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.
Keywords: Linear equation, One-dimensional double-base linear array, Path difference, Phase ambiguity, Phase comparison method, Phase difference localization equation, Phase shift.