在三维（3D）运动目标无源定位系统中，无模糊定位最少需要4个观测站。而传统的两步加权最小二乘（TSWLS）及其改进的闭式算法至少需要5个观测站进行求解，当减少一个观测站时，这些闭式算法往往无法提供可靠解。针对这一问题，提出一种最小化观测站数目的到达时间差（TDOA）与到达频率差（FDOA）定位算法。该算法是一种闭式解法并且能够在三维场景下仅使用4个观测站进行定位。该算法分为两步：第1步分离传统的TSWLS算法中未知参数空间，建立了一组新的等式，并且利用加权最小二乘（WLS）算法得到目标位置与速度的初始值；第2步利用泰勒级数展开算法将中间变量线性化，对目标位置和速度初始值进一步校正。理论分析证明了在适当的噪声水平下该算法能够达到克拉美罗界（CRLB）。此外，计算机仿真表明仅使用4个观测站时，该算法对于近场以及远场目标参数的估计精度在测量噪声较小时可以实现CRLB；并且还表明使用5个观测站估计时，该算法比TSWLS及其改进算法能更好地适应大的测量噪声。
In the Three-Dimension (3D) moving target passive positioning scenario, the minimum number of stations required for unambiguous localization is four. However, the traditional Two-Step Weighted Least Squares (TSWLS) and its improved closed-form algorithms require at least five stations. When reducing one station, these closed-form methods often cannot provide available solution. To solve this problem, a Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) based positioning algorithm which minimizes the number of stations is proposed in this paper. This algorithm is a closed-form solution and can locate the source with only four stations in the 3D scenario. This method has two steps:in the first step, a new set of equations is established by separating the unknown parameter space of the TSWLS from the known parameters, and the initial values of the source position and velocity are obtained by Weight Least Squares (WLS). In the second step, using taylor series expansion technique, the intermediates are linearized to further correct the initial values. Theoretical analysis proves that the proposed method can reach the Cramér-Rao Lower Bound (CRLB) at the appropriate noise level. In addition, simulations prove that both the far field and the near field parameter estimation accuracy of proposed method can achieve the CRLB at low noise level when only four stations are used.It also shows the positioning performance of the proposed algorithm can better adapt to large measurement noise than TSWLS or its improved algorithms when using five stations.
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