Uncovering Source Ranges From Range Differences Observed by Sensors at Unknown Positions: Fundamental Theory

Abstract

This paper establishes the theory for the fundamental problem of uncovering the ranges of unknown transmitting sources from the range differences acquired by a number of sensors whose positions are not available. Obtaining the range differences from ranges is direct and the reverse problem is non-trivial, or even may not be solvable. The work is a pioneering study to the problem with a concrete investigation. Through the construction of a range difference expansion matrix and the application of linear algebra, we derive first, the minimum number of sources and that of receivers necessary for the problem to be solvable, second, the sufficient condition on the rank of the expansion matrix to ensure the recovery of the ranges, and finally, the algebraic solution to obtain the ranges from range differences. Simulations validate the theory and apply the proposed range solution to joint source and sensor localization.