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Jamming-Resistant Receivers for the Massive MIMO Uplink

Tan Tai Do, Emil Björnson, Erik G. Larsson, S. Mohammad Razavizadeh

arXiv:1702.08800v1cs.ITmath.IT

TL;DR

Jamming during pilot transmission can contaminate massive MIMO channel estimates and impair uplink reception. The paper estimates the jammer channel using an unused pilot and builds jamming-resistant receivers from channel estimates. It reports that the receivers reduce jamming effects and improve performance, including under stronger jamming powers.

  • Problem

    Pilot-phase jamming contamination can correlate the legitimate-channel estimate with the jammer and degrade massive MIMO uplink performance.

    Source: A. Channel Estimation · p. 8

  • Method

    The scheme estimates the jamming channel using an unused pilot and uses legitimate- and jamming-channel estimates to construct receive filters.

    Source: §I · p. 3, §VII · p. 26

  • Results

    The proposed filters can greatly reduce jamming effects and improve system performance, and the scheme works well or even better as jamming powers increase.

    Source: §VII · p. 27

  • Takeaways & Limitations

    A ZF-type filter focused on nulling the jammer is favorable for massive MIMO with jamming, and judicious power allocation can substantially improve the proposed receivers.

    Source: §VII · p. 27

Abstract

from arXiv · show

We design a jamming-resistant receiver scheme to enhance the robustness of a massive MIMO uplink system against jamming. We assume that a jammer attacks the system both in the pilot and data transmission phases. The key feature of the proposed scheme is that, in the pilot phase, we estimate not only the legitimate channel, but also the jamming channel by exploiting a purposely unused pilot sequence. The jamming channel estimate is used to constructed linear receive filters that reject the impact of the jamming signal. The performance of the proposed scheme is analytically evaluated using asymptotic properties of massive MIMO. The optimal regularized zero-forcing receiver and the optimal power allocation are also studied. Numerical results are provided to verify our analysis and show that the proposed scheme greatly improves the achievable rates, as compared to conventional receivers. Interestingly, the proposed scheme works particularly well under strong jamming attacks, since the improved estimate of the jamming channel outweighs the extra jamming power.

I. INTRODUCTION

Jamming pilot contamination can make massive MIMO uplink receivers amplify interference, motivating filters that use estimates of both the legitimate and jamming channels. The paper frames this approach in a system where a jammer attacks during both pilot and data transmission.

I. INTRODUCTION

  • The paper proposes jamming-resistant receivers that use legitimate- and jamming-channel estimates to address pilot contamination, a challenge that can significantly reduce achievable rates.The study also analyzes RZF regularization and user power allocation.Source: §I · pp. 2–3

II. PROBLEM SETUP

  • The setup is a single-user uplink with a multi-antenna base station and single-antenna user and jammer, and the jammer attacks in both transmission phases.The paper notes that its methodology can be generalized to multiple legitimate users.Source: §II · pp. 4–5

A. Pilot Phase

  • During the pilot phase, the user sends a pilot while the jammer chooses a sequence without knowing the user's current pilot.The system preserves at least one unused pilot sequence orthogonal to the user's pilot.Source: A. Pilot Phase · p. 5

B. Data Transmission Phase

  • During data transmission, the base station applies a linear receive filter to the received signal, which contains the user's payload and the jammer's interference.The filter is selected to amplify the desired signal while mitigating jamming.Source: B. Data Transmission Phase · pp. 5–7

III. CHANNEL ESTIMATION AND JAMMING-RESISTANT RECEIVER DESIGN

The receiver design uses estimates of both channels because jamming pilot contamination correlates the legitimate-channel estimate with the jammer. An unused pilot can isolate the jammer's signal for estimating its channel.

III. CHANNEL ESTIMATION AND JAMMING-RESISTANT RECEIVER DESIGN

  • The proposed receiver is built from estimates of both the legitimate and jamming channels, rather than relying only on the legitimate-channel estimate.This addresses the correlation between the legitimate-channel estimate and jammer that can make MRC amplify jamming.Source: §III · p. 7, A. Channel Estimation · p. 8

A. Channel Estimation

  • The legitimate-channel estimate is obtained from the received pilot signal correlated with the user's pilot, using an LMMSE estimate when the pilot overlap is unknown but its distribution is known.The BS needs the jammer's pilot-phase power times its large-scale fading, p_jβ_j, for this estimate.Source: A. Channel Estimation · pp. 7–8
  • Projecting the received pilot signal onto an unused pilot orthogonal to the user's pilot eliminates the user pilot and leaves the jammer signal and noise.The resulting observation is used to estimate the jamming channel.Source: A. Channel Estimation · p. 8
  • The paper uses one randomly selected unused pilot to estimate the jamming channel and leaves potential improvements to pilot selection or combining for future work.With multiple unused pilots, selection or combining could improve the estimate.Source: A. Channel Estimation · p. 9

B. Jamming-Resistant Receiver

The proposed RZF receiver uses estimated legitimate and jamming channels, with regularization controlling the trade-off between desired-signal amplification and residual interference. The MMSE-type filter has an RZF form but is not the true MMSE filter under the stated jamming contamination.

B. Jamming-Resistant Receiver

1) MMSE-type receiver:

  • The MMSE-type filter corresponds to RZF with µ = σ²_e/q_j, but correlated equivalent noise from pilot contamination means it is not the true MMSE filter and may not be conventionally optimal.The paper calls it “MMSE-type” for this reason.Source: 1) MMSE-type receiver · p. 10

2) ZF-type receiver:

The ZF-type receiver is obtained as the zero-regularization limit of the RZF receiver and focuses on nulling the estimated jamming channel. The performance analysis also considers optimal RZF regularization and extremely strong jamming.

2) ZF-type receiver:

  • As µ →0, the RZF receiver becomes a ZF-type receiver that focuses on nulling the estimated jamming signal.The receiver uses a projection-matrix expression; because channel estimation is imperfect, it nulls the estimated jamming channel but does not generally null the true channel.Source: 2) ZF-type receiver · pp. 10–11

IV. PERFORMANCE ANALYSIS

  • The performance analysis evaluates the proposed receivers, derives the optimal RZF regularization factor, and considers extremely strong jammers.The section’s stated scope includes receiver performance and regularization choice under strong jamming.Source: §IV · p. 11

A. Large-scale Approximations

The section derives large-scale approximations for the effective SINR and examines how antenna count, receiver choice, and jamming-pilot power affect performance. It also establishes asymptotic robustness to inter-user interference under the stated pilot assumptions.

A. Large-scale Approximations

  • The effective SINR of the RZF receiver admits a closed-form large-scale approximation that almost surely converges to its true value as M tends to infinity.The BS can estimate δ1 and δ2 closely when M is large, allowing it to evaluate the RZF effective SINR with high accuracy.Source: A. Large-scale Approximations · pp. 11, 14
  • As M grows, channel-uncertainty and additive-noise power terms scale with M, while desired-signal and jamming terms scale with M^2.The former terms are therefore negligible relative to the desired and jamming terms in the large-M regime.Source: A. Large-scale Approximations · p. 13
  • For large M, the effective SINR with any RZF regularization factor converges to a finite limit independent of µ.The asymptotic expression includes a scaling factor from the jamming-channel estimation scheme, and the achievable rate increases with jamming pilot power above a certain level.Source: A. Large-scale Approximations · p. 15
  • When jamming pilot power exceeds a threshold, its improved jamming-channel estimate outweighs degraded legitimate-channel estimation, so achievable rate increases with that pilot power.The text attributes the benefit to improved jamming-channel estimation as jamming pilot power rises.Source: A. Large-scale Approximations · p. 15
  • With orthogonal user pilots and the stated receiver design, the proposed receiver is robust to inter-user interference as M tends to infinity and can be used in a multiuser setup.The effective SINR gains an additional inter-user interference term in the multiuser case.Source: A. Large-scale Approximations · p. 15

B. Optimal RZF Receiver

For finite antenna counts, the RZF regularization factor affects performance. The analysis finds that the effective SINR is maximized at zero regularization, making the ZF-type receiver optimal among the considered RZF choices.

B. Optimal RZF Receiver

  • At µ →0, the effective SINR reaches its maximum, so the ZF-type receiver is the optimal RZF receiver.The paper states that this receiver outperforms RZF receivers with non-zero µ, including the MMSE-type receiver.Source: B. Optimal RZF Receiver · p. 16

C. Extremely Strong Jamming

Under extremely strong jamming, the analysis considers the case where pilot and data jamming powers grow jointly, alongside cases where only one grows. The proposed ZF-type receiver works well in the joint-growth case because greater pilot power improves the jamming-channel estimate, though jamming remains.

C. Extremely Strong Jamming

  • If jamming pilot power stays constant while jamming data power tends to infinity, the RZF effective SINR converges to zero for every µ.Source: C. Extremely Strong Jamming · p. 17
  • If jamming pilot power tends to infinity while jamming data power stays constant, the effective SINR converges to zero for µ > 0, but to a non-zero finite value for µ = 0.Source: C. Extremely Strong Jamming · p. 17
  • The proposed ZF-type receiver works well under extremely strong jamming when jamming pilot and data powers increase jointly.The improving jamming-channel estimate enables better rejection as pilot power rises, although jamming remains because data power also grows.Source: C. Extremely Strong Jamming · pp. 16–17

V. POWER ALLOCATION

The infinite-antenna allocation maximizes asymptotic effective SINR under the user’s average power constraint. Its solution depends on pilot length and motivates a simple heuristic for finite-antenna systems.

A. Power Allocation for Infinite M

  • For infinite antennas, the user allocates pilot- and data-phase powers to maximize asymptotic effective SINR under an average power constraint.The optimization uses ρasy as its objective.Source: A. Power Allocation for Infinite M · pp. 17–18, §V · p. 17
  • The optimal asymptotic allocation depends only on pilot length τ, and divides the total transmit energy equally between pilot and data phases.The equal division is stated for a very large number of antennas.Source: A. Power Allocation for Infinite M · pp. 18–19
  • The asymptotic allocation is designed for infinite antennas, yet its performance loss is relatively small when used with finite antennas.It can therefore serve as a simple heuristic that does not depend on jammer powers or signal structure.Source: A. Power Allocation for Infinite M · p. 19

B. Power Allocation for Finite M

For finite antennas, the paper optimizes power allocation using a ZF-type effective SINR, which is the optimal RZF receiver. The exact solution requires jamming-sequence information, so the paper also gives a sub-optimal alternative for unknown sequences.

B. Power Allocation for Finite M

  • The finite-antenna allocation is optimized for the ZF-type receiver, identified as the optimal RZF receiver.The allocation maximizes the achievable rate using the ZF-type effective SINR.Source: B. Power Allocation for Finite M · p. 19
  • The exact finite-antenna allocation requires knowledge of the jamming sequence or its correlation with the pilot sequences.Without that knowledge, its achievable rate is an upper bound corresponding to perfect knowledge of the sequence.Source: B. Power Allocation for Finite M · p. 20
  • When the jamming sequence is unknown, the paper proposes a sub-optimal allocation by replacing δ1 and δ2 with their mean values.This allocation is consistent with the asymptotically optimal allocation, but unlike it depends on antenna count and jamming powers.Source: B. Power Allocation for Finite M · p. 21
  • Instantaneous estimates of δ1 and δ2 arrive only after the pilot phase, so they cannot inform an allocation that must be predetermined.The proposed alternative uses mean values instead.Source: B. Power Allocation for Finite M · p. 21

VI. NUMERICAL ANALYSIS

The numerical analysis compares linear receivers and checks analytical rates against simulations. The proposed ZF-type filter outperforms the MMSE-type filter and remains effective under strong jamming.

VI. NUMERICAL ANALYSIS

  • The numerical study averages achievable rates over 10,000 jammer-sequence realizations in a coherence block of 200 symbols.The comparison includes conventional MRC, which does not use the jamming-channel estimate.Source: §VI · pp. 21–22

A. Performance with Different Linear Receivers

  • The ZF-type receive filter outperforms the MMSE-type filter, and the analytical rates are close to simulations and become asymptotically tight as antenna count grows.The comparison uses achievable rates versus BS antenna count at SNR values of 0, 5, and 10 dB.Source: A. Performance with Different Linear… · p. 22, Figure 2 · p. 22
  • The ZF-type receiver performs well under strong jamming, while conventional MRC and MMSE-type achievable rates approach zero as both jamming powers grow.With jamming data power tending to infinity, ZF-type rate converges to a non-zero value when jamming pilot power grows proportionally.Source: Figure 3 · p. 23

B. Impact of Jamming Powers

The proposed ZF-type receiver remains effective under strong jamming, and can perform better as jamming pilot power rises. Power allocation and estimates of the quantities in (31) also support strong achievable rates.

B. Impact of Jamming Powers

  • When jamming data power grows without bound and pilot power grows proportionally, ZF-type rates converge to a non-zero value, unlike conventional MRC and MMSE-type rates, which approach zero.The scenario fixes legitimate user transmit powers at 5 dB and increases both jamming powers together.Source: B. Impact of Jamming Powers · p. 23
  • As jamming pilot power increases in the second scenario with fixed jamming data power, conventional MRC achievable rate decreases, while the proposed scheme, especially ZF-type, still works well under strong jamming attacks.Source: B. Impact of Jamming Powers · p. 23, Figure 4 · p. 24
  • Higher jamming pilot power can improve ZF-type performance when better jamming-channel estimates outweigh degraded legitimate-channel estimates.The proposed filters use estimates of both channels.Source: B. Impact of Jamming Powers · p. 24

C. Impact of Power Allocation

  • Achievable rates peak at particular pilot-to-data power ratios, and the asymptotically optimal ratio is close to the finite-antenna optima.The paper suggests using the asymptotic allocation as a heuristic for large but finite antenna counts.Source: C. Impact of Power Allocation · p. 25, Figure 5 · p. 25
  • The asymptotically optimal allocation substantially increases achievable rate, while the proposed sub-optimal allocation comes close to the upper bound.Fig. 6 compares different allocations using the ZF-type receiver, with conventional MRC included for comparison.Source: C. Impact of Power Allocation · p. 25, Figure 6 · p. 26

VII. CONCLUSION

  • The (31) estimates have small NMSEs that approach zero as M grows, and rates calculated from those estimates closely match rates using actual values.The δ2 estimate is especially good and independent of user pilot power; the authors expect the system can realize the rate in (5) without perfect knowledge of sj.Source: C. Impact of Power Allocation · p. 26, Figure 7 · p. 27

APPENDIX A

The appendix establishes that the RZF effective SINR is maximized as its regularization factor tends to zero. It also examines the behavior under two high-jamming regimes.

APPENDIX A

  • The proof of Corollary 1 rewrites (50) using γj = τpjδ2βj + σ2.It then uses γj ≥ σ2 to show the relevant expression is non-negative.Source: Appendix A · p. 28
  • The proof shows ρRZF ≤ ρZF for µ ≥ 0, so the RZF effective SINR reaches its maximum as µ tends to zero.The argument uses γj = τpjδ2βj + σ2 to establish non-negativity.Source: Appendix A · p. 28

APPENDIX B

B. ρRZF converges to a non-zero finite value when µ = 0 and pj = λqj →∞:

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