHANNEL TRANSMISSION SCENARIO . 1 2 .. . M

1 p1,1 p2,1 .. . pM,1

2 p1,2 p2,2 .. . pM,2

3 p1,3 p2,3 .. . pM,3

4 p1,4 p2,4 .. . pM,4

... ... ... .. . ...

K p1,K p2,K .. . pM,K

where λ∗j denotes the optimal value. This implies that the subgradient method finds an ǫ-suboptimal point within a finite number of steps. The value ǫ is a decreasing function of the step size. In the simulation, for convenience of implementation, we use constant step size. Moreover, if the diminishing step size rule, e.g., the square summable but not summable step size, is applied, the algorithm is guaranteed to converge to the optimal value. IV.

MULTIPLE CHANNEL TRANSMISSION

In the preceding section, we have considered single channel transmission for both SUs and PUs. In this section, we study a more general scenario, in which both SUs and PUs transmit over parallel channels such as orthogonal frequency division multiplexing (OFDM) channels. We assume that there are M sub-channels shared by PUs and SUs. We now define the following notation for our model: The channel response from SUi to the BS on the mth sub-channel is denoted by hm,i , and the power allocated to this sub-channel is denoted by pm,n . The signal model is the same as (1) except that each item in (1) is replaced with its corresponding sub-channel parameters. The interference power threshold of PUj for the mth sub-channel [m] is denoted by Pj for m = 1, . . . , M . In the multi-channel transmission scenario, the weighted sum rate optimization problem needs to determine not only the power allocated to each SU but also the power allocated to each sub-channel. We present a power allocation example in Table I, where the row and column indices represent the subchannel and SU indices, respectively. The individual transmit power constraint of SUi requires that the sum of all the items of the ith column in Table I is less than P¯i , which is the maximum power of SUi . The interference power constraint of PUj on the mth sub-channel requires that the interference power to PUj is lower than the threshold, i.e., the interference power to PUj caused by the mth row in Table I is less than [m] Pj . Therefore, Problem 1 can be extended to the multichannel transmission scenario as follows. Problem 3: max pm,i

subject to:

K X

i=1 M X

wi

M X

rm,i ,

(29)

m=1

pm,i ≤ P¯i , i = 1, . . . , K,

m=1 K X

[m]

gm,i,j pm,i ≤ Pj

(30)

, m = 1. . .M, j = 1. . .N,

i=1

(31)

 Pi H where rm,i = log |I + n=1 hm,n hm,n pm,n |/|I +  Pi−1 H n=1 hm,n hm,n pm,n | denotes the rate of SUi on the mth sub-channel, and gm,i,j denotes the power gain from the SUi to PUj on the mth sub-channel. Since the objective function is concave and constraints (30) and (31) are both linear, Problem 3 is a convex optimization problem. The dual decomposition approach used in Algorithm 1 can be applied to Problem 3. To do so, we transform Problem 3 into the following form: Problem 4: min max

λi ,θm,j pm,i

M X K X

m=1 k=1

k X ∆k log I + hm,i hH m,i pm,i i=1

K M N X M K X  X X  X [m] − λi sm,i−P¯i − θm,j qm,i,j − Pj i=1

m=1

j=1 m=1

i=1

subject to: pm,i ≤ sm,i , gm,i,j pm,i ≤ qm,i,j , i = 1, . . . , K, m = 1, . . . , M, j = 1, · · · , N, where λi and θm,j are auxiliary variables, relating to the peak transmit power constraint of SUi , and the interference power constraint for PUj on the mth sub-channel, respectively. Analogously to Lemma 1, the following lemma shows the equivalence between Problem 3 and Problem 4. Lemma 3: Problem 3 and Problem 4 achieve the same optimal value. We omit the proof for Lemma 3 since it is similar to the proof of Lemma 1. To solve Problem 4, we apply a two-loop iterative algorithm, in which the inner loop is used to compute the optimal power allocation for fixed λi and θm,j , and the outer one is used to compute the optimal λi and θm,j via a subgradient algorithm. The inner loop itself is iterative. In each iteration step, pm,i is sequentially updated, i.e., one pm,i is updated at one time with other pm,i s being fixed. Specifically, we update pm,i with the solution of the following maximization problem: max

pm,i ,sm,i ,qm,i,j

−λi

M X

K X

k=1

m=1

k X ∆k log I + hm,i hH p m,i m,i i=1

N K  X X  [m] sm,i − P¯i − θm,j qm,i,j − Pj , j=1

i=1

subject to: pm,i ≤ sm,i , gm,i,j pm,i ≤ qm,i,j , j = 1, · · · , N. (32) This (N + 1)-constraint problem can be solved via the decoupling algorithm discussed in Section III. By solving the problem (32), we update one pm,i in one iteration step of the inner loop. The iterative process cyclically searches through all the entries in the Table I until a fixed point is achieved. The convexity guarantees that the inner loop converges to the optimal point for fixed λi and θm,j . Until the inner loop converges, λi and θm,j are updated via a subgradient algorithm. It follows from the argument used in Lemma Psame M 2 that λi should increase if m=1 sm,i > P¯i , and vice versa, PK [m] and θm,j should increase if i=1 gm,i,j pm,i > Pj , and vice versa.

7

For the sake of clarity, we summarize the algorithm for solving Problem 4 as follows. Algorithm 3: (For C-MAC With Multi-Channel Transmission)

Algorithm 1 Maxdet algorithm 16

(0)

V. S IMULATION R ESULTS In this section, we provide examples to illustrate the effectiveness of the proposed algorithms. For simplicity, we assume that all SUs are located at the same distance, l1 , from the BS, (n) and the same distance l2 , from the PUn . For the single PU case, we omit the superscript and simply use l2 . Suppose that the same path loss model is used to describe the transmissions from the SUs to the BS and to the PUs, and the path loss exponent is 4. The elements of the matrix H are assumed to be circularly symmetric complex Gaussian (CSCG) RVs with means zero and variances one, and the power gain factor (n) from SUi to PUn can be modeled as gi,n = (l1 /l2 )4 |αi,n |2 , where αi,n is also modeled as a CSCG RV with mean zero and variance one. The noise covariance matrix at the BS is chosen as the identity matrix, the individual transmit power and interference power are defined in dB relative to the noise power, and Pt is chosen to be 0 dB. Fig. 2 compares the results obtained by Algorithm 1 and the MAXDET algorithm, which is based on an interior point algorithm. In this example, we choose K = Nr = 3, N = 1, l2 /l1 = 2, and w1 = w2 = w3 = 1. The transmit

14 Sum−rate (bps/Hz)

1) Initialization: λi , i = 1, . . . , K, θm,j , m = 1, . . . , M, j = 1, · · · , N , l = 1, 2) repeat a) initialization: pm,i , ∀i, ∀m, n = 1, b) repeat, i) for ∀i, ∀m update pm,i by the N + 1 constraint problem (32), update sm,i = pm,i and qm,i,j = gm,i,j pm,i , ii) n = n + 1, c) until it converges. (l) (l) d) for ∀i, ∀m, ∀j, update λi and θm,j via (l+1) a subgradient algorithm as follows λi = PM (l) (l+1) (l) λi + t( m=1 sm,i − P¯i ), θm,j = θm,j + PK [m] t( k=1 qm,k,j − Pj ), e) l = l + 1, (l) PM 3) stop when |λi ( m=1 sm,i − P¯i )| < ǫ for i = 1, . . . , K (l) PK [m] and |θm,j ( k=1 qm,k,j − Pj )| < ǫ for m = 1, . . . , M and j = 1, · · · , N are satisfied simultaneously, where n and l are the indices of the inner and outer iteration (l) (l) steps, respectively, and λi and θm,j are the estimates of λi and θm,j respectively at the lth outer iteration step. Remark 3: Orthogonal frequency division multiple access (OFDMA) is a multiple access scheme that exclusively assigns each subchannel to no more than one user. By exploiting the KKT conditions, necessary and sufficient conditions under which OFDMA becomes the optimal scheme to achieve the sum capacity have been derived in [25]. Moreover, it has been shown that OFDMA is an optimal scheme in the low SNR regime. It can be shown using an similar derivation that, under certain conditions such as low SNR, OFDMA is also an optimal scheme to achieve the sum capacity in the CR settings.

12

10

8

6

4

0

5

10

15 20 Individual Power (dB)

25

30

Fig. 2. Comparison of the achievable sum-rate obtained by Algorithm

1 and the MAXDET algorithm (K = 3, N = 1, w1 = w2 = w3 and Nr = 3).

25 Algorithm in [12] Algorithm 1 20

Sum−rate (bps/Hz)

(0)

18

15

10

5

0 −15

−10

−5

0

5

10

15

20

25

30

35

Average Peak Received Power (dB)

Fig. 3. Comparison of the achievable sum rate obtained by Algorithm

1 and the algorithm in [10] (K = Nr = 3, N = 1, and w1 = w2 = w3 ).

power constraint ranges from 2 dB to 30 dB. From Fig. 2, it can be seen that the two results obtained by the two algorithms coincide. This is because both algorithms can reach the globally optimal solution of the optimization problem. In Fig. 3, we compare the results obtained by Algorithm 1 and the algorithm in [10]. In [10], a zero-forcing based decision feedback equalizer (ZF-DFE) is applied to solve the same problem. But the ZF-DFE is a suboptimal solution even though it is close to the optimal one when the transmit power is very high or very low. Fig. 3 shows the achievable sumrate with respect to the transmit power constraint, ranging from −15 dB to 35 dB. As can be seen from the figure, the achievable sum-rate obtained by Algorithm 1 is not less than one obtained by the algorithm in [10]. Moreover, in the highand low-transmit-power constraint regimes, the two curves coincide since the ZF-DFE solutions approach the optimal solutions in these regimes. In the next example, we consider the convergence behavior (1) of Algorithm 2 with K = Nr = 3, N = 2, l2 /l1 = (2) l2 /l1 = 2, w1 = 2, and w2 = w3 = 1. The transmit power constraint for each SU is assumed to be 16 dB. In Fig. 4 (a),

8

14

12 two PUs exist one PU exists

13.5

Weighted sum−rate (bps/Hz)

Weighted Sum−rate (bps/Hz)

11

13

12.5

12

10 9 8 7 6 5 4

11.5

0

20

40

60

80

100

120

140

160

180

Number of Iterations

5

10

15

20

25

30

Fig. 5. Comparison of the achievable weighted sum-rate for the case

with one PU and two PUs (K = Nr = 3, w1 = 2, and w2 = w3 = 1).

2 PU1’s interference PU ’s interference

1.8

2

1.6 Interference Power (dB)

0

Individual Power (dB)

(a) Achievable rate versus iteration numbers.

mization problem under individual transmit power constraints and interference power constraints. In each iteration step of the proposed algorithm, the two-constraint problem can be decoupled into two single-constraint subproblems. A globally optimal solution can be obtained by solving the decoupled single-constraint subproblems iteratively. Moreover, the proposed algorithm can be extended to the case with multiple PUs and to the case with multiple channel transmission.

1.4 1.2 1 0.8 0.6 0.4

3

0

20

40

60

80 100 120 Number of Iterations

140

160

180

(b) Interference powers versus iteration numbers. Fig. 4. Convergence behavior of Algorithm 2 (K = Nr = 3, N = 2, w1 = 2, and w2 = w3 = 1).

the achievable rate is plotted versus the outer loop step. In Fig. 4(b), the interference powers of PU1 and PU2 are plotted versus the outer loop step. It can be observed that the final interference powers for both PUs are less than or equal to the threshold. Finally, we consider the influence of the PU interference power constraint. In the C-MAC network, we choose K = Nr = 3, w1 = 2, w2 = w3 = 1, and l2 /l1 = 2. Fig. 5 shows the achievable weighted sum rate versus the transmit power constraint in the cases of one PU and two PUs. It can be observed that the two curves coincide in the low individual transmit power regime. This is because in the low individual power regime the transmit power constraints dominate the achievable rate. Alternatively, in the high individual transmit power regime, the achievable weighted sum rate of the case with two PUs is less than that of the case with one PU. This is because the extra PU interference power constraint reduces the number of degrees of freedom at the transmitter. VI. C ONCLUSIONS This paper has proposed an efficient iterative algorithm to optimally solve the SIMO C-MAC weighted sum rate maxi-

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Lan Zhang (S’07) received the B.E. degree and the M.Eng. degree in Electronics Engineering from University of Electronic Science and Technology of China, Chengdu, China, in 2001 and 2004, respectively. From 2004 to 2005, he worked as a senior technical associate in ZTE. He is currently working toward his Ph.D. degree at the Department of Electrical and Computer Engineering in National University of Singapore. His research interests include cognitive radio, MIMO, and information theory.

Yan Xin (S’00, M’03) received the B.E. degree in electronics engineering from Beijing University of Technology, Beijing, China, in 1992, the M. Sc. degree in mathematics, the M.Sc. degree in electrical engineering, and the Ph.D. degree in electrical engineering from University of Minnesota, Minneapolis, in 1998, 2000, and 2003, respectively. Since 2004, he has been with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, where he is now an assistant professor. His current research interests include MIMO communication, network information theory, and cognitive radio. Dr. Xin was the co-recipient of the 2004 IEEE Marconi Paper Prize Award in wireless communications.

Ying-Chang Liang (SM’00) is now Senior Scientist in the Institute for Infocomm Research (I2R), Singapore, where he has been leading the research activities in the area of cognitive radio. He also holds adjunct associate professorship positions in Nanyang Technological University (NTU) and National University of Singapore (NUS), both in Singapore, and adjunct professorship position with University of Electronic Science and Technology of China (UESTC). He has been teaching graduate courses in NUS since 2004. From Dec 2002 to Dec 2003, Dr Liang was a visiting scholar with the Department of Electrical Engineering, Stanford University, CA, USA. His research interest includes cognitive radio, dynamic spectrum access, reconfigurable signal processing for broadband communications, space-time wireless communications, wireless networking, information theory and statistical signal processing. Dr Liang is now an Associate Editor of IEEE Transactions on Vehicular Technology. He was an Associate Editor of IEEE Transactions on Wireless Communications from 2002 to 2005, Lead Guest-Editor of IEEE Journal on Selected Areas in Communications, Special Issue on Cognitive Radio: Theory and Applications, and Guest-Editor of COMPUTER NETWORKS Journal (Elsevier) Special Issue on Cognitive Wireless Networks. He received the Best Paper Awards from IEEE VTC-Fall1999 and IEEE PIMRC2005, and 2007 Institute of Engineers Singapore (IES) Prestigious Engineering Achievement Award. Dr Liang has served for various IEEE conferences as technical program committee (TPC) member. He was Publication Chair of 2001 IEEE Workshop on Statistical Signal Processing, TPC Co-Chair of 2006 IEEE International Conference on Communication Systems (ICCS2006), Panel CoChair of 2008 IEEE Vehicular Technology Conference Spring (VTC2008Spring), TPC Chair of 3rd International Conference on Cognitive Radio Oriented Wireless Networks and Communications (CrownCom2008), Deputy Chair of 2008 IEEE Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN2008), TPC Chair of DySPAN2010, and Co-Chair, Thematic Program on Random matrix theory and its applications in statistics and wireless communications, Institute for Mathematical Sciences, National University of Singapore, 2006. Dr Liang is a Senior Member of IEEE. He holds six granted patents and more than 15 filed patents.

H. Vincent Poor (S’72, M’77, SM’82, F’87) received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at UrbanaChampaign. Since 1990 he has been on the faculty at Princeton, where he is the Michael Henry Strater University Professor of Electrical Engineering and Dean of the School of Engineering and Applied Science. Dr. Poor’s research interests are in the areas of stochastic analysis, statistical signal processing and their applications in wireless networks and related fields. Among his publications in these areas are the recent book MIMO Wireless Communications (Cambridge University Press, 2007), coauthored with Ezio Biglieri, et al., and the forthcoming book Quickest Detection (Cambridge University Press, 2009), coauthored with Olympia Hadjiliadis. Dr. Poor is a member of the National Academy of Engineering, a Fellow of the American Academy of Arts and Sciences, and a former Guggenheim Fellow. He is also a Fellow of the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. In 1990, he served as President of the IEEE Information Theory Society, and in 2004-07 he served as the Editor-in-Chief of the IEEE Transactions on Information Theory. He was the recipient of the 2005 IEEE Education Medal. Recent recognition of his work includes the 2007 IEEE Marconi Prize Paper Award, the 2007 Technical Achievement Award of the IEEE Signal Processing Society, and the 2008 Aaron D. Wyner Award of the IEEE Information Theory Society.

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