Algorithms Of Passive Localization Using Time And Frequency Differences Of Arrival

Passive source localization strongly supports electronic reconnaissance and electronic countermeasures.It is widely used in radar,navigation,sonar,surveillance,wireless communication,distributed sensor network,etc.Compared with the active location system,the passive location system has the advantages of better concealment,stronger anti-interference ability and further detection distance.Moreover,compared with the simple single passive location system,the multi-station passive location system can comprehensively utilize the more information from the multiple observations,then finds the preciser source location.In this paper,we focus on the algorithms of the passive source localization,on the basis of the Time Difference Of Arrival(TDOA),the Frequency Difference Of Arrival(FDOA)and TDOA,and the TDOA in the presence of the station errors.Since the TDOA and FDOA equations are highly nonlinear and the corresponding objective functions are nonconvex,the localization problem can not be solved in closed form.The localization algorithms can be classified into linear and nonlinear methods.Untill now,most of the existing algorithms focus on the linearization methods.These algorithms linearize the TDOA and FDOA nonlinear equations to estimate the source location.These algorithms have a small amount of calculation,and their estimated accuracy can reach the Cramer-Rao Bound(CRLB)under low noise level.However,since linearizing the nonlinear equations must bring up the loss of performance,all the linear algorithms will include a breakdown point,i.e.,when the noise power rises to a certain level,the location error will quickly increase and the location accuracy will deviate from CRLB.In this paper,we use the nonlinear algorithms to solve the TDOA localization problem.These algorithms have better performance comparing to the linear algorithms,and their location accuracy is higher in large noise environments.Since the nonlinear algorithm is easy to diverge when the initial value is too far from an optimal solution,we propose a Modified Newton algorithm(MNT).This algorithm is unsensitive to the selection of initial value,and can still converge to a solution when a bad initial value is chosen.For the joint TDOA and FDOA localization problem,we propose a two-step Newton algorithm.This algorithm makes the problem of the initial selection of the position and velocity variables be converted into the problem of the initial selection of only the position variables.This procedure reduces the difficulty of the initial value selection.For the TDOA localization problem in the presence of the station position errors,we propose a two-step Newton algorithm.This algorithm reduces the large calculations of the high dimensional Hessian matrix inversion caused by the station position errors.This algorithm also speeds up the downtrend of the objective function.The main research contents and achievements of this paper include the following aspects:1.When the initial value of a TDOA iterative algorithm is bad,it is easy to cause the ill-condition Hessian matrix which leads to the iteration divergence,then the location results can not converge to the optimal location.Based on the Newton(NT)method,the Modified Newton method(MNT)is proposed to modify the ill-condition Hessian matrix by the regularization theory.The regularization theory methods are divided into Tikhonov(TI),Diagonal Singular Value Decomposition(DSVD),etc.The regularization parameter which controls the properties of the regularized solution is determined by the L-curve method.Like the NT method,the Taylor Series(TS)method can also be modified,Then the Modified TS(MTS)method is proposed.The MNT and MTS algorithms are robust to the choice of the initial values compared with the NT and TS,and they also have better performance.Compared with the NT,TS algorithms and the classical algorithms,they have better location accuracy in low SNR environments.2.When the source and the receiver station have a relative motion,the FDOA observation information is needed to be combined with the TDOA to estimate the position and speed of the source.Therefore,a two-step Newton algorithm is proposed based on the traditional Newton method.The first step does not consider the source speed,only the source position is solved,and the iterative divergence problem can be overcomed by the regularization theory.The second step estimates the source position and speed together,and also refine the source position estimate at the same time.The two-step method is used to make selecting the initial value of the posotion ans speed variables be turned into choosing the initial value of position variables.The two step method is robust to both the location and the velocity initial values,and it has a better accuracy compared to the classicall algorithms.3.In practical application,the location of the receiver station is not entirely known,the existence of the receiver position error will have a greater impact on the TDOA location accuracy,and even the CRLB will rise with the sensor position error.When Newton method is used to solve the source and station location,the algorithm not only has an effect on the initial value,but also faces the problem of high dimensional Hessian matrix inversion and slow downtrend of the objective function.The two-step method is proposed.The first step is to assume that the stations are free from error.In this case,the source position is solved.The source position is used as the initial value.In the second step,the position errors of the stations are considered,and the positions of the source and the stations are estimated jointly.The algorithm is robust to the initial chosen,and also speeds up the convergence rate,reduces the computation of the high dimensional Hessian matrix inversion.4.The new source of the multiple model Probability Hypothesis Density(PHD)of the Particle Filter(PF)is proposed.With the source location and velocity information,the proposed algorithm initialize the new particle at each time.In this algorithm,the state of the motion source is estimated by combining the localization and tracking algorithm.The algorithm not only uses the observation information,but also uses the state association information to realize the high precision estimate of the source location and velocity.