4或5个传感器下，3维声源定位可精确求解，无需迭代或近似。
Time Difference of Arrival Source Localization: Exact Linear Solutions for the General 3D Problem
- 基于4或5个传感器的到达时间差，直接代数求解3维位置。
- 无噪声时计算结果精确到数值误差，5传感器方案无需符号歧义判断。
- 适合需要快速高精度定位的实时系统，如声学监测、无人机追踪。
到达时间差（TDOA）问题在4个或5个传感器且单个声源位于三维空间的情况下，存在完全代数的精确解，不涉及最小二乘投影或线性化。解法仅使用TDOA信息，不依赖频率差或角度信息，通过笛卡尔坐标系下的向量代数实现，具有代数透明性。5传感器解法无需处理符号歧义，4传感器解法则需解决一个符号歧义。数值实验表明，在无噪声条件下计算结果精确至数值误差；极少数定位失败情况由无先验条件下的符号歧义误判引起。因此，该方法在速度和精度上具备显著实际应用价值。
原文摘要 · Abstract (English)
The time difference of arrival (TDOA) problem admits exact, purely algebraic solutions for the situation in which there are 4 and 5 sensors and a single source whose position is to be determined in 3 dimensions. The solutions are exact in the sense that there is no least squares operation (i.e., projection) involved in the solution. The solutions involve no linearization or iteration, and are algebraically transparent via vector algebra in Cartesian coordinates. The solution with 5 sensors requires no resolution of sign ambiguities; the solution with 4 sensors requires resolution of one sign ambiguity. Solutions are effected using only TDOA and not, e.g., frequency difference of arrival (FDOA) or angle of arrival (AOA). We first present the 5-sensor solution and then follow with the 4-sensor scenario. Numerical experiments are presented showing the performance of the calculations in the case of no noise, before closing with conclusions. Performance of the calculations is exact within numerical error, and in the small fraction of cases in which source localization does not occur, it is driven by misidentification in resolution of sign ambiguity without priors. We therefore believe the calculations have substantial practical utility for their speed and exactness.
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