Radar signal PRI sequence reconstruction method based on trend filtering

Abstract

The invention discloses a radar signal PRI sequence reconstruction method based on trend filtering, and the method comprises the steps: firstly constructing a PRI sequence filtering model, proposing a maximum posterior probability estimation problem, then updating model parameters through employing an E-M algorithm, solving the maximum posterior probability estimation problem, obtaining PRI sequence filtering results, fusing the obtained PRI sequence filtering results, and carrying out the reconstruction of the PRI sequence. And finally, performing interpolation correction on the PRI sequence, and outputting a final reconstruction result of the PRI sequence. The PRI sequence probability model provided by the method has better capability of expressing radar signal PRI features, can effectively reconstruct radar signal PRI sequence patterns containing abnormal values, remarkably improves PRI sequence reconstruction performance, has important significance in the field of radar signal receiving and analysis, and has good application prospects. The method is suitable for PRI parameter reconstruction of a non-ideal detection signal sequence under false pulse and leakage pulse conditions.

Description

Technical Field

[0001] The invention belongs to the technical field of radar signal reception and analysis, and in particular relates to a radar signal PRI sequence reconstruction method based on trend filtering. Background Art

[0002] Multifunction radar (MFR) mainly refers to a radar system that can perform different tasks and support multiple functions. It has a flexible and variable working mode, so it has been widely used. The variability of multifunction radar behavior also brings huge challenges to third-party signal reception and analysis. Compared with the constant and group-variable PRI signals used by traditional radars, the signal parameters emitted by multifunction radars are agile and show more complex time structures and change patterns. The accurate extraction of complex signal parameters of multifunction radars helps the receiver to infer the behavioral intentions of the transmitter's radar, thereby guiding the adversarial equipment to decide on the optimal adversarial strategy and gain adversarial advantages. Therefore, the relevant technology has important research significance and practical value.

[0003] Due to the influence of electromagnetic environment and thermal noise inside the receiving device, the radar pulse sequence intercepted by the receiver contains a large number of missed pulses and false pulses. At present, the reconstruction of radar signal PRI sequence mainly starts from the perspective of formal language, mining frequent items appearing in PRI sequence. The document "Sequential extraction and recognition of pulsegroup structure for multi-function radar, IET Radar, Sonar&Navigation, 2022, 16(4): 678-691" proposes a multi-function radar regular grammar and establishes a corresponding hierarchical automaton to extract and identify the pulse group PRI structure. The document "A sequential estimation method for repetition frequency parameters of pulse Doppler radar, Modern Radar, 2024, 46(02): 103-111" realizes the sequential estimation of group-variable PRI sequence through trial integer division and other techniques.

[0004] The above method mainly focuses on the extraction of the PRI variation law of the pulse group in the radar signal sequence, and only reconstructs the group-varied PRI at the complete sequence level, and fails to involve the reconstruction of the complete PRI sequence under more modulation modes. Therefore, the radar signal PRI sequence reconstruction technology still needs to be further improved. Summary of the invention

[0005] In order to solve the above technical problems, the present invention provides a radar signal PRI sequence reconstruction method based on trend filtering, which can effectively reconstruct the radar signal PRI sequence pattern containing abnormal values.

[0006] The technical solution adopted by the present invention is: a radar signal PRI sequence reconstruction method based on trend filtering, the specific steps are as follows:

[0007] Step 1: Construct a PRI sequence filtering model and propose the maximum a posteriori probability estimation problem;

[0008] In the confrontation scenario, a PRI model for receiving radar signals is constructed, and the expression is as follows:

[0009] y=x+ε

[0010] in, Represents the true value of the radar signal PRI sequence, Represents independent errors, including: noise and outliers caused by false pulses and missed pulses, Indicates the measurement result of PRI sequence; [·] T represents the transpose of a matrix, represents the real number domain, and N represents the sequence length.

[0011] According to the PRI model of the radar signal received by the receiver, the radar signal PRI sequence filtering model is constructed, and the maximum a posteriori probability estimation problem is proposed. The expression is as follows:

[0012]

[0013] in, represents the radar signal PRI sequence obtained by maximum a posteriori probability estimation, P(x|y) represents the posterior probability of the PRI sequence, P(y|x) represents the likelihood function, and P(x) represents the prior probability.

[0014] Then the PRI sequence error probability distribution P(ε) is modeled as a mixed Gaussian model, expressed as follows:

[0015]

[0016] Among them, M represents the number of Gaussian distributions in the Gaussian mixture model, Indicates that the mean is 0 and the variance is η m A single Gaussian distribution, π m represents the weight of the mth Gaussian distribution in the Gaussian mixture distribution, satisfying

[0017] According to the probability distribution P(ε) of the PRI sequence error, the likelihood function P(y|x) of the PRI sequence is expressed as follows:

[0018]

[0019] Among them, x represents the true value of the radar signal PRI sequence, Π represents the set of weights of each component in the Gaussian mixture distribution, Σ represents the set of variances of each component in the Gaussian mixture distribution, Indicates x n is the mean, η m is the Gaussian probability density function of the variance.

[0020] Then the prior probability P(x) of the PRI sequence is modeled as a Gibbs distribution, expressed as follows:

[0021]

[0022] Where Z represents the normalization factor, φ λ (·) represents the sparsity inducing function, θ represents a small positive value, D represents a matrix, [Dx] i Represents the i-th element of the vector Dx. For different PRI modulation types, the prior function P(x) is constructed respectively, and different matrices D are used.

[0023] Based on the above PRI sequence likelihood function P(y|x) and prior function P(x), the optimization objective function expression of the maximum a posteriori probability estimation problem is as follows:

[0024]

[0025] in, represents the objective function to be optimized, Θ={x,Π,Σ} represents the set of unknown parameters to be estimated, Π represents the set of Gaussian distribution weights π, and Σ represents the set of Gaussian distribution variances η.

[0026] Step 2: Based on step 1, use the EM algorithm to update the model parameters, solve the maximum a posteriori probability estimation problem, and obtain the PRI sequence filtering result;

[0027] Based on the PRI sequence filtering model constructed in step one, the PRI sequence measurement result y containing various errors is input, and the EM algorithm is used to solve the estimation problem described in step one.

[0028] First, introduce the hidden variable z into the likelihood function P(y|Θ) described in step 1 to construct the likelihood function containing the hidden variable. The expression is as follows:

[0029]

[0030] Then, based on the likelihood function of the introduced latent variable z, according to Bayes' theorem, z is solved in the E step of the EM algorithm. mn The conditional expectation of β nm The calculation process of latent variable conditional expectation is as follows:

[0031]

[0032] Then, based on the hidden variable z obtained in the E step of the EM algorithm mn The conditional expectation of β nm , solve the M steps of the EM algorithm and obtain the estimated value of the parameter set Θ in the tth iteration {x (t+1) ,Π (t+1) ,Σ (t+1)}. The update process expression of the parameter collection is as follows:

[0033]

[0034] x (t+1) =((Λ (t) ) 2 +D T (W (t) ) 2 D) -1 (Λ (t) ) 2 y

[0035] in, x (t+1) They represent the Gaussian mixture model weight, Gaussian mixture model variance and the estimated results of the radar signal PRI sequence in t rounds of iterations. (t) , W (t) They are all diagonal matrices, and their diagonal element expressions in the tth iteration are as follows:

[0036]

[0037] Among them, φ λ ′(·) represents φ λ The derivative of (·).

[0038] The E step and the M step of the EM algorithm are iterated repeatedly until the algorithm reaches a specified number of iterations or the model converges.

[0039] Wherein, in the step 1, a plurality of different matrices D are used in the PRI sequence prior probability to construct a plurality of different filtering models, and then a plurality of PRI sequence filtering results will be obtained.

[0040] Step 3: Based on step 2, the obtained PRI sequence filtering results are integrated;

[0041] Based on the multiple PRI sequence filtering results obtained in step 2, multiple filtering results generated by different matrices D are fused. It is assumed that C different filtering results are generated, and the fusion process is as follows:

[0042] First, based on the latent variable conditional expectation β obtained in step 2nm The variance η of each Gaussian distribution in the Gaussian mixture model m , calculate the variance of the PRI sequence measurement results, the expression is as follows:

[0043]

[0044] Where M represents the number of Gaussian distributions in the Gaussian mixture model, and the subscript n represents the nth element of the PRI sequence.

[0045] Then, define the output of the cth filter model as x (c) , where the nth element is recorded as The corresponding variance is When fusing the nth elements of each filter model, The corresponding weight The expression is as follows:

[0046]

[0047] Finally, the fusion result expression is as follows:

[0048]

[0049] in, It represents the nth element of the fusion result of the PRI sequence filtering model, and C represents the number of filtering results.

[0050] Step 4: Based on the fusion result obtained in step 3, the PRI sequence is interpolated and corrected, and the final reconstruction result of the PRI sequence is output;

[0051] Based on step three, the obtained PRI sequence filter model fusion result is interpolated and corrected.

[0052] First, according to the PRI sequence measurement result y and the PRI sequence filtering fusion result Building a “pseudo index” I p , the pulse indexes in the reconstructed PRI sequence are moved from the integer domain to the actual positions in the real domain. The “pseudo-index” iterative construction expression is as follows:

[0053]

[0054] Among them, I p (i) represents the i-th element of the pseudo index, x i ′ represents the i-th element of the PRI sequence measurement result, Represents the i-th element of the PRI sequence filter model fusion result.

[0055] Then, a "reconstructed index" is constructed based on the "pseudo index" R, which is the position where interpolation is performed on the integer domain, and its length is the result of rounding down the maximum value in the "pseudo-index" The expression is as follows:

[0056]

[0057] Finally, the PRI sequence filtering fusion results obtained from steps 1 to 3 are And the pulse position I in the fusion result p , interpolate at the position of "reconstruction index" to obtain the PRI sequence interpolation correction result That is, the final reconstruction result of the radar signal PRI sequence, expressed as follows:

[0058]

[0059] in, It represents the interpolation correction result of PRI sequence, and Interp represents the interpolation function.

[0060] Beneficial effects of the present invention: The method of the present invention first constructs a PRI sequence filtering model, proposes a maximum a posteriori probability estimation problem, then uses the EM algorithm to update the model parameters, solves the maximum a posteriori probability estimation problem, obtains the PRI sequence filtering result, fuses the obtained PRI sequence filtering results, and finally interpolates and corrects the PRI sequence to output the final reconstruction result of the PRI sequence. The PRI sequence probability model proposed by the method of the present invention has a better ability to express the PRI characteristics of radar signals, can effectively reconstruct the PRI sequence pattern of radar signals containing abnormal values, and significantly improves the PRI sequence reconstruction performance. It is of great significance in the field of radar signal reception and analysis, and is suitable for PRI parameter reconstruction of non-ideal detection signal sequences under false pulse and leakage pulse conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 This is a flow chart of a radar signal PRI sequence reconstruction method based on trend filtering.

[0062] Figure 2 Schematic diagram of the fusion of radar signal PRI sequence filtering results in an embodiment of the present invention.

[0063] Figure 3 Schematic diagram of reconstruction error of group-varied PRI sequence under different error levels (different leakage pulse rates) in an embodiment of the present invention.

[0064] Figure 4 Schematic diagram of reconstruction error of sinusoidal modulation PRI sequence under different error levels (different leakage pulse rates) in an embodiment of the present invention. DETAILED DESCRIPTION

[0065] The method of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0066] like Figure 1 As shown, a flow chart of a radar signal PRI sequence reconstruction method based on trend filtering of the present invention, the specific steps are as follows:

[0067] Step 1: Construct a PRI sequence filtering model and propose the maximum a posteriori probability estimation problem;

[0068] In the confrontation (passive sensing) scenario, the PRI model of the receiver receiving the radar signal is constructed, and the expression is as follows:

[0069] y=x+ε

[0070] in, Represents the true value of the radar signal PRI sequence, Represents independent errors, including: noise and outliers caused by false pulses and missed pulses, Indicates the measurement result of PRI sequence; [·] T represents the transpose of a matrix, represents the real number domain, and N represents the sequence length.

[0071] According to the PRI model of the radar signal received by the receiver, the radar signal PRI sequence filtering model is constructed, and the maximum a posteriori probability (MAP) estimation problem is proposed. The expression is as follows:

[0072]

[0073] in, represents the radar signal PRI sequence obtained by maximum a posteriori probability estimation, P(x|y) represents the posterior probability of the PRI sequence, P(y|x) represents the likelihood function, which can be derived from the probability distribution P(ε) of the PRI sequence error vector, and P(x) represents the prior probability.

[0074] Then the PRI sequence error probability distribution P(ε) is modeled as a mixed Gaussian model, expressed as follows:

[0075]

[0076] Among them, M represents the number of Gaussian distributions in the Gaussian mixture model, Indicates that the mean is 0 and the variance is η m A single Gaussian distribution, π m represents the weight of the mth Gaussian distribution in the Gaussian mixture distribution, satisfying

[0077] According to the probability distribution P(ε) of the PRI sequence error, the likelihood function P(y|x) of the PRI sequence is expressed as follows:

[0078]

[0079] Among them, x represents the true value of the radar signal PRI sequence, Π represents the set of weights of each component in the Gaussian mixture distribution, Σ represents the set of variances of each component in the Gaussian mixture distribution, Indicates x n is the mean, η m is the Gaussian probability density function of the variance.

[0080] Then the prior probability P(x) of the PRI sequence is modeled as a Gibbs distribution, expressed as follows:

[0081]

[0082] Where Z represents the normalization factor, φ λ (·) represents the sparsity inducing function, θ represents a small positive value, D represents a matrix, [Dx] i Represents the i-th element of the vector Dx. For different PRI modulation types, the prior function P(x) is constructed respectively, and different matrices D are used.

[0083] Sparseness inducing function φ λ (·) The expression is defined as follows:

[0084]

[0085] Among them, λ represents the weight coefficient.

[0086] Based on the above PRI sequence likelihood function P(y|x) and prior function P(x), the optimization objective function expression of the maximum a posteriori probability estimation problem is as follows:

[0087]

[0088] in, represents the objective function to be optimized, Θ={x,Π,Σ} represents the set of unknown parameters to be estimated, Π represents the set of Gaussian distribution weights π, and Σ represents the set of Gaussian distribution variances η.

[0089] Step 2: Based on step 1, use the EM algorithm to update the model parameters, solve the maximum a posteriori probability estimation problem, and obtain the PRI sequence filtering result;

[0090] Based on the PRI sequence filtering model constructed in step one, the PRI sequence measurement result y containing various errors is input, and the EM algorithm is used to solve the estimation problem described in step one.

[0091] First, introduce the hidden variable z into the likelihood function P(y|Θ) described in step 1 to construct the likelihood function containing the hidden variable. The expression is as follows:

[0092]

[0093] Then, based on the introduction of hidden variables z The likelihood function of z is solved in the E step of the EM algorithm according to Bayes' theorem. mn The conditional expectation of β nm The calculation process of latent variable conditional expectation is as follows:

[0094]

[0095] Then, based on the hidden variable z obtained in the E step of the EM algorithm mn The conditional expectation of β nm , solve the M steps of the EM algorithm and obtain the estimated value of the parameter set Θ in the tth iteration {x (t+1) ,Π (t+1) ,Σ (t+1)}. The update process expression of the parameter collection is as follows:

[0096]

[0097] in, x (t+1) They represent the Gaussian mixture model weight, Gaussian mixture model variance and the estimated results of the radar signal PRI sequence in t rounds of iterations. (t) , W (t) They are all diagonal matrices, and their diagonal element expressions in the tth iteration are as follows:

[0098]

[0099]

[0100] Among them, φ λ ′(·) represents φ λ The derivative of (·).

[0101] The E step and the M step of the EM algorithm are iterated repeatedly until the algorithm reaches a specified number of iterations or the model converges.

[0102] Wherein, in the step 1, a plurality of different matrices D are used in the PRI sequence prior probability to construct a plurality of different filtering models, and then a plurality of PRI sequence filtering results will be obtained.

[0103] Step 3: Based on step 2, the obtained PRI sequence filtering results are integrated;

[0104] In this embodiment, the radar signal PRI sequence filtering results are fused as follows: Figure 2 As shown. Based on the multiple PRI sequence filtering results obtained in step 2, the multiple filtering results generated by different matrices D are fused. It is assumed that C different filtering results are generated, and the fusion process is as follows:

[0105] First, based on the latent variable conditional expectation β obtained in step 2 nm The variance η of each Gaussian distribution in the Gaussian mixture model m , calculate the variance of the PRI sequence measurement results, the expression is as follows:

[0106]

[0107] Where M represents the number of Gaussian distributions in the Gaussian mixture model, and the subscript n represents the nth element of the PRI sequence.

[0108] Then, define the output of the cth filter model as x (c ), where the nth element is recorded as The corresponding variance is When fusing the nth elements of each filter model, The corresponding weight The expression is as follows:

[0109]

[0110] Finally, the fusion result expression is as follows:

[0111]

[0112] in, It represents the nth element of the fusion result of the PRI sequence filtering model, and C represents the number of filtering results.

[0113] Step 4: Based on the fusion result obtained in step 3, the PRI sequence is interpolated and corrected, and the final reconstruction result of the PRI sequence is output;

[0114] Based on step three, the obtained PRI sequence filter model fusion result is interpolated and corrected.

[0115] First, according to the PRI sequence measurement result y and the PRI sequence filtering fusion result Building a “pseudo index” I p , the pulse indexes in the reconstructed PRI sequence are moved from the integer domain to the actual positions in the real domain. The “pseudo-index” iterative construction expression is as follows:

[0116]

[0117] Among them, Ip (i) represents the i-th element of the pseudo index, x i ′ represents the i-th element of the PRI sequence measurement result, Represents the i-th element of the PRI sequence filter model fusion result.

[0118] Then, a "reconstructed index" is constructed based on the "pseudo index" R , which is the position where interpolation is performed on the integer domain, and its length is the result of rounding down the maximum value in the "pseudo-index" The expression is as follows:

[0119]

[0120] Finally, the PRI sequence filtering fusion results obtained from steps 1 to 3 are And the pulse position I in the fusion result p , interpolate at the position of "reconstruction index" to obtain the PRI sequence interpolation correction result That is, the final reconstruction result of the radar signal PRI sequence, expressed as follows:

[0121]

[0122] in, represents the PRI sequence interpolation correction result, and Interp represents the interpolation function. In this embodiment, linear interpolation is used as the PRI sequence interpolation algorithm.

[0123] This embodiment also carried out simulation experiments to verify the method of the present invention. Figure 3 , 4 shown.

[0124] The radar signal group-variable PRI sequence data set is established. For constant and group-variable PRI, the first-order difference matrix is ​​used as the matrix D, and its definition expression is as follows:

[0125]

[0126] The PRI measurement error is 1%, the range of the missed pulse rate is 0 to 0.3, and the step is 0.03. The generated PRI sequence data set is input into the method of the present invention and the reconstruction method based on sequential estimation respectively, and the PRI sequence reconstruction mean absolute percentage error (MAPE) is obtained as follows: Figure 3 shown.

[0127] from Figure 3 It can be seen from the figure that the method of the present invention is significantly better than the existing method. When the missing pulse rate does not exceed 0.21, the MAPE error of the PRI sequence reconstruction is always less than 2%.

[0128] A radar signal sinusoidal modulation PRI sequence data set is established. For sinusoidal modulation PRI, the Fourier matrix is ​​used as the matrix D, and its definition expression is as follows:

[0129]

[0130] in, N represents the length of the input PRI sequence x, and K represents the number of FFT points. The PRI measurement error is 1%, and the range of the missed pulse rate is 0 to 0.15, with a step of 0.015. The generated PRI sequence data set is input into the method of the present invention and the HP filter model respectively, and the PRI sequence reconstruction MAPE error obtained is as follows: Figure 4 shown.

[0131] from Figure 4 It can be seen that when there is no leakage pulse in the signal sequence, the method of the present invention has similar performance to that of the HP filter; but when leakage pulse appears in the signal sequence, the PRI sequence reconstruction error of the HP filter increases rapidly, while the error of the method of the present invention increases slowly. When the leakage pulse rate is ≤0.09, the error does not exceed 4%. Therefore, the method of the present invention is significantly better than the existing method.

[0132] In summary, the above simulation experiment results show that the method of the present invention can significantly improve the accuracy of PRI sequence reconstruction compared with the existing methods, thereby verifying that the method of the present invention is practical and effective in reconstructing the PRI characteristics of radar signals. The PRI sequence probability model proposed by the method of the present invention has a better ability to express the PRI characteristics of radar signals, can effectively reconstruct the PRI sequence pattern of radar signals containing abnormal values, and significantly improves the PRI sequence reconstruction performance. It is of great significance in the field of radar signal reception and analysis, and is suitable for the PRI parameter reconstruction of non-ideal detection signal sequences under false pulse and leakage pulse conditions.

[0133] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, the present invention may have various changes and variations. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of the claims of the present invention.

Claims

1. A radar signal PRI sequence reconstruction method based on trend filtering, the specific steps are as follows: Step 1: Construct a PRI sequence filtering model and propose the maximum a posteriori probability estimation problem; In the confrontation scenario, a PRI model for receiving radar signals is constructed, and the expression is as follows: y=x+ε in, Represents the true value of the radar signal PRI sequence, Represents independent errors, including: noise and outliers caused by false pulses and missed pulses, Indicates the measurement result of PRI sequence; [·] T represents the transpose of a matrix, represents the real number domain, N represents the sequence length; According to the PRI model of the radar signal received by the receiver, the radar signal PRI sequence filtering model is constructed, and the maximum a posteriori probability estimation problem is proposed. The expression is as follows: in, represents the radar signal PRI sequence obtained by maximum a posteriori probability estimation, P(x|y) represents the posterior probability of the PRI sequence, P(y|x) represents the likelihood function, and P(x) represents the prior probability; Then the PRI sequence error probability distribution P(ε) is modeled as a mixed Gaussian model, expressed as follows: Among them, M represents the number of Gaussian distributions in the Gaussian mixture model, Indicates that the mean is 0 and the variance is η m A single Gaussian distribution, π m represents the weight of the mth Gaussian distribution in the Gaussian mixture distribution, satisfying According to the probability distribution P(ε) of the PRI sequence error, the likelihood function P(y|x) of the PRI sequence is expressed as follows: Among them, x represents the true value of the radar signal PRI sequence, Π represents the set of weights of each component in the Gaussian mixture distribution, Σ represents the set of variances of each component in the Gaussian mixture distribution, Indicates x n is the mean, η m is the Gaussian probability density function of the variance; Then the prior probability P(x) of the PRI sequence is modeled as a Gibbs distribution, expressed as follows: Where Z represents the normalization factor, φ λ (·) represents the sparsity inducing function, θ represents a small positive value, D represents a matrix, [Dx] i represents the i-th element of the vector Dx; for different PRI modulation types, the prior function P(x) is constructed respectively, and different matrices D are used; Based on the above PRI sequence likelihood function P(y|x) and prior function P(x), the optimization objective function expression of the maximum a posteriori probability estimation problem is as follows: in, represents the objective function to be optimized, Θ = {x, Π, Σ} represents the set of unknown parameters to be estimated, Π represents the set of Gaussian distribution weights π, and Σ represents the set of Gaussian distribution variances η; Step 2: Based on step 1, use the EM algorithm to update the model parameters, solve the maximum a posteriori probability estimation problem, and obtain the PRI sequence filtering result; Based on the PRI sequence filtering model constructed in step 1, the PRI sequence measurement result y containing various errors is input, and the EM algorithm is used to solve the estimation problem described in step 1; First, introduce the hidden variable z into the likelihood function P(y|Θ) described in step 1 to construct the likelihood function containing the hidden variable. The expression is as follows: Then, based on the likelihood function of the introduced latent variable z, according to Bayes' theorem, z is solved in the E step of the EM algorithm. mn The conditional expectation of β nm ; The calculation process expression of latent variable conditional expectation is as follows: Then, based on the hidden variable z obtained in the E step of the EM algorithm mn The conditional expectation of β nm , solve the M steps of the EM algorithm and obtain the estimated value of the parameter set Θ in the tth iteration {x (t+1) ,Π (t+1) ,Σ (t+1) }; The update process expression of the parameter collection is as follows: in, x (t+1) They represent the weight of Gaussian mixture model, the variance of Gaussian mixture model and the estimation result of radar signal PRI sequence in t rounds of iteration respectively; Λ (t) , W (t) They are all diagonal matrices, and their diagonal element expressions in the tth iteration are as follows: Among them, φ λ ′(·) represents φ λ The derivative of (·); Repeating the E step and the M step of the EM algorithm until the algorithm reaches a specified number of iterations or the model converges; Wherein, in the PRI sequence prior probability of step 1, multiple different matrices D are used to construct multiple different filtering models, and multiple PRI sequence filtering results will be obtained; Step 3: Based on step 2, the obtained PRI sequence filtering results are integrated; Based on the multiple PRI sequence filtering results obtained in step 2, multiple filtering results generated by different matrices D are fused; it is assumed that C different filtering results are generated, and the fusion process is as follows: First, based on the latent variable conditional expectation β obtained in step 2 nm The variance η of each Gaussian distribution in the Gaussian mixture model m , calculate the variance of the PRI sequence measurement results, the expression is as follows: Where M represents the number of Gaussian distributions in the Gaussian mixture model, and the subscript n represents the nth element of the PRI sequence; Then, define the output of the cth filter model as x( c ), where the nth element is recorded as The corresponding variance is When fusing the nth elements of each filter model, The corresponding weight The expression is as follows: Finally, the fusion result expression is as follows: in, represents the nth element of the fusion result of the PRI sequence filtering model, and C represents the number of filtering results; Step 4: Based on the fusion result obtained in step 3, the PRI sequence is interpolated and corrected, and the final reconstruction result of the PRI sequence is output; Based on step three, the obtained PRI sequence filter model fusion result is interpolated and corrected; First, according to the PRI sequence measurement result y and the PRI sequence filtering fusion result Building a "pseudo index" I p , the pulse index in the reconstructed PRI sequence is moved from the integer domain to the actual position in the real domain; the "pseudo-index" iterative construction expression is as follows: Among them, I p (i) represents the i-th element of the pseudo index, x i ′ represents the i-th element of the PRI sequence measurement result, Represents the i-th element of the fusion result of the PRI sequence filter model; Then, a "reconstructed index" is constructed based on the "pseudo index". R , which is the position where interpolation is performed on the integer domain, and its length is the result of rounding down the maximum value in the "pseudo-index" The expression is as follows: Finally, the PRI sequence filtering fusion results obtained from steps 1 to 3 are And the pulse position I in the fusion result p , interpolate at the position of "reconstruction index" to obtain the PRI sequence interpolation correction result That is, the final reconstruction result of the radar signal PRI sequence, expressed as follows: in, It represents the interpolation correction result of PRI sequence, and Interp represents the interpolation function.

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