IEEE ICC 2013 - Wireless Communications Symposium
Clustering Method for CoMP with Limited Backhaul Data Transfer Using Convex Relaxation Jian Zhao∗, Tony Q. S. Quek∗† , and Zhongding Lei∗ ∗ Institute
for Infocomm Research, A*STAR 1 Fusionopolis Way, #21-01 Connexis (South Tower), Singapore 138632 Email: {jzhao, leizd}@i2r.a-star.edu.sg † Singapore University of Technology and Design (SUTD), Singapore 138632 Email:
[email protected]
Abstract—Joint transmission (JT) is an attractive coordinated multipoint (CoMP) downlink transmission technique in multicell networks. It offers high system throughput but needs to distribute each user data to multiple base stations (BSs), which can lead to huge backhaul signaling overhead when the number of users is large. In this paper, we address the problem of jointly optimizing the BS assignment for each user data and the BS transmit beamformer design subject to per-BS power constraints and given quality-of-service (QoS) requirements. Our aim is to minimize the backhaul user data transfer from the data center to the BSs for the JT approach. We formulate such a problem into a sparsity-maximization problem. However, this problem is NP-hard. We propose to tackle it by first applying convex relaxation to an equivalent form of the original problem and improving the obtained results by solving a series of convex 1 norm minimization problems. Compared to full BS cooperation, the proposed method saves about 13%–35% backhaul user data transfer in our simulations for moderate number of users and QoS requirement.
I. I NTRODUCTION Coordinated multipoint (CoMP) transmission has been incorporated into the latest 3GPP LTE-Advanced Releases to combat the inter-cell interference (ICI) and improve the signal quality of cell-edge users in multi-cell wireless systems [1]. Coordinated scheduling/beamforming (CS/CB) and joint transmission (JT) are the two major approaches in CoMP [2]. In the JT approach, the user data for each mobile station (MS) is first distributed to multiple cooperating base stations (BSs) and then transmitted simultaneously from them. In terms of system throughput improvement, the JT approach is attractive. However, this is at the expense of high signaling overhead in the backhaul. In a low-mobility environment, the amount of user data to be transported in the backhaul significantly outweighs that of the channel state information (CSI). In order to alleviate the backhaul requirement, it is desirable to distribute the user data only to a subset of the cooperating BSs. This is referred to as BS clustering. Different design methods for the CS/CB approach can be found in [3]–[6]. The authors of [3] discussed coordinated beamformer design to minimize the maximum BS antenna power. Distributed beamformer design methods to minimize the sum BS transmit power subject to given quality-of-service (QoS) requirements were proposed in [4]. In [5], each BS maximizes the rate for intra-cell users and cancels inter-cell in-
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terference in a decentralized manner. Optimal BS beamformer design to maximize the minimum achievable QoS measure for multiple users were proposed and analyzed in [6], [7]. The JT approach was considered in [8]–[10]. Fully synchronous transmission from all cooperating BSs was discussed in [8]. The author of [9] proposed a zero-forcing scheme, which requires all the users’ data to be distributed on all the BSs. BS clustering was considered in [10] using heuristic methods. The authors of [11] considered the backhaul constraint and proposed a rate-splitting scheme for a two-BS scenario. All those works assign the BSs to each MS in a predefined manner. In this paper, we consider BSs with multiple antennas and address the problem of jointly designing the BS clusters and the transmit beamformers subject to per-BS power constraints at BSs and given QoS requirements at each MS. Our aim is to minimize the backhaul user data transfer from the data center to the BSs for the JT approach. We show that such a problem can be cast into an 0 -norm minimization problem that minimize the nonzero elements of a “data routing matrix”. Unfortunately, this problem is NP-hard. Inspired by recent development in compressive sensing [12], we convert the problem into an equivalent form and obtain a suboptimal solution using convex relaxation. We show the conditions under which the obtained solution is sparse. The obtained solution is used as an initial starting point and a centralized algorithm based on reweighted 1 -norm minimization is proposed to further improve the results, which requires to solve a series of 1 -norm minimization problems that are convex. The proposed algorithm is guaranteed to converge, at least to a local optimal solution of the original problem. Simulations show that the proposed algorithm outperforms our previously proposed heuristic algorithm [13], as well as the full BS cooperation method. The remainder of the paper is organized as follows: in Section II, we introduce the system model and formulate our joint clustering and beamforming problem into an 0 -norm minimization problem. The algorithm based on the reweighted 1 -norm minimization is presented in Section III. Section IV provides the convergence behavior and the simulation results in a multi-cell scenario. Finally, conclusions are drawn in Section V. Notation: we use bold uppercase letters to denote matri-
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The SINR is an important QoS metric. SINR requirements have to be satisfied for the successful transmission of data. We denote the SINR requirement for the ith MS as γi . To minimize the signaling overhead in the backhaul, we want to distribute the user data only to the minimum number of cooperating BSs, while satisfying the SINR constraint for each MS. We denote the set of BS indices that the user data xi is distributed to as Bi , where Bi ⊆ {1, · · · , B}. Thus, the sum of user data rate Rsum transferred through the backhaul is given by K |Bi | (3) Rsum = R ·
06
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%DFNKDXO
06
%6 [ Z% [. Z.%
06 .
%6 %
Fig. 1.
i=1
CoMP downlink transmission scenario
ces and bold lowercase letters to denote vectors. CN (0, σ 2 ) denotes a circularly symmetric complex normal zero mean random variable with variance σ 2 . (·)T and (·)H stand for the transpose and conjugate transpose, respectively. vec(A) is a column vector composed of the entries of A taken columnwise. C, R and Z stand for the complex, the real and the integer numbers, respectively. {wij } denotes the set made of wij , ∀i, j. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model We consider the CoMP downlink transmission as shown in Fig. 1. There are B BSs cooperating to transmit to K MSs using the same time and frequency channel. Here xi ∈ C denotes the complex data symbol for the ith MS, where i ∈ 2 {1, · · · , K}, and E |xi | = 1. The data center has access to the data of all the MSs, and the backhaul is the channel connecting the data center to the BSs. The control information and user data at each BS is obtained directly from the data center via the backhaul. We assume that each BS is equipped with M antennas and each MS has a single antenna. The transmit beamformer for the ith MS on the jth BS, where i ∈ {1, · · · , K} and j ∈ {1, · · · , B}, is denoted as wij ∈ CM . The transmit power constraint at the jth BS is Pj . The channel from the jth BS to M the ith MS is denoted as hH ij , where hij ∈ C . The received signal yi at the ith MS can be expressed as yi =
B j=1
hH ij wij xi +
B K
hH ij wkj xk + ni .
(1)
k=i j=1
The first term on the right-hand side of (1) represents the received useful signal, the second term represents the interference, and ni ∼ CN (0, σ 2 ) is the additive white Gaussian noise (AWGN) at the ith MS. The signal-to-interference-plus-noise ratio (SINR) at the ith MS can then be written as 2 B w j=1 hH ij ij (2) SINRi = 2 . B H σ2 + K k=i j=1 hij wkj
where |Bi | denotes the cardinality of the set Bi , and R is the backhaul data transmission rate. For the given SINR requirement, we obtain from K (3) that minimizing Rsum is equivalent to minimizing i=1 |Bi |. B. Problem Formulation To capture the data transfer in the backhaul, we define a , where the (i, j)-th element data routing matrix P ∈ RK×B + of P is given by H [P]ij = wij wij (4) and it represents the power allocated at the jth BS for the data xi . The matrix P also shows which MSs’ data are routed from the data center to each of the BSs. When [P]ij = 0, the data xi is not routed to BS j. When [P]ij > 0, we say there exists a link between the jth BS and the ith MS, and the data center needs to distribute xi to the BS j with [P]ij > 0. Therefore, K i=1 |Bi | is equivalent to counting the number of nonzero elements in P. Mathematically, we can formulate the Rsum minimization problem in (3) as: min
{wij }
subject to
vec (P)0 SINRi ≥ γi , ∀i ∈ {1, . . . , K} K H i=1 wij wij ≤ Pj , ∀j ∈ {1, . . . , B}
(5)
where ·0 represents the 0 -norm, which denotes the number of nonzero elements of a vector. Note that the optimization problem (5) needs to jointly determine the BS subsets for the routing of each user’s data, as well as to design the transmit beamformers at the BSs. Remark 1: The problem (5) is a combinatorial optimization problem and is NP-hard. The proof for a related problem where all the constraints are linear can be found in [14]. Using similar arguments as in [14], [15], the problem (5) is NP-hard. One solution to (5) is by exhaustive enumeration. That is, for each number S ∈ Z, where 0 ≤ S ≤ KB, we must check all the KB possible S assignments of zero elements in P. For each assignment, we must search for the {wij } satisfying the constraints of (5). In the end, we pick out the maximum S with feasible {wij }. However, the complexity of exhaustive enumeration grows exponentially with the size of P, which is not applicable in real-world applications.
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The problem (5) is a cardinality minimization problem. One approach to solve such a problem is by applying the 1 norm heuristic [16], i.e., replacing the 0 -norm in the objective function with the 1 -norm. This is convex relaxation because the 1 -norm is the convex envelope of the 0 -norm. However, simply substituting the objective function of (5) by vec (P)1 does not produce sparse solutions in general. Therefore, we provide in the next section an equivalent formulation of (5), where sparse solutions can be obtained.
Algorithm 1 Centralized reweighted 1 -norm minimization algorithm
3:
III. R EWEIGHTED 1 -N ORM M INIMIZATION
min
{wij },{sij }
subject to
(n+1)
4:
vec (S)0 SINRi ≥ γi , ∀i ∈ {1, . . . , K} wij 2 ≤ sij , ∀i, j K 2 i=1 sij ≤ Pj , ∀j ∈ {1, . . . , B}
where [S]ij = sij . The first and second constraints are secondorder cone constraints, the third constraint is a sum-of-squares constraint, and all those constraints are convex. The problem (P0 ) is equivalent to (5). Applying the 1 -norm heuristic to (P0 ), we obtain (P1 ) : min i,j sij {wij },{sij }
subject to
Constraints of (P0 ).
Now the objective function becomes linear and (P1 ) becomes a convex optimization problem. Applying the techniques developed in [12], [17], sparse solutions of {sij } can be obtained from (P1 ). The following proposition states which sij is obtained with zero vlaue. Proposition 1: The solutions of {sij } in (P1 ) is obtained at s∗ij = 0 if ρij < 1, where ρij is the dual variable associated with the constraint “wij 2 ≤ sij ” in (P1 ), ∀i ∈ {1, · · · , K} and j ∈ {1, · · · , B}. Proof: Refer to [18]. The solutions of (P1 ) are suboptimal solutions of (P0 ) due to convex relaxation. Improved solutions can be obtained by applying a weighting factor on each sij in the objective function of (P1 ). The weighting factors should be chosen such that the solutions with large sij values are penalized. A good choice of the weighting βij in the (n + 1)-th iteration factor (n) (n+1) (n) = 1/ δ + sij , where sij is the solution of sij is βij in the nth iteration, and δ is a small nonnegative parameter to ensure numerical stability. The original idea of reweighted 1 norm minimization was proposed in [19] to enhance the data acquisition in compressive sensing. Algorithm 1 summarizes our centralized method to solve (P0 ). Furthermore, we have the following proposition for Algorithm 1: Proposition 2: Algorithm 1 converges at least to a local optimal solution of (P0 ). Proof: See the Appendix. The reweighted 1 -norm minimization algorithm achieves better sparsity results than (P1 ) because although the 1 -norm
Update the weighting factor: βij
We observe that [P]ij 0 ⇔ wij 2 0. By introducing slack variables sij , the problem (5) can be equivalently formulated as (P0 ) :
(0)
Set the iteration counter n = 0 and βij = 1, ∀i ∈ {1, · · · , K} and j ∈ {1, · · · , B}. 2: Solve the weighted 1 -norm minimization problem (n+1) (n+1) (n) = arg min i,j βij sij wij , sij subject to: Constraints of (P0 ). 1:
=
1 (n) , δ + sij
∀i, j.
Calculate the relative error: vec S(n) − S(n−1) 2 . r= vec S(n) 2 Terminate if r < or n ≥ Nmax , where is a predefined threshold and Nmax is the specified maximum number of iterations. Otherwise, set n = n + 1 and go to step 2.
is the best convex relaxation of the 0 -norm, some concave functions can offer tighter approximation. In the proof, the choice of the weighting factors is explained by using the logsum surrogate function of the 0 -norm [19], [20]. Because the log-sum penalty function is concave, the global minimum solution may not always be attained. It is important to choose a suitable starting point and the solution of the unweighted 1 -norm minimization (P1 ) offers a good initialization for the reweighted 1 minimization algorithm [19], [21]. Furthermore, we observe in the simulation that the resulting sparsity of {sij } using Algorithm 1 typically converges within ten iterations. The biggest improvement in sparsity comes from the first several iterations and the iterations afterwards offers marginal improvement in sparsity. Therefore, Algorithm 1 can be terminated after the first several iterations, depending on the requirement in the sparsity of the final output. IV. S IMULATION R ESULTS A. Convergence Behavior The convergence behavior of the reweighted 1 -norm minimization algorithm is simulated for a multi-cell network with equal number of BSs and MSs, i.e., B = K = 5. The number of transmit antennas at each BS is M = 3. The SINR requirement for each MS receiver is set to γi = 6.02dB, ∀i ∈ {1, · · · , K}. The noise variances at the MSs are normalized to 1, i.e., σ 2 = 1. Furthermore, the per-BS transmit power constraint Pj is set according to Pj /σ 2 = 10dB, ∀j ∈ {1, · · · , B}. A randomly generated channel is used in the simulation. The convergence behavior for the reweighted 1 -norm minimization method, i.e., Algorithm 1, is shown in Fig. 2, where −4 the parameter δ is set to be 10−2 , 10−3 and 10 K, respectively. The number of active links corresponds to i=1 |Bi | in (3).
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20 δ = 0.0001 δ = 0.001 δ = 0.01
18
%6
Number of active links
16
%6
14
12 %6
10
8
Fig. 3. 6
0
2
4
6
8
10 12 Iterations
14
16
18
Fig. 2. Convergence behavior for the reweighted 1 -norm minimization algorithm, B = K = 5. M = 3 antennas at each BS, the SINR requirement γi = 6.02dB, ∀i ∈ {1, · · · , K}.
Fig. 2 shows that the resulting number of active links decreases with the number of iterations. The first several iterations lead to the biggest improvement. As iterations go on, there is no further improvement after the tenth iteration. In the first iteration, i.e., in the unweighted case (P1 ), twenty links are required to be active. That means KB − 20 = 5 entries in the data routing matrix P are zeros. This shows that (P1 ) generates a sparse solution. In the final iteration, only six links are required to be active. Compared to full cooperation, which requires all the links to be active, we save 1 − 6/25 = 76% of the user data transfer in the backhaul. In general, a small δ leads to better convergence behavior. This is because smaller δ approximates the 0 -norm better. However, the convergence for the different δ values does not differ much in Fig. 2. When δ = 10−2 , only one more active link is required at the sixth iteration compared to the cases of δ = 10−3 and 10−4 . In the following simulations, we choose δ to be 10−2 for Algorithm 1. B. Cellular Network Simulations We consider a three-cell network as shown in Fig. 3. The inter-cell distance between neighboring BSs is 0.5km. The transmission power constraint at each BS is 30dBm. The transmit antenna gain at each BS is 5dB. The pathloss model from the BS to the MS is L(dB) = 128 + 37.6 · log10 D,
Cellular network simulation scenario
20
(6)
where D is in the unit of km. The log-normal shadowing parameter is 10dB. The available transmission spectrum is 10MHz and the noise figure at each MS is 10dB. The power of the noise plus out-of-cooperating-cell interference is set to be -83.98dBm. The users are randomly and uniformly distributed within a disk of 100m radius at the center of the triangle formed by the three BSs. We perform 50 channel realizations
for each user location and 20 different user locations are chosen in each simulation. Only those channel realizations that can support the required SINR requirement are admitted. We compare the proposed reweighted 1 -norm minimization method with the channel strength based clustering (CSBC) method [13] and the full cooperation method. The CSBC method is similar to the proposed method, but it removes the links according to the channel strength. Full cooperation needs to distribute all the user data to all the BSs. We set the maximum number of iterations Nmax for Algorithm 1 to be 20 and the relative error threshold to be 10−4 . The SINR requirement for different MSs are the same, i.e., γi = γ, ∀i ∈ {1, · · · , K}. Fig. 4 shows the average number of active links for the simulation scenario when the number of antennas per BS is fixed to M = 4. We consider two cases: one with a fixed SINR requirement and varying number of simultaneously served users; the other with a fixed number of users but varying SINR requirement. Fig. 4(a) shows the result with a fixed SINR requirement and the SINR requirement is set to γ = 10.3dB. This is the reference SINR for 16QAM transmission with spectral efficiency of 2.41bps/Hz [22]. This system can serve up to BM = 12 single-antenna MSs based on a rule of thumb estimate. Full cooperation requires a linear increase in the backhaul user data transfer with respect to the number of MSs in Fig. 4(a). When K ≤ 4, the system has far less users than its serving capacity. So the average number of active links for the proposed method and CSBC all approximately equals K. The number of required active links increases with K for all the shown methods. When K = 8, the proposed method and CSBC require about 15.4 and 18.2 active links, respectively. In this case, the proposed method saves about 1 − 15.4/24 = 35.8% of the backhaul user data transfer compared to full cooperation. When K = 10, the system capacity is almost reached. The proposed method requires about 26 active links and CSBC requires 28.1 active links. About 13.3% of the backhaul user data transfer can be saved by the proposed method.
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12
30
Proposed CSBC Full coop
11 25
Number of active links
Number of active links
10 20
15
10
9 8 7 6
5
0
Proposed CSBC Full coop 2
3
4
5
6 7 Number of MSs
8
9
5 4
10
(a) SINR requirement γ = 10.3dB. Number of users K changes. Fig. 4.
4
6
8
10 12 14 SINR requirement (dB)
16
18
(b) K = 4 MSs. SINR requirement γ at each MS changes.
Simulation results for B = 3 BSs, M = 4 antennas per BS.
The simulation results for a system with a fixed number of users K = 4 is shown in Fig. 4(b). The SINR requirement is shown as the x-axis. The number of active links for the full cooperation method is constant and equals BK in the whole SINR range. When the SINR requirement γ ≤ 8.1dB, the number of active links for the proposed method and CSBC are all about 4 in Fig. 4(b), which equals K. As γ increases, the performance gap between the proposed method and CSBC also increases when γ ≥ 8.1dB. When γ = 18.7dB, the proposed method only requires 7.9 active links. In this case, it saves about 1 − 7.9/12 = 34.2% of the backhaul user data transfer compared to full cooperation. The CSBC method requires 9.9 links to be active, which is 25.3% more than that of the proposed method. V. C ONCLUSION We considered the problem of minimizing the user data transfer in the backhaul subject to SINR and per-BS power constraints in the CoMP JT downlink transmission scenario. We showed that it can be formulated into an 0 -norm minimization problem. However, such a problem is combinatorial and NP-hard. Furthermore, the solution of such a problem needs to jointly determine the BS choice for user data and the BS beamformers. Inspired by recent results in compressive sensing, we applied convex relaxation to an equivalent form of the original problem and we showed the conditions under which the obtained solution is sparse. Furthermore, we proposed a centralized approach that improves the obtained solution using the reweighted 1 -norm minimization method, which requires to solve a series of convex 1 -norm minimization problems. The proposed algorithm is guaranteed to converge at least to a local optimal solution of the original problem. Simulations showed that the proposed method can significantly reduce the user data distribution to BSs in the backhaul. Compared to full cooperation, the proposed method
saves the backhaul user data transfer by 13%–35% for moderate number of users and QoS requirement in our simulations. It also outperforms one of our previously proposed CSBC algorithm. A PPENDIX P ROOF OF P ROPOSITION 2 For a given nonnegative real vector x ∈ RM + , its 0 -norm can be approximated as [20] x0 = lim
δ→0
M log(1 + |xi | δ −1 ) i=1
log(1 + δ −1 )
(7)
If we choose δ to be sufficiently small and replace the objective function of (P0 ) with the RHS of (7), we obtain the following problem by neglecting some constant numbers min i,j log(sij + δ) {wij },{sij } (8) subject to Constraints of (P0 ). The cost function is a log-sum function. It is concave and (n) (n) below its tangent. Specifically, let wij , sij denote the solution in the nth iteration. Due to the concavity of the log function and sij ≥ 0 for feasible solutions, the first-order approximation of log(sij + δ) yields
1 (n) (n) sij − sij . (9) log(sij + δ) ≤ log(sij + δ) + (n) sij + δ The RHS of (9) majorizes the log function, and can be used as a surrogate function for the original cost function in (8). Specifically, replacing the RHS of (9) into (8), the solution we get in the (n + 1)-th iteration is (n+1) (n+1) s = arg min i,j (n)ij wij , sij sij +δ (10) subject to: Constraints of (P0 ).
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We can see that the weighting factor of sij in the objective (n) function of (10) is same to the βij in Algorithm 1. By iteratively minimizing the surrogate function, local optimal solutions can be obtained, which is along the idea of the majorization-minimization method [23]. (n) (n+1) For sij and sij , we have i,j
(n+1)
log(sij
(n)
(n+1)
(n)
+ δ) ≤ log(sij + δ) + (n)
≤ log(sij + δ) +
(n)
sij − sij (n) sij
+δ
=
i,j
sij
(n)
− sij
(n)
sij + δ (n)
log(sij + δ).
(11)
The two inequalities follow (9) and (10), respectively. So (n) i,j log(sij + δ) is a monotonically decreasing sequence, lower bounded by KB log(δ). Therefore, the optimal objective value of (8) must converge. ACKNOWLEDGMENT
[15] E. Matskani, N. Sidiropoulos, Z. Luo, and L. Tassiulas, “Convex approximation techniques for joint multiuser downlink beamforming and admission control,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2682–2693, Jul. 2008. [16] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [17] M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 68, no. 1, pp. 49–67, Feb. 2006. [18] J. Zhao, T. Q. S. Quek, and Z. Lei, “Coordinated multipoint transmission with limited backhaul data transfer,” IEEE Trans. Wireless Commun., submitted. [19] E. Cand`es, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted 1 minimization,” Journal of Fourier Analysis and Applications, vol. 14, no. 5, pp. 877–905, Dec. 2008. [20] B. Sriperumbudur, D. Torres, and G. Lanckriet, “A majorizationminimization approach to the sparse generalized eigenvalue problem,” Machine learning, vol. 85, no. 1, pp. 3–39, Oct. 2011. [21] M. Fazel, H. Hindi, and S. Boyd, “Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices,” in Proc. American Control Conference, Denver, CO, Jun. 2003, pp. 2156–2162. [22] A. Ghosh and R. Ratasuk, Essentials of LTE and LTE-A. Cambridge University Press, 2011. [23] D. Hunter and K. Lange, “A tutorial on MM algorithms,” The American Statistician, vol. 58, no. 1, pp. 30–37, 2004.
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