Joint Power Allocation and Beamforming with Users ... - IEEE Xplore

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

Joint Power Allocation and Beamforming with Users Selection for Cognitive Radio Networks Via Discrete Stochastic Optimization ∗ Depart.

Renchao Xie∗† , F. Richard Yu∗ , and Hong Ji†

of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada Laboratory of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing, P.R. China

† Key

Abstract—In this paper, we study the problem of mutual interference cancellation among secondary users (SUs) and interference control to primary users (PUs) in spectrum sharing underlay cognitive radio networks (CRNs). Multiple antennas are used at the secondary base station (SBS) to form multiple beams towards to individual SUs, and a set of SUs are selected to adapt to the beams. For the interference control to PUs, we study power allocation among SUs to guarantee the interference to PUs below a tolerable level while maximizing SUs’ QoS. Based on these conditions, the problem of joint power allocation and beamforming with SUs selection is studied. Specifically, we emphasize on the condition of imperfect channel sensing. And we formulate the problem as a discrete stochastic optimization problem, then an efficient algorithm based on a discrete stochastic optimization method is proposed to solve the joint power allocation and beamforming with SUs selection problem. The proposed algorithm has fast convergence rate. Finally, simulation results are presented to demonstrate the performance of the proposed scheme.

I. I NTRODUCTION Cognitive radio [1] is viewed as a smart technology to allow secondary users (SUs) to flexibly utilize the spectrum resource licensed to primary users (PUs). There are two main spectrum sharing strategies: spectrum sharing overlay and spectrum sharing underlay [2]. For the former case, the SUs sense the radio spectrum environment and opportunistically utilize the temporarily vacant spectrum holes. And for the latter case, the SUs could coexist with PUs on the condition that the interference to PUs is below a tolerable level. In the spectrum sharing underlay scenario, some works have been done to maintain the interference to the PUs below a tolerable level and null the mutual interference among SUs while maximizing the SUs’ quality of service (QoS). The authors of [3] consider the joint transmit precoding and power control strategy to avoid the interference to PUs and optimize the SU’s QoS, and the effective interference channel (EIC) is introduced to measure the interference effect to PUs. A distributed beamforming and rate allocation in a decentralized multi-antennas cognitive radio networks is considered in [4]. Then based on the decoding method utilized at the secondary users, the authors solve the optimization This work was jointly supported by State Key Program of National Natural Science of China (Grant No. 60832009), the National Natural Science Foundation for Distinguished Young Scholar (Grant No. 61001115), Natural Science Foundation of Beijing, China (Grant No. 4102044), and the Natural Sciences and Engineering Research Council of Canada.

problem to maximize the minimum weighted rate under the three scenarios respectively. Authors of [5] formulate the problem of resource allocation and beamforming as a mixedinteger programming problem, and a branch and bound (BnB) method is proposed to find the optimal solution. To the best of our knowledge, most of previous works mainly focus on beamforming vectors design or joint beamforming and resource allocation under the assumption of perfect channel sensing. The work in this paper is different from previous works. We study the problem of joint power allocation and beamforming with SUs selection in cognitive radio networks (CRNs) with imperfect channel sensing. We consider that the secondary base station (SBS) configures multiple antennas for spectrum sharing underlay CRNs, which can form multiple “beams” towards to individual SUs. In addition, we study the problem of joint power allocation and beamforming with SUs selection in CRNs to null interference while maximizing the sum rate. Particularly, we emphasize on the condition of imperfect channel sensing. Because it’s hard to know the perfect channel information due to the hardware limitation, short sensing time, and network connectivity issues in practical CRNs [2], [6], [7]. The remainder of the paper is organized as follows. In Section II, the cognitive radio system model with imperfect channel sensing is given. Then the joint power allocation and zero-forcing beamforming with SUs selection algorithm under imperfect channel sensing is proposed in Section III. In Section IV, some simulation results are illustrated to demonstrate the performances of proposed scheme. Finally, we conclude this study in Section V. II. S YSTEM D ESCRIPTION We consider a simple cognitive radio system including a secondary network and a primary network, which allows SUs to coexist with PUs on the condition that the interference to PUs is below a tolerable level. We assume that the SUs and PUs are operated in a time-slotted system. In each time slot, there is a SBS to control the admission and resource allocation for K SUs’ requesting services. Suppose that there are M antennas at the SBS and a single antenna for each SU. Similarly, there is also a primary base station (PBS) to communicate with PUs, and each PU has the maximum interference threshold constraint. Both the PBS and the PUs have single antenna. Without loss of generality, we assume

978-1-4244-9268-8/11/$26.00 ©2011 IEEE


that the PBS communicates with only one PU with the interference threshold κp constraint in each time slot. Let hk,m be channel gain between antenna m and SU k. Note that hk = [hk,1 , hk,2 , ..., hk,M ], and the channel matrix H is ∗

H = [h∗1 , h∗2 , ..., h∗M ] ,

(1)

∗

where (·) denotes the conjugate transpose operation. Let g = [g1,p , g2,p , . . . , gK,p ] denote the channel gain from PBS to SU k. Here, we assume that the channel matrix H has statistical independence distribution, thus we have rank (H) = min (M, K) with probability 1 [8]. We also assume that there are a larger number of SUs requesting to admit (i.e., M < K). Therefore, we have rank (H) = M . To null the mutual interference among SUs while maximizing the SUs’ QoS performance, we can take advantage of antenna array to form multiple “beams” towards to individual SUs, and we should select a set X of M SUs to admit in each time slot due to at most M “beams” vectors. Let T T wk = [wk,1 , wk,2 , ..., wk,M ] ((·) denotes transpose) be the beamforming weight vector for the SU k, and the beamforming weight matrix can be expressed as w = [w1 , w2 , ..., wK ]. Therefore the received signal for the SU k can be written as yk =

K √ √ pi hk wi xi + pp gk,p xp + nk .

(2)

i=1

The received signal at the PU receiver is expressed as yp =

K √

pi hp wi xi +

√

pp gp xp + np ,

(3)

i=1

where hp = [hp,1 , hp,2 , . . . , hp,M ] and gp denote the channel gain from SBS to PU and from PBS to PU, respectively, xi and xp are the modulated symbols for SU i and PU, nk and np are the noise random variables with zero-mean and variances σk2 and σp2 , respectively. Therefore, based on (2) and (3), the output signal-tointerference-and-noise ratio (SINR) of kth SU is SIN Rk = i=k

2

|hk wk | pk 2

2

|hk wi | pi + |gk,p | pp + σk2

,

(4)

2

|hp wk | pk ,

M

max

pk ,w,k∈X k=1

B log2 (1 + SIN Rk ).

Subject to : 2 wk pk ≤ Ptot , k 2 |hp wk | pk ≤ κp ,

where B is the transmit bandwidth. The first constraint condition denotes that the transmit power to SUs should satisfy the total power constraint, and the second inequality denotes that the interference to PUs should be below a tolerable level. III. J OINT P OWER A LLOCATION AND B EAMFORMING WITH SU S S ELECTION A LGORITHM A. Power Allocation and Beamforming Under the Assumption of Known SUs Selection Suppose M SUs are selected from K SUs, the user subset is denoted as X, and channel estimate information is known. The mutual interference between SUs can be nulled by selecting appropriate beamforming weighting vectors for SUs. Here, we adopt the zero-forcing beamforming [8]– [10] that transforms the broadcast channel into multi-parallel independent and orthogonal sub-channels. The beamforming vectors are selected to satisfy hk wi = 0 for i = k. One easy way to choose the beamforming matrix w that gives the zero-interference is as follows [8], [10]. w = H∗ [X](H[X]H∗ [X])

k

where pk is the transmit power from SBS to SU k, and pp is the transmit power from PBS to PU. Therefore, our interest is to select a subset with M SUs from total K SUs and maximize the sum rate of the secondary network under the total power constraint and interference constraint. The

−1

,

(7)

which is the Moore-Penrose pseudo inverse of the channel H. By the beamforming, the SBS could form multiple “beams” towards to the individual SUs and the condition 2 |h k wi | = 0 is satisfied, therefore the SINR in (4) i=k becomes 2 |hk wk | pk SIN Rk = . (8) 2 |gk,p | pp + σk2 Based on the (7) and (8), we can do the power allocation for SUs and rewrite the optimization problem in (6) as M

pk ,k∈X k=1

(5)

(6)

k

max

and the interference received by PU from SBS is κp =

optimization problem can be formulated as follows:

Blog2 (1 + γk pk ).

Subject to : 2 wk pk ≤ Ptot , k 2 |hp wk | pk ≤ κp ,

(9)

k

2

| where γk = |g |hk|2wpk +σ 2 . By observing the optimization p k,p k problem in (9), the objective function and constraints satisfy the convex optimization condition. Therefore, solving (9) is


and then exhaustive search to find X ∗ = max rL (H [Xp ]).

equivalent to solve the Lagrangian dual problem [11]

Xp ∈Φ

M

Blog2 (1 + γk pk )+ L (λ, β, pk ) = k=1 M M 2 2 λ Ptot − wk pk + β κp − |hp wk | pk , k=1

k=1

(10) where λ and β are the Lagrangian multiplier factors. Thus we can use the Karush-Kuhn-Tucker (KKT) [11] conditions for the (10) and get the power allocation for SU k as + 1 B/ln 2 , ∀k = 1, ..., M, (11) pk = 2 2 − γ k λwk + β|hp wk | +

where [a] = max (a, 0). The power allocation in (11) is similar to the conventional water-filling solution [12]. However, the difference is that the same water level μ is used in the conventional water-filling algorithm, while (11) is water-filling problem with multi water levels. B. Optimal Joint Power Allocation and Beamforming with SUs Selection Based on the above discussion, now we firstly present the SUs selection strategy, then we will describe the optimal joint power allocation and beamforming with SUs selection based on the discrete stochastic optimization algorithm. Assume that a SUs subset is X = K {SU (1), SU (2), . . . , SU (M )}. Let the set of all P = CM possible SUs subsets be as Φ = {X1 , X2 , . . . , XP }. Then, the SUs selection process is described as follows. The SBS selects one of SUs subsets to maximize the sum rate M Rk , where Rk = B log2 (1 + SIN Rk ). Therefore, R= k=1

the problem of SUs selection can be formulated as X ∗ = arg max R (H [X]) , X∈Φ

(12)

where X ∗ denotes the optimal SUs subset. However, due to imperfect channel sensing, only the noisy estimate of objective function can be obtained. Therefore, suppose that at an iteration time l, the SBS obtains the estimate of and the SBS selects a sub-channel gain the channel H, [l, X] and computes the relative noisy estimate subset H of the capacity function R (H [X]) denoted as r (l, H [X]). If each r (l, H [X]) is an unbiased estimate of R (H [X]), r (l, H [X]) , l = 1, 2, . . . is then a sequence of i.i.d. random variables. Then, the problem of SUs selection can be rewritten as the following discrete stochastic optimization problem X ∗ = arg max R(H(Xp )) = arg max E{r(l, H[Xp ])} . Xp ∈Φ

Xp ∈Φ

(13) To solve (13), an inefficient method is to compute L estimates of the objective functions for each of SUs subset and compute the empirical average which approximates the exact value of the objective function. That is, we compute for each Xp ∈ Φ, L

rL (H [Xp ]) =

1 r (l, H [Xp ]) , L l=1

(14)

Since for any fixed Xp ∈ Φ, r (l, H [Xp ]) is an i.i.d. sequence of random variables, by the strong law of large numbers, rL (H [Xp ]) → E {r (l, H [Xp ])} almost surely as L → ∞. Therefore, based on the above discussion about the SUs selection strategy and the given power allocation and beamforming scheme in Subsection III-A, now we can present the joint power allocation and beamforming with SUs selection based on exhaustive search algorithm as follows. For each SUs combination, the SBS computes the beamforming matrix, does the power allocation and computes the objective function as (14). Then output the optimal SUs selection, beamforming matrix and power allocation with maximum objective function. Although the exhaustive search algorithm in principle finds the optimal solution, it is highly inefficient and induces significant computation complexity. Therefore, we propose another suboptimal algorithm based on the aggressive discrete stochastic approximation algorithm, which is well studied in [13] and [14]. We assume that ξ = {e1 , e2 , ..., eP }, where ep denotes the (P × 1) vector with a one in the pth position and zeros elsewhere. At each iteration, the (P × 1) probability vector π [l] = {π [l, 1] , π [l, 2] , ..., π [l, P ]} is updated, which represents the state occupation probabilities with element (l) be the SUs π [l, p] ∈ [0, 1] and p π [l, p] = 1. Let X subset chosen at the iteration l. For notational simplicity, we map the sequence of subsets X (l) to the sequence {D [l]} ∈ ξ, where D [l] = ep if X (l) = Xp , p ∈ 1, . . . , P . Therefore, the joint power allocation and beamforming with SUs selection based on the aggressive discrete stochastic approximation algorithm is as follows. Algorithm 1: Joint Power Allocation and Beamforming with SUs Selection Based on Aggressive Discrete Stochastic Approximation Algorithm 1) Each SU estimates the channel state information and feedbacks it to the SBS. 2) The a SUs combination X (1) ∈ Φ, and set

SBS(1)selects = 1, π [1, X] = 0 for all X = X (1) ; π 1, X 3) For l = 1 . . . , do a) Given X (l) combined with another uniformly chosen (l) ∈ Φ\X (l) at iteration time l, the SBS gets the X beamforming matrix and does the power allocation,

(l) ; and computes r l, H X (l) , r l, H X

(l) , set X (l+1) = X (l) , b) If r l, H X (l) π l + 1, X , the SBS sets d) If π l + 1, X X ∗(l+1) = X (l+1) ; otherwise sets X ∗(l+1) = X (l) . e) Set l ← l + 1 end for;


4) Output the optimal SUs selection, beamforming matrix and power allocation. The Algorithm 1 proposed above can find an SUs selection, beamforming matrix and power allocation to maximize the sum rate with fast convergence. To prove the convergence of Algorithm 1 proposed above, we firstly present a sufficient condition based on [13] as follows. Lemma 1 (Sufficient convergent conditions) The Algorithm 1 converges to the global maximizer of the objective function R (H [X]),

if the independent observations , r [l, H [X]] satisfy the following r [l, H [X ∗ ]], r l, H X conditions

i=1

I{Yi ≤y} is the indicator function, which is equal to 1 if Yi ≤ y and equal to 0 otherwise. Therefore relying on the Kolmogorov-Smirnov test, we can verify that the estimated objective Fn (y) follows a Gaussian distribution value F (y) = N μ, σ 2 within a significance level of 5%. As [14], suppose the estimated objective value in each iteration approximate the Gaussian distribution, i.e., r [l, H [Xp ]] ∼ 2 N μr(H[Xp ]) , σr(H[X . We have μr(H[Xp ]) > μr(H[Xq ]) p ]) if r (H [l, Xp ]) > r (H [l, Xq ]). Therefore, we can rewrite the condition (16) as

90

1

120

60

30

180

0

210

330

240

300 270

Fig. 1. Mutual interference cancellation between SUs through beamforming. (We assume that there are 4 antennas at the SBS and 6 SUs. SUs are randomly located on the unit circle around the SBS.)

10 9.5 9 Sum Rate (bits/s/Hz)

Pr{r[l, H[X ∗ ]] > r[l, H[X]]} > Pr{r[l, H[X]] > r[l, H[X ∗ ]]}.

(16)

> Pr r[l, H [X]] > r l, H X . Pr r[l, H[X ∗ ]] > r l, H X (17) Based on the discussion in [13], we know that if the above two conditions are satisfied, the sequence X (l) is a homogeneous irreducible and aperiodic Markov chain with state space Φ. It spends more time on X ∗ than other states. Then the condition (16) states that it’s more possible to move into the global optimum X ∗ from any other states than in the other direction, and condition (17) states that it has more chance to go to the optimal state X ∗ than to the other state when the state is not in a optimal state X ∗ . Now relying on the sufficient condition, we can prove the convergence of Algorithm 1. Theorem 1 (Global convergence). If the iteration is sufficient, Algorithm 1 converges to the global maximizer. Proof: Suppose the mean and variance of the objective function for r [l, H [X]] are E {r [l, H [X]]} = μr[H[X]] 2 and V ar {r [l, H [X]]} = σr[H[X]] , respectively. We can get the statistical distribution of r [l, H [X]] through simulation method. Let the empirical distribution function Fn (y) = n 1 I{Yi μr(H[Xq ])−μr(H[Xp ]) . Therefore, condition (16) is satisfied due to the same variance. Similar to the above, condition (17) can also be proved. Thus the convergence is proved. IV. S IMULATION R ESULTS AND D ISCUSSIONS

In this section, we use computer simulations to demonstrate the performance of proposed scheme. There are 6 SUs randomly located around the SBS, and the number of transmit antennas at the SBS is 4. The total transmit power at SBS is 1W . The bandwidth of spectrum licensed to primary networks is 20MHZ, and the interference threshold Pr{r[l,H[Xp ]]−r[l,H[Xq ]]>0}>Pr{r[l,H[Xq ]]−r[l,H[Xp ]]>0}. to primary networks is −4dB. We follow [14] for the radio (18) channel model. In the simulation results, we use “exhaustive The inequality (18) is equivalent to search scheme” to represent the exhaustive search algo rithm, “proposed scheme” to represent the aggressive discrete 2 2 Pr N μr(H[Xp ]) −μr(H[Xq ]) , σr(H[Xp ]) +σr(H[Xq ]) > 0 stochastic approximation scheme in Algorithm 1, “perfect 2 2 > 0 . CSI” to represent the resource scheduling and beamforming +σr(H[X > Pr N μr(H[Xq ]) −μr(H[Xp ]) , σr(H[X q ]) p ]) (19) scheme based on perfect channel state information, “existing


0.06

10 9

0.05 8 Sum Rate (bits/s/Hz)

Interference (W)

0.04

0.03 Interference Threshold Interference Output for Proposed Scheme 0.02

Perfect CSI Exhaustive Search Scheme Proposed Scheme Existing Scheme

7 6 5 4

0.01 3 0

0

0.2

Fig. 3.

0.4

0.6

0.8 1 Total Power (W)

1.2

1.4

1.6

2 0.2

1.8

Interference output versus the total power.

scheme” to represent the power allocation and beamforming with random SUs selection. In Fig. 1, mutual interference cancellation among SUs through beamforming is shown. From the figure, we can observe that the SUs are selected to adapt the number of antennas, then the SBS forms the “beams” towards to these selected SUs. This means that the mutual interference among the selected SUs is nulled by beamforming. In Fig. 2, the convergence of proposed scheme over the number of time slots is studied. From the figure, we can see that the proposed scheme and the exhaustive search scheme converge to the optimal value as the number of time slot increases. This is because that the estimated objective functions of the exhaustive search scheme and the proposed scheme are unbiased estimates of the objective function. As the number of time slot increases, the estimated value approximates to the optimal value by the strong law of large numbers. Fig. 3 shows the performance of total interference to PU. From the figure, we can observe that the total interference to PU approximates to the interference threshold as the total power increases. This is because the interference constraint dominates when the total power constraint increases. In Fig. 4, we evaluate the sum rate for the secondary network over total power constraints. The sum rates in the exhaustive search and proposed scheme are approximate to the perfect CSI scheme and better than existing scheme. This is because the existing scheme takes the random SUs selection and the perfect CSI scheme does resource allocation under the deterministic channel condition. However, the perfect CSI scheme assumes that the channel information is known, which may not be realistic in practice due to imperfect channel sensing. V. C ONCLUSIONS In this paper, we have studied the problem of joint power allocation and beamforming with SUs selection in spectrum sharing underlay CRNs under the condition of imperfect channel sensing. A discrete stochastic optimization algorithm

Fig. 4.

0.3

0.4

0.5 0.6 0.7 Total Power (W)

0.8

0.9

1

Sum rate for cognitive radio networks versus the total power.

has been proposed to null the interference while maximizing the sum rate. We have analyzed the convergence of the proposed scheme and its tracking capability. Using computer simulations, we have demonstrated that the proposed scheme can obtain good performance under the condition of imperfect channel sensing. R EFERENCES [1] J. Mitola III, Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio. Ph.D Thesis, KTH Royal Institute of Technology, Stockholm, Sweden, 2000. [2] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [3] R. Zhang, F. Gao, and Y.-C. Liang, “Cognitive beamforming made practical effective interference channel and learning-throughput tradeoff,” IEEE Trans. Commun., vol. 58, no. 2, pp. 706–718, Feb. 2010. [4] A. Tajer, N. Prasad, and X. Wang, “Beamforming and rate allocation in MSIO cognitive radio networks,” IEEE Trans. Signal Proc., vol. 58, no. 1, pp. 362–377, Jan. 2010. [5] K. Cumanan, R. Krishna, L. Musavian, and S. Lambotharan, “Joint beamforming and user maximization techniques for cognitive radio networks based on branch and bound method,” IEEE Trans. Wireless Commun., vol. 9, no. 10, pp. 3082–3092, Oct. 2010. [6] F. R. Yu, M. Huang, and H. Tang, “Biologically inspired consensusbased spectrum sensing in mobile ad hoc networks with cognitive radios,” IEEE Networks, pp. 26–30, June 2010. [7] Z. Li, F. R. Yu, and M. Huang, “A distributed consensus-based cooperative spectrum sensing in cognitive radios,” IEEE Trans. Veh. Tech., 2010. [8] G. Dimi´c and N. D. Sidiropoulos, “On downlink beamforming with greedy user selection: Performance analysis and a simple new algorithm,” IEEE Trans. Signal Proc., vol. 53, no. 10, pp. 3857–3868, Oct. 2005. [9] Q. H. Spencer and M. Haardt, “Capacity and downlink transmission algorithms for a multi-user MIMO channel,” in Proc. 36th Asilomar Conf. Signals, Syst., Pacific Grove, CA, Nov. 2002. [10] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528–541, Mar. 2006. [11] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U. K : Cambridge Univ. Press, 2004. [12] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley & Sons, 1991. [13] S. Andradottir, “A global search method for discrete stochastic optimization,” SIAM J. Optimiz., vol. 6, no. 2, pp. 513–530, May 1996. [14] I. Berenguer, X. Wang, and V. Krishnamurthy, “Adaptive MIMO antenna selection via discrete stochastic optimization,” IEEE Trans. Signal Proc., vol. 53, no. 11, pp. 4315–4329, Nov. 2005.