Research On Direct Position Determination Methods And Theories For Multi-station Co-Location System Assited By Transponders

The most widely used positioning method is two-step method,such as TDOA(Time Difference Of Arrival),FDOA(Frequency Difference Of Arrival),and DOA(Direction of Arrival)etc.When the SNR(signal-to-noise ratioSNR)is low,the two-channel signal cannot obtain enough information to calculate the parameters in the first step,and the two-step methods cannot find the emitters at all.In addition,the two-step positioning method,which disassembles the original optimization problem into two-step optimization,will theoretically result in performance loss.DPD(Direct Position Determination)method combines two-step positioning into one-step positioning,constructs cost function related to the emitter positions,and directly estimates the emitter positions.DPD can significantly improve positioning performance at low SNR scenarios and obtain higher positioning accuracy in theory than the two-step method.In order to locate multiple emitters with long distance and low SNR,this paper designs a positioning system for NLoS(Non-Line-of-Sight)emitters.In this system,the signal of the NLoS emitter is transmitted through the transponder installed on the aerial platform,the positioning range is expanded,and more observation equations about the emitters are obtained by the artificial multipath,to improve the availability and positioning accuracy of the positioning system.Due to the existence of multipath,the time-frequency difference information can not be accurately extracted from the signals of two receiving stations,so the two-step method cannot find the emitters in this scenario.DPD has been studied mainly for single-path propagation,and DPD performance analysis of different cost functions is still taken by numerical simulation.In this paper,in order to locate the NLoS emitters in a low SNR scenario,a GDPD(General DPD)model with multipath propagation is established,and the performances of cost functions for the GDPD model are studied.Besides,the global optimization performance of different cost functions and it’s solving algorithm are analyzed and developed.In addition,the close-formed of the asymptotic distribution of positioning error are derived and compared.The main contributions of this paper include:(1)A transponder-assisted positioning system for NLoS emitters is proposed and a signal model for GDPD is established.In traditional positioning systems,various methods are developed to eliminate the influence of multipath on positioning performance,such as equalization technology,UWB(Ultra-Wide Band)technology.In this paper,a transponder-assisted cooperative location system using artificial multipath to improve the location performance is proposed.Compared with single-path positioning system,this positioning system enlarges the positioning range by transmitting signals,constructs more observation equations,and obtains higher positioning accuracy and greater positioning availability than single-path system.Based on the positioning principle of the transponder-assisted NLoS positioning system,A General Direct Positioning Determination(GDPD)signal model with multi-emitter,multi-transponder,multireceiving station,multi-path propagation,multi-time observation and Doppler shift is extracted,and the functional relationship between the received signals and the positions of the emitter is constructed.The GDPD signal model is compatible with the existing of DPD models,and laid the foundation for the research of GDPD theory and methods.(2)Three methods for solving GDPD model and corresponding algorithm are proposed.In a direction-finding application or in a single path scenario,the length of the array manifold vector is always 1,and the inner product of the array manifold vector and the subspace coordinate axis can correctly represent the parallel or vertical relationship between the two vectors.But in a multipath propagation GDPD application,because of the multipath,there are two paths share the same propagation delay and Doppler shift,which leads to the singularity of the signal phase matrix in the array manifold matrix.When the path attenuation coefficient is optimized,the array manifold vector length of the GDPD model cannot be guaranteed to be 1.Therefore,the inner product operator cannot accurately quantify the geometric relationship between the array manifold vector and the subspace in the GDPD model,and then cannot use the inner product operator to optimize the position parameters of the array manifold vector.In order to quantify accurately the overlap relation between the array manifold space and the signal subspace and the orthogonal relation between the array manifold space and the noise subspace,it is necessary to design the correct operator to represent the linear and orthogonal relation between the array manifold vector and the subspace coordinate axis.In this paper,three solutions are proposed: the first is to normalize the length of array manifold,and scale the length of array manifold vector to the standard length when quantize the relationship between two vectors;the second is to restrict the length of array manifold,excluding the array manifold vector whose length is not 1 from the feasible region;the third is to add the non-negative constraints manually to eliminate the possibility which causes the array manifold vector not to be 1 regions.The Monotone Iterative Algorithm for fractional optimization,the Interior Point method for nonlinear constrained quadratic convex optimization and the Active Set Algorithm for linear mixed quadratic convex optimization are designed respectively.Finally,numerical simulations verify the effectiveness of the three solutions and its algorithm of the GDPD model.(3)A USF(Unified Subspace Fitting)framework of cost function for GDPD model is constructed,eleven cost functions are derived from USF,and closed-form expressions for the asymptotic distributions of positioning errors of cost functions are derived.Based on the USF framework,three sub-frameworks are proposed according to different subspace fitting objects: SSF(Signal Subspace Fitting),NSF(Noise Subspace Fitting)and NNSF(Normalized Noise Subspace Fitting).From the perspective of dimension,11 typical cost functions are given.SSF and NNSF are extensions of array manifold vector normalized methods,and NSF is extensions of manifold vector length and non-negative constraint methods.The relationship between the estimation error of different cost functions and the increase of the number of snapshots is analyzed theoretically.Under the USF framework,all kinds of cost functions contain eigenvectors of the correlation matrix of the received signal.The randomness of the received signal leads to the randomness of the eigenvectors,and further leads to the randomness of the optimization results.Based on the optimization theory and the conclusion of the second-order moments of the stochastic distribution of eigenvectors,the closed-form expressions of the asymptotic distribution of the estimation errors of various cost functions are analyzed theoretically.Based on the closed-form expressions,the weighting matrix is optimized to obtain the OWSSF(Optimal Weighted Signal Subspace Fitting)cost function.Finally,the performances of cost functions and CRLB(Cramér–Rao Lower Bound)are compared by numerical simulation.(4)A calibration model and method of GDPD method with model errors are given.When the model parameters such as transponder position,processing delay and mixing frequency drift have errors,A wide-band pseudo-random reference signal is designed to correct the system model errors.In order to solve the problem of low SNR of the reference signal and high complexity of parameter estimation,three accelerated algorithms are proposed.In order to improve the accuracy of parameter estimation and model error estimation,a super-resolution time-frequency difference signal model is proposed.Finally,a numerical simulation is given to verify the feasibility of the scheme.