Rank Properties for Matrices Constructed From Time Differences of Arrival

Time Difference of Arrival (TDOA) positioning has been an active research over the years due to its rich amounts of applications. Improving the quality of the TDOA measurements, especially under harsh scenarios, is crucial to achieve better localization accuracy and higher noise tolerance. Increasing the TDOA quality has been traditionally based on the statistical estimation point of view. Nevertheless, TDOA exhibits interesting algebraic properties, which when exploited can further enhance the measurements. This paper develops two algebraic properties of TDOA in the situation where multiple sources and sensors are present, such as sensor network and joint source-sensor localizations. The algebraic properties are new and have not appeared in the literature. They are in terms of the rank behaviors of two matrices that are formed by the TDOA themselves and their squared values. The development starts from an established property of Time of Arrival and applies determinant and vector space manipulations to obtain the concrete results. An example is included to illustrate the use of the algebraic properties for reducing the noise in TDOAs.