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：我们考虑基于距离差测量的信号源定位问题。首先，我们给出了关于测量噪声和传感器部署的一些易于检验的条件，以确保模型的渐近可辨识性，并证明了最大似然（ML）估计量的相合性和渐近正态性。随后，我们设计了一种具有与ML估计量相同渐近性质的估计方法。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最小值且Hessian矩阵正定。因此，在得到相合估计后，执行局部迭代（如高斯-牛顿（GN）算法）是可行的。其中涉及的主要问题在于获得初步的相合估计。为此，我们通过代数运算和约束松弛构造了一个线性最小二乘问题，并给出了闭式解。我们进一步致力于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（即相合）估计。注意到该偏差是噪声方差的函数，我们进一步设计了一种涉及$3$次多项式求根的相合噪声方差估计量。基于初步的相合位置估计，我们证明单步GN迭代足以实现与ML估计量相同的渐近性质。仿真结果验证了所提算法在大样本情形下的优越性。