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With the development of space technology, high-precision passive localization based on small satellite platform plays an important role in civil and military fields and has broad application prospects.[1] The development trend of space electronic reconnaissance technology is to increase the measurement information of target and adopt joint positioning method to improve localization accuracy and increase reconnaissance range.[2] In recent years, several time-difference-of-arrival (TDOA)/frequency-difference-of-arrival (FDOA) localization methods with two or three satellites have been developed.[3]-[5] There are also literatures of multi-satellite TDOA/FDOA localization methods.[6]-[9] Compared to the existing passive localization systems, four-satellite localization using TDOA/FDOA shows more flexible constellation configuration and higher accuracy. In this article, we first derive a four-satellite TDOA/FDOA joint localization algorithm, and then present a geometric dilution of precision (GDOP)-based localization error analysis, from which we give some useful conclusions for guiding the real application of satellite passive localization.
Principle of four satellites using TDOA
Suppose that the model of a three-dimensional TDOA localization system (as shown in [Figure 1]) consists of one main satellite and three sub-satellites. [Formula Omitted: See PDF] is the main satellite, and [Formula Omitted: See PDF] are the three sub-satellites, respectively. [Formula Omitted: See PDF] ([Formula Omitted: See PDF]) is the coordinates of the i th sub-satellite. [Formula Omitted: See PDF] is the location of the target radiation source to be estimated.
Figure 1.
Principle of four-satellite localization system using TDOA.
[Image omitted: See PDF.]
Based on the geometry between the target and four satellites, we can obtain
[Formula Omitted: See PDF]
where [Formula Omitted: See PDF] is the radial distance between the main satellite and the target; [Formula Omitted: See PDF] is the radial distance between the sub-satellite i and the target; and [Formula Omitted: See PDF] is the distance difference ([Formula Omitted: See PDF]). [Equation (2)] can be obtained by simplifying [equation (1)]
[Formula Omitted: See PDF]
where [Formula Omitted: See PDF]. The matrix representation of [equation (2)] can be given by
[Formula Omitted: See PDF]
where
[Formula Omitted: See PDF]
If the three sub-satellites are not in line with the main satellite, [Formula Omitted: See PDF]. The estimated location of target can be given by the least squares (LS) solution
[Formula Omitted: See PDF]
Next, we analyze the accuracy of...