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Pilot Contamination and Precoding in Multi-Cell TDD Systems

Jubin Jose, Alexei Ashikhmin, Thomas L. Marzetta, Sriram Vishwanath

arXiv:0901.1703v2cs.IT

TL;DR

The paper addresses how non-orthogonal uplink training corrupts CSI and precoding in multi-cell TDD systems. It characterizes pilot contamination mathematically and develops a multi-cell MMSE-based precoder, with reported gains over certain single-cell methods.

  • Problem

    Non-orthogonal uplink training can corrupt a base station’s channel estimate through channels from users in other cells, limiting reliable precoding in multi-cell TDD systems.

  • Method

    The paper derives closed-form achievable-rate expressions and develops a multi-cell MMSE-based precoding method that depends on the assigned training sequences.

  • Results

    The analysis shows that achievable rates can saturate with the number of base-station antennas, while the proposed precoding reduces intra-cell and inter-cell interference.

  • Takeaways & Limitations

    The proposed method provides a closed-form linear precoding approach for mitigating pilot contamination without requiring the coordination used by joint-cell precoding techniques.

Abstract

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This paper considers a multi-cell multiple antenna system with precoding used at the base stations for downlink transmission. For precoding at the base stations, channel state information (CSI) is essential at the base stations. A popular technique for obtaining this CSI in time division duplex (TDD) systems is uplink training by utilizing the reciprocity of the wireless medium. This paper mathematically characterizes the impact that uplink training has on the performance of such multi-cell multiple antenna systems. When non-orthogonal training sequences are used for uplink training, the paper shows that the precoding matrix used by the base station in one cell becomes corrupted by the channel between that base station and the users in other cells in an undesirable manner. This paper analyzes this fundamental problem of pilot contamination in multi-cell systems. Furthermore, it develops a new multi-cell MMSE-based precoding method that mitigate this problem. In addition to being a linear precoding method, this precoding method has a simple closed-form expression that results from an intuitive optimization problem formulation. Numerical results show significant performance gains compared to certain popular single-cell precoding methods.

I. INTRODUCTION

The paper studies CSI acquisition and precoding in multi-cell MIMO TDD systems, where non-orthogonal uplink pilots create pilot contamination. It characterizes the resulting performance impact and develops a multi-cell MMSE-based mitigation method.

  • Motivation: TDD systems obtain downlink CSI through uplink training and channel reciprocity, avoiding explicit feedback.The paper identifies CSI acquisition and subsequent precoding design as central goals for multi-cell MIMO TDD systems.
  • Motivation: Orthogonal pilots require at least K × L symbols, so large multi-cell systems and short coherence times necessitate non-orthogonal pilots.Mobility limits the available training duration, making long orthogonal sequences infeasible.
  • Pilot contamination: Non-orthogonal training causes pilot contamination: a base station’s channel estimate becomes polluted by users in other cells.Consequently, the assumption that a precoder is uncorrelated with out-of-cell channels is invalid.
  • Analysis: The paper derives closed-form achievable-rate expressions to quantify how pilot contamination affects multi-cell performance.The analysis shows that achievable rates can saturate as the number of base-station antennas M increases.
  • Mitigation: It develops a multi-cell MMSE-based linear precoder that depends on assigned training sequences and does not require base-station coordination.The method has a simple closed-form expression from an intuitive optimization formulation and reduces intra-cell and inter-cell interference.
  • Evaluation: Numerical results show significant gains over certain popular single-cell precoding methods, including zero-forcing precoding.The proposed method is designed for multiple users in every cell.

A. Related Work

Prior work largely studies single-cell CSI feedback, training, estimation error, and precoding, often under perfect or otherwise favorable CSI assumptions. This paper focuses on the less-studied multi-cell TDD setting, demonstrates pilot contamination, and proposes a mitigation method.

  • Prior assumptions: Existing precoding and low-complexity methods often assume perfect CSI at the base station and users.This assumption differs from the channel-training setting examined here.
  • Single-cell FDD work: Single-cell FDD research studies limited feedback, quantization, beamforming, scheduling, and feedback scaling for MIMO downlink transmission.One cited result requires feedback per user to grow linearly with SNR in dB to obtain the full MIMO BC multiplexing gain.
  • Single-cell TDD work: Single-cell TDD research accounts for training and estimation error in achievable rates and studies scheduling and linear MMSE alternatives to zero forcing.These works include lower bounds on sum capacity and heterogeneous-user settings.
  • This paper: This paper’s contribution is to understand pilot contamination in multi-cell MIMO TDD systems and develop a new precoding method to mitigate it.The paper emphasizes that multi-cell TDD systems have been comparatively poorly studied relative to FDD systems.

B. Organization

The paper organizes its analysis around a multi-cell TDD system model, communication scheme, pilot contamination analysis, precoding method, numerical results, and conclusions. The model assumes reciprocal, block-fading channels and OFDM sub-bands, while omitting detailed OFDM features.

  • Organization: The paper proceeds from the system model and communication scheme to pilot-contamination analysis, a new precoding method, numerical results, and concluding remarks.
  • System model: The system contains L cells, each with one M-antenna base station and K single-antenna users, with average powers p_f and p_r at base stations and users.
  • System model: Path-loss and shadowing are represented by slowly varying β values, so specific cell-layout and shadowing details are irrelevant to the paper's abstraction.
  • System model: The channel model uses reciprocal forward and reverse links, block fading over T symbols, independent Gaussian fading variables, and additive i.i.d. CN (0, 1) noise.
  • Model scope: The model does not directly incorporate frequency-selective fading; OFDM permits applying it per sub-band, while cyclic-prefix details are omitted.

III. COMMUNICATION SCHEME

The communication scheme has separate uplink-training and downlink-data phases. Base stations estimate channels from user pilots and use transmit precoding to send data at achievable rates.

  • Communication phases: The communication scheme consists of uplink training followed by data transmission.
  • Uplink training: During uplink training, users transmit training pilots and base stations obtain channel estimates.
  • Data transmission: During data transmission, base stations transmit data to users through transmit precoding.
  • Performance measure: The paper provides achievable data rates for a given precoding method.

A. Uplink Training

At each coherence interval, users across all cells transmit training sequences, and each base station uses its received pilot signal to estimate channels.

  • Pilot transmission: At the beginning of every coherence interval, all users in all cells transmit training sequences.
  • Received training signal: The l-th base station receives a τ-length pilot vector at each antenna, with the received-signal model defined using pilot, channel, and noise quantities.
  • Channel representation: D_jl is a diagonal matrix containing the large-scale channel coefficients β_jl1 through β_jlK.
  • Channel representation: The channel matrix is formed from the small-scale fading coefficients h_jlkm for the users and antennas associated with the link.
  • Channel estimation: The base station computes the MMSE estimate of each channel from its received signal and collects estimates for all users in the matrix Ĥ_l.

B. Downlink Transmission

In downlink transmission, each base station maps estimated CSI and user symbols into a linear precoded signal. The received user signals include the intended transmission, interference, and noise, while uplink pilot contamination persists across settings using non-orthogonal pilots.

  • Precoding: The l-th base station forms its M × K linear precoding matrix as A_l = f(Ĥ_l), where f specifies the chosen precoding method.
  • Precoded transmission: The transmitted signal is A_l q_l, where q_l contains the information symbols for users in cell l and the power constraint is imposed through E[q_l q_l†].
  • Received signal: Users in the j-th cell receive a noisy signal vector generated by the downlink transmission from the base stations.
  • Received signal: The received signal of the k-th user is expressed using the precoding columns and additive noise, with z_jk denoting the corresponding noise element.
  • Method scope: The considered precoding is linear for low online complexity, excludes nonlinear precoding and forward-link training, and leaves users without channel knowledge.
  • Pilot contamination: Pilot contamination from uplink training with non-orthogonal pilots is present in all the considered settings.

C. Achievable Rates

The paper derives achievable rates for linear precoding by modeling effective noise as worst-case uncorrelated Gaussian noise. The resulting rate expression depends on the precoding method through moments of the effective channel and interference.

  • Achievable rates are obtained by rewriting the received signal as an effective channel plus uncorrelated noise.The analysis uses a worst-case Gaussian noise argument with the same variance.
  • The achievable-rate set is expressed using C(θ) = log2(1 + θ).
  • The rate expression applies to any linear precoding method and varies through effective-channel and interference moments.

IV. PILOT CONTAMINATION ANALYSIS

The pilot-contamination analysis characterizes how non-orthogonal training couples precoding with channels to users in other cells. For one user per cell and a common training sequence, it derives exact achievable-rate expressions and shows that performance can saturate as antennas increase.

  • Pilot contamination arises from correlation between a base station’s precoding vector and channels to users in other cells.The setting captures the primary effect of pilot contamination in the analyzed system.
  • For one user per cell using the same training sequence, the paper derives a closed-form achievable-rate expression.Theorem 3 provides the expression for downlink transmission, and it is exact for any antenna count M.
  • Performance saturates with M because interference grows like the intended signal as the number of antennas increases.
  • Pilot contamination can be very significant when cross-gains are of the same order as direct gains.The analysis suggests frequency/time and pilot reuse techniques to reduce relative cross-gains.
  • The exact rate formula can guide frequency/time reuse choices for a given antenna count and other system parameters.

V. MULTI-CELL MMSE-BASED PRECODING

The paper develops multi-cell MMSE-based precoding by incorporating training-sequence allocation and inter-cell interference into precoder design. It formulates an intuitive optimization problem and obtains a closed-form solution.

  • The optimization is decentralized because different base stations receive different training signals.
  • Single-cell precoding methods do not account for training-sequence allocation, which can increase inter-cell interference under pilot contamination.
  • The multi-cell MMSE objective balances intended-cell errors against interference imposed on users in other cells.Parameter γ controls the relative weights of these two terms.
  • The paper derives a closed-form expression for the optimal multi-cell MMSE-based precoding matrix.
  • The proposed precoding primarily targets maximizing the minimum rate across users, while sum-rate use can incorporate power control and scheduling.The numerical comparisons in this paper omit power control.

VI. NUMERICAL RESULTS

The numerical study evaluates zero-forcing, GPS, and multi-cell MMSE precoding in a four-cell system with reused training sequences. Multi-cell MMSE shows significant advantages across a range of cross-gain settings and over popular single-cell methods.

  • The experiment uses L = 4 cells, M = 8 antennas, K = 2 users per cell, and training length τ = 4.
  • Training sequences are orthogonal within the first two cells and reused respectively in the third and fourth cells.
  • Multi-cell MMSE precoding provides a significant advantage over zero-forcing across a wide range of a and b values.
  • Figure 3 compares zero-forcing and multi-cell MMSE precoding for different cross-gain values a and b.The comparison uses the minimum rate achieved by all users as the performance metric.
  • The numerical results show significant performance gains over popular single-cell precoding methods.Figure 4 compares GPS and multi-cell MMSE as a function of antenna count M.

VII. CONCLUSION

The paper characterizes pilot contamination in multi-cell TDD systems and develops a distributed multi-cell MMSE precoder that accounts for training sequences. Analytical and numerical results show rate saturation under contamination and improved performance over popular single-cell methods.

  • Pilot contamination correlates a base station’s precoding matrix with channels to users in other cells when non-orthogonal pilots are used.This corrupts the channel estimates used for precoding.
  • Achievable rates saturate with the number of base station antennas in the presence of pilot contamination.The paper concludes that appropriate frequency/time reuse techniques are needed to overcome this saturation effect.
  • The proposed multi-cell MMSE precoder depends on the training sequences assigned to users and is obtained from an optimization problem.Its objective combines same-cell signal mean-square error with interference caused at users in other cells.
  • The proposed method reduces both intra-cell and inter-cell interference, similarly to joint-cell precoding techniques.Unlike joint-cell methods, the approach is distributed and explicitly accounts for the assigned training sequences.
  • Numerical results show that the proposed method outperforms popular single-cell precoding methods, including zero-forcing precoding.
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