A Closed-Form Noise-Sensitivity Asymmetry for Causal Branch Selection in Minimal-Array TDoA Localization

Abstract

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.