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IEEE Wireless Communications Letters

Kernel Density Estimation-Based Secrecy Performance Analysis of Cooperative Beamforming for Arbitrary Distributed 3-D Networks

Jijun Hwang, Byungha You, Seok-Chul Kwon, Haejoon Jung

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Analog cooperative beamforming (ACB) improves physical-layer security (PLS) by enabling phase-coordinated transmissions among spatially distributed nodes without exchanging channel state information (CSI). However, previous studies have relied on predefined node distributions and two-dimensional (2D) network models, limiting the practical applicability of ACB. Motivated by this limitation, we propose a three-dimensional (3D) ACB-based PLS framework that predicts the secrecy performance of ACB under arbitrary node distributions by explicitly considering both LoS and NLoS channel components. Using kernel density estimation (KDE), we estimate the unknown spatial node distribution and derive closed-form expressions for the secrecy rate by analyzing the array gain statistics at the legitimate receiver and the eavesdropper.We verify the proposed framework through simulations and conduct a detailed analysis to examine the impact of network parameters, such as cluster size and the number of nodes.

DOI
10.1109/lwc.2026.3666437

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Analog cooperative beamforming (ACB) improves physical-layer security (PLS) by enabling phase-coordinated transmissions among spatially distributed nodes without exchanging channel state information (CSI). However, previous studies have relied on predefined node distributions and two-dimensional (2D) network models, limiting the practical applicability of ACB. Motivated by this limitation, we propose a three-dimensional (3D) ACB-based PLS framework that predicts the secrecy performance of ACB under arbitrary node distributions by explicitly considering both LoS and NLoS channel components. Using kernel density estimation (KDE), we estimate the unknown spatial node distribution and derive closed-form expressions for the secrecy rate by analyzing the array gain statistics at the legitimate receiver and the eavesdropper.We verify the proposed framework through simulations and conduct a detailed analysis to examine the impact of network parameters, such as cluster size and the number of nodes.
With the rapid increase in connectivity and complexity of 6G-enabled Internet-of-Things (IoT) networks, security has become a critical concern due to the integration of various devices. Analog cooperative beamforming (ACB)-based physical layer security (PLS) algorithms have been investigated to effectively cope with such a security issue in massive IoT networks with limited hardware capabilities. However, in the existing work on ACB-based PLS, IoT nodes are typically assumed to follow Gaussian or uniform distributions, which limits the applicability in general IoT networks with arbitrary node distributions. For this reason, we propose a novel approach to analyze the secrecy rate of the ACB-based PLS scheme with arbitrarily distributed nodes using the non-parametric kernel method, where a kernel function is introduced to better estimate the actual node distribution from survey data.
A wireless sensor network (WSN) refers to a network of sensor nodes that collaboratively work to sense, monitor, and control their surrounding environments. As WSNs are integrated into the Internet of Things, it is crucial to protect the network against malicious security attacks considering its wide applicability, such as military monitoring, healthcare, and civilian applications. Thus, in this paper, we consider a physical-layer security technique exploiting analog cooperative beamforming (ACB), where multiple sensor nodes create a virtual antenna array (VAA) and locally adapt their phases. As WSNs inherently have clustered topology, we model the VAA elements' locations by Gaussian distributions. The secrecy capacity of the ACB with the Gaussian-distributed elements is derived in a closed-form expression. The theoretical and numerical results indicate that the ACB-based schemes provide a better secrecy rate compared to the conventional co-located antenna array-based schemes. In addition, the ACB with Gaussian-distributed elements can better suppress side-lobe, which causes undesired information leakage to certain directions, compared to the ACB with uniformly distributed elements. Furthermore, we investigate the impacts of fading channel and phase estimation error in the ACB.
As the importance to guarantee confidentiality against security attacks by eavesdroppers has attracted significant attention, there has been much effort to enhance the security performance through cooperative beamforming (CB) in the context of physical layer security (PLS). In particular, analog CB (ACB), which requires vastly lower overhead compared to digital CB, can provide significantly higher secrecy rate compared to the PLS methods with co-located array. In this letter, we focus on ACB with open-loop architecture that requires low complexity, and consider phase offsets across multiple virtual antenna array (VAA) elements for ACB, which are caused by position estimation error of VAA. We investigate the impact of the phase offsets in ACB in terms of the secrecy throughput loss.
With the rapid proliferation of the Internet of Things (IoT), small devices in various applications generate, process, and exchange security and safety-critical data as well as privacy-sensitive information. However, due to limited hardware capabilities of such small devices, the IoT networks are inherently vulnerable to security attacks. In this context, the analog cooperative beamforming (ACB)-based physical layer security (PLS) technique can be a suitable solution to guarantee security in the IoT networks by creating a virtual antenna array with multiple IoT devices. Nevertheless, such techniques suffer from poor secrecy performance, when an eavesdropper is located within the main beam toward the legitimate receiver. Motivated by this limitation, in this article, we propose a fully distributed null-steering beamformer, which is suitable for IoT networks following decentralized architecture. Through analysis and simulation, we show that the proposed PLS algorithm outperforms the existing ACB-based PLS scheme. Furthermore, we show that there exists an optimal degree of nulling that maximizes the secrecy performance, which can be estimated based on the analytical results.
In the sixth-generation (6G) communications, how to deploy and manage massively connected Internet-of-Things (IoT) nodes will be one of the key technical challenges, because 6G is expected to provide 10 times higher connection, compared to 5G. At the same time, due to the sharp growth in connected devices and newly adopted technologies, learning-based attacks and big data breaches are expected to occur more frequently. With the advances of quantum computing in the future, conventional cryptography-based security protocols may be obsolete in the future wireless networks, which makes physical layer security (PLS) an attractive alternative or complement for secure communications. In this context, cooperative beamforming (CB)-based PLS schemes are known to be effective solutions to guarantee high secrecy rate with IoT devices, which have limited power and hardware complexity. However, the existing CB-based PLS algorithms suffer from extremely low secrecy rate, in case that eavesdroppers are close to the intended receiver. To overcome such critical issue, in this paper, we propose a CB-based PLS with artificial noise (AN) injection, which can be realized in a fully distributed manner to minimize the overhead in the IoT networks with a large number of devices. We analyze the array factor using the virtual antenna array (VAA) created by the proposed PLS algorithm. Then, the secrecy rate is derived in a closed-form expression, which can be used to optimize the performance for given system parameters in both the absence and presence of channel state information (CSI) error. The proposed scheme provides considerably higher secrecy rate compared to the conventional CB-based PLS schemes, when an eavesdropper exists near to the intended receiver. Furthermore, through simulation and numerical results, we show that the secrecy rate of the proposed scheme can be maximized by adjusting the ratio between the data beamformer and AN injection beamformer components. As a result, the proposed method shows a performance improvement of up to two times compared to the conventional CB-based PLS schemes, in terms of the secrecy rate. Such performance gain increases as the angular location of Eve becomes closer to that of Bob, which corresponds to the most vulnerable situation of the conventional CB-based PLS algorithms.
In this article, we consider the secure uplink transmissions from ground nodes to an unmanned aerial vehicle (UAV) receiver through analog collaborative beamforming (ACB). When the location information of the UAV receiver is available, ACB can be realized in a fully distributed manner, since each transmitting node can locally adjust its phase. ACB-based physical layer security (PLS) algorithm, where the virtual antenna array (VAA) elements are randomly selected at each transmit time unit, can effectively protect against eavesdropping attacks by creating artificial noise to the directions except the legitimate receiver (Bob). However, when Bob is mounted on the UAV, it can suffer from array gain loss due to jittering effects caused by airflow turbulence and mechanical vibration. Motivated by this issue, in this article, we investigate the impact of the UAV jitter on the secrecy performance. We derive the theoretical expressions of the secrecy rate both in the absence and presence of the jittering effects. Further, the simulation results show that the ACB-based PLS cannot guarantee non-zero secrecy rate even with the radial distance perturbations on the order of a few centimeters, especially when an eavesdropper (Eve) exists in close proximity to Bob.