Two-timescale joint power control and beamforming design with applications to cell-free massive MIMO

TL;DR

This paper addresses scalable uplink power control and beamforming for cell-free massive MIMO and XL-MIMO by introducing a two-timescale, long-term optimization framework. It leverages the use-and-forget (UatF) bound to formulate tractable ergodic-rate-based objectives and connects the resulting SINR optimization to a concave MMSE problem under information constraints, enabling globally optimal fixed-point algorithms. The proposed algorithms jointly optimize the power vector and the beamforming functions, with convergence guarantees, and are validated through extensive simulations across coordinated small cells, distributed cell-free, and centralized cell-free architectures, showing clear performance gains over traditional short-term methods. The findings highlight the importance of joint long-term power control and beamforming design, particularly in cell-free networks where beamforming is the dominant performance bottleneck, and they provide a versatile framework that can accommodate distributed processing and various CSI-sharing constraints. The work offers practical implications for scalable 6G deployments, including decreased signaling overhead and improved uniform quality of service across users.

Abstract

In this study we derive novel optimal algorithms for joint power control and beamforming design in modern large-scale MIMO systems, such as those based on the cell-free massive MIMO and XL-MIMO concepts. In particular, motivated by the need for scalable system architectures, we formulate and solve nontrivial two-timescale extensions of the classical uplink power minimization and max-min fair resource allocation problems. In our formulations, we let the beamformers be functions mapping partial instantaneous channel state information (CSI) to beamforming weights, and we jointly optimize these functions and the power control coefficients based on long-term statistical CSI. This long-term approach mitigates the severe scalability issues of competing short-term iterative algorithms in the literature, where a central controller endowed with global instantaneous CSI must solve a complex optimization problem for every channel realization, hence imposing very demanding requirements in terms of computational complexity and signaling overhead. Moreover, our approach outperforms the available long-term approaches, which do not jointly optimize powers and beamformers. The obtained optimal long-term algorithms are then illustrated and compared against existing short-term and long-term algorithms via numerical simulations in a cell-free massive MIMO setup with different levels of cooperation.

Figures (6)

• Figure 1: Illustration of the differences between the proposed and existing approaches for the particular case of perfect CSI and max-min SINR criterion. In Fig. (a), a complex optimization problem is solved for every $t$th channel realization $\mathbf{H}[t]$ to find jointly optimal instantaneous powers $\mathbf{p}[t]$ and beamformers $\mathbf{V}[t]=(\mathbf{v}_1[t],\ldots, \mathbf{v}_K[t])$. In contrast, in Fig. (b), a single optimization problem is solved to find jointly optimal long-term powers $\mathbf{p}$ and beamforming function $f$ based on channel statistics. The obtained power vectors are kept fixed for all channel realizations, while the beamforming weights evolve over each channel realization as $\mathbf{V}[t]=f(\mathbf{H}[t])$. Fig. (c) depicts a typical long-term power control approach in the literature, where the function $f$ is fixed a priori (e.g., to maximum-ratio combining or zero-forcing), and it is not jointly optimized with $\mathbf{p}$. We note that, although introduced here for illustration purposes, in this work we do not use the functional notation $f$ and the channel realization index $t$, but we use a completely equivalent stochastic formulation.
• Figure 2: Convergence behaviour of (a) the fixed-point iterations for computing a solution to Problem \ref{['eq:QoS']}; and (b) the normalized fixed-point iterations for computing a solution to Problem \ref{['eq:maxmin']} under different information constraints.
• Figure 3: Empirical CDF of the ergodic rates achieved by the proposed optimal solution to Problem \ref{['eq:maxmin']} under multiple user drops and different information constraints, evaluated using (a) the UatF bound in \ref{['eq:uatf']}; and (b) its upper bound \ref{['eq:coh']} based on coherent decoding.
• Figure 4: Comparison of centralized cell-free schemes: (a) Empirical CDF of the ergodic rates \ref{['eq:coh']} achieved by the proposed optimal solution to Problem \ref{['eq:maxmin']} under a centralized cell-free information constraint, and by the competing techniques in Section \ref{['ssec:centr']}; and (b) instantaneous rates (as defined for the short term approach) across users and channel realizations.
• Figure 5: Comparison of distributed cell-free schemes. Empirical CDF of the ergodic rates \ref{['eq:coh']} achieved by the proposed optimal solution to Problem \ref{['eq:maxmin']} under a distributed cell-free information constraint, and by the competing techniques in Section \ref{['ssec:distr']}.

Theorems & Definitions (20)

• Remark 1
• Remark 2
• Remark 3
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4