CN114690116B - Passive target positioning method and system based on TDOA and FDOA measurement - Google Patents

Abstract

A passive target positioning method and system based on TDOA and FDOA measurement, the method comprises the following steps: acquiring measurement information and receiving station position information; converting the position information of the receiving station into a geocentric and geocentric fixed coordinate system; establishing a measurement equation according to the measurement information and the position information of the receiving station under the geocentric and geodetic fixed coordinate system; carrying out deformation processing on the measurement equation by combining prior information; and solving the target position vector by using the measurement equation after deformation processing to obtain an estimated value of the target coordinate. The invention introduces the prior range information of the target, converts the prior range information into an approximate value of the height coordinate of the target to be positioned after processing the prior range information, and avoids the ambiguity of positioning the three-station TDOA and the two-station TDOA-FDOA because of the acquisition of the additional coordinate. In the process of solving the target coordinates, the method directly brings the parameter-containing coordinate result to the theoretical definition constraint solving parameter, avoids the introduction of additional errors, and reduces the final positioning error.

Description

Passive target positioning method and system based on TDOA and FDOA measurements

Technical Field

The present invention belongs to the technical field of target positioning, and in particular relates to a passive target positioning method and system based on TDOA and FDOA measurements.

Background Art

Passive positioning technology refers to a technology in which an observation station does not actively radiate electromagnetic signals, but determines its position in space by measuring the parameters of electromagnetic signals emitted by external radiation sources or measuring their visible light and infrared parameters. The external radiation source signals that can be used by the working platform are generally divided into two types: those from the target itself ^[1] or those from television, radio, etc.

Compared with active positioning systems, passive positioning systems have many advantages ^[2] : (1) strong survivability. Since the observation station does not radiate electromagnetic waves to the target, the chance of being discovered by the enemy is reduced; (2) strong anti-interference ability. Since the observation station is not easily discovered by the enemy, the enemy cannot impose targeted interference; (3) wide airspace coverage and high signal interception probability ^[3] ; (4) light weight, small size and low cost. Since it does not emit high-power electromagnetic wave signals to the target, there is no need to build a huge high-power transmitter. These advantages improve the combat effectiveness and survivability of passive positioning systems. Using passive positioning systems to locate or track target radiation sources has become an important way to achieve covert detection and precision strikes.

In a passive positioning system, an observation station can locate a target by measuring the energy intensity of the signal emitted by the radiation source, the angle of arrival (AOA) and the time of arrival (TOA) of the signal arriving at the observation station, or by measuring the time difference (TDOA) of the signal arriving at multiple receiving stations ^[4] . When there is relative motion between the radiation source and the receiving station, the target can also be located by using the frequency difference (FDOA) of the signal arriving at multiple receiving stations. The positioning algorithm based on TDOA measurement is to calculate the time difference of arrival at different observation stations by combining the arrival time of the target emission source signal measured by multiple receiving stations ^[5] , and then construct a positioning equation about the target position state to achieve passive positioning of space targets. In the positioning system based on arrival time difference, each receiving station does not need complex attitude measurement equipment, but only a single receiving channel. In addition, the current arrival time measurement instruments are very accurate, so the positioning method based on TDOA measurement can achieve higher positioning accuracy. The positioning algorithm based on the frequency difference of arrival (FDOA) is suitable for positioning scenarios where there is relative motion between the receiving station and the target radiation source ^[6] . By combining the radiation source signals detected by multiple receiving stations, the frequency difference of the radiation source signals arriving at different receiving stations is obtained, thereby constructing a positioning equation group about the state parameters of the target radiation source. The positioning equation group is solved to obtain the state information of the target radiation source. Similar to the positioning system based on the time difference of arrival, each receiving station in this positioning system does not need an attitude measurement device, but only a single receiving channel.

Since the target positioning algorithm based on the above two measurements was proposed, it has been a hot topic in the field of positioning research. A series of studies have been conducted on the solution methods of the positioning algorithm, including Taylor expansion method, least squares polynomial solution method ^[7][8] and various weighted methods ^[9] . However, in practical applications, due to cost and other issues, insufficient number of observation stations may lead to positioning ambiguity. The research on this issue is not very sufficient at present. When the number of observation stations is insufficient, some prior information of the positioning target can usually be obtained (for example, the target height is 0 or can be measured in advance with an altimeter). At this time, due to the lack of observation stations and other issues, only the estimation effect of the x and y coordinates of the space target is often concerned. In the literature ^[10] , this problem has been studied, but it is necessary to know the height of the target in advance and establish the constraint relationship in the Earth-centered Earth-fixed coordinate system. Combined with the radius of the earth, this method will introduce large errors due to the approximation of the radius of the earth.

References

[1] Zeng Hui, Zeng Fangling. Ambiguity and unsolvable problems of time difference positioning of three space stations[J]. Journal of Terahertz Science and Electronic Information, 2010, 8(2): 139-142.

[2] Feng Tianjun. Research on dual-aircraft time difference-frequency difference combined positioning[D]. National University of Defense Technology.

[3] Dong Quan, Wang Hong, Wang Shunan. Dual-station time difference and frequency difference positioning technology[C]. China Electronics Society, 2017.

[4]Fang B T.Simple solutions for hyperbolic and related positionfixes[J].Aerospace&Electronic Systems IEEE Transactions on,1990,26(5):748-753.

[5] Yan Zhenya, Chen Yi, Chen Xinnian, et al. Analysis of positioning accuracy of time difference and frequency difference among three stations[J]. Modern Radar, 2018, 40(9):4.

[6] Zhu Huajin, Zhang Yang, E Mei, et al. Time difference and frequency difference joint positioning algorithm based on constrained weighted least squares [J]. Telecommunication Technology, 2013, 53(4):7.

[7]Friedlander, B.A passive localization algorithm and its accuracy analysis[J]. IEEE Journal of Oceanic Engineering, 1987, 12(1):234-245.

[8] Smith, J, Abel. Closed-form least-squares source location estimation from range-difference measurements [J]. Acoustics, Speech and Signal Processing, IEEE Transactions on, 1987, 35(12): 1661-1669.

[9]Chan Y T, Ho K C. A simple and efficient estimator for hyperboliclocation[J]. IEEE Transactions on Signal Processing, 2002, 42(8): 1905-1915.

[10]Ho K C, Chan Y T.Geolocation of a known altitude object from TDOAand FDOA measurements[J]. IEEE Transactions on Aerospace and ElectronicSystems,1997,33(3):770-783.

Summary of the invention

The purpose of the present invention is to address the problems in the above-mentioned prior art and provide a passive target positioning method and system based on TDOA and FDOA measurements, using signals from the target itself as external radiation source signals, including two positioning systems using TDOA measurement alone and combining TDOA and FDOA measurements, which can reduce positioning errors.

In order to achieve the above object, the present invention has the following technical solutions:

In a first aspect, a passive target positioning method based on TDOA and FDOA measurements is provided, comprising the following steps:

Obtain measurement information and receiving station location information;

Convert the receiving station location information into the Earth-centered Earth-fixed coordinate system;

Establishing a measurement equation based on the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system;

Deforming the measurement equation in combination with prior information;

The target position vector is solved using the measurement equation after deformation processing to obtain the estimated value of the target coordinates.

As a preferred solution of the passive target positioning method of the present invention, the measurement information includes the TDOA measurement value t _i1 and the FDOA measurement value The receiving station location information includes the latitude and longitude coordinates of the observation station The receiving station location information is converted to the Earth-centered Earth-fixed coordinate system as follows:

Where R is the radius of curvature of the reference ellipsoid, e is the first eccentricity of the ellipsoid, The radius of the major axis of the ellipsoid is a=6378137m, and the radius of the minor axis is b=6356752m.

As a preferred solution of the passive target positioning method of the present invention, the step of establishing the measurement equation according to the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system includes:

Assume that the position and velocity of the N receiving stations _si after coordinate conversion are x _si = [x _si , y _si , z _si ] ^T and i＝1,2,...,N, and these receiving stations are not on the same straight line or plane. The position and velocity of the unknown radiation source are expressed as x＝[x,y,z] ^T and Then the distance between the radiation source x and the receiving station _si is:

Select receiving station _s1 as the reference receiving station. The measurement equations of TDOA and FDOA between the radiation source transmitting signal and the i-th receiving station and the reference receiving station are expressed as follows:

Among them, _ti and _t1 are the time when the signal starts from the radiation source and arrives at the receiving stations _si and _s1 respectively, c is the propagation speed of electromagnetic waves, _f0 is the frequency of the signal emitted by the target radiation source, _ri is the distance from the target radiation source to each receiving station, is the rate of change of the distance from the target radiation source to each receiving station, _vi represents the radial relative velocity between the receiving station and the target, and the calculation expression is as follows:

As a preferred solution of the passive target positioning method of the present invention, the linearization process of the measurement equation is performed before the deformation process of the measurement equation is performed in combination with the prior information, including:

Convert the TDOA measurement information into the distance difference RDOA between the radiation source and the two receiving stations:

according to and The equivalent equation to obtain the RDOA equation is as follows:

The above equation is derived with respect to time t to obtain the linearized FDOA positioning equation:

As a preferred solution of the passive target positioning method of the present invention, the step of deforming the measurement equation in combination with prior information includes: obtaining a coordinate range of the target source area according to the receiving station position information in the earth-centered earth-fixed coordinate system, assuming it is [x _min , y _min , z _min ] ~ [x _max , y _max , z _max ], taking As a priori z coordinate, the equivalent equation of the RDOA equation is transformed into the following form:

In the formula, x _i1 =x _si -x _s1 , y _i1 =y _si -y _s1 , z _i1 =z _si -z _s1 ;

If the target motion is not considered, assuming Then the linearized FDOA positioning equation is transformed into the following form:

make x _i1 = x _si - x _s1 , y _i1 = y _si - y _s1 , z _i1 = z _si - z _s1 . After obtaining the priori z coordinate of the target to be determined, the FDOA positioning equation in the above formula is transformed into the following form:

As a preferred solution of the passive target positioning method of the present invention, the step of solving the target position vector using the deformed measurement equation includes: if only TDOA measurement is used to complete the positioning, the RDOA equations determined by different sensor pairs are written into a matrix form and then solved; when there is a relative motion between the observation station and the target source, the Doppler effect caused by the relative motion between the radiation source and the receiving station is used to estimate the frequency difference of the signal arriving at different receiving stations, and the RDOA and FDOA equations determined by different sensor pairs are combined into a matrix form and then solved.

As a preferred solution of the passive target positioning method of the present invention, if there are only three receiving stations and there is no relative motion between the receiving stations and the target, the RDOA equations determined by different sensor pairs are written in matrix form as follows:

h＝Gx _a

The matrix vectors are:

x _a =[x,y] ^T

x _a ＝G ^-1 h

From the above formula, we can get the calculation results of the target x and y coordinates. Since r ₁ is an unknown quantity, we can express the target x and y coordinates as a function of r ₁ , and then substitute the result including r ₁ into the definition formula. A quadratic equation about r ₁ is obtained. The value of r ₁ is obtained by solving the equation, and r ₁ is substituted into the calculation result again to obtain the estimated values of the target x and y coordinates.

As a preferred solution of the passive target positioning method of the present invention, the value of r ₁ is a positive real root. If both solutions of the equation are positive real roots, an ideal solution is determined by prior information.

As a preferred solution of the passive target positioning method of the present invention, if there are only two receiving stations and there is relative motion between the receiving station and the target, the prior range of the target and the approximate value of the z coordinate of the target are obtained, and the RDOA equation and the FDOA equation measured by the two stations are jointly written into a matrix form as follows:

From the above formula, we can get the calculation results of the target x, y coordinates. Since r ₁ and are functions of the unknown variable x, so r ₁ and They are also unknown quantities. The x and y coordinates of the target are expressed as a function of r ₁ ;

Since the target is stationary, To guide, Written as a function of r ₁ , the derivative is written as the multiplication of coordinate vectors as follows:

Assume x′ _s1 = x _s1 (1:2,:), Then transform the above formula into:

Will Written as a function of r ₁ , the final form is as follows:

Substitute the result of the above formula into the matrix, and the calculated target position coordinate x contains only one unknown parameter _r1 . Substitute the x coordinate containing the unknown parameter _r1 into the constraint formula _r12 =( _xxs1 ) ^T ( _xxs1 ), ^and get a quartic equation about _r1 . Solve the quartic equation to get the value of _r1 , and then substitute the value of _r1 into the parameter-containing result to get the estimated values of the target x and y coordinates.

In a second aspect, a passive target positioning system based on TDOA and FDOA measurements is provided, comprising:

An information acquisition module, used to obtain measurement information and receiving station location information;

A coordinate conversion module is used to convert the receiving station location information into an Earth-centered Earth-fixed coordinate system;

A measurement equation building module, used to build a measurement equation based on the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system;

A measurement equation deformation module, used for deforming the measurement equation in combination with prior information;

The target position estimation module is used to solve the target position vector using the measurement equation after deformation processing to obtain the estimated value of the target coordinates.

Compared with the prior art, the present invention has at least the following beneficial effects:

In the three-station TDOA and two-station TDOA-FDOA spatial target positioning algorithms, it is often necessary to use an altimeter to obtain the approximate value of the target's height coordinate, and then use the TDOA or FDOA measurement equation to solve the target's two-dimensional position coordinates. Otherwise, the three-station TDOA and two-station TDOA-FDOA positioning is an ambiguous problem that cannot obtain a definite solution. The present invention introduces the target prior range information, and converts the range prior information into an approximate value of the target height coordinate z after processing. Because of the acquisition of this additional z coordinate, the ambiguity of the three-station TDOA and two-station TDOA-FDOA positioning is avoided. In the process of solving the target coordinates, the present invention directly brings the parameter-containing coordinate results to the theoretically defined constraint solution parameters, avoiding the introduction of additional errors, thereby reducing the final positioning error and improving the estimation effect of passive target positioning.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a flow chart of a passive target positioning method based on TDOA and FDOA measurements according to the present invention;

Fig. 2 3TDOA positioning error percentage under different noise levels;

Fig. 3 3TDOA positioning error percentage diagram under different height errors;

Fig. 4 is a comparison of 3TDOA errors when the noise level is σ ² =10 ^-6 ;

Fig. 5 3TDOA error comparison diagram when the noise level σ ² =10 ^-2 ;

Figure 6 Comparison of 3TDOA positioning errors under different noise levels;

Fig. 7 2TDOA-FDOA positioning error percentage at different noise levels;

Fig. 8 2TDOA-FDOA positioning error percentage diagram under different height errors;

Fig. 9 2TDOA-FDOA error comparison diagram when the noise level σ ² =10 ^-6 ;

Fig. 10 2TDOA-FDOA error comparison diagram when the noise level σ ² =10 ^-3 ;

Figure 11 Comparison of 2TDOA-FDOA positioning errors under different noise levels.

DETAILED DESCRIPTION

In order to make the objectives, technical solutions and advantages of the present invention more clear, the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

Referring to FIG. 1 , the passive target positioning method based on TDOA and FDOA measurement of the present invention includes the following steps:

S1. Obtain measurement information and receiving station location information (latitude, longitude and height);

S2, converting the receiving station location information into an Earth-centered Earth-fixed coordinate system;

In the commonly used TDOA/FDOA positioning algorithm, the Cartesian coordinates x, y, and z of the target and the observation station are generally used for calculation and solution. However, in many engineering applications, the latitude and longitude information of the observation station is often given. Therefore, when the latitude and longitude coordinates of the observation station are obtained, After that, they should all be converted to Cartesian coordinates (x, y, z) in the Earth-centered Earth-fixed coordinate system. The specific conversion formula is as follows:

Where R is the radius of curvature of the reference ellipsoid, e is the first eccentricity of the ellipsoid, The radius of the major axis of the ellipsoid is a=6378137m, and the radius of the minor axis is b=6356752m.

S3, establishing a measurement equation based on the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system;

Based on the observation station coordinates in the Earth-centered Earth-fixed coordinate system obtained in step S1, the TDOA measurement value t _i1 and the FDOA measurement value t i2 are obtained. After that, the measurement equation about the position of the target source to be determined is established. It is assumed that the position and speed of the N receiving stations _si after the coordinate transformation are x _si = [x _si , y _si , z _si ] ^T and It is assumed that these receiving stations are not on the same straight line or plane. The position and velocity of the radiation source to be determined are expressed as x = [x, y, z] ^T and Then the distance between the radiation source _x and the receiving station _si is:

Select receiving station _s1 as the reference receiving station. The measurement equations of TDOA and FDOA between the radiation source transmitting signal and the i-th receiving station and the reference receiving station can be expressed as:

Among them, _ti and _t1 are the time when the signal starts from the radiation source and arrives at the receiving stations _si and _s1 respectively, c is the propagation speed of electromagnetic waves, _f0 is the frequency of the signal emitted by the target radiation source, _ri is the distance from the target radiation source to each receiving station, is the rate of change of the distance from the target radiation source to each receiving station, _vi represents the radial relative velocity between the receiving station and the target, and the calculation expression is as follows:

Since the above measurement equation is nonlinear and difficult to solve, the TDOA measurement information is converted into the distance difference RDOA between the radiation source and the two receiving stations by linearizing both sides of equation (3) by multiplying the signal propagation speed c:

Then move the right side r ₁ of equation (6) to the left side of the equation, then square both sides of the equation and use the relationship and The equivalent positioning equation of TDOA, RDOA equation, is as follows:

By taking the derivative of the above formula with respect to time t, the linearized FDOA positioning equation is as follows:

S4, deforming the measurement equation in combination with prior information;

In actual engineering applications, for the three-station TDOA positioning or the dual-station TDOA-FDOA joint positioning problem, due to the insufficient number of observation stations, a priori latitude and longitude high area of the target source is usually given. After the latitude and longitude high area is transformed into the Earth-centered Earth-fixed coordinate system, a coordinate range of the target source area can be obtained. Assuming that the coordinate range is [x _min ,y _min ,z _min ]～[x _max ,y _max ,z _max ], take As the priori z coordinate, the RDOA equation of equation (7) above can be transformed into the following form:

in x _i1 =x _si -x _s1 , y _i1 =y _si -y _s1 , z _i1 =z _si -z _s1 .

If we do not consider the motion of the target, that is, assume Then the above formula (8) can be written as follows:

make x _i1 = x _si - x _s1 , y _i1 = y _si - y _s1 , z _i1 = z _si - z _s1 . After obtaining the prior z coordinate of the target to be determined, the FDOA equation in formula (10) above can be further written as follows:

S5, solving the target position vector using the measurement equation after deformation processing to obtain an estimated value of the target coordinates;

Normally, if only TDOA measurement is used to complete positioning, it is only necessary to write the RDOA equations determined by different sensor pairs into a matrix form and then solve them. When there is relative motion between the observation station and the target source, such as when the positioning system is installed on a mobile platform, the Doppler effect caused by the relative motion between the radiation source and the receiving station can be used to estimate the frequency difference (FDOA) of the signal arriving at different receiving stations, and then combine TDOA and FDOA to estimate the radiation source state parameters. In this case, it is necessary to combine the RDOA and FDOA equations determined by different sensor pairs into a matrix form and then solve them.

The passive target positioning method proposed in the present invention is aimed at solving the problem of estimating the [x, y, z] coordinates of three-dimensional coordinates when the number of observation stations is insufficient in actual engineering applications. At the same time, in order to improve the calculation efficiency, a linear least squares analytical solution is used for solution, so it is necessary to introduce an additional variable r ₁ in the linear solution process. Therefore, in the solution process, the method proposed in the present invention first regards r ₁ as a known quantity, and then expresses the coordinates of the target source to be determined as a function of r ₁ , and then brings the coordinate vector containing the unknown parameter r ₁ into the definition formula r ₁ ² =(xx _s1 ) ^T (xx _s1 ) to obtain a high-order equation about r ₁ , solves the high-order equation to obtain an estimated value of r ₁ , and finally replaces the obtained estimated value of r ₁ to obtain the estimated value of the final target coordinates.

The method of the present invention includes two positioning systems: using TDOA measurement alone and combining TDOA and FDOA measurement.

For three-dimensional space target positioning, if only TDOA measurements can be obtained, generally, for targets in space, at least four receiving stations are required to complete positioning, because four receivers can determine three TDOA measurement equations to determine the x, y, z coordinates of the space target. If there are only three stations, only two TDOA equations can be determined, and the exact position of the target in space cannot be estimated. However, if the a priori range in which the target may appear is known, and the approximate value of the target height coordinate is determined through the known a priori range, and then the two-dimensional coordinates [x, y] of the target are estimated, the positioning result of the space target [x, y, z] based on the three-station TDOA can be achieved.

Based on the deformation measurement equation (Equation (9)) given in step S4, the equations determined by different sensor pairs are written in matrix form as follows:

h＝Gx _a (12)

The matrix vectors are:

x _a = [x, y] ^T (13)

x _a ＝G ^-1 h (16)

From the above equations (12) to (16), we can get the calculation results of the target x and y coordinates. However, since _r1 is an unknown quantity, the target x and y coordinates can only be expressed as a function of _r1 . Then, the result containing _r1 obtained in equation (16) can be substituted into the definition equation _r12 = (xx _s1 ) ^T (xx _s1 ) to get a quadratic equation about _r1 . ^By solving this equation, we can get the value of _r1 (usually there is only one positive real root. If both solutions are positive real roots, an ideal solution can be determined by prior information). At this time, the obtained _r1 can be substituted into the calculation result containing parameters obtained for the first time to get the estimated values of the target x and y coordinates. Since the coordinate z has been estimated by prior information, all the coordinates [x, y, z] have been obtained.

The method of the present invention first treats the unknown variable r ₁ as a known variable, then expresses the coordinates of the target source as a function of r ₁ , and then substitutes the coordinate vector containing the unknown parameter r ₁ into the definition formula at the beginning for solution. The difference between this method and the method in the literature ^[10] is that the relevant definition formula of r ₁ is directly used to solve r ₁ , eliminating the error caused by using the earth's radius, so the positioning accuracy is higher.

Compared with three-station TDOA positioning, dual-station can only obtain one TDOA equation, which makes space positioning impossible to achieve. However, when there is relative motion between the transmitter and the receiver, an arrival frequency difference (FDOA) measurement equation can also be obtained. Combining these two measurement equations with prior height information can complete the joint positioning of dual-station TDOA and FDOA for space targets.

When the prior range of the target and the approximate z-coordinate of the target are obtained, the RDOA equation (9) and the FDOA equation (11) measured by the two stations can be combined into a matrix form as follows:

From equations (17) to (20), we can obtain the calculation results of the target x, y coordinates by the dual-station TDOA-FDOA positioning algorithm. However, due to _the are functions of the unknown variable x, so r ₁ and They are also unknown quantities. At this time, we can only express the x and y coordinates of the target as a function of r _1. The following describes how to solve these two unknown parameters and obtain the final positioning result.

Since the target is stationary, Take the derivative, so that we can Written as a function of r ₁ , after taking the derivative, it is written as the multiplication of coordinate vectors as follows:

Since the above formula (17) is only an estimate of the target x, y coordinates, that is, x _a = x(1:2,:), if we assume that x′ _s1 = x _s1 (1:2,:), Then the above formula (21) can be written as:

Combining (17) to (22), we can Written as a function of r ₁ , the final form is as follows:

At this time, substitute the result of the above formula into formula (19). The calculated target position coordinate x contains only _one unknown parameter r1. Substitute the x coordinate containing the unknown parameter _r1 into the constraint formula _r12 =( ^xxs1 ) ^T ( _xxs1 ), and you can get a quartic equation about _{r1 .} Solve the quartic equation to get the value of _r1 , and then substitute it into the result containing the parameter of formula (19) to get the final estimated coordinate of the target.

The basic idea of the dual-station TDOA and FDOA joint positioning method is the same as that of the three-station TDOA positioning method. First, the distance r ₁ from the target to the reference observation station is taken as a known parameter and then the r ₁ is solved by back-tracking to the definition formula. Then the solved r ₁ value is brought into the vector containing the parameters to be estimated to solve the final position coordinates of the target. However, compared with the TDOA positioning method, the dual-station TDOA-FDOA positioning introduces additional unknown variables. Therefore, the method of the present invention further uses the derivative formula to Expressed as a function of r ₁ , the problem is finally transformed into solving a system of equations containing the additional variable r ₁ .

The performance of the method proposed in the present invention is verified by simulation experiments as follows.

Environment: MATLAB R2019b under WIN10 system;

Evaluation indicators:

The mean square error RMSE is defined as:

Among them, x and They represent the true value of the target position coordinates to be determined and the estimated value of the i-th Monte Carlo simulation respectively, and L is the total number of Monte Carlo simulations.

The error percentage is defined as:

After obtaining the RMSE of the experiment, it is compared with the average Euclidean distance from the target to different observation stations to obtain the error percentage.

Simulation experiment 1 (three-station TDOA positioning):

First, assume that the latitude and longitude coordinates of the three observation stations are The target's true latitude and longitude are set to [75.945.353000]. In engineering applications, the target's prior information range is generally given. Here, it is assumed that the target's z coordinate is known, and the target and the sensor are all stationary. Observation station 1 is used as the reference station. According to the TDOA definition formula, two pairs of TDOA values measured by three observation stations are created, namely t ₂₁ and t ₃₁ . Then, different levels of TDOA measurement noise (the noise is a Gaussian white noise with a mean of 0 and a variance of σ ² ) are added. The experimental results are shown in Figure 2. All conditions remain unchanged as above, and the measurement noise variance is set to σ ² =10 ^-4 , that is, the TDOA measurement noise variance is 100ns. Under different height errors, the estimation results are shown in Figure 3.

As shown in Figure 2, when the measurement error is very small, the error of the three-station TDOA positioning algorithm based on prior range information is very small. As the noise level increases, the estimated error slowly increases. However, assuming that the target z coordinate is accurately known, the overall positioning error percentage is very small, and the positioning result is relatively ideal. Figure 3 shows the positioning results under different height errors (i.e., adding an error of plus or minus 1000 meters to the actual z coordinate) when the noise level is constant. When the error is 0 meters, the estimated error percentage is almost zero. As the height error increases, the positioning error continues to increase, but the overall positioning error has always been at a low level.

The comparison results with the method in reference ^[10] are shown in Figures 4, 5 and 6.

Figures 4 and 5 show the positioning errors of the two algorithms based on three-station TDOA measurement under different height errors when the noise level is constant. Range represents the method based on prior range information proposed by the present invention, and height represents the positioning method based on height prior information proposed in document ^[10] . It can be seen from the experimental results that the estimation error of the method proposed by the present invention is always smaller than the error in document ^[10] . Especially when the height error is small, the superior performance of the scheme proposed by the present invention is more obvious. Figure 6 shows the trend of the positioning error of the two algorithms changing with noise when there is no height error. It can be seen from the figure that the error of the method proposed by the present invention changes more obviously with height. This is because the method in document ^[10] itself relies on the approximate elliptical model of the earth and has a large inherent error. Although the positioning error is not sensitive to the height error, it always has a high positioning error.

Simulation experiment 2 (dual-station TDOA-FDOA joint positioning):

First, assume that the longitude and latitude coordinates of the two observation stations are The target’s true latitude and longitude are set to [75.945.353000], which is consistent with the prior information of the three-station TDOA experiment. Assume that the target’s z coordinate is known and the target is stationary, but there is relative motion between the target and the observation station. The velocity vector of the observation station is expressed as According to the definition of TDOA and FDOA, the actual TDOA and FDOA measurement values are given, that is, t ₂₁ and Under different measurement noise levels (TDOA noise is Gaussian white noise with zero mean and variance σ ² , and FDOA noise is Gaussian white noise with zero mean and variance 0.1σ ² ), the experimental results are shown in Figure 7 as the noise changes.

All conditions remain the same as above. Set the TDOA measurement noise variance σ ² =10 ^-4 (ie, the TDOA measurement noise variance is 100ns), and the FDOA measurement noise variance is 0.1σ ² . The estimation results are shown in FIG8 under different height errors.

As can be seen from Figure 7, the estimation error of the dual-station TDOA-FDOA positioning error based on the prior range information increases slowly as the noise level increases. Due to the introduction of FDOA measurement information, the overall estimation error percentage is smaller than the above three-station TDOA positioning. Figure 8 shows the positioning results under different height errors (i.e., adding an error of plus or minus 1000 meters to the actual z coordinate) when the noise level is constant. Similar to the three-station TDOA positioning error result graph, when the error is 0 meters, the estimation error percentage is almost zero. As the height error increases, the positioning error continues to increase, but the overall positioning error has always been at a low level.

The comparison results with the method in reference ^[10] are shown in Figures 9, 10 and 11. It can be seen from Figures 9 and 10 that under the same noise level, the dual-station TDOA-FDOA positioning method proposed in the present invention has the same change trend as the method in reference ^[10] , but the estimation error of the method proposed in the present invention is always smaller than that of the method in reference ^[10] , that is, the estimation performance is very good; it can also be seen from Figure 11 that as the noise increases, the estimation errors of the two methods are increasing, but it is obvious that the estimation error of the scheme proposed in the present invention is smaller.

In the two positioning schemes based on prior range information proposed by the present invention, the values that need to be set subjectively are the TDOA measurement noise and the FDOA measurement noise, as well as the number of Monte Carlo simulations set during the experiment. Under normal circumstances, the TDOA measurement error is small, generally between 30ns and 150ns, and the FDOA error is generally set to 0.01 to 0.1 times the TDOA positioning error; regarding the number of Monte Carlo simulations during the experiment, the more the better, but for the sake of simplicity of calculation, for each specific parameter of each experiment, the present invention sets the number of Monte Carlo simulations to 1000. For the positioning method in the literature [ ¹⁰ ], in addition to setting the above parameters, it is also necessary to set the radius of the earth in the constraint condition model, which is generally selected from the short semi-axis of the earth to the long semi-axis of the earth, that is, between 6356752m and 6378137m. The simulation experiment of the present invention selects the radius of the earth r _g = 6371km based on the actual longitude and latitude of the target. In actual engineering applications, the setting of this value can be given an approximate value based on the prior longitude and latitude range of the target.

Another embodiment of the present invention further provides a passive target positioning system based on TDOA and FDOA measurements, comprising:

An information acquisition module, used to obtain measurement information and receiving station location information;

A coordinate conversion module is used to convert the receiving station location information into an Earth-centered Earth-fixed coordinate system;

A measurement equation building module, used to build a measurement equation based on the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system;

A measurement equation deformation module, used for deforming the measurement equation in combination with prior information;

The target position estimation module is used to solve the target position vector using the measurement equation after deformation processing to obtain the estimated value of the target coordinates.

It will be appreciated by those skilled in the art that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

The present invention is described with reference to the flowchart and/or block diagram of the method, device (system), and computer program product according to the embodiment of the present invention. It should be understood that each process and/or box in the flowchart and/or block diagram, as well as the combination of the process and/or box in the flowchart and/or block diagram can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device produce a device for implementing the functions specified in one or more processes in the flowchart and/or one or more boxes in the block diagram.

These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory produce a manufactured product including an instruction device that implements the functions specified in one or more processes in the flowchart and/or one or more boxes in the block diagram.

These computer program instructions may also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, whereby the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one or more processes in the flowchart and/or one or more boxes in the block diagram.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the relevant field should understand that the specific implementation methods of the present invention can still be modified or replaced by equivalents, and any modifications or equivalent replacements that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims (7)

1. A passive target positioning method based on TDOA and FDOA measurement, characterized by comprising the following steps: Obtain measurement information and receiving station location information; Convert the receiving station location information into the Earth-centered Earth-fixed coordinate system; Establishing a measurement equation based on the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system; Deforming the measurement equation in combination with prior information; The target position vector is solved by using the measurement equation after deformation processing to obtain the estimated value of the target coordinates; The step of establishing the measurement equation according to the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system comprises: Assume that the position and velocity of the N receiving stations _si after coordinate conversion are x _si = [x _si , y _si , z _si ] ^T and Moreover, these receiving stations are not on the same straight line or plane. The position and velocity of the radiation source to be determined are respectively expressed as x = [x, y, z] ^T and Then the distance between the radiation source x and the receiving station _si is: Select receiving station _s1 as the reference receiving station. The measurement equations of TDOA and FDOA between the radiation source transmitting signal and the i-th receiving station and the reference receiving station are expressed as follows: Among them, _ti and _t1 are the time when the signal starts from the radiation source and arrives at the receiving stations _si and _s1 respectively, c is the propagation speed of electromagnetic waves, _f0 is the frequency of the signal emitted by the target radiation source, _ri is the distance from the target radiation source to each receiving station, is the rate of change of the distance from the target radiation source to each receiving station, _vi represents the radial relative velocity between the receiving station and the target, and the calculation expression is as follows: The measurement equation is linearized before being deformed in combination with prior information, including: Convert the TDOA measurement information into the distance difference RDOA equation between the radiation source and the two receiving stations: According to r _i ² ＝(xx _si ) ^T (xx _si ) and r ₁ ² ＝(xx _s1 ) ^T (xx _s1 ), the equivalent equation of the RDOA equation is as follows: Derivative the above equation with respect to time t, we get the linearized FDOA positioning equation: The step of deforming the measurement equation in combination with the prior information includes: obtaining a coordinate range of the target source area according to the receiving station position information in the earth-centered earth-fixed coordinate system, assuming it is [x _min , y _min , z _min ] ~ [x _max , y _max , z _max ], taking As a priori z coordinate, the equivalent equation of the RDOA equation is transformed into the following form: In the formula, x _i1 =x _si -x _s1 , y _i1 =y _si -y _s1 , z _i1 =z _si -z _s1 ; Ignoring the target motion, assume Then the linearized FDOA positioning equation is transformed into the following form: make x _i1 = x _si - x _s1 , y _i1 = y _si - y _s1 , z _i1 = z _si - z _s1 . After obtaining the priori z coordinate of the target to be determined, the FDOA positioning equation in the above formula is transformed into the following form: 2. The passive target positioning method based on TDOA and FDOA measurement according to claim 1, characterized in that the measurement information includes the TDOA measurement value t _i1 and the FDOA measurement value The receiving station location information includes the latitude and longitude coordinates of the observation station The receiving station location information is converted to the Earth-centered Earth-fixed coordinate system as follows: Where R is the radius of curvature of the reference ellipsoid, e is the first eccentricity of the ellipsoid, The radius of the major axis of the ellipsoid is a=6378137m, and the radius of the minor axis is b=6356752m. 3. According to claim 1, the passive target positioning method based on TDOA and FDOA measurements is characterized in that the step of solving the target position vector using the deformed measurement equation includes: if only TDOA measurement is used to complete the positioning, the RDOA equations determined by different sensor pairs are written into a matrix form and then solved; when there is relative motion between the observation station and the target source, the Doppler effect caused by the relative motion between the radiation source and the receiving station is used to estimate the frequency difference of the signal arriving at different receiving stations, and the RDOA and FDOA equations determined by different sensor pairs are combined into a matrix form and then solved. 4. The passive target positioning method based on TDOA and FDOA measurements according to claim 3 is characterized in that if there are only three receiving stations and there is no relative motion between the receiving stations and the target, the RDOA equations determined by different sensor pairs are written in matrix form as follows: h＝Gx _a The matrix vectors are: x _a =[x,y] ^T x _a ＝G ^-1 h From the above formula, we get the calculation results of the target x and y coordinates. Since _r1 is an unknown quantity, the target x and y coordinates are expressed as a function of _r1 . Then the result containing _r1 is ^substituted into the definition formula _r12 =( _xxs1 ) ^T ( _xxs1 ), and a quadratic equation about _r1 is obtained. By solving the equation, the value of _r1 is obtained. _r1 is substituted into the calculation result again to obtain the estimated values of the target x and y coordinates. 5. The passive target positioning method based on TDOA and FDOA measurements according to claim 4 is characterized in that the value of r ₁ is a positive real root, and if both solutions of the equation are positive real roots, an ideal solution is determined by prior information. 6. The passive target positioning method based on TDOA and FDOA measurement according to claim 3 is characterized in that if there are only two receiving stations and there is relative motion between the receiving station and the target, when the prior range of the target and the approximate value of the z coordinate of the target are obtained, the RDOA equation and the FDOA equation measured by the two stations are jointly written into a matrix form as follows: From the above formula, we can get the calculation results of the target x, y coordinates. Since r ₁ and are functions of the unknown variable x, so r ₁ and They are also unknown quantities. The x and y coordinates of the target are expressed as a function of r ₁ ; Since the target is stationary, To guide, Written as a function of r ₁ , the derivative is written as the multiplication of coordinate vectors as follows: Assume x′ _s1 = x _s1 (1:2,:), Then transform the above formula into: Will Written as a function of r ₁ , the final form is as follows: Substitute the result of the above formula into the matrix, and the calculated target position coordinate x contains only one unknown parameter _r1 . Substitute the x coordinate containing the unknown parameter _r1 into the constraint formula _r12 =( _xxs1 ) ^T ( _xxs1 ), ^and get a quartic equation about _r1 . Solve the quartic equation to get the value of _r1 , and then substitute the value of _r1 into the parameter-containing result to get the estimated values of the target x and y coordinates. 7. A passive target positioning system based on TDOA and FDOA measurement, characterized by comprising: An information acquisition module, used to obtain measurement information and receiving station location information; A coordinate conversion module is used to convert the receiving station location information into an Earth-centered Earth-fixed coordinate system; A measurement equation building module, used to build a measurement equation based on the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system; A measurement equation deformation module, used for deforming the measurement equation in combination with prior information; The target position estimation module is used to solve the target position vector using the measurement equation after deformation processing to obtain the estimated value of the target coordinates; The step of establishing the measurement equation according to the measurement information and the receiving station position information in the Earth-centered Earth-fixed coordinate system comprises: Assume that the position and velocity of the N receiving stations _si after coordinate conversion are x _si = [x _si , y _si , z _si ] ^T and Moreover, these receiving stations are not on the same straight line or plane. The position and velocity of the radiation source to be determined are respectively expressed as x = [x, y, z] ^T and Then the distance between the radiation source x and the receiving station _si is: Select receiving station _s1 as the reference receiving station. The measurement equations of TDOA and FDOA between the radiation source transmitting signal and the i-th receiving station and the reference receiving station are expressed as follows: Among them, _ti and _t1 are the time when the signal starts from the radiation source and arrives at the receiving stations _si and _s1 respectively, c is the propagation speed of electromagnetic waves, _f0 is the frequency of the signal emitted by the target radiation source, ri _is the distance from the target radiation source to each receiving station, is the rate of change of the distance from the target radiation source to each receiving station, _vi represents the radial relative velocity between the receiving station and the target, and the calculation expression is as follows: The measurement equation is linearized before being deformed in combination with prior information, including: Convert the TDOA measurement information into the distance difference RDOA equation between the radiation source and the two receiving stations: According to r _i ² ＝(xx _si ) ^T (xx _si ) and r ₁ ² ＝(xx _s1 ) ^T (xx _s1 ), the equivalent equation of the RDOA equation is as follows: Derivative the above equation with respect to time t, we get the linearized FDOA positioning equation: The step of deforming the measurement equation in combination with the prior information includes: obtaining a coordinate range of the target source area according to the receiving station position information in the earth-centered earth-fixed coordinate system, assuming it is [x _min , y _min , z _min ] ~ [x _max , y _max , z _max ], taking As a priori z coordinate, the equivalent equation of the RDOA equation is transformed into the following form: In the formula, x _i1 =x _si -x _s1 , y _i1 =y _si -y _s1 , z _i1 =z _si -z _s1 ; Ignoring the target motion, assume The linearized FDOA positioning equation is transformed into the following form: make x _i1 = x _si - x _s1 , y _i1 = y _si - y _s1 , z _i1 = z _si - z _s1 . After obtaining the priori z coordinate of the target to be determined, the FDOA positioning equation in the above formula is transformed into the following form: