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 that involves $3$-order polynomial rooting. Based on the preliminary consistent location estimate, 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：我们考虑基于距离差测量的信号源定位问题。首先，我们给出一些易于检验的条件，涉及测量噪声和传感器部署，以确保模型的渐近可辨识性，并证明最大似然估计量的一致性和渐近正态性。然后，我们设计了一种与最大似然估计量具有相同渐近性质的估计器。具体而言，我们证明了负对数似然函数收敛到一个函数，该函数在真实源位置处具有唯一的最小值和正定的黑塞矩阵。因此，在得到一致估计后，执行局部迭代（例如高斯-牛顿算法）是可行的。主要问题是获得一个初步的一致估计。为此，我们通过代数运算和约束松弛构建了一个线性最小二乘问题，并得到了闭式解。然后，我们重点推导并消除了线性最小二乘估计量的偏差，从而得到渐近无偏（因此一致）的估计量。注意到偏差是噪声方差的函数，我们进一步设计了一个一致噪声方差估计量，该估计量涉及三阶多项式求根。基于初步的一致位置估计，一步高斯-牛顿迭代足以达到与最大似然估计量相同的渐近性质。仿真结果证明了我们提出的算法在大样本情况下的优越性。