Robust waveform design method for extended targets in bandwidth-constrained environments for SAR

Abstract

This invention discloses a robust SAR waveform design method for extended targets under bandwidth-constrained environments, belonging to the field of signal processing. The method includes: S1, selecting signal-to-clutter ratio (SCR) as the optimization criterion, establishing an error model with auxiliary knowledge, applying multiple constraints, and constructing an optimization problem for robust SAR waveform design; S2, solving the inner-layer minimization problem to derive closed-form solutions for the target and background scattering characteristic statistics; S3, constructing a sub-optimization problem related to the waveform, decoupling the numerator and denominator of the objective function; S4, transforming the sub-problems in the iteration process into semidefinite programming problems; S5, determining whether the stopping condition is met; if yes, proceeding to S6; otherwise, continuing iteration; S6, determining whether the optimality condition is met; if yes, the solution process ends; otherwise, using semidefinite relaxation and rank-one decomposition to solve the sub-problems obtained in S3 until the stopping condition is met. This invention can significantly improve the SCR of SAR images and effectively highlight the scattering characteristics of extended targets.

Description

Technical Field

[0001] This invention relates to the field of signal processing technology, and in particular to a robust waveform design method for SAR targeting extended targets in bandwidth-constrained environments. Background Technology

[0002] Synthetic Aperture Radar (SAR) is an active microwave imaging system that, with its advantages of all-weather, all-day operation and high resolution, is widely used in various fields such as national defense and civilian applications. The performance of SAR waveforms plays a significant role in improving image quality, and the optimization design of SAR waveforms has always been one of the main themes in this research field.

[0003] Existing SAR waveforms possess large bandwidth characteristics, and after matched filtering, correlation functions with narrow main lobes and low side lobes can be formed, thus ensuring the resolution and detection performance of point targets. Today, with the significant improvement in SAR spatial resolution, SAR images can capture more details, causing most high-value targets to appear as extended targets in images, thus creating conditions for refined perception of extended targets. However, since extended targets cannot be considered as a linear superposition of point targets, point target models cannot accurately describe the characteristics of extended targets. This leads to a mismatch between existing SAR waveforms for point target detection and the application requirements for refined perception of extended targets, resulting in poor refined perception performance of extended targets, especially for faint targets. Therefore, there is an urgent need to optimize the design of SAR waveforms based on extended target scattering models.

[0004] Although research on radar waveform design for extended targets is nothing new, the design process requires the incorporation of auxiliary knowledge about the target. However, the problem of performance degradation caused by inaccurate estimation of this auxiliary knowledge has not been fully resolved. Meanwhile, SAR waveforms typically require large bandwidth to ensure high resolution, and the coexistence of broadband systems with other radio frequency equipment faces challenges in environments with limited spectrum resources.

[0005] Therefore, designing SAR waveforms with auxiliary knowledge error tolerance, compatibility with other radio frequency devices, and high resolution and low sidelobes based on extended target models has important scientific significance and application value. Summary of the Invention

[0006] To improve the imaging and observation performance of extended targets of interest by SAR, this invention proposes a robust waveform design method for SAR in bandwidth-constrained environments. This method can relax the estimation accuracy of auxiliary knowledge, achieve coexistence with other radio frequency equipment in the observation scenario, and at the same time ensure the high resolution and low sidelobe characteristics of SAR in the range direction.

[0007] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0008] A robust waveform design method for SAR targeting extended targets in a bandwidth-constrained environment includes the following steps:

[0009] S1. Select the signal-to-clutter ratio in the SAR echo domain as the optimization criterion, construct an error model of auxiliary knowledge, apply spectral constraints, energy constraints and similarity constraints, and construct an optimization problem for SAR robust waveform design.

[0010] S2, solve the minimization problem of the inner layer, and derive the closed-form solution of the statistical quantities of target scattering characteristics and background scattering characteristics;

[0011] S3, construct a sub-optimization problem about the waveform, and use the Dinkelbach algorithm to decouple the numerator and denominator of the fractional objective function;

[0012] S4, based on Lagrange duality and Schul complement theorem, transforms the subproblems in the iterative process into semidefinite programming problems;

[0013] S5: Determine if the stopping condition is met. If yes, proceed to S6; otherwise, continue iterating.

[0014] S6. Determine whether the optimality condition is met. If yes, the solution process ends. If not, use semidefinite relaxation and rank-one decomposition algorithms to solve the subproblems obtained in S3 until the stopping condition is met, and output the solution results.

[0015] A further improvement to the technical solution of the present invention is that S1 specifically includes the following:

[0016] Based on the extended target scattering model to characterize the signal-to-clutter ratio (SCR) in the SAR echo domain; using x(t) to represent the radar transmitted signal, G(t) and B(t) to represent the target scattering characteristics and background scattering characteristics, respectively, and N(t) to represent noise, the received echo signal Y(t) at any azimuth and time is expressed as:

[0017]

[0018] By constructing a Toeplitz matrix multiplication to implement the convolution operator, the discrete signal model is:

[0019] y = Gx + Bx + n (2)

[0020] In the formula, G and B represent the Toeplitz matrices formed after discretizing G(t) and B(t), respectively. Let R represent the discretized vectors of x(t), Y(t), and N(t), respectively; G and R B Let G and B be the correlation matrices, respectively. Then the SCR of the SAR range direction is expressed as:

[0021]

[0022] In the formula, R G and R B It is supplementary knowledge estimated from historical data, empirical models, and electromagnetic measurement methods; therefore, its error model is established as follows:

[0023]

[0024] In the formula, and For an ideal, error-free correlation matrix, ε G and ε B This is used to limit the error range; to improve the robustness of the optimized waveform against auxiliary knowledge errors, the objective function is to maximize the SCR corresponding to the worst auxiliary knowledge, i.e.:

[0025]

[0026] To ensure the waveform can coexist with other RF systems, spectral constraints are applied:

[0027] x H R I x≤E I (6)

[0028] In the formula, E I This is the energy threshold that other radio frequency devices within the frequency band can tolerate, and γ q ≥0 indicates the weight assigned to the interfering device, Ω q The (m,l) elements are:

[0029]

[0030] In the formula, f1 q and Let x represent the frequency band range of the q-th radio frequency device; to ensure that x also has high resolution characteristics, a similarity constraint is introduced, and a maximization-minimization problem is established.

[0031]

[0032] In the formula, E x ε represents the energy of the maximum transmitted waveform, and ε is used to limit the similarity between the transmitted waveform and the reference waveform c.

[0033] A further improvement to the technical solution of the present invention is that, in S2, it specifically includes:

[0034] Introduce two auxiliary variables ΔR G , Make The minimization problem of the inner layer is equivalent to:

[0035]

[0036] The objective function in the above equation is equivalently transformed into:

[0037]

[0038] Based on this, we obtain the following information regarding ΔR. G and ΔR B The linear programming problem, namely:

[0039]

[0040] In the formula, I is an all-one matrix; the above linear programming problem has a closed-form solution, i.e. and

[0041] A further improvement to the technical solution of this invention lies in: S3 specifically includes: forming an optimization problem about x based on the closed-form solution obtained from auxiliary knowledge.

[0042]

[0043] The Dinkelbach algorithm is used to decouple the numerator and denominator of the fractional objective function through an iterative process. Assuming the iteration reaches k steps, the following quadratic constrained quadratic programming problem is obtained:

[0044]

[0045] Through iteration Able to form a monotonically increasing sequence of objective functions Furthermore, if each iteration can obtain... The global optimal solution can be obtained when the iteration stops. The global optimal solution.

[0046] A further improvement to the technical solution of this invention lies in the fact that, in S4, it specifically includes: constructing a sub-problem. To solve the dual problem, we first write out the Lagrange function, which is:

[0047]

[0048] In the formula, A1 = A2 = I, A3 = R I , b0=b1=b3=0, b2=-c, c0=0, c1=-E x c3 = -E I , Therefore, the dual problem is derived:

[0049]

[0050] In the formula, Let A(λ) represent the range space of matrix A, and

[0051]

[0052] In the formula, A(λ) (1) Denotes the pseudo-inverse matrix of A(λ);

[0053] Based on Schur complement theorem The equivalent transformation is to a semidefinite programming problem:

[0054]

[0055] Solving for x (k) =-A(λ) * ) (1) b(λ * ), For convex problems, the interior point method is used to solve them in polynomial time.

[0056] A further improvement to the technical solution of this invention lies in: in S5, the stopping condition for the iteration process is set as follows:

[0057] f(x (k) )-f(x (k-1) )≤e (18)

[0058] In the formula, e represents the minimum allowable change of the objective function f(x) in two consecutive iterations; if the stopping condition is met, the iteration terminates.

[0059] A further improvement to the technical solution of the present invention is that, in S6, it specifically includes:

[0060] Based on the complementary relaxation theorem, a sufficient condition for global optimality is derived as follows: Let λ be the solution when the stopping condition is met; if the optimality condition is met, it indicates that the duality gap is 0, and the solution can be obtained in each iteration. The global optimal solution; at this point, it can be proven that when the stopping condition is met, if e is arbitrarily small, then x at the time of iteration stopping... (k) for The global optimal solution; if If the conditions are not met, then solve using an algorithm based on SDR. At this point, it requires obtaining the result at the cost of greater computational complexity. The global optimal solution; The SDR form is:

[0061]

[0062] In the formula, This is a convex problem; the solution yields... Then, for X * To perform rank-one factorization, we introduce the following rank-one factorization theorem:

[0063] Suppose X * The dimension is greater than or equal to 3, and B1, B2, B3, and B4 are related to X. * Hermitian matrices of the same dimension, X * Let rank[X] be a positive semidefinite matrix to be decomposed. * If ] = r, then:

[0064] 1) If r≥3, can we find a range space to which a nonzero vector s belongs in polynomial time? Make

[0065] s H A i s = tr[B i X], i=1,2,3,4 (20)

[0066] 2) If r = 2, for any Not belonging to There exists a vector s belonging to space Make

[0067] s H A i s = tr[B i X], i=1,2,3,4 (21)

[0068] 3) If r = 1, then eigenvalue decomposition can be performed directly;

[0069] According to the rank-one factorization theorem, let B2 = E, B4 = R I Therefore, X * It can achieve rank-one factorization in polynomial time, and also guarantee... Achieve global optimum.

[0070] The technological advancements achieved by this invention due to the adoption of the above technical solutions are as follows:

[0071] 1. This invention allows multiple radio frequency devices to coexist in a bandwidth-limited environment, and can significantly improve the SCR of extended targets, thereby improving the performance of fine-grained sensing of extended targets.

[0072] 2. This invention relaxes the estimation accuracy of auxiliary knowledge, creating conditions for the transformation of SAR waveform design from theoretical demonstration to engineering implementation.

[0073] 3. The solution algorithm proposed in this invention has global convergence, and the solution method based on Lagrange duality has lower time complexity than the SDR algorithm, saving running time and making it more suitable for large-scale waveform design applications. Attached Figure Description

[0074] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0075] Figure 1 This is a flowchart of a robust SAR waveform design method for extended targets under bandwidth-constrained environments provided by the present invention;

[0076] Figure 2 This is an algorithm flowchart of a robust waveform design method for extended targets under bandwidth-constrained environments provided in this embodiment of the invention.

[0077] Figure 3(a) shows the sequence of objective functions corresponding to different parameters ε. Line chart;

[0078] Figure 3(b) shows different parameters E I The corresponding objective function sequence Line chart;

[0079] Figure 4 Indicate different R G and R B Schematic diagrams of the corresponding robust and non-robust objective functions;

[0080] Figure 5(a) shows the values ​​corresponding to different ε. The minimum eigenvalue curve;

[0081] Figure 5(b) shows different E I corresponding The minimum eigenvalue curve;

[0082] Figure 6 It is a different E I Energy spectral density (ESD) curve of the optimized waveform;

[0083] Figure 7 This compares the solutions based on Lagrange duality with those based on the SDR algorithm. Time complexity comparison chart;

[0084] Figure 8(a) shows a SAR image generated by the algorithm proposed in this invention;

[0085] Figure 8(b) shows the SAR image generated from the suboptimal waveform;

[0086] Figure 8(c) shows the SAR image generated from the Chirp signal;

[0087] Figure 8(d) is a SAR image generated from the NLFM waveform;

[0088] Figure 9 This is a diagram illustrating the target detection probability and false alarm probability. Detailed Implementation

[0089] It should be noted that the terms "comprising" and "having" and any variations thereof in the specification, claims and accompanying drawings of this invention are intended to cover non-exclusive inclusion. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are explicitly listed, but may include other steps or units that are not explicitly listed or that are inherent to such processes, methods, products or devices.

[0090] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments:

[0091] like Figure 1-2 As shown, a robust waveform design method for SAR targeting extended targets in a bandwidth-constrained environment can relax the estimation accuracy of auxiliary knowledge, enabling coexistence with other radio frequency devices in the observation scenario, while ensuring high resolution and low sidelobe characteristics in the SAR range direction. The method specifically includes the following steps:

[0092] S1. Select the signal-to-clutter ratio (SCR) in the SAR echo domain as the optimization criterion, construct an error model of auxiliary knowledge, and apply spectral constraints, energy constraints and similarity constraints to construct an optimization problem for SAR robust waveform design.

[0093] Specifically, this includes: SCR (Scattering Reflection) in the SAR echo domain based on an extended target scattering model. Let x(t) represent the radar transmitted signal, G(t) and B(t) represent the target scattering characteristics and background scattering characteristics, respectively, and N(t) represent noise. Then, the received echo signal Y(t) at any azimuth and time is expressed as:

[0094]

[0095] By constructing a Toeplitz matrix multiplication to implement the convolution operator, the discrete signal model is:

[0096] y = Gx + Bx + n (2)

[0097] In the formula, G and B represent the Toeplitz matrices formed after discretizing G(t) and B(t), respectively. Let R represent the discretized vectors of x(t), Y(t), and N(t), respectively. G and R B Let G and B be the correlation matrices, respectively. Then the SCR of the SAR range direction is expressed as:

[0098]

[0099] In the formula, R G and R B This error model is typically derived from auxiliary knowledge estimated using historical data, empirical models, electromagnetic measurements, etc., and is therefore established as follows:

[0100]

[0101] In the formula, and For an ideal, error-free correlation matrix, ε G and ε B This is used to limit the error range. To improve the robustness of the optimized waveform to auxiliary knowledge errors, the objective function is to maximize the SCR corresponding to the worst auxiliary knowledge, i.e.:

[0102]

[0103] To ensure the waveform can coexist with other RF systems, spectral constraints are applied:

[0104] x H R I x≤E I (6)

[0105] In the formula, E I This is the energy threshold that other radio frequency devices within the frequency band can tolerate, and γ q ≥0 indicates the weight assigned to the interfering device, Ω q The (m,l) elements are:

[0106]

[0107] In the formula, f1 q and Let x represent the frequency band range of the q-th radio frequency device; to ensure that x also has high resolution characteristics, a similarity constraint is introduced, and a maximization-minimization problem is established.

[0108]

[0109] In the formula, E x ε represents the energy of the maximum transmitted waveform, and ε is used to limit the similarity between the transmitted waveform and the reference waveform c.

[0110] S2, solve the minimization problem of the inner layer, and derive the closed-form solution of the statistical quantities of target scattering characteristics and background scattering characteristics;

[0111] Specifically, this includes: introducing two auxiliary variables ΔR. G , Make The minimization problem of the inner layer is equivalent to:

[0112]

[0113] The objective function in the above equation can be equivalently transformed into:

[0114]

[0115] Based on this, we can obtain the following information about ΔR. G and ΔR B The linear programming (LP) problem, namely:

[0116]

[0117] In the formula, I is an all-one matrix; the above linear programming problem has a closed-form solution, i.e. and

[0118] S3, construct a sub-optimization problem about the waveform, and use the Dinkelbach algorithm to decouple the numerator and denominator of the fractional objective function;

[0119] Specifically, this includes: formulating an optimization problem about x based on closed-form solutions derived from auxiliary knowledge.

[0120]

[0121] The Dinkelbach algorithm is used to decouple the numerator and denominator of the fractional objective function through an iterative process. Assuming k iterations, the following quadratic constrained quadratic programming (QCQP) problem is obtained:

[0122]

[0123] Possible proof: through iteration Able to form a monotonically increasing sequence of objective functions Furthermore, if each iteration can obtain... The global optimal solution can be obtained when the iteration stops. The global optimal solution.

[0124] S4, based on Lagrange duality and Schul complement theorem, transforms the subproblems in the iterative process into semidefinite programming (SDP) problems;

[0125] Specifically, this includes: constructing subproblems To solve the dual problem, we first write out the Lagrange function, which is:

[0126]

[0127] In the formula, A1 = A2 = I, A3 = R I , b0=b1=b3=0, b2=-c, c0=0, c1=-E x c3 = -E I , Therefore, the dual problem can be derived:

[0128]

[0129] In the formula, Let A(λ) represent the range space of matrix A, and

[0130]

[0131] In the formula, A(λ) (1) Denotes the pseudo-inverse matrix of A(λ);

[0132] Based on Schur complement theorem The equivalent transformation is to a semidefinite programming (SDP) problem:

[0133]

[0134] Solving for x (k) =-A(λ) * ) (1) b(λ * ), For convex problems, the interior point method is used to solve them in polynomial time.

[0135] S5: Determine if the stopping condition is met. If yes, proceed to S6; otherwise, continue iterating.

[0136] Specifically, it includes:

[0137] Set the stopping condition for the iteration process as follows:

[0138] f(x (k) )-f(x (k-1) )≤e (18)

[0139] In the formula, e represents the minimum allowable change of the objective function f(x) in two consecutive iterations; if the stopping condition is met, the iteration terminates.

[0140] S6. Determine whether the optimality condition is met. If yes, the solution process ends. If not, use the semidefinite relaxation (SDR) and rank-one decomposition algorithm to solve the subproblem obtained in S3 until the stopping condition is met, and output the solution result.

[0141] Specifically, based on the complementary relaxation theorem, a sufficient condition for global optimality can be derived as follows: This is the solution for λ when the stopping condition is met. If the optimality condition is met, it indicates that the duality gap is 0, and the solution can be obtained in each iteration. The global optimal solution. It can then be proven that when the stopping condition is met, if e is arbitrarily small, then x at the time of iteration stopping... (k) for The globally optimal solution. If If the conditions are not met, then solve using an algorithm based on SDR. At this point, it requires obtaining the result at the cost of greater computational complexity. The global optimal solution. The SDR form is:

[0142]

[0143] In the formula, This is a convex problem; the solution yields... Then, for X * To perform rank-one factorization, we introduce the following rank-one factorization theorem:

[0144] Suppose X * The dimension is greater than or equal to 3, and B1, B2, B3, and B4 are related to X. * Hermitian matrices of the same dimension, X * Let rank[X] be a positive semidefinite matrix to be decomposed. * If ] = r, then:

[0145] 1) If r≥3, a range space to which a nonzero vector s belongs can be found in polynomial time. Make

[0146] s H A i s = tr[B i X], i=1,2,3,4 (20)

[0147] 2) If r = 2, for any Not belonging to There exists a vector s belonging to space Make

[0148] s H A i s = tr[B iX],i=1,2,3,4 (21)

[0149] 3) If r = 1, then eigenvalue decomposition can be performed directly.

[0150] According to the rank-one factorization theorem, let B2 = E, B4 = R I Therefore, X * It can achieve rank-one factorization in polynomial time, and can also guarantee... Achieve global optimum.

[0151] Example

[0152] A. Theoretical Derivation and Verification

[0153] The theoretical derivation involved in this invention is verified through numerical experiments, including the convergence of the proposed algorithm, the robustness of the waveform to auxiliary knowledge errors, the optimality conditions, and the effectiveness of the constraints.

[0154] 1) Convergence of the proposed algorithm

[0155] Set N=120, e=10 -4 E x =E0=N, The reference signal x0 is the Chirp signal. By setting different ε or E... I Obtain different feasible domains and construct Solve the problem Obtain the initial iteration point x (0) Start the solution process The iterative steps.

[0156]

[0157] Calculate the objective function value for each iteration, such as Figures 3(a)-3(b) As shown in the figure, the objective function sequence is monotonically increasing, indicating that the solution algorithm has convergence. f(x) (k) After three iterations, the algorithm stabilizes and triggers the stopping condition, indicating that the proposed algorithm converges quickly. Furthermore, by increasing the parameters ε or E... I This expands the feasible region, thereby increasing the objective function value, which is consistent with expectations.

[0158] 2) Waveform robustness

[0159] To evaluate the robustness of the optimized waveform to auxiliary knowledge errors, calculations were performed for different R... G and R B The objective function values ​​of the samples. Randomly select 100 different R values. G and R BThe proposed algorithm is used to calculate the objective function value of robust waveforms. Simultaneously, the proposed algorithm is used to design non-robust waveforms, i.e., waveforms that do not consider R0. G and R B Uncertainty, for different R G and R B The objective function value of the sample is as follows Figure 4 As shown. Since optimizing the waveform involves maximizing R... G and R B Robustness is achieved by taking the worst-case f(x) on the uncertainty set, thus the objective function value of the optimized waveform remains at a high level, and the objective function value corresponding to different samples is relatively stable. Furthermore, the objective function value of the robust waveform in the worst-case scenario is significantly higher than that of the non-robust waveform, further demonstrating that the SAR robust waveform design method for extended targets under bandwidth-constrained environments provided by this invention can guarantee good robustness of the optimized waveform.

[0160] 3) Optimality condition

[0161] Whether the global optimal solution to the waveform can be obtained by solving the Lagrange dual problem, thereby achieving global convergence with lower computational complexity, requires verification of the optimality condition, i.e., verification. It is equivalent to eig min [·] represents the smallest eigenvalue of the matrix. Changing the parameters ε and E... I The value of , constructing different feasible regions, and giving The smallest eigenvalue, such as Figures 5(a)-5(b) As shown in the figure. It can be seen from the figure that regardless of the feasible region, If the condition is always true, it means that the optimal conditions are met and global convergence can be achieved with low computational complexity.

[0162] 4) Band compatibility

[0163] To verify the frequency band compatibility of the optimized waveform, assume two RF devices operate in the normalized frequency bands Ω1 = [0.27, 0.37] and Ω2 = [0.27, 0.37] respectively, and the radar system coexists with them in the frequency band [0, 1]. Different E values ​​are set... I To limit the energy distribution of the waveform in the stopband, the energy spectral density (ESD) of the optimized waveform is given, such as... Figure 6 As shown. Here, the ESD of the reference signal x0 is used as the benchmark. It can be seen that, through spectral constraints, the ESD of the optimized waveform can be significantly attenuated within the stopband, preventing transmitted energy from mixing into the operating frequency bands of other devices, thus avoiding resource waste and mutual interference. Moreover, if E I The smaller the value, the less energy is distributed in the stopband, which is consistent with expectations and indicates that the waveform designed in this invention allows multiple radio frequency devices to work simultaneously in a bandwidth-constrained environment.

[0164] 5) Computational complexity

[0165] As mentioned earlier, although the solution can also be obtained using an SDR-based algorithm... However, this increases time and memory overhead. To compare the complexity of Lagrange duality-based and SDR-based solution methods, different numbers of sampling points are set for the waveforms, and the CPU runtime of the two algorithms is given, such as... Figure 7 As shown, Figure 7 This indicates that the solution algorithm based on Lagrange duality has significantly lower complexity, especially when the number of waveform sampling points is large. This algorithm can greatly reduce CPU running time, which is in line with expectations.

[0166] B. Practical Applications of SAR

[0167] To verify whether waveform optimization can improve the imaging quality of extended targets by SAR, a simplified airborne SAR system was simulated to acquire SAR images of typical extended targets, and performance analysis was conducted. The main parameters of the SAR system are shown in Table 1.

[0168] Table 1 Main parameters of the SAR system

[0169]

[0170]

[0171] Existing SAR waveforms are introduced for performance comparison, including suboptimal waveforms, chirp signals, and NLFM signals. First, ensuring that the different waveforms have the same transmission energy and bandwidth, a Boeing 737 passenger aircraft is used as the target. The target impulse response G(t) and background scattering characteristics B(t) are filled into a specified area to generate the scattering characteristics of the observation scene. Echo signals are obtained through SAR system observation, and then SAR images are generated using the RD imaging algorithm, such as... Figures 8(a)-8(d) As shown, compared to other images, the SAR image corresponding to this invention presents a better rendering effect for extended targets, exhibiting greater contrast between the target and the background. Because this invention uses SCR as the objective function, the optimized waveform achieves a 3.5dB improvement in SCR compared to the Chirp signal.

[0172] Since SCR enhancement can improve target detection performance, target detection experiments were conducted using the obtained SAR images. The amplitude of the SAR image signal was sequentially iterated and set as a detection threshold; a binary image was generated based on each threshold, and the detection probability P was calculated. D And the false alarm probability P FA ,like Figure 9 As shown.

[0173] Therefore, increasing the SCR improves SAR image quality, leading to more ideal target detection results. This embodiment verifies that the waveform proposed in this invention has better imaging and observation capabilities for extended targets under bandwidth-limited environments.

[0174] In summary, this invention applies energy constraints, similarity constraints, and spectral constraints, maximizing the signal-to-clutter ratio (SCR) corresponding to the worst-case auxiliary knowledge, thus highlighting the scattering characteristics of extended targets and improving the performance of refined target perception in SAR images. It derives closed-form solutions for the statistical quantities of target and background scattering characteristics, transforming the minimax problem into a non-convex quadratic constrained quadratic programming problem. Using the Dinkelbach algorithm and constructing a dual problem, it decomposes the non-convex quadratic constrained quadratic programming problem into a series of solvable semidefinite programming problems. It provides sufficient conditions for global convergence and, when these conditions are not met, provides supplementary solution methods to obtain the global optimal solution with greater computational complexity. This invention can improve the SCR of extended targets in spectrally constrained environments while relaxing the estimation accuracy of auxiliary knowledge, laying the foundation for the engineering application of optimized waveforms.

[0175] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A robust waveform design method for SAR targeting extended targets under bandwidth-constrained environments, characterized in that, Includes the following steps: S1. Select the signal-to-clutter ratio in the SAR echo domain as the optimization criterion, construct an error model of auxiliary knowledge, apply spectral constraints, energy constraints and similarity constraints, and construct an optimization problem for SAR robust waveform design. S2, solve the minimization problem of the inner layer, and derive the closed-form solution of the statistical quantities of target scattering characteristics and background scattering characteristics; S3, construct a sub-optimization problem about the waveform, and use the Dinkelbach algorithm to decouple the numerator and denominator of the fractional objective function; S4, based on Lagrange duality and Schul complement theorem, transforms the subproblems in the iterative process into semidefinite programming problems; S5: Determine if the stopping condition is met. If yes, proceed to S6; otherwise, continue iterating. S6. Determine whether the optimality condition is met. If yes, the solution process ends. If not, use semidefinite relaxation and rank-one decomposition algorithms to solve the subproblems obtained in S3 until the stopping condition is met, and output the solution results.

2. The robust waveform design method for extended targets under bandwidth-constrained environments according to claim 1, characterized in that, S1 specifically includes the following: Based on the extended target scattering model to characterize the signal-to-clutter ratio (SCR) in the SAR echo domain; using x(t) to represent the radar transmitted signal, G(t) and B(t) to represent the target scattering characteristics and background scattering characteristics, respectively, and N(t) to represent noise, the received echo signal Y(t) at any azimuth and time is expressed as: By constructing a Toeplitz matrix multiplication to implement the convolution operator, the discrete signal model is: y = Gx + Bx + n (2) In the formula, G and B represent the Toeplitz matrices formed after discretizing G(t) and B(t), respectively. Let R represent the discretized vectors of x(t), Y(t), and N(t), respectively; G and R B Let G and B be the correlation matrices, respectively. Then the SCR of the SAR range direction is expressed as: In the formula, R G and R B It is supplementary knowledge estimated from historical data, empirical models, and electromagnetic measurement methods; therefore, its error model is established as follows: In the formula, and For an ideal, error-free correlation matrix, ε G and ε B This is used to limit the error range; to improve the robustness of the optimized waveform against auxiliary knowledge errors, the objective function is to maximize the SCR corresponding to the worst auxiliary knowledge, i.e.: To ensure the waveform can coexist with other RF systems, spectral constraints are applied: In the formula, E I This is the energy threshold that other radio frequency devices within the frequency band can tolerate, and Ω represents the weight assigned to the interfering device. q The (m,l) elements are: In the formula, f1 q and Let x represent the frequency band range of the q-th radio frequency device; to ensure that x also has high resolution characteristics, a similarity constraint is introduced, and a maximization-minimization problem is established. In the formula, E x ε represents the energy of the maximum transmitted waveform, and ε is used to limit the similarity between the transmitted waveform and the reference waveform c.

3. The robust waveform design method for extended targets under bandwidth-constrained environments according to claim 2, characterized in that, Specifically, S2 includes: Introduce two auxiliary variables Make The minimization problem of the inner layer is equivalent to: The objective function in the above equation is equivalently transformed into: Based on this, we obtain the following information regarding ΔR. G and ΔR B The linear programming problem, namely: In the formula, I is an all-one matrix; the above linear programming problem has a closed-form solution, i.e. and 4. The robust waveform design method for extended targets under bandwidth-constrained environments according to claim 3, characterized in that, In S3, specifically, it includes: forming an optimization problem about x based on the closed-form solution obtained from auxiliary knowledge: The Dinkelbach algorithm is used to decouple the numerator and denominator of the fractional objective function through an iterative process. Assuming the iteration reaches k steps, the following quadratic constrained quadratic programming problem is obtained: Through iteration Able to form a monotonically increasing sequence of objective functions Furthermore, if each iteration can obtain... The global optimal solution can be obtained when the iteration stops. The global optimal solution.

5. The robust waveform design method for extended targets under bandwidth-constrained environments according to claim 4, characterized in that, In S4, this specifically includes: the construction of subproblems. To solve the dual problem, we first write out the Lagrange function, which is: In the formula, A1 = A2 = I, A3 = R I , b0=b1=b3=0, b2=-c, c0=0, c1=-E x c3 = -E I , Therefore, the dual problem is derived: In the formula, Let A(λ) represent the range space of matrix A, and In the formula, A(λ) (1) Denotes the pseudo-inverse matrix of A(λ); Based on Schur complement theorem The equivalent transformation is to a semidefinite programming problem: Solving For convex problems, the interior point method is used to solve them in polynomial time.

6. The robust waveform design method for extended targets under bandwidth-constrained environments according to claim 5, characterized in that, In S5, the stopping condition for the iteration process is set as follows: f(x (k) )-f(x (k-1) )≤e (18) In the formula, e represents the minimum allowable change of the objective function f(x) in two consecutive iterations; if the stopping condition is met, the iteration terminates.

7. A robust waveform design method for extended targets under bandwidth-constrained environments according to claim 5 or 6, characterized in that, In S6, specifically including: Based on the complementary relaxation theorem, a sufficient condition for global optimality is derived as follows: Let λ be the solution when the stopping condition is met; if the optimality condition is met, it indicates that the duality gap is 0, and the solution can be obtained in each iteration. The global optimal solution; at this point, it can be proven that when the stopping condition is met, if x is arbitrarily small, then x at the time of iteration stops. (k) for The global optimal solution; if If the conditions are not met, then solve using an algorithm based on SDR. At this point, it requires obtaining the result at the cost of greater computational complexity. The global optimal solution; The SDR form is: In the formula, This is a convex problem; the solution yields... Then, for X * To perform rank-one factorization, we introduce the following rank-one factorization theorem: Suppose X * The dimension is greater than or equal to 3, and B1, B2, B3, and B4 are related to X. * Hermitian matrices of the same dimension, X * Let rank[X] be a positive semidefinite matrix to be decomposed. * If ] = r, then: 1) If r≥3, can we find a range space to which a nonzero vector s belongs in polynomial time? Make s H A i s=tr[B i X], i=1,2,3,4 (20) 2) If r = 2, for any Not belonging to There exists a vector s belonging to space Make s H A i s=tr[B i X], i=1,2,3,4 (21) 3) If r = 1, then eigenvalue decomposition can be performed directly; According to the rank-one factorization theorem, let B4 = R I Therefore, X * It can achieve rank-one factorization in polynomial time, and also guarantee... Achieve global optimum.

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