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Performance of Cell-Free Massive MIMO with Rician Fading and Phase Shifts

Özgecan Özdogan, Emil Björnson, Jiayi Zhang

arXiv:1903.07335v2cs.IT

TL;DR

The paper addresses UL and DL spectral efficiency in cell-free massive MIMO when Rician LoS phases vary with mobility and phase noise. It derives phase-aware and phase-unaware estimators with closed-form SE analyses for UL two-layer decoding and DL transmission modes. LSFD improves UL SE, coherent DL transmission generally performs better, and phase-information losses depend on pilot length and pilot contamination.

  • Problem

    Rician-fading analyses often neglect LoS phase shifts, although mobility and phase noise can change phase while leaving amplitude largely unchanged.

  • Method

    The paper models LoS phase as random and derives phase-aware MMSE, non-aware LMMSE, and LS estimators with closed-form UL and DL SE expressions.

  • Results

    LSFD improves UL SE, coherent transmission generally outperforms non-coherent DL transmission, and phase-knowledge losses vary with pilot length and contamination.

  • Takeaways & Limitations

    Pilot length should account for phase shifts in high-mobility or low-quality-hardware scenarios, while tested low-contamination losses were 13.4% for coherent and 2.4% for non-coherent transmission.

Abstract

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In this paper, we study the uplink (UL) and downlink (DL) spectral efficiency (SE) of a cell-free massive multiple-input-multiple-output (MIMO) system with Rician fading channels. The phase of the line-of-sight (LoS) path is modeled as a uniformly distributed random variable to take the phase-shifts due to mobility and phase noise into account. Considering the availability of prior information at the access points (APs), the phase-aware minimum mean square error (MMSE), non-aware linear MMSE (LMMSE), and least-square (LS) estimators are derived. The MMSE estimator requires perfectly estimated phase knowledge whereas the LMMSE and LS are derived without it. In the UL, a two-layer decoding method is investigated in order to mitigate both coherent and non-coherent interference. Closed-form UL SE expressions with phase-aware MMSE, LMMSE, and LS estimators are derived for maximum-ratio (MR) combining in the first layer and optimal large-scale fading decoding (LSFD) in the second layer. In the DL, two different transmission modes are studied: coherent and non-coherent. Closed-form DL SE expressions for both transmission modes with MR precoding are derived for the three estimators. Numerical results show that the LSFD improves the UL SE performance and coherent transmission mode performs much better than non-coherent transmission in the DL. Besides, the performance loss due to the lack of phase information depends on the pilot length and it is small when the pilot contamination is low.

I. INTRODUCTION

The paper analyzes cell-free massive MIMO with Rician fading while modeling LoS phase shifts caused by mobility and phase noise. It derives channel estimators and spectral-efficiency expressions under different phase-information assumptions.

  • System context: Cell-free massive MIMO uses distributed APs that jointly serve fewer UEs through fronthaul-coordinated network MIMO.The APs spatially multiplex UEs using locally obtained CSI.
  • Motivation and model: LoS phases are modeled as i.i.d. random variables in each coherence block, alongside AP-UE-specific Rician means and variances.The model captures phase changes that can occur without amplitude changes during mobility or because of phase noise.
  • Contributions: Phase-aware MMSE, non-aware LMMSE, and LS channel estimators are derived for different levels of prior information.MMSE assumes phase knowledge, LMMSE uses large-scale fading parameters, and LS uses no prior information.
  • Contributions: The paper derives closed-form UL and DL SE expressions using two-layer UL decoding and coherent or non-coherent DL transmission.UL uses MR combining in the first layer and LSFD in the second layer; DL uses MR precoding.
  • System model: The system model considers single-antenna APs and UEs, TDD operation, channel reciprocity, and UL pilot-based channel estimation.Pilot observations are processed locally at each AP, with pilot reuse creating sets of UEs sharing sequences.

B. LMMSE Channel Estimator

The LMMSE estimator is designed for APs that know channel statistics but not the instantaneous LoS phase. Its estimate and error retain tractable second-order statistics for subsequent processing.

  • Estimator definition: The LMMSE estimator applies when channel statistics are available but the LoS phase is completely unknown at the AP.The relevant statistics are the LoS magnitude and NLoS variance.
  • Estimator statistics: The LMMSE estimate and estimation error are uncorrelated random variables with specified second-order statistics.The paper characterizes the error variance and estimate covariance for analysis.
  • Vector formulation: The scalar LMMSE formulation is also written in vector form for the channel estimates and their covariance matrices.A detailed derivation is provided in Appendix A.
  • Comparison with LS: The paper contrasts LMMSE with LS estimation, which requires neither prior phase information nor channel statistics.The LS estimator is non-Bayesian and minimizes the pilot-observation residual.

IV. UPLINK DATA TRANSMISSION

Uplink decoding combines local MR processing with CPU-level LSFD weighting. The LSFD coefficients use slowly varying large-scale fading to reduce inter-user interference and maximize the resulting SE bound.

  • First-layer decoding: Each AP uses an estimated channel as the local MR combining scalar for detecting the desired UE.The considered estimates include phase-aware MMSE, LMMSE, and LS estimates.
  • Second-layer decoding: The CPU performs second-layer decoding by applying LSFD coefficients to the locally decoded signals from all APs.This architecture is illustrated by the two-layer decoding technique.
  • LSFD principle: LSFD weights balance AP contributions using large-scale fading, reducing the risk that weak or interference-dominated signals amplify interference.Large-scale fading mainly reflects AP-UE distance and shadowing and varies more slowly than small-scale fading.
  • SE analysis: The UL achievable SE is obtained with the use-and-then-forget bound and an effective SINR for the LSFD receiver.The bound averages desired-signal, interference, and noise terms over their randomness.
  • LSFD optimization: For fixed pilot and data powers, the maximizing LSFD vector follows from the effective SINR's generalized Rayleigh quotient.The resulting solution applies across channel distributions and beamforming schemes, while their parameter values change.

A. Uplink Spectral Efficiency with the Phase-aware MMSE Estimator

The uplink analysis derives spectral-efficiency expressions using two-layer decoding with LSFD for phase-aware MMSE, LMMSE, and LS estimators. LSFD can compensate for the poorer LS estimator when optimal statistics-based processing is available, although that processing is not practical for LS.

  • A. Uplink Spectral Efficiency with the Phase-aware MMSE Estimator: Phase-aware MMSE, LMMSE, and LS estimators are incorporated into closed-form uplink SE expressions with LSFD receivers.The derivations use the effective SINR and its Rayleigh quotient form for optimizing the LSFD vector.
  • A. Uplink Spectral Efficiency with the Phase-aware MMSE Estimator: The maximizing LSFD receiver vector is obtained by reformulating each effective SINR as a Rayleigh quotient.This formulation is stated for both the phase-aware MMSE and LMMSE cases.
  • A. Uplink Spectral Efficiency with the Phase-aware MMSE Estimator: With the optimal LSFD vector, the LS and LMMSE SINR expressions are proportional, and the resulting SE expressions are identical.This equivalence follows because the LS and LMMSE estimators differ only by a scaling factor depending on large-scale fading coefficients.
  • A. Uplink Spectral Efficiency with the Phase-aware MMSE Estimator: LSFD requires only large-scale fading coefficients, not channel realizations, so it adds no additional information requirement when paired with LMMSE or MMSE estimation.The optimal LSFD vector is not practically computable with LS because the required channel statistics are unknown.
  • A. Uplink Spectral Efficiency with the Phase-aware MMSE Estimator: Without LSFD, the LS estimator performs poorly in cell-free massive MIMO and therefore provides a conservative lower bound on performance.The paper contrasts this with the compensating effect of optimal LSFD.

V. COHERENT DOWNLINK TRANSMISSION

The coherent downlink mode has all APs transmit the same data symbol to each UE using MR precoding. Its ergodic downlink capacity is lower-bounded through the UatF bound, producing effective SINR expressions for the three estimators.

  • V. COHERENT DOWNLINK TRANSMISSION: Each AP transmits the same downlink data symbol to each UE, enabling coherent joint transmission.Setting selected transmit powers to zero also covers serving only subsets of UEs.
  • V. COHERENT DOWNLINK TRANSMISSION: MR precoding uses the channel estimate, with per-AP power coefficients selected to satisfy the downlink power constraint.The transmitted signal is formed from the estimated channel and power-control coefficients.
  • V. COHERENT DOWNLINK TRANSMISSION: The ergodic downlink capacity is lower-bounded using the UatF bound, with expectations taken over all sources of randomness.The resulting bound is expressed through an effective SINR for UE k.
  • V. COHERENT DOWNLINK TRANSMISSION: The effective coherent downlink SINR is computed for phase-aware MMSE, LMMSE, and LS channel estimators.The three estimator-specific expressions are derived from the same UatF-based formulation.

A. Coherent Downlink Spectral Efficiency with the Phase-aware MMSE Estimator

The coherent downlink analysis evaluates estimator-specific effective SINRs under MR precoding and inserts their expectation terms into the common UatF-based expression. The corresponding non-coherent mode avoids AP phase synchronization but has a smaller signal term than coherent transmission.

  • A. Coherent Downlink Spectral Efficiency with the Phase-aware MMSE Estimator: For phase-aware MMSE, the required expectation terms are substituted into the common coherent downlink SINR expression.The resulting closed-form SINR is given after inserting the estimator-specific terms.
  • A. Coherent Downlink Spectral Efficiency with the Phase-aware MMSE Estimator: The same coherent downlink SINR construction is repeated for LMMSE and LS estimators using their respective expectation terms.Each substitution yields an estimator-specific effective SINR expression.
  • VI. NON-COHERENT DOWNLINK TRANSMISSION: Non-coherent transmission lets APs send different data symbols, reducing the need for phase synchronization among APs.The mode is analyzed with MR precoding for phase-aware MMSE, LMMSE, and LS estimators.
  • VI. NON-COHERENT DOWNLINK TRANSMISSION: The non-coherent SINR numerator sums squared AP contributions, whereas the coherent case places the summation inside the square.Consequently, coherent transmission has the larger signal term but requires synchronized, cooperating APs.

A. Non-coherent Downlink Spectral Efficiency with the Phase-aware MMSE Estimator

For non-coherent downlink transmission, the LMMSE and LS estimators yield the same spectral efficiency because access points do not cooperate. The corresponding expectations are therefore computed separately for each antenna.

  • The LMMSE and LS estimators produce the same downlink SINR at UE k.
  • Their equal spectral efficiency follows from the absence of cooperation between access points in non-coherent transmission.
  • The expectations are calculated individually for each antenna.

VII. NUMERICAL RESULTS

Numerical evaluation validates the closed-form expressions and compares estimators, decoding schemes, pilot lengths, UE loads, and coherent versus non-coherent DL transmission. LSFD improves UL SE broadly, while coherent DL generally provides higher SE but is more sensitive to missing phase knowledge.

  • Uplink: Two-layer decoding improves UL SE for all estimators, with LS benefiting most because it is more vulnerable to interference.The maximizing LSFD vector makes LS and LMMSE coincide in the reported setting.
  • Uplink: 6.9% and 24.8% performance losses occur without phase knowledge at τp = 20 and τp = 5, respectively, in the tested UL scenarios.The larger loss occurs with shorter pilots and higher pilot contamination.
  • Downlink: Coherent DL transmission provides higher SE in almost all tested cases except with LS estimation.Its performance gap relative to phase-unaware estimators is larger because coherent transmission is sensitive to phase errors.
  • Downlink: Fewer UEs produce higher SEs for both coherent and non-coherent DL transmission and across the tested estimators.The reported comparison attributes this to reduced interference and pilot contamination when pilot length is fixed.

APPENDIX A DERIVATION OF THE LMMSE ESTIMATOR

This appendix derives the LMMSE estimator, its mean-square error, estimator statistics, and the expectation terms needed for the paper’s SE expressions.

  • Estimator derivation: The LMMSE estimator and its mean-square error are obtained from the received pilot signal.The derivation uses the desired channel component in the pilot observation.
  • Estimator statistics: The derivation computes the estimator’s mean and variance, followed by the LS estimator’s mean, variance, and estimation-error statistics.These statistics support the subsequent SE calculations.
  • SE expressions: Inserting the required expectation results gives the UL and DL SE results.The appendix connects the computed estimator statistics to the closed-form spectral-efficiency expressions.

APPENDIX D PROOF OF UL AND DL SE WITH LMMSE ESTIMATOR

This appendix proves the expectation identities required for the LMMSE-based UL and DL SE expressions by handling dependence, independence, and zero-mean cases across AP and UE indices.

  • Dependence structure: Pilot-contaminating estimators at the same AP are not independent, so their joint expectations require separate calculations.The proof explicitly treats same-AP dependence before combining the cases.
  • Expectation cases: The proof evaluates denominator expectations for all AP and UE combinations, distinguishing pilot-sharing cases from independent non-sharing cases.Terms with mutually independent zero-mean factors vanish in the non-sharing cases.
  • Independence simplifications: Independence across different APs and uncorrelated zero-mean estimates simplify several cross terms to zero.These properties are invoked repeatedly when evaluating the LMMSE moments.
  • Assembly: The separately evaluated cases are combined into matrix-form expressions that complete the proof.The appendix concludes after assembling the expectation results and applying the stated independence relations.

APPENDIX F PROOF OF DL SE WITH NON-COHERENT TRANSMISSION

The appendix derives the non-coherent DL SE by sequentially detecting AP contributions, treating residual terms as uncorrelated noise, and summing the resulting rates.

  • Sequential detection: UE k first detects AP 1 using the average channel, then sequentially detects each later AP after subtracting previously detected signals.This establishes the two-stage sequential detection structure used in the proof.
  • Effective channel: Each detected AP contribution is represented by a deterministic average channel plus a desired signal term and uncorrelated noise.The noise power is used to form the per-stage effective detection expression.
  • SE result: The total non-coherent DL spectral efficiency is obtained by combining the sequential detection terms and cancelling common numerator and denominator factors.The resulting expression is shown to equal the paper’s non-coherent DL SE formula.
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