Planar localization of radio-frequency or acoustic sources with two receivers

In the localization of electromagnetic or acoustic emitters, generally, when a pulse is radiated from a source, the wave will arrive to two receivers at different times. One of the advantages of measuring these time differences of arrival or TDOA is that it is not required a common clock as in other localization techniques based on the time of arrival of the pulse to the receiver. With only two sensors, all the possible points in the plane that would give the same TDOA describe a hyperbola. Using an independent third receiver and calculating the intersection of the three hyperbolas will give the position of the source. Therefore, planar localization of emitters using multilateration techniques can be solved at least with three receivers. This paper presents a method to locate sources in a plane with only two receivers reducing the number of acquisition channels and hence, the cost of the equipment. One of the receivers is in a fixed position and the other describes a circumference around the first one. The TDOA are measured at different angles completing a total turn and obtaining a periodic function, angle versus TDOA, that has all the geometric information needed to locate the source. The paper will show how to derive this function analytically with the distance from the fixed receiver to the source and a bearing angle as parameters. Then, it will be demonstrated that it is possible to fit the curve with experimental measurements to obtain the parameters of the position of the source.