We consider signal source localization from range-difference measurements. First, we give some readily-checked conditions on measurement noises and sensor deployment to guarantee the asymptotic identifiability of the model and show the consistency and asymptotic normality of the maximum likelihood (ML) estimator. Then, we devise an estimator that owns the same asymptotic property as the ML one. Specifically, we prove that the negative log-likelihood function converges to a function, which has a unique minimum and positive-definite Hessian at the true source's position. Hence, it is promising to execute local iterations, e.g., the Gauss-Newton (GN) algorithm, following a consistent estimate. The main issue involved is obtaining a preliminary consistent estimate. To this aim, we construct a linear least-squares problem via algebraic operation and constraint relaxation and obtain a closed-form solution. We then focus on deriving and eliminating the bias of the linear least-squares estimator, which yields an asymptotically unbiased (thus consistent) estimate. Noting that the bias is a function of the noise variance, we further devise a consistent noise variance estimator which involves $3$-order polynomial rooting. Based on the preliminary consistent location estimate, we prove that a one-step GN iteration suffices to achieve the same asymptotic property as the ML estimator. Simulation results demonstrate the superiority of our proposed algorithm in the large sample case.

翻译：我们研究基于距离差测量的信号源定位问题。首先，给出关于测量噪声和传感器部署的易于检验的条件，以保障模型渐近可辨识性，并证明极大似然估计量的一致性和渐近正态性。其次，设计一种与极大似然估计量具有相同渐近性质的估计方法。具体而言，证明负对数似然函数收敛至一个在真实信号源位置处具有唯一最小值和正定黑塞矩阵的函数，因此，在获得一致估计后执行局部迭代（例如高斯-牛顿算法）是可行的。主要挑战在于获取初步的一致估计。为此，通过代数运算和约束松弛构建线性最小二乘问题，并得到闭式解。进而推导并消除线性最小二乘估计量的偏差，从而得到渐近无偏（即一致）估计。注意到偏差是噪声方差的函数，进一步设计一种涉及三阶多项式求根的一致噪声方差估计器。基于初步一致位置估计，证明单步高斯-牛顿迭代足以达到与极大似然估计量相同的渐近性质。仿真结果表明，在大样本情况下所提算法具有优越性。