An asymptotically unbiased 3D source localisation method based on frequencies and angles measurements

Based on frequency of arrival (FOA) and angle of arrival (AOA) measurements, the localisation of a moving source using a number of stationary stations is discussed. A closed-form solution and bias reduction method using FOA and AOA measurements will be presented to quickly and accurately estimate target parameters, including location and velocity. The closed-form solution is implemented using two-stage weighted least squares, which constructs a pseudolinear equation by introducing auxiliary variables to perform linear estimation. In the process of linearisation, the authors utilise AOA measurements to simplify the FOA pseudolinear equation and reduce the number of auxiliary parameters. This means that fewer stations are needed to estimate the target parameters. However, the use of the computationally attractive pseudolinear formulation will introduce a non-ignorable localisation bias if the measurements are not sufficiently accurate. To solve the above problem, a quadratic constraint on least squares minimisation is considered in the bias reduction method. Under moderate Gaussian noise, theoretical analysis and simulation results show that the root mean square error of proposed method can significantly reduce positioning deviation and asymptotically approach the Cramer–Rao Lower Bound. Keywords Radar, Radar detection, Doppler shift, Parameter estimation.