Joint Design Method of SAR Waveform and Range Filter Based on FEKO

Abstract

The present invention relates to a joint design method of SAR waveform and range filter based on FEKO, including: Step 1, obtaining the scattering characteristics of the target; Step 2, establishing a discretized signal model of the SAR one-dimensional range profile based on the target scattering model; Step 3, analyzing the objective function characterizing the joint design of the SAR waveform and the range filter; Step 4, constructing an optimization problem of the SAR waveform and the range filter; Step 5, separating the variables of the optimization problem to obtain a sub-optimization problem P1 containing only a single variable, decoupling its fractional objective function, and equivalently transforming it into a semidefinite programming problem P3; Step 6, relaxing the rank-one constraint of P3, introducing a penalty term for iteration until the global optimal solution of the sub-optimization problem P1 is obtained; Step 7, repeating Steps 5-6 until the global optimal solution of the sub-problem P5 of another variable is obtained; Step 8, alternately solving the sub-problems P1 and P5, and after the iteration process converges, outputting the calculation results of the SAR waveform and the range filter.

Description

Technical Field

[0001] The present invention relates to the technical field of signal processing, and particularly to a joint design method of SAR waveforms and range filters based on FEKO. Background Art

[0002] With the progress of basic technologies such as transmission, reception, and processing, as well as engineering implementation technologies related to the carrier platform, the spatial resolution of synthetic aperture radar (SAR) has been significantly improved, making the fine perception of high-value targets one of the urgent needs in the SAR field. Optimizing the design of transmit waveforms and receive filters is an effective way to enhance the radar's detection ability. In nature, organisms such as bats and dolphins with ultrasonic radars can adjust waveforms and echo processing methods according to different detection targets and environments, thereby enhancing their perception ability of complex environments and achieving risk avoidance, prey capture, and action route optimization.

[0003] Under the assumption of point targets, traditional SAR generally uses linear frequency modulation signals and matched filters to achieve target observation. Although it can obtain relatively ideal spatial resolution and sidelobe level, it does not distinguish between targets and background scatterers, that is, all scatterers within the beam range are regarded as "targets", making many high-value targets submerged by strong clutter backgrounds and difficult to be highlighted. Therefore, it is necessary to propose a joint design method for SAR waveforms and range filters for high-value targets of interest to meet the application requirements of target fine perception.

[0004] Although research on designing radar waveforms for typical targets has been common for a long time, such research usually pre-gives the scattering characteristics of the targets, only focusing on the mathematical modeling and solution methods of waveform design optimization problems, and does not explain how the target scattering characteristics are obtained as auxiliary knowledge. Constructing auxiliary knowledge through electromagnetic scattering characteristic analysis is the primary condition for target-driven radar waveform design. Therefore, it has important application value to use the electromagnetic calculation software FEKO to analyze the target scattering characteristics, establish the auxiliary knowledge of SAR waveform design, and then propose a joint design method for waveforms and filters. Summary of the Invention

[0005] In order to solve the problems existing in the above-mentioned prior art, the purpose of the present invention is to propose a joint design method of SAR waveforms and range filters based on FEKO for high-value targets that are difficult to be highlighted under strong clutter backgrounds by analyzing the scattering characteristics of the targets of interest based on the FEKO platform. On the premise of ensuring that the waveforms have characteristics such as constant modulus, high resolution, and low peak sidelobe ratio, the information acquisition ability of SAR for certain types of targets is enhanced.

[0006] To achieve the above purpose, the present invention provides the following solutions:

[0007] Joint Design Method of SAR Waveform and Range Filter Based on FEKO, including:

[0008] Step 1: Obtain the scattering characteristics of typical targets based on the FEKO platform;

[0009] Step 2: Establish a signal model of the SAR one-dimensional range profile based on the scattering model and discretize it;

[0010] Step 3: Take the mutual information between the scattering characteristics of the target and the SAR one-dimensional range profile as the optimization criterion, and analytically characterize the objective function for the joint design of the SAR waveform and range filter;

[0011] Step 4: Impose constant modulus constraint and cross-correlation function constraint conditions; the constant modulus constraint ensures that the SAR achieves the maximum transmission power, and the correlation function constraint ensures high range resolution and low sidelobe level in the range direction of the SAR. Combining the objective function and constraint conditions, construct a mathematical model for the joint design optimization problem of the SAR waveform and range filter;

[0012] Step 5: Based on the sequential iteration method, separate variables to obtain a sub-optimization problem P1 that only contains the SAR waveform variable; the range filter decouples the fractional objective function of the sub-optimization problem P1 to form a sub-problem P2, and then transforms it into a semidefinite programming problem P3;

[0013] Step 6: Relax the rank-one constraint in the sub-problem P3, add a penalty term to gradually reduce the rank of the semidefinite programming variable during the iteration process, and obtain the solution X after the rank is reduced to 1. * , perform eigenvalue decomposition on X * to obtain the optimal solution of the sub-problem P2. Continuously iterate the sub-problem P2, and finally the global optimal solution of the sub-optimization problem P1 can be obtained;

[0014] Step 7: Substitute the global optimal solution of the sub-problem into the original optimization problem to obtain a sub-optimization problem regarding the range filter Repeat steps 5 - 6 until the global optimal solution of the sub-optimization problem is obtained;

[0015] Step 8: Alternately solve the sub-optimization problem of the SAR waveform variable and the sub-problem of the range filter After the iteration process converges, obtain the solution results of the SAR waveform range filter.

[0016] Optionally, obtaining the scattering characteristics of typical targets includes:

[0017] Establish a target model based on the geometric shape and material parameter information of the target;

[0018] Mesh generation is performed according to the details of the target model and the wavelength of the incident wave, and the excitation source parameters are set to start the calculation task to obtain the electromagnetic scattering data of the target;

[0019] Among them, the excitation source parameters include: incident direction, frequency, incident angle, and polarization mode.

[0020] Optionally, a signal model of the SAR one-dimensional range profile is established based on the scattering model, including:

[0021] The SAR echo is processed by range filtering to obtain a one-dimensional range profile:

[0022]

[0023] Among them, is the sampling value of the SAR one-dimensional range profile at any time e, r is the range filter, G(e) and B(l) respectively represent the Toeplitz matrices formed after discretizing the target scattering characteristic G(t) and the background scattering characteristic B(t), x is the discretized SAR transmit waveform, n(e) is the discretized noise sequence, and (·) H represents the conjugate transpose of a matrix or vector.

[0024] Optionally, the objective function characterizing the joint design of the SAR waveform and the range filter:

[0025] Taking the mutual information between the target scattering characteristic and the SAR one-dimensional range profile as the optimization criterion, the objective function characterizing the joint design of the SAR waveform and the range filter is analyzed:

[0026]

[0027] R G = E[G(l)G * (l)];

[0028] R n = E[n(l)n H (l)];

[0029]

[0030] A x = E[Q(l)xx H Q H (l)];

[0031] Q(l) = G(l) + B(l);

[0032] Among them, The equivalent form of the mutual information quantity, E[·] is the mathematical expectation, and (·) * represents the conjugate of a vector or scalar.

[0033] Optionally, a mathematical model for jointly designing and optimizing the SAR waveform and the range filter is constructed, including:

[0034] Add a constant modulus constraint condition to the SAR waveform, and at the same time limit the cross-correlation function CCF of the SAR waveform and the range filter to have a narrow main lobe and a low peak sidelobe ratio by forcing the peak energy of the CCF main lobe to be greater than a first preset threshold and forcing the peak energy of all sidelobes to be lower than a second preset threshold; on the premise that the waveform and filter energies are constant, raise the main lobe of the CCF and reduce the sidelobes;

[0035] Integrate the objective function and the constraint conditions to construct a mathematical model for the joint design and optimization problem of the SAR waveform and the range filter:

[0036]

[0037] where f(x,r) is the objective function obtained by ignoring the constant term in , and respectively represent the value sets of i and j, δ1 is the first preset threshold, δ2 is the second preset threshold, x(i) represents the i-th element of the waveform vector, J r is the Toeplitz matrix formed by the range filter r, W N is a matrix with all elements being 0 except for the element W(N,N)=1, will form the CCF, W j is a matrix with all elements being 0 except for the element W(j,j)=1, E r represents the energy of the range filter r.

[0038] Optionally, isolate the sub-optimization problem of the SAR waveform variable For decouple the fractional objective function to form a sub-problem and then transform it into a semidefinite programming problem including:

[0039] Use the sequential iteration method to separate the variables in the optimization problem and establish an iteration process At the k-th iteration, set the range filter r = r (k-1) , and r (k-1) is the value of r at the (k - 1)-th iteration, thereby obtaining a sub-optimization problem only about the SAR waveform:

[0040]

[0041] where, is the matrix generated by r (k-1) ;

[0042] Decouple the fractional objective function in to obtain the optimization problem x (m) , and start the iterative process Let m represent the number of iterations, then:

[0043]

[0044] where x (m) represents the decision variable of the waveform at the m-th iteration, F(x (m) ) = α(x (m) , r (k-1) ) - εβ(x (m) , r (k -1) ), ε = f(x (m-1) , r (k-1) ), α(x (m) , r (k-1) ) and β(x (m) , r (k-1) ) are the numerator and denominator of f(x, r (k-1) ) respectively;

[0045] Introduce the auxiliary variable X = x (m) x (m)H , and equivalently transform into the SDP form

[0046]

[0047] where rank[·] represents the rank of a matrix, tr[·] represents the trace of a matrix, Λ i represents a matrix with all elements being 0 except Λ(i, i) = 1, and X ± 0 means X is positive semi-definite.

[0048] Optionally, obtaining the global optimal solution of the sub-problem includes:

[0049] Relax the rank-one constraint in the sub-problem , introduce a penalty term and gradually reduce X through iteration, and start the iterative process where the optimization problem for iteration is:

[0050]

[0051] where w > 1 is the penalty term, t represents the number of iterations, s (t) is the slack variable, and V (t-1) is X (t-1) at the (t - 1)-th step of iteration.The matrix composed of the eigenvectors corresponding to the N - 1 eigenvalues except the largest eigenvalue, I N-1 represents the N - 1 dimensional identity matrix, and set the iteration stopping condition:

[0052] s (t) < δ;

[0053] where δ is a very small positive number;

[0054] When the stopping condition of the iterative process is satisfied, the global optimal solution X of the optimization problem will be obtained * ;

[0055] Perform eigenvalue decomposition on the solution X * and set the iteration stopping condition:

[0056] f(x (m) ,r (k-1) ) - f(x (m-1) ,r (k-1) ) ≤ e;

[0057] where e represents the minimum change in the objective function value allowed for two adjacent iterations in the iterative process ;

[0058] When the stopping condition of the iterative process is satisfied, the global optimal solution of the sub - optimization problem will be obtained.

[0059] Optionally, obtaining the global optimal solution of the sub - optimization problem of the range filter includes:

[0060] Substitute the global optimal solution x of the problem (k) , that is, the k - th iteration of , into the objective function f(x, r), and obtain the sub - problem about the filter variable r:

[0061]

[0062] where,

[0063]

[0064] is the Toeplitz matrix formed by x (k) ; Similarly, obtain the global optimal solution of the sub - optimization problem of the range filter: still use the Dinkelbach algorithm to decouple the problem The objective function is then converted into an equivalent SDP form, and finally, through an iterative method, the rank of the SDP variables is gradually reduced until the global optimal solution is obtained.

[0065] Optionally, obtaining the solution result of the SAR waveform range filter includes:

[0066] Alternately solving the sub-problems of the SAR waveform and the filter sub-problem until the iterative process converges, and setting the stopping condition of

[0067]

[0068] where e′ is the minimum allowable change in the objective function for two adjacent iterations, and K is the maximum number of iterations;

[0069] When the stopping condition of is not satisfied, then repeat steps 5 - 7, and finally output the solution results of the SAR waveform and the filter.

[0070] The beneficial effects of the present invention are:

[0071] The present invention focuses on the joint optimization design of the SAR waveform and the range filter. By using FEKO to give the electromagnetic scattering characteristics of typical targets, a transmit / receive joint optimization problem is established, and a solution method is proposed, which can effectively enhance the detection ability of SAR for high-value targets.

[0072] The solution method proposed by the present invention can solve optimization problems with polynomial time complexity, and at the same time has characteristics such as a fractional non-convex objective function, constant modulus constraints, non-convex quadratic equality / inequality constraints, etc., and obtains the global optimal solution.

[0073] The joint design method of the SAR waveform and the range filter proposed by the present invention can be applied to the optimization design of other radar systems, providing ideas and a preliminary theoretical basis for the simultaneous optimization of the radar transmitter and receiver. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the embodiments. The drawings described below are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0075] Figure 1 is a flowchart of the joint design method of the SAR waveform and the range filter based on FEKO according to the embodiment of the present invention;

[0076] Figure 2 Schematic diagram of the system model of the one-dimensional range profile of SAR in the embodiment of the present invention;

[0077] Figure 3 For the slack variable s in the embodiment of the present invention (t) Schematic diagram of the change of the slack variable s with the number of iterations t; where (a) shows the change of the slack variable s with the number of iterations t when δ2 = 20 and different values of δ1 are taken, (t) and (b) shows the change of the slack variable s with the number of iterations t when δ1 = 350 and different values of δ2 are taken; (t)

[0078] Figure 4 For the objective function f(x (m) , r (k-1) ) sequence with the change of the number of iterations m; where (a) shows the change of the objective function f(x (m) , r (k-1) ) sequence with the number of iterations m when δ2 = 20 and different values of δ1 are taken, and (b) shows the change of the objective function f(x (m) , r (k-1) ) sequence with the number of iterations m when δ1 = 350 and different values of δ2 are taken;

[0079] Figure 5 For the execution of the alternating iteration process in the embodiment of the present invention to obtain the relationship diagram between the number of iterations k and the objective function value f(x (k) , r (k) ); where (a) shows the relationship between the number of iterations k and the objective function value f(x (k) , r (k) ) obtained by executing the alternating iteration process when δ2 = 20 and different values of δ1 are taken, and (b) shows the relationship between the number of iterations k and the objective function value f(x (k) , r (k) ) obtained by executing the alternating iteration process when δ1 = 350 and different values of δ2 are taken;

[0080] Figure 6 CCF diagrams when different threshold values are taken in the embodiment of the present invention; where (a) is the CCF when δ2 = 20 and different values of δ1 are taken, and (b) is the CCF when δ1 = 350 and different values of δ2 are taken;

[0081] Figure 7 Real and imaginary parts of the time-domain waveform diagrams when different threshold values are taken in the embodiment of the present invention; where (a) are the real and imaginary parts of the time-domain waveform when δ2 = 20 and different values of δ1 are taken, and (b) are the real and imaginary parts of the time-domain waveform when δ1 = 350 and different values of δ2 are taken; ​

[0082] Figure 8 The SNR of the embodiment of the present invention and different waveforms Schematic diagram; wherein, (a) is the SNR and different waveforms when SCR = 5dB (b) is the SCR and different waveforms when SNR = 5dB

[0083] Figure 9 Schematic diagram of CCF for comparing different waveforms in the embodiment of the present invention; wherein, (a) is the CCF for comparing different waveforms, and (b) is the real part and imaginary part of different waveforms;

[0084] Figure 10 Schematic diagram of establishing an aircraft model based on FEKO in the embodiment of the present invention;

[0085] Figure 11 Schematic diagram of SAR image in the embodiment of the present invention; wherein, (a) is the real SAR image, (b) is the SAR image corresponding to the algorithm proposed by the present invention, and (c) is the SAR image corresponding to the Chirp signal. Detailed implementation manners

[0086] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0087] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below in conjunction with the drawings and specific implementation manners.

[0088] Such as Figure 1As shown in the figure, this embodiment discloses a joint design method for SAR waveforms and range filters based on FEKO, including: Step 1, obtaining the scattering characteristics of typical targets based on the FEKO platform; Step 2, establishing a signal model of the SAR one-dimensional range profile based on the scattering model and discretizing it; Step 3, taking the mutual information between the scattering characteristics of the target and the SAR one-dimensional range profile as the optimization criterion, and analyzing and characterizing the objective function for the joint design of SAR waveforms and range filters; Step 4, imposing constant modulus constraints and CCF constraint conditions; the constant modulus constraint ensures that the SAR achieves the maximum transmission power, and the correlation function constraint ensures high resolution and low sidelobe level in the range direction of the SAR. Combining the objective function and the constraint conditions, a mathematical model for the joint design optimization problem of SAR waveforms and range filters is constructed; Step 5, separating variables based on the sequential iteration method to obtain a sub-optimization problem P1 that only contains SAR waveform variables; the range filter decouples the fractional objective function of the sub-optimization problem P1 to form a sub-problem P2, and then converts it into a semidefinite programming problem P3; Step 6, relaxing the rank-one constraint in the sub-problem P3, adding a penalty term to gradually reduce the rank of the semidefinite programming variables during the iteration, and obtaining the solution X after the rank is reduced to 1. * Performing eigen-decomposition on X * will enable the optimal solution of the sub-problem P2 to be obtained. By continuously iterating the sub-problem P2, the global optimal solution of the sub-problem P1 can ultimately be obtained; Step 7, substituting the global optimal solution of the sub-problem P1 into the original optimization problem P0 to obtain a sub-optimization problem P5 regarding the range filter, and repeating Steps 5-6 until the global optimal solution of this problem P5 is obtained; Step 8, alternately solving the SAR waveform sub-problem P1 and the range filter sub-problem P5. After the iteration process converges, the solution results of the SAR waveform range filter are obtained.

[0089] In this embodiment, aiming at the difficulty of existing SAR waveforms and matched filters in highlighting high-value targets under strong clutter backgrounds, the scattering characteristics of the target of interest are analyzed based on the FEKO platform, and a joint design method for SAR waveforms and range filters is proposed. On the premise of ensuring that the waveform has characteristics such as constant modulus, high resolution, and low peak sidelobe ratio, the information acquisition ability of the SAR for a certain type of target is enhanced, and the implementation process is as Figure 1 shown, specifically including the following steps:

[0090] Step 1: Analyze the scattering characteristics of the target based on FEKO;

[0091] First, use the modeling tool of FEKO to create the geometry of the target and establish the target model. Ensure that the size and shape of the model conform to the actual physical characteristics. You can import the CAD model or draw it manually as needed. The target modeling should be as accurate as possible to approximate the real target characteristics. For complex electrically large targets, geometric simplification is also required to reduce the computational burden. Configure appropriate material properties for each component module in the model, including conductivity, permittivity, and permeability. Here, you can select materials from the existing material library or define new material properties. If it is a stealth target with coating, the electromagnetic parameters of the surface need to be defined.

[0092] Secondly, perform mesh division on the geometric model, dividing the model into multiple small units. The mesh size depends on the details of the model and the wavelength of the incident wave. The finer the mesh division, the higher the computational accuracy, but the greater the computational burden. Therefore, appropriate mesh optimization needs to be carried out in different regions of the model, refining the mesh in key regions (such as corners and tips) to improve the computational accuracy, while coarsening the mesh in simple regions to reduce the computational burden.

[0093] Then, set the excitation source parameters, including parameters such as the incident direction, frequency, incident angle, and polarization mode. Select an appropriate solution method according to the mesh quantity, computational resources, frequency range, etc. FEKO supports multiple solution methods, such as the finite-difference time-domain method (FDTD), the method of moments (MoM), and the multi-spectrum method, etc. These methods have different speeds and accuracies and are suitable for different application scenarios.

[0094] Finally, start the calculation task, monitor the calculation status, observe the change of the residual with the number of iterations, and ensure the convergence of the calculation process. After the calculation is completed, output the electromagnetic scattering data of the target as auxiliary knowledge for SAR waveform design.

[0095] Step 2: Establish a discretized signal model for the one-dimensional range of SAR;

[0096] The signal acquisition process of the SAR one-dimensional range image is as Figure 2 shown. The SAR transmitted waveform x(t) interacts with the target scattering characteristic G(t) and the background scattering characteristic B(t) to generate an echo Y(t) interfered by noise N(t). After being processed by the receiving filter r(t), the one-dimensional range image is expressed as:

[0097]

[0098] where represents the convolution operation. Using x = [x0 x1…x N-1 T and r = [r0 r1…r N-1 T ​​respectively represent the discretized SAR transmit waveform and the range filter, and \(n(l)=[n(l)\ n(l + 1)\ \cdots\ n(l + N - 1)]\) T represents the discrete noise, \(G(l)\) and \(B(l)\) respectively represent the Toeplitz matrices formed after discretization of the target scattering characteristic \(G(t)\) and the background scattering characteristic \(B(t)\) for implementing the convolution operation. Then, at a given time \(l\), the one-dimensional range image of the SAR is represented in the discretized form as:

[0099]

[0100] Step 3: Select the mutual information between the target scattering characteristic and the one-dimensional range image as the optimization criterion, and analyze and represent the objective function for the joint design of the SAR waveform and the range filter;

[0101] The essence of radar detecting a target is to obtain the target scattering information as much as possible. The mutual information can quantitatively describe how much target (information source) information can be obtained by a communication or radar system. Therefore, the mutual information is selected as the optimization criterion. Under the Gaussian hypothesis, according to the definition of the mutual information, the mutual information between the target scattering characteristic and the one-dimensional range image can be expressed as:

[0102]

[0103] In the formula, \(f(G)\), and are respectively the probability density function of \(G(l)\), the probability density function of and the joint probability density function of \(G(l)\) and is the square of the modulus of the Pearson correlation coefficient, expressed as:

[0104]

[0105] In the formula, \(E[\cdot]\) represents the mathematical expectation. After simplification, the analytical expression of the objective function can be obtained:

[0106]

[0107] where \(R G =E[G(l)G * (l)]\), \(R n =E[n(l)n H (l)]\), and \(Q(l)=G(l)+B(l)\).

[0108] Step 4: Apply the constant modulus constraint and the CCF constraint conditions, and establish a mathematical model for the optimization problem in combination with the objective function;

[0109] Constraint conditions need to be introduced to ensure the engineering feasibility of the SAR waveform and the high-resolution and low-sidelobe characteristics in the range direction. First, add the constant modulus constraint condition, that is, |x(i)|, i = 0, 1, … N-1, which can ensure that the amplifier of the radar transmitter works in the best state, and then achieve the maximum transmission power. Second, in order to ensure the high resolution and low sidelobe level in the SAR range direction, the CCF of the SAR waveform x and the receiving filter r should have a narrow main lobe and a low peak sidelobe ratio, and this requirement can be achieved through two constraint conditions, that is:

[0110]

[0111] Here, W N (N, N) = 1, and the other elements in W N are 0; Generate the CCF of the waveform x and the range filter r, and:

[0112]

[0113] By constructing W N The main lobe of the CCF can be selected, and the peak power of the main lobe is forced to be greater than a certain threshold δ1. This constraint condition makes the main lobe narrower by increasing the peak power of the main lobe of the CCF, and then achieves high resolution. At the same time, by forcing the peak power of all sidelobes to be lower than a certain threshold δ2, the sidelobes are suppressed, that is:

[0114]

[0115] Finally, introduce the energy constraint r of the range filter H r = E r , to increase the main lobe and reduce the sidelobe on the premise of ensuring the CCF energy remains unchanged. Combining with the objective function, an optimization problem is established

[0116]

[0117] In the formula, f(x, r) is Ignoring the constant term The obtained objective function. And Represent the value sets of i and j respectively.

[0118] Step 5: Use the sequential iteration method to decompose the binary problem into sub-problems containing only one variable and convert them into equivalent SDP problems;

[0119] First, separate variables based on the sequential iteration method and start the iteration process k represents the number of iterations, and given the range filter r = r (k-1), at the k-th iteration, the sub-problem regarding the variable x can be obtained:

[0120]

[0121] In the formula, is the matrix generated by r (k-1) , similar to J r , the objective function is expressed as:

[0122]

[0123] where, A r = E[Q H (e)r (k-1) r (k-1)H Q(e)].

[0124] Then, the Dinkelbach algorithm is used to decouple the numerator and denominator of the objective function, thereby transforming the fractional problem into a quadratically constrained quadratic programming problem. Start the iterative process Let m denote the number of iterations, and the sub-problem is obtained:

[0125]

[0126] where, the objective function is expressed as:

[0127] F(x (m) ) = α(x (m) , r (k-1) ) - εβ(x (m) , r (k-1) ) (13)

[0128] where, ε = f(x (m-1) , r (k-1) ), α(x (m) , r (k-1) ) and β(x (m) , r (k-1) ) are the numerator and denominator of f(x, r (k-1) ), respectively. It can be proved that through iteration a monotonically increasing sequence of the fractional objective function can be obtained At this time, although the objective function has been simplified, the constraint conditions are still non-convex and need to be further transformed.

[0129] Finally, an auxiliary variable is introduced to equivalently transform the waveform sub-problem into the SDP form. Define X = x (m) x (m)H , the equivalent SDP form of

[0130]

[0131] In the formula, rank[·]=1 indicates that the rank of X is 1. At this time, Except for the rank-one constraint, the objective function and other constraint conditions are all convex functions.

[0132] Step 6: Relax the rank-one constraint and use the iterative method to gradually reduce the rank of X until it is 1.

[0133] Since the auxiliary variable X is a positive semi-definite Hermitian matrix, X can necessarily be eigenvalue decomposed. The rank of X being 1 is equivalent to X having only 1 non-zero eigenvalue, and the remaining N - 1 eigenvalues are all 0. Based on this, a penalty term is added to force, during the iterative process, the other N - 1 small eigenvalues except the largest eigenvalue of X to be continuously reduced until they are reduced to 0. At this time, the rank of X is 1. Start the iterative process Let t be the number of iterations, then can be solved by continuous iteration to complete the solution, and is:

[0134]

[0135] In the formula, w>1 is the penalty term, s (t) is the slack variable, V (t-1) is the matrix composed of the eigenvectors corresponding to the N - 1 small eigenvalues of X except the largest eigenvalue when iterating to the (t - 1)-th step. Thus, (t-1) is a convex SDP problem and can be solved by using the solving tool CVX. Set the stopping condition for the iterative process as: The stopping condition for the iterative process

[0136] s (t) <δ (16)

[0137] where δ is a very small positive number greater than 0. When this stopping condition is satisfied, it indicates that the solution X of * has N - 1 eigenvalues close to 0. At this time, X * can be considered to have a rank of 1.

[0138] When the stopping condition of the said iteration is not satisfied, then the relaxed rank-one constraint is reintroduced until the stopping condition of the said iteration is satisfied. When the stopping condition of the said iteration is satisfied, then the solution X * is obtained;

[0139] It can be proved that: By iterating the innermost sub-problem the global optimal solution of the problem can be obtained. And if P3 has global convergence, it can be guaranteed Global convergence is achieved, and then it is proved that the global optimal solution can be obtained. The proof process mainly includes: First, it is proved that through a finite number of iterations, the auxiliary variable s in (t) can be made 0, and at this time the rank of X * is 1; Then it is proved that when the rank of X * is 1, and are equivalent; Finally, it is proved that the global optimal solution obtained by eigenvalue decomposition of the global optimal solution of is the global optimal solution of , and then the global optimal solution of is obtained. The stopping condition of the iterative process can be set as:

[0140] f(x (m) ,r (k-1) ) - f(x (m-1) ,r (k-1) ) ≤ e (17)

[0141] where e represents the minimum allowable change in the objective function value between two adjacent iterations in the iterative process . If the stopping condition is satisfied, it indicates that the objective function no longer has a significant increment as the number of iterations increases. After the iterative process converges, the global optimal solution of is obtained

[0142] When the stopping condition of the said iteration is not satisfied, then repeat step 5 until the stopping condition of the said iteration is satisfied. When the stopping condition of the said iteration is satisfied, then obtain the global optimal solution of the said sub-optimization problem .

[0143] Step 7: Solve the sub-problem of the range filter r in the iterative process ;

[0144] Substitute the optimal solution of the problem into the objective function f(x, r), and obtain the sub-problem about the variable r:

[0145]

[0146] where

[0147] where is the Toeplitz matrix formed by x (k) .

[0148] Similarly, in line with the solution idea of the SAR waveform sub-problem, the Dinkelbach algorithm is still used to decouple the problem objective function, and then it is converted into an equivalent SDP form. Finally, the rank of the SDP variable is gradually reduced through an iterative method until it is 1. The global optimal solution of the range filter r can also be obtained by adopting the above process.

[0149] Step 8: Alternately solve the sub-problems of the waveform and the filter and until the stopping condition of the iterative process is met, and the solution results are output

[0150] Set the stopping condition of the iterative process as when there is no significant increment in the objective function between two adjacent iterations after updating x and r, and when the number of iterations reaches the maximum value at the same time, stop the iteration, that is:

[0151]

[0152] In the formula, e′ is the minimum allowable change in the objective function between two adjacent iterations, and K represents the maximum number of iterations in the alternating iterative process.

[0153] When the stopping condition of the iterative L1 is not met, repeat steps 5 - 7 until the stopping condition of the iterative L1 is met, and obtain the solution results of the SAR waveform and the solution results of the range filter.

[0154] This embodiment verifies the theoretical derivation and practical application involved in this embodiment through numerical experiments, including the convergence of the algorithm, the effectiveness of the constraint conditions, and the information acquisition ability for the target to be measured. And by simulating a simple airborne SAR system, it is verified whether this embodiment can improve the observation performance of the imaging radar for typical targets.

[0155] 1. Verification of algorithm convergence:

[0156] Set the sequence length N = 20, the penalty term w = 2, and the energy E r of the receiving filter r -2 = N. The stopping condition parameters e and δ of the iterative processes L1, L2, and L3 are both set to 10 (k＝0) . In the iterative process L1, initialize x (k＝0) and r (m＝0) as Chirp signals and matched filters; in the iterative processes L2 and L3, initialize x (k＝0) = x (t＝0) and X (m＝0) = x (m＝0)H. Execute the iterative process L3 in Step 6 to obtain the slack variable s (t) The variation with the number of iterations t is as Figure 3 (a)-(b) shown.

[0157] It can be seen that the slack variable s (t) continually decreases as the number of iterations t increases, indicating that as the penalty term w t increases, the rank of the SDP variable continuously decreases until the stopping condition is triggered, thus indicating that the iterative process L3 in Step 6 has convergence. s (t) Finally, it decreases to a number close to 0, which will cause the N - 1 small eigenvalues of X to gradually decrease to 0, and the rank of X to gradually decrease to 1, thereby proving that the global optimal solution of Problem P3 can be obtained by iterating Problem P4, which is consistent with the theoretical derivation.

[0158] Subsequently, execute the iterative process L2 in Step 5 to obtain the sequence of the objective function f(x (m) ,r (k-1) ) varying with the number of iterations m, as Figure 4 (a)-(b) shown. The stopping condition can be triggered after only three iterations, proving that the convergence speed of the algorithm is relatively fast. At the same time, as δ1 or δ2 increases, a larger feasible region is obtained, and the value of the objective function f(x (m) ,r (k-1) ) will also increase accordingly.

[0159] Finally, set different δ1 and δ2 to generate different feasible regions, execute the iterative process L1, and obtain the relationship between the value of the objective function f(x (k) ,r (k) ) and the number of iterations k as Figure 5 (a)-(b) shown. It can be seen that: as the number of iterations k increases, the objective function f(x, r) monotonically increases, and the objective function can converge to the limit point after only 3 or 4 iterations, indicating that the proposed solution method can solve the optimization problem At the same time, when the feasible region is increased, the overall objective function will increase, which is in line with the expectation.

[0160] 2. Verification of constraint conditions:

[0161] To ensure the resolution and sidelobe level of the CCF of the SAR waveform and the range - direction filter, constraint conditions for the main - lobe peak and sidelobe peak of the CCF are added, and a constant - modulus constraint is added to achieve high transmit power. To verify whether the above - mentioned constraint conditions can play the expected role, set different δ1 and δ2, and give the CCF between x and r, as Figure 6 (a)-(b) shown, and give the real and imaginary parts of the time - domain waveform as Figure 7 (a)-(b) shown.

[0162] It can be seen that by selecting an appropriate value of δ1, the main lobe height can be adjusted, and increasing or decreasing δ2 can also control the sidelobe level. Specifically, the larger δ1 is, the higher the main lobe peak is, and the smaller δ2 is, the lower the sidelobe level is, indicating that the constraint conditions for the CCF in this embodiment are effective and can achieve high resolution and low sidelobe level. In addition, all sampling points of the SAR waveform designed in this embodiment are located on the unit circles of the real and imaginary parts, indicating that the design result meets the condition of constant modulus value and is in line with expectations.

[0163] 3. Target information acquisition ability:

[0164] Fix δ1 = 370 and δ2 = 8, and solve the original problem through the algorithm proposed in this embodiment solution To verify whether this embodiment can effectively improve the information acquisition ability for SAR observed targets, it is compared with existing radar waveforms and filters, including the classical Chirp signal and the matched filter, the NLFM signal and the matched filter, the joint design scheme (both the waveform and the filter satisfy the similarity constraint), and the sub-optimal waveform and the matched filter. Different SCRs and SNRs are set, and the corresponding as Figure 8 (a)-(b) are shown.

[0165] It can be seen from this that the objective function value of the scheme proposed in this embodiment is better than other schemes. Compared with the Chirp signal, the objective function value can be maximally increased by 0.24. At the same time, the CCF and the modulus values of the time-domain waveforms of the above schemes are compared, as Figure 9 (a)-(b) are shown.

[0166] Since the bandwidths of the above waveforms and filters are similar, the main lobe widths of their CCFs are similar. However, the scheme proposed in this embodiment can obtain a peak sidelobe ratio of -16.65 dB, which is 3.3 dB lower than the sidelobe of the classical Chirp signal. In addition, compared with the waveforms of the joint design and the sub-optimal waveform with significantly fluctuating modulus values, the waveform designed in this embodiment is similar to the Chirp signal and the NLFM signal and has the property of constant modulus.

[0167] 4. Application in SAR:

[0168] To verify whether the method proposed in this embodiment can be applied to SAR, SAR images of typical targets are obtained by simulating a simple airborne SAR system. The basic parameters of the airborne SAR system are shown in Table 1.

[0169] Table 1

[0170]

[0171]

[0172] Select the B-2 bomber as the target to be measured, and establish an aircraft model based on FEKO as shown in Figure 10 . Obtain the radar cross section (RCS) of each frequency component of the target through Step 1 as the auxiliary knowledge for SAR joint design. Based on the real SAR image for verification, the SAR image of a certain airport is used as the backscattering coefficient to form the scene to be measured, and the scattering coefficient of the B-2 is generated through ISAR imaging. Assume that a B-2 is parked on the airport, the echo signal is obtained through the observation of the SAR system, and then the SAR image is generated through the RD imaging algorithm, as shown in Figure 11 (a)-(c).

[0173] Compared with the image of the chirp signal, the SAR image corresponding to this embodiment has a greater contrast between the B-2 bomber target and the background, has the best visual effect, and the contour of the aircraft target is clear, which will be more helpful for subsequent applications such as target detection and recognition of the SAR image.

[0174] In summary, the joint method of SAR waveform and range filter based on FEKO proposed in this embodiment gives a solution for obtaining auxiliary knowledge in radar waveform design, establishes an optimization problem for transmit / receive joint design, designs a targeted solution method, can effectively enhance the information acquisition ability of SAR for high-value targets, and at the same time can ensure the characteristics of constant modulus value, high resolution and low sidelobe of the SAR waveform.

[0175] The embodiments described above are only descriptions of the preferred embodiments of the present invention, and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of the present invention shall fall within the protection scope determined by the claims of the present invention.

Claims

1. A joint design method of SAR waveform and range filter based on FEKO, characterized in that Including: Step 1: Obtain the scattering characteristics of typical targets based on the FEKO platform; Step 2: Establish a signal model of the SAR one-dimensional range profile based on the scattering model and perform discretization; Step 3: Take the mutual information between the scattering characteristics of the target and the SAR one-dimensional range profile as the optimization criterion, and analyze and represent the objective function for the joint design of the SAR waveform and the range filter; Step 4: Impose constant modulus constraints and cross-correlation function constraints; the constant modulus constraint ensures that the SAR achieves the maximum transmission power, and the correlation function constraint ensures high range resolution and low sidelobe level in the range direction of the SAR. Combining the objective function and the constraints, construct a mathematical model for the joint design optimization problem of the SAR waveform and the range filter; Step 5: Separate variables based on the sequential iteration method to obtain a sub-optimization problem that only contains SAR waveform variables The range filter decouples the fractional objective function of the sub-optimization problem to form sub-problems and then transforms them into semidefinite programming problems Step 6, relaxation sub-problem For the rank-one constraint in * , add a penalty term to gradually reduce the rank of the semidefinite programming variable during the iteration. After the rank is reduced to 1, the solution X * is obtained. Perform eigenvalue decomposition on X , then the optimal solution of the sub-problem can be obtained. Continuously iterate the sub-problem Finally, the global optimal solution of the sub-optimization problem can be obtained; Step 7: Substitute the global optimal solution of the sub-problem into the original optimization problem to obtain a sub-optimization problem for the range filter Repeat Steps 5 - 6 until the global optimal solution of the sub-optimization problem is obtained; Step 8: Alternately solve the sub-optimization problems of the SAR waveform variables and the range filter sub-problem After the iterative process converges, obtain the solution result of the range filter of the SAR waveform.

2. The joint design method of SAR waveform and range filter based on FEKO according to claim 1, characterized in that Obtain the scattering characteristics of typical targets, including: Establish a target model based on the geometric shape and material parameter information of the target; Perform mesh division according to the details of the target model and the wavelength of the incident wave, and set the excitation source parameters to start the calculation task to obtain the electromagnetic scattering data of the target; Among them, the excitation source parameters include: incident direction, frequency, incident angle, and polarization mode.

3. The joint design method of SAR waveform and range filter based on FEKO according to claim 1, characterized in that Establish a signal model of the SAR one-dimensional range profile based on the scattering model, including: The SAR echo is processed by range filtering to obtain a one-dimensional range profile: Among them, is the sampling value of the SAR one-dimensional range profile at any moment, r is the range filter, and and respectively represent the Toeplitz matrices formed after discretizing the target scattering characteristic G(t) and the background scattering characteristic B(t) vectors, x is the discretized SAR transmit waveform, is the discretized noise sequence, (·) H represents the conjugate transpose of a matrix or vector.

4. The joint design method of SAR waveform and range filter based on FEKO according to claim 1, characterized in that The objective function representing the joint design of the SAR waveform and the range filter: Take the mutual information between the target scattering characteristics and the SAR one-dimensional range profile as the optimization criterion, and analyze and represent the objective function for the joint design of the SAR waveform and the range filter: Among them, The equivalent form of mutual information, E[·] is the mathematical expectation, (·) * Denotes the conjugate of a vector or scalar.

5. The joint design method of SAR waveform and range filter based on FEKO according to claim 1, characterized in that Construct the mathematical model for the joint design optimization problem of the SAR waveform and the range filter, including: Add a constant modulus constraint condition to the SAR waveform, and at the same time limit the cross-correlation function CCF between the SAR waveform and the range filter to have a narrow main lobe and a low peak sidelobe ratio, by forcing the peak energy of the CCF main lobe to be greater than the first preset threshold and forcing the peak energy of all sidelobes to be lower than the second preset threshold; on the premise of constant energy of the waveform and the filter, raise the main lobe of the CCF and reduce the sidelobes; Integrate the objective function and the constraints to construct a mathematical model for the joint design optimization problem of the SAR waveform and the range filter: where f(x,r) is obtained by ignoring the constant term in the objective function, and respectively represent the value sets of i and j, δ1 is the first preset threshold, δ2 is the second preset threshold, x(i) represents the i-th element of the waveform vector, J r is the Toeplitz matrix formed by the range filter r, W N is a matrix with all elements equal to 0 except for the element W(N,N)=1, will form the CCF, W j is a matrix with all elements equal to 0 except for the element W(j,j)=1, E r represents the energy of the range filter r.

6. The joint design method of SAR waveform and range filter based on FEKO according to claim 1, characterized in that Sub-optimization problem of isolating SAR waveform variables For the fractional objective function of is decoupled to form sub-problems and then transformed into a semidefinite programming problem Including: Separate the variables in the optimization problem using the sequential iteration method to establish the iteration process Set the range filter \(r = r^{(k)}\) at the \(k\) -th iteration (k-1) , and \(r^{(k - 1)}\) (k-1) is the value of \(r\) at the \((k - 1)\) -th iteration, thus obtaining a sub - optimization problem only about the SAR waveform: Among them, is the matrix generated by r (k-1) ​ Decouple the fractional objective function in to obtain the optimization problem x (m) , and start the iterative process Let m denote the number of iterations, then: where x ( m ) represents the decision variable of the waveform when iterated to the m-th time, F(x (m) ) = α(x (m) , r (k-1) ) - εβ(x (m) , r (k-1) ), ε = f(x (m-1) , r (k-1) ), and α(x (m) , r (k-1) ) and β(x (m) , r (k-1) ) are the numerator and denominator of f(x, r (k-1) ), respectively; Introduce the auxiliary variable X = x (m) x (m)H , and equivalently transform it into the SDP form where rank[·] represents the rank of a matrix, tr[·] represents the trace of a matrix, and Λ i denotes a matrix with all elements being 0 except for the element Λ(i,i) = 1, and X ± 0 indicates that X is positive semi - definite.

7. The joint design method of SAR waveform and range filter based on FEKO according to claim 6, characterized in that Obtain sub-problems with the globally optimal solutions, including: Relaxed subproblem For the rank-one constraint in , introduce a penalty term to gradually reduce X through iteration and start the iterative process The iterative optimization problem is as follows: where w > 1 is the penalty term, t represents the number of iterations, s (t) is the slack variable, V (t-1) is the matrix composed of the eigenvectors corresponding to N - 1 eigenvalues except the maximum eigenvalue when iterating to the (t - 1)-th step of X (t-1) , and I N-1 represents the (N - 1)-dimensional identity matrix. Set the stopping condition of the iteration as follows: Among them, δ is a very small positive number; When the stopping condition of the iterative process is satisfied, the global optimal solution X of the optimization problem * ; Decompose the solution X * into eigenvectors and set the iteration stopping condition: f(x (m) ,r (k-1) ) - f(x (m-1) ,r (k-1) ) ≤ e; where \(e\) represents the iteration process and \(\Delta\) represents the minimum change in the objective function value allowed between two adjacent iterations; When the stopping condition of the iterative process is satisfied, the global optimal solution of the sub-optimization problem will be obtained.

8. The joint design method of SAR waveform and range filter based on FEKO according to claim 1, wherein Obtain the global optimal solution of the sub-optimization problem of the range filter, including: ​ The problem of the global optimal solution x (k) , that is at the k-th iteration, substitute it into the objective function f(x, r) to obtain a sub-problem regarding the filter variable r: Among them, is x (k) the formed Toeplitz matrix; similarly, obtaining the global optimal solution of the range filter sub-optimization problem is: still using the Dinkelbach algorithm to decouple the problem of the objective function, then converting it into an equivalent SDP form, and finally gradually reducing the rank of the SDP variable through an iterative method until obtaining the global optimal solution.

9. The joint design method of the SAR waveform and the range filter based on FEKO according to claim 1, wherein Obtain the solution results of the SAR waveform range filter, including: Separate sub-problems of the SAR waveform and the filter sub-problem are alternately solved until the iterative process converges, and the stopping condition for setting is as follows: Among them, e′ is the minimum change in the objective function allowed for two adjacent iterations, and K is the maximum number of iterations; When the stop condition is not met, steps 5-7 are repeated, and finally the SAR waveform and the calculation results of the filter are output.

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