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Robust Beamforming for Security in MIMO Wiretap Channels with Imperfect CSI

Amitav Mukherjee, A. Lee Swindlehurst

arXiv:1009.2274v1cs.IT

TL;DR

The paper addresses secure MIMO transmission when eavesdropper CSI is unavailable and transmitter CSI is imperfect. It allocates power to meet Bob’s SINR target and uses remaining power for artificial interference, with robust beamforming recovering a substantial portion of perfect-CSI performance.

  • Problem

    The paper asks how to reduce interception risk without eavesdropper CSI while maintaining a prescribed SINR at the desired receiver.

  • Method

    It uses single-stream beamforming, power minimization for the desired receiver’s SINR target, artificial interference, and CSI-error statistics with second-order perturbation analysis.

  • Results

    Robust beamforming restores Bob’s SINR near its desired value and provides non-zero secrecy capacity across all tested QoS values, recovering a reasonable fraction of perfect-CSI performance.

  • Takeaways & Limitations

    Allocating only the power needed for the desired QoS leaves more resources for jamming unknown passive eavesdroppers, while robust designs mitigate CSI-error degradation.

Abstract

from arXiv · show

In this paper, we investigate methods for reducing the likelihood that a message transmitted between two multiantenna nodes is intercepted by an undetected eavesdropper. In particular, we focus on the judicious transmission of artificial interference to mask the desired signal at the time it is broadcast. Unlike previous work that assumes some prior knowledge of the eavesdropper's channel and focuses on maximizing secrecy capacity, we consider the case where no information regarding the eavesdropper is available, and we use signal-to-interference-plus-noise-ratio (SINR) as our performance metric. Specifically, we focus on the problem of maximizing the amount of power available to broadcast a jamming signal intended to hide the desired signal from a potential eavesdropper, while maintaining a prespecified SINR at the desired receiver. The jamming signal is designed to be orthogonal to the information signal when it reaches the desired receiver, assuming both the receiver and the eavesdropper employ optimal beamformers and possess exact channel state information (CSI). In practice, the assumption of perfect CSI at the transmitter is often difficult to justify. Therefore, we also study the resulting performance degradation due to the presence of imperfect CSI, and we present robust beamforming schemes that recover a large fraction of the performance in the perfect CSI case. Numerical simulations verify our analytical performance predictions, and illustrate the benefit of the robust beamforming schemes.

I. INTRODUCTION

Wireless broadcast transmissions are vulnerable to passive interception, motivating physical-layer security beyond encryption. This paper studies artificial-interference beamforming when eavesdropper CSI is unavailable and primary-channel CSI may be imperfect.

  • Wireless broadcasts allow passive eavesdroppers to obtain transmitted information without detection, while encryption introduces computational and key-management challenges.
  • Wiretap-channel research developed secrecy capacity, but MIMO secrecy-rate analysis generally requires the eavesdropper’s CSI or its distribution.
  • The paper instead minimizes transmit power needed to guarantee desired-receiver QoS and uses the remainder for artificial interference against unknown eavesdroppers.
  • Imperfect CSI can leak artificial interference into the desired receiver and make eavesdropper-CSI-based techniques highly sensitive to small perturbations.
  • The study uses single-stream beamforming and received SINR as the desired receiver’s QoS metric, rather than relying solely on secrecy capacity.

A. Artificial Interference

The paper models artificial interference as a spatial noise-like signal that protects a data stream while preserving the desired receiver’s QoS. It contrasts SINR-based evaluation with secrecy-capacity approaches that require information about the eavesdropper.

  • Transmission model: Artificial interference splits Alice’s transmission into a scalar data stream z and an Na × 1 jamming vector z′.The data and jamming components are controlled through the interference covariance, information beamformer, and power fraction ρ.
  • Scope: A complete secrecy strategy also requires a secrecy codebook with secret and randomization subcodebooks, which this paper does not jointly design.The beamforming methods represent a spatial analogue of assigning secret and random messages to separate precoders.
  • Performance metrics: Secrecy-capacity optimization generally requires the eavesdropper’s channel or its distribution, which may be unavailable for passive eavesdroppers.Distribution-based optimization can still require Eve’s antenna count and relative channel strength.
  • Performance metrics: Without information about Eve’s channel, the secrecy-capacity maximization problem is ill-posed, so the paper uses SINR for a single data stream.Linear-beamforming SINR indicates the relative ability of Bob and Eve to determine the transmitted signal.

III. FIXED-SINR BEAMFORMING WITH PERFECT CSI

With perfect CSI, the proposed fixed-SINR strategy meets Bob’s target while allocating remaining transmit power to artificial interference that is orthogonal at Bob. Its jamming directions use the channel’s spatial structure without requiring Eve’s CSI.

  • Unknown Eavesdropper CSI: The strategy first specifies Bob’s target SINR, minimizes the information-power fraction ρ, and assigns the remaining power to orthogonal jamming.If the desired link cannot support the target with total power P, it is treated as an outage.
  • Unknown Eavesdropper CSI: Alice uses the right singular vector of Hba with largest singular value, while Bob uses wb = Hbat to minimize power for target SINR S.The largest singular value is denoted by σ1.
  • Unknown Eavesdropper CSI: The jamming signal uniformly spreads remaining transmit power across the Na −1 right singular vectors associated with Hba’s smallest singular values.These directions produce no interference at Bob under the perfect-CSI design.
  • Unknown Eavesdropper CSI: Bob’s optimal receive beamformer is the maximal ratio combiner because the orthogonal jamming leaves him with white noise.Eve’s optimal beamformer presumes knowledge of Alice’s information beamformer and interference covariance.
  • Unknown Eavesdropper CSI: When Eve’s interference channel is full rank, typically with Alice having more antennas than Eve, the resulting Eve SINR follows the full-rank case; rank deficiency arises, for example, when Eve has more antennas than Alice.The paper distinguishes these cases analytically.

RRH ,

The paper compares unknown-Eve-CSI artificial interference with a full-CSI strategy based on generalized eigenvectors. It also notes a regime where Eve’s SINR remains nonzero despite vanishing Bob noise.

  • Unknown Eavesdropper CSI: R is an orthonormal basis for the subspace orthogonal to HeaQ′z.
  • Unknown Eavesdropper CSI: As σb approaches zero while σb/σe remains approximately O(1), Eve’s SINR generally remains nonzero.
  • Known Eavesdropper CSI: With perfect Eve CSI, the optimal comparison strategy uses full transmit power and minimizes Eve’s SINR subject to Bob’s SINR equaling S.
  • Known Eavesdropper CSI: The constrained known-CSI solution is the generalized eigenvector associated with the largest generalized eigenvalue, scaled to satisfy Bob’s SINR when power is sufficient.When Ne < Na, the beamformer lies in Eve’s channel nullspace and Eve’s SINR is zero; the smallest generalized eigenvalue is then numerically preferable.

IV. IMPACT OF IMPERFECT CSI

The paper models CSI uncertainty at Alice and analyzes how mismatched beamformers and power allocation degrade Bob’s SINR. A second-order SVD perturbation analysis expresses this degradation using CSI-error statistics and motivates robust designs.

  • Perfect transmitter CSI is impractical because of estimation error, quantized feedback, and channel mobility.
  • The CSI error is modeled as a zero-mean random matrix with a specified covariance, and the analysis uses SVD perturbations of Hba.The derivation assumes a fat or square channel matrix; for Nb > Na, the transpose and left singular vectors are used instead.
  • The naive scheme combines erroneous data/artificial-noise power allocation, mismatched artificial-noise covariance, and mismatched transmit and receive beamformers.Bob’s fixed receive beamformer no longer cancels the artificial interference, producing the bulk of the SINR degradation.
  • A second-order perturbation analysis yields an approximate average SINR for Bob based on expected received signal, noise, and interference powers.The approximation is valid to the order assumed by the perturbation analysis and is evaluated in simulations.
  • The analytical SINR expression reduces to the perfect-CSI result when the singular-vector and singular-value perturbations vanish.
  • For i.i.d. CSI errors, the covariance-dependent expressions simplify because G equals σ2I.

V. ROBUST BEAMFORMING APPROACHES

The robust beamforming approaches use CSI-error statistics to estimate artificial-interference effects and compensate for SINR degradation. The paper considers separate FDD and TDD information-sharing scenarios.

  • Knowledge of the CSI-error covariance lets Bob calculate artificial-interference effects and incorporate them into a maximum-SINR receive beamformer.
  • The FDD case gives Bob awareness of Alice’s quantized CSI, whereas in TDD Alice and Bob independently estimate the channel and lack each other’s exact CSI.

A. Robust Beamforming - FDD Case

In the FDD case, Bob knows the CSI information fed back to Alice and can reconstruct Alice’s interference covariance and information beamformer. He then computes the receive beamformer and resulting SINR accordingly.

  • With a mismatched transmit beamformer, Bob’s received interference-plus-noise component must be computed from the imperfect-CSI signal model.
  • Bob can determine the exact artificial-interference covariance Qint because he knows the CSI perturbation value fed back to Alice.
  • Bob can also determine Alice’s exact information beamformer and use it to calculate the optimal receive beamformer that maximizes SINR.
  • The resulting performance metric is Bob’s SINR under the reconstructed FDD transmit and receive beamformers.

B. Robust Beamforming - TDD Case

In the TDD case, Bob lacks Alice’s exact CSI-dependent transmission parameters, so he estimates the interference covariance and transmit beamformer from CSI-error statistics. The resulting SINR accounts for covariance mismatch.

  • Bob does not know the exact artificial-interference covariance or transmit beamformer used by Alice in the TDD case.
  • Using the CSI-error covariance and second-order perturbation analysis, Bob computes expected interference covariance and estimates Alice’s transmit beamformer.
  • Bob calculates his receive beamformer from the estimated interference-plus-noise covariance matrix.
  • Because the estimated and actual interference covariances differ, Bob’s resulting SINR must be evaluated using the TDD mismatch.

VI. SIMULATION RESULTS

The simulations evaluate SINR performance across antenna configurations and CSI assumptions, including unknown, perfect, and perturbed eavesdropper CSI. They show that perfect eavesdropper CSI can substantially help when Eve has fewer antennas, but much of that benefit disappears under CSI perturbations.

  • Simulation setup: The simulations average 3000 independent trials using unit-variance Gaussian channel matrices, equal Bob and Eve noise power, and transmit power P = 100 (20dB).When Bob’s target SINR cannot be met, all power is assigned to Bob and none to artificial interference.
  • Simulation setup: Figure 1 compares Eve’s SINR as her antenna count varies from 1 to 20 for Alice and Bob arrays of 4 or 8 antennas, with Bob’s target SINR set to 20dB.The available transmit power was sufficient to meet Bob’s target in all 3000 trials for this simulation.
  • Eavesdropper CSI conditions: The comparison includes unknown ECSI using artificial noise, perfect ECSI using a generalized eigenvector approach, and imperfect ECSI using the same approach without accounting for perturbations.The imperfect-ECSI channel was generated with γ = 0.05, corresponding to a perturbation of about -13dB.
  • Effects of eavesdropper CSI: For Ne < {Na, Nb}, perfect ECSI can theoretically drive Eve’s SINR to zero, whereas its gain is smaller when Ne ≥ {Na, Nb}.For Na = Nb = 4, the gain when Ne ≥ {Na, Nb} is less than 2dB.
  • Effects of eavesdropper CSI: Even relatively small ECSI perturbations can erase much of the perfect-CSI benefit, making it preferable to ignore ECSI for small Ne rather than use its perturbed version.The simulations therefore show strong sensitivity to inaccurate eavesdropper-channel information.

B. SINR Degradation Analysis

Imperfect CSI substantially degrades SINR, while robust beamforming restores Bob’s performance and preserves positive secrecy capacity across the evaluated desired-SINR range.

  • SINR approximation accuracy: Second-order SINR approximations remain accurate up to about σH = −10dB for both 2- and 5-antenna configurations.The simulations compare naive perturbation approximations with Monte Carlo SINR results.
  • CSI-induced degradation: 6dB of SINR is lost by Bob at σH = 0.1 when Na = 10, demonstrating strong sensitivity to CSI perturbations.The loss occurs despite the perturbation being relatively small.
  • Robust beamforming results: At σH = −10dB, naive schemes place Bob 15-17dB below target and 6-7dB below Eve, whereas robust receive beamforming restores Bob near target.The evaluation uses Na = Nb = Ne = 5 antennas and compares measured SINR for Bob and Eve.
  • Power-allocation trade-off: Robust beamforming increases Eve’s SINR because uncancelled artificial interference requires additional desired-signal power, leaving less jamming power available.Eve’s performance is most degraded in the FDD case, where Bob knows Alice’s transmission scheme exactly.
  • Secrecy capacity: Robust strategies provide non-zero secrecy capacity for all desired-SINR values and recover a reasonable fraction of the perfect-Eve-CSI performance.In the naive case, secrecy capacity is zero because Eve’s SINR always exceeds Bob’s under the stated receiver assumptions.
  • Sensitivity to CSI errors: FDD robust beamforming loses little performance up to σH = -15dB, while TDD degrades at a somewhat lower perturbation threshold.At σH = −10dB, positive secrecy capacity can remain even when average Bob and Eve SINR values are approximately equal.
  • Overall findings: The proposed schemes perform well for moderate CSI errors, but sufficiently large channel mismatch can eliminate the secrecy advantage of artificial noise.The paper validates the analytical degradation analysis with simulations and reports recovery of a large fraction of lost SINR.

APPENDIX

The appendix develops second-order perturbation expressions for singular values and beamforming-related quantities, then rewrites the required expectations using CSI-error covariance statistics.

  • Perturbation expansion: The appendix approximates the perturbation in Vs up to second order in ∆Hba.The derivation uses results from prior perturbation analysis.
  • Derivation simplification: Circular symmetry of ∆Hba is used to simplify the resulting perturbation expressions.The simplification is applied after expressing the relevant terms through the covariance statistics.
  • Singular-value analysis: The perturbation of the singular values Σs is likewise approximated through second-order terms.The expressions introduce intermediate matrices and perturbation quantities such as P2.
  • CSI-error statistics: Expected values needed by the SINR analysis are expressed using second-order statistics of ∆Hba.The covariance Cij represents the covariance of the ith and jth columns of ∆Hba.
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