Consider advanced posterior cramer-rao lower bound related to target measurement uncertainty state

Abstract

The application discloses an advanced posterior Cramer-Rao lower bound considering target measurement uncertainty state correlation, comprising the following steps: step one, determining a target state equation of a target motion process and a measurement equation of a double-base radar when target tracking is performed; step two, after the target state equation and the sensor measurement equation are obtained, a Fisher information matrix of target state estimation of the target at time k is recorded as J k , and the lower bound of the target state estimation at time k is the inverse matrix of J k , that is, the advanced posterior Cramer-Rao lower bound; step three, calculating a conditional measurement information matrix; step four, substituting the calculated conditional measurement information matrix into the advanced posterior Cramer-Rao lower bound. The method focuses on the influence of target missed detection and measurement noise on the PCRLB under the influence of the target-radar geometric position under the basis of clutter interference. Simulation experiments verify that, compared with other PCRLB methods, the method of the patent can obtain a more accurate lower bound under the condition of highly uncertain double-base radar measurement information.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of target tracking, and relates to unbiased estimation optimal performance under highly uncertain measurement information of bistatic radar, in particular to an advanced posterior Cramer-Rao lower bound (APCRLB) considering state-dependent measurement uncertainty of a target. BACKGROUND

[0002] The posterior Cramer-Rao lower bound (PCRLB) provides a reference lower bound for unbiased estimation performance of an unknown parameter vector in a clutter and missed detection environment. The PCRLB can not only serve as a performance reference for filters in positioning and tracking, but also as an objective function in sensor management. In the modeling and derivation of the PCRLB, the state-dependent measurement noise and detection probability are usually not considered, and the measurement information matrix only contains an information reduction factor. Subsequently, some scholars considered a state-dependent detection probability model in the derivation of the PCRLB, and obtained an additional information impact parameter in the measurement information matrix, which can obtain a more accurate lower bound in a scenario with large bandwidth and moderate signal-to-noise ratio. However, in the existing derivation of the PCRLB, the measurement noise covariance is usually modeled as independent of the geometric position relationship between the target and the radar, and the PCRLB established according to the constant measurement noise covariance has the problem of overestimation of the lower bound. SUMMARY

[0003] In view of the deficiencies of the prior art, the application provides an advanced posterior Cramer-Rao lower bound (APCRLB) considering state-dependent measurement uncertainty of a target. On the basis of clutter interference, the influence of target detection and measurement noise on the PCRLB under the influence of the geometric position relationship between the target and the radar is considered, and an additional signal-to-noise ratio gain factor is obtained on the basis of the information reduction factor in the measurement information matrix, so that the problem of overestimation of the PCRLB under highly uncertain measurement information of the bistatic radar can be effectively solved.

[0004] The application discloses an advanced posterior Cramer-Rao lower bound considering state-dependent measurement uncertainty of a target, and has the characteristics that the following steps are included.

[0005] Step one, in target tracking, the target state equation of the target motion process and the measurement equation of the bistatic radar are determined.

[0006] Step two, after obtaining the target state equation and the sensor measurement equation, the Fisher information matrix of the target state estimation of the target at time k is denoted as J k , and the lower bound of the state estimation of the target at time k is the inverse matrix of J k , that is, the advanced posterior Cramer-Rao lower bound, which is defined as follows:

[0007]

[0008] where E{·} is the expectation function, is an unbiased estimate of the target state at time k;

[0009] Step three, the expression of the advanced posterior CRLB is obtained is part of the solution The conditional measurement information matrix is calculated

[0010] Step four, the conditional measurement information matrix is calculated and substituted into to obtain the advanced posterior CRLB.

[0011] As a preference, the target state equation is described by an additive noise equation:

[0012] x k+1 = f(x k ) + υ k

[0013] where x k+1 and x k are the target states at time k+1 and k respectively, f(·) is a linear function, υ k is additive Gaussian white noise, υ k ~ N(0, Q k ), and Q k is the process noise covariance.

[0014] As a preference, the measurement equation of the bistatic radar at time k is:

[0015] z k = h k (x k , x T,k , x R,k ) + w k (x k , x T,k , x R,k )

[0016] where z k is the measurement of the target by the bistatic radar at time k, x T,k is the state of the radar transmitting station, x R,k is the state of the radar receiving station, h k is a function known about (x k , x T,k , x R,k ), and w k is a function known about (x k , x T,k,x R,k Measurement noise, w k (x k ,x T,k ,x R,k )~N(0,R k (x k ,x T,k ,x R,k )), R k (x k ,x T,k ,x R,k ) represents the bistatic radar measurement noise covariance related to geometric location.

[0017] Preferably, in step two,

[0018] If f(·) in the target state equation is a linear function, then the Fisher information matrix J of the target state estimation is... k Written as:

[0019]

[0020] In the formula, F is the target state equation f(·). The new measurement information at time k is defined as follows:

[0021]

[0022] In the formula, For the Jacobi operator, Given a conditional measurement information matrix, p(Z(k)|x k Let Z(k) be the probability density function of the target measurement vector Z(k) of the measurement model at time k, which includes missed detections and clutter:

[0023]

[0024] In the formula, m k Let k be the number of clutter at time k. middle The measurement originates from the target; the rest is clutter. p(z) k |x k The expression for ) is as follows:

[0025]

[0026] Let M be the number of target measurements at time k. k =m k Then the number of target measurements at time k is m. k The probability of time for:

[0027]

[0028] where λ is the spatial density of clutter, λ = MP FA V, M is the total number of resolution cells in the measurement space, MP FA is the expected number of clutter in the measurement space, V is the volume of the measurement space, P D (x k ) is the detection probability, P FA is the false alarm rate of the radar, N FA is the number of false alarms;

[0029] d(x k , m k ) is the number of measurements M k = m k at time k and the probability that one of the measurements is generated by the target is described as:

[0030]

[0031] As a preferred, the conditional measurement information matrix in step three is calculated as follows:

[0032]

[0033] To get the following three settings are needed; first, set R k (x k ) as:

[0034]

[0035] where x k variable is separated from the measurement noise covariance matrix of signal-to-noise ratio;

[0036] Second, set the measurement noise covariance R k (x k ) as a diagonal matrix, which has:

[0037]

[0038] where σ1, σ2 and σ3 are the standard deviations of the range, velocity and angle measurement noise, respectively;

[0039] Third, set P D (x k ) and R k (x k ) as constants within one sensor sampling period;

[0040] Through the above settings, we can calculate ​

[0041]

[0042] where the expressions for A, B, C are defined as:

[0043]

[0044] where H k (x k ) is the Hessian matrix of the measurement function h(x k );

[0045] Computing the conditioned measurement information matrix The expression is:

[0046]

[0047] where A is an odd function of the arguments , B and C are even functions of the arguments , AA T , BB T , CC T , BC T and CB T are even functions of the arguments , AB T , AC T , BA T and CA T are odd functions of the arguments , and the integrals of AB T , AC T , BA T and CA T all vanish when integrated over the measurement space V, and BC T = CB T , using this property to simplify:

[0048]

[0049] where the information reduction factor and the signal-to-noise gain factor are defined as:

[0050]

[0051] The simplifications and are written as and Γ1(x k ,x T,k ,x R,k ,m k ), Γ2(x kx T,k x R,k m k ) and Γ3(x k x T,k x R,k m k ) are denoted as Γ1, Γ2 and Γ3, respectively; where the signal-to-noise ratio gain factor Γ1, Γ2 and Γ3 in the above equations are defined as follows:

[0052]

[0053]

[0054] As a preferred, the step three further comprises a simplified information reduction factor and a signal-to-noise ratio gain factor BRIEF DESCRIPTION OF DRAWINGS

[0055] Figure 1 Fig. 1 is a target trajectory and receiver position map for an example;

[0056] Figure 2 Fig. 5 is a plot of different posterior CRLBs for an example;

[0057] Figure 3 Fig. 7 is a plot of detection probability variation for an example;

[0058] Figure 4 Fig. 9 is a plot of measurement noise standard deviation variation for an example;

[0059] Figure 5 Fig. 11 is a plot of inverse versus different signal-to-noise ratio constants for an example;

[0060] Figure 6 Fig. 13 is a plot of detection probability variation for an example;

[0061] Figure 7 Fig. 15 is a plot of measurement noise standard deviation variation for an example;

[0062] Figure 8 Fig. 17 is a plot of inverse versus different false alarm rates for an example;

[0063] Figure 9 Fig. 19 is a plot of detection probability variation for an example; DETAILED DESCRIPTION

[0064] The present application will be further explained with reference to the accompanying drawings wherein:

[0065] The present application provides a method for calculating an advanced posterior CRLB considering the correlation of target measurement uncertainty states:

[0066] Step one: when target tracking is performed, the target state equation of the target motion process can be described by an additive noise equation:

[0067] x k+1 = f(x k ) + υ k

[0068] where x k+1 and x k are the states of the target at k+1 and k, respectively, f(·) is a linear function, υ k is additive Gaussian white noise, υ k ~ N(0, Q k ), and Q k is the process noise covariance.

[0069] The measurement equation of the dual-base radar at k is:

[0070] z k = h k (x k , x T,k , x R,k ) + w k (x k , x T,k , x R,k )

[0071] where z k is the measurement of the target by the dual-base radar at k, x T,k is the state of the radar transmitting station, x R,k is the state of the radar receiving station, h k is a function known about (x k , x T,k , x R,k ), w k is measurement noise about (x k , x T,k , x R,k ), w k (x k , x T,k , x R,k ) ~ N(0, R k (x k , x T,k , x R,k )), and R k (x k , x T,k , x R,k ) is the dual-base radar measurement noise covariance related to the geometric position.

[0072] Step two: after obtaining the target state equation and sensor measurement equation, the Fisher information matrix of the target state estimation at time k is denoted as J k , the lower bound (APCRLB) of the state estimation at time k is J k , that is, the advanced posterior Cramer-Rao lower bound of the present patent, which is defined as follows:

[0073]

[0074] In the formula, E{·} is an expectation function, is an unbiased estimation of the target state at time k. Assuming that f(·) in the target state equation is a linear function, the Fisher information matrix J k of the target state estimation is written as:

[0075]

[0076] In the formula, F is the target state equation f(·), , which represents new measurement information at time k, and is defined as follows:

[0077]

[0078] In the formula, is a Jacobian operator, is a conditional measurement information matrix, and p(Z(k)|x k is a probability density function of the target measurement vector Z(k) of the measurement model containing missed detection and clutter at time k:

[0079]

[0080] In the formula, m k is the number of clutters at time k, comes from the target, and the rest of the measurements are clutters, and the expression of p(z k |x k ) is as follows:

[0081]

[0082] Suppose that there are M k target measurements at time k, and m k , then the number of target measurements at time k is m k , and the probability of m at time k is:

[0083]

[0084] In the formula, the expression of the spatial density λ of the clutter is λ=MP FA ​ / V, M is the total number of resolution cells of the radar in the measurement space, MP FA Let V be the expected number of clutter particles in the measurement space, and P be the volume of the measurement space. D (x k P represents the detection probability. FA N represents the radar false alarm rate. FA This represents the number of false alarms.

[0085] d(x k ,m k M represents the number of target measurements at time k. k =m k And there is a metric that measures the probability of being generated by the target, described as:

[0086]

[0087] Step 3: In Step 2, we obtained the expression for the advanced posterior Cramérault lower bound. yes Part of the solution The conditional measurement information matrix needs to be calculated first.

[0088]

[0089] In order to obtain The following three settings are required. First, set R... k (x k Set as:

[0090]

[0091] In the formula, does not contain x k Measurement noise covariance matrix for variable separation signal-to-noise ratio.

[0092] Second, measure the noise covariance R. k (x k If we define it as a diagonal matrix, then:

[0093]

[0094] In the formula, σ1, σ2 and σ3 are the standard deviations of distance, speed and angle measurement noise, respectively.

[0095] Third, P D (x k ) and R k (x k It is set to be a constant within a sensor sampling period.

[0096] The above settings can be used to calculate... obtained:

[0097]

[0098] where the expressions for A, B, C are defined as:

[0099]

[0100] where H k (x k ) is the Jacobian matrix of the measurement function h(x k ).

[0101] Computing the conditioned measurement information matrix The expression:

[0102]

[0103] where A is an odd function of the arguments , B and C are even functions of the arguments , AA T , BB T , CC T , BC T and CB T are even functions of the arguments , AB T , AC T , BA T and CA T are odd functions of the arguments , and the integrals of AB T , AC T , BA T and CA T all vanish when integrated over the measurement space V, BC T = CB T , using this property to simplify:

[0104]

[0105] where the information reduction factor and the signal-to-noise gain factor are defined as:

[0106]

[0107] The simplifications and are written as and Γ1(x k ,x T,k ,x R,k ,mk ), Γ2(x k ,x T,k ,x R,k ,m k ) and Γ3(x k ,x T,k ,x R,k ,m k Simplified as Γ1, Γ2, and Γ3. Wherein, the signal-to-noise ratio gain factor... Γ1, Γ2, and Γ3 are defined as follows:

[0108]

[0109] Step 4: Obtained in Step 3 After the expression, for Information reduction factor To further simplify the calculation, Make the following changes:

[0110]

[0111] Through the above transformation, the measurement space A is mapped to the measurement space. Information reduction factor Conduct on The expected expansion is then performed, and the following two hypotheses are proposed to simplify the information reduction factor:

[0112] Assumption 1: Assume that the measurement error of the target is within the threshold of g times the standard deviation of the target measurement, that is:

[0113]

[0114] In the formula, h = 1, 2, 3, z[1], z[2] and z[3] are the radar range measurement, Doppler measurement and angle measurement, respectively, and g is usually taken as 3 or 4. When g = 3, it means that the target measurement has a 99% probability of falling within 3σ. h Within.

[0115] Assumption 2: Assume that the measurement dimensions are mutually orthogonal. Using this assumption, the measurement space can be represented as follows: This measurement space The volume is V g = (2g) 3 σ1σ2σ3, and can use the assumed P D P FA d(m) k )and Represented as:

[0116]

[0117] The information reduction factor after unfolding Using assumption 1 and assumption 2, we have:

[0118]

[0119] In which, using assumption 1 and assumption 3, we have is expressed as:

[0120]

[0121] Using assumption 1 and assumption 2, the integral interval is a symmetric interval, independent of each other, with and as the independent variables, the function has the following properties: when i≠j, the function to be integrated is equivalent to the multiplication of two independent odd functions, and the integral result is 0; when i=j, i=1,...,m k , the function to be integrated is equivalent to the multiplication of two identical odd functions, and each integral result is the same. We can use this property to simplify the above formula:

[0122]

[0123] Make the following transformation to

[0124]

[0125] Through the above transformation, the measurement space is mapped and transformed into the measurement space The volume of the measurement space is (2g) 3 , and using assumption 2 to perform differential substitution, the steps are as follows:

[0126]

[0127] Using the new measurement substitution transformation and substituting assumption 2, the information reduction factor is further simplified:

[0128]

[0129] In which, the expression of is:

[0130]

[0131] In the function to be integrated with and as the independent variables, are independent of each other, and when These parts of integral, they are all odd functions on the interval of integration, the final integral result is 0, finally can be Described as:

[0132]

[0133] When the normalized Matrix in The integral is calculated respectively, the function expression of the three in the integrand is consistent, the upper and lower limits of integration are consistent, so the final calculation result is also the same, then Can be written as:

[0134]

[0135] In the formula, I3 is a 3x3 unit matrix, further simplify the formula, the matrix Convert to scalar There is:

[0136]

[0137] The final expression of the multiplication reduction factor is:

[0138]

[0139] In the formula, the expression of β(x k ,m k ) is:

[0140]

[0141] Step five: after getting the expression of In step three, the signal-to-noise ratio gain factor In Further simplified calculation, through the transformation of The signal-to-noise ratio gain factor About The expected expansion, and using Transformation and setting 2, hypothesis 1, hypothesis 2 to simplify:

[0142]

[0143] In the formula, And The expression is:

[0144]

[0145]

[0146] In the formula,

[0147] The following specific experimental verification is given in order to further illustrate the effectiveness of the present application:

[0148] The experimental verification of the present application shows that the APCRLB method derived in this chapter is lower than PCRLB and EFIM, and the improvement degree of APCRLB relative to the other two PCRLBs is explored in different signal-to-noise ratio constants and different false alarm rates. As shown in Figure 1 , in the two-dimensional Cartesian coordinate system, the transmitting station of the bistatic radar is located at the coordinate origin, the receiving station is located at [300m, 0m] T , the target moves at a uniform speed from left to right, and the initial state is [170m, 45m, 2.5m / s, 0m / s] T . In order to increase the contribution of the signal-to-noise ratio gain factor in APCRLB, the simulation in this section selects the signal parameter with larger measurement noise and the signal-to-noise ratio constant under higher clutter density, and keeps the target and the transmitting station and the receiving station at a closer distance to make the signal-to-noise ratio derivative matrix value larger, and the simulation scene parameters are shown in Tables 1 and 2.

[0149] Table 1 Factor simulation parameter table

[0150]

[0151]

[0152] Table 2 Posterior Cramer-Rao lower bound simulation parameter table

[0153]

[0154] As shown in Figure 3 , 4 , the detection probability and measurement noise standard deviation change diagram when comparing different posterior Cramer-Rao lower bounds. When the target moves from left to right, the bistatic distance decreases, and the signal-to-noise ratio increases as a whole, so the measurement noise covariance decreases, and the detection probability increases. As can be seen from Figure 2 , different posterior Cramer-Rao lower bounds achieve the expected effect. PCRLB does not consider the derivative of the time-varying detection probability and the measurement noise covariance when deriving, and lacks the signal-to-noise ratio gain term, so the lower bound is the lowest; EFIM does not consider the time-varying derivative of the measurement noise covariance when deriving, and lacks the measurement noise covariance gain term in the signal-to-noise ratio gain term, so the lower bound is between PCRLB and APCRLB.

[0155] The inverse of the measurement information matrix under different signal-to-noise ratio constants is selected on the basis of the last experiment, the target coordinate position is [180m, 45m] T , and the false alarm rate P FA=0.01, exploring the effect of changes in the signal-to-noise ratio constant on the inverse of the measurement information matrix of different posterior Cramer-Rao, such as Figure 6 , 7 The graph shows the changes in detection probability and measurement noise standard deviation as the signal-to-noise ratio (SNR) constant changes. When the SNR constant increases, the measurement noise covariance decreases, and the detection probability increases. Figure 5 The diagram shows the trace changes of the inverse matrix of the measurement information from different posterior Cramero measurements as the signal-to-noise ratio constant changes. The signal-to-noise ratio gain factor can be analyzed by combining the derived formulas and simulation results. At low signal-to-noise ratio (SNR) constants, Γ1 is the main contributor, while Γ2 and Γ3 contribute almost nothing; therefore, the values ​​of EFIM and APCRLB are essentially the same. At medium and high SNR constants, Γ1 contributes almost nothing, while Γ2 and Γ3 are the main contributors. A significant difference in the values ​​of EFIM and APCRLB can be observed at medium SNR constants. However, at high SNR constants, due to the ideal measurement noise covariance and detection probability, the overall environmental complexity decreases, and the SNR gain factor increases. The gain is greatly reduced, so the values ​​of EFIM, APCRLB, and PCRLB begin to converge.

[0156] The inverse comparison of the measurement information matrix under different false alarm rates is based on the comparison of different posterior Cramero lower bounds, with the target coordinate position selected as [180m, 45m]. T While maintaining the signal-to-noise ratio constant R0 = 420, we explored the effect of changes in the false alarm rate on the inverse of the measurement information matrix of different posterior Cramerro. Figure 9 This indicates that the detection probability of bistatic radar increases as the false alarm rate decreases. According to the clutter model formula, the clutter density in the environment decreases as the false alarm rate decreases. Figure 8 This indicates that the gap between EFIM, PCRLB, and APCRLB increases as the false alarm rate rises. This is because as the false alarm rate increases, the number of clutter in the environment increases rapidly, while the detection probability increases more slowly, making the overall environment relatively more complex.

[0157] The simulation experiments described above verify that under conditions of high uncertainty in bistatic radar measurement information, the newly derived APCRLB is more accurate than other existing PCRLBs. It can be well applied in target tracking and information fusion, for example, providing a reference lower bound for the unbiased estimation performance of unknown parameter vectors in cluttered and missed detection environments, serving as a performance reference for filters in localization and tracking, and acting as an objective function in sensor management, significantly improving the performance of target tracking algorithms. In practical applications, it can be used in autonomous driving technology to improve the target tracking accuracy of millimeter-wave radar and enhance the safety performance of autonomous driving; it can also be used in missile strikes to improve missile hit probability, etc.

Claims

1. An advanced posterior Cramer-Rao lower bound considering the correlation of target measurement uncertainty states, characterized in that, Includes the following steps: Step 1: When tracking a target, determine the target state equation and the measurement equation of the bistatic radar during the target motion process. Step 2: After obtaining the target state equation and sensor measurement equation, denote the Fisher information matrix of the target state estimate at time k as J. k Then the lower bound of the state estimate of the target at time k is J. k The inverse matrix of , i.e., the advanced posterior Cramer-Rao lower bound, is defined as follows: In the formula, E{·} is the expectation function. This is an unbiased estimate of the target state at time k; Construct the probability density function of the target measurement vector in a measurement model that includes missed detections and clutter; Step 3: The expression for the advanced posterior Cramérault lower bound is obtained. yes Part of the solution Calculate the conditional measurement information matrix The measurement information matrix yields an additional signal-to-noise ratio gain factor based on the information reduction factor. Step 4: Calculate the conditional measurement information matrix. Substitution Obtain the advanced a posteriori Clamello lower bound.

2. The advanced posterior Cramer-Rao lower bound considering the uncertainty state of target measurement according to claim 1, characterized in that, The target state equation is described by an additive noise equation: x k+1 =f(x k )+υ k In the formula, x k+1 x k Let f(·) represent the target states at times k+1 and k, respectively, and let υ be a linear function. k For additivity Gaussian white noise, υ k ~N(0,Q) k ), Q k Let be the process noise covariance.

3. The advanced posterior Cramer-Rao lower bound considering the uncertainty of target measurement as described in claim 2, characterized in that, The measurement equation for the bistatic radar at time k is: z k =h k (x k ,x T,k ,x R,k )+w k (x k ,x T,k ,x R,k ) In the formula, z k For the bistatic radar measurement of the target at time k, x T,k For the status of the radar transmitting station, x R,k The status of the radar receiving station, h k For about (x) k ,x T,k ,x R,k The known function, w k For about (x) k ,x T,k ,x R,k Measurement noise, w k (x k ,x T,k ,x R,k )~N(0,R k (x k ,x T,k ,x R,k )), R k (x k ,x T,k ,x R,k ) represents the bistatic radar measurement noise covariance related to geometric location.

4. The advanced posterior Cramer-Rao lower bound considering the uncertainty of target measurement as described in claim 3, characterized in that, In step two If f(·) in the target state equation is a linear function, then the Fisher information matrix J of the target state estimation is... k Written as: In the formula, F is the target state equation f(·). The new measurement information at time k is defined as follows: In the formula, For the Jacobi operator, Given a conditional measurement information matrix, p(Z(k)|x k Let Z(k) be the probability density function of the target measurement vector Z(k) of the measurement model at time k, which includes missed detections and clutter: In the formula, m k Let k be the number of clutter at time k. middle The measurement originates from the target, while the rest is clutter, p(z) k |x k The expression for ) is as follows: Let M be the number of target measurements at time k. k =m k Then the number of target measurements at time k is m. k The probability of time for: In the formula, the spatial density λ of clutter is expressed as λ = MP FA / V, M is the total number of resolution cells of the radar in the measurement space, MP FA Let V be the expected number of clutter particles in the measurement space, and P be the volume of the measurement space. D (x k P represents the detection probability. FA N represents the radar false alarm rate. FA This represents the number of false alarms. d(x k ,m k M represents the number of target measurements at time k. k =m k And there is a metric that measures the probability of being generated by the target, described as:

5. The advanced posterior Cramer-Rao lower bound considering the uncertainty state of target measurement according to claim 4, characterized in that, In step three, the conditional measurement information matrix The calculation method is as follows: In order to obtain The following three settings are required; first, set R... k (x k Set as: In the formula, does not contain x k Measurement noise covariance matrix for variable-separated signal-to-noise ratio; Second, measure the noise covariance R. k (x k If we define it as a diagonal matrix, then: In the formula, σ1, σ2, and σ3 are the standard deviations of distance, velocity, and angle measurement noise, respectively; Third, P D (x k ) and R k (x k The value is set to be a constant within a sensor sampling period; The above settings can be used to calculate... have to: In the formula, the expressions for A, B, and C are defined as follows: In the formula, H k (x k ) is the measurement equation h(x k Jacobian matrix of ) Calculate the conditional measurement information matrix expression: In the formula, A is about the independent variable. The odd function, where B and C are about the independent variable. even function, AA T BB T CC T BC T and CB T For the independent variable an even function, AB T AC T BA T and CA T For the independent variable The odd function, when integrated over the entire measurement space V, has the following relationship with respect to AB. T AC T BA T and CA T The integrals all result in 0, BC T =CB T This property simplifies the process: In the formula, the information reduction factor and signal-to-noise ratio gain factor The definition is as follows: Will and Simplified notation and Γ1(x k ,x T,k ,x R,k ,m k ), Γ2(x k ,x T,k ,x R,k ,m k ) and Γ3(x k ,x T,k ,x R,k ,m k Simplified as Γ1, Γ2, and Γ3; where Γ1 is the signal-to-noise ratio gain factor. Γ1, Γ2, and Γ3 are defined as follows:

6. The advanced posterior Cramer-Rao lower bound considering the uncertainty of target measurement as described in claim 5, characterized in that, Step three also includes a simplified information reduction factor. and signal-to-noise ratio gain factor

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