TDOA based direct positioning maximum likelihood estimator and the cramer-rao bound

The maximum likelihood estimator (MLE) and its performance for the localization of a stationary emitter using a network of spatially separated passive stationary sensors is presented. The conventional approach for localization using multiple sensors is to first estimate the time differences of arrival (TDOAs) independently between pairs of sensors and then find the location of the emitter using the intersection point of the hyperbolas defined by these TDOAs. It has recently been shown that this two-step approach is suboptimal and an alternate direct position determination (DPD) approach has been proposed. In the work presented here we take the DPD approach to derive the MLE and show that the MLE outperforms the conventional two-step approach.We analyze the two commonly occurring cases of signal waveform unknown and signal waveform known with unknown transmission time. This paper covers a wide variety of transmitted signals such as narrowband or wideband, lowpass or bandpass, etc. Sampling of the received signals has a quantization-like effect on the location estimate and so a continuous time model is used instead.We derive the Fisher information matrix (FIM) and show that the proposed MLE attains the Cramer-Rao lower bound (CRLB) for high signal-to-noise ratios (SNRs).