Optimal Weighted Subspace Fitting-based Direct Position Determination with HF/VHF 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-based Direct Position Determination with HF/VHF 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-based Direct Position Determination with HF/VHF Collaboration[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT260001

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Optimal Weighted Subspace Fitting-based Direct Position Determination with HF/VHF Collaboration

doi: 10.11999/JEIT260001 cstr: 32379.14.JEIT260001

Institute of Information Engineering, Information Engineering University, Zhengzhou 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: 2026-01-04
Accepted Date: 2026-03-27
Rev Recd Date: 2026-03-26

Available Online: 2026-04-22

Abstract

Abstract

Objective Passive localization is essential for target detection, navigation, and track tracking, particularly in military applications involving maritime and aerial targets. These targets often transmit across multiple frequency bands, including shortwave High Frequency(HF) and Very High Frequency (VHF). Existing localization methods largely rely on single-band approaches or two-step positioning techniques. Single-band methods underutilize the positional information available across different bands, while two-step methods lose information during intermediate parameter estimation (e.g., Direction-Of-Arrival (DOA); Time-Difference-Of-Arrival (TDOA)), reducing localization accuracy. Collaborative fusion of HF signals (via ionospheric reflection) and VHF signals (via Doppler effects from moving arrays) has been rarely addressed. To overcome low positioning accuracy and limited spatial resolution in over-the-horizon multi-target scenarios, this study proposes a novel collaborative Direct Position Determination (DPD) method designed to integrate the complementary strengths of HF and VHF signals, enhancing localization precision and robustness in complex electromagnetic environments. Methods An Optimal Weighted Subspace Fitting (OWSF) DPD algorithm is proposed. Comprehensive signal propagation models are established for heterogeneous observation platforms (Fig. 1). HF signal propagation is modeled using a two-dimensional DOA framework based on ionospheric reflection, incorporating azimuth and elevation angles to handle nonlinear over-the-horizon propagation. VHF signals are modeled using a space-time extended signal framework for a moving Unmanned Aerial Vehicle (UAV), exploiting Doppler effects to create a virtual large-aperture array that captures both one-dimensional angle and Frequency-Of-Arrival (FOA) information. Unlike traditional methods that process each band separately, the OWSF algorithm constructs a unified cost function that fuses the signal and noise subspaces of both HF and VHF data using optimal weighting matrices, balancing the contributions of different signal qualities. Target positions are then estimated by minimizing this cost function via grid search or Newton iteration. The Cramér-Rao Bound (CRB) under Earth-ellipsoid constraints is derived to provide the theoretical performance limit. Results and Discussions Simulations are conducted in a centralized processing scenario, where HF stations and UAV VHF signals are transmitted to a central station for joint processing (Fig. 2). The simulation involves three stationary targets and a collaborative system comprising HF stations and a UAV (Fig. 3, Table 2, Table 3). Performance comparisons demonstrate that the OWSF method consistently outperforms traditional two-step positioning methods and single-system DPD methods (DOA-only or FOA-only) in Root Mean Square Error (RMSE) (Fig. 4). When HF SNR is 5 dB lower than VHF SNR, OWSF exhibits superior robustness compared to Subspace Data Fusion (SDF) and Minimum Variance Distortionless Response (MVDR) methods, approaching the CRB at high SNR (Fig. 5). The impact of system parameters is further analyzed, showing that increasing the number of sampling points (Fig. 6) and array elements (Fig. 7) improves accuracy, particularly in low SNR regimes. Regarding spatial resolution, the OWSF algorithm generates sharper spectral peaks for distant targets and successfully resolves closely spaced targets that the SDF-DPD algorithm fails to distinguish (Fig. 8, Fig. 9). Conclusions The HF/VHF collaborative DPD method effectively integrates multidimensional observational information from ionospheric reflection and Doppler-based propagation. Simulation results demonstrate substantial improvements in localization accuracy, spatial resolution, and robustness, especially under low-SNR conditions or heterogeneous signal quality between bands. The derived CRB provides a solid theoretical benchmark, confirming that the method overcomes the limitations of single-band and two-step approaches. This approach offers a highly effective solution for over-the-horizon passive localization of multiple stationary targets.

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