A method for designing NLFM waveform against SAR combined jamming

Abstract

The application discloses a kind of NLFM waveform design methods for anti-SAR combined interference, first determine the generation model of NLFM waveform, then establish the waveform optimization model of anti-repeater deception jamming and the waveform optimization model of anti-narrowband radio frequency interference, combinedly establish the waveform optimization model of anti-combined interference, finally split into two nonlinear constraint optimization sub-problems by multi-objective constraint optimization model, solve optimization problem using augmented Lagrange genetic algorithm, obtain the NLFM waveform of anti-SAR combined interference.The method of the application can resist two kinds of interference types simultaneously by optimizing the waveform of transmitting end, including repeater deception jamming and narrowband radio frequency interference, using the designed NLFM waveform as transmitting signal, can effectively suppress the influence of combined interference on the basis of realizing imaging performance, improve the survivability and practical efficiency of SAR system, so that SAR system can be widely used in resource exploration, geological mapping and other fields.

Description

Technical Field

[0001] This invention belongs to the field of radar waveform design, specifically relating to an NLFM waveform design method for resisting SAR combined interference. Background Technology

[0002] Synthetic Aperture Radar (SAR), as an active microwave remote sensing device, has the capability of all-day, all-weather high-resolution imaging. Currently, it is widely used in important fields such as environmental monitoring, disaster monitoring, resource exploration, and geological mapping.

[0003] However, with the increasing sophistication of electronic countermeasures technology, the electromagnetic environment faced by Synthetic Aperture Radar (SAR) has become increasingly complex, and the forms of interference have become more diverse. During signal transmission and reception, SAR is susceptible to electromagnetic interference in space, which poses a serious threat to its survivability and practical effectiveness. On the one hand, as a broadband radar system, SAR is subject to unintentional interference from mobile communication signals, television signals, broadcast signals, and other signals within its operating frequency band—a phenomenon known as radio frequency interference (RFI). On the other hand, adversary jammers can intercept radar transmission signals and then generate false targets highly similar to real targets through storage, modulation, and forwarding, thereby affecting the radar system's detection performance—a deception jamming technique. Therefore, to effectively eliminate the impact of interference on SAR imaging, it is necessary to design radar transmission waveforms with anti-jamming capabilities.

[0004] The paper "Li, Tingjun, H. Yang, and ZO Zhou. RFI suppression in through-wall radar based on phase-coded stepped-frequency waveform." 2014 IEEE Radar Conference 0" designs a phase-coded stepped-frequency waveform based on the study of the initial phase of the transmitted signal, which improves the radar's anti-RF interference performance. The paper "Wang Mingjie, Shi Shengnan, Tang Zhihua, et al. Anti-RF interference waveform design based on ISL constraints. Electronic Technology and Software Engineering, 2021, 00(020): P.81-83" proposes a low-sidelobe waveform design method for anti-RF interference, achieving anti-interference while ensuring low-sidelobe output of matched filtering. The paper "Jin, Guodong, et al. The Design of Orthogonal Waveform Suiting for Synthetic Aperture Radar Imaging." 2019 6th Asia-Pacific Conference on Synthetic Aperture Radar (APSAR) IEEE, 2019" proposes a novel orthogonal NLFM waveform optimization scheme applicable to SAR imaging. The paper “Q.Xie, J.Yang, C.Liu and W.Li. Low Sidelobe Quasi-Orthogonal NLFM Waveforms with Reciprocating Frequency Modulation. IEEE Geoscience and Remote Sensing Letters, 2022, 1-5” proposes an orthogonal waveform optimization design method based on an alternating NLFM (nonlinear frequency modulation signal) model, which further broadens the waveform degrees of freedom and improves correlation performance.

[0005] The aforementioned anti-jamming waveform design research can only counteract single jamming patterns and cannot effectively combat combined jamming patterns, thus having certain limitations. However, due to the complex electromagnetic interference environment in space, SAR systems may simultaneously face the threat of combined interference such as radio frequency interference, suppression interference, and deception interference. Unlike single jamming patterns, combined interference has multiple interference effects, posing a greater threat to radar imaging and detection, and correspondingly, its countermeasure complexity is also higher. Therefore, researching a waveform design method to resist combined SAR interference is of great significance. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides an NLFM waveform design method for resisting combined SAR interference, solving the problem of anti-interference waveform optimization modeling and solution in scenarios where forwarding spoofing interference and narrowband radio frequency interference coexist.

[0007] The technical solution adopted in this invention is: an NLFM waveform design method for resisting SAR combined interference, the specific steps of which are as follows:

[0008] S1. Determine the generation model of the NLFM waveform;

[0009] The time-frequency relationship of the NLFM waveform is defined using a piecewise linear PWL function, where the instantaneous frequency of the NLFM waveform is composed of n+1 linear functions. The time-domain expression for the nonlinear frequency-modulated signal s(t) is then as follows:

[0010]

[0011] Where rect(·) represents a rectangular window function, t represents time, T represents the pulse width of the signal, and f(t) represents the instantaneous frequency function, expressed as follows:

[0012]

[0013] Where B represents the bandwidth of the signal, (B1, B2, ..., B n () represents a frequency point, and 0 ≤ (B1, B2, ..., B) n )≤B,(T1,T2,...,T n () represents the corresponding time point, and 0 ≤ (T1, T2, ..., T) n )≤T,(k0,k1,...,k n The slope of the piecewise function is represented by the following expression:

[0014]

[0015] Phase information of nonlinear frequency modulated signals By integrating equation (2), we obtain the following expression:

[0016]

[0017] in, It is a constant. If the initial phase of the signal is set to 0, then... The calculation expression is as follows:

[0018]

[0019] S2. Establish a waveform optimization model to resist forwarding spoofing interference;

[0020] Consider two sets of nonlinear frequency-modulated signals s1(t) and s2(t), whose cross-correlation r(t) is defined as follows:

[0021]

[0022] Where u represents the integration variable, and (*) represents conjugate.

[0023] The expression for the optimization problem of minimizing the integral sidelobe level (ISL) of waveform cross-correlation is as follows:

[0024]

[0025] The convolutional form of r(t) is as follows:

[0026]

[0027] in, Represents convolution. Let represent the complex conjugate of time-reflected s2(t), f represent the frequency, and R(f), S1(f), and S2(f) represent the Fourier transforms of r(t), s1(t), and s2(t), respectively.

[0028] According to Parseval's theorem, minimizing the time-domain energy is equivalent to minimizing the frequency-domain energy. Based on equations (7)-(9), the final optimization model expression for resisting forwarding spoofing interference is as follows:

[0029]

[0030] S3. Establish a waveform optimization model to resist narrowband radio frequency interference;

[0031] Consider two sets of nonlinear frequency-modulated signals, s1(t) and s2(t). The problem of designing waveforms to resist narrowband radio frequency interference is transformed into suppressing the energy of the waveform power spectrum in a specific frequency band. Assuming that narrowband radio frequency interference exists in the frequency band [f1, f2], then suppressing narrowband radio frequency interference is equivalent to suppressing the spectral energy of the waveform in the frequency band [f1, f2], as expressed below:

[0032]

[0033] Where f1 represents the lower bound of the frequency band and f2 represents the upper bound of the frequency band.

[0034] S4. Establish a waveform optimization model to resist combined interference;

[0035] Combining steps S2 and S3, the objective function expression for the waveform optimization problem against combined interference is as follows:

[0036]

[0037] Where λ1 and λ2 represent weighting factors.

[0038] The integral sidelobe ratio (ISLR) and peak sidelobe ratio (PSLR) of the matched filter output of a single waveform are used as metrics. Let the length of the nonlinear frequency modulated signal s(t) be N, and the pulse compression result after matched filtering be y(t), whose discretization expression is as follows:

[0039] y = [y 1-N ,…,y -δ ,…,y -1 ,y0,y1,…,y δ ,…y N-1 (13)

[0040] Among them, y k (1-N≤k≤N-1) represents the result of the autocorrelation of the signal s(t), |y0| represents the peak value of the main lobe, and δ represents the resolution cell width.

[0041] The peak-to-sidelobe ratio is defined as the ratio of the energy of the first sidelobe to the peak energy of the main lobe after the matched filter response. The integral-to-sidelobe ratio is defined as the ratio of the energy outside the resolution cell to the energy inside the resolution cell after the matched filter response, as shown in the following expressions:

[0042]

[0043] In summary, the final multi-objective constrained optimization model expression for the anti-combined interference waveform design is as follows:

[0044]

[0045] Where λ1 and λ2 represent weighting factors, and [a,b,c,d] represents the upper limit of constraints.

[0046] S5. Decompose the multi-objective constrained optimization model into two nonlinear constrained optimization sub-problems, and solve the optimization problem using the augmented Lagrange genetic algorithm;

[0047] S51. Divide the multi-objective constrained optimization model into two sub-problems;

[0048] Equation (16) is broken down into two optimization problems and solved one by one. First, the spectrum and pulse compression output characteristics of waveform s1(t) are optimized. Then, based on waveform s1(t), the spectrum and pulse compression output characteristics of waveform s2(t) and the cross-correlation performance of the two waveforms are optimized, as follows:

[0049] Optimization issue 1:

[0050]

[0051] Optimization issue 2:

[0052]

[0053] Where λ represents the weighting factor.

[0054] S52. Solve optimization problems using augmented Lagrange genetic algorithm;

[0055] First, the augmented Lagrange genetic algorithm combines the fitness function and nonlinear constraint function into a subproblem by introducing Lagrange multipliers and penalty factors, as expressed below:

[0056]

[0057] Wherein, vector x represents the optimization variable, i.e., the frequency points (B1, B2, ..., B...). n ) and time points (T1, T2, ..., T n ), where vector l represents the Lagrange multiplier estimate, l i These are its components, where vector s represents the non-negative offset, s i It is its component, f(x) represents the fitness function, that is, the objective function in optimization problems 1 and 2, c i (x) represents the nonlinear inequality constraint, i.e. the nonlinear constraint in optimization problems 1 and 2, and mt represents the number of nonlinear inequality constraints.

[0058] Next, the optimization problem shown in equation (19) is approximated by a genetic algorithm to satisfy the linear constraints and boundary conditions. When the subproblem is minimized to the required accuracy and the feasibility condition is met, the Lagrange estimate l is updated to form a new subproblem, and step S52 is repeated until the constraints are met and the difference in the change of the optimal fitness function is less than the set threshold.

[0059] S6. Based on steps S1-S5, the NLFM waveform resisting SAR combined interference is obtained, which is the optimized waveform.

[0060] The beneficial effects of this invention are as follows: The method of this invention first determines the generation model of the NLFM waveform, then establishes waveform optimization models against transponder-based spoofing interference and narrowband radio frequency interference, and combines them to establish a waveform optimization model against combined interference. Finally, the multi-objective constrained optimization model is decomposed into two nonlinear constrained optimization sub-problems, and the optimization problem is solved using the augmented Lagrange genetic algorithm to obtain the NLFM waveform resistant to combined SAR interference. This invention's method, through optimized design of the transmitter waveform, can simultaneously combat two types of interference, including transponder-based spoofing interference and narrowband radio frequency interference. Using the designed NLFM waveform as the transmitted signal, it can effectively suppress the influence of combined interference while achieving imaging performance. It solves the problem of anti-interference waveform optimization modeling and solving in scenarios where transponder-based spoofing interference and narrowband radio frequency interference coexist, improving the survivability and practical performance of the SAR system, enabling the SAR system to be widely used in resource exploration, geological mapping, and other fields. Attached Figure Description

[0061] Figure 1 This is a flowchart of an NLFM waveform design method for resisting SAR combined interference according to the present invention.

[0062] Figure 2 This is a schematic diagram of the time-frequency relationship of the NLFM waveform based on a piecewise linear function in an embodiment of the present invention.

[0063] Figure 3 This is a schematic diagram of a combination of forwarding spoofing and narrowband radio frequency interference in an embodiment of the present invention.

[0064] Figure 4 This is a performance diagram of the transmitted waveform in an embodiment of the present invention.

[0065] Figure 5 This is a point target imaging image in an embodiment of the present invention.

[0066] Figure 6 This is a comparison image of anti-combined interference imaging in an embodiment of the present invention. Detailed Implementation

[0067] The method of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0068] like Figure 1 The flowchart of an NLFM waveform design method for resisting SAR combined interference is shown below. The specific steps are as follows:

[0069] S1. Determine the generation model of the NLFM waveform;

[0070] NLFM waveforms can have their power spectral density modified by altering their time-frequency relationship to achieve better waveform performance. The time-frequency relationship of the NLFM waveform is defined using a piecewise linear (PWL) function. The instantaneous frequency of the NLFM waveform is composed of n+1 linear functions. A schematic diagram of the time-frequency relationship of the NLFM waveform based on the piecewise linear function is shown below. Figure 2 As shown. The time-domain expression for the nonlinear frequency-modulated signal s(t) is then as follows:

[0071]

[0072] Where rect(·) represents a rectangular window function, t represents time, T represents the pulse width of the signal, and f(t) represents the instantaneous frequency function, expressed as follows:

[0073]

[0074] Where B represents the bandwidth of the signal, (B1, B2, ..., B n () represents a frequency point, and 0 ≤ (B1, B2, ..., B) n )≤B,(T1,T2,...,T n () represents the corresponding time point, and 0 ≤ (T1, T2, ..., T) n )≤T,(k0,k1,...,k n The slope of the piecewise function is represented by the following expression:

[0075]

[0076] Phase information of nonlinear frequency modulated signals By integrating equation (2), we obtain the following expression:

[0077]

[0078] in, It is a constant. Assuming the initial phase of the signal is 0, to ensure the phase continuity of the nonlinear frequency modulated signal, then... The calculation expression is as follows:

[0079]

[0080] S2. Establish a waveform optimization model to resist forwarding spoofing interference;

[0081] The problem of designing waveforms to resist repeater-induced deception interference is essentially about designing a set of waveforms with orthogonality. Consider two sets of nonlinear frequency-modulated signals s1(t) and s2(t), whose cross-correlation r(t) is defined as follows:

[0082]

[0083] Where u represents the integration variable, and (*) represents conjugate.

[0084] The expression for the optimization problem of minimizing the integral sidelobe level (ISL) of waveform cross-correlation is as follows:

[0085]

[0086] The convolutional form of r(t) is as follows:

[0087]

[0088]

[0089] in, Represents convolution. Let represent the complex conjugate of time-reflected s2(t), f represent the frequency, and R(f), S1(f), and S2(f) represent the Fourier transforms of r(t), s1(t), and s2(t), respectively.

[0090] According to Parseval's theorem, minimizing the time-domain energy is equivalent to minimizing the frequency-domain energy. Based on equations (7)-(9), the final optimization model expression for resisting forwarding spoofing interference is as follows:

[0091]

[0092] S3. Establish a waveform optimization model to resist narrowband radio frequency interference;

[0093] Consider two sets of nonlinear frequency-modulated signals s1(t) and s2(t). The problem of designing waveforms to resist narrowband radio frequency interference can be transformed into suppressing the power spectrum energy of the waveform in a specific frequency band. Assuming that narrowband radio frequency interference exists in the frequency band [f1, f2], then suppressing narrowband radio frequency interference is equivalent to suppressing the spectral energy of the waveform in the frequency band [f1, f2], as expressed below:

[0094]

[0095] Where f1 represents the lower bound of the frequency band and f2 represents the upper bound of the frequency band.

[0096] The schematic diagram of the combined relay spoofing and narrowband radio frequency interference in this embodiment is shown below. Figure 3 As shown, the SAR imaging simulation parameters and interference simulation parameters are shown in Tables 1 and 2.

[0097] Table 1

[0098]

[0099] Table 2

[0100]

[0101] S4. Establish a waveform optimization model to resist combined interference;

[0102] Combining steps S2 and S3, the objective function expression for the waveform optimization problem against combined interference is as follows:

[0103]

[0104] Where λ1 and λ2 represent weighting factors.

[0105] Due to the imaging performance requirements of synthetic aperture radar, the transmitted waveform must have good pulse compression output characteristics. Therefore, the integral sidelobe ratio (ISLR) and peak sidelobe ratio (PSLR) of the matched filter output of a single waveform are considered as the evaluation indicators.

[0106] Let the length of the nonlinear frequency modulated signal s(t) be N, and the pulse compression result after matched filtering be y(t), whose discretization expression is as follows:

[0107] y = [y 1-N ,…,y -δ ,…,y -1 ,y0,y1,…,y δ ,…y N-1 (13)

[0108] Among them, y k (1-N≤k≤N-1) represents the result of the autocorrelation of the signal s(t), |y0| represents the peak value of the main lobe, and δ represents the resolution cell width.

[0109] The peak-to-sidelobe ratio is defined as the ratio of the energy of the first sidelobe to the peak energy of the main lobe after the matched filter response. The integral-to-sidelobe ratio is defined as the ratio of the energy outside the resolution cell to the energy inside the resolution cell after the matched filter response, as shown in the following expressions:

[0110]

[0111] In summary, the final multi-objective constrained optimization model expression for the anti-combined interference waveform design is as follows:

[0112]

[0113] Where λ1 and λ2 represent weighting factors, and [a,b,c,d] represents the upper limit of constraints.

[0114] S5. Decompose the multi-objective constrained optimization model into two nonlinear constrained optimization sub-problems, and solve the optimization problem using the augmented Lagrange genetic algorithm;

[0115] S51. Divide the multi-objective constrained optimization model into two sub-problems;

[0116] To reduce the difficulty and increase the speed of solving the problem, equation (16) is split into two optimization problems and solved one by one. First, the spectrum and pulse compression output characteristics of waveform s1(t) are optimized. Then, based on waveform s1(t), the spectrum and pulse compression output characteristics of waveform s2(t) and the cross-correlation performance of the two waveforms are optimized, as follows:

[0117] Optimization issue 1:

[0118]

[0119] Optimization issue 2:

[0120]

[0121] Where λ represents the weighting factor.

[0122] S52. Solve optimization problems using augmented Lagrange genetic algorithm;

[0123] First, the augmented Lagrange genetic algorithm combines the fitness function and nonlinear constraint function into a subproblem by introducing Lagrange multipliers and penalty factors, as expressed below:

[0124]

[0125] Wherein, vector x represents the optimization variable, i.e., the frequency points (B1, B2, ..., B...). n ) and time points (T1, T2, ..., T n ), where vector l represents the Lagrange multiplier estimate, l i These are its components, where vector s represents the non-negative offset, s i It is its component, f(x) represents the fitness function, that is, the objective function in optimization problems 1 and 2, c i (x) represents the nonlinear inequality constraint, i.e. the nonlinear constraint in optimization problems 1 and 2, and mt represents the number of nonlinear inequality constraints.

[0126] Next, the optimization problem shown in equation (19) is approximated by a genetic algorithm to satisfy the linear constraints and boundary conditions. When the subproblem is minimized to the required accuracy and the feasibility condition is met, the Lagrange estimate l is updated to form a new subproblem, and step S52 is repeated until the constraints are met and the difference in the change of the optimal fitness function is less than the set threshold.

[0127] S6. Based on steps S1-S5, the NLFM waveform resisting SAR combined interference is obtained, i.e., the optimized waveform. The waveform performance obtained in this embodiment is as follows: Figure 4 As shown. Figure 4 (a) shows the autocorrelation performance of the waveforms (the peak-to-side-lobe ratio of waveform s1(s1(t)) in the figure is -20.04dB, and the peak-to-side-lobe ratio of waveform s2(s2(t)) is -20.00dB). Figure 4 (b) shows the cross-correlation performance between waveforms (the cross-correlation sidelobe level of waveforms s1 and s2 in the figure is -29.42dB). Figure 4 (c) shows the spectral performance of the waveforms (the stopband level of waveform s1 in the figure is -24.47dB, and the stopband level of waveform s2 is -22.56dB). From Figure 4 As can be seen, the peak sidelobe level of the waveform autocorrelation reaches -20dB, the peak sidelobe level of the waveform crosscorrelation reaches -29dB, and the spectral stopband level of the waveform reaches -22dB. The waveform designed by the method of this invention has good sidelobe performance, orthogonality performance, and bandgap performance.

[0128] In this embodiment, to verify the imaging performance of the NLFM waveform designed by the method of the present invention, SAR imaging adopts a front-side-view stripe structure, and the radar signal adopts an alternating transmission mode. Point target imaging simulation is performed using the waveform obtained in step S6. The imaging results are as follows: Figure 5 As shown. Figure 5 (a) is a two-dimensional cross-sectional view of the point target. Figure 5 (b) is a range profile of the point target (the peak sidelobe ratio is -20.00 dB, the integral sidelobe ratio is -10.08 dB, and the resolution is 1.68 m). Figure 5 (c) is an azimuth profile of the point target (peak sidelobe ratio is -13.37 dB, integral sidelobe ratio is -9.94 dB, and resolution is 1.34 m). From Figure 5 As can be seen, the peak sidelobe ratio of the point target range profile is -20.00dB, and the integral sidelobe ratio is -10.08dB. The waveform designed by the method of this invention has good imaging performance.

[0129] In this embodiment, to verify the anti-interference effect of the NLFM waveform designed by the method of the present invention, the waveform obtained in step S6 is used to perform combined interference scene imaging simulation, and the imaging effect of the LFM waveform is compared. The imaging results are as follows. Figure 6 As shown, Figure 6 (a) is an image of LFM waveform anti-interference imaging. Figure 6 (b) is an image of the NLFM waveform resisting combined interference of the present invention. As can be seen from the figure, the method of the present invention can effectively suppress the effects of repeater spoofing and narrowband radio frequency combined interference while ensuring imaging performance.

[0130] In summary, the method of this invention can simultaneously combat two types of interference, including repeater spoofing interference and narrowband radio frequency interference, by optimizing the waveform design at the transmitting end. By using the designed NLFM waveform as the transmitted signal, it can effectively suppress the influence of combined interference while achieving imaging performance. It solves the problem of anti-interference waveform optimization modeling and solving in the scenario of coexistence of repeater spoofing interference and narrowband radio frequency interference, improves the survivability and practical performance of SAR system, and enables SAR system to be widely used in resource exploration, geological mapping and other fields.

[0131] 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. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of the claims of the invention.

Claims

1. A method for designing NLFM waveforms to resist combined SAR interference, the specific steps of which are as follows: S1. Determine the generation model of the NLFM waveform; The time-frequency relationship of the NLFM waveform is defined using a piecewise linear PWL function. The instantaneous frequency of the NLFM waveform is determined by... The signal is composed of segmental linear functions; therefore, the time domain of the nonlinear frequency modulated signal is... The expression is as follows: （1）； in, This represents the rectangular window function. Indicates time, Indicates the pulse width of the signal. The instantaneous frequency function is expressed as follows: （2）； in, Indicates the bandwidth of the signal. Indicates frequency point, and , This indicates the corresponding point in time, and , The slope of the piecewise function is expressed by the following relationship: （3）； Phase information of nonlinear frequency modulated signals By integrating equation (2), we obtain the following expression: （4）； in, It is a constant; if the initial phase of the signal is set to 0, then... The calculation expression is as follows: （5）； S2. Establish a waveform optimization model to resist forwarding spoofing interference; Consider two sets of nonlinear frequency-modulated signals and Their interrelation The expression is defined as follows: （6）； in, Represents the integral variable. Indicates conjugate; The expression for the optimization problem of minimizing the integral sidelobe level (ISL) of waveform cross-correlation is as follows: （7）； in, The convolutional form is expressed as follows: （8）； （9）； in, Represents convolution. express The complex conjugate of time reversal, Indicates frequency, , , They represent , , Fourier transform; According to Parseval's theorem, minimizing the time-domain energy is equivalent to minimizing the frequency-domain energy. Based on equations (7)-(9), the final optimization model expression for resisting forwarding spoofing interference is as follows: （10）； S3. Establish a waveform optimization model to resist narrowband radio frequency interference; Consider two sets of nonlinear frequency-modulated signals and The problem of waveform design to resist narrowband radio frequency interference is transformed into suppressing the energy of the waveform power spectrum in a specific frequency band, which is set within the frequency band range. If narrowband radio frequency interference exists, then suppressing narrowband radio frequency interference is equivalent to suppressing the waveform in the frequency band. The spectral energy is expressed as follows: （11）； in, Indicates the lower bound of the frequency band. Indicates the upper bound of the frequency band; S4. Establish a waveform optimization model to resist combined interference; Combining steps S2 and S3, the objective function expression for the waveform optimization problem against combined interference is as follows: （12）； in, and Indicates the weighting factor; The integral sidelobe ratio (ISLR) and peak sidelobe ratio (PSLR) of the single waveform matched filter output are used as metrics; a nonlinear frequency modulated signal is set. The length of the pulse is N, and the pulse compression result after matched filtering is: Its discretization expression is as follows: （13）； in, Indicates signal The results of autocorrelation , Indicates the peak value of the main lobe. Indicates the width of the resolution unit; The peak-to-sidelobe ratio is defined as the ratio of the energy of the first sidelobe to the peak energy of the main lobe after the matched filter response. The integral-to-sidelobe ratio is defined as the ratio of the energy outside the resolution cell to the energy inside the resolution cell after the matched filter response, as shown in the following expressions: （14）； （15）； In summary, the final multi-objective constrained optimization model expression for the anti-combined interference waveform design is as follows: （16）； in, and Indicates the weighting factor. Indicates the upper limit of the constraint; S5. Decompose the multi-objective constrained optimization model into two nonlinear constrained optimization sub-problems, and solve the optimization problem using the augmented Lagrange genetic algorithm; S51. Divide the multi-objective constrained optimization model into two sub-problems; Equation (16) is broken down into two optimization problems and solved one by one, i.e., first optimize the waveform. The spectrum and pulse compression output characteristics, and then in the waveform Based on this, optimize the waveform The spectral density and pulse compression output characteristics, as well as the cross-correlation performance of the two waveforms, are as follows: Optimization issue 1: （17）； Optimization issue 2: （18）； in, Indicates the weighting factor; S52. Solve optimization problems using augmented Lagrange genetic algorithm; First, the augmented Lagrange genetic algorithm combines the fitness function and nonlinear constraint function into a subproblem by introducing Lagrange multipliers and penalty factors, as expressed below: （19）； Where, vector This represents the optimization variable, i.e., frequency point. and time point ,vector This indicates the Lagrange multiplier estimate. It is its components, vector Indicates a non-negative offset. It is its weight. This represents the fitness function, which is the objective function in optimization problems 1 and 2. This represents nonlinear inequality constraints, i.e., the nonlinear constraints in optimization problems 1 and 2. Indicates the number of nonlinear inequality constraints; Next, the optimization problem shown in equation (19) is approximately minimized using a genetic algorithm, such that the linear constraints and boundary conditions are satisfied; when the subproblem is minimized to the required accuracy and the feasibility conditions are met, the Lagrange estimate is updated. This creates a new subproblem, and step S52 is repeated until the constraints are met and the difference in the change of the optimal fitness function is less than the set threshold. S6. Based on steps S1-S5, the NLFM waveform resisting SAR combined interference is obtained, which is the optimized waveform.

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