Optimal Weighted Subspace Fitting Direct Position Determination Method with HF/UHF Collaboration

YANG Gao-yuan; YIN Jie-xin; WANG Ding; YANG Bin

doi:10.11999/JEIT260001

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YANG Gao-yuan, YIN Jie-xin, WANG Ding, YANG Bin. Optimal Weighted Subspace Fitting Direct Position Determination Method with HF/UHF Collaboration[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT260001

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YANG Gao-yuan, YIN Jie-xin, WANG Ding, YANG Bin. Optimal Weighted Subspace Fitting Direct Position Determination Method with HF/UHF Collaboration[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT260001

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Optimal Weighted Subspace Fitting Direct Position Determination Method with HF/UHF Collaboration

doi: 10.11999/JEIT260001 cstr: 32379.14.JEIT260001

Institute of Information Engineering, Information Engineering University, Zhengzhou, Henan 450001, China

Funds: The National Natural Science Foundation of China (61901526, 62171469, 62071029), The Youth Talent Recruitment Project of the China Association for Science and Technology (2022-JCJQ-QT-028), The Outstanding Youth Science Foundation of Henan Province (242300421174)

Received Date: 2026-01-01
Accepted Date: 2026-03-27
Rev Recd Date: 2026-03-26

Available Online: 2026-04-22

Abstract

Abstract

Objective Passive localization technology plays an indispensable role in target detection, navigation, and track tracking, particularly in military applications involving maritime and aerial targets. These targets often utilize complex communication systems covering multiple frequency bands, such as Shortwave (HF) and Ultra-Shortwave (UHF). Existing localization methods mainly rely on single-frequency bands or Two-Step positioning approaches. However, single-band methods fail to fully exploit the diverse position feature information contained in different signals, while Two-Step methods suffer from information loss during the intermediate parameter estimation process (e.g., DOA, TDOA), leading to reduced accuracy. Furthermore, current Direct Position Determination (DPD) research rarely addresses the collaborative fusion of HF signals (utilizing ionospheric reflection) and UHF signals (utilizing Doppler effects from moving arrays). To address the challenges of low positioning accuracy and poor spatial resolution in over-the-horizon multi-target scenarios, this study aims to propose a novel collaborative DPD method. This method is designed to integrate the complementary advantages of HF and UHF bands, thereby significantly enhancing localization precision and robustness in complex electromagnetic environments. Methods To achieve high-precision localization, this paper proposes an Optimal Weighted Subspace Fitting (OWSF) DPD method. First, comprehensive signal propagation models are established for the heterogeneous observation platforms (Fig. 1). For HF signals, a two-dimensional Direction of Arrival (DOA) model is constructed based on ionospheric reflection, incorporating azimuth and elevation angle information to handle non-linear over-the-horizon propagation. For UHF signals, a space-time extended signal model is developed for a moving Unmanned Aerial Vehicle (UAV) platform. This model utilizes the station's motion to exploit the Doppler effect, effectively creating a virtual large-aperture array to capture both one-dimensional angle and Frequency of Arrival (FOA) information. Secondly, unlike traditional methods that process bands separately, the proposed OWSF algorithm constructs a unified cost function. This function fuses the signal subspaces and noise subspaces of both HF and UHF observed data using optimal weighting matrices to balance the contributions of different signal qualities. The target position is then estimated by minimizing this cost function via a grid search or Newton iteration method. Additionally, the theoretical performance limit is established by deriving the Cramér-Rao Bound (CRB) for the positioning estimation error under the geometric constraints of the Earth ellipsoid. Results and Discussions Numerical simulations were conducted in a centralized processing scenario where data is transmitted to a central station (Fig. 2) to validate the effectiveness and performance of the proposed method. The simulation setup involved three stationary targets and a collaborative system comprising shortwave stations and a UAV (Fig. 3, Table 2, Table 3). Performance comparisons demonstrate that the proposed OWSF method consistently outperforms traditional Two-Step positioning methods and single-system DPD methods (DOA-only or FOA-only) in terms of Root Mean Square Error (RMSE) (Fig. 4). Specifically, under conditions where the Signal-to-Noise Ratio (SNR) of the HF signal is 5dB lower than that of the UHF signal, the OWSF algorithm exhibits superior robustness compared to existing algorithms such as the Subspace Data Fusion (SDF) and Minimum Variance Distortionless Response (MVDR) methods, closely approaching the theoretical CRB at high SNR levels (Fig. 5). The impact of system parameters was also analyzed, showing that increasing the number of sampling points (Fig. 6) and array elements (Fig. 7) improves accuracy, particularly in low SNR regimes. Furthermore, regarding spatial resolution, the OWSF algorithm generates sharper spectral peaks for distant targets and successfully resolves closely spaced targets that the conventional SDF-DPD algorithm fails to distinguish (Fig. 8, Fig. 9). Conclusions This paper presents a robust HF/UHF collaborative Direct Position Determination method based on Optimal Weighted Subspace Fitting. By mathematically modeling the ionospheric reflection for HF signals and the space-time Doppler characteristics for UHF signals, the method effectively fuses multi-dimensional observational information. Simulation results verify that the new method significantly improves positioning accuracy and spatial resolution compared to existing algorithms, particularly in scenarios with low SNR or unequal signal quality between bands. The derivation of the CRB provides a solid theoretical benchmark for the system. The proposed approach successfully overcomes the bottlenecks of single-band limitations and the information loss inherent in two-step methods, proving to be a highly effective solution for over-the-horizon passive localization of multiple stationary targets.

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