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Beamforming Design Via GNN in mmWave Cell-Free Massive MIMO Using Sub-6 GHz CSI
Sina Tavakolian, Abolfazl Zakeri, Ahmed Alkhateeb, Markku Juntti, Nhan Thanh Nguyen
TL;DR
MmWave CFmMIMO beamforming normally depends on costly mmWave CSI acquisition, while beamforming from solely sub-6 GHz CSI remains unexplored in this setting. The paper uses a graph-based GNN with customized message passing to learn sum-rate-maximizing digital beamformers, achieving performance close to full-mmWave-CSI benchmarks and robust operation with partial sub-6 GHz CSI.
Problem
MmWave CSI acquisition entails significant training overhead, and beamforming for CFmMIMO using solely sub-6 GHz CSI remains unexplored.
Method
A GNN models the CFmMIMO system as a wireless graph and uses customized message passing over BS–UE links to learn digital beamformers from sub-6 GHz CSI.
Results
At Pb = 5 W, the proposed method achieves 15.18 bps/Hz versus 15.15 bps/Hz for MMSE, 13.95 bps/Hz for ZF, and 10.59 bps/Hz for MRT.
Takeaways & Limitations
The learned beamformers remain close to full-mmWave-CSI benchmarks, generalize across network topologies, and remain robust with partial sub-6 GHz CSI.
Abstract
from arXiv · showhide
Beamforming methods in millimeter-wave (mmWave) cell-free massive multiple-input multiple-output (CFmMIMO) systems require accurate channel state information (CSI), whose acquisition entails significant training overhead. This paper shows that fully digital cell-free mmWave beamforming can be effectively learned from sub-6 GHz CSI using a graph neural network (GNN). Specifically, we represent a CFmMIMO system as a wireless graph, and the GNN is trained to approximate beamformers that maximize the downlink sum-rate based on the available sub-6 GHz CSI. A message-passing mechanism is proposed to capture inter-user interference and inter-base-station cooperation across different network topologies. Simulation results demonstrate that the proposed sub-6 GHz-assisted GNN-based beamformer achieves competitive and often superior sum-rate performance compared to classical baselines that rely on full mmWave CSI.
I. INTRODUCTION
CFmMIMO beamforming at mmWave frequencies needs CSI, but acquiring mmWave CSI creates substantial training overhead. The paper addresses this gap by learning beamformers from more efficiently estimated sub-6 GHz CSI with a GNN.
- CFmMIMO jointly serves UEs through distributed BSs connected to a CPU, while mmWave operation requires beamforming because of severe propagation loss and large antenna arrays.
- Sub-6 GHz CSI can be estimated more efficiently than mmWave CSI and has been explored as out-of-band side information for beam selection and beam prediction.
- Prior GNN-based approaches assume fixed desired/interfering links or partial mmWave CSI, assumptions that do not apply to jointly served CFmMIMO systems.
- The proposed GNN learns beamformers from only uplink sub-6 GHz CSI, using message passing to capture interference and BS cooperation across dynamic network topologies.
A. System Model
The system is a dual-band CFmMIMO downlink in which distributed BSs jointly serve single-antenna UEs using mmWave beamformers, with channels generated by a spatially consistent geometric model.
- The system contains B distributed BSs connected to a CPU and jointly serving U UEs, with co-located sub-6 GHz and mmWave antenna arrays at each BS.
- Downlink data transmission occurs in the mmWave band, while each UE uses a single antenna at both frequency bands.
- The geometric channel model is applied to sub-6 GHz uplink and mmWave downlink channels, with parameters generated by 3D ray tracing that preserves cross-band spatial consistency.
- Each BS precodes its data-symbol vector with a beamformer matrix, producing the transmitted vector that contributes to each UE’s received signal.
- The received signal includes additive white Gaussian noise, desired transmission, and inter-UE interference components.
- Each UE’s achievable rate is log2(1 + γu), and the system sum-rate aggregates the downlink rates across UEs.
B. Problem Formulation
The paper formulates fully digital beamforming as sum-rate maximization under per-BS power constraints, then learns the unavailable-CSI mapping from uplink sub-6 GHz CSI.
- The design objective is to choose fully digital beamformers that maximize the downlink sum-rate.
- Each BS has a power budget Pb, enforced as an equality because sum-rate maximization often uses the fully available transmit power.
- Directly solving the beamforming problem requires unavailable mmWave downlink CSI, whereas the CPU has only uplink sub-6 GHz CSI.
- The paper learns a mapping from available sub-6 GHz CSI to digital beamformers that maximizes sum-rate while satisfying each BS’s power constraint.
III. PROPOSED GNN-BASED FRAMEWORK
The proposed framework represents CFmMIMO as a typed wireless graph whose BS–UE edge features come from sub-6 GHz channels and whose nodes use learnable embeddings.
- The wireless graph contains BS and UE nodes, with edges representing all communication links between BSs and UEs.
- Each BS–UE edge receives a feature vector extracted from the corresponding sub-6 GHz channel.
- BS and UE nodes are assigned learnable d-dimensional embedding vectors optimized jointly with the GNN parameters.
A. GNN Architecture
The proposed GNN represents the CFmMIMO system through BS–UE edge features and iteratively refines them with attention-based message passing. It then uses contextual edge representations to generate and power-normalize beamformers.
- Architecture overview: The architecture uses three phases: initialization, message passing, and readout.These phases progressively construct, refine, and decode graph representations into beamforming vectors.
- Initialization (Phase 1): Initialization sets BS and UE node states to learnable embeddings and initializes edge states from sub-6 GHz channel features.A learnable affine mapping projects channel features into the d-dimensional embedding space.
- Message Passing (Phase 2): Message passing updates each BS–UE edge by aggregating information from neighboring edges sharing its BS or UE.This edge-centric design captures both BS-side and UE-side interactions rather than aggregating only into node embeddings.
- Message Passing (Phase 2): Attention coefficients weight the relevance of neighboring edges during BS-side and UE-side aggregation.Trainable projections and attention parameters transform features before the messages are aggregated.
- Message Passing (Phase 2): The model fuses BS-side and UE-side messages, then updates edge states with a residual rule across T stacked message-passing layers.The residual component linearly transforms the current edge state, while repeated layers progressively refine edge representations.
- Readout (Phase 3): Readout summarizes neighboring edge features, maps the resulting contextual vector through an MLP, and normalizes the beamformers to satisfy per-BS power constraints.The contextual vector combines the direct BS–UE relation with mean-aggregated information from other edges sharing the BS or UE.
B. Offline Training and Online Operation
Offline training optimizes the GNN directly for downlink sum-rate using mmWave CSI only to evaluate the training loss. During deployment, the pretrained model uses only sub-6 GHz CSI and produces beamformers in one forward pass, avoiding explicit mmWave CSI acquisition.
- Offline Training: The GNN is trained self-supervised by minimizing negative downlink sum-rate rather than using labelled beamforming targets.The loss is L = −Rsum, and mmWave CSI is required only during training to evaluate achievable rates.
- Offline Training: Training constructs graphs from sub-6 GHz CSI, refines their states through T message-passing layers, and generates power-normalized beamformers before computing the loss.Model parameters are updated over minibatches and epochs using the resulting communication performance.
- Online Operation: At inference, only sub-6 GHz CSI is required; the pretrained GNN generates beamforming vectors for a new graph through a single forward pass.This deployment procedure does not require mmWave CSI as an input.
- Online Operation: The GNN forward-pass complexity scales with the number of BS–UE links and their neighborhoods.Message passing and aggregation dominate the computational cost.
- Online Operation: The approach reduces system-level training and feedback overhead by inferring beamformers from sub-6 GHz CSI instead of explicitly acquiring mmWave CSI.The stated reduction concerns CSI acquisition and feedback at deployment rather than elimination of GNN inference cost.
IV. NUMERICAL RESULTS
The evaluation examines training convergence, robustness to partial sub-6 GHz CSI, and sum-rate performance against beamforming baselines using DeepMIMO-based simulations.
- Convergence Behavior: Training loss decreases rapidly during the first few epochs and then stabilizes for Pb = 1 W and Pb = 3 W, indicating stable convergence.The behavior is shown for two representative per-BS transmit powers.
- Partial CSI at Inference: Increasing available sub-6 GHz antennas from 3 to 8 raises sum-rate from 11.10 bps/Hz to 12.95 bps/Hz at Pb = 3 W.Using 3 antennas achieves about 85.7% of the performance obtained with all sub-6 GHz antennas.
- Sum-Rate Comparison: At Pb = 5 W, the proposed beamformer achieves 15.18 bps/Hz versus 15.15 bps/Hz for MMSE, 13.95 bps/Hz for ZF, and 10.59 bps/Hz for MRT.The proposed method consistently outperforms ZF and MRT while remaining close to MMSE.
- Sum-Rate Comparison: At Pb = 5 W, GNN-induced power allocation increases MMSE sum-rate from 13.66 to 15.15 bps/Hz, a 10.9% improvement.The comparison uses MMSE with uniform power allocation as the reference.
V. CONCLUSION
The paper concludes that a graph-based framework can learn fully digital CFmMIMO beamformers using only uplink sub-6 GHz CSI. Numerical results support generalization across topologies, robustness to partial CSI, and performance close to full-mmWave-CSI benchmarks.
- Conclusion: The proposed framework represents CFmMIMO as a graph and learns beamformers through customized message passing over BS–UE links.The method targets fully digital downlink beamforming using only uplink sub-6 GHz CSI.
- Conclusion: Numerical results indicate good generalization across different network topologies and robustness under partial sub-6 GHz CSI at inference.These properties are reported alongside the sum-rate comparison with full-mmWave-CSI schemes.
- Conclusion: The proposed approach achieves sum-rate performance close to fully digital benchmark schemes that rely on full mmWave CSI.The conclusion identifies hybrid beamforming architectures as future extension work.