Distributed Robust Multicell Coordinated Beamforming With Imperfect CSI: An ADMM Approach

Abstract

Multi-cell coordinated beamforming (MCBF), where multiple base stations (BSs) collaborate with each other in the beamforming design for mitigating the inter-cell interference, has been a subject drawing great attention recently. Most MCBF designs assume perfect channel state information (CSI) of mobile stations (MSs); however CSI errors are inevitable at the BSs in practice. Assuming elliptically bounded CSI errors, this paper studies the robust MCBF design problem that minimizes the weighted sum power of BSs subject to worst-case signal-to-interference-plus-noise ratio (SINR) constraints on the MSs. Our goal is to devise a distributed optimization method that can obtain the worst-case robust beamforming solutions in a decentralized fashion, with only local CSI used at each BS and little backhaul signaling for message exchange between BSs. However, the considered problem is difficult to handle even in the centralized form. We first propose an efficient approximation method in the centralized form, based on the semidefinite relaxation (SDR) technique. To obtain the robust beamforming solution in a decentralized fashion, we further propose a distributed robust MCBF algorithm, using a distributed convex optimization technique known as alternating direction method of multipliers (ADMM). We analytically show the convergence of the proposed distributed robust MCBF algorithm to the optimal centralized solution and its better bandwidth efficiency in backhaul signaling over the existing dual decomposition based algorithms. Simulation results are presented to examine the effectiveness of the proposed SDR method and the distributed robust MCBF algorithm.

Optimal Resource Allocation in Coordinated Multi-Cell Systems

“…The uncertainty originates from a variety of sources; for example, imperfect channel estimation, feedback quantization, inadequate channel reciprocity, and delays in CSI acquisition on fading channels. It is common to have an additive error model [9,17,26,50,239,241,242,257,286,289,328] with…”

“…The system cannot account for any error; the stochastic error vector˜ k could potentially cancel out the nominal vector as˜ k = − h k or be very large (the distribution is even unbounded for Rayleigh fading channels). This is often handled by only considering a subset of error vectors, the uncertainty set, that has high probability of containing the error [9,17,26,50,239,241,242,257,286,289,328]. If the design of these sets is explicitly included in the resource allocation (e.g., optimization with acceptable outage probabilities), it seems that conservative approximations 1 of each user's performance are required to achieve tractable problem formulations [50,241,289].…”

“…As single-stream beamforming is always sufficient under perfect CSI, we can however expect it to also work well when the uncertainty is small. Recent work in [51,251,239] proves that single-stream beamforming is indeed sufficient for single-cell and coordinated beamforming scenarios with per-transmitter power constraints and some minor technical assumptions (e.g., a generous bound on the amount of uncertainty or uniqueness of the solution). Therefore, we expect single-stream beamforming to be sufficient also under channel uncertainty.…”

“…Coordinated beamforming is such an example, because the transmitted signals over each channel component are independent. This is also fulfilled when multiple base stations send multiple independent signals to a given user [180,239], but for simplicity this subsection concentrates on coordinated beamforming.…”

“…The modified cost function will not affect the feasibility, but simplifies the analysis and explains why we prove the sufficiency of single-stream beamforming and not also necessity. The dual problem is [239,251]. Observe that the first part of Y k is positive definite while −…”