Research On Multi-station TDOA/FDOA Localization Methods Via Convex Optimization

The multi-station time-frequency difference positioning technology has the advantages of strong concealment,high positioning accuracy,and wide application fields,making this type of passive positioning technology attracts more and more people's attention.However,because the mathematical relationships between the observations such as timefrequency difference and the target state parameters to be estimated are not only non-linear,but also non-convex,the amount of calculation required to directly solve the corresponding maximum likelihood estimation problem is very large,and it is difficult to guarantee real-time positioning.Among the commonly used positioning solutions,the grid search method has a higher computational complexity,and the amount of calculation increases rapidly with the improvement of the positioning accuracy;due to the local convergence,the iterative method generally requires a suitable initial value estimation;because the quadratic term of noises are ignored in the closed solution,the positioning accuracy is poor when the time-frequency difference measurement errors are large.Therefore,it is necessary to study the time-frequency difference positioning technology based on convex optimization that has global convergence and can fully utilize the nonlinear constraint relationship between parameters.The convex optimization positioning technology is to use a convex positioning problem to approximate the original non-convex positioning problem by introducing some nuisance variables or relaxation terms,and then use the interior point method to solve it.Therefore,the main difference between different convex optimal positioning methods is the convex positioning problem to be solved.Based on the analysis of the original non-convex maximum likelihood estimation problem,this paper studies two approximate positioning problem models,and gives corresponding convex optimization positioning methods.The main work includes:(1)The convex optimal positioning method based on maximum likelihood estimation is studied.The advantages and disadvantages of the second-order cone relaxation method and semi-definite relaxation method are analyzed theoretically,and the convex positioning problem is solved using the CVX toolbox.The simulation results show that the second order cone relaxation method needs to introduce a penalty term,and the penalty factor has a greater impact on the positioning accuracy,and the semi-definite relaxation method introduces more nuisance variables and relaxation terms,which increase the computational complexity.(2)The convex optimal positioning method based on pseudo-linear least squares is studied,and its performance is compared with the two-step weighted least squares method.Theoretical analysis and simulation experiments show that because the pseudo-linear least squares problem model ignores the quadratic term of noise,both methods have a threshold effect when the measurement error is large,but the performance of the convex optimal positioning method is better than the two-step weighted minimum Double multiplication.(3)Aiming at the problems existing in the pseudo-linear least squares problem model,a stochastic robust least squares problem model is studied,and a convex optimal positioning method based on stochastic robust least squares is given.Simulation results show that compared with the first two convex optimization positioning methods,the positioning accuracy of the proposed method is close to the convex optimization positioning method based on maximum likelihood estimation,but the algorithm complexity is much smaller than the latter;the algorithm complexity of that is similar to the convex optimal positioning method based on pseudo-linear least squares,but the positioning accuracy is better than the latter.