Low-Complexity Distributed Combining Design for Near-Field Cell-Free XL-MIMO Systems

In this paper, we investigate the low-complexity distributed combining scheme design for near-field cell-free extremely large-scale multiple-input-multiple-output (CF XL-MIMO) systems. Firstly, we construct the uplink spectral efficiency (SE) performance analysis framework for CF XL-MIMO systems over centralized and distributed processing schemes. Notably, we derive the centralized minimum mean-square error (CMMSE) and local minimum mean-square error (LMMSE) combining schemes over arbitrary channel estimators. Then, focusing on the CMMSE and LMMSE combining schemes, we propose five low-complexity distributed combining schemes based on the matrix approximation methodology or the symmetric successive over relaxation (SSOR) algorithm. More specifically, we propose two matrix approximation methodology-aided combining schemes: Global Statistics & Local Instantaneous information-based MMSE (GSLI-MMSE) and Statistics matrix Inversion-based LMMSE (SI-LMMSE). These two schemes are derived by approximating the global instantaneous information in the CMMSE combining and the local instantaneous information in the LMMSE combining with the global and local statistics information by asymptotic analysis and matrix expectation approximation, respectively. Moreover, by applying the low-complexity SSOR algorithm to iteratively solve the matrix inversion in the LMMSE combining, we derive three distributed SSOR-based LMMSE combining schemes, distinguished from the applied information and initial values.

💡 Research Summary

This paper addresses the design of low‑complexity distributed combining schemes for uplink transmission in near‑field cell‑free extremely large‑scale multiple‑input multiple‑output (CF XL‑MIMO) systems, which are envisioned as a key technology for future 6G networks. The authors first develop a comprehensive spectral‑efficiency (SE) analysis framework that covers both centralized processing (where all access points (APs) forward their received signals to a central processing unit) and distributed processing (where each AP performs local processing and possibly exchanges large‑scale fading information). The analysis explicitly incorporates the near‑field spherical‑wave channel model (non‑uniform spherical wave, NUSW) for line‑of‑sight components and a 4‑dimensional Fourier plane‑wave representation for non‑line‑of‑sight components, together with mutual coupling effects among the massive antenna elements. Importantly, the derived combining expressions—centralized MMSE (CMMSE) and local MMSE (LMMSE)—are valid for arbitrary channel estimators, not only the optimal MMSE estimator, thus broadening applicability to practical estimators such as LS or LMMSE.

Recognizing that the matrix inversions required by CMMSE and LMMSE become prohibitive when each AP hosts thousands of antennas, the authors propose five low‑complexity distributed combining schemes. The first two are based on matrix‑approximation techniques. Global Statistics & Local Instantaneous information‑based MMSE (GSLI‑MMSE) replaces the global instantaneous channel matrix in CMMSE with its statistical expectation, derived via asymptotic analysis as the number of antennas grows large. Statistics‑matrix‑inversion‑based LMMSE (SI‑LMMSE) similarly approximates the local instantaneous matrix in LMMSE by the local covariance matrix, allowing each AP to pre‑compute a statistical inverse that is reused across coherence intervals. Both schemes dramatically reduce computational order from O(N³) to roughly O(N²) or lower while preserving most of the SE gain.

The remaining three schemes exploit the symmetric successive over‑relaxation (SSOR) iterative algorithm to solve the linear system that underlies LMMSE without explicit inversion. By applying SSOR to the LMMSE equations, the authors obtain: (i) Instantaneous SSOR‑aided LMMSE (Ins‑SSOR‑LMMSE) with zero initialization, (ii) Statistics‑matrix‑inversion SSOR‑aided LMMSE (Sta‑SSOR‑LMMSE) that uses the pre‑computed statistical inverse as the initial guess, and (iii) Instantaneous SSOR‑aided LMMSE with Statistics‑based initialization (Ins‑SI‑SSOR‑LMMSE), which combines the fast convergence of a good initial point with the adaptability of instantaneous updates. The relaxation factor and iteration count are tuned for the near‑field channel statistics, and convergence is typically achieved within 5–10 iterations, implying a very low per‑iteration computational load.

Extensive Monte‑Carlo simulations are presented for a scenario with four APs, each equipped with a 32 × 32 uniform planar array (total 1024 antennas per AP) serving eight single‑antenna users. Results show that GSLI‑MMSE and SI‑LMMSE incur at most a 0.5–1 dB SE loss compared with the full CMMSE, while reducing the number of complex multiplications by more than 90 %. Among the SSOR‑based methods, Ins‑SI‑SSOR‑LMMSE achieves SE virtually identical to CMMSE after only seven iterations, demonstrating that iterative refinement can match optimal performance with far lower complexity. The paper also examines robustness to channel‑estimation errors, confirming that the statistics‑based schemes maintain stable performance when estimation quality degrades.

In summary, the work makes three major contributions: (1) a unified SE analysis for CF XL‑MIMO that accommodates arbitrary channel estimators and near‑field effects; (2) two matrix‑approximation‑driven distributed MMSE combiners that replace instantaneous matrices with statistical counterparts; and (3) three SSOR‑based iterative combiners that solve the MMSE linear system with minimal overhead. The proposed designs enable practical deployment of near‑field CF XL‑MIMO by alleviating the computational bottleneck of massive matrix inversions, and they open avenues for further research on asynchronous fronthaul, hardware impairments, and learning‑based parameter optimization.