- The paper establishes a robust multi-stream precoding framework for massive MIMO LEO satellite systems that leverages statistical CSI to overcome challenges from high Doppler shifts and channel aging.
- The methodology employs inter-user interference suppression, majorization theory, and the Lanczos algorithm to achieve near-optimal performance with reduced computational complexity.
- Numerical results validate that sCSI-based designs approach iCSI upper bounds, offering scalable high-throughput transmission with minimal degradation even with outdated CSI.
Robust Multi-Stream Precoding in Massive MIMO Satellite Systems with Statistical CSI
Problem Motivation and System Assumptions
This work addresses the design of robust downlink (DL) multi-stream precoding in the context of massive multiple-input multiple-output (mMIMO) low Earth orbit (LEO) satellite communication systems, especially when only statistical channel state information (sCSI) is available at the satellites. Accurate instantaneous CSI (iCSI) is notoriously difficult to obtain in this scenario due to severe path loss, high Doppler shifts, and rapid channel aging. The primary aim is to enable high-capacity, multi-stream transmission from multi-antenna satellites to multi-antenna user terminals (UTs) via coherent multi-satellite (SAT) cooperation, limited to statistical CSI acquisition.
The system considered combines clustered SAT cooperation, each cluster coordinated through a central "super-SAT", with multi-antenna UTs, facilitating simultaneous multi-stream and multi-user transmission. A universal delay and Doppler precompensation technique is adopted to mitigate the effects of large and variable propagation delays and Doppler shifts. The SAT-UT channel is modeled as Rician, with rank-1 per SAT-UT link, but full-rank achieved through SAT cooperation.
Channel Modeling and the Impact of SAT Cooperation
The work provides a precise spatio-temporal and frequency channel model for the SAT-UT link, explicitly capturing:
- Array geometries at both SATs and UTs (planar arrays)
- Path-specific Doppler and delay effects, with line-of-sight (LoS) dominant but including non-LoS (NLoS) scattering
- The significant impact and near-independence of inter-SAT interference due to propagation delay diversity
It is demonstrated that only via multi-SAT cooperation and multi-antenna UTs can multi-stream transmission per UT be reliably supported. Single-SAT links, with rank-1 structure, are inherently limited to single-stream. Statistical channel characteristics—first- and second-order moments—are derived for the composite multi-SAT, multi-UT scenario.
Locally Optimal Precoding under iCSI and sCSI
Precoding and receiver designs are formulated for both iCSI- and sCSI-informed scenarios under two models: total power constraint (TPC) and per-antenna power constraint (PAPC).
Under the weighted minimum mean squared error (WMMSE) framework, the locally optimal iCSI-based precoder-receiver pairs are iteratively computed, yielding MMSE-optimal structures equivalent to maximizing sum-rate. A block coordinate descent algorithm is employed, with the receiver, auxiliary weights, and precoder alternately optimized. This serves as a performance upper bound.
For sCSI-based design, closed-form expressions for the achievable rate and MSE are derived using Jensen’s inequality and the properties of second-order channel statistics. The entire iCSI-based optimization framework is mapped into the sCSI scenario by substituting the instantaneous channel with its statistical equivalent.
Robust and Low-Complexity Precoding with Statistical CSI
Given the cubic complexity in the number of SAT antennas inherent in iterative optimal WMMSE methods, a key contribution is the derivation and analysis of robust, low-complexity precoders under sCSI. The construction employs the following principles:
- Inter-user interference suppression: The precoder for each UT is constrained to the null space of aggregated steering vectors of all other UTs for each SAT, thus ensuring strong decoupling at the physical layer and simplifying the receiver.
- Majorization theory-based analysis: The optimal structure of the power allocation matrix within the reduced-dimensional null space is theoretically characterized using majorization and Schur-concavity to guarantee optimal diagonalization for both MSE minimization and sum-rate maximization objectives.
- Lanczos algorithm for eigen-decomposition: To reduce complexity in extracting the required principal eigenmodes, the Lanczos algorithm is used, targeting only the dominant modes that contribute to the effective signal subspace.
These techniques collectively provide linear or quadratic scaling (with antenna number) for the crucial steps in robust precoding matrix computation, without significant loss in achievable rate compared to the full-complexity solution.
Extensive simulations, parameterized for Starlink LEO orbital geometry and standard 3GPP channel models, substantiate the robust sCSI-based approaches:
- Gap with iCSI-optimality: With a sufficiently large number of SAT antennas, sCSI-based low-complexity designs approach iCSI-based upper bounds, especially as the SAT/UT antenna ratio increases.
- Algorithmic scalability: The Lanczos-based method retains near-optimal performance with a fraction of the computation, making real-time adaptation feasible.
- Effect of system parameters:
- The performance gap between optimal and reduced-complexity schemes shrinks rapidly as per-SAT antenna numbers increase.
- Multi-stream gain increases with the number of cooperative SATs.
- The algorithms exhibit strong robustness to Rician factor variations and phase compensation errors.
- Performance degradation due to outdated sCSI is minimal for update periods up to 100 ms.
- Fairness aspects: Although multi-stream precoding maximizes average throughput and robustness, it does not uniformly maximize rate fairness (e.g., 5% rate percentile). Weighted designs can be adopted for fairness-critical applications.
Theoretical and Practical Implications
This work provides a comprehensive framework for robust, low-complexity, multi-stream precoding in massive MIMO LEO SAT networks constrained to sCSI. Theoretically, it extends majorization-based design principles to large-scale, non-terrestrial, cooperative MIMO channels and rigorously establishes the optimal structures under both power constraints and channel uncertainty.
Practically, the results demonstrate that industry-scale, high-throughput SAT systems can support multi-stream services to multi-antenna devices (e.g., smartphones, IoT nodes) without requiring frequent, costly, or infeasible downlink CSI feedback or pilot transmission. The proposed low-complexity algorithms are directly applicable to real-world deployments, provided sufficient SAT payload antenna resources and moderate inter-SAT coordination are available. The architectures and algorithms readily adapt to emerging 6G scenarios emphasizing ubiquitous global connectivity.
Future directions may include multi-hop non-terrestrial networks—with both SAT and aerial layers—quantization/feedback-constrained sCSI refinement, advanced fairness and QoS control, and cross-layer scheduling with statistical precoding integration.
Conclusion
This paper provides a rigorous and practically viable solution to robust multi-stream downlink precoding in cooperative massive MIMO LEO satellite systems utilizing only statistical CSI. Key contributions include the analysis and design of scalable, low-complexity, and theoretically optimal precoders for both TPC and PAPC regimes under highly uncertain CSI. The results confirm that, with sufficient per-SAT antenna resources, sCSI-based robust designs offer performance close to that of classical iCSI approaches, making them highly promising for next-generation non-terrestrial mMIMO deployments (2604.08925).