This paper investigates the localization of a moving source in position and velocity, by observing the emitted frequency from the source that is subject to the Doppler shift effect at a number of stationary sensors. Previous attempts rely on exhaustive grid search or numerical polynomial optimization to obtain a solution. We shall propose a constrained optimization to formulate the localization problem, which enables the problem to be solved efficiently using the linear optimization method to reach a closed-form solution or the semi-definite relaxation technique to achieve a noise resilient estimate. The algorithms are developed for the single-time and multiple-time observations. The presence of errors in the source frequency and the sensor positions are considered. The non-cooperative scenario where the source frequency is completely not known is also addressed. Analysis validates the proposed closed-form solution in reaching the Cramer-Rao Lower Bound accuracy under Gaussian noise over the small error region. Simulations support the performance of the proposed solutions.

中文翻译：

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通过频率测量定位移动源

本文通过观察受多个静止传感器多普勒频移效应影响的源发射频率，研究了移动源在位置和速度方面的定位。以前的尝试依赖于详尽的网格搜索或数值多项式优化来获得解决方案。我们将提出一种约束优化来制定定位问题，这使得可以使用线性优化方法有效地解决问题以达到封闭形式的解决方案或半定松弛技术以实现噪声弹性估计。这些算法是为单次和多次观测而开发的。考虑了源频率和传感器位置中误差的存在。还解决了源频率完全未知的非合作场景。分析验证了所提出的封闭形式解决方案在小误差区域上的高斯噪声下达到 Cramer-Rao 下界精度。模拟支持所提出的解决方案的性能。