The General Solution of the Phase Difference Positioning Equation

For passive positioning technology, not only the signal transmitting time of the target is not known at all, but also the initial phase of the signal is not known. Therefore, the phase comparison method cannot be used. But if we just use geometry, starting from the basic definition of phase shift detection of distance, it is explained in mathematical form that, because phase shift measurement has the fuzziness of period, the number of wavelengths contained in the observed quantity itself is an unknown variable to be determined, so the positioning equation based on phase shift measurement is unsolvable if we only use existing analytical methods. If the unknown quantity representing the period ambiguity is regarded as an undetermined quantity, the solution of the phase difference localization equation can still be obtained formally. As a mathematical basis, this chapter gives a linear solution method of passive positioning equation based on path difference measurement in two dimensional plane. In the passive positioning problem based on path difference measurement in two-dimensional plane, it is generally necessary to use at least 3 or more measuring stations to collect data to get the path difference between the radiation source and each measuring station. The existing method is to make use of these distance differences to form a set of nonlinear hyperbolic equations about the position of the radiation source, and the coordinate position of the radiation source can be obtained by solving the hyperbolic equations. The author's existing research results show that for the plane multi-station positioning problem, the linear equations can be obtained if the auxiliary equations are constructed by using the existing plane geometric relations on the basis of the path difference measurement.