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