FDA-MIMO Target Parameter Estimation Method Based on Reconstruction and Dimensionality Reduction for Root Finding

Abstract

The present invention discloses an FDA-MIMO target parameter estimation method based on reconstructed dimensionality reduction and root finding, which relates to the technical field of target positioning. By utilizing the DOA and distance coupling characteristics of the transmission direction vector and reconstructing the 2D-MUSIC spatial spectrum function of the noise subspace, and using dimensionality reduction transformation and polynomial root finding techniques to obtain the distance-related transmission direction vector and DOA estimation values, and finally performing least squares fitting on the distance-related transmission direction vector to obtain the distance estimation value. The present invention realizes target DOA estimation by aggregating the DOA information in the transmission direction vector and the reception direction vector, avoiding performance degradation caused by insufficient information utilization. At the same time, the distance estimation value is obtained by using the distance-related transmission direction vector, avoiding decoupling errors and achieving high-precision distance estimation. The present invention greatly reduces the computational complexity while ensuring the accuracy of DOA and distance estimation, is more suitable for real-time estimation of target parameters, and is conducive to engineering implementation.

Description

Technical Field

[0001] The present invention relates to the technical field of target positioning, and specifically relates to an FDA-MIMO target parameter estimation method based on reconstructed dimensionality reduction and root finding. Background Art

[0002] The FDA-MIMO radar combines the range-related characteristics of the FDA (Frequency Diverse Array) array and the spatial diversity of the MIMO (Multiple Input Multiple Output) radar, and has the potential to realize joint detection in the angle domain and range domain, interference suppression, target parameter estimation, and beamforming, and has broad application prospects in the fields of mobile communication, medical imaging, and radar signal processing.

[0003] When estimating the target parameters of the FDA-MIMO radar, in addition to the radar system architecture, the target parameter estimation algorithm is an important factor affecting the target estimation performance. The current target parameter estimation algorithms can be roughly divided into search algorithms, algorithms based on rotational invariance, and compressed sensing algorithms based on sparse characteristics. The most typical representative of the spectral peak search algorithm is the MUSIC (Multiple Signal Classification) algorithm. This algorithm uses the orthogonality between the direction vector and the signal subspace to achieve refined search for parameters. The search accuracy is proportional to the computational complexity. In multi-dimensional parameter estimation, the computational complexity increases significantly with the improvement of the search accuracy, and it cannot meet the requirements of real-time estimation of target parameters. The most typical representative of the algorithm based on rotational invariance is the ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) algorithm. This algorithm uses the equivalent transformation relationship between the array manifold and the signal subspace and the rotational invariance between arrays to estimate the target parameters. However, since only partial information is used for parameter estimation, the parameter estimation performance is poor and it cannot meet the requirements of accurate estimation of target parameters. The compressed sensing algorithms based on sparse characteristics use the sparse characteristics of the target in the spatial domain, and obtain the position of the target parameters in the sparse dictionary grid by constructing a sparse dictionary and a compressed sensing model, and then further obtain the target parameter estimation value. The grid accuracy of the sparse dictionary is proportional to the parameter estimation accuracy and the computational complexity, and this type of algorithm can generally only be used for one-dimensional parameter estimation and cannot achieve joint estimation of multiple parameters.

[0004] Therefore, it is necessary to propose a new method for estimating the target angle and range to solve the above problems. Summary of the Invention

[0005] Aiming at the above deficiencies in the prior art, a method for estimating target parameters of FDA-MIMO based on reconstruction and dimensionality reduction to find roots provided by the present invention solves the problems of high complexity in the prior art and inability to perform accurate multi-dimensional and multi-parameter joint estimation.

[0006] In order to achieve the above invention purpose, the technical solution adopted by the present invention is as follows:

[0007] A method for estimating target parameters of FDA-MIMO based on reconstruction and dimensionality reduction to find roots includes the following steps:

[0008] S1. Construct the transmitting end and the receiving end of the FDA-MIMO radar as uniform linear arrays, and transmit electromagnetic waves that satisfy uniform frequency offsets through the transmitting end;

[0009] S2. Obtain the electromagnetic waves reflected by each target through the receiving end, perform matched filtering on them, and obtain the output signal;

[0010] S3. Process the eigenvalues of the output signal covariance matrix by the step ratio method to obtain the number of targets, and divide the eigenvectors of the output signal covariance matrix according to the number of targets to obtain the noise subspace;

[0011] S4. Utilize the coupling characteristics of the transmitting direction vectors of the FDA-MIMO radar and the noise subspace to reconstruct the 2D-MUSIC spatial spectrum function, so that the DOA and distance information are separated, and the 2D-MUSIC is a two-dimensional MUSIC (Multiple Signal Classification) algorithm;

[0012] S5. Perform dimensionality reduction transformation and polynomial root finding on the reconstructed 2D-MUSIC spatial spectrum function to obtain the distance-related transmitting direction vector and DOA estimation values;

[0013] S6. Perform least squares fitting on the distance-related transmitting direction vector to obtain the distance estimation value.

[0014] Further, in the step S1, the transmitting end includes M array elements, the receiving end includes N array elements, and the array element spacing is d;

[0015] The frequencies of the electromagnetic waves transmitted by each array element at the transmitting end are:

[0016] f m = f0 + (m - 1)Δf

[0017] where f m is the frequency of the electromagnetic wave transmitted by the m-th array element at the transmitting end, f0 is the reference frequency, Δf is the basic frequency offset, and Δf << f0.

[0018] Further, the step S3 includes the following sub-steps:

[0019] S31. Calculate the covariance matrix of the output signal;

[0020] S32. Perform eigenvalue decomposition on the covariance matrix of the output signal to obtain its respective eigenvalues and corresponding eigenvectors, and sort the eigenvalues in descending order;

[0021] S33. Use the step ratio method to obtain the target number based on the respective eigenvalues of the covariance matrix of the output signal;

[0022] S34. Select the eigenvectors corresponding to the eigenvalues of the covariance matrix of the output signal whose sequence numbers are greater than the target number to form the noise subspace.

[0023] Furthermore, in the step S31, the covariance matrix of the output signal is calculated through the following formula:

[0024]

[0025] where R is the covariance matrix of the output signal, L is the number of snapshots, x(l) is the l-th snapshot of the output signal, and x H (l) is the conjugate transpose of the l-th snapshot of the output signal.

[0026] Furthermore, in the step S32, the eigenvalues of the covariance matrix of the output signal obtained by eigenvalue decomposition are successively λ1 to λ N×M , with a total of N×M eigenvalues; and λ1≥λ2≥…≥λ N×M-1 ≥λ N×M .

[0027] Furthermore, in the step S33, the target number is obtained by the following formula using the step ratio method:

[0028]

[0029]

[0030] where K is the target number, μ i is the i-th step ratio, λ i is the i-th eigenvalue of the covariance matrix of the output signal, λ i+1 is the (i + 1)-th eigenvalue of the covariance matrix of the output signal, i ∈ [1, N×M - 1], is the function for obtaining the numerical value of the sequence number i when μ i is maximized.

[0031] Furthermore, in the step S4, the 2D-MUSIC spatial spectrum function constructed using the noise subspace is:

[0032]

[0033] Among them, f 2D-MUSIC (θ, r) is the 2D-MUSIC spatial spectrum function, a r (θ) is the receiving direction vector related to DOA, a t (θ, r) is the transmitting direction vector related to DOA-distance, θ is DOA, r is the distance, E n is the noise subspace, is the conjugate transpose of E n , [·] H is the conjugate transpose operation, is the Kronecher product operation;

[0034] The 2D-MUSIC spatial spectrum function reconstructed by using the coupling characteristic of the transmitting direction vector is:

[0035]

[0036] Among them, a t (r) is the transmitting direction vector related to the distance, is the conjugate transpose of a t (r), a t (θ) is the transmitting direction vector related to DOA, diag(·) is the diagonal matrix composed of the elements of the vector in the brackets.

[0037] Furthermore, the step S5 includes the following sub-steps:

[0038] S51. Perform a dimensionality reduction transformation on the reconstructed 2D-MUSIC spatial spectrum function to obtain an expression of the transmitting direction vector related to the distance:

[0039]

[0040] Among them, e1 is the unit vector, is the conjugate transpose of e1, Q -1 (θ) is the inverse matrix of the parameter matrix Q(θ),

[0041] S52. According to the transmitting direction vector related to the distance, reconstruct the objective function of 2D-MUSIC as:

[0042]

[0043] Among them, is the θ value corresponding to the minimum function value, det(·) is the operation of calculating the determinant of the matrix, compan(·) 1，1 is the operation of calculating the element in the first row and first column of the adjoint matrix of the matrix;

[0044] S53. Construct the parameter matrix Q(θ) as the symbolic variable parameter matrix Q(z) through the symbolic variable z:

[0045]

[0046] where z = e j2πd sin(θ) / λ , e is the base of the natural logarithm, j is the imaginary part identifier, π is the pi, sin(·) is the sine function, and λ is the wavelength of the transmitted electromagnetic wave;

[0047] S54. Substitute the symbolic variable parameter matrix Q(z) into the reconstructed 2D-MUSIC objective function, perform polynomial root finding, and obtain K root values

[0048] S55. According to the K root values Solve to obtain the DOA estimation value of each target through the following formula:

[0049]

[0050] where is the DOA estimation value of the k-th target, arcsin[·] is the arcsine function, and angle(·) is the function for solving the phase angle.

[0051] Furthermore, the method in step S6 is: Substitute the DOA estimation values of each target into the distance-related transmission direction vector expression, and perform least squares fitting on it through the following formula to obtain the distance estimation value:

[0052]

[0053] where is the distance estimation value of the k-th target, is the distance-related transmission direction vector estimation result obtained by substituting the DOA estimation value of the k-th target, and [·] T+ is the generalized inverse vector of the vector transpose.

[0054] The beneficial effects of the present invention are:

[0055] (1) By aggregating the DOA information in the transmission direction vector and the reception direction vector, the present invention avoids the performance degradation caused by insufficient information utilization. At the same time, the distance estimation value is directly obtained by using the distance-related transmission direction vector, avoiding the error transmission effect brought by manual decoupling, and realizing high-precision distance estimation.

[0056] (2) The present invention does not involve any spectral peak search process, can greatly reduce the algorithm complexity and improve the system efficiency on the premise of ensuring the parameter estimation accuracy. Description of the Drawings

[0057] Figure 1 Flow chart of a method for estimating target parameters of FDA - MIMO based on reconstructed dimensionality reduction and root finding provided by an embodiment of the present invention;

[0058] Figure 2 Scatter plot of parameter estimation results of an embodiment of the present invention;

[0059] Figure 3 Comparison chart of computational complexity between an embodiment of the present invention and other methods;

[0060] Figure 4 Comparison chart of the root - mean - square error of DOA varying with signal - to - noise ratio between an embodiment of the present invention and other methods;

[0061] Figure 5 Comparison chart of the root - mean - square error of distance varying with signal - to - noise ratio between an embodiment of the present invention and other methods;

[0062] Figure 6 Comparison chart of the root - mean - square error of DOA varying with the number of snapshots between an embodiment of the present invention and other methods;

[0063] Figure 7 Comparison chart of the root - mean - square error of distance varying with the number of snapshots between an embodiment of the present invention and other methods. Detailed implementation manners

[0064] The following describes the detailed implementation manners of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the detailed implementation manners. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions made using the concept of the present invention are within the scope of protection.

[0065] As Figure 1 shown, in an embodiment of the present invention, a method for estimating target parameters of FDA - MIMO based on reconstructed dimensionality reduction and root finding includes the following steps:

[0066] S1. Construct the transmitting end and receiving end of the FDA - MIMO radar as uniform linear arrays, and transmit electromagnetic waves that satisfy uniform frequency offsets through the transmitting end.

[0067] In step S1, the transmitting end includes M array elements, the receiving end includes N array elements, and the array element spacing is d;

[0068] The frequencies of the electromagnetic waves transmitted by each array element at the transmitting end are:

[0069] f m = f0+(m - 1)Δf

[0070] where f mLet \(f_m\) be the frequency of the electromagnetic wave transmitted by the \(m\)-th array element at the transmitter, \(f_0\) be the reference frequency, and \(\Delta f\) be the basic frequency offset, where \(\Delta f\ll f_0\).

[0071] S2. Obtain the electromagnetic waves reflected by each target at the receiver, perform matched filtering on them, and obtain the output signal.

[0072] Assume there are \(K\) far-field independent targets (\(\theta\) k , \(r\) k ) in the noise environment, where \(k = 1,2,\cdots,K\). Then the output signal \(x(t)\) is:

[0073]

[0074] where \(a\) r \((\theta_1)\) to \(a\) r \((\theta\) K ) are the DOA-related receiving direction vectors of each target, \(a\) t \((\theta\) K , \(r\) K ) is the DOA-distance-related transmitting direction vector of the \(K\)-th target, is the Kronecker product operation, \(\theta_1\) to \(\theta\) K are the DOAs of each target, \(r_1\) to \(r\) K are the distances of each target, \(K\) is the number of targets, and \(n(t)\) is circular Gaussian white noise with a mean of 0 and a variance of \(\delta\) 2 and is independent of the signal vector \(s(t)\).

[0075] S3. Process the eigenvalues of the output signal covariance matrix by the step ratio method to obtain the number of targets, and divide the eigenvectors of the output signal covariance matrix according to the number of targets to obtain the noise subspace.

[0076] Step S3 includes the following sub-steps:

[0077] S31. Calculate the covariance matrix of the output signal through the following formula:

[0078]

[0079] where \(R\) is the covariance matrix of the output signal, \(L\) is the number of snapshots, \(x(l)\) is the \(l\)-th snapshot of the output signal, and \(x\) H \((l)\) is the conjugate transpose of the \(l\)-th snapshot of the output signal.

[0080] S32. Perform eigenvalue decomposition on the output signal covariance matrix to obtain its respective eigenvalues and corresponding eigenvectors, and sort the eigenvalues in descending order.

[0081] The eigenvalues of the output signal covariance matrix obtained by eigenvalue decomposition are \(\lambda_1\) to \(\lambda\) N×M, a total of N×M eigenvalues; and λ1≥λ2≥…≥λ N×M-1 ≥λ N×M .

[0082] S33. Through the following formula, using the step ratio method, according to each eigenvalue of the output signal covariance matrix, obtain the number of targets:

[0083]

[0084]

[0085] where K is the number of targets, μ i is the i-th step ratio, λ i is the i-th eigenvalue of the output signal covariance matrix, λ i+1 is the (i + 1)-th eigenvalue of the output signal covariance matrix, i ∈ [1, N×M - 1], is the function for obtaining the numerical value of the sequence number i when μ i is the largest.

[0086] S34. Select the eigenvectors corresponding to the eigenvalues of the output signal covariance matrix whose sequence numbers are greater than the number of targets to form the noise subspace.

[0087] S4. Utilize the transmission direction vector coupling characteristic of the FDA - MIMO radar and the noise subspace to reconstruct the 2D - MUSIC spatial spectrum function, so that the DOA and distance information are separated, where 2D - MUSIC is the 2 - dimensional MUSIC (Multiple Signal Classification) algorithm.

[0088] The 2D - MUSIC spatial spectrum function constructed using the noise subspace is:

[0089]

[0090] where f 2D-MUSIC (θ, r) is the 2D - MUSIC spatial spectrum function, a r (θ) is the receiving direction vector related to DOA, a t (θ, r) is the transmitting direction vector related to DOA - distance, θ is the DOA, r is the distance, E n is the noise subspace, is the conjugate transpose of E n , [·] H is the conjugate transpose operation, is the Kronecher product operation;

[0091] The 2D - MUSIC spatial spectrum function reconstructed using the transmission direction vector coupling characteristic is:

[0092]

[0093] wherein, a t (r) is the distance-related transmission direction vector, is the conjugate transpose of a t (r), a t (θ) is the DOA-related transmission direction vector, and diag(·) is a diagonal matrix composed of the elements of the vector within the parentheses.

[0094] S5. Perform dimensionality reduction transformation and polynomial root finding on the reconstructed 2D-MUSIC spatial spectrum function to obtain the distance-related transmission direction vector and the DOA estimation value.

[0095] Step S5 includes the following sub-steps:

[0096] S51. Perform dimensionality reduction transformation on the reconstructed 2D-MUSIC spatial spectrum function to obtain the expression of the distance-related transmission direction vector:

[0097]

[0098] wherein, e1 is a unit vector, is the conjugate transpose of e1, Q -1 (θ) is the inverse matrix of the parameter matrix Q(θ),

[0099] S52. According to the distance-related transmission direction vector, reconstruct the objective function of 2D-MUSIC as:

[0100]

[0101] wherein, is the θ value corresponding to the minimum function value, det(·) is the operation of calculating the determinant of a matrix, and compan(·) 1，1 is the operation of calculating the element in the first row and first column of the adjoint matrix of a matrix;

[0102] S53. Construct the parameter matrix Q(θ) as a symbolic variable parameter matrix Q(z) through the symbolic variable z:

[0103]

[0104] wherein, z = e j2πd sin(θ) / λ , e is the base of the natural logarithm, j is the imaginary part identifier, π is the pi, sin(·) is the sine function, and λ is the wavelength of the transmitted electromagnetic wave;

[0105] S54. Substitute the symbolic variable parameter matrix Q(z) into the reconstructed 2D-MUSIC objective function, perform polynomial root finding, and obtain K root values

[0106] In this embodiment, through the above operations, the solution of the objective function is converted into the polynomial root-finding operation of det(Q(z)) = 0.

[0107] S55. According to the K root values Through the following formula, the DOA estimation values of each target are solved and obtained:

[0108]

[0109] where is the DOA estimation value of the k-th target, arcsin[·] is the arcsine function, and angle(·) is the function for solving the phase angle.

[0110] S6. Perform least squares fitting on the distance-related transmission direction vectors to obtain the distance estimation values.

[0111] In this embodiment, the method of step S6 is: substitute the DOA estimation values of each target into the distance-related transmission direction vector expression, and perform least squares fitting on it through the following formula to obtain the distance estimation values:

[0112]

[0113] where is the distance estimation value of the k-th target, is the distance-related transmission direction vector estimation result obtained by substituting the DOA estimation value of the k-th target, and [·] T+ is the generalized inverse vector of the vector transpose.

[0114] The present invention will be further described below in conjunction with the simulation experiment results of MALTAB:

[0115] To evaluate the present invention, consider an FDA-MIMO radar system. The transmitting and receiving ends are uniform linear arrays, with the number of array elements being M = 6 and N = 8 respectively. The reference frequency f0 = 10 GHz, the base frequency offset Δf = 300 KHz, the number of targets is K = 3, and the parameters of each target are (θ1, r1) = (10°, 200 m), (θ2, r2) = (20°, 220 m), (θ3, r3) = (30°, 240 m). To evaluate the performance of different algorithms, the root mean square error (RMSE) is introduced, that is

[0116]

[0117] where represents the arrival angle (DOA) or distance estimation value of the k-th target during the p-th Monte Carlo simulation, and β krepresents the true DOA or distance value of the k-th target, and P represents the number of Monte Carlo simulations.

[0118] Figure 2 is the scatter plot of the estimated results of the target parameters of the present invention. Among them, the signal-to-noise ratio is SNR = 5 dB, the number of snapshots is L = 400, and the number of Monte Carlo simulations is P = 200. Observe Figure 2 It can be seen that the estimated values of the three targets are concentrated at the positions of their respective true target values, which indicates that the present invention can estimate the DOA and distance parameter values of the three targets and has high accuracy. Figure 2 The results verify the reliability of the present invention in the joint estimation of target DOA and distance.

[0119] Figure 3 is the comparison graph of the computational complexity of the present invention and other methods with the change of the number of receiving array elements. The comparison algorithms include the traditional 2D-MUSIC algorithm, RD-MUSIC algorithm, ESPRIT algorithm, and Cramér-Rao bound (CRB). Among them, the search accuracy of the 2D-MUSIC algorithm is θ d = 0.01°, r d = 0.1 m, the DOA search accuracy of the RD-MUSIC algorithm is θ d = 0.01°, and the number of snapshots is L = 200. Observe Figure 3 It can be seen that the computational complexity of this method is much lower than that of the traditional 2D-MUSIC algorithm and RD-MUSIC algorithm, and is basically the same as that of the ESPRIT algorithm, which can meet the requirements of real-time estimation of target parameters and is more conducive to engineering implementation. Figure 3 The results verify the effectiveness of the present invention in the joint estimation of target DOA and distance.

[0120] Figures 4-5 is the comparison graph of the root mean square error of DOA and distance between the present invention and other methods with the change of signal-to-noise ratio. Among them, the number of snapshots is L = 200, and the number of Monte Carlo simulations is P = 600. Figures 6-7 is the comparison graph of the root mean square error of DOA and distance between the present invention and other methods with the change of the number of snapshots. Among them, the signal-to-noise ratio SNR = 10 dB, and the number of Monte Carlo simulations is P = 600. Figures 4-7 The comparison algorithms in Figures 4-7It can be seen that with the increase of the signal-to-noise ratio or the number of snapshots in the method of the present invention, the estimation errors of the DOA and distance parameters gradually decrease, the estimation performance gradually improves, and it is always superior to the ESPRIT algorithm. At the same time, the RMSE curves of the method of the present invention and the traditional 2D-MUSIC algorithm and RD-MUSIC algorithm are basically overlapped, indicating that their performances are basically the same. Even in the case of low complexity, the method of the present invention is superior to the traditional 2D-MUSIC algorithm and RD-MUSIC algorithm. And from Figure 3 it can be seen that the computational complexity of the method of the present invention is much lower than that of the traditional 2D-MUSIC algorithm and RD-MUSIC algorithm. Therefore, the method of the present invention is more suitable for real-time estimation of target parameters. Figures 4-7 The results effectively verify the superiority of the method of the present invention in the joint estimation of target angle and distance.

[0121] In summary, the present invention utilizes the DOA and distance coupling characteristics of the transmission direction vector and the noise subspace to reconstruct the 2D-MUSIC spatial spectrum function, and uses the dimensionality reduction transformation and polynomial root-finding technology to obtain the distance-related transmission direction vector and DOA estimation values. Finally, the least squares fitting is performed on the distance-related transmission direction vector to obtain the distance estimation value. The present invention realizes the target DOA estimation by aggregating the DOA information in the transmission direction vector and the reception direction vector, avoiding the performance degradation caused by insufficient information utilization. At the same time, the distance estimation value is obtained by using the distance-related transmission direction vector, avoiding the decoupling error and realizing high-precision distance estimation. While ensuring the DOA and distance estimation accuracy, the present invention greatly reduces the computational complexity, is more suitable for real-time estimation of target parameters, and is conducive to engineering implementation.

Claims

1. A method for estimating target parameters of FDA - MIMO based on root - finding with reconstruction and dimensionality reduction, characterized in that, It includes the following steps: S1. Construct the transmitting end and receiving end of the FDA-MIMO radar as uniform linear arrays, and transmit electromagnetic waves that satisfy uniform frequency offset through the transmitting end; S2. Obtain the electromagnetic waves reflected by each target through the receiving end, perform matched filtering on them, and obtain the output signal; S3. Process the eigenvalues of the output signal covariance matrix by the step ratio method to obtain the number of targets, and divide the eigenvectors of the output signal covariance matrix according to the number of targets to obtain the noise subspace; S4. Utilize the coupling characteristic of the transmitting direction vector of the FDA-MIMO radar and the noise subspace to reconstruct the 2D-MUSIC spatial spectrum function, so that its DOA and distance information are separated; S5. Perform dimensionality reduction transformation and polynomial root finding on the reconstructed 2D-MUSIC spatial spectrum function to obtain the distance-related transmitting direction vector and DOA estimation values; S6. Perform least squares fitting on the distance-related transmitting direction vector to obtain the distance estimation value; The step S5 includes the following sub-steps: S51. Perform dimensionality reduction transformation on the reconstructed 2D-MUSIC spatial spectrum function to obtain the expression of the distance-related transmitting direction vector: where, e1 is a unit vector, is the conjugate transpose of e1, Q -1 (θ) is the inverse matrix of the parameter matrix Q(θ), S52. According to the distance-related transmitting direction vector, reconstruct the objective function of 2D-MUSIC as: Among them, to obtain the θ value corresponding to the minimum function value, det(·) is the operation of calculating the determinant of a matrix, and compan(·) 1，1 is the operation of calculating the element in the first row and first column of the adjoint matrix of a matrix; S53. Through the symbolic variable z, construct the parameter matrix Q(θ) as the symbolic variable parameter matrix Q(z): where z = e j2πd sin(θ) / λ , e is the base of the natural logarithm, j is the imaginary part identifier, π is the pi, sin(·) is the sine function, and λ is the wavelength of the transmitted electromagnetic wave; S54. Substitute the symbolic variable parameter matrix Q(z) into the reconstructed 2D-MUSIC objective function, perform polynomial root finding, and obtain K root values. S55. According to K root values The DOA estimation values of each target are obtained by solving the following formula: Among them, is the DOA estimation value of the k-th target, arcsin[·] is the arcsine function, and angle(·) is the function for solving the phase angle.

2. The method for estimating target parameters of FDA - MIMO based on root - finding with reconstruction and dimensionality reduction according to claim 1, characterized in that, In the step S1, the transmitting end includes M array elements, the receiving end includes N array elements, and the array element spacing is d; The frequencies of the electromagnetic waves transmitted by each array element at the transmitting end are: f m = f0 + (m - 1)Δf where, f m is the frequency of the electromagnetic wave transmitted by the m-th array element at the transmitting end, f0 is the reference frequency, and Δf is the basic frequency offset, where Δf << f0.

3. The method for estimating target parameters of FDA - MIMO based on root - finding with reconstruction and dimensionality reduction according to claim 2, characterized in that, The step S3 includes the following sub-steps: S31. Calculate the covariance matrix of the output signal; S32. Perform eigenvalue decomposition on the output signal covariance matrix to obtain its respective eigenvalues and corresponding eigenvectors, and sort the eigenvalues in descending order; S33. Adopt the step ratio method to obtain the number of targets according to the respective eigenvalues of the output signal covariance matrix; S34. Select the eigenvectors corresponding to the eigenvalues of the output signal covariance matrix with serial numbers greater than the number of targets to form the noise subspace.

4. The method for estimating target parameters of FDA - MIMO based on root - finding with reconstruction and dimensionality reduction according to claim 3, characterized in that, The covariance matrix of the output signal in the step S31 is calculated by the following formula: where R is the covariance matrix of the output signal, L is the number of snapshots, x(l) is the l-th snapshot of the output signal, and x H (l) is the conjugate transpose of the l-th snapshot of the output signal.

5. The FDA-MIMO target parameter estimation method based on reconstructed dimensionality reduction and root finding according to claim 4, wherein, The eigenvalues of the output signal covariance matrix obtained by eigenvalue decomposition in the step S32 are successively λ1 to λ N×M , with a total of N×M eigenvalues; and λ1≥λ2≥…≥λ N×M-1 ≥λ N×M .

6. The FDA-MIMO target parameter estimation method based on reconstructed dimensionality reduction and root finding according to claim 5, wherein, The number of targets in the step S33 is obtained by the following formula using the step ratio method: where K is the target number, μ i is the i-th step ratio, λ i is the i-th eigenvalue of the output signal covariance matrix, λ i+1 is the (i + 1)-th eigenvalue of the output signal covariance matrix, i ∈ [1, N × M - 1], is the function for obtaining the numerical value of the sequence number i when μ i is at its maximum.

7. The FDA-MIMO target parameter estimation method based on reconstructed dimensionality reduction and root finding according to claim 6, wherein, In the step S4, the 2D-MUSIC spatial spectrum function constructed by using the noise subspace is: Among them, f 2D-MUSIC (θ, r) is the 2D-MUSIC spatial spectrum function, a r (θ) is the receiving direction vector related to DOA, a t (θ, r) is the transmitting direction vector related to DOA-distance, θ is the DOA, r is the distance, E n is the noise subspace, is the conjugate transpose of E n , [·] H is the conjugate transpose operation, is the Kronecher product operation; The 2D-MUSIC spatial spectrum function reconstructed by utilizing the coupling characteristic of the transmitting direction vector is: where a t (r) is the distance-related transmission direction vector, is the conjugate transpose of a t (r), a t (θ) is the DOA-related transmission direction vector, and diag(·) is a diagonal matrix composed of the elements of the vector in the parentheses.

8. The FDA-MIMO target parameter estimation method based on reconstructed dimensionality reduction and root finding according to claim 7, wherein, The method of the step S6 is: Substitute the DOA estimation values of each target into the expression of the distance-related transmitting direction vector, and perform least squares fitting on it through the following formula to obtain the distance estimation value: Among them, is the distance estimation value of the k-th target, is the distance-related transmission direction vector estimation result obtained by substituting the DOA estimation value of the k-th target, is the generalized inverse vector of the vector transpose.

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