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Massive MIMO with Imperfect Channel Covariance Information

Emil Björnson, Luca Sanguinetti, Merouane Debbah

arXiv:1612.04128v3cs.ITmath.IT

TL;DR

Covariance information is needed for MMSE channel estimation, but practical estimates are imperfect and perfect knowledge is often assumed despite limited observations and changing statistics. The paper proposes covariance-estimation approaches and derives spectral-efficiency bounds under imperfect covariance knowledge. Its results show that a small number of extra pilots can outperform LS estimation, while covariance knowledge is not critical to achieve substantial performance.

  • Problem

    Covariance information supports MMSE channel estimation, but practical estimates are imperfect and perfect covariance knowledge is questionable when observations are limited and statistics change over time.

    Source: §I · p. 1, A. Channel Estimation: Perfect… · p. 2

  • Method

    The paper evaluates direct covariance estimation and a proposed two-stage Via Q approach, derives a spectral-efficiency bound for arbitrary combining, and gives a closed-form MRC expression.

    Source: A. Covariance Matrix Estimation · p. 3, C. Spectral Efficiency · pp. 3–4

  • Results

    With Via Q, 10 extra pilots outperform LS; 95% of MMSE-estimator SE requires 100 pilots for MRC and 250 for RZF.

    Source: §IV · p. 5

  • Takeaways & Limitations

    Covariance knowledge is not critical, but it largely improves performance, especially with interference suppression such as RZF; a few tens of extra pilots can outperform LS with MRC or RZF.

    Source: §V · p. 5

    The tightest capacity bound for linear combining requires MMSE estimation and perfect covariance knowledge, so imperfect covariance requires an alternative bound.

    Source: B. Spectral Efficiency: Perfect… · p. 2

Abstract

from arXiv · show

This work investigates the impact of imperfect statistical information in the uplink of massive MIMO systems. In particular, we first show why covariance information is needed and then propose two schemes for covariance matrix estimation. A lower bound on the spectral efficiency (SE) of any combining scheme is derived, under imperfect covariance knowledge, and a closed-form expression is computed for maximum-ratio combin- ing. We show that having covariance information is not critical, but that it is relatively easy to acquire it and to achieve SE close to the ideal case of having perfect statistical information.

I. INTRODUCTION

Massive MIMO uses large antenna arrays to spatially multiplex many user equipments, but fading requires frequent channel re-estimation. Covariance information supports MMSE estimation and other tasks, yet is often assumed perfect despite limited observations and changing statistics.

  • Arrays with hundreds of steerable antennas enable spatial multiplexing of tens of user equipments per cell in uplink and downlink.Source: §I · p. 1
  • Channel vectors must be re-estimated frequently across time and frequency because of channel fading.Source: §I · p. 1
  • Covariance matrices characterize spatial channel properties and support MMSE channel estimation, resource allocation, and pilot-contamination suppression.In practice, their dimensions grow with antenna count and their statistics change over time, making perfect knowledge questionable.Source: §I · p. 1

II. SYSTEM MODEL

The system is an uplink massive MIMO network with multiple cells, each served by a multi-antenna base station. Channels are correlated Rayleigh fading, while their covariance matrices describe spatial correlation and pathloss and vary more slowly than the channels.

  • The uplink system has L cells, each with a base station equipped with M antennas and K single-antenna user equipments.Source: §II · p. 1
  • The channel covariance matrix describes spatial channel correlation and average pathloss across spatial directions.Source: §II · p. 1
  • Covariance matrices remain constant over the transmission bandwidth and change more slowly than channel vectors; they are assumed constant for τ_s channel blocks.The passage notes measurements suggesting roughly two orders of magnitude slower variation and says τ_s can be on the order of thousands.Source: §II · p. 1

A. Channel Estimation: Perfect Covariance Information

The base station estimates channels using reused, orthogonal pilots and an MMSE estimator that depends on channel statistics. Because the needed covariance information is not known a priori, it must be estimated.

B. Spectral Efficiency: Perfect Covariance Information

With perfect covariance information, linear receive combining supports a spectral-efficiency capacity lower bound based on the instantaneous SINR and pilot overhead. The tightest available bound requires MMSE estimation and therefore perfect covariance knowledge.

III. UPLINK PERFORMANCE WITH IMPERFECT CHANNEL COVARIANCE INFORMATION

The section studies covariance estimation over intervals when channel statistics remain fixed, allowing estimation observations to be collected across many coherence blocks. In the stated mobile scenario, those intervals span 25,000 blocks, so extra covariance-estimation pilots can be used in a fraction of them while keeping overhead low.

III. UPLINK PERFORMANCE WITH IMPERFECT CHANNEL COVARIANCE INFORMATION

  • The channel statistics are assumed fixed over system bandwidth Bs and time interval Ts, which contain τs coherence blocks.The section uses this interval to evaluate the consequences of imperfect covariance estimates.Source: §III · p. 2
  • In the example with Bs = 10 MHz and Ts = 0.5 s, the statistics remain fixed for τs = 25,000 coherence blocks.The example uses Bc = 200 kHz, Tc = 1 ms, and a 2 GHz carrier frequency, supporting 135 km/h mobility.Source: §III · p. 2
  • Extra pilots for covariance estimation can be placed in a fraction of those blocks while keeping overhead low.The passage points to Fig. 1 for an illustration and motivates investigating how extra pilots aid covariance estimation.Source: §III · p. 2

A. Covariance Matrix Estimation

The section estimates the pilot-signal covariance Q_jk from existing pilots and estimates the desired-user covariance R_jjk through either direct training or a proposed two-stage route via Q. Regularization addresses unreliable sample covariance structure, while the two R-estimation routes trade observation count against interference-subtraction error.

A. Covariance Matrix Estimation

  • The BS needs both R_jjk and Q_jk to compute the MMSE estimate of h_jjk, and these covariance matrices are approximated with sample covariances that may need regularization because they are M × M.Source: A. Covariance Matrix Estimation · p. 2

1) Estimation of Q_jk

  • Q_jk can be estimated from NQ received-pilot observations already used for channel estimation, so this covariance-estimation step requires no extra pilots.The sample variance converges almost surely to its true value as NQ grows, and its standard deviation decays as 1/sqrt(NQ).Source: A. Covariance Matrix Estimation · p. 2
  • Estimating Q_jk’s eigenstructure is harder than estimating individual entries because errors across its M^2 elements affect eigenvalues and eigenvectors used by the MMSE estimator.The method regularizes by combining the sample covariance with its diagonalized version, which retains the sample matrix’s main diagonal.Source: A. Covariance Matrix Estimation · p. 2
  • The regularized Q_jk estimate keeps the same main diagonal for every η, while η scales down off-diagonal elements and makes the estimate full rank for η < 1, even when NQ < M.At η = 0, the estimate is diagonal and ignores correlations between channel elements.Source: A. Covariance Matrix Estimation · pp. 2–3

2) Estimation of R_jjk

  • “R direct” obtains NR noisy observations of h_jjk using extra orthogonal pilots, forms a sample covariance, and requires regularization for robustness when NR is small.The described training phase uses NRKL extra pilots, with NR pilots for each UE.Source: A. Covariance Matrix Estimation · p. 3
  • The proposed “Via Q” estimate of R_jjk can use more observations of h_jjk than “R direct” when NQ > NR, typically the practical case, but incurs perturbation from imperfect interference-covariance subtraction.The estimate uses observations of received pilot signals from pilot-contaminating UEs, then combines the resulting sample estimate with a diagonalized estimate using regularization factor µ.Source: A. Covariance Matrix Estimation · p. 3

B. Channel Estimation

The section treats regularized covariance estimates as true matrices to form an approximate MMSE channel estimate, then evaluates its MSE. The regularization factors can be selected to minimize that MSE, while extra covariance-training pilots must be reused sparsely across the network.

B. Channel Estimation

  • Treating the regularized estimates Rhat_jjk(µ) and Qhat_jk(η) as true covariances yields an approximate MMSE estimate of h_jjk.Source: B. Channel Estimation · p. 3
  • Extra pilots used for covariance estimation cannot be unique across every cell in practice and can instead be reused very sparsely in the network.Source: B. Channel Estimation · p. 3
  • The regularization factors µ and η can be selected to minimize the MSE, which is computed for a deterministic W_jk.The MSE computation assumes the channel realizations used to estimate the covariance matrices are not used for data transmission.Source: B. Channel Estimation · p. 3

C. Spectral Efficiency

The use-and-then-forget bound evaluates spectral efficiency without requiring MMSE channel estimates, so it applies with any channel estimator and combining scheme. For MRC, the paper derives a closed-form SE expression for arbitrary channel estimators.

C. Spectral Efficiency

  • The use-and-then-forget bound rewrites the received combined signal to form a capacity lower bound whose expectations are over channel realizations.The bound accounts for pilot overhead and can be used with arbitrary combining choices.Source: C. Spectral Efficiency · p. 3
  • For MRC, the paper obtains a closed-form SE expression with an arbitrary channel estimator.The expression follows from computing the bound’s expectations for deterministic W_jk.Source: C. Spectral Efficiency · pp. 3–4

IV. NUMERICAL EVALUATION

The numerical evaluation compares channel estimators in a seven-cell uplink scenario, measuring covariance-estimation accuracy and center-cell sum SE. The proposed “Via Q” approach outperforms direct covariance estimation in MSE, while covariance estimates from relatively few extra pilots can improve SE over LS.

IV. NUMERICAL EVALUATION

  • The evaluation uses a seven-cell setup with 10 UEs per cell, 300 m inter-BS distance, and 100 antennas at the center-cell BS.The channel covariance model is a one-ring model with a 20° angular interval.Source: §IV · p. 4
  • “Via Q” largely outperforms “R direct” in normalized MSE at every tested NR, and a few hundred samples bring its performance close to MMSE.NR = 10 suffices for both approaches to outperform LS; “Via Q” uses NQ = 10NR observations to estimate R_jjk.Source: §IV · p. 4, Figure 3 · p. 4
  • NR = 10 extra pilots makes “Via Q” outperform LS in SE; reaching 95% of MMSE-estimator SE requires NR = 100 with MRC and NR = 250 with RZF.“R direct” needs 2.5 times as many extra pilots as “Via Q” to reach 95% with RZF.Source: Figure 4 · p. 5, §IV · p. 5

V. CONCLUSIONS

The paper develops covariance estimation and performance analysis for massive MIMO under imperfect covariance knowledge. Its results show that covariance knowledge can improve performance, particularly with interference suppression, while requiring only a few tens of extra pilots to outperform LS estimation.

V. CONCLUSIONS

  • A few tens of extra pilots suffice to outperform the LS estimator with either MRC or RZF.The LS estimator requires no covariance information.Source: §V · p. 5
  • Covariance knowledge is not critical, but largely improves performance, especially with interference suppression schemes such as RZF.Source: §V · p. 5
  • The proposed covariance estimation approach outperforms previous solutions under the same pilot overhead.Source: §V · p. 5
  • The paper derives MSE and SE expressions for any combining scheme under imperfect covariance information and computes a closed-form SE expression for MRC.Source: §V · p. 5
  • The appendix evaluates expectations using channel statistics, exploiting independence and treating the dependent case through an independent residual.For i = k, v_jk − W_jk h_jli and h_jli are independent.Source: §V · p. 5

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