Passive localization from Doppler-shifted frequency measurements

The position and speed of a moving source that radiates a pure tone can be determined by measurements of Doppler-shifted frequencies at five or more different locations. The equations relating to the measured frequencies to the localization parameters are nonlinear, and there is no closed-form solution. Solving nonlinear equations by a grid search, whereby all possible combination of source parameters are entered into the equation to see if they constitute a solution, would require a five-dimensional search. A more efficient search scheme that uses intermediate variables which are products of some of the unknowns is introduced. The search dimension is two. The resultant intermediate equations become linear and are easier to solve: the major calculation at each grid point requires only an inversion of a 3×3 matrix. The Cramer-Rao bound for localization is given, together with simulation results that verify the method