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arXiv:2503.20223

Phase-Only Zero-Forcing for Secure Wireless Communication in Multi-User Systems

Hong, Jordan, Safwan Jamal, Ashish J. Khisti

Published Mar 26, 2025• 0 citations
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Abstract

Artificial noise (AN) transmission is a physical layer security technique in multi-antenna wireless communication systems. Synthetic noise is broadcast to all receivers except designated legitimate users via beamforming in the legitimate users' null space. We consider AN transmission employing a single RF chain and analog beamforming, where beamforming vectors maintain constant magnitude while allowing arbitrary phases. Our primary objective is to design a constant-magnitude vector capable of nullifying multiple users' channel vectors simultaneously. To tackle this zero-forcing problem, we propose a novel successive partition zero-forcing (SPZF) scheme, which transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability. The SPZF scheme can be generalized to any number of users, but our analysis focuses on the two-user case. Theoretical analysis reveals that our proposed SPZF scheme can attain arbitrarily low outage probability in the limit of large number of transmit antenna. We present three partition algorithms (random, iterative, and genetic) to minimize the outage probability. The outage probabilities and secrecy rates of the three partition algorithms are compared via numerical simulations. We find that the more advanced partition algorithms (iterative and genetic) achieve higher secrecy rates than the random algorithm, particularly under conditions of high signal-to-noise ratio (SNR), large number of eavesdroppers, or small number of transmit antennas.

DOI
10.48550/arxiv.2503.20223

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Artificial noise (AN) transmission is a physical layer security technique in multi-antenna wireless communication systems. Synthetic noise is broadcast to all receivers except designated legitimate users via beamforming in the legitimate users' null space. We consider AN transmission employing a single RF chain and analog beamforming, where beamforming vectors maintain constant magnitude while allowing arbitrary phases. Our primary objective is to design a constant-magnitude vector capable of nullifying multiple users' channel vectors simultaneously. To tackle this zero-forcing problem, we propose a novel successive partition zero-forcing (SPZF) scheme, which transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability. The SPZF scheme can be generalized to any number of users, but our analysis focuses on the two-user case. Theoretical analysis reveals that our proposed SPZF scheme can attain arbitrarily low outage probability in the limit of large number of transmit antenna. We present three partition algorithms (random, iterative, and genetic) to minimize the outage probability. The outage probabilities and secrecy rates of the three partition algorithms are compared via numerical simulations. We find that the more advanced partition algorithms (iterative and genetic) achieve higher secrecy rates than the random algorithm, particularly under conditions of high signal-to-noise ratio (SNR), large number of eavesdroppers, or small number of transmit antennas.
Artificial noise (AN) transmission can enhance the secrecy in multiantenna wireless channels by superimposing judiciously constructed synthetic noise signals over information signals during transmission. Motivated by the large-scale multiple-input multiple-output systems, we study AN transmission under a constraint that all elements (weights) in each beamforming vector must have a constant magnitude, but can have arbitrary phases. For the special case of one AN beamforming vector and one legitimate receiver, we derive a necessary and sufficient condition for finding a beamforming vector in the null space of the legitimate receiver's channel and provide a geometric interpretation. For the independent identically distributed Rayleigh fading channel, we derive an approximate expression for the probability of failing to find such a beamforming vector, and show that it decreases exponentially in the square of the number of transmit antennas. For the general case, we propose a numerical algorithm for obtaining a set of mutually orthogonal AN beamforming vectors in the null space of all the receivers. Our approach involves reducing the problem to an unconstrained nonlinear programming problem, which is then solved using the Gauss-Newton method. We show numerically that our proposed algorithm performs significantly better than a heuristic relaxation approach. Finally, for a multiantenna system with multiple RF chains, we show numerically that the secrecy rate achieved by our proposed approach is close to that achieved by the AN transmission with unconstrained beamforming, when the number of transmit antennas is sufficiently large.
The security of a wireless channel between a transmitter and legitimate user(s) in the presence of eavesdroppers can be improved by the addition of artificial noise (AN) transmission. In the case where the channel state information of the eavesdroppers is not known, some AN schemes transmit multiple uncorrelated noise signals via distinct orthogonal beamforming vectors. The vectors are designed to avoid affecting the legitimate links, and mutual orthogonality increases the likelihood that the AN signals are received at the eavesdroppers with high power. Applications in massive-MIMO, where hardware is constrained, motivate us to design vectors whose elements have arbitrary phases but equal magnitudes [1].This work proposes an algorithm that generates multiple such phase-only beamforming vectors via a single optimization problem and significantly increases the number of vectors which can be found compared to the previous work [2]. Consequently, our results demonstrate substantially higher lower-bounds on maximum achievable secrecy rate via this approach compared to the previous work [2] for similar scenarios.
This paper investigates the physical layer security in a multiple-input-single-output multicast channel, where a common message is transmitted to multiple receivers and only statistical channel state information of the eavesdropper is available. We consider transmitting some artificial noise (AN) along with the information-bearing signal to improve secrecy. To find the optimal AN design, we formulate a secrecy outage probability (SOP) constrained secrecy rate maximization problem. For solving this problem, we adopt a novel transformation of the intractable SOP constraint and establish an efficient algorithm which solves the transformed problem to global optimality. We also propose a low-complexity refinement method to reduce the conservatism incurred by the transformation and an efficient algorithm to find a suboptimal solution when the information-bearing signal is transmitted using single-stream beamforming. Our solution is found to be a valuable generalization of the traditional null-space AN scheme. According to our simulation results, when the number of receivers is less than that of transmit antennas, distributing AN uniformly in the null space of the legitimate channel vectors is near-optimal. When there is no valid null space, transmitting AN will interfere with some legitimate receivers, and the secrecy performance can also be improved with carefully designed AN.
This paper studies the artificial noise (AN) aided secure transmission design in a multiple-input- single-output multicast channel where a common message is transmitted to multiple receivers and only statistical channel state information of the eavesdropper is available. In this scenario, there may be no valid null space which allows the AN not to affect the legitimate receivers, and the traditional null-space AN scheme may not be applicable. To find the optimal design, we formulate a secrecy outage probability (SOP) constrained secrecy rate maximization problem which applies to an arbitrary number of the receivers. We adopt a novel approximation of the intractable SOP constraint and establish an efficient algorithm which solves the approximated problem to global optimality. The effectiveness of AN as well as the proposed solution is confirmed by simulation results. It is shown that the AN aided approach performs much better than pure beamforming even when there is no valid null space.
In this paper, we consider multiple-input single-output multi-eavesdropper cognitive radio networks (MISOME-CRNs), where a secondary user (SU) aims to transmit confidential information to a legitimate SU receiver in the presence of a primary user (PU) and a multi-antenna passive eavesdropper in fast fading environments. For this system setting, we study beamforming with artificial noise (AN) for the SU to achieve confidential communications. We consider designing an AN-assisted optimal beamforming scheme, denoted cognitive beamforming (CB), which maximizes the ergodic secrecy rate. Moreover, we propose two suboptimal beamforming schemes, namely, scaled beamforming (SB) and projected beamforming (PB). We develop an analytical framework to assess the performance of the proposed schemes in a unified manner. First, we analyze the achievable ergodic secrecy rate of the three schemes. Second, we derive the optimal power allocation for the information and AN signals that maximizes the achievable ergodic secrecy rate in the large-antenna regime. Third, we study the performance of the three schemes in terms of the secrecy outage probability. Using numerical simulations, we validate our analytical results and show that CB achieves the best performance among the three proposed schemes. In addition, we provide insights into the effect of various system parameters on the secrecy performance. In particular, we show that the interference threshold at the PU plays an important role in the beamforming design.
This paper studies the artificial noise (AN) aided beamforming design in the multiple-input, single-output, and multiple-eavesdropper wiretap channel from the perspective of secrecy outage. Unlike many existing works, which directly adopted the traditional null-space AN scheme, we start with a general assumption on the structure of the transmit signal, and seek to find the optimal structure by solving a secrecy outage probability (SOP) constrained secrecy rate maximization problem. By generalizing several existing conclusions regarding a single-antenna eavesdropper, we prove that the null-space AN scheme is indeed optimal for an arbitrary number of the eavesdropper’s antennas from the perspective of secrecy outage. Furthermore, the optimal power allocation between the information-bearing signal and the AN is found under both general conditions and the conservative assumption that the eavesdropper can be arbitrarily close to the transmitter or his receiver noise power can be arbitrarily low. It is also found that the increase in the eavesdropper’s antenna number is equivalent to requiring a smaller SOP under the conservative assumption. The results in this paper can help generalize existing design solutions for single-antenna eavesdroppers to multi-antenna eavesdroppers.