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Artificial-Noise-Aided Secure Transmission with Directional Modulation based on Random Frequency Diverse Arrays
Jinsong Hu, Shihao Yan, Feng Shu, Jiangzhou Wang, Jun Li, Yijin Zhang
TL;DR
The paper addresses secure transmission when an eavesdropper’s location is unavailable, including cases where the eavesdropper shares the legitimate user’s direction. It proposes RFDA-DM-AN with random frequency allocation and artificial noise, derives an ESC lower bound for power allocation, and reports higher secrecy than PA- and LFDA-based schemes, with the bound converging to ESC as antenna count increases.
Problem
Prior DM security studies generally assume the eavesdropper is in a different direction, leaving the same-direction and unknown-location case insufficiently addressed.
Method
RFDA-DM-AN randomly allocates antenna frequencies, beamforms useful signals toward Bob, places artificial noise in Bob’s null space, and derives an ESC lower bound for power allocation.
Results
RFDA-DM-AN significantly outperforms PA-DM-AN and LFDA-DM-AN in average ESC, while its exact ESC approaches the asymptotic result as N increases.
Takeaways & Limitations
The lower bound supports efficient useful-signal/AN power allocation and precisely matches ESC when the number of transmit antennas is sufficiently large.
Abstract
from arXiv · showhide
In this paper, we propose a novel directional modulation (DM) scheme based on random frequency diverse arrays with artificial noise (RFDA-DM-AN) to enhance physical layer security of wireless communications. Specifically, we first design the RFDA-DM-AN scheme by randomly allocating frequencies to transmit antennas, thereby achieving two-dimensionally (i.e., angle and range) secure transmissions, and outperforming the state-of-the-art one-dimensional (i.e., angle) phase array (PA) based DM scheme. Then we develop the closed-form expression of a lower bound on the ergodic secrecy capacity (ESC) of our RFDA-DM-AN scheme. Based on the theoretical lower bound derived, we further optimize the transmission power allocation between the useful signal and artificial noise (AN) in order to enhance the ESC. Simulation results show that 1) our RFDA-DM-AN scheme achieves a higher secrecy capacity than that of the PA based DM scheme, 2) the lower bound derived is shown to approach the ESC as the number of transmit antennas N increases and precisely matches the ESC when N is sufficiently large, and 3) the proposed optimum power allocation achieves the highest ESC compared with other power allocations in the RFDA-DM-AN.
I. INTRODUCTION
Directional modulation secures communications by preserving a desired-direction constellation while distorting signals elsewhere, but conventional approaches face implementation limits and may fail when an eavesdropper shares the user’s direction. RFDA with artificial noise is introduced to add range-dependent protection and support secrecy analysis and power allocation.
- Motivation: Directional modulation projects the intended signal toward a predetermined spatial direction while distorting its constellation elsewhere.This characteristic reduces the probability of interception and motivates DM for physical-layer security.
- Prior approaches: RF-frontend DM offers flexibility limitations and high constellation-design complexity, motivating baseband implementations.Prior work also developed orthogonal-vector and robust baseband approaches.
- Security gap: Conventional PA-based DM studies assume the eavesdropper is in a different direction from the legitimate user.That assumption is problematic when the eavesdropper is passive and its location is unavailable.
- Security gap: LFDA adds controllable direction and range, but its coupled direction-range behavior can create multiple locations with identical received signals.This leaves a limitation when legitimate user and eavesdropper share a direction but differ in range.
- Proposed approach: RFDA-DM-AN randomly assigns antenna frequencies and transmits artificial noise outside the desired direction to enhance physical-layer security.The scheme targets useful-signal SNR at the desired direction while interfering with eavesdroppers.
- Contributions: The paper derives an ESC lower bound, uses it for useful-signal/AN power allocation, and examines continuous and discrete uniform frequency assignments.The lower bound matches ESC when the antenna count is sufficiently large.
A. Random Frequency Diverse Array
The RFDA differs from a phased array by assigning frequency increments across antenna elements, with random increments controlling a two-dimensional angle-range response. Its steering and phase relations incorporate carrier frequency, element spacing, target angle, and target range.
- RFDA structure: RFDA assigns each antenna element a carrier frequency increment, unlike a phased array.The n-th element uses fn = fc + kn∆f, with kn normally random.
- RFDA structure: The random coefficient kn determines a mapping rule for assigning carrier frequencies across transmit elements.A continuous uniform distribution is one example of such a rule.
- Geometry: For a far-field target, the n-th element’s range is approximated using the target range, angle, element index, and ULA spacing.The model assumes a uniform linear array and uses the array geometric center as phase reference.
- Phase response: The RFDA phase shift determines an array radiation pattern that depends on both target range and frequency increment.Under N∆f ≪ fc and near-half-wavelength spacing, the third phase term is negligible.
- Steering response: The steering vector is formed for a specific location (θ, R), making the RFDA response location-dependent.This location dependence supports angle-and-range directional modulation.
B. Directional Modulation with Artificial Noise
The RFDA-DM-AN transmitter beamforms the useful signal toward Bob while placing artificial noise in Bob’s channel null space. Bob can recover the intended symbol, whereas Eve receives distorted amplitude, phase, and constellation components.
- System model: The system is a MISO wiretap channel with Alice using N antennas and Bob and Eve using single antennas.Bob’s location is known to Alice, while Eve’s location is unavailable and may be anywhere.
- Signal design: The transmitted signal combines a modulated symbol and artificial noise, with α controlling power allocation between them.The symbol has average power constraint E[|x|^2] = 1.
- Signal design: The useful-signal beamforming vector is selected to maximize Bob’s SNR using Bob’s RFDA steering vector.This design uses Bob’s known angle-range location.
- Artificial noise: The artificial-noise vector lies in the null space of Bob’s steering vector, preventing interference to Bob.The construction uses h^H(θB, RB)w = 0.
- Received signals: Bob can restore the original signal without knowing the random frequency-mapping rule, while Eve is assumed not to know that rule.At Eve, nonorthogonality of the steering vectors and AN distort the signal’s amplitude, phase, and constellation.
III. SECRECY PERFORMANCE OF THE RFDA-DM-AN SCHEME
The paper evaluates RFDA-DM-AN secrecy using ESC, derives a lower bound, optimizes useful-signal/AN power allocation from that bound, and compares continuous and discrete random frequency allocations.
- Evaluation framework: ESC is the secrecy metric, obtained by averaging secrecy capacity over Eve’s random SINR induced by random frequency allocation.Bob’s SNR does not depend on the RFDA frequency allocation, whereas Eve’s SINR does.
- Analysis procedure: The analysis first determines ESC, then derives a lower bound and uses it to determine optimal power allocation between the useful signal and artificial noise.Two random frequency-allocation strategies are subsequently examined.
A. Ergodic Secrecy Capacity
The paper defines ESC by averaging secrecy capacity over the randomness induced by RFDA frequency allocation, and then extends it to average over Eve’s possible locations. This metric supports optimizing the useful-signal/AN power-allocation parameter.
- Secrecy capacity is defined as the positive part of Bob’s capacity minus Eve’s capacity, {0, CB−CE}+.
- 18th? Under the path-loss-free model, CB ≥ CE is guaranteed because Alice maximizes Bob’s SNR.
- ESC averages instantaneous secrecy capacity over γE because Eve’s SNR depends on the random frequency allocation while Bob’s SNR does not.
- For unknown Eve location, the paper further averages secrecy capacity over all possible locations in the assumed threat region.An annular region centered on Bob is given as an example.
- The power-allocation parameter α is selected to maximize the location-averaged ESC, using a closed-form lower-bound expression because the exact expression is mathematically intractable.
B. A Lower Bound on the Ergodic Secrecy Capacity
The paper derives a lower bound on RFDA-DM-AN’s ESC to make transmit-power optimization tractable. The bound applies to arbitrary antenna counts and approaches the ESC as N increases, becoming asymptotically exact.
- Theorem 1 provides a lower bound on RFDA-DM-AN’s ESC to facilitate power allocation between the useful signal and AN.
- The derivation uses the cross-correlation coefficient between Bob’s and Eve’s steering vectors, with its random-frequency expectation characterized through the MGF of kn.
- Jensen’s inequality converts the ESC expression into the stated lower bound for arbitrary values of N.
- As N → ∞, distance concentration makes the relevant random correlation converge to its mean, so the lower bound approaches the ESC.
- The paper gives an asymptotic ESC expression for the RFDA-DM-AN scheme when N → ∞.
- PA-DM-AN and LFDA-DM-AN secrecy capacities are derived as benchmarks for evaluating RFDA-DM-AN.
C. Continuous and Discrete Uniform Frequency Allocations
The lower-bound framework supports both continuous-uniform and discrete-uniform random frequency allocations. Their corresponding MGFs are substituted into the general bound to obtain each allocation’s secrecy performance.
- The lower bound is valid for any random frequency allocation whose MGF exists.
- For continuous-uniform allocation, kn is modeled as a continuous uniform random variable over a bounded interval determined by the available frequency bandwidth.
- For discrete-uniform allocation, kn takes values from a finite uniformly distributed set, with K denoting the number of possible values.
- Substituting the continuous- and discrete-uniform MGFs into Theorem 1 yields the corresponding lower bounds on secrecy capacity for both frequency-allocation strategies.
IV. NUMERICAL RESULTS
The numerical results compare RFDA-DM-AN with PA-DM-AN and LFDA-DM-AN, examine its ESC lower bound and power allocation, and evaluate continuous versus discrete frequency allocation. RFDA-DM-AN provides stronger secrecy, while the lower bound becomes accurate as antenna count increases.
- RFDA-DM-AN significantly outperforms PA-DM-AN and LFDA-DM-AN in secrecy performance.PA-DM-AN has zero secrecy capacity when Eve is in Bob’s direction, while LFDA-DM-AN achieves much lower secrecy than RFDA-DM-AN.
- RFDA decouples range and angle correlations, preventing Eve from selecting locations that guarantee zero ESC.Such zero-capacity locations exist for PA-DM-AN and LFDA-DM-AN but not for RFDA-DM-AN.
- The gap between the ESC and its lower bound decreases as the number of transmit antennas N increases.At N = 256, the lower bound is reported to become closely aligned with the ESC.
- The lower bound remains close enough to the ESC to determine transmit power allocation between the useful signal and AN when N is not very large.For N = 16, the optimal α determined from the lower bound is still close to that determined from the ESC.
- As N increases, the optimal α approaches one, so AN is unnecessary when Alice can form an ultranarrow beam toward Bob.The exact ESC also approaches the asymptotic ESC as N increases.
- Continuous uniform frequency allocation outperforms discrete uniform allocation in average ESC.Average ESC increases with μB, while the optimal α increases as μB decreases and reaches α = 1 at sufficiently low transmit power.
V. CONCLUSION
The paper concludes that RFDA-DM-AN improves physical-layer security through random frequency allocation and artificial noise. Its ESC lower bound supports power allocation, approaches the ESC for large antenna arrays, and the scheme outperforms PA-DM-AN and LFDA-DM-AN.
- RFDA-DM-AN is proposed to enhance physical layer security in wireless communications.
- The derived ESC lower bound enables efficient allocation of transmit power between the useful signal and AN.
- The asymptotic ESC precisely matches the derived lower bound when N is sufficiently large.
- RFDA-DM-AN significantly outperforms PA-DM-AN and LFDA-DM-AN in average ESC.