A complementary transmit waveform design method for realizing SAR target enhancement

Abstract

The application discloses a complementary transmission waveform design method for realizing SAR target enhancement, first utilizes prior target and clutter power spectrum density to model transmission waveform design as a non-convex constraint optimization problem, then utilizes an inexact alternating direction penalty method to decompose non-convex constraint problem solving into two sub-problems, utilizes a spectrum constraint complementary sequence and a maximum-minimization framework to solve the two sub-problems, and finally obtains a transmission waveform capable of meeting SAR imaging performance and improving signal-to-clutter ratio. The method of the application obtains the SAR transmission waveform by utilizing environmental prior information and a spectrum complementary constraint sequence, adopts a complementary waveform set design method to design and optimize the whole transmission waveform set, so as to realize the improvement of the signal-to-clutter ratio of the imaging result while guaranteeing the SAR imaging performance, effectively improve the target signal-to-clutter ratio under the condition of meeting the imaging performance index, and realize the enhancement of the target and the weakening of the clutter.

Description

Technical Field

[0001] This invention belongs to the field of radar technology, specifically relating to a complementary transmission waveform design method for SAR target enhancement. Background Technology

[0002] Synthetic Aperture Radar (SAR) is an all-weather, all-day, high-resolution imaging system, and acquiring target information is one of the main purposes of SAR imaging. However, clutter caused by complex geographical environments can obscure targets, making target detection difficult. Existing SAR transmission waveforms and imaging methods treat targets and clutter equally, employing uniform waveforms and processing methods, making it difficult for targets to stand out from background clutter. Radar waveforms, as an important controllable resource, have the potential to enhance target imaging by utilizing the differences between clutter and targets and designing waveforms that match targets and mismatch clutter.

[0003] The paper "Optimization of the receive filter and transmit sequence for active sensing, IEEE Transactions on Signal Processing, vol.60, no.4, pp.1730-1740, 2012" optimizes the transmit sequence and receive filter under peak-to-average power ratio (PAR) constraints to minimize the mean square error (MSE) between the target scattering coefficient and the estimation under clutter and interference. However, this method is mainly for radar target detection tasks, not target imaging, so it cannot be directly applied to SAR.

[0004] The paper "Cognitive SAR waveform design method based on joint optimization criteria, 2019 6th Asia-Pacific Conference on Synthetic Aperture Radar (APSAR), 2019, pp.1-4" proposes a frequency domain amplitude design method for the transmitted waveform that can maximize the signal-to-clutter-to-noise ratio using prior environmental information. However, this method does not consider the constant mode constraint of the transmitted waveform in the time domain and therefore cannot be directly applied to SAR systems.

[0005] The waveform design methods described above can only optimize a single waveform and cannot improve the signal-to-clutter ratio of the imaging results while ensuring SAR imaging performance. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a complementary transmit waveform design method for SAR target enhancement, which solves the problem that other waveform design methods cannot effectively balance improving the signal-to-clutter ratio and ensuring SAR imaging performance.

[0007] The technical solution adopted in this invention is: a complementary transmit waveform design method for SAR target enhancement, the specific steps of which are as follows:

[0008] S1. Obtain the target power spectral density H(f) and clutter power spectral density P in the imaging scene based on known prior information. c (f);

[0009] Wherein, the known prior information refers to the target area and clutter area in the imaging scene, and f represents the frequency.

[0010] S2. Using the target power spectral density H(f) and clutter power spectral density P obtained in step S1 c (f) Establishing a waveform optimization problem to improve signal-to-noise ratio;

[0011] S3. Solve the optimization problem obtained in step S2 to obtain a convergent sidelobe complementary waveform that can improve the signal-to-clutter ratio of SAR.

[0012] Furthermore, step S2 is specifically as follows:

[0013] Let the transmitted waveform be x(t), where t represents a fast time variable. Then, the expression for the transmitted waveform spectrum constraint to achieve the maximum signal-to-clutter-to-noise ratio is as follows:

[0014]

[0015] Where X(f) represents the Fourier transform of x(t); P n (f) represents the power spectral density of the ambient noise, which is treated as Gaussian white noise, i.e., P n (f)=1,λ x This represents the Lagrange multiplier.

[0016] Since equation (1) cannot be directly used as the amplitude of the transmitted waveform spectrum, a joint optimization strategy is proposed, which utilizes the scenario prior information from step S1 to mark the power spectral density H(f) and clutter power spectral density P. c (f) Design the transmit waveform spectral amplitude |X that can optimize SAR imaging performance and improve signal-to-clutter ratio. sopt (f)|:

[0017]

[0018] Where a1, a2, and b represent non-negative real numbers. b represents the scaling factor that makes the energy spectral density satisfy the energy constraint. a2 represents the threshold for removing excessive peaks in the energy spectral density to reduce sidelobes. a1 represents the magnitude by which the designed waveform's spectrum is boosted. X sopt (f) indicates that the waveform spectrum amplitude constraint of SCR can be improved.

[0019] Define η as the slow-time variable, x η (t) represents the change in the signal transmitted between pulses. Let the sequence set... x represents η Discrete form of (t).

[0020] Where M represents the number of subsequences, Let N represent the transmission sequence at the m-th pulse, and N represent the number of coded sub-pulses. and Let represent an N×1 complex matrix.

[0021] We introduce a weighted complementary integral sidelobe level (WCISL) metric for a set of sequences, expressed as follows:

[0022]

[0023] in, Indicates non-negative weights. x represents m The aperiodic autocorrelation function at time delay k.

[0024] Then the discrete Fourier transform of x is y = F H x.

[0025] in,(·) H Let represent the conjugate transpose of a matrix, and It is a DFT matrix, f n This represents the nth column of matrix F.

[0026] The spectral constraint is then considered as follows:

[0027]

[0028] Where q(l,m) represents the m-th X at the l-th frequency point. sopt (f) spectral amplitude,

[0029] Considering the constant modulus constraint, the expression is as follows:

[0030]

[0031] Combining equations (3), (4), and (5), the resulting expression for the optimization problem is as follows:

[0032]

[0033] The waveform design problem is modeled as an optimization problem as shown in equation (6).

[0034] Furthermore, step S3 is specifically as follows:

[0035] S31. Decompose the optimization problem obtained in step S2 into two sub-problems;

[0036] Problem (6) is decomposed into two tractable subproblems using the Inaccurate Alternating Direction Penalty Framework (IADPM).

[0037] Introducing two auxiliary variables, y(l,m) and z, the expression is as follows:

[0038]

[0039]

[0040] in,(·) T Represents the transpose of a matrix, 0 N-1 This represents an N-1×1 zero matrix.

[0041] The expression for problem (6) is rewritten as follows:

[0042]

[0043] Where, r z (k)=z H J k z, k=1-L,...,L-1, L=M(2N-1)-1,J k Let represent an L×L Topulitz matrix, where the k-th diagonal element is 1 and the rest are 0.

[0044] The augmented Lagrange function expression for problem (9) is as follows:

[0045]

[0046] in, and Let represent the multiplier vector and the penalty parameter, respectively, and Re{·} denote the real part of the complex number. Y = [y1,...,y M ],

[0047] Problem (9) is decomposed into two subproblems using an algorithm within an imprecise alternating direction penalty framework. The algorithm determines the direction in an alternating iterative manner. Y, and and the constraints in problem (9) to minimize use Y (t) , and To represent the inaccurate alternating direction penalty algorithm Y, and The result of the t-th iteration.

[0048] S32. Solve the two subproblems iteratively using the alternating multiplier method;

[0049] S321, Solving Iterative Solutions

[0050] Set the initial iteration count variable t = 0.

[0051] Using the augmented Lagrangian function expressed by equation (10), for a given... By solving equation (11), the expression is as follows:

[0052]

[0053] definition I represents the identity matrix, z (i) This represents the result of the i-th iteration of the max-min framework.

[0054] in, Represents a 2L×2L FFT matrix. Indicates taking Diag(x) represents a diagonal matrix with x as its diagonal element. in:

[0055]

[0056]

[0057]

[0058]

[0059]

[0060]

[0061] Ignore the constant term and let The solution to equation (11) satisfies the following form:

[0062]

[0063] Finally, the iteration results are obtained. Determine if the conditions are met and solve. The termination condition is set to reaching the maximum number of iterations. After satisfying the condition, proceed with iterative solutions for Y. (t+1) If the condition is not met, then repeat step S321 iteratively to solve the problem. Until the termination condition is met.

[0064] S322, Solving the iterative Y (t+1) ;

[0065] Using the augmented Lagrangian function expressed by equation (10), Y (t+1) The iteration is obtained by solving the following problem:

[0066]

[0067] The solution expression for optimization problem (19) is as follows:

[0068]

[0069] in,

[0070] S323, Iteration and

[0071] Lagrange multiplier vectors and penalty parameters Iterating through equations (21) and (22) respectively, the expressions are as follows:

[0072]

[0073] in, 0 < δ 1,c <1,δ 2,c >1 but close to 1,

[0074]

[0075] Where v represents a user-defined positive integer,

[0076] S324. Let t = t + 1, and determine whether the termination condition is met, setting it to the maximum number of iterations reached. If the condition is not met, repeat steps S321-S323 until the termination condition is met; finally, if the termination condition is met, a convergent sidelobe complementary waveform that can improve the signal-to-clutter ratio of SAR is obtained.

[0077] The beneficial effects of this invention are as follows: The method of this invention first models the design of the transmitted waveform as a non-convex constrained optimization problem using the prior power spectral density of the target and clutter. Then, it decomposes the solution of the optimization problem into two sub-problems using the inaccurate alternating direction penalty method. These two sub-problems are solved using a complementary spectral constraint sequence and a minimization framework. Finally, a transmitted waveform that satisfies SAR imaging performance while improving the signal-to-clutter ratio is obtained. This invention obtains the SAR transmitted waveform by utilizing prior environmental information and a complementary spectral constraint sequence. It then optimizes the entire transmitted waveform set using a complementary waveform set design method to improve the signal-to-clutter ratio of the imaging results while ensuring SAR imaging performance. Under the condition of meeting imaging performance indicators, it effectively improves the target signal-to-clutter ratio, achieving both target enhancement and clutter reduction effects.

[0078] The method of this invention utilizes the degrees of freedom provided by the azimuth multipulse sequence of SAR imaging to eliminate high sidelobes of the waveform after azimuth focusing, effectively improving the signal-to-clutter ratio while ensuring SAR imaging performance. It avoids the drawback of existing single-waveform design methods, which result in waveforms with high autocorrelation sidelobes. The SAR transmit waveform set designed using this method has low sidelobes, effectively improving the signal-to-clutter ratio while ensuring SAR imaging performance. It solves the problems of existing radar transmit linear frequency modulated signals failing to achieve target enhancement and other transmit waveforms designed based on improving the signal-to-clutter ratio failing to guarantee SAR imaging performance. Therefore, this transmit waveform can be widely used in SAR mapping, remote sensing, reconnaissance, and other fields. Attached Figure Description

[0079] Figure 1 This is a flowchart of a complementary transmission waveform design method for SAR target enhancement according to the present invention.

[0080] Figure 2 This is a schematic diagram of the geometric configuration of SAR imaging in an embodiment of the present invention.

[0081] Figure 3 This is a comparison chart of the waveform imaging results of the existing linear frequency modulated waveform, the single waveform design method for minimizing the weighted peak sidelobe level ratio, and the method of the present invention in an embodiment of the present invention.

[0082] Figure 4 This is a comparison chart of the three waveforms and the autocorrelation performance of the Hanning window in the embodiments of the present invention. Detailed Implementation

[0083] This invention is primarily verified using simulation experiments; all steps and conclusions were verified through simulation in Matlab 2020. The method of this invention will be further described below with reference to the accompanying drawings and embodiments.

[0084] like Figure 1 The flowchart shown is a method for designing complementary transmit waveforms to enhance SAR targets according to the present invention. The specific steps are as follows:

[0085] S1. Obtain the target power spectral density H(f) and clutter power spectral density P in the imaging scene based on known prior information. c (f) (Scene prior information);

[0086] Wherein, the known prior information refers to the target area and clutter area in the imaging scene, and f represents the frequency.

[0087] S2. Using the target power spectral density H(f) and clutter power spectral density P obtained in step S1 c (f) Establishing a waveform optimization problem to improve signal-to-noise ratio;

[0088] S3. Solve the optimization problem obtained in step S2 to obtain a convergent sidelobe complementary waveform that can improve the signal-to-clutter ratio of SAR.

[0089] In this embodiment, the geometric configuration diagram of SAR imaging in step S1 is as follows: Figure 2 As shown in Table 1, the simulation system parameters are as follows.

[0090] Table 1

[0091]

[0092] In this embodiment, step S2 is specifically as follows:

[0093] Let the transmitted waveform be x(t), where t represents a fast time variable. Then, the expression for the transmitted waveform spectrum constraint to achieve the maximum signal-to-clutter-to-noise ratio is as follows:

[0094]

[0095] Where X(f) represents the Fourier transform of x(t); P n (f) represents the power spectral density of environmental noise. Generally, the noise can be considered as Gaussian white noise, i.e., P0. n (f)=1,λ x This represents the Lagrange multiplier.

[0096] However, due to the fluctuating H(f) and P c (f) will result in high sidelobes of the autocorrelation function of X(f), which will seriously affect the imaging performance of SAR. Therefore, equation (1) cannot be directly used as the amplitude of the transmitted waveform spectrum. This embodiment proposes a simple joint optimization strategy, which uses the scene prior information in step S1 to mark the power spectral density H(f) and clutter power spectral density P. c(f) Design the transmit waveform spectral amplitude |X that can optimize SAR imaging performance and improve signal-to-clutter ratio. sopt (f)|:

[0097]

[0098] Where a1, a2, and b represent non-negative real numbers. b represents the scaling factor that makes the energy spectral density satisfy the energy constraint. a2 represents the threshold for removing excessive peaks in the energy spectral density to reduce sidelobes. a1 represents the magnitude of the boosting of the designed waveform spectrum, which increases the similarity between the energy spectral density and the linear frequency modulated signal to ensure the range resolution of SAR imaging. sopt (f) indicates that the waveform spectrum amplitude constraint of SCR can be improved.

[0099] By utilizing the azimuth focusing characteristics of SAR, the relationship between signal-to-noise ratio and high sidelobes of the transmitted waveform can be eliminated by transmitting waveforms that vary in azimuth.

[0100] Define η as the slow-time variable, x η (t) represents the change in the signal transmitted between pulses. Let the sequence set... x represents η Discrete form of (t).

[0101] Where M represents the number of subsequences, Let N represent the transmission sequence at the m-th pulse, and N represent the number of coded sub-pulses. and Let N×1 be a complex matrix. To eliminate sidelobes, a set of weighted complementary integral sidelobe levels (WCISL) measures of sequence is introduced, expressed as follows:

[0102]

[0103] in, Indicates non-negative weights. x represents m The aperiodic autocorrelation function at time delay k.

[0104] Then the discrete Fourier transform of x is y = F H x.

[0105] in,(·) H Let represent the conjugate transpose of a matrix, and It is a DFT matrix, f n This represents the nth column of matrix F.

[0106] To ensure SAR imaging performance while matching energy spectral density, the spectral constraint is considered as follows:

[0107]

[0108] Where q(l,m) represents the m-th X at the l-th frequency point. sopt (f) spectral amplitude, Furthermore, to avoid waveform degradation or distortion when the SAR system generates transmitted waveforms, constant mode constraints must also be considered, as expressed below:

[0109]

[0110] Combining equations (3), (4), and (5), the resulting expression for the optimization problem is as follows:

[0111]

[0112] The waveform design problem is modeled as an optimization problem as shown in equation (6).

[0113] In this embodiment, step S3 is specifically as follows:

[0114] S31. Decompose the optimization problem obtained in step S2 into two sub-problems;

[0115] To solve the above non-convex optimization problem, the Inaccurate Alternating Direction Penalty Framework (IADPM) is used to decompose problem (6) into two tractable subproblems.

[0116] Introducing two auxiliary variables, y(l,m) and z, the expression is as follows:

[0117]

[0118]

[0119] in,(·) T Represents the transpose of a matrix, 0 N-1 This represents an N-1×1 zero matrix.

[0120] Therefore, the expression for problem (6) is rewritten as follows:

[0121]

[0122] Where, r z (k)=z H J k z, k=1-L,...,L-1, L=M(2N-1)-1,J k Let represent an L×L Topulitz matrix, where the k-th diagonal element is 1 and the rest are 0.

[0123] The augmented Lagrange function expression for problem (9) is as follows:

[0124]

[0125] in, and Let represent the multiplier vector and the penalty parameter, respectively, and Re{·} denote the real part of the complex number. Y = [y1,...,y M ],

[0126] Problem (9) is decomposed into two subproblems using an algorithm within an imprecise alternating direction penalty framework. The algorithm determines the direction in an alternating iterative manner. Y, and and the constraints in problem (9) to minimize use Y (t) , and To represent the inaccurate alternating direction penalty algorithm Y, and The result of the t-th iteration.

[0127] S32. Solve the two subproblems iteratively using the alternating multiplier method;

[0128] S321, Solving Iterative Solutions

[0129] Set the initial iteration count variable t = 0.

[0130] Using the augmented Lagrangian function expressed by equation (10), for a given... By solving equation (11), the expression is as follows:

[0131]

[0132] definition I represents the identity matrix, z (i) This represents the result of the i-th iteration of the max-min framework.

[0133] in, Represents a 2L×2L FFT matrix. Indicates taking Diag(x) represents a diagonal matrix with x as its diagonal element. in:

[0134]

[0135]

[0136]

[0137]

[0138]

[0139]

[0140] Ignore the constant term and let The solution to equation (11) satisfies the following form:

[0141]

[0142] Finally, the iteration results are obtained. Determine if the conditions are met and solve. The termination condition is set to reaching the maximum number of iterations. After satisfying the condition, proceed with iterative solutions for Y. (t+1) If the condition is not met, then repeat step S321 iteratively to solve the problem. Until the termination condition is met.

[0143] S322, Solving the iterative Y (t+1) ;

[0144] Using the augmented Lagrangian function expressed by equation (10), Y (t+1) The iteration is obtained by solving the following problem:

[0145]

[0146] The solution expression for optimization problem (19) is as follows:

[0147]

[0148] in,

[0149] S323, Iteration and

[0150] Lagrange multiplier vectors and penalty parameters Iterating through equations (21) and (22) respectively, the expressions are as follows:

[0151]

[0152] in, 0 < δ 1,c <1,δ 2,c >1 but close to 1,

[0153]

[0154] Where v represents a user-defined positive integer,

[0155] S324. Let t = t + 1, and determine whether the termination condition is met, setting it to the maximum number of iterations reached. If the condition is not met, repeat steps S321-S323 until the termination condition is met; finally, if the termination condition is met, a convergent sidelobe complementary waveform that can improve the signal-to-clutter ratio of SAR is obtained.

[0156] In this embodiment, the method of the present invention further includes step S4, which involves changing the transmitted waveform to a designed waveform for imaging simulation, as follows:

[0157] In step S4, the transmitted waveform in the SAR simulation imaging experiment is changed to the convergent sidelobe complementary waveform designed in step S3, and the simulation imaging experiment is performed. The results are compared with the imaging results of existing linear frequency modulated signals and single waveform design methods that minimize the weighted peak sidelobe level. Figure 3 As shown.

[0158] in, Figure 3 (a) shows the imaging results of the linear frequency modulated signal. Figure 3 (b) The imaging results of the waveform designed to minimize the weighted peak-to-sidelobe ratio algorithm. Figure 3 (c) shows the imaging results of the method of the present invention.

[0159] In this embodiment, the three waveforms and the autocorrelation performance of the Hanning window are compared as follows: Figure 4 As shown in Table 2, the main lobe width, peak sidelobe level ratio, and integral sidelobe level ratio of the three transmitted waveforms were calculated respectively. The numerical comparison of the imaging performance of the three waveforms is shown in Table 3. The signal-to-noise ratio (SNR) of the same scene under different transmitted waveforms (SNR of the imaging results of the three waveforms) is calculated in Table 3.

[0160] Table 2

[0161]

[0162] Table 3

[0163]

[0164] from Figure 3 , Figure 4As shown in Tables 2 and 3, the method of this invention can effectively improve the signal-to-clutter ratio of the imaging results while ensuring SAR imaging performance. This method employs a complementary SAR waveform set design approach to design a transmit waveform that improves the signal-to-clutter ratio while suppressing sidelobes. Unlike existing single-waveform design methods, this invention performs joint optimization of the transmit waveform set, and the designed waveform with complementary sidelobes varies between pulses.

[0165] In summary, the method of this invention first obtains prior information about the scene, then uses this prior information to design a transmit waveform spectral amplitude that balances signal-to-clutter ratio (SNR) improvement and SAR imaging performance. Next, it designs the transmit waveform using spectral amplitude constraints and constant mode constraints, ultimately obtaining a set of transmit waveforms with complementary sidelobes. This method utilizes the degrees of freedom provided by the azimuth multipulse sequence in SAR imaging to eliminate high sidelobes in the waveform after azimuth focusing, effectively improving the SNR while maintaining SAR imaging performance. It avoids the drawback of existing single-waveform design methods, which result in waveforms with high autocorrelation sidelobes. The SAR transmit waveform set designed using this method has low sidelobes, effectively improving the SNR while ensuring SAR imaging performance, thus enabling the widespread application of this transmit waveform in SAR mapping, remote sensing, reconnaissance, and other fields.

[0166] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A complementary transmit waveform design method for SAR target enhancement, the specific steps of which are as follows: S1. Obtain the target power spectral density in the imaging scene based on known prior information. clutter power spectral density ; in, The known prior information refers to the target area and clutter area in the imaging scene. Indicates frequency; S2. Using the target power spectral density obtained in step S1 clutter power spectral density Establish a waveform optimization problem to improve signal-to-noise ratio; S3. Solve the optimization problem obtained in step S2 to obtain the convergent sidelobe complementary waveform that can improve the signal-to-clutter ratio of SAR. Step S2 is as follows: Let the transmitted waveform be , If we denote a fast-time variable, then the expression for the transmit waveform spectrum constraint to achieve the maximum signal-to-clutter-to-noise ratio is as follows: （1）； in, express Fourier transform; The power spectral density of ambient noise is represented by the noise being treated as Gaussian white noise, i.e. , Represents the Lagrange multiplier; Since equation (1) cannot be directly used as the amplitude of the transmitted waveform spectrum, a joint optimization strategy is proposed, which uses the scenario prior information from step S1 to mark the power spectral density. clutter power spectral density Design the transmit waveform spectral amplitude that can optimize SAR imaging performance and improve signal-to-clutter ratio. : （2）； in, , and Represents a non-negative real number; This represents the scaling factor that makes the energy spectral density satisfy the energy constraint. This indicates the threshold for removing excessive peaks in the energy spectral density to reduce sidelobes; This indicates the magnitude by which the designed waveform spectrum is raised; This indicates that it can improve the waveform spectrum amplitude constraint of SCR; definition Represents a slow-time variable. Represents the change in signal transmitted between pulses; let the sequence set express Discrete form; in, Indicates the number of subsequences. Indicates the first The transmission sequence at each pulse, Indicates the number of coded sub-pulses. and , express Complex matrices; We introduce a weighted complementary integral sidelobe level (WCISL) metric for a set of sequences, expressed as follows: （3）； in, Indicates non-negative weights. express In time delay The non-periodic autocorrelation function at time; Then we can obtain The discrete Fourier transform is ; in, Let represent the conjugate transpose of a matrix, and It is a DFT matrix. express The first of the matrix List; The spectral constraint is then considered as follows: （4）； in, Indicates the first The frequency point of the first indivual Spectral amplitude, ; Considering the constant modulus constraint, the expression is as follows: （5）； Combining equations (3), (4), and (5), the resulting expression for the optimization problem is as follows: （6）； The waveform design problem is modeled as an optimization problem as shown in equation (6); Step S3 is as follows: S31. Decompose the optimization problem obtained in step S2 into two sub-problems; The imprecise alternating direction penalty framework (IADPM) is used to decompose equation (6) into two tractable subproblems; Introduce two auxiliary variables and The expression is as follows: （7）； （8）； in, Represents the transpose of a matrix. express The zero matrix; Then the expression for equation (6) can be rewritten as follows: （9）； in, , , , Represent a The Toplitz matrix of dimension , its The diagonal elements are 1, and the rest are 0; The augmented Lagrange function expression of equation (9) is as follows: （10）； in, and These represent the multiplier vector and the penalty parameter, respectively. Represents the real part of a complex number; ; The algorithm, using an inaccurate alternating direction penalty framework, decomposes equation (9) into two subproblems; the algorithm determines the direction in an alternating iterative manner. , , and And the constraints in equation (9) to minimize ;use , , and To represent the inaccurate alternating direction penalty algorithm , , and No. The result of the next iteration; S32. Solve the two subproblems iteratively using the alternating multiplier method; S321, Solving Iterative Solutions ; Set the initial iteration count variable ; Using the augmented Lagrangian function expressed by equation (10), for a given... , By solving equation (11), we obtain the following expression: （11）； definition , Represents the identity matrix. The first step in representing the max-min framework is... The result of the second iteration ; in, Represent a The FFT matrix, Indicates taking The sum of all rows of the matrix and 1 to 1 List, Indicates Let be a diagonal matrix with diagonal elements, defined ,in: （12）； （13）； （14）； （15）； （16）； （17）； Ignore the constant term and let The solution to equation (11) satisfies the following form: （18）； Finally, the iteration results are obtained. Determine if the solution is satisfied. The termination condition is set to reaching the maximum number of iterations. After satisfying the requirements, proceed with iterative solutions. If the condition is not met, then repeat step S321 iteratively to solve the problem. Until the termination condition is met; S322, Solving Iterative Solutions ; Using the augmented Lagrangian function expressed by equation (10), The iteration is obtained by solving the following problem: （19）； The solution expression for optimization equation (19) is as follows: （20）； in, ; S323, Iteration and ; Lagrange multiplier vectors and penalty parameters Iterating through equations (21) and (22) respectively, the expressions are as follows: （21）； in, , But it is close to 1. ; （22）； in, Represents a user-defined positive integer. , , ; S324, Order Determine if the termination condition is met, and set it to the maximum number of iterations reached. If the condition is not met, repeat steps S321-S323 until the termination condition is met; finally, if the termination condition is met, a convergent sidelobe complementary waveform that can improve the signal-to-clutter ratio of SAR is obtained.

Images