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Linear Precoding of Data and Artificial Noise in Secure Massive MIMO Systems

Jun Zhu, Robert Schober, Vijay K. Bhargava

arXiv:1505.00330v2cs.IT

TL;DR

The paper studies secure downlink transmission in multi-cell massive MIMO when the BS lacks eavesdropper CSI and secrecy is affected by pilot contamination and multi-cell interference. It analyzes linear data and artificial-noise precoders, then proposes polynomial alternatives whose simulated performance closely approaches selfish RCI data and null-space AN precoding.

  • Problem

    The paper addresses how to enhance secrecy in multi-cell massive MIMO with unavailable eavesdropper CSI, pilot contamination, and multi-cell interference.

  • Method

    The paper analyzes selfish and collaborative ZF/RCI data precoders and random or null-space AN precoders, and optimizes polynomial precoders using free probability and random matrix theory.

  • Results

    The proposed polynomial data and AN precoders closely approach the performances of selfish RCI data and null-space AN precoders, respectively.

  • Takeaways & Limitations

    The results provide insights for designing secure multi-cell massive MIMO systems while balancing precoding complexity and secrecy performance.

Abstract

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In this paper, we consider secure downlink transmission in a multi-cell massive multiple-input multiple-output (MIMO) system where the numbers of base station (BS) antennas, mobile terminals, and eavesdropper antennas are asymptotically large. The channel state information of the eavesdropper is assumed to be unavailable at the BS and hence, linear precoding of data and artificial noise (AN) are employed for secrecy enhancement. Four different data precoders (i.e., selfish zero-forcing (ZF)/regularized channel inversion (RCI) and collaborative ZF/RCI precoders) and three different AN precoders (i.e., random, selfish/collaborative null-space based precoders) are investigated and the corresponding achievable ergodic secrecy rates are analyzed. Our analysis includes the effects of uplink channel estimation, pilot contamination, multi-cell interference, and path-loss. Furthermore, to strike a balance between complexity and performance, linear precoders that are based on matrix polynomials are proposed for both data and AN precoding. The polynomial coefficients of the data and AN precoders are optimized respectively for minimization of the sum mean squared error of and the AN leakage to the mobile terminals in the cell of interest using tools from free probability and random matrix theory. Our analytical and simulation results provide interesting insights for the design of secure multi-cell massive MIMO systems and reveal that the proposed polynomial data and AN precoders closely approach the performance of selfish RCI data and null-space based AN precoders, respectively.

I. INTRODUCTION

The paper addresses secrecy in multi-cell massive MIMO with unavailable eavesdropper CSI, where pilot contamination and inter-cell interference complicate protection. It compares linear data and AN precoders and proposes polynomial designs to balance secrecy performance and complexity.

  • Motivation: Pilot contamination impairs channel estimates and limits massive MIMO performance, while multi-cell interference hampers protection of Bob and degradation of Eve’s channel.These effects motivate secrecy analysis beyond single-cell settings.
  • Motivation: Passive eavesdroppers prevent Alice and Bob from learning Eve’s CSI, motivating artificial noise and linear precoding for secrecy enhancement.AN is intended to degrade Eve’s channel while limiting impairment to Bob’s channel.
  • Prior limitations: MF data precoding suffers a large information-rate loss as the number of MTs increases, motivating secrecy analysis of ZF and RCI precoders.The paper also notes that ZF and RCI improve performance at higher complexity than MF.
  • Contributions: Closed-form asymptotic ergodic secrecy-rate expressions compare MF, selfish and collaborative ZF/RCI, and random, selfish and collaborative null-space AN precoders.The analysis includes uplink estimation, pilot contamination, multi-cell interference, and path-loss.
  • Contributions: The paper studies selfish and collaborative data and AN precoders, trading local-cell CSI requirements against inter-cell interference and AN leakage.Collaborative designs use CSI to MTs in all cells and incur limited additional overhead and complexity because the extra CSI can be estimated locally.
  • Contributions: Polynomial data and AN precoders are proposed to avoid large-scale matrix inversions, with coefficients obtained using free probability and closely approaching selfish RCI and null-space AN performance.The polynomial designs target lower computational complexity and potential stability issues.

II. SYSTEM MODEL AND PRELIMINARIES

The system model is a multi-cell TDD massive MIMO downlink with single-antenna MTs, passive multi-antenna eavesdroppers, linear data and AN precoding, and uplink channel training. Selfish designs use local-cell CSI, whereas collaborative designs use CSI from all cells to reduce inter-cell interference and AN leakage.

  • System Model: The model contains M cells, each with one N_T-antenna BS, K single-antenna MTs, and potentially an N_E-antenna passive eavesdropper.The downlink uses frequency reuse factor one, so all BSs share the same spectrum.
  • Eavesdropper Model: The eavesdropper is passive and its CSI is unavailable, so each BS generates AN to mask information and prevent eavesdropping.Neither BSs nor MTs are assumed to know which MT the eavesdropper targets.
  • System Model: Each BS transmits data and artificial noise through precoding matrices F_n and A_n, whose efficient design is the paper’s main scope.The AN matrix has rank L≤N_T, representing the signal-space dimensions used to jam the eavesdropper.
  • Precoding Designs: Selfish precoders use only estimated in-cell CSI, whereas collaborative precoders use estimated CSI between the local BS and MTs in all cells.Collaborative designs aim to avoid excessive interference and AN leakage to other cells.
  • Precoding Designs: Collaborative designs require more channel-estimation overhead but may not outperform selfish designs because CSI is imperfect and spatial degrees of freedom are limited.The comparison therefore depends on estimation quality and system loading.
  • Channel Estimation: Uplink training uses mutually orthogonal pilots within each cell and the same pilot sequences across cells, causing pilot contamination.Path-loss matrices are assumed perfectly known at the BS for MMSE small-scale fading estimation.

C. Ergodic Secrecy Rate

The paper evaluates ergodic secrecy rate as the difference between legitimate-user and eavesdropper capacities, using tractable bounds under a pessimistic eavesdropper model. The analysis links secrecy to AN dimensionality and develops low-complexity polynomial precoders for asymptotic comparison.

  • Secrecy-Rate Metric: Ergodic secrecy rate is the adopted metric, bounded using the legitimate MT rate and the eavesdropper’s channel capacity.The MT rate is lower bounded through an SINR expression.
  • Secrecy-Rate Bounds: The eavesdropper is pessimistically assumed to cancel all in-cell and out-of-cell interfering MT signals except the target signal, yielding a secrecy-rate lower bound.This lower bound is achievable if the eavesdropper accesses the interfering users’ data.
  • Asymptotic Analysis: A necessary condition for nonzero secrecy rate is α≤ML/N_T, so larger AN rank L lets the BS tolerate more eavesdropper antennas.The stated condition arises from invertibility of the relevant matrix in the large-system analysis.
  • Asymptotic Analysis: Under approximate orthogonality between data vectors and the AN subspace, the eavesdropper-capacity upper bound depends on AN dimensionality L rather than the exact precoder structures.The legitimate-user rate remains affected by both data and AN precoders.
  • Asymptotic Analysis: For N_T→∞, the tractable secrecy-rate lower bound is reported to be applicable and tight for values of α permitting a non-vanishing secrecy rate.The result supports analytical comparison of the considered precoder combinations.
  • Polynomial Precoding: The paper analyzes existing data and AN designs asymptotically and proposes low-complexity polynomial matrix-expansion precoders.These designs are intended to avoid the large-scale inversions associated with ZF, RCI, and null-space AN precoding.

III. LINEAR DATA PRECODERS FOR SECURE MASSIVE MIMO

This section analyzes selfish and collaborative ZF/RCI data precoders in the asymptotic secure massive MIMO regime, including AN leakage in the SINR. It also introduces polynomial data precoding to avoid matrix inversion while optimizing coefficients through asymptotic MSE minimization.

  • Linear ZF/RCI precoders: The analysis derives received-SINR expressions for selfish and collaborative ZF/RCI data precoders when K and N_T grow with finite β and α.AN leakage is represented in the SINR denominator and is analyzed separately for different AN precoders.
  • Linear ZF/RCI precoders: Selfish precoders use only in-cell CSI, whereas collaborative precoders also require inter-cell CSI to suppress interference to out-of-cell users.Collaborative designs require more antennas: N_T > K for SZF and N_T > MK for CZF.
  • Linear ZF/RCI precoders: The preferred data precoder depends on system parameters because collaborative designs can improve overall performance but require additional CSI and antennas.The paper explicitly leaves the choice between selfish and collaborative precoding dependent on the considered system parameters.
  • Polynomial data precoder: Matrix inversion gives ZF/RCI precoders higher performance than simple MF precoding but imposes high computational complexity for massive dimensions.The complexity concern motivates a low-complexity alternative that avoids matrix inversion.
  • Polynomial data precoder: The proposed selfish POLY data precoder uses optimized real-valued polynomial coefficients that minimize asymptotic average user MSE.The coefficients are constant and do not depend on instantaneous channel estimates; they are obtained using free probability and random matrix theory.

C. Computational Complexity of Data Precoding

This section compares data-precoding complexity over a coherence interval and explains the computational advantages of polynomial precoding. POLY precoding avoids matrix inversion, while collaborative designs incur substantially greater complexity than selfish designs.

  • Complexity accounting: Data-precoding complexity is measured in floating-point operations per coherence interval, with τ training symbols and T − τ data symbols.The interval includes generating one precoding matrix and producing T − τ precoded vectors.
  • Collaborative data precoding: Collaborative data precoders add at most a factor of M^3 in complexity relative to selfish data precoders.The comparison follows from the stated FLOP expressions for selfish and collaborative ZF/RCI designs.
  • Selfish data precoding: The POLY data precoder requires (T −τ) ((I + 1)(2K −1)N_T + I(2N_T −1)K) FLOPs using Horner’s rule.This expression accounts for the overall complexity of polynomial data precoding.
  • Polynomial data precoding: POLY data-precoding savings relative to SZF/SRCI increase with K for a fixed coherence length T.POLY precoders also avoid stability issues that can arise from large matrix inverses in fixed-point implementations.

2) CNS AN Precoder:

This section develops collaborative null-space and polynomial AN precoders, analyzing their leakage, feasibility, performance regime, and complexity. POLY AN precoding minimizes asymptotic average leakage while avoiding the matrix inversion required by null-space designs.

  • CNS AN precoder: The collaborative null-space AN precoder lies in the null space of estimated channels between the BS and all MK mobile terminals.Its rank is L = N_T − MK and it exists only if β < 1/M.
  • CNS AN precoder: CNS AN produces the same PAN as SNS AN while suppressing AN leakage to the estimated channels of users across cells.The equality of PAN is stated explicitly for CNS and SNS AN precoders.
  • AN leakage and performance: When τpτ → 0, S/CNS and random AN precoders have the same qQ and PAN and therefore similar SINR performance for a given mobile terminal.For τpτ > 0, S/CNS precoders cause less AN leakage and improve SINR relative to random AN, at higher complexity.
  • POLY AN precoder: The proposed selfish POLY AN precoder optimizes polynomial coefficients to minimize asymptotic average AN leakage to users in the local cell.The design uses L = N_T − K and is motivated by the complexity of matrix inversion in null-space precoding.
  • POLY AN precoder: Theorem 2 gives the optimal coefficient vector for the AN polynomial structure by minimizing asymptotic average leakage.The optimization is expressed through moment-based quantities including Σ and ω.
  • AN complexity: CNS AN requires 0.5((MK)^2 +MK)(2N_T −1)+(MK)^3 +(MK)^2 +MK +N_T(N_T +MK)(2MK −1)+(2N_T −1)N_T(T −τ) FLOPs.Random AN requires only (2N_T −1)N_T(T −τ) FLOPs because it uses vector-matrix multiplications.

V. COMPARISON OF LINEAR DATA AND AN PRECODERS

This section compares MF, SZF, and CZF data precoding under a simplified path-loss model, focusing on SINR and the effects of loading, pilot energy, AN leakage, cell count, and inter-cell interference. The preferred precoder changes with system conditions rather than being universally fixed.

  • Setup: The comparison evaluates MF, SZF, and CZF data precoders under a simplified path-loss model with inter-cell interference factor ρ.The model yields a = 1 +(M −1)ρ and c = 1 +(M −1)ρ^2.
  • Number of MTs: For lightly loaded systems, β → 0, all three data precoders have similar performance.The comparison is based on received SINR for a fixed AN precoder.
  • Number of MTs: SZF is advantageous over MF and CZF is advantageous over SZF only below respective maximum mobile-terminal thresholds.The thresholds decrease with AN leakage and cell count but increase with channel-estimation resources and quality.
  • Number of MTs: Increasing ρ decreases the SZF-over-MF threshold but increases the CZF-over-SZF threshold.Thus, inter-cell interference affects the two pairwise comparisons in opposite directions under the simplified model.
  • Pilot energy: MF, SZF, and CZF are preferable in successive pilot-energy ranges separated by (p_ττ)_SZF>MF and (p_ττ)_CZF>SZF.As β increases, more pilot energy is required for SZF and CZF to become beneficial; beyond β_MF, MF is always preferable.
  • Pilot energy: Beyond β_SZF, SZF is always preferable to CZF regardless of pilot energy p_ττ.The thresholds depend on the simplified path-loss parameters and inter-cell interference factor.

B. Comparison of SNS, CNS, and MF AN Precoding

The analysis compares how AN precoders affect eavesdropper capacity, mobile-terminal leakage, and secrecy-rate feasibility across system loads. CNS is favored in lightly loaded settings, while SNS or random AN can become preferable as loading increases.

  • Mechanisms: AN precoders affect secrecy through both the eavesdropper-capacity term L and mobile-terminal leakage ˜Q.The eavesdropper-capacity upper bound decreases with L, while mobile-terminal SINRs decrease with ˜Q.
  • Mechanisms: ˜Q_random ≥ ˜Q_SNS ≥ ˜Q_CNS, so lower leakage generally improves the mobile-terminal rate for a fixed data precoder.This ordering is combined with the secrecy-rate expressions to compare AN designs.
  • Lightly loaded networks: The CNS AN precoder is expected to outperform SNS and random AN in lightly loaded networks because it achieves smaller ˜Q.For small β and M, L is approximately N_T for the considered AN precoders, making leakage the decisive difference.
  • Heavily loaded networks: In heavily loaded networks, CNS performance is compromised by its small L, so SNS and even random AN can achieve larger secrecy thresholds.The relevant threshold is α_s, the maximum α admitting a non-zero secrecy rate.
  • Eavesdropper capacity: The eavesdropper capacity decreases as β increases, benefits from larger α, and is largest for CNS AN because the precoders have different L values.The analytical upper bound is reported as tight for N_T = 200 under the considered conditions.

B. Ergodic Secrecy Rate for Conventional Linear Data Precoders

The paper evaluates analytical and simulated secrecy rates for conventional linear data and AN precoders under lightly and densely loaded multi-cell conditions. Collaborative precoding helps in lightly loaded networks, whereas dense interference can favor selfish designs and change the preferred power allocation and AN strategy.

  • Analytical validation: Analytical results provide tight lower bounds for simulated ergodic secrecy rates across the considered conventional precoders.The evaluations use expressions for SRCI, CRCI, MF, SZF, and CZF precoding.
  • Data precoders: RCI data precoders outperform ZF data precoders for both selfish and collaborative strategies, but the gap diminishes as N_T increases.This comparison applies to both selfish and collaborative designs.
  • Lightly loaded network: In the lightly loaded network, collaborative designs outperform selfish designs, while CZF and SZF provide large gains over MF.For N_T = 400, the analysis gives K_SZF>MF ≈250 and K_CZF>SZF ≈60.
  • Power allocation: The secrecy-rate curves versus φ are concave with a single maximum, while φ = 0 or φ = 1 eliminates secrecy by transmitting only AN or no AN, respectively.For 0 < φ < 1, a positive secrecy rate may result depending on system parameters and precoding schemes.
  • Power allocation: The optimal φ increases with β and with data-precoder effectiveness, because heavier loading reduces per-user data power and stronger precoding benefits from more data power.For less effective data precoders, allocating more power to AN is more beneficial.

E. Low-Complexity POLY Data and AN Precoders

The POLY data and AN precoders offer lower-complexity alternatives whose secrecy performance approaches conventional selfish RCI and SNS null-space precoders as polynomial order increases. Their tradeoffs depend on network loading, interference, and the number of users.

  • Simulation setup: Across the evaluated schemes, ergodic secrecy rate increases monotonically with pilot energy because more accurate channel estimates improve performance.The simulations numerically optimize φ for each simulation point.
  • POLY data precoding: Increasing the polynomial order I quickly improves POLY data-precoder secrecy rates toward SRCI performance.Convergence is faster in dense networks, where interference limits the performance differences among data precoders.
  • POLY AN precoding: Increasing the polynomial order J quickly brings POLY AN performance close to SNS AN performance.In dense networks, performance differences among AN precoders are smaller, and random AN incurs only a small loss relative to SNS AN.
  • Complexity-performance tradeoff: Collaborative data and AN precoding provide moderate performance gains over selfish strategies but substantially increase complexity, especially for large K.The figures show secrecy rate and computational complexity as functions of the number of users in a cell.
  • Complexity-performance tradeoff: POLY data with I = 1 outperforms MF while having substantially lower complexity than SRCI in the considered setting.For large I, POLY data also has lower complexity than SRCI for large K and avoids stability issues associated with large-scale matrix inversions.
  • Complexity-performance tradeoff: POLY AN with J = 5 achieves almost the same performance as SNS AN with substantially lower complexity.With small K, Horner’s-scheme implementation can make POLY AN less complex than random AN.
  • Design insights: CNS AN is preferable in lightly loaded systems, whereas SNS AN is preferable in heavily loaded systems because CNS lacks sufficient degrees of freedom against the eavesdropper.For large eavesdropper antenna counts, MF data precoding is reported as preferable to SZF and CZF when only small positive secrecy rates are achievable.
  • Design insights: The proposed POLY data and AN precoders approach SRCI data and SNS AN performance with only a few polynomial terms.The paper presents them as practical options offering a compromise between complexity and performance.

APPENDIX

The appendix derives asymptotic expressions for received-signal, interference, and variance terms used in the SINR analysis. It applies large-system limits, pilot-contamination structure, free-probability results, and matrix decompositions to complete the stated propositions and coefficient optimization.

  • Asymptotic SINR derivation: The appendix decomposes the received SINR into effective signal, intra-cell interference, and inter-cell interference terms.Pilot contamination makes each base station’s data-precoding matrix depend on channel vectors to users in all cells.
  • Asymptotic SINR derivation: As NT tends to infinity with constant β, normalized quantities converge to deterministic limits used to obtain the SINR expression.The derivation invokes cited random-matrix results for convergence of Xnk and the vanishing of Ank.
  • POLY data optimization: Free-probability results and asymptotic trace relations simplify the terms in the POLY data objective and lead to the theorem expression.The proof uses freeness between the relevant matrices and large-system convergence arguments.
  • POLY data optimization: The POLY data-precoder optimization is rewritten using an eigen-decomposition and a Vandermonde matrix containing powers of the eigenvalues.The resulting Lagrangian conditions yield the optimal coefficient vector μopt under the stated constraint.
  • POLY data optimization: Differentiating the data-precoder Lagrangian with respect to μ and the constraint multiplier produces the optimal coefficient conditions.Substitution and simplification of the resulting expressions completes the derivation of Theorem 1.

C. Proof of Theorem 2

The proof of Theorem 2 simplifies the POLY AN objective under its constraint and derives the optimal AN polynomial coefficients through Lagrangian stationarity.

  • POLY AN optimization: The POLY AN objective is simplified using the constraint in (29) and the same asymptotic approach used earlier.The resulting expression is then rewritten before forming the Lagrangian.
  • POLY AN optimization: The optimal coefficient vector νopt is obtained by differentiating the Lagrangian with respect to ν and setting the gradient to zero.Further simplification yields the result stated in Theorem 2.
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