Two-stage Constrained Weighted Least Squares method for Multistatic Passive Localization of a Moving Object Under Unknown Transmitter Position and Velocity
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摘要: 针对发射站位置与速度未知时的运动目标定位问题，该文提出一种低复杂度的两阶段定位算法。首先，利用直接路径时间迟延和多普勒频移量测构建一个基于约束加权最小二乘(CWLS)模型获得发射站的位置和速度。然后，将发射站位置和速度估计值代入直接路径和间接路径联合模型，建立基于CWLS的目标位置和速度估计模型，利用拟牛顿法求解目标的位置和速度。理论分析和仿真结果显示所提算法定位性能达到克拉美罗下界(CRLB)。
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关键词:
- 多基外辐射源定位 /
- 运动目标 /
- 未知发射站位置和速度 /
- 约束加权最小二乘 /
- 拟牛顿法
Abstract:Objective This study addresses target localization in multistatic passive radar systems under conditions where the transmitter’s position and velocity are unknown. Multistatic passive radar systems utilize covert deployment, exhibit resilience to jamming, and provide wide-area coverage. Conventional localization techniques rely on precise transmitter state information, which is often unavailable due to the mobility of transmitters mounted on dynamic platforms. Environmental disturbances can further introduce inaccuracies in position and velocity measurements. In non-cooperative scenarios, direct acquisition of transmitter state information is typically infeasible. Existing localization methods, such as the Two-Step Weighted Least Squares (TSWLS) approach, exhibit a threshold effect under high noise conditions, while the Semi-Definite Programming (SDP) method achieves Cramér-Rao Lower Bound (CRLB) accuracy but incurs excessive computational costs, limiting real-time applicability. To address these challenges, a localization algorithm is formulated that enables high-precision tracking of moving targets under uncertain transmitter conditions while maintaining relatively low computational complexity. Methods Multistatic passive radar localization systems employ two receiving channels: the reference channel, which receives direct signals from the transmitter, and the surveillance channel, which captures signals reflected from the target. Delay-Doppler cross-correlation between the reflected and reference signals enables the measurement of time delay and Doppler shift. The time delay from the reference channel corresponds to the distance between the transmitter and receivers, denoted as the Direct Range (DR), while the associated Doppler shift represents the Direct Range Rate (DRR). Similarly, the bistatic time delay from the surveillance channel corresponds to the sum of the distances between the target, transmitter, and receivers, referred to as the Bistatic Range (BR), with the associated Doppler shift representing the Bistatic Range Rate (BRR). A two-stage localization algorithm is proposed for estimating the positions and velocities of both the target and transmitter. In the first stage, a Constrained Weighted Least Squares (CWLS) problem is formulated using DR and DRR measurements to estimate the transmitter’s position and velocity. In the second stage, the estimated transmitter state is incorporated into the DR/DRR and BR/BRR measurements to construct a new CWLS problem. This problem is then solved using the Quasi-Newton method to determine the position and velocity of the moving target. Results and Discussions Compared with traditional localization approaches that rely solely on indirect path information (BR/BRR), incorporating direct path information (DR/DRR) for joint estimation improves target localization accuracy when the transmitter’s position and velocity are unknown ( Figure 1 ). The performance of the proposed two-stage localization algorithm is evaluated through Monte Carlo simulations and compared with the TSWLS method, the SDP approach, and the CRLB. Estimation accuracy is assessed using the Mean Squared Error (MSE), while algorithm complexity is evaluated based on runtime. In scenarios with only four receivers, the TSWLS algorithm fails to provide accurate estimates, whereas the proposed algorithm maintains localization performance, with deviations occurring only when noise reaches 25 dB (Figure 2 ). When five receivers with uncorrelated noise are used, the TSWLS algorithm deviates from the CRLB and exhibits a threshold effect at a measurement noise level of 10 dB. At 30 dB, the proposed algorithm reduces the MSE for target position estimation by approximately 7 m2 compared to the TSWLS algorithm, slightly outperforming the SDP algorithm, and reduces the MSE for target velocity estimation by approximately 10 (m/s)2, approaching the localization accuracy of the SDP algorithm (Figure 3 ). When 5 receivers with correlated noise are used, the TSWLS algorithm begins to deviate from the CRLB at a noise variance of 15 dB and exhibits significant performance degradation at 30 dB. Under these conditions, the proposed algorithm reduces the MSE for target position estimation by approximately 5 m2 compared to the TSWLS algorithm, slightly outperforming the SDP algorithm, and reduces the MSE for target velocity estimation by approximately 7 (m/s)2, achieving localization accuracy comparable to the SDP algorithm (Figure 4 ). While the SDP algorithm has higher computational complexity and longer runtime, the proposed algorithm achieves a shorter runtime while maintaining localization accuracy, demonstrating good real-time performance (Table 1 ).Conclusions This study investigates the localization of a moving target in a multistatic passive radar system when the transmitter’s position and velocity are unknown. By leveraging time delay and Doppler frequency shift measurements from both direct and indirect paths, a quadratic constraint model is formulated and iteratively solved using the Quasi-Newton method. Simulation results demonstrate that the proposed algorithm can achieve CRLB accuracy even under high-noise conditions. Compared with localization algorithms based on joint transmitter and target estimation using TSWLS and SDP, the proposed algorithm achieves lower computational complexity and enables three-dimensional moving target localization with only four receivers. The proposed method, designed for a single-transmitter multiple-receiver system, can be directly extended to multiple-transmitter multiple-receiver configurations. Additionally, this method assumes time synchronization between the transmitter and receivers. Future research will focus on extending multistatic passive radar localization to scenarios where the transmitter is not synchronized with the receivers. -
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