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Optimal Bilinear Equalizer Beamforming Design for Cell-Free Massive MIMO Networks with Arbitrary Channel Estimators

Zhe Wang, Jiayi Zhang, Hao Lei, Dusit Niyato, Bo Ai

TL;DR

The paper tackles CF mMIMO with arbitrary statistics-based channel estimators over Rician fading, addressing the need for low-complexity beamforming. It develops an optimal bilinear equalizer (OBE) framework that uses combining/precoding matrices derived from channel statistics rather than instantaneous CSI, and derives closed-form UL and DL spectral efficiency expressions under this framework. A central contribution is the distributed OBE design that yields explicit optimal weights and SINR expressions, along with DL BE precoding inspired by UL OBE, all applicable to multi-antenna APs and pilot-contaminated scenarios. The results show that OBE beamforming outperforms conventional schemes (e.g., LMMSE, LRZF) and, in Rayleigh fading, the UL SE with OBE is largely independent of the chosen channel estimator, with practical implications for reducing estimation complexity in CF mMIMO systems.

Abstract

This paper studies the distributed optimal bilinear equalizer (OBE) beamforming design for both the uplink and downlink cell-free massive multiple-input multiple-output networks. We consider arbitrary statistics-based channel estimators over spatially correlated Rician fading channels. In the uplink, we derive the achievable spectral efficiency (SE) performance and OBE combining schemes with arbitrary statistics-based channel estimators and compute their respective closed-form expressions. It is insightful to explore that the achievable SE performance is not dependent on the choice of channel estimator when OBE combining schemes are applied over Rayleigh channels. In the downlink, we derive the achievable SE performance expressions with BE precoding schemes and arbitrary statistics-based channel estimators utilized and compute them in closed form. Then, we obtain the OBE precoding scheme leveraging insights from uplink OBE combining schemes.

Optimal Bilinear Equalizer Beamforming Design for Cell-Free Massive MIMO Networks with Arbitrary Channel Estimators

TL;DR

The paper tackles CF mMIMO with arbitrary statistics-based channel estimators over Rician fading, addressing the need for low-complexity beamforming. It develops an optimal bilinear equalizer (OBE) framework that uses combining/precoding matrices derived from channel statistics rather than instantaneous CSI, and derives closed-form UL and DL spectral efficiency expressions under this framework. A central contribution is the distributed OBE design that yields explicit optimal weights and SINR expressions, along with DL BE precoding inspired by UL OBE, all applicable to multi-antenna APs and pilot-contaminated scenarios. The results show that OBE beamforming outperforms conventional schemes (e.g., LMMSE, LRZF) and, in Rayleigh fading, the UL SE with OBE is largely independent of the chosen channel estimator, with practical implications for reducing estimation complexity in CF mMIMO systems.

Abstract

This paper studies the distributed optimal bilinear equalizer (OBE) beamforming design for both the uplink and downlink cell-free massive multiple-input multiple-output networks. We consider arbitrary statistics-based channel estimators over spatially correlated Rician fading channels. In the uplink, we derive the achievable spectral efficiency (SE) performance and OBE combining schemes with arbitrary statistics-based channel estimators and compute their respective closed-form expressions. It is insightful to explore that the achievable SE performance is not dependent on the choice of channel estimator when OBE combining schemes are applied over Rayleigh channels. In the downlink, we derive the achievable SE performance expressions with BE precoding schemes and arbitrary statistics-based channel estimators utilized and compute them in closed form. Then, we obtain the OBE precoding scheme leveraging insights from uplink OBE combining schemes.

Paper Structure

This paper contains 10 sections, 5 theorems, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Uplink Data Transmission
  4. Uplink Spectral Efficiency Analysis
  5. Optimal Bilinear Equalizer Combining Design
  6. Downlink Data Transmission
  7. Numerical Results
  8. Conclusions
  9. Proof of Theorem \ref{['thm_distributed_closed']}
  10. Proof of Corollary \ref{['Rayleigh_max']}

Key Result

Theorem 1

When applying the BE-structure combining scheme $\mathbf{v}_{mk}=\mathbf{W}_{mk}\hat{\mathbf{g}}_{mk}$, the UL achievable SE expressions can be calculated in closed-form $\overline{\mathrm{SE}}_{k}^{\mathrm{ul}}=\frac{\tau _u}{\tau _c}\log _2( 1+\overline{\mathrm{SINR}}_{k}^{\mathrm{ul}} )$, where $ $\xi _{mkl}=\sqrt{p_l}\tau _p\mathrm{tr}( \mathbf{W}_{mk}^{H}\bar{\mathbf{G}}_{mlk} ) \mathrm{tr}(

Figures (3)

  • Figure 1: Average UL SE against $K$ under different combining schemes over Rician fading channels with $M=40$ and $N=4$. The legend "A-B" denotes the "B" combining-based scheme over the "A" channel estimator.
  • Figure 2: CDF of the UL SE per UE under OBE combining schemes over the Rician and Rayleigh fading channels with $M=40$, $K=10$, and $N=4$.
  • Figure 3: Average DL SE against $N$ under different precoding schemes over Rician fading channels with $M=20$ and $K=10$. The legend "A-B" denotes the "B" precoding-based scheme over the "A" channel estimator.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Corollary 2
  • Theorem 3
  • Remark 4
  • Remark 5