CN105722214A - GSM-R interference source positioning method covering base station position errors - Google Patents

Abstract

The invention belongs to the technical field of the positioning of interference sources of the railway wireless communication network, and relates to a GSM-R interference source positioning method covering base station position errors. The method mainly comprises the steps of acquiring n-1 arrival time difference measured values and n arrival angle measured values from a GSM-R network by using a positioning algorithm based on the signal arrival time difference and a positioning algorithm based on the signal arrival angle; constructing a model of geometrical relationship between interference sources and mobile base stations; under the condition of ignoring the position error of the mobile base station, acquiring the estimated value of the coordinate and the error vector of the interference source; constructing a position error weighting matrix; and via the position error weighting matrix, reconstructing a covariance matrix of the error vector, and acquiring the accurate positioning result of the interference source. The method has the advantages that the influence of the position error of the mobile base station on the interference source positioning result is eliminated, the positioning accuracy of the interference source is further improved, and the shortage that the position errors of the mobile base stations are not considered in the existing railway wireless communication network interference source positioning method is covered.

Description

A kind of interference source localization method of the GSM-R considering base station location error

Technical field

The invention belongs to GSM-R (Global System for Mobile Communications Railway, railway Wireless communication networks) interference source field of locating technology, relate to a kind of interference source localization method for GSM-R.

Background technology

Along with the development of China Railway, the Railway Mobile Radio system of existing analog system falls due to technology Afterwards, equipment is outmoded, function singleness, can not meet the demand of modernized railway development, therefore, GSM-R based on GSM standard (Global System for Mobi le Communications-Railway) has obtained quick development.GSM-R is one Planting the radio digital communication system exclusively for railway communication design, this system adds scheduling, control etc. on the basis of GSM standard Communication function, and possess the feature used under high velocity environment.The continuous maturation of GSM-R technology, accelerates China railways information-based The paces built, provide guarantee reliably for the development and raising competitiveness promoting China railways.

In the range of GSM-R frequency band operates mainly in 885-889/930-934MHz, 4MHz bandwidth altogether, this frequency range was once ferrum Road GSM-R and China Mobile's public mobile communication system are shared by region, are exclusively enjoyed this frequency range by GSM-R system now.Due to GSM Network is arranged net relatively early, after GSM-R frequency range exclusively enjoys, although the method that GSM takes supplement frequency range, but being total to of GSM and GSM-R network Whole having been resolved is not had, so the signal source of China Mobile GSM supplement frequency range still can be to GSM-R network with covering problem Produce interference, moreover, remaining signal as UNICOM, telecommunications and radio set etc. are launched is also possible to cause with frequency Disturb and Intermodulation Interference, and illegal undelegated base station or antenna will also result in co-channel interference, due to the ground of GSM-R network Shape is complicated, the problem that network internal there is also co-channel interference.These wireless interference signals all can affect GSM-R normally receive and Launch, bring the problems such as poor, the channel congestion of call drop, speech quality to the mobile communication of base station coverage area, thus cause railway The work efficiency drop of operation, brings secret worry even to the development of the safe operation of railway, the life security of passenger and country.Cause This, need the efficient GSM-R interference source location algorithm of development badly, eliminate the interference source shadow to Measurement of Railway Radio Communication System physically Ring, it is ensured that the safe operation of Railroad Communication System.

The location algorithm of GSM-R network interference source is broadly divided into ranging localization algorithm and range-free localization algorithm two class, non- Location algorithm mainly includes centroid algorithm, Dv-Hop algorithm and APIT algorithm.Owing to non-ranging algorithm can only provide rough grade Positioning service, has therefore developed location algorithm based on range finding.Location algorithm based on range finding is by Measurement Network interior joint Between distance or orientation angle and signal intensity calculate location node coordinate.Main measurement technology has AOA (arrival Of angle angle of arrival), RSS (received signal strength received signal strength), TOA (time of Arrival time of advent), TDOA (time difference of arrival step-out time) etc..At present, calculate in these location In method, the most not occurring there may be the situation of site error in view of movement station, therefore the present invention has developed a kind of consideration shifting The railway wireless communication net interference source localization method of dynamic station location error, further increases the positioning precision to interference source, more Mend this deficiency not considering mobile station location error in existing railway wireless communication net interference source localization method.

Summary of the invention

The present invention seeks to for shortcoming present in existing GSM-R network interference source location technology with not enough, with position Put error weighting matrix, eliminate the impact on positioning result of the mobile station location error, develop a kind of consideration mobile station location by mistake The railway wireless communication net interference source localization method of difference.

The technical scheme is that the interference source localization method of a kind of GSM-R considering base station location error, its feature It is, comprises the following steps:

Step 1: obtain n-1 TDOA and n AOA measured value, builds geometric equation group.

The position coordinates assuming n movement station in railway wireless communication net is (x_i,y_i), i=1,2 ..., n, interference source Position coordinates be (x, y), builds equation below group to interference source and movement station:

Wherein r_iFor the distance value of interference source to i-th movement station, r_i,1For interference source to i-th movement station distance with To the range difference of the 1st movement station, θ_iRadian for interference source to i-th movement station.

Formula (1) is launched abbreviation can obtain:

Wherein, x_i,1=x_i-x₁, y_i,1=y_i-y₁,

Being write formula (2) as matrix form is:

Y=GZ (3)

Wherein,

Step 2: ignore mobile station location error, is carried out according to a preliminary estimate position of interference source coordinate.

In the set up matrix equation of step 1, r_iAnd θ_iFor distance value and the radian value of actual measurement, there is certain measurement error, As follows:

Wherein,WithFor real distance value and radian value,WithIt is respectively the range error and radian error measured, Obeying average is the Gauss distribution of zero, and variance is respectivelyWith

By formula (4) substitution formula (3) can obtain residual error it is:

E=Y-GZ (5)

Formula (5) is asked maximum likelihood estimator, and the position coordinates that can obtain interference source is worth according to a preliminary estimate:

Z=(G^Tcov(e)^-1G)^-1G^Tcov(e)^-1Y (6)

Wherein, cov (e) is the covariance matrix of error vector:

Cov (e)=E (ee^T) (7)

Launch to obtain by formula (5):

Wherein,

From formula (3)Therefore to formula (8) carry out Taylor expansion and ignore high-order term, can obtain:

E can be obtained by formula (9)_ie_j, e_n+i-1e_n+j-1,WithExpected value be:

By formula (10), (11), (12), (13) substitution formula (7) can obtain the covariance matrix of error vector and be:

Wherein, B_θIn containing unknown quantity x and y, first can carry out according to a preliminary estimate with TDOA, then estimated value is substituted into B_θ, it may be assumed that

Wherein,

Step 3: be worth according to a preliminary estimate by position of interference source coordinate, builds site error weighting matrix.

The position of interference source coordinate obtained by formula (6) is worth according to a preliminary estimate, does not accounts for the site error of movement station, actual The position coordinates of upper movement station is to carry out the error that the existence that positions is certain, therefore for improving the positioning accurate of interference source by GPS Degree, need to build site error weighting matrix according to the value according to a preliminary estimate of position of interference source coordinate, eliminates site error to location knot The impact of fruit.

First, the movement station of band site error is modeled:

Wherein,WithFor site error, the separate and Gauss distribution of obedience zero-mean, standard deviation is respectively σ_xWith σ_y。

It is updated to formula (16) in formula (5) launch to obtain:

Wherein

From formula (3)Therefore to formula (17), (18) Taylor expansion also ignores high-order term, can obtain:

By formula (19), (20) can obtain e_ie_j, e_n+i-1e_n+j-1,WithExpected value be:

By formula (21), (22), (23), (24) understand site error weighting matrix:

B in formula (25) and (26)_rxAnd B_ryThere is unknown quantity x and y, the position of interference source required by available step 2 Chinese style (6) Coordinate is worth replacement according to a preliminary estimate.

Step 4: eliminate mobile station location error, be accurately positioned interference source.

Obtained the site error weighting matrix eliminating mobile station location error by step 3, therefore restructural error is vowed The covariance matrix of amount, optimizes interference source positioning result further.

By formula (21), (22), (23), (24), (25), (26), (27), the error vector covariance square that (28) can reconstruct Battle array is:

Error vector covariance matrix formula (29) of reconstruct is substituted in formula (6), can obtain eliminating mobile station location error Position of interference source coordinate estimated value.

Position of interference source coordinate owing to being tried to achieve by formula (6) does not considers x, y, r₁Between dependency, therefore can enter One-step optimization interference source positioning result.

First, it is assumed that x, y, r₁Estimation difference be μ₁,μ₂,μ₃, then the estimated value tried to achieve by formula (6) is represented by:

[z]₁=x+ μ₁, [z]₂=y+ μ₂, [z]₃=r₁+μ₃ (30)

Error vector can be constructed as follows by formula (30):

E'=Y'-G'Z'(31)

Wherein

Formula (30) is updated to formula (31), and ignores quadratic term and can obtain:

Then the covariance matrix of e' is:

Cov (e')=E (e'e'^T)=D{cov (Z) } D+Q (33)

Wherein, D=diag{ [2 (x-x₁) 2(y-y₁) 2r₁]}≈diag{[2([z]₁-x₁) 2([z]₂-y₁) 2 [z]₃], , cov (Z) is the covariance matrix of the estimated value that formula (6) is tried to achieve, and can be solved by perturbation analysis method, and definition Δ is that error is disturbed Dynamic component, then formula (6) can be write as:

Due toTherefore launched by formula (34) and ignore high-order term to obtain:

Formula (35) abbreviation is transplanted:

Due toSo can be obtained by formula (36):

Can be obtained cov (Z) by formula (37) is:

Maximal possibility estimation is asked to obtain formula (31):

The position coordinates estimated value of the interference source after finally can being tried to achieve optimization by formula (39) is:

Beneficial effects of the present invention is, the present invention utilizes TDOA/AOA mixed positioning algorithm that interference source is carried out coarse positioning and obtains The value according to a preliminary estimate arrived, constructs site error cost matrix, eliminates mobile station location error to interference source positioning result Impact, further increases the positioning precision of interference source, compensate in existing railway wireless communication net interference source localization method Do not consider this deficiency of mobile station location error.

Accompanying drawing explanation

The interference source positioning flow figure of Fig. 1 present invention.

The movement station of Fig. 2 present invention and interference source coordinate random placement figure.

The position of interference source coordinate setting design sketch of Fig. 3 present invention.

The position of interference source coordinate setting Error Graph of Fig. 4 present invention.

Detailed description of the invention

With embodiment, the present invention is described in detail below in conjunction with the accompanying drawings

As it is shown in figure 1, the flow process of the present invention is: the TDOA first movement station from railway wireless communication net obtained and AOA measured value builds geometric equation group, and position of interference source coordinate is carried out according to a preliminary estimate by the site error ignoring movement station afterwards, Then the value according to a preliminary estimate by obtaining builds site error weighting matrix, finally eliminates mobile station location error and enters interference source Row is accurately positioned.

Embodiment

As in figure 2 it is shown, on railway wireless communication network two dimensional surface, deploy 8 movement stations for measuring interference source TDOA and AOA value, coordinate be respectively (0,0), (0,5000), (1250,2500), (2500,5000), (2500,0), (3750, 2500), (5000,5000), (5000,0), with (x_i,y_i), i=1,2 ..., 8 represent, wherein No. 1 movement station is service station.Dry Wherein, in this embodiment, position of interference source coordinate is (3650.3839,3833.9651) to the source random distribution of disturbing, with (x, y) table Show.

Step 1: obtain 7 TDOA and 8 AOA measured values, builds geometric equation group.

First movement station and interference source are modeled, build as follows by 7 TDOA obtained and 8 AOA measured values Geometric equation group:

Wherein r_iFor the distance value of interference source to i-th movement station, r_i,1For interference source to i-th movement station distance with To the range difference of the 1st movement station, θ_iRadian for interference source to i-th movement station.

Formula (41) is launched abbreviation can obtain:

Wherein, x_i,1=x_i-x₁, y_i,1=y_i-y₁,

Being write formula (42) as matrix form is:

Y=GZ (43)

Wherein,

Step 2: ignore mobile station location error, is carried out according to a preliminary estimate position of interference source coordinate.

In the set up matrix equation of step 1, r_iAnd θ_iFor distance value and the radian value of actual measurement, there is certain measurement error, As follows:

Wherein,WithFor real distance value and radian value,WithIt is respectively the range error and radian error measured, Obeying average is the Gauss distribution of zero, and standard deviation is respectively σ_r=10 and σ_θ=0.02.

By formula (44) substitution formula (43) can obtain residual error it is:

E=Y-GZ (45)

The covariance matrix of formula (45) is:

Wherein, B_θIn containing unknown quantity x and y, first can carry out according to a preliminary estimate with TDOA, then estimated value is substituted into B_θ, it may be assumed that

Wherein,

The position of interference source coordinate of mobile station location error can be ignored by formula (45) and formula (46) and be worth according to a preliminary estimate and be:

Step 3: be worth according to a preliminary estimate by position of interference source coordinate, builds site error weighting matrix.

The movement station of band site error is modeled:

WhereinWithFor site error, separate and obedience average is the Gauss distribution of zero, and standard deviation is respectively σ_x= 3 and σ_y=3, with site error mobile station location coordinate diagram as shown in Figure 3.

Being substituted in formula (45) by formula (49) band, can try to achieve site error weighting matrix is:

WhereinX and y is respectively the true coordinate value of movement station and interference source, for obtaining the weighting of site error Matrix,WithThe mobile station location coordinate x of available band site error_iAnd y_iReplace, the interference source position that x and y is obtained by formula (48) Put coordinate setting estimated value to replace.

Step 4: eliminate mobile station location error, be accurately positioned interference source

For eliminating the impact on interference source positioning result of the mobile station location error, need reconstructed error vectors covariance matrices, The error vector covariance matrix that then can be reconstructed by formula (50), (51), (52), (53) is:

Error vector covariance matrix formula (54) of reconstruct is substituted in formula (48), can obtain eliminating mobile station location error Position of interference source coordinate estimated value be:

The covariance matrix of formula (55) is:

The position of interference source coordinate tried to achieve by formula (55) does not considers x, y, r₁Between dependency, therefore can be further Optimize interference source positioning result.

Assume x, y, r₁Estimation difference be μ₁,μ₂,μ₃, then the estimated value tried to achieve by formula (55) is represented by:

[z]₁=x+ μ₁, [z]₂=y+ μ₂, [z]₃=r₁+μ₃ (57)

Error vector can be constructed as follows by formula (57):

E'=Y'-G'Z'(58)

Wherein

Then the covariance matrix of formula (58) is:

Cov (e')=E (e'e'^T)=D{cov (Z) } D+Q (59)

Wherein, D=diag{ [2 (x-x₁) 2(y-y₁) 2r₁]}≈diag{[2([z]₁-x₁) 2([z]₂-y₁) 2 [z]₃],。

The estimated value that can be obtained formula (58) by maximal possibility estimation is:

The position coordinates of the interference source after can being tried to achieve optimization by formula (60) is:

Formula (61) is final interference source positioning result, locating effect figure as it is shown on figure 3, as can be seen from Figure 3 this The true coordinate value of bright interference source positioning result closely interference source.

In order to further illustrate the locating effect of the inventive method, give at Fig. 4 and measure noise circumstance in different distance Under position error effect, in Fig. 4 range measurement noise error obey zero-mean Gauss distribution, standard deviation scope be 10～ 20m, as can be seen from Figure 4 there is less position error in the inventive method.

From above-mentioned the result it can be seen that the position of interference source coordinate using the inventive method to estimate has the least location The margin of error, location is preferably.

Claims (5)

1. the interference source localization method of the GSM-R considering base station location error, it is characterised in that comprise the following steps:

A. location algorithm based on signal step-out time and location algorithm based on direction of arrival degree are used, from GSM-R network N-1 step-out time measured value of middle acquisition and n angle of arrival measured value；Wherein, the number of movement station during n is GSM-R network； Build the geometrical relationship model between interference source and movement station；

B., according to the geometrical relationship model obtained in step a, under conditions of ignoring mobile station location error, interference source position is obtained Put coordinate estimated value and error vector；

C. according to the position of interference source coordinate estimated value obtained in step b and error vector, site error weighting matrix is built；

D. by site error weighting matrix, the covariance matrix of reconstructed error vector, obtain interference source positioning result.

The interference source localization method of a kind of GSM-R considering base station location error the most according to claim 1, its feature exists In, the concrete construction method of geometrical relationship model described in step a is:

The position coordinates assuming n movement station in GSM-R network is (x_i,y_i), i=1,2 ..., n, the position of interference source is sat It is designated as that (x, y), wherein, subscript i refers to the sequence number of movement station；Build following geometric equation group:

$\left\{\begin{array}{c}{r}_{i,1}=r_i-r_1=\sqrt{{(x-x_i)}^2+{(y-y_i)}^2}-\sqrt{{(x-x_1)}^2+{(y-y_1)}^2}\\ r_i^2={(r_{i,1}+r_1)}^2=r_{i,1}^2+2r_{i,1}+r_1^2,i=2,\mathrm{...},n\\ {\mathrm{tan\θ}}_i=\frac{{\mathrm{sin\θ}}_i}{{\mathrm{cos\θ}}_i}=\frac{y-y_i}{x-x_i},i=1,2,\mathrm{...},n\end{array}\right.$

Wherein r_iFor the distance value of interference source to i-th movement station, r_i,1For the distance of interference source to i-th movement station and to the 1st The range difference of individual movement station, θ_iRadian for interference source to i-th movement station；

This deployable abbreviation of geometric equation group is:

$\left\{\begin{array}{c}{x}_{i,1}x+y_{i,i}y+r_{i,1}{r}_1=\frac{1}{2}(k_i-k_1-r_{i,1}^2)\\ {\mathrm{sin\θ}}_{i}x-{\mathrm{cos\θ}}_{i}y=x_i{\mathrm{sin\θ}}_i-y_i{\mathrm{cos\θ}}_i\end{array}\right.$

Wherein, x_i,1=x_i-x₁, y_i,1=y_i-y₁,

Being write as matrix form is:

Y=GZ；

Wherein, $G=\left\{\begin{array}{ccc}{x}_{2,1}& y_{2,1}& r_{2,1}\\ .& .& .\\ .& .& .\\ .& .& .\\ x_{n,1}& y_{n,1}& r_{n,1}\\ {\mathrm{sin\θ}}_1& -{\mathrm{cos\θ}}_1& 0\\ .& .& .\\ .& .& .\\ .& .& .\\ {\mathrm{sin\θ}}_n& -{\mathrm{cos\θ}}_n& 0\end{array}\right\},Y=\frac{1}{2}\left\{\begin{array}{c}{k}_2-k_1-r_{2,1}^2\\ .\\ .\\ .\\ k_n-k_1-r_{n,1}^2\\ 2x_1{\mathrm{sin\θ}}_1-2y_1{\mathrm{cos\θ}}_1\\ .\\ .\\ .\\ 2x_n{\mathrm{sin\θ}}_n-2y_n{\mathrm{cos\θ}}_n\end{array}\right\},Z=\left\{\begin{array}{c}x\\ y\\ r_1\end{array}\right\}.$

The interference source localization method of a kind of GSM-R considering base station location error the most according to claim 2, its feature exists In, obtain position of interference source coordinate estimated value and error vector described in step b method particularly includes:

Assume that the distance value surveyed and radian value and real relation between distance value and radian value are:

$\left\{\begin{array}{c}{r}_i={\hat{r}}_i+n_{r_i}\\ {\mathrm{\θ}}_i={\widehat{\mathrm{\θ}}}_i+n_{{\mathrm{\θ}}_i}\end{array}\right.$

Wherein,WithFor real distance value and radian value,WithIt is respectively the range error and radian error measured, clothes Being the Gauss distribution of zero from average, variance is respectivelyWith

It is e=Y-GZ that associate(d) matrix Y=GZ can obtain error vector e；

After ignoring mobile station location error, position of interference source coordinate estimated value is the maximum likelihood estimator of e=Y-GZ, can obtain: Z =(G^Tcov(e)^-1G)^-1G^Tcov(e)^-1Y, wherein cov (e) is the covariance matrix of error vector: cov (e)=E (ee^T)。

The interference source localization method of a kind of GSM-R considering base station location error the most according to claim 3, its feature exists In, build site error weighting matrix described in step c method particularly includes:

The movement station of band site error is modeled:

$x_i={\hat{x}}_i+n_{x_i},y_i={\hat{y}}_i+n_{y_i}$

Wherein, wherein,WithFor site error, the separate and Gauss distribution of obedience zero-mean, standard deviation is respectively σ_xWith σ_y；

Will $x_i={\hat{x}}_i+n_{x_i},y_i={\hat{y}}_i+n_{y_i}$ Substitute into e=Y-GZ can obtain:

$\begin{array}{c}{e}_{n+i-1}=x_i{\mathrm{sin\θ}}_i-y_i{\mathrm{cos\θ}}_i-{\mathrm{xsin\θ}}_i+{\mathrm{ycos\θ}}_i\\ =({\hat{x}}_i-x+n_{x_i})\sin ({\widehat{\mathrm{\θ}}}_i+n_{{\mathrm{\θ}}_i})-({\hat{y}}_i-y+n_{y_i})\cos ({\widehat{\mathrm{\θ}}}_i+n_{{\mathrm{\θ}}_i})\\ =({\hat{x}}_i-x+n_{x_i})(\sin {\widehat{\mathrm{\θ}}}_i{\mathrm{cosn}}_{{\mathrm{\θ}}_i}+{\mathrm{sinn}}_{{\mathrm{\θ}}_i}\cos {\widehat{\mathrm{\θ}}}_i)-({\hat{y}}_i-y+n_{y_i})(\cos {\widehat{\mathrm{\θ}}}_i{\mathrm{cosn}}_{{\mathrm{\θ}}_i}-\sin {\widehat{\mathrm{\θ}}}_i{\mathrm{sinn}}_{{\mathrm{\θ}}_i})\\ =\[({\hat{x}}_i-x)\sin {\widehat{\mathrm{\θ}}}_i-({\hat{y}}_i-y)\cos {\widehat{\mathrm{\θ}}}_i\]{\mathrm{cosn}}_{{\mathrm{\θ}}_i}+(n_{x_i}\sin {\widehat{\mathrm{\θ}}}_i-n_{y_i}\cos {\widehat{\mathrm{\θ}}}_i){\mathrm{cosn}}_{{\mathrm{\θ}}_i}\\ +\[({\hat{x}}_i-x)\cos {\widehat{\mathrm{\θ}}}_i+({\hat{y}}_i-y)\sin {\widehat{\mathrm{\θ}}}_i\]{\mathrm{sinn}}_{{\mathrm{\θ}}_i}+(n_{x_i}\cos {\widehat{\mathrm{\θ}}}_i+n_{y_i}\sin {\widehat{\mathrm{\θ}}}_i){\mathrm{sinn}}_{{\mathrm{\θ}}_i}\\ i=1,2,\mathrm{...},n\end{array}$

Wherein $n_{x_{i+1,1}}=n_{x_{i+1}}-n_{x_1},n_{y_{i+1,1}}=n_{y_{i+1}}-n_{y_1};$

Can be obtained by Y=GZ $\frac{1}{2}(k_i-k_1-{\hat{r}}_{i,1}^2)-x_{i,1}x+y_{i,1}y+{\hat{r}}_{i,1}{r}_1,({\hat{x}}_i-x)sin{\widehat{\mathrm{\θ}}}_i=({\hat{y}}_i-y)cos{\widehat{\mathrm{\θ}}}_i,$ Therefore can obtain:

$\begin{array}{c}{e}_i={\hat{x}}_{i+1}{n}_{x_{i+1}}+{\hat{y}}_{i+1}{n}_{y_{i+1}}-{\hat{x}}_1{n}_{x_1}-{\hat{y}}_1{n}_{y_1}-{\hat{r}}_{i+1}(n_{r_{i+1}}-n_{r_1})-(n_{x_{i+1}}-n_{x_1})x-(n_{y_{i+1}}-n_{y_1})y\\ =-{\hat{r}}_{i+1}(n_{r_{i+1}}-n_{r_1})+({\hat{x}}_{i+1}-x)n_{x_{i+1}}+({\hat{y}}_{i+1}-y)n_{y_{i+1}}-({\hat{x}}_1-x)n_{x_1}-({\hat{y}}_1-y)n_{y_1}\\ i=1,2,\mathrm{...},n-1\end{array}$

$\begin{array}{c}{e}_{n+i-1}=(n_{x_i}\sin {\widehat{\mathrm{\θ}}}_i-n_{y_i}\cos {\widehat{\mathrm{\θ}}}_i)(1-\frac{n_{{\mathrm{\θ}}_i}^2}{2!}+\mathrm{...}+{\left(-1\right)}^n\frac{n_{{\mathrm{\θ}}_i}^{2n}}{\left(2n\right)!})\\ +\[(x_i-x)\cos {\widehat{\mathrm{\θ}}}_i-(y_i-y)\sin {\widehat{\mathrm{\θ}}}_i\](n_{{\mathrm{\θ}}_i}-\frac{n_{{\mathrm{\θ}}_i}^3}{3!}+\mathrm{...}+{\left(-1\right)}^n\frac{n_{{\mathrm{\θ}}_i}^{2n+1}}{(2n+1)!})\\ +(n_{x_i}\cos {\widehat{\mathrm{\θ}}}_i+n_{y_i}\sin {\widehat{\mathrm{\θ}}}_i)(n_{{\mathrm{\θ}}_i}-\frac{n_{{\mathrm{\θ}}_i}^3}{3!}+\mathrm{...}+{\left(-1\right)}^n\frac{n_{{\mathrm{\θ}}_i}^{2n+1}}{(2n+1)!})\\ =\[(x_i-x)\cos {\widehat{\mathrm{\θ}}}_i+(y_i-y)\sin {\widehat{\mathrm{\θ}}}_i\]n_{{\mathrm{\θ}}_i}+n_{x_i}\sin {\widehat{\mathrm{\θ}}}_i-n_{y_i}\cos {\widehat{\mathrm{\θ}}}_i\\ i=1,2,\mathrm{...},n\end{array}$

E can be obtained_ie_j, e_n+i-1e_n+j-1,WithExpected value be:

$\begin{array}{c}E\left(e_i{e}_j\right)=E({\hat{r}}_{i+1}{\hat{r}}_{j+1}{n}_{r_1}^2+{\left({\hat{x}}_1-x\right)}^2{n}_{x_1}^2+{\left({\hat{y}}_1-y\right)}^2{n}_{y_1}^2)\\ ={\hat{r}}_{i+1}{\hat{r}}_{j+1}{\mathrm{\σ}}_r^2+{({\hat{x}}_1-x)}^2{\mathrm{\σ}}_x^2+{({\hat{y}}_1-y)}^2{\mathrm{\σ}}_y^2\\ i,j=1,2,\mathrm{...},n-1,i\≠j\end{array}$

$\begin{array}{c}E\left(e_i^2\right)=E\[{(-{\hat{r}}_{i+1}(n_{r_{i+1}}-n_{r_1})+({\hat{x}}_{i+1}-x)n_{x_{i+1}}+({\hat{y}}_{i+1}-y)n_{y_{i+1}}-({\hat{x}}_1-x)n_{x_1}-({\hat{y}}_1-y\left)n_{y_1}\right)}^2\]\\ =E({\hat{r}}_{i+1}^2{n}_{r_{i+1}}^2+{\hat{r}}_{i+1}^2{n}_{r_i}^2+{\left({\hat{x}}_{i+1}-x\right)}^2{n}_{x_{i+1}}^2+{\left({\hat{y}}_{i+1}-y\right)}^2{n}_{y_{i+1}}^2+{\left({\hat{x}}_1-x\right)}^2{n}_{x_1}^2+{\left({\hat{y}}_1-y\right)}^2{n}_{y_1}^2)\\ =2{\hat{r}}_{i+1}^2{\mathrm{\σ}}_r^2+\[{({\hat{x}}_{i+1}-x)}^2+{({\hat{x}}_1-x)}^2\]{\mathrm{\σ}}_x^2+\[{({\hat{y}}_{i+1}-y)}^2+{({\hat{y}}_1-y)}^2\]{\mathrm{\σ}}_y^2\\ i=1,2,\mathrm{...},n-1\end{array}$

$\begin{array}{c}E\left(e_{n+i-1}{e}_{n+j-1}\right)=E\left\{\begin{array}{c}\[(\left(x_i-x\right)cos{\widehat{\mathrm{\θ}}}_i+\left(y_i-y\right)sin{\widehat{\mathrm{\θ}}}_i)n_{{\mathrm{\θ}}_i}+n_{x_i}\sin {\widehat{\mathrm{\θ}}}_i-n_{y_i}\cos {\widehat{\mathrm{\θ}}}_i\]\·\\ \[(\left(x_j-x\right)cos{\widehat{\mathrm{\θ}}}_j+\left(y_j-y\right)sin{\widehat{\mathrm{\θ}}}_j)n_{{\mathrm{\θ}}_j}+n_{x_j}\sin {\widehat{\mathrm{\θ}}}_j-n_{y_j}\cos {\widehat{\mathrm{\θ}}}_j\]\end{array}\right\}=0\\ i,j=1,2,\mathrm{...},n,i\≠j\end{array}$

$\begin{array}{c}E\left(e_{n+i-1}^2\right)=E\left\{{\[\left(\right(x_i-x)\cos {\widehat{\mathrm{\θ}}}_i+(y_i-y\left)\sin {\widehat{\mathrm{\θ}}}_i\right)n_{{\mathrm{\θ}}_i}+n_{x_i}\sin {\widehat{\mathrm{\θ}}}_i-n_{y_i}\cos {\widehat{\mathrm{\θ}}}_i\]}^2\right\}\\ {\[(x_i-x)\cos {\widehat{\mathrm{\θ}}}_i+(y_i-y)\sin {\widehat{\mathrm{\θ}}}_i\]}^{2}E\left(n_{{\mathrm{\θ}}_i}^2\right)+{\left(\sin {\widehat{\mathrm{\θ}}}_i\right)}^{2}E\left(n_{x_i}^2\right)+{\left(\cos {\widehat{\mathrm{\θ}}}_i\right)}^{2}E\left(n_{y_i}^2\right)\\ ={\[(x_i-x)\cos {\widehat{\mathrm{\θ}}}_i+(y_i-y)\sin {\widehat{\mathrm{\θ}}}_i\]}^2{\mathrm{\σ}}_{\mathrm{\θ}}^2+{\left(\sin {\widehat{\mathrm{\θ}}}_i\right)}^2+{\left(\cos {\widehat{\mathrm{\θ}}}_i\right)}^2{\mathrm{\σ}}_y^2\\ i=1,2,\mathrm{...},n\end{array}$

Can obtain site error weighting matrix is:

$B_{rx}={\left[\begin{array}{ccc}\[{({\hat{x}}_2-x)}^2+{({\hat{x}}_1-x)}^2\]{\mathrm{\σ}}_x^2& ...& {({\hat{x}}_1-x)}^2{\mathrm{\σ}}_x^2\\ .& .& .\\ .& .& .\\ .& .& .\\ {({\hat{x}}_1-x)}^2{\mathrm{\σ}}_x^2& ...& \[{({\hat{x}}_n-x)}^2+{({\hat{x}}_1-x)}^2\]{\mathrm{\σ}}_x^2\end{array}\right]}_{(n-1)\×(n-1)}$

$B_{ry}={\left[\begin{array}{ccc}\[{({\hat{y}}_2-y)}^2+{({\hat{y}}_1-y)}^2\]{\mathrm{\σ}}_y^2& ...& {({\hat{y}}_1-y)}^2{\mathrm{\σ}}_y^2\\ .& .& .\\ .& .& .\\ .& .& .\\ {({\hat{y}}_1-y)}^2{\mathrm{\σ}}_y^2& ...& \[{({\hat{y}}_n-y)}^2+{({\hat{y}}_1-y)}^2\]{\mathrm{\σ}}_y^2\end{array}\right]}_{(n-1)\×(n-1)}$

$B_{\mathrm{\θ}x}=diag\left\{\left[\begin{array}{ccc}{\left(sin{\widehat{\mathrm{\θ}}}_1\right)}^2{\mathrm{\σ}}_x^2& ...& {\left(sin{\widehat{\mathrm{\θ}}}_n\right)}^2{\mathrm{\σ}}_x^2\end{array}\right]\right\}$

$B_{\mathrm{\θ}y}=diag\left\{\left[\begin{array}{ccc}{\left(cos{\widehat{\mathrm{\θ}}}_1\right)}^2{\mathrm{\σ}}_y^2& ...& {\left(cos{\widehat{\mathrm{\θ}}}_n\right)}^2{\mathrm{\σ}}_y^2\end{array}\right]\right\}.$

The interference source localization method of a kind of GSM-R considering base station location error the most according to claim 4, its feature exists In, go back step d method particularly includes:

The error vector covariance matrix that can be reconstructed by step c is:

$\mathrm{cov}\left(e\right)=\left[\begin{array}{cc}{B}_r+B_{rx}+B_{ry}& 0\\ 0& B_{\mathrm{\θ}}+B_{\mathrm{\θ}x}+B_{\mathrm{\θ}y}\end{array}\right]$

The error vector covariance matrix formula of reconstruct is substituted into Z=(G^Tcov(e)^-1G)^-1G^Tcov(e)^-1Y can eliminate movement The position of interference source coordinate estimated value of station location error, the position of interference source coordinate now obtained does not considers x, y, r₁Between phase Guan Xing；Eliminate x, y, r₁Between the method for correlation error be:

Assume x, y, r₁Estimation difference be μ₁,μ₂,μ₃, by Z=(G^Tcov(e)^-1G)^-1G^Tcov(e)^-1The estimated value that Y tries to achieve can It is expressed as:

[z]₁=x+ μ₁, [z]₂=y+ μ₂, [z]₃=r₁+μ₃

Can instrument error vector e':e'=Y'-G'Z'

Wherein $Y^{\′}=\left[\begin{array}{c}{({\[Z\]}_1-x_1)}^2\\ {({\[Z\]}_2-y_1)}^2\\ {\[Z\]}_3^2\end{array}\right],G^{\′}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\end{array}\right],e^{\′}=\left[\begin{array}{c}{e_1}^{\′}\\ {e_2}^{\′}\\ {e_3}^{\′}\end{array}\right]Z^{\′}=\left[\begin{array}{c}{(x-x_1)}^2\\ {(y-y_1)}^2\end{array}\right]$

By [z]₁=x+ μ₁, [z]₂=y+ μ₂, [z]₃=r₁+μ₃Substitute in e'=Y'-G'Z' and ignore quadratic term and can obtain:

e₁'=([Z]₁-x₁)²-(x-x₁)²=(x+ μ₁-x₁-n_x1)²-(x-x₁-n_x1)²≈2(x-x₁)μ₁

e₂'=([Z]₂-y₁)²-(y-y₁)²=(y+ μ₂-y₁-n_y1)²-(y-y₁-n_y1)²≈2(y-y₁)μ₂

$\begin{array}{c}{e_3}^{\′}={\[Z\]}_3^2-{(x-x_1)}^2-{(y-y_1)}^2={(r_1+{\mathrm{\μ}}_3)}^2-{(x-x_1-n_{x1})}^2-{(y-y_1-n_{y1})}^2\\ \≈2r_1{\mathrm{\μ}}_3+2(x-x_1)n_{x1}+2(y-y_1)n_{x1}\end{array}$

Then the covariance matrix of e' is: cov (e')=E (e'e'^T)=D{cov (Z) } D+Q；

Wherein, D=diag{ [2 (x-x₁) 2(y-y₁) 2r₁]}≈diag{[2([z]₁-x₁) 2([z]₂-y₁) 2[z]₃], $Q=diag\left\{\left[\begin{array}{ccc}0& 0& 4{(x-x_1)}^2{\mathrm{\σ}}_x^2+4{(y-y_1)}^2{\mathrm{\σ}}_y^2\end{array}\right]\right\}\≈diag\left\{\left[\begin{array}{ccc}0& 0& 4{({\[z\]}_1-x_1)}^2{\mathrm{\σ}}_x^2+4{({\[z\]}_2-y_1)}^2{\mathrm{\σ}}_y^2\end{array}\right]\right\}$ , cov (Z) it is Z=(G^Tcov(e)^-1G)^-1G^Tcov(e)^-1The covariance matrix of the estimated value that Y tries to achieve, can be asked by perturbation analysis method Solving, definition Δ is error disturbance component, then Z=(G^Tcov(e)^-1G)^-1G^Tcov(e)^-1Y can be write as:

$\left(\right({\hat{G}}^T+{\mathrm{\ΔG}}^T\left)\mathrm{cov}{\left(e\right)}^{-1}\right(\hat{G}+\mathrm{\Δ}G\left)\right)(\hat{Z}+\mathrm{\Δ}Z)=({\hat{G}}^T+{\mathrm{\ΔG}}^T)\mathrm{cov}{\left(e\right)}^{-1}(\hat{Y}+\mathrm{\Δ}Y)$

Due toTherefore further spread out and ignore high-order term and can obtain:

${\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\mathrm{\Δ}Z+{\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\mathrm{\Δ}G\hat{Z}={\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\mathrm{\Δ}Y$

After abbreviation it is:

$\mathrm{\Δ}Z={\left({\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\right)}^{-1}{\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}(\mathrm{\Δ}Y-\mathrm{\Δ}G\hat{Z})$

Due to $e=\mathrm{\Δ}Y-\mathrm{\Δ}G\hat{Z},$ So by $\mathrm{\Δ}Z={\left({\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\right)}^{-1}{\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}(\mathrm{\Δ}Y-\mathrm{\Δ}G\hat{Z})$ Can obtain:

$\mathrm{\Δ}Z={\left({\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\right)}^{-1}{\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}e$

Can obtain cov (Z) is:

$\begin{array}{c}\mathrm{cov}\left(Z\right)=E\[{\mathrm{\ΔZ\ΔZ}}^T\]\\ ={\left({\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\right)}^{-1}{\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}E\left({\mathrm{ee}}^T\right)\mathrm{cov}{\left(e\right)}^{-1}\hat{G}{\left({\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\right)}^{-1}\\ ={\left({\hat{G}}^T\mathrm{cov}{\left(e\right)}^{-1}\hat{G}\right)}^{-1}\end{array}$

E'=Y'-G'Z' maximal possibility estimation is asked to obtain:

$\begin{array}{c}{Z}^{\′}=\arg \min \left\{{(Y^{\′}-G^{\′}{Z}^{\′})}^T\mathrm{cov}{\left(e^{\′}\right)}^{-1}(Y^{\′}-G^{\′}{Z}^{\′})\right\}\\ ={\left(G^{\′T}\mathrm{cov}{\left(e^{\′}\right)}^{-1}{G}^{\′}\right)}^{-1}{G}^{\′T}\mathrm{cov}{\left(e^{\′}\right)}^{-1}{Y}^{\′}\end{array}$

The position coordinates estimated value finally trying to achieve interference source is:

$Z^{\′\′}=\left[\begin{array}{c}x\\ y\end{array}\right]=\±\sqrt{Z^{\′}}+\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right].$