α-Fair Multistatic ISAC Beamforming for Multi-User MIMO-OFDM Systems via Riemannian Optimization

Abstract: This paper proposes an $α$-fair multistatic integrated sensing and communication (ISAC) framework for multi-user multi-input multi-output (MIMO)-orthogonal frequency division multiplexing (OFDM) systems, where communication users act as passive bistatic receivers to enable multistatic sensing. Unlike existing works that optimize aggregate sensing metrics and thus favor geometrically advantageous targets, we minimize the $α$-fairness utility over per-target Cramér--Rao lower bounds (CRLBs) subject to per-user minimum data rate and transmit power constraints. The resulting non-convex problem is solved via the Riemannian conjugate gradient (RCG) method with a smooth penalty reformulation. Simulation results validate the effectiveness of the proposed scheme in achieving a favorable sensing fairness--communication trade-off.

• The paper introduces an α-fair beamforming framework that balances sensing accuracy and communication fairness using per-target CRLBs.
• It employs a Riemannian conjugate gradient method on a complex sphere manifold to jointly optimize user rates and sensing performance efficiently.
• Simulation results show that the proposed approach achieves superior estimation fidelity and computational scalability over conventional SDR-based methods.

α-Fair Multistatic ISAC Beamforming for Multi-User MIMO-OFDM Systems via Riemannian Optimization

Introduction and Motivation

This paper introduces a novel α-fair multistatic ISAC (Integrated Sensing and Communication) beamforming approach tailored for multi-user MIMO-OFDM systems, wherein each communication user operates as a passive bistatic receiver to enable distributed multistatic sensing. The central motivation is to address the inherent resource allocation tension between sensing and communication, especially in multistatic networks that leverage spatial diversity across geographically dispersed nodes. Unlike prior works, which primarily optimize aggregate sensing metrics and consequently favor geometrically advantageous targets, this paper proposes a flexible utility function based on α-fairness over per-target CRLBs (Cramér-Rao lower bounds), capturing a broad spectrum of fairness-efficiency trade-offs in target parameter estimation.

System Model and Performance Metrics

The proposed ISAC system contains a single base station (BS), multiple communication users, and multiple sensing targets. The signal model builds upon MIMO-OFDM, with joint communication and sensing signals transmitted via dedicated beamforming vectors $\mathbf{v}_{k,i}$ (user-specific) and $\mathbf{w}_{i}$ (sensing signal), respectively, on each subcarrier. The communication data rate per user is explicitly calculated from the achievable SINR across subcarriers, while the sensing performance for each target is rigorously characterized via the Fisher Information Matrix (FIM) and the CRLB for delay and Doppler estimation, leveraging both monostatic BS observations and bistatic user reflections.

Crucially, the multistatic FIM for a target aggregates contributions from the BS and all cooperating users, thereby increasing the estimation fidelity by exploiting spatial diversity. The optimization objective, parameterized by the fairness coefficient $\alpha$, interpolates between sum-optimal ($\alpha=0$) and max-min fair ($\alpha\to\infty$) formulations, allowing the system designer to prioritize either aggregate accuracy or equitable target estimation. Constraints are imposed on minimum user data rates and total transmit power, ensuring communication QoS.

Optimization Methodology

The joint transmit beamforming problem is highly nonconvex due to the complex relationships between CRLB, user rate constraints, and power budget. To circumvent the limitations of traditional semidefinite relaxation (SDR) approaches, which impose rank-one approximations resulting in suboptimal feasible solutions, this paper leverages a Riemannian conjugate gradient (RCG) method on the complex sphere manifold. The rate constraints are incorporated into the objective via a smooth penalty function, facilitating unconstrained optimization over the manifold defined by the transmit power constraint.

The algorithmic framework includes:

• Penalty-based reformulation of rate constraints to enable gradient-based optimization.
• Euclidean and Riemannian gradient calculations for the composite α-fair objective and penalty terms.
• Iterative search direction updates via the Polak–Ribiere method, tangent space projection, and retraction operations.
• Efficient line-search-based step size selection to ensure convergence within manifold geometry.

This solution eliminates the rank-relaxation loss of SDR and is computationally more scalable, with per-iteration complexity scaling linearly in the number of antennas and subcarriers.

Numerical Results and Empirical Insights

Simulations for realistic mmWave MIMO-OFDM settings (20 BS antennas, 10 users, 10 targets, 2048 subcarriers, 100 MHz bandwidth, 30 dBm transmit power) validate key claims:

• The RCG algorithm converges swiftly (≈50 iterations) for both objective (sum CRLB) and constraint (data rate), with larger penalty parameters enforcing rate feasibility reliably.
• Adjustment of α yields controllable fairness-efficiency trade-offs: small α minimizes aggregate CRLB but introduces high per-target disparity, while large α reduces the gap between minimum and maximum CRLBs, elevating the worst-case but improving fairness.
• The proposed multistatic approach consistently outperforms both monostatic and conventional multistatic (SDR/SCA) schemes in sum and max CRLB across all communication rate requirements.
• Scalability analysis demonstrates that increasing the number of users systematically strengthens multistatic FIM and lowers CRLBs, confirming the benefit of bistatic diversity. Even under stringent rate constraints, the proposed scheme maintains superior estimation performance relative to baselines.
• The RCG-based solution exhibits markedly lower computational complexity versus interior-point SDR solvers, enabling practical deployment in large-scale systems.

Theoretical and Practical Implications

By formalizing the α-fairness-CRLB framework and demonstrating scalable optimization on the complex sphere manifold, the paper advances ISAC beamforming for multi-user MIMO-OFDM systems. The primary theoretical contribution is the generalization of sensing fairness objectives, enabling continuous control between sum-optimal and min-max fairness extremes via a tunable α parameter. On the practical front, the adoption of distributed multistatic sensing via passive user participation, combined with direct manifold optimization, facilitates robust joint communication and sensing in next-generation wireless networks. This enables geographically flexible sensing with strict communication QoS guarantees—a crucial requirement for emerging 6G and IoT scenarios.

The framework opens paths for further research on adaptive fairness parameter selection, distributed or federated manifold optimization, and integration with other ISAC resource allocation protocols. Extending the RCG approach to other nonconvex multi-objective formulations may further accelerate computational wireless intelligence.

Conclusion

This paper proposes a flexible, computationally efficient α-fair multistatic ISAC beamforming paradigm for multi-user MIMO-OFDM systems, leveraging Riemannian optimization to achieve robust sensing fairness-communication trade-offs (2603.29717). The approach demonstrates superior sensing accuracy and fairness over conventional monostatic and SDR-based multistatic schemes, with strong scalability and practical feasibility for large-scale deployments.