Method for tracking maneuvering target in clutter based on waveform selection

Abstract

The application discloses a method for tracking a maneuvering target in clutter based on waveform selection, and is implemented according to the following steps: step 1, establishing a waveform model; step 2, establishing a state and measurement model; step 3, establishing a clutter model; step 4, establishing a filtering model; and step 5, a waveform scheduling algorithm based on fractional Fourier transform, which realizes minimization of a posteriori estimation error through dynamic waveform selection. The application reduces the calculation complexity of the tracking algorithm and improves the target tracking precision and the accuracy of the state estimation of the maneuvering target through the improved filtering algorithm and the dynamic waveform selection.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of radar tracking, and relates to a method for tracking a maneuvering target in clutter based on waveform selection. BACKGROUND

[0002] A radar transmits electromagnetic waves to a target object and receives a reflected echo of the electromagnetic waves, thereby obtaining information of the target. Due to the closed-loop feedback of the receiver to the transmitter, a cognitive radar changes a waveform to adapt to a dynamically changing environment, so as to improve the tracking performance of the whole system. With the increasingly complex detection environment in the air and on the ground, the enhanced maneuverability of the target, and the gradually improved ability of a signal processor, the requirements for target tracking technology are continuously improved. The existing target tracking technology still has problems of high computational complexity and poor tracking precision in tracking a target in a clutter environment and tracking a maneuvering target.

[0003] The existing document "Jiantao Wang, Yuliang Qin, Hongqiang Wang, et al. Dynamic waveform selection for manoeuvering target tracking in clutter, IET Radar Sonar Navig., 2013, Vol. 7, Iss. 7, pp. 815-825" discloses a tracking problem in a nonlinear system and a clutter environment, and uses a particle filter to process a nonlinear transformation. However, the existing document has problems of high computational complexity and inconvenience for practical application. Mainly, the existing research selects an optimal waveform by traversing calculation of each parameter, which leads to a high computational burden, so that the waveform selection algorithm cannot be practically applied in engineering, and the selection of the waveform affects the measurement error of the target, and further affects the tracking precision of the target. SUMMARY

[0004] The application aims to provide a method for tracking a maneuvering target in clutter based on waveform selection, and solves the problems of high computational complexity and low precision in the prior art.

[0005] The technical solution adopted by the application is that the method for tracking a maneuvering target in clutter based on waveform selection is implemented according to the following steps:

[0006] Step 1, establishing a waveform model;

[0007] Step 2, establishing a state and measurement model;

[0008] Step 3, establishing a clutter model;

[0009] Step 4, establishing a filtering model;

[0010] Step 5: The waveform scheduling algorithm based on fractional Fourier transform minimizes the posterior estimation error through dynamic waveform selection.

[0011] The invention is further characterized in that,

[0012] Step 1 is implemented in the following steps:

[0013] Step 1.1: Establish a transmission waveform model in a narrowband environment;

[0014]

[0015] Among them, E T It is the energy of the signal waveform, f c It is the carrier frequency. It is the complex envelope of the transmitted pulse;

[0016] Step 1.2: Establish the received waveform model;

[0017]

[0018] Among them, E R τ represents the energy of the received signal; n(t) is the added white noise; τ represents the time delay; r is the distance between the target and the radar; Represents the speed of the target's movement, and c represents the speed of light;

[0019] When the waveform time-bandwidth product satisfies the narrowband condition, s R (t) is considered as:

[0020]

[0021] Step 2 is implemented in the following steps:

[0022] Step 2.1: Establish the state model to build the MCS model;

[0023]

[0024]

[0025] Among them, X k Let [x] be the target state vector. k ,y k ]、 and These represent the position, velocity, and acceleration in the x and y directions, respectively. It is the average value of the first-order acceleration; h(·) is the nonlinear transformation function; z k The target measurement matrix; process noise W k It is zero-mean Gaussian white noise with variance Q. k=2ασ a 2 q cs ;F ACS and U ACS The specific form is as follows:

[0026]

[0027]

[0028] Where T is the sampling interval; ε k This represents the waveform selected at time k, while R k Subject to ε k The effect of a specific waveform on R k Represented as R k (ε k );

[0029] Step 2.2: Establish the measurement model as a linear measurement model;

[0030] Assuming the target moves in a two-dimensional plane, and distance, velocity, and direction are measured simultaneously, the nonlinear measurement model is as follows:

[0031] z k =h(X) k )+V k (6)

[0032]

[0033] Among them, z k For the target measurement matrix, the measurement noise V k It is zero-mean Gaussian white noise with variance R. k ;r k , θ k These represent the distance between the target and the radar, the radial velocity of the target, and the azimuth angle of the target, respectively.

[0034] Step 3 is implemented in the following steps:

[0035] The measured value of time k under clutter conditions is expressed as:

[0036]

[0037] Where, m k It represents the total number of targets measured by the radar at time k;

[0038] Step 3.1, z i k This includes distance, speed, and angle information, assuming the number of false alarms follows an expected value of ρV. k The Poisson distribution has the following false alarm probability:

[0039]

[0040] Where ρ is the density of the erroneous measurement, and V k To verify the gate volume;

[0041] Step 3.2: Assuming that the clutter is uniformly distributed within the verification gate, and under the assumption that only noise and the target exist, the test statistic follows an exponential distribution.

[0042] Using the peak value of the ambiguity function to estimate the time delay and Doppler shift, the detection probability at time k is:

[0043]

[0044] Where P f η represents the expected probability of a false alarm. k The signal-to-noise ratio at time k is represented.

[0045] Step 4 is implemented in the following steps:

[0046] Step 4.1: Establish the MPDA model;

[0047] Assume β i k The measured value z at time k i k The association probability, β 0 k If the probability of no association from the target measurement is given, then the posterior error covariance matrix in MPDAF is:

[0048]

[0049] in It is the Kalman filter gain; S k+1 =HP k+1|k H T +R k+1 It is the new information covariance matrix; P k+1|k It is the prediction covariance matrix; α is the influence factor of the filtering error covariance matrix:

[0050]

[0051] Where P d It is the detection probability, and P g It is the threshold probability; c Γ It is an influence factor, representing the correlation gate pair information covariance matrix S. k+1 The impact;

[0052] When the measurement dimension is three-dimensional:

[0053]

[0054] Where γ is the association threshold, and γ = g 2 , g is the associated region The defined error function;

[0055] R k+1 Corresponding to a specific waveform ε k+1 (or waveform parameters), P k+1|k+1 Also with ε k+1 Correspondingly, it is defined as P k+1|k+1 (ε k+1 When the modified Riccator equation is used to estimate P k+1|k+1 :

[0056]

[0057] Where q2 is a scalar between 0 and 1, depending on the clutter density ρ and the correlation threshold V. k+1 The detection probability P at time k+1 dk+1 ;q2 is approximately fitted as:

[0058]

[0059] Where n Z =3 and V k+1 =(4π / 3)g 3 |S k+1 | 1 / 2 ;

[0060] Step 4.2: Establish the MPDA-SCKF model;

[0061] Before performing data processing, define R as matrix A. T The upper triangular matrix obtained by QR decomposition, the QR decomposition of matrix A is expressed as S = Tria(A) = R T Where matrix S is a lower triangular matrix, the MPDAF-SCKF processing flow is as follows:

[0062] Step 4.2.1, Time Update;

[0063] Step 4.2.2, Measurement update.

[0064] The specific steps for time updating in step 4.2.1 are as follows:

[0065] Step 1.1: Factorization

[0066] P k|k =S k|k (S k|k ) T (18)

[0067] Step 1.2: Calculate the volume point and propagation volume point

[0068]

[0069]

[0070] Where X i k|k and X i* k+1|k These are the volume point and the propagation volume point, respectively; m is the total number of volume points, and n is the dimension of the state vector X, satisfying the condition m = 2n; ξ i =(m / 2) 1 / 2 [1], [1] is the set of points obtained by permuting or inverting unit vectors in n-dimensional space;

[0071]

[0072] Step 1.3: Calculate the mean of the state prediction and the square root of the covariance of the state prediction error;

[0073]

[0074]

[0075] The weighted centrality matrix is:

[0076]

[0077] The specific steps for measurement updating in step 4.2.1 are as follows:

[0078] Step 2.1: Calculate the volume point and perform a nonlinear transformation:

[0079]

[0080]

[0081] Step 2.2: Calculate the measurement prediction mean

[0082]

[0083] Step 2.3: Calculate the square root coefficients of the residual (innovation) covariance matrix:

[0084]

[0085] The weighted center matrix is:

[0086]

[0087] Step 2.4: Calculate the residual covariance matrix:

[0088]

[0089] Step 2.5: Calculate the cross-covariance matrix between the state and the measurement.

[0090]

[0091] The weighted centrality matrix is:

[0092]

[0093] Step 2.6: Calculate the filter gain:

[0094]

[0095] Step 2.7: Calculate the square root coefficients of the corresponding error covariance matrix and the posterior estimated error covariance matrix:

[0096]

[0097]

[0098] Step 2.8: Update the corresponding state matrix and the corresponding error covariance matrix:

[0099] If no measurement result is accurate, the following formula is used for updating:

[0100]

[0101] S k+1|k+1 =Chol(P k+1|k+1 (37)

[0102] Otherwise, use the following formula:

[0103]

[0104]

[0105] Assume that the state error covariance matrix at time kT is P k|k .

[0106] Step 5 consists of the following steps:

[0107] Step 5.1: Calculate the lower bound of Clamelloe.

[0108] Step 5.2: Optimized waveform selection based on fractional Fourier transform.

[0109] Step 5.1 The specific steps are as follows:

[0110] Measurement noise covariance matrix R k+1 The transmitted waveform parameter ε at time k+1 k+1 Correlation (i.e., the waveform chosen at time k), i.e., A(τ,f) d ) is the ambiguity function of the transmitted waveform s(t), that is:

[0111]

[0112] Regarding time delay τ and Doppler shift f d The Fisher information matrix is ​​as follows:

[0113]

[0114] Where η is the signal-to-noise ratio; R k+1 and A(τ,f) d The relationship between them is:

[0115] R k+1 (ε k+1 ) = TJ -1 (ε k+1 )T (42)

[0116] Where T = diag(c / 2, c / (2f) c Time delay τ and Doppler frequency shift f d Fisher information matrix regarding distance and velocity; J -1 (ε k+1 ) is the Cramer-Rao lower bound for a selected waveform under unbiased estimation;

[0117] Equation (40) shows that the selected waveform at time k determines R. k+1 The optimal waveform is selected by minimizing the estimation error at time k+1.

[0118]

[0119] The choice of waveform at time k affects the measurement error covariance matrix and also the state estimation error at time k+1. By selecting the optimal waveform through waveform scheduling, the performance of target tracking under clutter conditions can be greatly improved.

[0120] Step 5.2 The specific steps are as follows:

[0121] Assume the basic transmitted waveform is S0(t), and the ambiguity function is A0(τ,f). d The Fisher information matrix is ​​J0, the corresponding measurement noise covariance matrix is ​​R0, and the fractional factor in the fractional Fourier transform is... It is applied to the basic transmission waveform to achieve orthogonality between the measurement error ellipse and the state error ellipse;

[0122] The fractional Fourier transform is viewed as a rotation operation of a coordinate system. When the fractional Fourier transform is used with parameters... The basic transmitted waveform, the waveform's ambiguity function is obtained through... Rotation yields a new waveform, characterized by the following features:

[0123]

[0124]

[0125] J k+1 and R k+1 These are the Fisher information matrix and covariance matrix obtained after rotation, respectively; It is a rotation matrix that satisfies Since the fuzzy function is independent of the angle dimension, there is no rotational transformation in the angle dimension. Orthogonalization is achieved through rotational transformation.

[0126] Assume the state error covariance matrix at time k is P k+1|k The fractional factors are:

[0127]

[0128] Where v P (i) and v R (i) are matrices P and R respectively. k The largest eigenvalue corresponds to the i-th element in the eigenvector, where:

[0129]

[0130]

[0131] Will Substituting into formula (45), we get R. k+1 Then, the posterior state error covariance matrix P is obtained. k+1|k+1 The iteration is used for the selection of the next waveform.

[0132] The beneficial effects of this invention are as follows: Based on the improved current statistical (MCS) model, it utilizes the improved probabilistic data association filter (MPDAF) and the square-root capillary Kalman filter (SCKF) to handle clutter and nonlinear transformations, reducing filtering estimation errors and improving filtering accuracy, while also possessing the advantage of low computational complexity. Furthermore, a waveform scheduling algorithm based on fractional Fourier transform is proposed, minimizing posterior estimation errors through dynamic waveform selection. This invention reduces the computational complexity of the tracking algorithm by improving the filtering algorithm and dynamic waveform selection, while simultaneously improving target tracking accuracy. It provides stable tracking of maneuvering targets under clutter interference conditions. When the target maneuvers, this invention exhibits strong adaptability and good tracking performance, improving the accuracy of maneuvering target state estimation. Attached Figure Description

[0133] Figure 1(a) shows the present invention. The waveform generated at that time;

[0134] Figure 1(b) shows the present invention. The waveform generated at that time;

[0135] Figure 1(c) is an illustration of the present invention. The waveform generated at that time;

[0136] Figure 2(a) is a non-orthogonal ellipse plot of the prediction error of the present invention;

[0137] Figure 2(b) is an orthogonal diagram of the prediction error ellipse of the present invention;

[0138] Figure 3 This is a flowchart of the adaptive MPDA-SCKF processing operation of the present invention;

[0139] Figure 4 This is the acceleration diagram of the maneuvering target of this invention;

[0140] Figure 5(a) is a plot of the root mean square error of the location;

[0141] Figure 5(b) is the root mean square error diagram of velocity;

[0142] Figure 5(c) is the root mean square error of acceleration.

[0143] Figure 6 This is a rotation angle diagram of the Fourier transform of the waveform of this invention. Detailed Implementation

[0144] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0145] Example 1

[0146] This invention relates to a method for tracking maneuvering targets in clutter based on waveform selection, which is implemented according to the following steps:

[0147] Step 1: Establish a waveform model;

[0148] Step 2: Establish the state and measurement model;

[0149] Step 3: Establish a clutter model;

[0150] Step 4: Establish the filtering model;

[0151] Step 5: The waveform scheduling algorithm based on fractional Fourier transform minimizes the posterior estimation error through dynamic waveform selection.

[0152] Example 2

[0153] This invention relates to a method for tracking maneuvering targets in clutter based on waveform selection, which is implemented according to the following steps:

[0154] Step 1: Establish a waveform model;

[0155] Step 1.1: Establish a transmission waveform model in a narrowband environment;

[0156]

[0157] Among them, E T It is the energy of the signal waveform, f c It is the carrier frequency. It is the complex envelope of the transmitted pulse;

[0158] Step 1.2: Establish the received waveform model;

[0159]

[0160] Among them, E R τ represents the energy of the received signal; n(t) is the added white noise; τ represents the time delay; r is the distance between the target and the radar; Represents the speed of the target's movement, and c represents the speed of light;

[0161] When the waveform time-bandwidth product satisfies the narrowband condition, s R (t) is considered as:

[0162]

[0163] Step 2: Establish the state and measurement model;

[0164] Step 2.1: Establish a state model

[0165] The MCS model is established as follows:

[0166]

[0167]

[0168] Among them, X k Let [x] be the target state vector. k ,y k ]、 and These represent the position, velocity, and acceleration in the x and y directions, respectively. It is the average value of the first-order acceleration; h(·) is the nonlinear transformation function; z k The target measurement matrix; process noise W k It is zero-mean Gaussian white noise with variance Q. k =2ασ a 2 q cs ;F ACS and U ACS The specific format is as follows:

[0169]

[0170]

[0171] Where T is the sampling interval; ε k This represents the waveform (or waveform parameters) selected at time k, while R... k Subject to ε k The effect of a specific waveform on R k Represented as R k (ε k ).

[0172] Step 2.2: Establish a measurement model

[0173] Assuming the target moves in a two-dimensional plane, and distance, velocity, and direction are measured simultaneously, the nonlinear measurement model is as follows:

[0174] z k =h(X) k )+V k (6)

[0175]

[0176] Among them, z k For the target measurement matrix, the measurement noise V k It is zero-mean Gaussian white noise with variance R. k ;r k , θ k These represent the distance between the target and the radar, the target's radial velocity, and the target's azimuth angle, respectively.

[0177] Step 3: Establish a clutter model

[0178] The measured value of time k under clutter conditions is expressed as:

[0179]

[0180] Where, m k It represents the total number of targets measured by the radar at time k;

[0181] Step 3.1, z i k This includes distance, speed, and angle information, assuming the number of false alarms follows an expected value of ρV. k The Poisson distribution has the following false alarm probability:

[0182]

[0183] Where ρ is the density of the erroneous measurement, and V k To verify the gate volume;

[0184] Step 3.2: Assuming that the clutter is uniformly distributed within the verification gate, and under the assumption that only noise and the target exist, the test statistic follows an exponential distribution.

[0185] Using the peak value of the ambiguity function to estimate the time delay and Doppler shift, the detection probability at time k is:

[0186]

[0187] Where P f η represents the expected probability of a false alarm. k The signal-to-noise ratio at time k is represented.

[0188] Step 4: Establish the filtering model

[0189] Step 4.1: Establish the MPDA model

[0190] Assume β i k The measured value z at time k i k The association probability, β 0 k If the probability of no association from the target measurement is given, then the posterior error covariance matrix in MPDAF is:

[0191]

[0192] in It is the Kalman filter gain; S k+1 =HP k+1|k H T +R k+1 It is the new information covariance matrix; P k+1|k It is the prediction covariance matrix; α is the influence factor of the filtering error covariance matrix:

[0193]

[0194] Where P d It is the detection probability, and P g It is the threshold probability; c Γ It is an influence factor, representing the correlation gate pair information covariance matrix S. k+1 The impact.

[0195] When the measurement dimension is three-dimensional:

[0196]

[0197] Where γ is the association threshold, and γ = g 2 , g is the associated region The error function is defined.

[0198] R k+1 Corresponding to a specific waveform ε k+1 (or waveform parameters), P k+1|k+1 Also with ε k+1 Correspondingly, it is defined as P k+1|k+1 (ε k+1 When the modified Riccator equation is used to estimate P k+1|k+1 :

[0199]

[0200] Where q2 is a scalar between 0 and 1, depending on the clutter density ρ and the correlation threshold V. k+1 The detection probability P at time k+1 dk+1 The approximate fit of q2 is:

[0201]

[0202] Where n Z =3 and V k+1 =(4π / 3)g 3 |S k+1 | 1 / 2 .

[0203] Step 4.2: Establish the MPDA-SCKF model;

[0204] Before performing data processing, define R as matrix A. TThe upper triangular matrix obtained by QR decomposition, the QR decomposition of matrix A is expressed as S = Tria(A) = R T Where matrix S is a lower triangular matrix, such as Figure 3 As shown, the processing flow of MPDAF-SCKF is as follows:

[0205] Step 4.2.1, Time Update

[0206] Assume that the state error covariance matrix at time kT is P k|k .

[0207] Step 1.1: Factorization

[0208] P k|k =S k|k (S k|k ) T (18)

[0209] Step 1.2: Calculate the volume point and propagation volume point

[0210]

[0211]

[0212] Where X i k|k and X i* k+1|k These are the volume points and the propagation volume points, respectively. m is the total number of volume points, and n is the dimension of the state vector X, satisfying the condition m = 2n. ξ i =(m / 2) 1 / 2 [1], [1] is the set of points obtained by permuting or inverting the unit vectors in n-dimensional space.

[0213]

[0214] Step 1.3: Calculate the mean of the state prediction and the square root of the covariance of the state prediction error.

[0215]

[0216]

[0217] The weighted centrality matrix is:

[0218]

[0219] Step 4.2.2, Measurement Update

[0220] Step 2.1: Calculate the volume point and perform a nonlinear transformation:

[0221]

[0222]

[0223] Step 2.2: Calculate the measurement prediction mean

[0224]

[0225] Step 2.3: Calculate the square root coefficients of the residual (innovation) covariance matrix:

[0226]

[0227] The weighted center matrix is:

[0228]

[0229] Step 2.4: Calculate the residual covariance matrix:

[0230]

[0231] Step 2.5: Calculate the cross-covariance matrix between the state and the measurement.

[0232]

[0233] The weighted centrality matrix is:

[0234]

[0235] Step 2.6: Calculate the filter gain:

[0236]

[0237] Step 2.7: Calculate the square root coefficients of the corresponding error covariance matrix and the posterior estimated error covariance matrix:

[0238]

[0239]

[0240] Step 2.8: Update the corresponding state matrix and the corresponding error covariance matrix:

[0241] If no measurement result is accurate, the following formula is used for updating:

[0242]

[0243] S k+1|k+1 =Chol(P k+1|k+1 (37)

[0244] Otherwise, use the following formula:

[0245]

[0246]

[0247] Step 5: The waveform scheduling algorithm based on fractional Fourier transform minimizes the posterior estimation error through dynamic waveform selection.

[0248] Step 5.1: Calculate the lower bound of Clamelloe.

[0249] Measurement noise covariance matrix R k+1 The transmitted waveform parameter ε at time k+1 k+1 Correlation (i.e., the waveform chosen at time k), i.e., A(τ,f) d ) is the ambiguity function of the transmitted waveform s(t), that is:

[0250]

[0251] Regarding time delay τ and Doppler shift f d The Fisher information matrix is ​​as follows:

[0252]

[0253] Where η is the signal-to-noise ratio. R k+1 and A(τ,f) d The relationship between them is:

[0254] R k+1 (ε k+1 ) = TJ -1 (ε k+1 )T (42)

[0255] Where T = diag(c / 2, c / (2f) c Time delay τ and Doppler frequency shift f d Fisher's information matrix regarding distance and velocity. J -1 (ε k+1 ) is the Clamer-Rao lower bound for a selected waveform under unbiased estimation.

[0256] Equation (40) shows that the selected waveform at time k determines R. k+1 .

[0257] Therefore, the optimal waveform is selected by minimizing the estimation error at time k+1:

[0258]

[0259] The choice of waveform at time k affects the measurement error covariance matrix and also the state estimation error at time k+1. Therefore, by selecting the optimal waveform through waveform scheduling, the performance of target tracking under clutter conditions can be significantly improved.

[0260] Step 5.2: Optimized waveform selection based on fractional Fourier transform

[0261] Assume the basic transmitted waveform (a rectangular pulse) is S0(t), and the ambiguity function is A0(τ,f). d The Fisher information matrix is ​​J0, and the corresponding measurement noise covariance matrix is ​​R0. Fractional factors in the fractional Fourier transform. It is applied to the basic transmission waveform to achieve orthogonality between the measurement error ellipse and the state error ellipse.

[0262] As shown in Figures 1(a)-(c), for different In the case of the obtained waveform, the fractional Fourier transform is regarded as a rotation operation of the coordinate system. When the fractional Fourier transform is used with parameters... The basic transmitted waveform, the waveform's ambiguity function is obtained through... Rotation yields a new waveform. The characteristics of the resulting waveform are:

[0263]

[0264]

[0265] J k+1 and R k+1 These are the Fisher information matrix and the (measurement noise) covariance matrix obtained after rotation, respectively. It is a rotation matrix that satisfies Since the fuzzy function is independent of the angle dimension, there is no rotation transformation in the angle dimension. Orthogonality is achieved through rotation transformation. Therefore, this is called the error ellipse orthogonalization method. As shown in Figure 2, Figure 2(a) shows the case where the prediction error ellipse and the measurement error are not orthogonal; while Figure 2(b) shows the case where the two error ellipses are orthogonal. When the two error ellipses are orthogonal, their overlap area is minimized, resulting in the minimum tracking error. (Rotation angle) Using formula (44), we can obtain the state error covariance matrix at time k as P. k+1|k The fractional factors are:

[0266]

[0267] Where v P (i) and v R (i) are matrices P and R respectively. k The largest eigenvalue corresponds to the i-th element in the eigenvector. Where:

[0268]

[0269]

[0270] Will Substituting into formula (45), we get R. k+1 Then, the posterior state error covariance matrix P is obtained. k+1|k+1 Iteration is used for the next waveform selection.

[0271] Example 3

[0272] This invention presents a method for tracking maneuvering targets in clutter based on waveform selection. It establishes an MPDAF (Multi-Purpose Diffusing Array) model, laying the foundation for the next step of building an MPDA-SCKF (Multi-Purpose Diffusing Array) model. The modified Ricardian equation is used as the state error covariance matrix for the next step. In the PDAF (Dynamic Persistent Waveform Analysis), the modified Ricardian equation serves as an approximate solution under conditions of large-area correlation. In contrast, the MPDAF's modified Ricardian equation provides an exact solution and does not require large-area correlation. Therefore, the MPDAF is used as the update tracker. An error ellipse orthogonalization method is employed; when two error ellipses are orthogonal, their overlap is minimized, resulting in the minimum tracking error. This method offers low computational cost and intuitive physical meaning.

[0273] From the figure

[0274] As can be seen from Figures 5(a) to (c), the proposed algorithm exhibits excellent performance under both strong and weak maneuvers. In particular, under the condition of sudden state transition, the proposed algorithm has the lowest root mean square error and the fastest convergence speed.

[0275] like Figure 6 The diagram illustrates the rotation angle of the waveform Fourier transform in the proposed algorithm, where the optimal waveform is selected dynamically. This achieves minimal tracking accuracy and effectively improves tracking performance. In contrast, the two filters with a fixed waveform cannot effectively utilize waveform scheduling, resulting in lower tracking performance compared to the proposed algorithm.

[0276] Table 1 shows the average error, tracking loss rate, and algorithm runtime.

[0277] Table 1. Average Error, Tracking Loss Rate, Running Time

[0278]

[0279] This invention achieves the minimum average error and tracking loss rate. Furthermore, the tracking loss rates of IMM-PDA-PF, MCS-PDA-SCKF-WWS, and the proposed algorithm are very close. In comparison, the tracking loss rate of the two filters with fixed waveforms is twice that of the proposed algorithm. Therefore, waveform scheduling can indeed improve tracking performance in cluttered environments.

[0280] Simulations were conducted in a high signal-to-noise ratio environment, and all algorithms exhibited relatively low tracking loss rates. Furthermore, it can be observed that the IMM-PDA-PF algorithm not only has a larger average error but also the longest running time, three times that of the proposed algorithm. In comparison, the proposed algorithm maintains effectiveness while achieving the lowest average error. The slightly longer running time of the proposed algorithm, based on two filters with a fixed waveform, is due to waveform scheduling. However, the tracking performance is effectively improved.

[0281] In particular, compared with fixed waveforms Compared to MCS-MPDA-SCKF, the proposed algorithm improves position estimation accuracy by 32.46%, velocity estimation accuracy by 25.62%, and acceleration estimation accuracy by 10.37%. The runtime increases by 13.23%. Compared to a fixed waveform... Compared to MCS-MPDA-SCKF, the proposed algorithm improves position estimation accuracy by 34.83%, velocity estimation accuracy by 35.73%, and acceleration estimation accuracy by 24.17%. However, the runtime increases by 12.98%. Simulation results show that, compared to the fixed waveform method, this invention has a lower trajectory tracking loss ratio and higher tracking accuracy. Furthermore, compared to the two existing methods, this invention has the advantages of simple structure and high accuracy.

[0282] This invention addresses the problems of nonlinear measurement and target tracking in cluttered environments. Based on the MCS motion model, it combines MPDAF and SCKF into a new filter, enabling dynamic waveform selection through effective waveform configuration. Compared to filters with fixed waveforms, this invention achieves low RMSE, low ME, and TLP while maintaining a reasonable increase in computational complexity, and features a simpler structure and higher estimation accuracy.

Claims

1. A method for tracking maneuvering targets in clutter based on waveform selection, characterized in that, The specific steps are as follows: Step 1: Establish a waveform model; Step 2: Establish the state and measurement model; Step 3: Establish a clutter model; Step 4: Establish the filtering model; Step 5: The waveform scheduling algorithm based on fractional Fourier transform minimizes the posterior estimation error through dynamic waveform selection; It also includes: Step 2.1, establishing a state model to establish an improved current statistical MCS model; Step 2.2, establishing a measurement model as a nonlinear measurement model; based on the improved current statistical MCS model, using an improved probabilistic data interconnection filter (MPDA) and a square root capacitive Kalman filter to handle clutter and nonlinear transformations.

2. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 1, characterized in that, Step 1 is implemented in the following steps: Step 1.1: Establish a transmission waveform model in a narrowband environment; （1） in, It is the energy of the signal waveform. It is the carrier frequency. It is the complex envelope of the transmitted pulse; Step 1.2: Establish the received waveform model; （2） in, It is the energy received from the signal; It is added white noise; Represents latency; It is the distance between the target and the radar; Represents the speed of the target's movement, and , Represents the speed of light; When the waveform time-bandwidth product satisfies the narrowband condition View as: （3）。 3. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 2, characterized in that, Step 2 is implemented in the following steps: Step 2.1: Establish the state model to build the MCS model; （4） （5） in, Let be the target state vector. , and They represent x Position, velocity, and acceleration in the y-direction; It is the average value of the first-order acceleration; process noise It is zero-mean Gaussian white noise with variance of ; and The specific form is as follows: （8） （9） in, Sampling interval; Indicates at time The selected waveform, and Received The effect of a specific waveform Represented as ; Step 2.2: Establish the measurement model as a nonlinear measurement model; Assuming the target moves in a two-dimensional plane, and distance, velocity, and direction are measured simultaneously, the nonlinear measurement model is as follows: （6） （7） in, It is a nonlinear transformation function; For the target measurement matrix, measurement noise It is zero-mean Gaussian white noise with variance of ; These represent the distance between the target and the radar, the radial velocity of the target, and the azimuth angle of the target, respectively.

4. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 3, characterized in that, Step 3 is implemented in the following steps: clutter conditions The measured value is expressed as: （10） in, Is the radar constantly The total number of measurement targets; Step 3.1 This includes distance, speed, and angle information, assuming the number of false alarms follows an expected value. The Poisson distribution has a false alarm probability of: （11） in, For the density of the erroneous measurement, and To verify the gate volume; Step 3.2: Assuming that the clutter is uniformly distributed within the verification gate, and under the assumption that only noise and the target exist, the test statistic follows an exponential distribution. The peak value of the fuzzy function is used to estimate the time delay and Doppler frequency shift at time t. The detection probability is: （12） in This represents the expected probability of a false alarm. Representative moment The signal-to-noise ratio.

5. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 4, characterized in that, Step 4 is implemented in the following steps: Step 4.1: Establish the MPDA model; Assumption It is a moment Measured values The probability of association, If the probability of no association from the target measurement is given, then the posterior error covariance matrix in MPDAF is: （13） in It is the Kalman filter gain; It is the new information covariance matrix; It predicts the covariance matrix; It is an influence factor of the filter error covariance matrix: （14） in It is the detection probability, and It is the threshold probability; It is an influence factor, representing the correlation gate pair information covariance matrix. The impact; When the measurement dimension is three-dimensional: （15） in For the correlation threshold, and , For related regions, The defined error function; Corresponding to a specific waveform Or waveform parameters, also with Correspondingly, it is defined as ; When the modified Richter equation is used to estimate : （16） in It is a scalar between 0 and 1, depending on the clutter density λ and the associated threshold. and Detection probability at time ; The approximate fit is: （17） in and ; Step 4.2: Establish the MPDA-SCKF model; Before performing data processing, define It is a matrix The upper triangular matrix obtained by QR decomposition, matrix The QR decomposition is represented as , where the matrix It is a lower triangular matrix. The processing flow of MPDAF-SCKF is as follows: Step 4.2.1, Time Update; Step 4.2.2, Measurement update.

6. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 5, characterized in that, The specific steps for time updating in step 4.2.1 are as follows: Step 1.1: Factorize; （18） Step 1.2: Calculate the volume point and the propagation volume point; （19） （20） in and These are the volume point and the propagation volume point, respectively. For the total number of volume points, It is a state vector The dimension of, and satisfying conditions; , Yes The set of points obtained by permuting or inverting unit vectors in a 3D space; （21） Step 1.3: Calculate the mean of the state prediction and the square root of the covariance of the state prediction error; （22） （23） The weighted centrality matrix is: （24）。 7. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 6, characterized in that, The specific steps for measurement update in step 4.2.1 are as follows: Step 2.1: Calculate the volume point and perform a nonlinear transformation: （25） （26） Step 2.2: Calculate the predicted mean of the measurements; （27） Step 2.3: Calculate the square root coefficients of the residual covariance matrix: （28） The weighted center matrix is: （29） Step 2.4: Calculate the residual covariance matrix: （30） Step 2.5: Calculate the cross-covariance matrix between the state and the measurement. （31） The weighted centrality matrix is: （32） Step 2.6: Calculate the filter gain: （33） Step 2.7: Calculate the square root coefficients of the corresponding error covariance matrix and the posterior estimated error covariance matrix: （34） （35） Step 2.8: Update the corresponding state matrix and the corresponding error covariance matrix: If no measurement result is accurate, the following formula is used for updating: （36） （37） Otherwise, use the following formula: （38） （39） Assuming in The state error covariance matrix at time step is .

8. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 7, characterized in that, The specific steps of step 5 are as follows: Step 5.1: Calculate the lower bound of Clamelloe. Step 5.2: Optimized waveform selection based on fractional Fourier transform.

9. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 8, characterized in that, The specific steps of step 5.1 are as follows: Measurement noise covariance matrix and Transmit waveform parameters at time Relevance, that is, at time The selected waveform, i.e. It is the transmitted waveform. The fuzzy function, namely: （40） Regarding latency and Doppler shift The Fisher information matrix is ​​as follows: （41） in It's the signal-to-noise ratio; and The relationship between them is: （42） in Delay and Doppler shift Fisher's information matrix regarding distance and speed; It is the Clamer-Rao lower bound for a selected waveform under unbiased estimation; Formula (40) shows that at time 10:00 The selected waveform determines By minimizing the time The optimal waveform is selected based on the estimation error. （43） time The choice of waveform affects the measurement error covariance matrix, and also affects the measurement at time t. State estimation error; by selecting the optimal waveform through waveform scheduling, the performance of target tracking under clutter conditions is greatly improved.

10. The method for tracking maneuvering targets in clutter based on waveform selection according to claim 9, characterized in that, The specific steps of step 5.2 are as follows: Assuming the basic transmitted waveform is The fuzzy function is The Fisher information matrix is The corresponding measurement noise covariance matrix is Fractional factors in fractional Fourier transform It is applied to the basic transmission waveform to achieve orthogonality between the measurement error ellipse and the state error ellipse; The fractional Fourier transform is viewed as a rotation operation of a coordinate system. When the fractional Fourier transform is used with parameters... The basic transmitted waveform, the waveform's ambiguity function is obtained through... Rotation yields a new waveform, characterized by the following features: （44） （45） in and These are the Fisher information matrix and covariance matrix obtained after rotation, respectively; It is a rotation matrix that satisfies Since the fuzzy function is independent of the angle dimension, there is no rotation transformation in the angle dimension. Orthogonalization is achieved through rotation transformation. Assuming time The state error covariance matrix at time is The fractional factors are: （46） in and They are matrices sum matrix The largest eigenvalue corresponds to the eigenvector of the eigenvector. There are n elements, of which: （47） （48） Will Substituting into formula (45), we get Then, the posterior state error covariance matrix is ​​obtained. The iteration is used for the selection of the next waveform.

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