This paper investigates the problem of active localization of a moving object in its initial position and velocity, using time delay only or with Doppler shift measurements acquired by a number of monostatic sensors. Each sensor has non-negligible motion during the observation period, causing it at different positions when it sends and receives the signal, with the separation proportional to the signal travel time in reaching the object and returning back. The object is not at the same position when it reflects the signals from various sensors due to its motion. Both motion effects lead to recursive model equations for time delay and Doppler shift, making the localization problem interesting and challenging. We shall derive the measurement model equations under this scenario, evaluate the Cramer-Rao lower bound (CRLB) of the estimation problem and analyze the proposed models by contrasting with the performance loss when ignoring the object and sensor motion effects. The Maximum Likelihood Estimators (MLEs) are next developed, using the Gauss-Newton or Quasi-Newton iterations. Algebraic solution for the special case of moving object non-moving sensors is derived and analyzed, and it can serve as an effective initialization of the iterative MLEs if sensor motion is present. Both the theoretical analysis and simulation studies corroborate the importance of taking the object and sensor motions into consideration during the observation period, when the relative velocity between the object and sensor is significant compared to the signal propagation speed, such as in an acoustic or underwater environment.

中文翻译：

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通过时间延迟和多普勒频移对运动中的传感器进行运动物体的定位


本文研究了仅使用时间延迟或使用多个单基地传感器获取的多普勒频移测量来主动定位运动物体的初始位置和速度的问题。每个传感器在观察期间都有不可忽略的运动，导致其在发送和接收信号时处于不同的位置，其间隔与信号到达物体和返回的传播时间成正比。当物体由于其运动而反射来自各个传感器的信号时，其位置并不相同。这两种运动效应都会导致时间延迟和多普勒频移的递归模型方程，使定位问题变得有趣且具有挑战性。我们将推导这种情况下的测量模型方程，评估估计问题的 Cramer-Rao 下界（CRLB），并通过与忽略物体和传感器运动影响时的性能损失进行对比来分析所提出的模型。接下来使用高斯牛顿或拟牛顿迭代开发最大似然估计器 (MLE)。推导并分析了运动物体非运动传感器特殊情况的代数解，如果存在传感器运动，它可以作为迭代 MLE 的有效初始化。理论分析和仿真研究都证实了在观测期间考虑物体和传感器运动的重要性，此时物体和传感器之间的相对速度与信号传播速度相比非常显着，例如在声学或水下环境中。