Validation gating is a fundamental component of classical Kalman-based tracking systems. Only measurements whose normalized innovation squared (NIS) falls below a prescribed threshold are considered for state update. While this procedure is statistically motivated by the chi-square distribution, it implicitly replaces the unconditional innovation process with a conditionally observed one, restricted to the validation event. This paper shows that innovation statistics computed after gating converge to gate-conditioned rather than nominal quantities. Under classical linear--Gaussian assumptions, we derive exact expressions for the first- and second-order moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimension-dependent contraction of the innovation covariance. The analysis is extended to NN association, which is shown to act as an additional statistical selection operator. We prove that selecting the minimum-norm innovation among multiple in-gate measurements introduces an unavoidable energy contraction, implying that nominal innovation statistics cannot be preserved under nontrivial gating and association. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.

翻译：验证门控是经典卡尔曼跟踪系统的基本组成部分。只有归一化创新平方（NIS）低于预设阈值的测量值才会被考虑用于状态更新。虽然这一过程在统计上由卡方分布驱动，但它隐含地将无条件的创新过程替换为一个有条件观测的过程，该过程被限制在验证事件内。本文证明，门控后计算的创新统计量收敛于门控条件下的量，而非名义量。在经典的线性-高斯假设下，我们推导了椭圆门控条件下创新的一阶和二阶矩的精确表达式，并证明门控会导致创新协方差发生确定性的、维度依赖的收缩。该分析被扩展到最近邻（NN）关联，证明其充当了额外的统计选择算子。我们证明，在多个门内测量值中选择最小范数创新会引入不可避免的能量收缩，这意味着在非平凡的门控和关联下无法保持名义创新统计量。二维情况下的闭式结果量化了这些效应的综合影响，并阐明了其实际意义。