An alternating projection array synthesis method based on lobe correction
By treating the beamlobes as a whole for correction in the alternating projection method, the problems of numerous iterations and slow iteration process are solved, achieving efficient array synthesis that is suitable for array antennas with arbitrary arrangements.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- UNIV OF ELECTRONICS SCI & TECH OF CHINA
- Filing Date
- 2026-03-31
- Publication Date
- 2026-06-16
AI Technical Summary
Existing alternating projection methods involve numerous iterations in array synthesis, and point-by-point correction strategies ignore the physical continuity within the lobe, resulting in slow iteration processes and a tendency for non-physical truncation, which affects engineering feasibility.
An alternating projection method based on lobe correction is adopted, which treats each lobe of the radiation pattern as a whole for correction. Iterative optimization is performed through constraints and matrix relationships across the entire region to reduce the number of iterations.
It significantly reduced the number of iterations, improved iteration efficiency, avoided the plateau effect during iteration, and enhanced engineering feasibility and overall efficiency.
Smart Images
Figure CN121965138B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of array synthesis, and more specifically to an alternating projection array synthesis method based on lobe correction. Background Technology
[0002] Low-sidelobe array synthesis aims to improve the antenna's anti-jamming performance by optimizing the excitation or arrangement of array elements to form a high-gain main beam in the target direction while strictly suppressing the radiation level in the sidelobe region. This technology has been widely used in wireless communication, radar systems, electronic countermeasures, and satellite navigation. As the performance requirements for array antennas in related technological fields gradually increase, low-sidelobe array synthesis methods will play an increasingly important role.
[0003] Traditional low-sidelobe synthesis methods, such as Chebyshev synthesis, Taylor synthesis, and various global optimization algorithms, each have their own applicable scope and limitations. While Chebyshev and Taylor synthesis methods are classic and efficient, they are typically only applicable to regular arrays and struggle with irregular arrays or complex beamforming requirements. Global optimization algorithms such as genetic algorithms and particle swarm optimization, despite their high flexibility, are unsuitable for large-scale array synthesis due to their high computational complexity and slow convergence speed. Therefore, the alternating projection method, as an efficient and flexible iterative optimization framework, has been widely studied. This method transforms the array synthesis problem into a mathematical problem of finding the intersection between the feasible set and the desired set. By repeatedly projecting onto the two sets, it gradually approximates a solution that simultaneously satisfies engineering feasibility and beam performance indicators.
[0004] In alternating projection methods, the design of the desired projection is particularly important. Existing methods generally employ a point-by-point correction strategy, which independently processes the amplitude and phase of each discrete sampling point on the array pattern. While this strategy is versatile, in practical applications it ignores the physical continuity between sampling points within the same lobe, often introducing non-physical truncations or dips within a single lobe, thus compromising the engineering feasibility of the beam. Furthermore, when the lobe region happens to cross the transition zone of a given beam specification, or when only a few sampling points in the lobe do not meet the constraints, the point-by-point correction strategy will lead to a slow iteration process and a significant plateau effect, severely impacting the overall iteration efficiency. These problems collectively cause existing alternating projection methods to require a large number of iterations to obtain the final solution. Summary of the Invention
[0005] To address the aforementioned problems or shortcomings, this invention provides an alternating projection array synthesis method based on lobe correction. By employing an alternating projection framework, each lobe of the radiation pattern is treated as a whole and corrected in the desired projection, which can significantly reduce the number of iterations compared to point-by-point correction.
[0006] A method for synthesizing alternating projection arrays based on beamline correction specifically includes the following steps:
[0007] Step 1: Data preprocessing: Extract the vector active element radiation pattern of each element in the given array, and convert and store it in the array manifold matrix.
[0008] First, the direction of the vector active element is obtained through full-wave simulation or experimental measurement, and then interpolated to... domain On each sampling point; then, The point sampling vector active element radiation pattern data is decomposed into common polarization and cross polarization directions; finally, the decomposed common polarization and cross polarization component data are multiplied by the spatial phase factor corresponding to each array element and sampling point, and stored in the array manifold matrix.
[0009] Step 2: Based on the given beam index, provide the full-area boundary function and constraints for the desired radiation pattern.
[0010] Given beam indices for a specified main lobe and side lobe region, and defining upper and lower boundary values or thresholds within the region; first, extend the upper and lower boundary values and thresholds into functions to support setting boundary values and thresholds that vary in different regions, and list the extended constraints; then, modify the extended functions and constraints into boundary functions and constraints for the entire region.
[0011] Step 3: Set the maximum number of iterations and the initial excitation vector, and calculate the initial array pattern vector.
[0012] Step 4: Perform desired projection based on lobe correction on the current array pattern vector to obtain the corrected array pattern vector.
[0013] First, calculate the region where each lobe is located in the bias matrix and the co-polarization and cross-polarization components of the current array pattern vector; then, count the maximum and minimum values of the bias matrix in each lobe region and their corresponding sampling points; finally, calculate the correction factors for co-polarization and cross-polarization on a lobe-by-lobe basis and perform the desired projection based on the lobe correction.
[0014] Step 5: Perform feasible projection on the corrected array pattern vector to obtain the feasible array pattern vector.
[0015] Step 6: Repeat steps 4 and 5 until the array pattern vector under the current excitation satisfies the full-area constraint conditions of the desired pattern given in step 2, or reaches the maximum number of iterations set in step 3. Output the array pattern vector and its corresponding excitation vector at this time.
[0016] Furthermore, step 1 specifically includes:
[0017] Consider one An arbitrary irregular planar array antenna in the global coordinate system The pattern of a point-sampled vector array can be represented as:
[0018] (1)
[0019] in, for Domain One sampling point, and For the first The pitch and azimuth angles corresponding to each sampling point. For the first The complex excitation of each array element includes amplitude and phase. For the first Vector active element radiation pattern of each array element. The imaginary unit, Let be the wave number in free space. For the first The spatial position of each array element for and linear combination, For the first The array element in the first Spatial phase factor at each sampling point.
[0020] The vector active element radiation pattern of each array element is obtained through full-wave simulation or experimental measurement, and then interpolated to... domain On each sampling point. Point-sampled vector active cell radiation pattern data can be decomposed into common polarization and cross-polarization directions, i.e. .
[0021] In the formula For common polarization components, For cross-polarization components, and Let be the unit vectors in the common polarization and cross polarization directions, respectively. Therefore, equation (1) can be expressed as:
[0022] (2)
[0023] In the formula, and These represent the common polarization and cross-polarization components of the vector array pattern, respectively. The following vectors and matrices are defined:
[0024] (3)
[0025] (4)
[0026] (5)
[0027] make , ;
[0028] The matrix product relationship of the common polarization and cross polarization components of the vector array pattern is: In the formula, This is the array pattern vector. For array manifold matrix, The excitation vector. The array manifold matrix. The elements in the vector active unit pattern are common polarized. and cross-polarization components Multiply by spatial phase factor get.
[0029] Furthermore, step 2 specifically includes:
[0030] Given beam parameters: (1) In the main lobe region Within, the upper and lower boundary values of the common polarization component are and (2) In the sidelobe region Within, the co-polarization component is below the threshold. (3) In the total sampling area That is, in step 1 The set of sampling points has cross-polarization components all below the threshold. .
[0031] The aforementioned beam indexes represent the constraints of the desired radiation pattern. The upper and lower boundary values can be expanded into functions that vary with the sampling points. and Support the addition of zero-depression regions in the sidelobe area. and its threshold The original sidelobe threshold is extended to the following function:
[0032] (6)
[0033] Meanwhile, to maintain consistency, the threshold for cross-polarization is expressed as a function. .
[0034] The extended constraints can be expressed as:
[0035] (7)
[0036] Considering the subsequent solution of the deviation pattern, the extended constraints are modified to the following full-domain form:
[0037] (8)
[0038] The upper and lower boundary functions for the common polarization and cross polarization components are:
[0039] (9)
[0040] (10)
[0041] (11)
[0042] (12)
[0043] Furthermore, step 3 specifically includes:
[0044] Set the maximum number of iterations. No less than 5,000 times.
[0045] Set the initial excitation vector And solve for the initial array pattern vector. .
[0046] The initial activation vector can be uniform, Chebyshev, or random, with no specific requirements; setting an appropriate initial activation vector can reduce the number of iterations and speed up convergence.
[0047] Furthermore, step 4 specifically involves:
[0048] The full-area constraints of the desired radiation pattern given in step 2 are simply referred to as the desired constraints. The deviation of the array radiation pattern vector from the desired constraints is defined as the deviation radiation pattern, expressed as:
[0049] (13)
[0050] Let the deviation matrix Boundary matrix This can be transformed into the following matrix relationship:
[0051] (14)
[0052] In this context, square brackets represent Boolean matrices.
[0053] Calculate the regions containing each lobe in the common-polarization and cross-polarization components of the current array pattern vector, assuming the total number of lobes is... The regions where each lobe is located are denoted as .
[0054] Statistical bias matrix in the region of each lobe The maximum and minimum values within the range and their corresponding sampling points:
[0055] (15)
[0056] In the formula, The maximum value, To be the minimum value, and These are the sampling points corresponding to the maximum and minimum values, respectively.
[0057] Define the desired projection based on lobe correction That is, the corrected array pattern vector To simplify the formula, let's denote... The expected projections of the co-polarization and cross-polarization components are expressed as:
[0058] (16)
[0059] In the formula, for phase, for The phase factor. and The first Correction factors for lobe co-polarization and cross-polarization;
[0060] (17)
[0061] In the formula, This is an overpressure factor used to accelerate convergence.
[0062] Furthermore, step 5 specifically includes:
[0063] The corrected array pattern vector obtained in step 4 The closest excitation vector is found using the least squares method:
[0064] (18)
[0065] (19)
[0066] in, for The left pseudo-inverse matrix. Superscript The superscript represents the conjugate transpose of a matrix. Represents the inverse of a matrix. It is an identity matrix. This is the regularization coefficient, used to improve computational stability. The corresponding achievable array pattern vector is Define feasible projections. ,Right now .
[0067] Furthermore, step 6 specifically involves:
[0068] Assume the number of iterations is The time array pattern vector satisfies the full-area constraint conditions of the desired pattern given in step 2, or the number of iterations reaches the maximum number of iterations set in step 3. The array pattern vector at this time is:
[0069] (20)
[0070] The corresponding activation vector is .
[0071] It should be noted that while transforming the correction unit from a "point" to a "lobe" is not formally complex, it has long been neglected in a systematic and in-depth study. The root causes can be summarized in the following three points:
[0072] First, the alternating projection method naturally discretizes the radiation pattern into a sequence of sampling points during mathematical modeling, thus naturally forming a mindset of processing point by point. This processing is very convenient in mathematical form, as the amplitude and phase constraints of each sampling point are independent, and the projection operation can be simplified to direct threshold truncation or scaling, making it simple to implement and computationally lightweight.
[0073] Secondly, from a historical application perspective, early research primarily focused on one-dimensional linear arrays or planar arrays with separable radiation patterns. In this case, the lobe structure was clear, the main lobe and each side lobe were well-defined, and point-by-point correction strategies could achieve acceptable engineering results. Therefore, there was no strong demand for overall optimization at the lobe level.
[0074] A deeper reason lies in the technical challenges of using "lobes" as processing units. Specifically, accurately defining and identifying each lobe region in the radiation pattern is no easy task. Especially in two-dimensional or irregular array synthesis, the complex beam geometry and varied structures of the main and side lobes make precise segmentation of lobe regions difficult. In contrast, the point-by-point processing strategy cleverly bypasses this identification problem by independently constraining discrete sampling points, making it more flexible and uniform in form. However, the cost of this "universality" is ignoring the inherent continuity of lobes, leading to slow convergence of the iterative process and the potential for non-physical oscillations within the lobes, which impairs the quality of the solution and its engineering feasibility.
[0075] In summary, this invention employs an alternating projection framework, which offers good versatility and is independent of array layout type. Furthermore, the alternating projection framework has low computational complexity and high synthesis efficiency. This invention considers the vector active element radiation pattern during synthesis, resulting in a highly accurate synthesized array radiation pattern. In the desired projection, this invention breaks away from the long-standing mindset of point-by-point processing in alternating projection methods, using "lobes" as the core processing unit. This ensures the continuity of lobes is not disrupted during correction while effectively overcoming the plateau effect caused by local fine-tuning in point-by-point correction. Specifically, if any point in a lobe violates the constraint conditions, a correction factor is calculated based on the maximum deviation within that lobe and then applied to all sampling points within that lobe. This invention summarizes this correction process as a consistency correction formula based on lobe correction for common polarization and cross-polarization components. Simultaneously, to avoid significantly increasing the additional computational load caused by this correction process in a single iteration, this invention converts the given beam index into constraints and boundary matrices for the entire region, and converts the expressions for the array radiation pattern, deviation radiation pattern, and feasible projection into matrix relationships. These matrix relationships and the consistency correction formulas based on lobe correction for the common polarization and cross polarization components provided in the desired projection all conform to the alternating projection framework and have high computational efficiency. This invention is applicable to array antennas with arbitrary arrangements and can be used for low sidelobe synthesis of large-scale array antennas. Compared with existing point-by-point correction alternating projection methods, it significantly reduces the number of iterations required for array synthesis. Attached Figure Description
[0076] Figure 1 The flowchart shows the overall process of the alternating projection array synthesis method based on beam lobe correction.
[0077] Figure 2 The common polarization patterns obtained in the examples and comparative examples are shown below;
[0078] Figure 3 The cross-polarization patterns obtained in the examples and comparative examples are shown below;
[0079] Figure 4 (a) is a diagram showing the amplitude distribution of the excitation obtained in the embodiment and the comparative example. Figure 4 (b) is a phase distribution diagram of the excitation obtained in the embodiment and the comparative example;
[0080] Figure 5 The curves show the change in the number of sampling points that do not meet the expected constraints during the iteration process for the examples and comparative examples. Detailed Implementation
[0081] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings.
[0082] The core of this invention lies in elevating the correction unit in alternating projection from a "point" to a "lobe" level, and verifying the effectiveness of this strategy in improving convergence efficiency. To clearly verify the advantages of this method and avoid getting bogged down in discussions about lobe identification problems in complex beamforms, the embodiments are selected for verification in a typical and structurally well-defined scenario: a one-dimensional linear array. Specifically, taking a 65-element linear array as an example, the desired common-polarization sidelobe level and cross-polarization level are below -55dB; and Within the null region, the common-polarization null level is below -80dB. Using the alternating projection array synthesis method based on lobe correction as described in this invention, the maximum number of iterations is set to 10,000, the overvoltage factor to 0.8, and the total number of sampling points to 1024. The solution process is as follows: Figure 1 As shown, the specific implementation steps are as follows:
[0083] Step 1: Data preprocessing.
[0084] First, the vector active element pattern of a 65-element linear array is obtained through full-wave simulation. Since this embodiment considers a along-wave pattern... The axially distributed linear array is synthesized, therefore the principal tangent direction of the vector active element radiation pattern is ( One-dimensional data Interpolation to of At each sampling point, in this embodiment Each sampling point is uniformly sampled. Then, The point-sampled vector active element radiation pattern data is decomposed into common-polarization and cross-polarization directions. Finally, the decomposed common-polarization and cross-polarization components are multiplied by the spatial phase factor corresponding to each array element and sampling point, and stored in the array manifold matrix. middle.
[0085] Step 2: Based on the given beam index, provide the full-area boundary function and constraints for the desired radiation pattern.
[0086] Given beam specifications: common-polarization sidelobe level and cross-polarization level below -55dB; in the null region Within this range, the zero-dip level is below -80 dB. The normalized electric field amplitudes corresponding to the two thresholds are respectively... and ,Right now , To ensure the completeness of the constraints, the lower boundaries of the common polarization and cross polarization components are supplemented. and the copolarization component in the main lobe region upper boundary In this embodiment, the array synthesis aims to obtain a pencil beam with ultra-low sidelobes and two null regions. The main beam broadens as the sidelobe level decreases during iteration. Therefore, the main lobe region is not specified in this embodiment. Instead, it is calculated in real time during the iteration process. Meanwhile, the total sampling area in this embodiment... That is, in step 1 of A set of uniformly sampled points. Due to the main lobe region. The sidelobe region needs to be calculated in real time during the iteration process. It also needs to be calculated and updated in real time during the iteration process.
[0087] In this embodiment, the full-area constraint condition for the desired radiation pattern is:
[0088] (8)
[0089] Substituting the parameters in the given beam index into equations (9)-(12), the full-area upper and lower boundary functions of the common polarization and cross polarization components in this embodiment are:
[0090] (twenty one)
[0091] (twenty two)
[0092] (twenty three)
[0093] (twenty four)
[0094] Step 3: Set the maximum number of iterations and the initial excitation vector, and calculate the initial array pattern vector.
[0095] In this embodiment, the maximum number of iterations is set to 10000, and the initial activation vector is set to a column vector consisting entirely of 1s. And solve for the initial array pattern vector. .
[0096] Step 4: Perform desired projection based on lobe correction on the current array pattern vector to obtain the corrected array pattern vector.
[0097] First, calculate the deviation matrix. as well as Regions where each lobe is located Then, statistics In each The maximum and minimum values within the range and their corresponding sampling points are determined; finally, the correction factor is calculated lobe-by-lobe. And perform the desired projection based on lobe correction. In this embodiment, the overpressure factor Set it to 0.8.
[0098] Step 5: Perform feasible projection on the corrected array pattern vector to obtain the feasible array pattern vector.
[0099] Calculate the left pseudo-inverse matrix And perform feasible projection. In this embodiment, the regularization coefficient Setting it to 0.01 applies a normalization constraint to the excitation obtained by the least squares method. A dynamic range constraint can also be applied if needed. The feasible projection after applying the normalized excitation constraint is modified as follows:
[0100] (25)
[0101] Step 6: Repeat steps 4 and 5 until the array pattern vector under the current excitation satisfies the full-area constraint conditions of the desired pattern given in step 2, or reaches the maximum number of iterations set in step 3. Output the array pattern vector and its corresponding excitation vector at this time.
[0102] The array synthesis was performed using the alternating projection method based on beamline correction proposed in this invention, and compared with the traditional alternating projection method, namely the alternating projection method based on point correction. The array radiation patterns obtained in the embodiments and comparative examples are shown below. Figure 2 and Figure 3 As shown, the sidelobe level and cross-polarization level of the common polarization reach -55dB, and the null level of the common polarization reaches -80dB, all of which meet the given beam specifications. Figure 4 (a) shows the amplitude distribution of the array element excitation obtained in the embodiment and the comparative example. Figure 4 (b) shows the phase distribution of the array element excitation obtained in the example and the comparative example. Figure 5 The figures show the change curves of the number of sampling points that do not meet the expected constraints during the iteration process in the examples and comparative examples. The number of sampling points that do not meet the expected constraints will be referred to as the target value. It can be observed that the alternating projection method based on lobe correction significantly outperforms the traditional method in terms of the rate of decrease of the target value, exhibiting a steeper decreasing trend in its convergence process. Specifically, the proposed method shows a higher convergence rate in the initial convergence phase, reducing the target value to a lower threshold with fewer iterations. More importantly, after reaching the lower threshold, the proposed method does not exhibit significant convergence stalling in the asymptotic convergence phase, avoiding the additional iteration overhead required by the traditional method during tail convergence. In terms of the number of iterations and synthesis time, the traditional alternating projection method requires 8643 iterations; while the proposed method only requires 3044 iterations, reducing the number of iterations by approximately 64.78%, fully demonstrating the significant improvement in convergence efficiency.
[0103] As can be seen from the above embodiments, the method proposed in this invention exhibits superior numerical performance in convergence efficiency, with its convergence curve displaying a more ideal monotonically decreasing characteristic and no significant plateau effect. This phenomenon indicates that the proposed method can maintain high convergence efficiency even when approaching the feasible region boundary, effectively alleviating the asymptotic convergence degradation problem commonly found in alternating projection algorithms. This invention is applicable to arbitrarily arranged array antennas and can be used for low sidelobe synthesis of large-scale array antennas. Compared with existing point-correction-based alternating projection methods, it significantly reduces the number of iterations required for array synthesis.
Claims
1. A method for synthesizing alternating projection arrays based on beamline correction, characterized in that, Specifically, the following steps are included: Step 1: Data preprocessing: Extract the vector active element radiation pattern of each element in the given array, and convert and store it in the array manifold matrix; First, the direction of the vector active element is obtained through full-wave simulation or experimental measurement, and then interpolated to... domain On each sampling point; then, The point sampling vector active element radiation pattern data is decomposed into common polarization and cross polarization directions; finally, the decomposed common polarization and cross polarization component data are multiplied by the spatial phase factor corresponding to each array element and sampling point, and stored in the array manifold matrix; Step 2: Based on the given beam index, provide the full-area boundary function and constraints for the desired radiation pattern; Given beam indexes for specified main lobe and side lobe regions, and defining upper and lower boundary values or thresholds within these regions; first, extend the upper and lower boundary values and thresholds into functions to support setting boundary values and thresholds that vary in different regions, and list the extended constraints; then, modify the extended functions and constraints into boundary functions and constraints for the entire region. Step 3: Set the maximum number of iterations and the initial excitation vector, and calculate the initial array pattern vector; Step 4: Perform desired projection based on lobe correction on the current array pattern vector to obtain the corrected array pattern vector; First, calculate the bias matrix and the regions where each lobe is located in the co-polarization and cross-polarization components of the current array pattern vector; Then, the maximum and minimum values of the statistical deviation matrix in each lobe region and their corresponding sampling points are calculated; finally, the correction factors for co-polarization and cross-polarization are calculated for each lobe, and the expected projection based on the lobe correction is performed. Step 5: Perform feasible projection on the corrected array pattern vector to obtain the realizable array pattern vector; Step 6: Repeat steps 4 and 5 until the array pattern vector under the current excitation satisfies the full-area constraint conditions of the desired pattern given in step 2, or reaches the maximum number of iterations set in step 3. Output the array pattern vector and its corresponding excitation vector at this time.
2. The alternating projection array synthesis method based on beamline correction as described in claim 1, characterized in that, Step 1 specifically involves: Consider one An arbitrary irregular planar array antenna in the global coordinate system The pattern of a point-sampled vector array can be represented as: （1） in, for Domain One sampling point, and For the first The pitch and azimuth angles corresponding to each sampling point in the direction; For the first The complex excitation of each array element includes amplitude and phase; For the first Vector active element radiation pattern of each array element; The imaginary unit, The wave number in free space; For the first The spatial position of each array element for and linear combination, For the first The array element in the first Spatial phase factor at each sampling point; The vector active element radiation pattern of each array element is obtained through full-wave simulation or experimental measurement, and then interpolated to... domain At each sampling point; Point-sampled vector active cell radiation pattern data can be decomposed into common polarization and cross-polarization directions, i.e. ; In the formula For common polarization components, For cross-polarization components, and Let be the unit vectors in the common polarization and cross polarization directions, respectively; therefore, equation (1) can be expressed as: （2） In the formula, and Let the common polarization and cross-polarization components of the vector array pattern be defined as follows: （3） （4） （5） make , ; The matrix product relationship of the common polarization and cross polarization components of the vector array pattern is: In the formula, This is the array pattern vector. For array manifold matrix, The excitation vector; array manifold matrix The elements in the vector active unit pattern are common polarized. and cross-polarization components Multiply by spatial phase factor get.
3. The alternating projection array synthesis method based on beam lobe correction as described in claim 2, characterized in that, Step 2 specifically involves: Given beam parameters: (1) In the main lobe region Within, the upper and lower boundary values of the common polarization component are and ; (2) In the sidelobe region Within, the co-polarization component is below the threshold. (3) In the total sampling area That is, in step 1 The set of sampling points has cross-polarization components all below the threshold. ; The aforementioned beam indexes represent the constraints of the desired radiation pattern. The upper and lower boundary values can be expanded into functions that vary with the sampling points. and Support the addition of zero-depression areas in the sidelobe region. and its threshold The original sidelobe threshold is expanded to the following function: （6） Meanwhile, to maintain consistency, the threshold for cross-polarization is expressed as a function. ; The extended constraints can be expressed as: （7） Considering the subsequent solution of the deviation pattern, the extended constraints are modified to the following full-domain form: （8） The upper and lower boundary functions for the common polarization and cross polarization components are: （9） （10） （11） （12）。 4. The alternating projection array synthesis method based on beam lobe correction as described in claim 3, characterized in that, Step 3 specifically involves: Set the maximum number of iterations. No less than 5000 times; Set the initial excitation vector And solve for the initial array pattern vector. ; The initial excitation vector can be uniform excitation, Chebyshev excitation, or random excitation.
5. The alternating projection array synthesis method based on beam lobe correction as described in claim 4, characterized in that, Step 4 specifically involves: The full-area constraints of the desired radiation pattern given in step 2 are simply referred to as the desired constraints. The deviation of the array radiation pattern vector from the desired constraints is defined as the deviation radiation pattern, expressed as: （13） Let the deviation matrix Boundary matrix This can be transformed into the following matrix relationship: （14） Where square brackets denote Boolean matrices; Calculate the regions containing each lobe in the common-polarization and cross-polarization components of the current array pattern vector, assuming the total number of lobes is... The regions where each lobe is located are denoted as ; Statistical bias matrix in the region of each lobe The maximum and minimum values within the range and their corresponding sampling points: （15） In the formula, The maximum value, To be the minimum value, and These are the sampling points corresponding to the maximum and minimum values, respectively; Define the desired projection based on lobe correction That is, the corrected array pattern vector To simplify the formula, let's denote... The expected projections of the co-polarization and cross-polarization components are expressed as: （16） In the formula, for phase, for Phase factor; and The first Correction factors for lobe co-polarization and cross-polarization; （17） In the formula, This is an overpressure factor used to accelerate convergence.
6. The alternating projection array synthesis method based on beamline correction as described in claim 5, characterized in that, Step 5 specifically involves: The corrected array pattern vector obtained in step 4 The closest excitation vector is found using the least squares method: （18） （19） in, for The left pseudo-inverse matrix; superscript The superscript represents the conjugate transpose of a matrix. Represents the inverse of a matrix. It is the identity matrix; The regularization coefficient is used. The corresponding achievable array pattern vector is Define feasible projections ,Right now .
7. The alternating projection array synthesis method based on beamline correction as described in claim 6, characterized in that, Step 6 specifically involves: Assume the number of iterations is The time array pattern vector satisfies the full-area constraint conditions of the desired pattern given in step 2, or the number of iterations reaches the maximum number of iterations set in step 3. The array pattern vector at this time is: （20） The corresponding activation vector is .