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Pilot Reuse for Massive MIMO Transmission over Spatially Correlated Rayleigh Fading Channels

Li You, Xiqi Gao, Xiang-Gen Xia, Ni Ma, Yan Peng

arXiv:1502.05433v1cs.IT

TL;DR

The paper addresses excessive pilot overhead in single-cell massive MIMO and develops pilot reuse for spatially correlated Rayleigh channels. It links covariance structure to angular power, derives robust transmission and statistical scheduling methods, and reports significant net-spectral-efficiency gains over orthogonal training.

  • Problem

    Conventional orthogonal training incurs pilot overhead proportional to the number of user-terminal antennas, motivating reduced-overhead training for massive MIMO.

  • Method

    The paper combines spatial-correlation and power-angle-spectrum analysis with pilot reuse, robust uplink/downlink MMSE transmission, and low-complexity statistical greedy pilot scheduling.

  • Results

    The proposed pilot-reuse scheme achieves significant performance gains over conventional orthogonal training in net spectral efficiency.

  • Takeaways & Limitations

    Pilot reuse is feasible when pilot-sharing users have non-overlapping angle-of-arrival intervals, while robust transmission accounts for reuse-induced channel-estimation error.

Abstract

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We propose pilot reuse (PR) in single cell for massive multiuser multiple-input multiple-output (MIMO) transmission to reduce the pilot overhead. For spatially correlated Rayleigh fading channels, we establish a relationship between channel spatial correlations and channel power angle spectrum when the base station antenna number tends to infinity. With this channel model, we show that sum mean square error (MSE) of channel estimation can be minimized provided that channel angle of arrival intervals of the user terminals reusing the pilots are non-overlapping, which shows feasibility of PR over spatially correlated massive MIMO channels with constrained channel angular spreads. Regarding that channel estimation performance might degrade due to PR, we also develop the closed-form robust multiuser uplink receiver and downlink precoder that minimize sum MSE of signal detection, and reveal a duality between them. Subsequently, we investigate pilot scheduling, which determines the PR pattern, under two minimum MSE related criteria, and propose a low complexity pilot scheduling algorithm which relies on the channel statistics only. Simulation results show that the proposed PR scheme provides significant performance gains over the conventional orthogonal training scheme in terms of net spectral efficiency.

I. INTRODUCTION

The paper proposes single-cell pilot reuse to reduce massive MIMO training overhead, exploiting spatially correlated channels and angular separation while accounting for reuse-induced estimation errors.

  • Conventional orthogonal training creates pilot overhead proportional to the number of user-terminal antennas, potentially becoming a system bottleneck.
  • Pilot reuse is feasible when users occupy nearly orthogonal spatial directions because realistic propagation concentrates channel power in limited spatial directions.
  • The paper develops robust uplink receivers, downlink precoders, and statistical pilot scheduling, with simulations showing significant net-spectral-efficiency gains over orthogonal training.
  • As the base-station antenna number tends to infinity, covariance eigenvectors are determined by array responses while eigenvalues depend on each channel’s power angle spectrum.
  • Channel-estimation MSE can be minimized when users reusing pilots have non-overlapping angle-of-arrival intervals.

III. PR FOR UL CHANNEL TRAINING

This section formulates uplink channel training with pilot reuse and derives conditions under which channel-estimation error is minimized.

  • Pilot reuse assigns orthogonal pilot sequences to multiple users, with the base station estimating all users sharing a pilot from their combined observation.
  • The channel-estimation error covariance defines the channel-estimation MSE, while pilot reuse can correlate estimation errors among users sharing pilots.
  • The minimum channel-estimation MSE is achieved under the condition specified by Theorem 1 for the users reusing pilots.
  • As the antenna number tends to infinity, channel-estimation MSE is minimized when the angle-of-arrival intervals of pilot-sharing users do not overlap.
  • At high training SNR, pilot noise vanishes and channel-estimation MSE tends to zero under the stated angular-separation condition.

IV. ROBUST UL/DL DATA TRANSMISSIONS

Pilot reuse can degrade channel estimation, motivating robust uplink and downlink data-transmission designs that incorporate channel-estimation error.

  • Robust uplink and downlink transmission is designed because pilot reuse can degrade channel-estimation performance.

A. Robust UL Data Transmission

The robust uplink design models channel-estimation error explicitly in the MMSE signal-detection problem and derives an optimal linear receiver.

  • The uplink received signal model includes estimated channels, channel-estimation errors, user data, additive noise, and uplink data-transmission SNR.
  • Uplink signal-detection MSE is defined by expectation over user data, additive noise, and channel-estimation error.
  • The optimal linear uplink receiver is obtained by minimizing the MMSE signal-detection criterion.
  • When the channel-estimation error covariance vanishes, the robust MMSE receiver reduces to the conventional receiver.

B. Robust DL Data Transmission

The paper formulates robust downlink transmission under pilot-reuse channel-estimation errors and derives an optimal MMSE-SD precoder. The precoder uses the received-signal model, power normalization, and channel reciprocity to support downlink data detection.

  • Signal model: The downlink received signal includes the transposed uplink channel, data signals, additive noise, and a power-constrained linear precoder.The downlink channel is the transpose of the uplink channel under TDD reciprocity.
  • MMSE-SD formulation: Downlink MSE-SD is defined by averaging over the data signal, noise, and channel-estimation error, with a scalar power-scaling parameter.The scalar parameter α accounts for potential power scaling at the user terminals.
  • Optimal precoding: The optimal downlink precoder is obtained by formulating and solving the MMSE-SD optimization problem.The solution is given in Theorem 3.
  • Interpretation: The robust MMSE precoder retains the structure of the conventional precoder while incorporating channel-estimation errors caused by pilot reuse.This structural correspondence is stated after Theorem 3.

C. UL-DL Duality

The robust uplink receiver and downlink precoder exhibit an MMSE duality under equal uplink and downlink SNRs. Pilot scheduling is then optimized under channel-estimation and signal-detection MSE criteria, although exhaustive search is computationally expensive.

  • UL-DL duality: If ρu = ρd, the optimal downlink precoder equals the optimal uplink receiver after power normalization, and their minimum MSE-SD values are equal.This establishes UL-DL MMSE duality within each TDD coherence block.
  • UL-DL duality: The duality reduces robust downlink-precoder computation because the precoder can be obtained from the robust uplink receiver with proper normalization.The result applies when pilot-assisted CSI acquisition and equal data-transmission SNRs are used.
  • Pilot scheduling: Pilot scheduling uses long-term statistical CSI to allocate pilots under two MMSE-related criteria: channel-estimation MSE and signal-detection MSE.The MMSE-SD objective averages performance achievable by robust receivers and precoders over channel and pilot-noise statistics.
  • Complexity limitation: For channel-estimation MSE scheduling, exhaustive search has complexity O(τ KM 3), making combinatorial optimization difficult as the user-terminal number grows.The paper separately identifies exponential exhaustive-search complexity as a practical limitation of pilot scheduling.
  • Pilot scheduling: Under the MMSE-SD criterion, exhaustive search evaluates candidate pilot-reuse patterns with complexity O(τ KM 3K2).A lower bound is introduced because the exact objective lacks a closed-form expression, and it is reported to be tight over a wide SNR region.
  • Optimality condition: When M →∞, the lower-bound average MSE-SD can be minimized under the same AoA-separation conditions that optimize channel training.With rigorous spatial separation, both pilot interference and data interference vanish; at high SNR, average MSE-SD approaches zero.

C. SGPS Algorithm

The statistical greedy pilot scheduling algorithm uses channel covariance statistics to construct pilot-reuse patterns with approximately orthogonal reused channels. It provides a lower-complexity alternative to exhaustive search.

  • Motivation: Exhaustive search becomes difficult to implement as the number of user terminals grows because its complexity is exponential.This motivates the low-complexity SGPS alternative.
  • Algorithm rationale: SGPS is motivated by the optimality conditions for channel estimation and data transmission, assigning reused pilots so their channel covariance matrices are as orthogonal as possible.The algorithm is designed specifically for the single-cell pilot-reuse setting.
  • Complexity: The SGPS complexity is O(M 2K3), compared with exhaustive-search complexities O(τ KM 3K) for MMSE-CE and O(τ KM 3K2) for MMSE-SD.The reduction follows from using no more than (K −1)K(K + 1)/6 orthogonality calculations, each costing O(M 2).
  • Inputs and output: The algorithm takes the user-terminal set, channel covariance information, orthogonal pilot set, and pilot length as inputs, and outputs a pilot-reuse pattern.Its decisions rely on statistical CSI rather than instantaneous channel realizations.
  • Scheduling procedure: SGPS first assigns orthogonal pilots to user terminals with similar channel covariance matrices, then assigns each remaining terminal its best pilot.The subsequent assignment seeks to maximize covariance orthogonality among terminals reusing each pilot.

VI. NUMERICAL RESULTS

Numerical experiments evaluate pilot-reuse transmission with a 128-antenna ULA and truncated-Laplacian channel power angle spectra. Robust reception outperforms conventional reception, while pilot scheduling materially affects data-transmission performance.

  • Simulation setup: The simulations use a 128-antenna half-wavelength-spaced ULA, a 120° user-terminal sector, and equal training and data SNRs.The mean channel AoAs are uniformly distributed over [−π/3, π/3], with ρp = ρu = ρd = ρ.
  • Receiver evaluation: The experiment compares robust and conventional receivers for K = 10, angular spread σ = 10°, pilot length τ = 5, and two pilot-reuse patterns.The robust receiver is evaluated using both true and covariance matrices estimated from 100 samples.
  • Receiver performance: The robust MMSE receiver outperforms the conventional receiver, especially at high SNR where pilot interference dominates.The conventional receiver is more sensitive to channel-estimation error, and increasing SNR can increase its MSE-SD.
  • Covariance estimation: Using estimated rather than true channel covariance matrices causes almost negligible average MSE-SD loss in the reported experiment.The estimated covariance matrices are obtained by averaging over 100 samples.
  • Scheduling and bounds: The lower bound on average MSE-SD is tight over a wide SNR region for different pilot-reuse patterns, and pilot scheduling is crucial to data-transmission performance.The comparison includes both pilot-reuse patterns considered in the experiment.

B. Performance of SGPS Algorithm

SGPS achieves MSE-CE and average MSE-SD performance close to exhaustive search across pilot lengths and SNRs, while the proposed PR scheme improves net spectral efficiency over conventional OT, especially with smaller angular spreads.

  • SGPS performance: SGPS closely approaches ES for MSE-CE across a wide SNR region and different pilot lengths.The comparison uses K = 10 and σ = 10°.
  • SGPS performance: SGPS closely approaches ES for average MSE-SD across a wide SNR region and different pilot lengths.The comparison uses K = 10 and σ = 10°.
  • Net spectral efficiency: The proposed PR scheme provides net spectral-efficiency gains over conventional OT in both uplink and downlink.The comparisons vary coherence block length and SNR for different angular spreads, with K = 10.
  • Net spectral efficiency: PR gains become larger as channel angular spread decreases, particularly when pilot interference or pilot overhead dominates.The high-SNR regime is pilot-interference dominated, whereas the small-coherence-block regime is pilot-overhead dominated.

APPENDIX A PROOF OF LEMMA 1

The appendix analyzes the MMSE-SD optimization, establishes existence of a global optimum, and identifies an optimal precoder through KKT-based reasoning. The objective is non-convex in (B, α), but compactness and continuity ensure a global minimum exists.

  • The robust receiver and precoder designs use MSE-SD objectives derived from channel-fading and pilot-noise statistics.
  • The simplified objective function is non-convex with respect to (B, α).
  • A global optimal solution exists for problem (33).The feasible set is compact and the objective is continuous over it.
  • The global optimum is sought among solutions satisfying the KKT necessary conditions.
  • The precoder given by (75) is optimal because its MSE-SD is smaller than the value K from the zero solution.

APPENDIX E PROOF OF LEMMA 2

The proof invokes matrix-valued Jensen’s inequality and then applies identities from the preceding development to obtain the stated result.

  • The proof invokes matrix-valued Jensen’s inequality.
  • The resulting relation uses equations (14) and (13).
  • The derivation concludes the proof of the lemma.

APPENDIX F PROOF OF THEOREM 4

The theorem proof uses the Schwartz inequality and covariance-matrix orthogonality to characterize when the matrix Ω is diagonal.

  • The Schwartz inequality gives a bound whose equality holds if and only if Ω is diagonal.
  • For users with i ≠ j and π_i = π_j, R_iR_j = 0 is used to show the off-diagonal elements of Ω vanish.
  • Under the same orthogonality condition, the diagonal elements of Ω are obtained using equation (56).
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