用于最小阵列到达时间差（TDoA）定位中因果分支选择的闭式噪声敏感性不对称性
A Closed-Form Noise-Sensitivity Asymmetry for Causal Branch Selection in Minimal-Array TDoA Localization
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中文总结 AI 辅助
研究最小阵列TDoA定位的分支选择模糊性问题，通过闭式分析得出不对称性符号条件，利用因果恒定记忆选择器，在多数操作点上物理根更敏感，外根规则实现高精度分支选择，而基于平滑性和连续性的方法在噪声下效果不佳。
中文摘要 AI 辅助
具有三个平面接收器的最小阵列到达时间差（TDoA）定位可简化为一个标量二次方程，其两个根都可能是可行的目标位置，这在最小几何结构中存在分支选择模糊性。传统上仅通过添加第四个接收器或角度传感器来消除这种模糊性，而牺牲了阵列的最小性。本文仅从测量中解决了这个问题。隐式微分表明两个根共享相同的敏感性分母，整个不对称性存在于分子中。闭式分析得出不对称性的精确符号条件，在可行内部，更敏感的根是外部的物理根。一个因果、恒定记忆选择器平滑每个根的可变性统计量，其期望值在噪声主导区域与每个根的敏感性成比例，并选择较大的根。在无量纲接收器图谱上，物理根在85%的双可行根操作点处更敏感，简单的外根规则在此内部实现了近0.99的分支选择精度。基于平滑性和连续性的自然替代方法在定时噪声下会反转且低于随机水平。
英文摘要
Minimal-array time-difference-of-arrival (TDoA) localization with three planar receivers reduces to a scalar quadratic whose two roots can both be feasible target positions, leaving a branch-selection ambiguity intrinsic to the minimal geometry. Because a minimal array is the cheapest, most deployable passive configuration, this ambiguity is conventionally broken only by adding a fourth receiver or an angle sensor, sacrificing the very minimality that motivates the array. This article resolves it from the measurements alone. Implicit differentiation shows that the two roots share an identical sensitivity denominator, so classical root conditioning cannot separate them and the entire asymmetry resides in the numerator. A closed-form analysis then yields an exact sign condition for the asymmetry, and over the feasible interior the more sensitive root is the outer, physical one, away from two algebraic degeneracy loci. A causal, constant-memory selector smooths a per-root variability statistic, whose expectation is shown to be proportional to the per-root sensitivity in the noise-dominated regime, and selects the larger one. Across a dimensionless receiver atlas the physical root is the more sensitive at a median of 85\% of two-feasible-root operating points (median sensitivity ratio 16), and, because the more sensitive root is the outer one, a simple outer-root rule attains near-0.99 branch-selection accuracy on this interior, matched online by a causal, constant-memory realization. The decisive empirical finding is that smoothness- and continuity-based disambiguation, the natural alternatives, invert under timing noise and fall below chance.
