Coordinated beamforming for sum rate maximization in multi-cell downlink systems
Introduction
The performance of wireless cellular systems is limited by the inter-cell cochannel interference (CCI), thus inter-cell interference reduction is urgently needed to improve the system performance especially for the cell-edge users [1], [2]. Recently, multi-point coordination technology has emerged as a promising way to address this issue. The main idea is that coordinated base stations (BSs) jointly design beamforming vectors and allocate transmit powers. Two basic problem formulations have been extensively studied due to its convexity, i.e., the problem of minimizing the total power consumption subject to individual signal-to-interference-plus-noise-ratio (SINR) requirements and that of maximizing the minimum SINR. To address the first problem, an efficient way is to recast the downlink problem into a dual uplink problem where the beamformers optimization at the BSs are decoupled [3], [4]. It was also found that the uplink–downlink duality can be used again to solve the second problem [5]. In particular, an approximate uplink–downlink duality was proposed in [5] to realize a distributed multi-cell beamforming scheme which only requires limited inter-cell communication and is able to reach the point on the Pareto boundary with max–min rate fairness. By exploiting the property of massive MIMO channels, in [6] a distributed coordinated power allocation method is developed to balance the weighted SINR in a multi-cell massive multiple input single output (MISO) downlink system. Alternatively, some other methods including alternating optimization approach were also developed to address the max–min optimization problem [7], [8], [9].
Besides the above optimization problems, the sum-rate maximization (SRMax) problem has also been intensively studied in recent years. It is well known that this problem is non-convex and therefore is difficult to tackle directly [10], [11]. It was revealed in [12], [13] that the original non-convex problem can be approximately solved using a sequence successive convex approximation if the BSs and the user terminals are equipped with single-antenna. However, extension to multi-antenna systems is not straightforward, since the coupling between the beamformers makes the optimization problem NP-hard. This issue can be partially addressed using the uplink–downlink duality. A virtual uplink problem, dual to the multi-cell multi-antenna downlink SRMax problem, was first derived in [14]. However, this dual problem is still difficult to solve due to the fact that it involves uncertain virtual uplink noise variances, resulting in high computational complexity. Alternatively, it was revealed that the SRMax problem can be solved by recasting into an equivalent weighted sum mean square error minimization (WMMSE) problem [15], [16], [17]. However, it needs in general dozens of iterations to reach a desired performance. Recently, the weighted SRMax (WSRMax) problem in multi-cell downlink systems is tackled based on a branch and bound technique by computing a sequence of asymptotically tight upper and lower bounds [18], [19], which is effective but its computational complexity is still high.
Contrary to previous works, in this paper, we tackle the SRMax problem in multi-cell MISO downlink systems by adopting a surrogate object function method which is widely used to solve some complex optimization problems [20], [21], [22]. We first investigate the SRMax problem subject to a total power constraint. To address the issue, the primal problem is transformed into a virtual uplink SRMax optimization problem where the optimal beamformers can be analytically obtained. For fixed beamformers, the virtual uplink power allocation problem is solved by exploiting a tunable-parameter convex approximation method. Based on that, an alternative optimization beamforming algorithm with provable convergence is proposed to solve the SRMax problem. Second, we extend the above idea to handle the SRMax problem subject to per-BS power constraints and develop two new methods by utilizing the separable and linear properties. Numerical simulations show that our proposed algorithms always converge to a stable point only with a few iterations.
The rest of this paper is organized as follows. The system model is described in Section 2. In Section 3, a coordinated multi-cell beamforming algorithm is proposed for multi-cell MISO downlink system subject to a total power constraint. Then, two coordinated multi-cell beamforming algorithms for multi-cell MISO downlink system subject to per-BS power constraints are given in Section 4. In Section 5, our proposed algorithms are briefly analyzed in terms of the computational complexity and the convergence. The simulation results are shown in Section 6 and conclusions are finally given in Section 7.
The following notations are used throughout this paper. Bold lowercase and uppercase letters represent column vectors and matrices, respectively. The superscript T, H and represent the transpose operator, the conjugate transpose operator, the Moore Penrose pseudo-inverse of matrix, respectively.
Section snippets
System model
Consider a multi-cell MISO downlink system consisting of K BSs each equipped with M transmit antennas and serving one single-antenna user. The K BSs simultaneously transmit to K mobile users. In order to suppress CCI, the beamformers and transmit powers at BSs are jointly designed. Denoting the ith BS and its served user as BS i and user i, respectively. The received signal of user i is written aswhere pi denotes the transmit power of BS i, denotes
Coordinated multi-cell beamforming for total power constraint
As we all know that a key technique in the solutions of the coordinated beamforming problems [3], [4], [5] is the idea of uplink–downlink duality. Motivated by this, here we resort to solve the SRMax problem (3) by using the jointly convex approximation approach and the uplink–downlink duality.
Coordinated multi-cell beamforming for per-BS power constraints
By far we show that the SRMax problem subject to a total power constraint can be solved using the uplink–downlink duality and the convex approximation. While in practical systems, it is more reasonable to consider per-BS power constraints that make the SRMax problem more intractable. Herein, we now focus on the SRMax problem subject to per-BS power constraints, given bywhere Pi denotes the individual power constraint of BS i. In what follows, we
Algorithm analysis
In this subsection, we investigate the convergence behavior of our proposed algorithms and briefly analyze their computational complexity.
Simulation results
In this section, the performance of the proposed multi-cell beamforming schemes is investigated via numerical simulations. We consider a cooperative cluster of K=3 hexagonal adjacent cell where each cell includes one multiple antenna BS and each BS serves a single user. The cell radius is set to be 500 m and each user has at least 400 m distance from its serving BS. The channel vector from BS j to user i is generated based on the formulation , where denotes the small
Conclusions
In this paper, an alternating optimization algorithm was first proposed to maximize the sum rate for multi-cell coordinated downlink systems subject to a total power constraint. Numerical results have illustrated that this algorithm converges to a stable point with a few iterations. Taking into account the per-BS power constraints, two low computational complexity algorithms were further proposed. We also demonstrated that the proposed algorithms always converge to a desired point. Finally,
Acknowledgments
This work was supported by National Natural Science Foundation of China under Grants 61271018, and 61372101, Research Project of Jiangsu Province under Grants BK20130019, BK2011597, and BE2012167, National Science and Technology Major Project of China under Grant 2013ZX03003006-002, Program for New Century Excellent Talents in University under Grant NCET-11-0088.
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