SAR anti-interference method based on multiple orthogonal frequency modulation waveform design

Abstract

According to the SAR anti-interference method based on the multiple orthogonal frequency modulation waveform design disclosed by the invention, the waveform agility number and orthogonality can be improved while the low sidelobe imaging performance is ensured; in addition, according to the method, a segmented high-degree-of-freedom frequency modulation coding structure is adopted, and the long-distance imaging requirement of a satellite-borne SAR system under the condition of a large time bandwidth product can be met. The method comprises the following steps: S1, constructing a multi-target signal optimization model for minimizing a peak sidelobe ratio and cross-correlation peak energy; s2, designing an improved adaptive genetic algorithm for solving a non-convex complex waveform optimization problem; s3, an MOFM waveform with accurate spectrum control capability is obtained through algorithm optimization; s4, effective suppression of main lobe active interference is realized by using the designed waveform; compared with a traditional linear frequency modulation waveform, the method has the advantages that the low sidelobe characteristic is kept, meanwhile, the orthogonal agility number is remarkably increased, and the method is suitable for the anti-interference imaging requirement of a modern spaceborne SAR system.

Description

Technical Field

[0001] This invention relates to the field of SAR waveform design for anti-interference, and more specifically, to a SAR anti-interference method based on multiple orthogonal frequency modulation waveform design. Background Technology

[0002] Synthetic Aperture Radar (SAR), as an active microwave imaging technology, plays an increasingly important role in Earth observation fields such as military reconnaissance, topographic mapping, and disaster monitoring due to its advantages of all-weather, all-day operation and high resolution. SAR improves azimuth resolution by constructing an equivalent long antenna array through coherent accumulation and signal processing of azimuth signals. Among these, the transmitted waveform is a key factor in spaceborne SAR systems, and its design must consider multiple technical constraints and performance requirements.

[0003] Existing waveforms are insufficient to simultaneously meet the imaging quality and anti-interference requirements of spaceborne SAR systems. The widely used Linear Frequency Modulation (LFM) signal suffers from high sidelobes, range ambiguity, and weak anti-interference performance. First, the power spectral density (PSD) function of the LFM signal is approximately rectangular, with a peak-to-sidelobe ratio of −13.3 dB, limiting imaging performance. Second, excessively wide antenna elevation beam coverage leads to spillover waves returning from outside the mapping zone, causing image quality degradation and range ambiguity. Finally, the LFM signal's limited waveform variety makes it easily intercepted and its parameters predictable, thus restricting the anti-interference performance of SAR systems.

[0004] Alternating Transmission Mode (ATM), based on waveform agility technology, offers a new dimension for addressing the range ambiguity problem and improving anti-interference capabilities of traditional waveforms. ATM enhances anti-interference capabilities while mitigating range ambiguity by alternately transmitting pseudo-orthogonal waveforms at different pulse repetition intervals (PRI). This method requires low cross-correlation peaks (CCP) between different waveforms, thereby suppressing interference signals into band-limited noise and improving anti-interference performance. The peak sidelobe ratio (PSLR) of each waveform determines the imaging quality. Compared to azimuth phase coding (APC) and similar pulse phase coding (PPC) methods, this method does not rely on increasing the pulse repetition frequency (PRF), avoiding the problem of a significant reduction in mapping bandwidth and demonstrating strong application potential in spaceborne SAR systems.

[0005] While phase-coded waveforms offer a good peak-to-side-lobe ratio and can generate pseudo-orthogonal waveform sets with excellent orthogonality, their specific sampling rate requirements significantly limit their application in spaceborne SAR systems. Phase-coded waveforms require a sampling rate that is an integer multiple of the bandwidth, leading to a surge in sampled data in systems with large Time-Bandwidth Products (TBPs), significantly increasing the complexity of real-time signal processing in spaceborne SAR systems. Furthermore, from a hardware implementation perspective, when the system bandwidth and the effective operating bandwidth of the RF devices are on the same order of magnitude, high-spectral-sidelobe components will exceed the device's passband range, causing transmit power loss, which phase-coded waveforms cannot meet due to their high spectral-sidelobe content.

[0006] Nonlinear Frequency Modulation (NLFM) signals possess a constant envelope and precisely control the spectral content, significantly reducing data size while maintaining a large TBP (Total Block Filter) and avoiding high spectral and autocorrelation sidelobes. NLFM signals construct their PSD (Power-Side Detection) function by altering the time-frequency relationship, providing a low-autocorrelation sidelobe intermediate frequency output without sacrificing signal-to-noise ratio (SNR), thus mitigating the high autocorrelation sidelobes of LFM signals and further improving image quality. Simultaneously, the precise control of the PSD function in NLFM signals generates a non-uniform rate of frequency change, concentrating spectral energy more in the main lobe region and reducing spectral sidelobes. The most representative NLFM signal generation method utilizes the Principle of Stationary Phase (POSP) to generate low-autocorrelation sidelobes based on a preset window function. Summary of the Invention

[0007] To address existing problems, this invention proposes a design and optimization method for Multi-Orthogonal Frequency Modulated (MOFM) waveform groups. The proposed MOFM waveform consists of multiple sub-pulse signals, and multiple MOFM waveforms form a MOFM waveform group. NLFM waveforms are selected and generated as sub-pulse signals. Through joint optimization of the time-width coefficient matrix and the frequency modulation coefficient matrix, the spectral content is precisely controlled. Combined with an Adaptive Genetic Algorithm (AGA), a pseudo-orthogonal waveform group design with high imaging quality and controllable quantity is achieved. Under the ATM-based imaging method, this improves the imaging quality of spaceborne SAR, effectively suppresses main lobe interference signals, and constructs a technical solution for novel anti-main lobe interference scenarios.

[0008] This invention discloses a SAR anti-interference method based on multiple orthogonal frequency modulation waveform design, comprising the following steps:

[0009] S1. Define a parameter matrix to jointly characterize the structure of a multi-orthogonal frequency modulation waveform group; the waveform group contains I waveforms, each waveform is composed of M subpulse segments; the parameter matrix includes a time-width coefficient matrix. With frequency modulation coefficient matrix ;in, The element, i.e., the first The first waveform Segment pulse duration factor Satisfy row normalization constraints Used to determine the duration of each sub-pulse as a percentage of the total pulse duration of the waveform. The proportion; elements Used to determine the direction of the frequency modulation slope of each sub-pulse.

[0010] S2. Based on the principle of stationary phase and combined with a preset Taylor window function, a nonlinear frequency modulation signal with constant envelope and optimized power spectral density is generated as the basic sub-pulse signal for constructing the multiple orthogonal frequency modulation waveforms.

[0011] S3. A multi-objective optimization model is established with the maximum value F1 of the peak-to-sidelobe ratio of all waveforms in the waveform group and the maximum value F2 of the cross-correlation peak value between any two different waveforms in the waveform group as joint optimization objectives.

[0012] S4. Using an adaptive genetic algorithm, guided by the joint optimization objective, optimize the time-width coefficient matrix. and the frequency modulation coefficient matrix Iterative optimization is performed; the adaptive genetic algorithm includes population initialization, fitness evaluation, selection, crossover, mutation, and adaptive parameter adjustment operations, ultimately outputting the optimized result. and Based on this, multiple orthogonal frequency modulated waveform groups with low sidelobes and low cross-correlation are generated.

[0013] Preferably, in step S1, the single MOFM waveform emitted by the transmitter can be represented as:

[0014]

[0015] In the formula, The center time of each sub-pulse signal satisfies , This represents the phase function of each sub-pulse signal. Furthermore, due to the bandwidth of each MOFM waveform... It is preset, so the bandwidth of each sub-pulse signal is 1. Similarly, due to the pulse duration of each MOFM waveform Consistency should also be maintained. The pulse duration satisfies the following equation

[0016]

[0017] As M increases, the number of segments of the segmented frequency modulation-based sub-pulse signal in each MOFM waveform increases, which means that the degrees of freedom of the multiple orthogonal waveform groups increase. However, the degrees of freedom and waveform performance are not strongly correlated, and a suitable optimization method is needed to meet the waveform design requirements. This requires establishing an optimization model that includes sub-pulse parameter correlation constraints and optimizing performance indicators by controlling the phase coupling characteristics of the segmented frequency modulation coding structure.

[0018] Let the parametric degrees of freedom of the MOFM waveform group be determined by the time-width coefficient matrix. With frequency modulation coefficient matrix Characterization, in which Satisfying the row normalization constraint shown in the following formula .

[0019] Preferably, in step S2, initial waveform selection refers to designing MOFM sub-pulse signals, including the type and frequency modulation direction of the sub-pulses, under determined parameter conditions, to provide the waveform optimization method as the starting state for the optimization process. The MOFM sub-pulses must satisfy the characteristics of constant bandwidth and uniform spectral energy. The NLFM waveform generated based on the POSP algorithm combined with a window function has adaptability. After the initial waveform is determined, the construction of the MOFM waveform group can be implemented based on an adaptive genetic algorithm optimization framework.

[0020] MOFM uses a POSP-based method to generate NLFM signals as initial waveforms. First, POSP allows the synthesis of suitable signal waveforms using a predefined PSD function for the signal shape. (Baseband radar signal) The analytical form can be expressed as

[0021]

[0022] In the formula, The duration of the pulse. For time variables, To describe the phase of signal modulation, Let be a rectangular window function. instantaneous frequency and modulation phase The differential relation is

[0023]

[0024] It is worth noting that the time-frequency curve f is a function of time t. Similarly, the instantaneous frequency modulation is the differential of f with respect to t.

[0025]

[0026] According to POSP, the PSD of a signal is related to the reciprocal of the instantaneous frequency modulation. The amplitude envelope of the PSD... It can be represented as

[0027]

[0028] in It is a constant. Different window functions have different sidelobe suppression performances. By using different window functions as the amplitude envelope of the power spectrum, NLFM signals with different characteristics can be obtained. Due to the requirements of SAR images on waveform autocorrelation sidelobes and the existence of systematic errors, it is necessary to compare the 3dB main lobe width performance and PSLR performance of the linear frequency modulated signal after window function spectral weighting.

[0029] The Chebyshev window function has the optimal 3dB main lobe width for the same sidelobe height, but its window function ends with a bulge, making it unsuitable for spectral weighting. The Taylor window function is the next best option; its performance is close to that of the Chebyshev window function, but it can better adapt to the requirements of spectral weighting.

[0030] Will Expressed as a function of instantaneous frequency f, It can be represented as The duration of the pulse is... , can be obtained ,in It is the bandwidth of the waveform. A constant. Should be This allows it to simultaneously meet the waveform's duration and bandwidth requirements. Therefore, It can be deduced as It can be concluded that the instantaneous frequency yes reciprocal According to the derivation The waveform can be expressed as MOFM waveforms retain the characteristics of NLFM signals while also possessing frequency modulation coding capabilities. This allows for precise definition of the spectral content, avoids the problem of high Doppler sidelobes, and utilizes a Taylor window-based approach. The generated NLFM signal is used as a sub-pulse signal and encoded into diverse MOFM initial waveforms. An adaptive genetic algorithm optimization framework is used to obtain a MOFM waveform group with low autocorrelation sidelobes and low cross-correlation peak energy.

[0031] Preferred: In step S3, the goal of the optimization problem is to design a device with better PSLR and CCP performance levels. and . No. The parameter configuration of a MOFM waveform is described from two dimensions: time domain structure and frequency domain structure. The time domain structure includes... The nonlinear frequency modulation (NLFM) sub-pulse, where the first... The duration of each sub-pulse can be expressed as: Frequency domain structure: (The rest of the text appears to be a fragment and requires further context for accurate translation.) The frequency modulation direction of each sub-pulse is from It is confirmed that its actual frequency modulation slope is

[0032]

[0033] Therefore, the waveform expression can be reformulated as

[0034]

[0035] The problem of designing MOFM waveform groups was broken down into designing ones with lower performance metrics. and MOFM waveform group, in which Corresponding to the worst value of PSLR within the MOFM waveform group, The objective function corresponding to the worst-case value of CCP within the MOFM waveform group can be expressed as:

[0036]

[0037] Indicates the first in the MOFM waveform group A complete waveform, Indicates the calculation of the first Peak-to-sidelobe ratio of the autocorrelation function of each waveform; Indicates the calculation of the first The and the first The cross-correlation peaks between the waveforms.

[0038] Depend on and The different MOFM waveform groups represented are all obtained through... and The performance of the MOFM waveform group was evaluated, and optimization was performed based on an adaptive genetic algorithm optimization framework.

[0039] Preferred: In step S4, the sidelobe suppression index for the MOFM waveform group and orthogonal performance index A multi-objective optimization problem is solved by designing an adaptive genetic algorithm to optimize the time-width coefficient matrix. and frequency modulation coefficient matrix Perform iterative optimization.

[0040] An adaptive genetic algorithm optimization framework is designed to achieve dynamic optimization of waveform parameters through matrix gene encoding and evolution mechanisms. Represents the maximum number of iterations. Indicates population size, The crossover rate (its calculation formula is...) represents the crossover rate. ,in For the number of individuals in the parent generation, (Number of offspring individuals) The mutation probability, This indicates the number of individuals to be reinitialized. The algorithm's adaptability is reflected in the dynamic adjustment of these parameters based on fitness values ​​during the iteration process. The algorithm employs an elite retention strategy to ensure that high-fitness genotypes are preserved and passed on to the next generation, enhancing its performance while maintaining the integrity of the MOFM waveform structure.

[0041] The main steps of the adaptive genetic algorithm optimization framework include: 1) System parameter initialization: setting waveform parameters, including the number of waveforms. Number of sub-pulse segments Waveform parameters ( , ) and algorithm parameters ( , , , , Initialize the iteration count. .

[0042] 2) Initialize the population: Construct the initial population The individual gene encoding is composed of a time-width coefficient matrix. and frequency modulation coefficient matrix Joint characterization.

[0043] 3) Define a multi-objective fitness function: The fitness of each individual in the population is evaluated, and fitness is defined as...

[0044]

[0045] in These are dynamic weighting coefficients, adjusted by... Achieving sidelobe suppression capability ( ) and waveform orthogonality ( The optimization involves a trade-off. Non-dominated sorting is implemented based on fitness, and a Pareto front solution set is constructed.

[0046] 4) Before selection Each elite individual serves as the parent generation, and offspring individuals are generated through a matrix block crossover strategy, as shown in the following formula.

[0047]

[0048]

[0049] in Represents element-wise product. This represents a vector concatenation operation. For random selection function, It is a non-zero indicator function. Represents the zero norm, Represents an integer range Discrete uniform distribution on, It is a zero-padding vector. Given a random mask matrix, its elements Subscript Indicates parental interpretation. Indicates offspring, , Represents the parent matrix's first generation. Row vector (1×M) represents the row vector of the first row. The time width coefficient of M sub-pulses in a waveform.

[0050] 5) Directed Mutation Mechanism. The mutation rate of parent individuals is dynamically adjusted by comparing the sum of parent fitness with the population average, while being limited to a preset range. For selected mutant individuals, their mutation rate is... Matrix and The random rows of the matrix are randomly reinitialized based on a uniform distribution. Then, Matrix passed through Norm normalization is applied to impose sparsity constraints.

[0051] 6) Adaptive adjustment of AGA parameters. The parameters of the AGA algorithm (including...) , and The remaining quantity will be dynamically adjusted based on performance optimization. Individuals are reinitialized. Specifically, as iterations proceed, elite individuals may get stuck in local optima. Therefore, the algorithm adaptively reduces... This is intended to promote the introduction of new genetic material, thereby enhancing population diversity and preventing premature convergence to suboptimal solutions.

[0052] Beneficial Effects: Compared with existing technologies, the beneficial effects of the proposed SAR anti-jamming method based on multiple orthogonal frequency modulation waveform design are as follows: The proposed SAR anti-jamming method based on multiple orthogonal frequency modulation waveform design can simultaneously improve SAR resolution and anti-jamming capability without increasing the complexity of the radar hardware system, and meet the pulse sampling frequency requirements of complex spaceborne SAR systems. Specifically, it is manifested in:

[0053] 1. By optimizing the design of multiple orthogonal frequency modulation (MOFM) waveform groups, the peak sidelobe ratio (PSLR) is generally better than -36 dB, achieving an improvement of approximately 22.7 dB in sidelobe suppression performance compared to the traditional linear frequency modulation (LFM) signal (-13.3 dB). This significantly compresses range sidelobes, effectively suppressing false targets and background noise in imaging, thereby obtaining higher contrast and cleaner SAR images.

[0054] 2. The designed MOFM waveform group exhibits excellent orthogonality, with a cross-correlation peak (CCP) below -25 dB. Within the framework of Alternating Transmit Mode (ATM), this low cross-correlation effectively suppresses active interference entering the radar main lobe (such as deception interference, coherent suppression interference, and intermittent sampling-forwarding interference) into broadband noise, significantly reducing the impact of interference on imaging quality and enhancing the system's survivability and robustness in complex electromagnetic environments.

[0055] 3. This invention uses nonlinear frequency modulation (NLFM) signals as the basic sub-pulse, inheriting their advantages of constant envelope and low spectral sidelobes, while avoiding stringent requirements on the operating bandwidth of RF devices and reducing the complexity of the transmitter hardware implementation. Simultaneously, this method does not rely on increasing the pulse repetition frequency (PRF) or special integer multiples of the sampling rate, maintaining a large time-bandwidth product (TBP) and long-range imaging capabilities while avoiding the problems of surging data volume and excessive real-time processing burden, making it particularly suitable for resource-constrained spaceborne SAR platforms.

[0056] 4. By introducing a time-width coefficient matrix and a frequency modulation coefficient matrix, and combining them with an improved adaptive genetic algorithm for joint optimization, this invention can precisely control the spectral content and temporal structure of waveform groups. This method can not only generate a considerable number of pseudo-orthogonal waveform groups to meet the requirements of high-dimensional agility, but also allows for flexible trade-offs and customized designs based on specific imaging performance indicators (PSLR) and anti-interference performance indicators (CCP), exhibiting a high degree of design freedom and application flexibility. Attached Figure Description

[0057] Figure 1 A schematic diagram of a SAR anti-interference method based on multiple orthogonal frequency modulation waveform design

[0058] Figure 2 is the normalized spectrum function of the linear frequency modulated signal after being weighted by the Taylor window function, where (a) is the normalized spectrum function of the linear frequency modulated signal after being weighted by the Taylor window function; and (b) is the normalized time-frequency function of the linear frequency modulated signal after being weighted by the Taylor window function.

[0059] Figure 3 It is the autocorrelation function of the linear frequency modulated signal after being weighted by the Taylor window function;

[0060] Figure 4 This is the time-frequency diagram of the initial MOFM waveform;

[0061] Figure 5 This is a schematic diagram of the adaptive genetic algorithm optimization framework;

[0062] Figure 6 It is the autocorrelation function of the MOFM waveform group;

[0063] Figure 7 It is the cross-correlation function of the MOFM waveform group;

[0064] Figure 8 It is the effect of suppressing deception interference from different waveform groups;

[0065] Figure 9 The coherent suppression interference suppression effect of different waveform groups is shown in (a) LFM waveform imaging under no interference conditions, (b) LFM waveform imaging under deception interference conditions, and (c) positive and negative LFM waveform group imaging under deception interference conditions.

[0066] Figure 10 Among them, (a) is the LFM waveform image under intermittent sampling and forwarding interference, (b) is the image composed of positive and negative LFM waveforms under intermittent sampling and forwarding interference, and (c) is the image composed of the proposed waveform under intermittent sampling and forwarding interference. Detailed Implementation

[0067] The following detailed description, with reference to the accompanying drawings, illustrates a SAR anti-interference method based on multiple orthogonal frequency modulation waveform design proposed in this invention. Without loss of generality, the specific implementation steps of this method are described below using a SAR system applying this invention as an example, such as... Figure 1 As shown.

[0068] Step S1: Define a parameter matrix that collectively characterizes the structure of a multi-orthogonal frequency modulation waveform group.

[0069] A single MOFM waveform emitted by the transmitter can be represented as:

[0070]

[0071] In the formula, The center time of each sub-pulse signal satisfies , This represents the phase function of each sub-pulse signal. Furthermore, due to the bandwidth of each MOFM waveform... It is preset, so the bandwidth of each sub-pulse signal is 1. Similarly, due to the pulse duration of each MOFM waveform Consistency should also be maintained. The pulse duration satisfies the following equation

[0072]

[0073] As the number of sub-pulse segments M increases, the number of segments of the segmented frequency modulation-based sub-pulse signal in each MOFM waveform will increase, which means that the degrees of freedom of the multiple orthogonal waveform group increases. However, the degrees of freedom and the performance of the waveform are not strongly correlated. A suitable optimization method is needed to meet the waveform design requirements. This requires establishing an optimization model that includes sub-pulse parameter correlation constraints and optimizing performance indicators by controlling the phase coupling characteristics of the segmented frequency modulation coding structure.

[0074] Let the parametric degrees of freedom of the MOFM waveform group be determined by the time-width coefficient matrix. With frequency modulation coefficient matrix Characterization, in which Satisfying the row normalization constraint shown in the following formula

[0075]

[0076] No. The duration of the segment pulse can be expressed as .

[0077] Frequency modulation coefficient matrix Its elements Used to determine the direction of the frequency modulation slope (positive or negative slope) of each sub-pulse.

[0078] Step S2: Based on the principle of stationary phase and combined with a preset Taylor window function, a nonlinear frequency modulation signal with constant envelope and optimized power spectral density is generated as the basic sub-pulse signal for constructing the multiple orthogonal frequency modulation waveforms.

[0079] Step S2, initial waveform selection, refers to designing MOFM sub-pulse signals, including the type and frequency modulation direction of the sub-pulses, under the given parameter conditions, to provide the waveform optimization method as the starting state for the optimization process. The MOFM sub-pulses must satisfy the characteristics of constant bandwidth and uniform spectral energy. The NLFM waveform generated based on the POSP algorithm combined with a window function has adaptability. After the initial waveform is determined, the construction of the MOFM waveform group can be implemented based on an adaptive genetic algorithm optimization framework.

[0080] MOFM uses a POSP-based method to generate NLFM signals as initial waveforms. First, POSP allows the synthesis of suitable signal waveforms using a predefined PSD function for the signal shape. (Baseband radar signal) The analytical form can be expressed as

[0081]

[0082] In the formula, The duration of the pulse. For time variables, To describe the phase of signal modulation, Let be a rectangular window function. instantaneous frequency and modulation phase The differential relation is

[0083]

[0084] It is worth noting that the time-frequency curve f is a function of time t. Similarly, the instantaneous frequency modulation is the differential of f with respect to t.

[0085]

[0086] According to POSP, the PSD of a signal is related to the reciprocal of the instantaneous frequency modulation. The amplitude envelope of the PSD... It can be represented as

[0087]

[0088] in It is a constant. Different window functions have different sidelobe suppression performances. By using different window functions as the amplitude envelope of the power spectrum, NLFM signals with different characteristics can be obtained. Due to the requirements of SAR images on waveform autocorrelation sidelobes and the existence of systematic errors, it is necessary to compare the 3dB main lobe width performance and PSLR performance of the linear frequency modulated signal after window function spectral weighting.

[0089] The Chebyshev window function has the optimal 3dB main lobe width for the same sidelobe height, but its window function ends with a bulge, making it unsuitable for spectral weighting. The Taylor window function is the next best option; its performance is close to that of the Chebyshev window function, but it can better adapt to the requirements of spectral weighting.

[0090] Will Expressed as a function of instantaneous frequency f, It can be represented as The duration of the pulse is... , can be obtained ,in It is the bandwidth of the waveform. A constant. Should be This allows it to simultaneously meet the waveform's duration and bandwidth requirements. Therefore, It can be deduced as It can be concluded that the instantaneous frequency yes reciprocal According to the derivation The waveform can be expressed as MOFM waveforms retain the characteristics of NLFM signals while also possessing frequency modulation coding capabilities. This allows for precise definition of the spectral content, avoids the problem of high Doppler sidelobes, and utilizes a Taylor window-based approach. The generated NLFM signal is used as a sub-pulse signal and encoded into diverse MOFM initial waveforms. An adaptive genetic algorithm optimization framework is used to obtain a MOFM waveform group with low autocorrelation sidelobes and low cross-correlation peak energy.

[0091] The normalized spectrum function, normalized time-frequency function, and autocorrelation function of the linear frequency modulated signal weighted by the Taylor window function are as follows: Figure 2 (a) and 2(b) and Figure 3 express.

[0092] Step S3: Establish a multi-objective optimization model with the maximum value F1 of the peak-to-sidelobe ratio of all waveforms in the waveform group and the maximum value F2 of the cross-correlation peak value between any two different waveforms in the waveform group as joint optimization objectives.

[0093] In step S3, the goal of the optimization problem is to design a device with better PSLR and CCP performance levels. and . No. The parameter configuration of a MOFM waveform is described from two dimensions: time domain structure and frequency domain structure. The time domain structure includes... The nonlinear frequency modulation (NLFM) sub-pulse, where the first... The duration of each sub-pulse can be expressed as: Frequency domain structure: (The rest of the text appears to be a fragment and requires further context for accurate translation.) The frequency modulation direction of each sub-pulse is from It is confirmed that its actual frequency modulation slope is

[0094]

[0095] Therefore, the waveform expression can be reformulated as

[0096]

[0097] The problem of designing MOFM waveform groups was broken down into designing ones with lower performance metrics. and MOFM waveform group, in which Corresponding to the worst value of PSLR within the MOFM waveform group, The objective function corresponding to the worst-case value of CCP within the MOFM waveform group can be expressed as:

[0098]

[0099] Depend on and The different MOFM waveform groups represented are all obtained through... and The performance of the MOFM waveform group was evaluated, and optimization was performed based on an adaptive genetic algorithm optimization framework.

[0100] Step S4: Using an adaptive genetic algorithm, guided by the joint optimization objective, optimize the time-width coefficient matrix. and the frequency modulation coefficient matrix Through iterative optimization, a multi-orthogonal frequency modulated waveform group with low sidelobes and low cross-correlation is finally generated.

[0101] In step S4, the sidelobe suppression index for the MOFM waveform group is... and orthogonal performance index A multi-objective optimization problem is solved by designing an adaptive genetic algorithm to optimize the time-width coefficient matrix. and frequency modulation coefficient matrix Perform iterative optimization. The time-frequency plot of the initial MOFM waveform is as follows: Figure 4 As shown.

[0102] An adaptive genetic algorithm optimization framework is designed to achieve dynamic optimization of waveform parameters through matrix gene encoding and evolution mechanisms. Represents the maximum number of iterations. Indicates population size, The crossover rate (its calculation formula is...) represents the crossover rate. ,in For the number of individuals in the parent generation, (Number of offspring individuals) The mutation probability, This indicates the number of individuals to be reinitialized. The algorithm's adaptability is reflected in the dynamic adjustment of these parameters based on the fitness value during the iteration process. This algorithm employs an elite retention strategy to ensure that high-fitness genotypes are preserved and passed on to the next generation, enhancing its performance while maintaining the integrity of the MOFM waveform structure. A schematic diagram of the adaptive genetic algorithm optimization framework is shown below. Figure 5 As shown.

[0103] The main steps of the adaptive genetic algorithm optimization framework include:

[0104] 1) System parameter initialization: Set waveform parameters, including the number of waveforms. Number of sub-pulse segments Waveform parameters ( , ) and algorithm parameters ( , , , , Initialize the iteration count. .

[0105] 2) Initialize the population: Construct the initial population The individual gene encoding is composed of a time-width coefficient matrix. and frequency modulation coefficient matrix Joint characterization.

[0106] 3) Define a multi-objective fitness function: The fitness of each individual in the population is evaluated, and fitness is defined as...

[0107]

[0108] in These are dynamic weighting coefficients, adjusted by... Achieving sidelobe suppression capability ( ) and waveform orthogonality ( The optimization involves a trade-off. Non-dominated sorting is implemented based on fitness, and a Pareto front solution set is constructed.

[0109] 4) Before selection Each elite individual serves as the parent generation, and offspring individuals are generated through a matrix block crossover strategy, as shown in the following formula.

[0110]

[0111]

[0112] in Represents element-wise product. This represents a vector concatenation operation. For random selection function, It is a non-zero indicator function. Represents the zero norm, index , Represents an integer range Discrete uniform distribution on, It is a zero-padding vector. Given a random mask matrix, its elements Subscript Indicates parental interpretation. Indicates offspring.

[0113] 5) Directed Mutation Mechanism. The mutation rate of parent individuals is dynamically adjusted by comparing the sum of parent fitness with the population average, while being limited to a preset range. For selected mutant individuals, their mutation rate is... Matrix and The random rows of the matrix are randomly reinitialized based on a uniform distribution. Then, Matrix passed through Norm normalization is applied to impose sparsity constraints.

[0114] 6) Adaptive adjustment of AGA parameters. The parameters of the AGA algorithm (including...) , and The remaining quantity will be dynamically adjusted based on performance optimization. Individuals are reinitialized. Specifically, as iterations proceed, elite individuals may get stuck in local optima. Therefore, the algorithm adaptively reduces... This is intended to promote the introduction of new genetic material, thereby enhancing population diversity and preventing premature convergence to suboptimal solutions.

[0115] The experiment designed a MOFM waveform group with four waveforms based on an adaptive genetic algorithm parameter optimization framework, with each waveform pulse duration... and bandwidth Set to 60 uniformly and 40MHz, where the NLFM sub-pulse signal is based on the Taylor window function ( The number of individuals in the population is generated. In the optimization parameters of the adaptive genetic algorithm, this includes... The crossover probability is 1024. The mutation probability is 0.5. The value was 0.05, and the population underwent 100 rounds of adaptive genetic algorithm iteration. A larger population size provides greater diversity, increasing the chance of finding the global optimum. A higher crossover probability leads to more gene recombination, increasing diversity among individuals, but also introduces fewer new gene combinations, potentially leading to local optima. A lower crossover probability may result in insufficient population diversity, affecting genetic performance. Mutation is used to introduce new gene combinations and prevent the population from prematurely converging to local optima. By optimizing key parameters such as the number of sub-pulse segments, pulse width coefficient, and frequency modulation coefficient, a pseudo-orthogonal waveform group with a differentiated frequency modulation coding structure was constructed. The ACF and CCF of the optimized MOFM waveform group with four pseudo-orthogonal waveforms are shown below. Figure 6 Experimental data show that the ACF sidelobe energy of the MOFM waveforms in this set exhibits a uniform distribution, with PSLR values ​​all below -36 dB. Compared to the PSLR value of -13.3 dB for traditional LFM signals, this represents an improvement of approximately 22.7 dB in sidelobe suppression performance. Simultaneously, the CCF energy distribution demonstrates good orthogonality, with all mismatch energies below -25 dB and no significant mismatch energy focusing phenomenon observed. This indicates that the optimized waveform set possesses excellent orthogonality, avoiding the problem of false targets even under pseudo-orthogonal conditions.

[0116] Based on a spaceborne SAR scenario, a surface target simulation of typical main lobe interference patterns is performed to verify the imaging quality and anti-interference performance of the proposed method. The method suppresses the echo energy of the interference signal to band-limited noise with the same energy intensity, thereby improving imaging quality. In a spaceborne SAR scenario, a MOFM waveform group with four waveforms effectively counters deception interference (…). Figure 7Coherent suppression interference ( Figure 8 ) and intermittent sampling and forwarding interference ( Figure 9 The suppression effect shows that this waveform set has the advantages of a large number of orthogonal waveforms, low interception probability, and strong anti-interference capability, and its interference suppression capability is significantly better than that of traditional waveforms. Experiments confirm that the optimized MOFM waveform set can significantly enhance the anti-interference performance of the spaceborne SAR system in complex electromagnetic environments, providing a reliable guarantee for high-resolution SAR imaging.

Claims

1. A SAR anti-interference method based on multiple orthogonal frequency modulation waveform design, characterized in that, Includes the following steps: S1. Define a parameter matrix to jointly characterize the structure of a multi-orthogonal frequency modulation waveform group; the waveform group contains I waveforms, each waveform is composed of M subpulse segments; the parameter matrix includes a time-width coefficient matrix. With frequency modulation coefficient matrix ;in, The element, i.e., the first The first waveform Segment pulse duration factor Satisfy row normalization constraints Used to determine the duration of each sub-pulse as a percentage of the total pulse duration of the waveform. The proportion; elements Used to determine the direction of the frequency modulation slope of each sub-pulse; S2. Based on the principle of stationary phase and combined with a preset Taylor window function, a nonlinear frequency modulation signal with constant envelope and optimized power spectral density is generated as the basic sub-pulse signal for constructing the multiple orthogonal frequency modulation waveforms. S3. A multi-objective optimization model is established with the maximum value of the peak-to-sidelobe ratio of all waveforms in the waveform group, F1, and the maximum value of the cross-correlation peak value between any two different waveforms in the waveform group, F2, as the joint optimization objectives. S4. Using an adaptive genetic algorithm, guided by the joint optimization objective, optimize the time-width coefficient matrix. and the frequency modulation coefficient matrix Iterative optimization is performed; the adaptive genetic algorithm includes population initialization, fitness evaluation, selection, crossover, mutation, and adaptive parameter adjustment operations, ultimately outputting the optimized result. and Based on this, multiple orthogonal frequency modulated waveform groups with low sidelobes and low cross-correlation are generated.

2. The method according to claim 1, characterized in that, Step S2 specifically includes the following steps: S2.1 Based on the stationary phase principle, the power spectral density of the signal is weighted using a Taylor window function with preset parameters; S2.

2. Based on the weighted power spectral density, synthesize a nonlinear frequency-modulated signal with a constant envelope; S2.

3. Use the nonlinear frequency modulation signal as the basic sub-pulse signal for constructing the multiple orthogonal frequency modulation waveform.

3. The method according to claim 1, characterized in that, Step S3 specifically includes the following steps: S3.1, Define the first optimization objective F1 as minimizing the maximum value of the peak-to-sidelobe ratio of all waveforms within the waveform group, i.e. ; Indicates the first in the MOFM waveform group A complete waveform, Indicates the calculation of the first The peak-to-side-lobe ratio of the autocorrelation function of each waveform; S3.2, Define the second optimization objective F2 as minimizing the maximum value of the cross-correlation peak value between any two different waveforms within the waveform group, i.e. ,in ; Indicates the calculation of the first The and the first The cross-correlation peaks between the waveforms; S3.3, with the joint optimization of F1 and F2 as the objective, a multi-objective optimization model is established. The objective of the model is to find the time-width coefficient matrix that optimizes F1 and F2. With frequency modulation coefficient matrix .

4. The method according to claim 3, characterized in that, Step S4 specifically includes the following sub-steps: S4.1 Initialize the population of the adaptive genetic algorithm, and set the time-width coefficient matrix. With the frequency modulation coefficient matrix Encoded as an individual within the population; S4.

2. Evaluate individuals in the population based on the fitness function, wherein the fitness function F is defined as follows: ,in These are dynamically adjustable weighting coefficients; S4.

3. Based on the evaluation results, perform selection, crossover, and mutation operations to generate a new generation of population; wherein, the crossover operation includes adjusting the time-width coefficient matrix. Matrix block crossover and frequency modulation coefficient matrix S4.

4. Iteratively execute steps S4.2 to S4.3, and adaptively adjust the algorithm parameters during the iteration until the termination condition is met, and output the optimized parameter matrix; S4.

5. Generate the final multiple orthogonal frequency modulation waveform group based on the optimized parameter matrix.

5. The method according to claim 1, characterized in that, In step S1, the time-width coefficient matrix The frequency modulation coefficient matrix .

6. The method according to claim 1, characterized in that, In step S1, the m-th segment sub-pulse signal of the i-th waveform in the waveform group The mathematical expression is: in, The center time of the sub-pulse is... The duration of this sub-pulse. , This represents the phase function of the sub-pulse.

7. The method according to claim 2, characterized in that, The Taylor window function with preset parameters has adjustable parameters. The sidelobe level is designed to be -37 dB.

8. The method according to claim 4, characterized in that, The adaptive genetic algorithm employs an elite retention strategy, and in step S4.4, it dynamically adjusts the crossover probability and mutation probability based on the evolutionary state of the population.

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