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Directional Modulation via Symbol-Level Precoding: A Way to Enhance Security

Ashkan Kalantari, Mojtaba Soltanalian, Sina Maleki, Symeon Chatzinotas, Björn Ottersten

arXiv:1606.04488v2cs.ITmath.IT

TL;DR

The analysis considers secure transmission against eavesdropping and examines a case with fewer eavesdropper antennas than transmit antennas (Ne < Nt). The paper designs directional-modulation precoders using user-channel and symbol information, with objectives and solution methods tailored to antenna configurations. In most cases, designed precoders impose considerably more SER on the eavesdropper than conventional precoding, and signal-level minimization consumes the same power as the benchmark scheme.

  • Problem

    The analysis considers secure transmission against eavesdropping and examines a case with fewer eavesdropper antennas than transmit antennas (Ne < Nt).

    Source: A. Motivation · p. 1, C. The Case of Strong Eavesdropper (Ne… · p. 8

  • Method

    The paper designs directional-modulation precoders using user-channel and symbol information, with objectives and solution methods tailored to antenna configurations.

    Source: C. The Case of Strong Eavesdropper (Ne… · p. 8, §VII · p. 14

  • Results

    In most cases, designed precoders impose considerably more SER on the eavesdropper than conventional precoding, and signal-level minimization consumes the same power as the benchmark scheme.

    Source: §VI · p. 11, §VII · p. 14

  • Takeaways & Limitations

    The reported results support directional modulation as a security-enhancing approach when the transmitter has more antennas than the eavesdropper, with large-scale arrays noted as a favorable setting.

    Source: §VII · p. 14

Abstract

from arXiv · show

Wireless communication provides a wide coverage at the cost of exposing information to unintended users. As an information-theoretic paradigm, secrecy rate derives bounds for secure transmission when the channel to the eavesdropper is known. However, such bounds are shown to be restrictive in practice and may require exploitation of specialized coding schemes. In this paper, we employ the concept of directional modulation and follow a signal processing approach to enhance the security of multi-user MIMO communication systems when a multi-antenna eavesdropper is present. Enhancing the security is accomplished by increasing the symbol error rate at the eavesdropper. Unlike the information-theoretic secrecy rate paradigm, we assume that the legitimate transmitter is not aware of its channel to the eavesdropper, which is a more realistic assumption. We examine the applicability of MIMO receiving algorithms at the eavesdropper. Using the channel knowledge and the intended symbols for the users, we design security enhancing symbol-level precoders for different transmitter and eavesdropper antenna configurations. We transform each design problem to a linearly constrained quadratic program and propose two solutions, namely the iterative algorithm and one based on non-negative least squares, at each scenario for a computationally-efficient modulation. Simulation results verify the analysis and show that the designed precoders outperform the benchmark scheme in terms of both power efficiency and security enhancement.

I. INTRODUCTION

The paper uses symbol-level directional modulation to enhance security in multi-user MIMO without requiring the transmitter to know the eavesdropper’s channel. It introduces precoder designs and solution methods, then sets out the system model they use.

A. Motivation

  • The secrecy-rate approach can require eavesdropper channel knowledge, lower transmission rates, and Gaussian signaling, motivating an alternative security approach.Directional modulation instead aims to increase the eavesdropper’s symbol error rate without requiring its channel state information.Source: A. Motivation · p. 1, B. Contributions · p. 2

B. Contributions

  • The paper designs symbol-level precoders for arbitrary numbers of users and derives a necessary condition for their existence.The precoder uses user channels and intended symbols to induce symbols on receiver antennas.Source: B. Contributions · p. 2
  • The designs address cases where the eavesdropper has fewer or more antennas than the transmitter, with objectives tailored to each configuration.The paper minimizes transmission power in the former case and maximizes eavesdropper SER in the latter, while keeping users’ SNR above a predefined threshold.Source: B. Contributions · p. 2
  • The precoder problems are formulated as linearly constrained quadratic programs and solved using an iterative algorithm or non-negative least squares.The iterative method uses a penalty approach and is shown to converge to the optimal point.Source: B. Contributions · p. 2

C. Additional Related Works to Directional Modulation

  • Prior directional-modulation work includes antenna switching, array-weight optimization, and artificial-noise methods, while digital symbol-level precoding targets constructive interference.Directional modulation applies array weights in the analog domain; digital symbol-level precoding designs the digital signal.Source: A. Motivation · pp. 1–2, C. Additional Related Works to… · p. 2

D. Organization

  • The paper is organized from system modeling and security analysis through precoder design, complexity analysis, simulations, and conclusions.The system model considers a multi-antenna transmitter, multiple multi-antenna users, and a multi-antenna eavesdropper.Source: D. Organization · p. 3, §II · p. 3

II. SIGNAL AND SYSTEM MODEL

  • The system assumes quasi-static block-fading channels and considers RF generation with either active power-amplifier and phase-shifter chains or parasitic antennas.With parasitic antennas, the far-field channel must be line of sight and its fading-channel CSI must be acquired.Source: §II · p. 3, Figure 2 · p. 3, Figure 3 · p. 3
  • The transmitter knows the users’ channels, while the eavesdropper is assumed to know both user and eavesdropper channels.The received signals are modeled using the transmit precoding vector, the corresponding channels, and additive white Gaussian noise.Source: §II · p. 4

III. SECURITY ANALYSIS OF DIRECTIONAL MODULATION

The security analysis examines whether an eavesdropper can estimate the transmit precoding vector and use it to recover user-side signals. It considers zero-forcing and MMSE approaches under different antenna and channel-knowledge conditions.

III. SECURITY ANALYSIS OF DIRECTIONAL MODULATION

  • The analysis assumes independent eavesdropper and user channels, full-rank eavesdropper channels, and an eavesdropper that knows both channels.Because the precoder depends on the symbols, the eavesdropper cannot directly calculate it and must estimate it to recover user signals.Source: §III · p. 4

A. Zero-Forcing Estimation

  • With zero forcing, the eavesdropper estimates the precoder and maps it through the user channels; it cannot estimate user-side signals when it has fewer antennas than the transmitter.The process can let the eavesdropper view signals from the users’ perspective, but increases its noise.Source: A. Zero-Forcing Estimation · p. 4

B. Minimum Mean-Square Error Estimation

  • MMSE estimation of the precoder requires its covariance matrix, which the eavesdropper can estimate by averaging across random symbol sequences and channel realizations.The method also uses the eavesdropper-noise covariance matrix.Source: B. Minimum Mean-Square Error Estimation · p. 4

C. Successive Interference Cancellation and Sphere Decoding

The paper argues that successive interference cancellation and sphere decoding do not directly fit this eavesdropping problem because the eavesdropper must estimate a continuous-valued precoder rather than finite-alphabet symbols. The analysis also relates eavesdropper SER to the transmitter–eavesdropper antenna-count difference.

C. Successive Interference Cancellation and Sphere Decoding

  • ZF-SIC and MMSE-SIC rely on detecting and canceling finite-alphabet symbols, whereas this eavesdropper must estimate the continuous-valued precoding vector.The same finite-alphabet mismatch is given for sphere decoding.Source: C. Successive Interference… · p. 4
  • The eavesdropper must estimate the precoding vector whether it targets one user’s symbols or all users’ symbols.Source: C. Successive Interference… · p. 5
  • As the transmitter–eavesdropper antenna-count difference grows, the imposed eavesdropper SER increases for both ZF and MMSE estimators.The paper notes that large-scale transmitter arrays make a larger antenna-count difference more probable.Source: C. Successive Interference… · p. 5

D. Brute-force and maximum likelihood Approach

The eavesdropper can recover the precoder by testing possible symbol combinations, but brute-force and maximum-likelihood strategies carry extremely high computational cost that grows exponentially with modulation order and antenna count.

D. Brute-force and maximum likelihood Approach

IV. OPTIMAL PRECODER DESIGN FOR DIRECTIONAL MODULATION

The precoder design treats Ne < Nt and Ne ≥ Nt as separate cases because the eavesdropper’s SER depends on the difference between Nt and Ne.

IV. OPTIMAL PRECODER DESIGN FOR DIRECTIONAL MODULATION

  • The design considers separate precoders for Ne < Nt and Ne ≥ Nt because the eavesdropper’s SER depends on the difference between Nt and Ne.Source: §IV · p. 5
  • For Ne < Nt, the design focuses on energy efficiency.Source: §IV · p. 5
  • For Ne < Nt, the authors also perform relaxed phase analysis.Source: §IV · p. 5

A. The Case of Strong Transmitter (Ne < Nt, Fixed Phase)

For fixed-phase M-PSK, the design minimizes transmit power while keeping each intended received symbol at its target phase and above an SNR threshold. The section gives a necessary antenna-count condition and two approaches for solving the resulting quadratic program.

A. The Case of Strong Transmitter (N_e < N_t, Fixed Phase)

  • The precoder minimizes total transmit power while fixing each received signal's phase to its intended M-PSK symbol and enforcing a minimum SNR.The design uses the uncoded-signal SNR; coded transmission with adaptive coding and modulation is outside the paper's scope.Source: A. The Case of Strong Transmitter (Ne… · pp. 5–6
  • When A H_U1 − H_U2 is full rank, feasibility requires N_t > N_U, or more than half as many transmit antennas as total receiver antennas.The general necessary condition is N_t > r′, where r′ is the rank of A H_U1 − H_U2.Source: A. The Case of Strong Transmitter (Ne… · p. 6
  • The necessary rank condition alone does not guarantee feasibility; the constraints must also have a nonempty intersection.The constraints require the real-imaginary relation for the intended phase and a received-signal threshold.Source: A. The Case of Strong Transmitter (Ne… · p. 6
  • The fixed-phase constraints become a convex linearly constrained quadratic program, and the authors propose iterative and non-negative least-squares solvers to reduce design time.The iterative algorithm monotonically converges to the optimal point; the non-negative least-squares formulation uses real-valued terms.Source: A. The Case of Strong Transmitter (Ne… · p. 7, 1) Iterative solution · p. 7

B. The Case of Strong Transmitter (Ne < Nt, Relaxed Phase)

The relaxed-phase design allows the received phase to vary within the M-PSK detection region rather than fixing it to a line. It uses the same proposed solvers as the fixed-phase formulation.

B. The Case of Strong Transmitter (N_e < N_t, Relaxed Phase)

C. The Case of Strong Eavesdropper (Ne ≥Nt)

When the eavesdropper has at least as many antennas as the transmitter, the design seeks to limit its decoding success by constraining received-signal levels. The section applies the iterative and non-negative least-squares approaches to this design.

C. The Case of Strong Eavesdropper (N_e ≥ N_t)

  • When N_e ≥ N_t, the design minimizes the sum power of the users' received signals, which is also the sum power of the eavesdropper's estimated signals.The stated motivation is that the eavesdropper's received SNR may be lower than the users' and may prevent successful decoding.Source: C. The Case of Strong Eavesdropper (Ne… · pp. 8–9
  • The design fixes each intended received phase and imposes signal-level constraints before converting the problem to a linearly constrained form.Source: C. The Case of Strong Eavesdropper (Ne… · p. 9
  • The section adapts both the iterative algorithm and non-negative least-squares formulation to solve the strong-eavesdropper design.The extension to M-QAM is noted as possible by changing the constraint to equality, but its derivation is outside the paper's scope.Source: C. The Case of Strong Eavesdropper (Ne… · p. 9

D. Benchmark Scheme

The benchmark applies zero forcing at the transmitter to remove inter-stream interference, with the same SNR-related scaling used for comparison. The eavesdropper's recovery options differ between the benchmark and the proposed symbol-dependent precoder.

D. Benchmark Scheme

  • The benchmark uses transmitter zero forcing to remove interference among symbol streams and matches its scaling factor β to the proposed design's √γ for a fair comparison.Source: D. Benchmark Scheme · p. 9
  • When N_e ≥ N_U, the eavesdropper can use zero forcing to derive the benchmark precoder and estimate symbols, but cannot do so for the proposed symbol-dependent precoder.In the benchmark, the eavesdropper can use knowledge of H_U to calculate the precoder.Source: D. Benchmark Scheme · pp. 9–10
  • With the benchmark, the eavesdropper's MMSE symbol-error rate depends on N_e − N_U, whereas the proposed method's depends on N_e − N_t.The authors argue security is more probable with their design because the base station usually has more antennas than the users.Source: D. Benchmark Scheme · p. 10

V. REMARKS ON COMPUTATIONAL COMPLEXITY

The complexity comparison considers non-negative least-squares solvers and the benchmark, with the relative cost depending on how often the channel changes, the number of user antennas, and solution accuracy.

V. REMARKS ON COMPUTATIONAL COMPLEXITY

  • The complexity analysis considers non-negative least squares solved by interior-point and fast projected-gradient algorithms, alongside the benchmark scheme.The excerpt gives benchmark complexity as an expression ending in O(NtNU).Source: §V · p. 10
  • Each designed precoder solves its optimization problem once per group of symbols, so one precoder can communicate NU symbols.The benchmark precoder can be reused while the channel remains fixed.Source: §V · p. 10
  • The comparison depends on channel-change rate, total user antennas, and the required accuracy of the non-negative least-squares solution.The benchmark precoder remains usable as long as the channel is fixed.Source: §V · p. 10

VI. SIMULATION RESULTS

Simulations use quasi-static Rayleigh fading and 8-PSK, evaluating strong-transmitter and strong-eavesdropper cases. Across scenarios, designed precoders raise eavesdropper error relative to ZF while meeting user decoding requirements, with power and security outcomes depending on antenna configuration.

VI. SIMULATION RESULTS

  • At Nt = 20 and NU = 20, iterative and non-negative least-squares design took 173.4 and 10.5 milliseconds, respectively, versus 999.3 milliseconds for CVX.Across the experiments, both methods were considerably faster than CVX.Source: §VI · p. 10
  • At Nt = NU, relaxed-phase power minimization consumes 2.5 dB less power than fixed-phase minimization, while the signal-level design has nearly the benchmark's power at Nt = NU = 10.As Nt exceeds NU, signal-level minimization uses more degrees of freedom and its power approaches power minimization.Source: §VI · p. 10, Figure 5 · p. 11
  • When Ne ≥ Nt, directional modulation with signal-level minimization is preferable for secure communication because it imposes more error on the eavesdropper while consuming the same power as the benchmark scheme.Source: §VI · p. 11
  • With signal-level minimization, user SNR stays at the decoding threshold while eavesdropper SNR remains below it, imposing maximum eavesdropper SER.The comparison is shown for Nt = 11, NU = 10, Ne = 16 and γ = 15.56 dB.Source: §VI · p. 11, Figure 10 · p. 12
  • As NU approaches Nt, power consumption rises as degrees of freedom decrease, and the power-minimization scheme's gap from the other designs widens.For close NU and Nt, MMSE lowers eavesdropper SER relative to ZF; as NU approaches Nt, their performance gets closer.Source: §VI · p. 12, Figure 11 · p. 12, Figure 12 · p. 13
  • As required SNR increases, the design imposes more eavesdropper SER than the benchmark, while ZF and MMSE performance converges.The power gap between power minimization and the other schemes is larger at low SNR; the user-SER difference between power and signal-level minimization stays almost constant as γ increases.Source: §VI · p. 13, Figure 14 · p. 13, Figure 15 · p. 13
  • In the line-of-sight scenario, SER falls sharply from 0.6 to 2 × 10^-3 at the five user locations.The users are at 10°, 50°, 110°, 260°, and 310° on a 4 m-radius circle.Source: §VI · p. 13, Figure 17 · p. 14
  • Increasing system dimension or modulation order greatly increases brute-force computation time at the eavesdropper.The figure compares average consumed time across system dimensions and modulation orders.Source: §VI · p. 14

VII. CONCLUSIONS

The conclusions emphasize symbol-level directional-modulation precoding for multiuser MIMO security and efficient algorithms for designing it. Simulations show increased eavesdropper SER and favorable power and security comparisons with conventional precoding.

VII. CONCLUSIONS

  • The brute-force eavesdropper's average computation time increases enormously with system dimension or modulation order.The reported comparison covers proposed precoders with different modulation orders and Ne = Nt.Source: Figure 18 · p. 14
  • The proposed iterative and non-negative least-squares methods reduce optimal-precoder design time, and numerical examples show lower power and/or higher eavesdropper SER than the benchmark.The authors report that in most cases, designed precoders impose considerable eavesdropper SER compared with conventional precoding.Source: §VII · p. 14
  • Signal-level minimization maintains maximum eavesdropper SER regardless of antenna count, and matches power minimization consumption when transmit and receive antenna counts differ sufficiently.The conclusion describes these properties as simulation findings.Source: §VII · p. 14
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