The resolution of the integer phase ambiguity among a group of spatially separated sensors is considered, where the smallest inter-sensor spacing exceeds half wavelength (λ/2) of a single-frequency signal. The sufficient condition that the integer phase ambiguities can be unambiguously determined is derived and verified numerically. The traditional receiver of a triplet is identified as a particular example of this general framework. An efficient strategy for averaging redundant virtual observations is outlined. The performance in terms of the success rate of ambiguity resolution is significantly improved due to the averaging process.

Localization with low cost and high accuracy is of utmost importance for most applications in wireless sensor networks (WSNs). Most existing range-free schemes have limitations in one or more areas, such as low accuracy, low scalability and threat from the malicious nodes. Secure sensor localization in WSNs has received considerable attention with the rise of Internet of things. In this paper, we propose an efficient and secure range-free localization scheme based on the outlier elimination and the vector refinement process. An outlier elimination method is introduced to the localization process by filtering inaccurate beacon nodes in the presence of the malicious beacon nodes, which can prevent the location attacks and improve the localization accuracy. A vector refinement feedback process is introduced to further improve the localization accuracy. Parameters that affect the localization accuracy are discussed. Simulation and experiment are carried out to evaluate the performance of the proposed outlier elimination vector refinement DV-hop with the presence of the malicious beacon nodes. All the results show that the localization accuracy is improved significantly compared to the DV-hop algorithm and the vector refinement DV-hop algorithm, for both of the cases of with and without the malicious beacon nodes.