A convex optimization method for the radiation pattern of an integrated phased array-radome
By rapidly acquiring short-circuit active radiation patterns and using the alternating direction multiplier method, the optimization challenges of phased array and radome systems are solved, achieving efficient and accurate radiation pattern synthesis. This method is applicable to arrays with different radome shapes and polarizations, reducing system complexity and cost.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2026-02-11
- Publication Date
- 2026-05-26
AI Technical Summary
Existing technologies suffer from problems such as large computational load, long time consumption, high cost, and difficulty in implementing optimization results when dealing with the interaction between phased arrays and radomes. Furthermore, they fail to effectively control radiation performance indicators under different scanning angles.
By treating the phased array and radome as a whole system, a fast electromagnetic algorithm is used to obtain the short-circuit active radiation pattern, a multi-constraint optimization model is constructed, and the alternating direction multiplier method is used to solve the model, thereby achieving synchronous synthesis of the sum and difference radiation patterns and reducing system complexity and cost.
It achieves fast and accurate radiation pattern synthesis, reduces data acquisition costs and time, improves the control accuracy of system radiation performance, is applicable to arrays with different dome shapes and polarizations, and reduces hardware complexity and manufacturing costs.
Smart Images
Figure CN122088087A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of radiation pattern technology, and in particular to a convex optimization method for an integrated phased array-radome radiation pattern. Background Technology
[0002] As a crucial protective component of phased arrays, radomes are widely used in engineering practice to resist the corrosive effects of harsh environments on phased arrays. However, their presence inevitably interferes with the electromagnetic radiation performance of the antenna. Specifically, this manifests as a series of problems such as aiming errors, gain attenuation, sidelobe level elevation, increased cross-polarization components, and even input impedance mismatch, severely impacting the overall performance of the phased array system.
[0003] Existing technologies for addressing the interaction between phased arrays and radomes often employ a "separate" optimization approach, treating them as independent systems and processing them separately to correct the radiation pattern. Hardware correction methods compensate for electrical performance losses by optimizing the inner wall profile and local wall thickness of the radome. However, this method is limited by the aerodynamic requirements of the radome's shape, resulting in extremely low design freedom and making it difficult to precisely control key indicators such as sidelobe levels and image lobes at different scanning angles. Phase compensation methods eliminate aiming errors by extracting the corrected phase of an ideal dipole array, but they ignore the complexity of the actual antenna structure and the mutual coupling effect between the array and the radome, leading to discrepancies between the compensation effect and the actual scenario.
[0004] To overcome the aforementioned technical bottlenecks, some existing solutions attempt to treat the phased array and radome as a unified system, compensating for the negative impact of the radome by adjusting the complex excitation vectors of the antenna elements, thereby correcting the radiation pattern. These solutions are based on Active Element Pattern (AEP) technology, using full-wave simulation to extract the AEP data of each element, then calculating the array pattern through vector synthesis, and combining intelligent optimization algorithms such as simulated annealing and genetic algorithms for pattern synthesis. However, this approach has significant drawbacks: full-wave simulation is computationally intensive and time-consuming, making it difficult to adapt to the optimization needs of large-scale arrays; simultaneously, obtaining the AEP of radome-covered antenna elements lacks efficient and accurate methods, and relying on full-wave simulation or experimental measurements is not only costly, but the practice of uniformly setting a 50-ohm port matching load during the calculation process introduces errors; furthermore, most existing optimization methods do not constrain the range of element excitation amplitudes, resulting in optimization results that, while meeting theoretical performance indicators, cannot be applied practically because they exceed the actual carrying capacity of the hardware. Summary of the Invention
[0005] The purpose of this invention is to propose a convex optimization method for the integrated radiation pattern of a phased array-radome. By rapidly acquiring short-circuit active radiation pattern data, constructing a multi-constraint optimization model, and solving it using the alternating direction multiplier method, the method achieves synchronous synthesis of sum and difference radiation patterns under multiple constraints, thereby improving the system's radiation performance and adapting to the application requirements of arrays with different radome shapes and polarizations.
[0006] To achieve the above objectives, this invention proposes a convex optimization method for the radiation pattern of an integrated phased array-radome, the specific steps of which are as follows: Step S1: Treat the phased array and radome as an N-port network, and obtain the overall short-circuit active radiation pattern SC-AEP of the array based on the fast electromagnetic algorithm; Step S2: Construct the upper and lower mask functions for the sum and difference patterns, and synthesize the overall pattern of the array based on the short-circuit active pattern; Step S3: Construct a multi-constraint pattern optimization model to minimize the excitation amplitude range while satisfying the upper and lower mask constraints of the sum and difference patterns. In particular, the multi-constraint pattern optimization model adopts a shared feeder network architecture for the sum and difference patterns, which significantly reduces the hardware complexity and implementation cost of the system. Step S4: The multi-constraint pattern optimization model is solved by using the alternating direction multiplier method. The constraint conditions are gradually satisfied through iterative updates, so as to achieve simultaneous integration of the sum pattern and the difference pattern by sharing the power supply network.
[0007] Preferably, in step S1, the short-circuit active radiation pattern is: the radiation pattern when a certain unit of the array is fed with a unit voltage, while all other unit ports are short-circuited, which is the short-circuit radiation pattern of that unit.
[0008] Preferably, in step S2, the upper and lower mask functions of the sum pattern are constructed based on three core indicators: maximum beam pointing, 3dB beamwidth, and sidelobe level; the upper and lower mask functions of the difference pattern are constructed based on two core indicators: zero point position and zero depth.
[0009] Preferably, in step S3, for simultaneous optimization of the sum and difference of the shared power supply network, the array is divided into a left half-plane and a right half-plane, the radiation patterns of the left half-plane and the right half-plane are calculated respectively, and an inverter is installed in the right half-plane. The switching between the sum channel and the difference channel is realized by controlling the on and off of the inverter.
[0010] Preferably, a multi-constraint pattern optimization model is constructed, as shown in the following formula: ; in, For the first n The incentive of each unit, For the mask function under the direction pattern, For mask functions on the direction pattern, This is a mask function for the difference direction pattern. For mask functions on the difference direction map, This is the short-circuit active pattern vector. Let them be the channel excitation vectors. It is a diagonal matrix. The pitch angle, It is the azimuth angle. n For unit numbering, N The number of units.
[0011] Preferably, a diagonal matrix This achieves the conversion of sum-channel excitation to difference-channel excitation, the diagonal matrix Half of the diagonal elements are 1 and the other half are -1, which realizes the inverse control of the left and right half-plane array units. The differential channel excitation is represented as: ; in, Differential channel excitation.
[0012] Preferred, let , , The original multi-constraint pattern optimization model is equivalent to: ; in, The minimum unit excitation magnitude, For the maximum unit excitation magnitude, To incentivize dynamic range ratio, To select a vector; make By introducing auxiliary variables, the equivalent multi-constraint pattern optimization model is represented as follows: ; in, The number of sampling points for the radiation pattern. For the first m The pitch angle of each sampling point For the first m The azimuth angle of each sampling point The first auxiliary variable for motivation n One element, For normalized incentives, For the auxiliary variables of the pattern m One element, The first auxiliary variable of the difference direction diagram m Each element.
[0013] Preferably, in step S4, the alternating direction multiplier method is used to solve the multi-constraint pattern optimization model. The specific steps are as follows: Step S41, for auxiliary variables , , Introduce the corresponding Lagrange multipliers respectively , , A quadratic penalty term is added to construct the augmented Lagrange function. The minimum value of the augmented Lagrange function is then calculated, as shown in the following formula: ; Step S42: Update variables and The remaining variables are treated as constant terms and ignored; let ,if It is known that the optimal solution is obtained by point-by-point projection. : ; in, For the first k +1 iterations of incentive auxiliary variables The n One element, for The n One element, For the first k +1 iterations , For the first k The next iteration , For the first k Lagrange multipliers in the next iteration , This is the iteration step size; choose Elements greater than 1 are used as the endpoints of the interval, and the endpoints are sorted in ascending order to obtain... ,variable The domain is divided into S+1 adjacent subintervals; within the s-th subinterval, the following conditions will be met. The elements constitute the set of valid elements. And the set remains unchanged within the interval; where, For the first k The iteration's incentive auxiliary variable is... N The amplitude of each element, For the first k Maximum endpoint value in the next iteration; Substituting the updated values into the augmented Lagrange function, each subinterval contains values related to the variable. Related problems involving finding the minimum value of quadratic functions: ; ; in, The coefficient of the quadratic term, The coefficient of the linear term, For constant terms; Compare the candidate solutions obtained in each small interval The solution that minimizes the objective function value is selected as the optimal solution for the current iteration. And complete the variable projection method. Update; Step S43: Update variables The remaining variables are treated as constant terms and ignored; variables The update is transformed into a least squares problem, as shown in the following formula: ; Set its first derivative to 0 to complete the update. The formula is as follows: ; in, For the first k The sum of auxiliary variables for the pattern in the next iteration, For the first k Lagrange multipliers in the next iteration , This is the short-circuit active radiation pattern matrix. For the first k Auxiliary variables for the difference pattern in the next iteration For the first k Lagrange multipliers in the next iteration H is the conjugate transpose of the matrix; Step S44: Update variables and The remaining variables are treated as constant terms and ignored; let , Perform point-by-point projection update, using the following formula: ; in, for The m One element, for The m One element, For the mask on the direction map m One element, for The m One element, for The m One element, The first mask on the difference direction map m One element; Step S45: Update variables The remaining variables are treated as constants and ignored; the value is updated by solving for the minimum of the quadratic function. The formula is as follows: ; ; Where R is the quadratic coefficient matrix and b is the linear coefficient vector. It is an N-order identity matrix; Step S46: Update the Lagrange multipliers , , The formula is as follows: ; Step S47: Repeat steps S43 to S46 until the iteration converges or the maximum number of iterations is reached, and output the final array excitation to achieve simultaneous synthesis of the array and the pattern and the difference pattern.
[0014] Preferably, the method is applicable to arrays with different dome shapes and polarizations. As long as the array short-circuit active radiation pattern data is obtained, radiation patterns of arbitrary excitation can be synthesized, and the balance between various performance indicators can be achieved by adjusting the constraints.
[0015] Therefore, this invention proposes a convex optimization method for the radiation pattern of an integrated phased array-radome, which has the following advantages: (1) The present invention obtains the short-circuit active radiation pattern (SC-AEP) through a fast electromagnetic algorithm. Only one electromagnetic calculation is needed to accurately characterize the mutual coupling effect between array elements and between the array and the radome, which meets the requirements for radiation pattern synthesis under arbitrary excitation conditions. Compared with traditional full-wave simulation or experimental measurement, it greatly reduces the cost, time consumption and error of data acquisition. (2) This invention constructs a multi-constraint optimization model covering the core indicators of the sum and difference patterns (maximum beam pointing, 3dB beamwidth, and sidelobe level of the sum pattern; zero point position and zero depth of the difference pattern). At the same time, it introduces the excitation amplitude range (DRR) constraint and combines the alternating direction multiplier method (ADMM) to decompose the complex problem into independent sub-problems for solution. While ensuring convergence, it achieves high-precision control of the pattern and avoids the optimization results from deviating from the actual hardware carrying capacity. (3) This invention is not limited to arrays of specific shape or polarization type. By adjusting the constraints, various performance indicators can be flexibly balanced. Moreover, the sum and difference channels share the power supply network, without the need for additional hardware, which significantly reduces the complexity and manufacturing cost of the power supply network, and takes into account both technical versatility and engineering feasibility.
[0016] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description
[0017] Figure 1 This is a flowchart of a convex optimization method for an integrated phased array-radome radiation pattern according to the present invention. Figure 2 This is a schematic diagram of the sum-difference shared power supply network in an embodiment of the present invention; Figure 3 This is a schematic diagram of the dipole array arrangement in an embodiment of the present invention; Figure 4 This is a schematic diagram comparing the array before and after optimization and the radiation pattern in an embodiment of the present invention; Figure 5 This is a schematic diagram comparing the difference radiation patterns before and after array optimization in an embodiment of the present invention. Detailed Implementation
[0018] To make the technical solutions, advantages, and objectives of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below. The described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the protection scope of the present invention.
[0019] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.
[0020] Example 1 like Figure 1 As shown, this invention provides a convex optimization method for the radiation pattern of an integrated phased array-radome, the specific steps of which are as follows: Step S1: Treat the phased array and radome as an N-port network, and obtain the overall short-circuit active radiation pattern of the array based on a fast electromagnetic algorithm; The short-circuit active radiation pattern is defined as the radiation pattern when a certain cell in the array is fed with a unit voltage while all other cell ports are short-circuited.
[0021] In this embodiment, the sum and difference radiation patterns of the array as a whole can be represented as: ; in, For the direction pattern, This is a difference direction pattern. It is a diagonal matrix. This is the short-circuit active pattern vector. The pitch angle, It is the azimuth angle; Step S2: Construct the upper and lower mask functions for the sum and difference patterns, and synthesize the overall pattern of the array based on the short-circuit active pattern; The top and bottom masking functions for the sum pattern are constructed based on three core indicators: maximum beam pointing, 3dB beamwidth, and sidelobe level; the top and bottom masking functions for the difference pattern are constructed based on two core indicators: zero point position and zero depth.
[0022] Step S3: Construct a multi-constraint pattern optimization model to minimize the excitation amplitude range while satisfying the upper and lower mask constraints of the sum and difference patterns. In particular, the multi-constraint pattern optimization model adopts a shared feeder network architecture for the sum and difference patterns, which significantly reduces the hardware complexity and implementation cost of the system. Construct a multi-constraint orientation pattern optimization model, as shown in the following formula: ; in, For the first n The incentive of each unit, For the mask function under the direction pattern, For mask functions on the direction pattern, This is a mask function for the difference direction pattern. For mask functions on the difference direction map, Let them be the channel excitation vectors. n For unit numbering, N The number of units; diagonal matrix This achieves the conversion of sum-channel excitation to difference-channel excitation, using a diagonal matrix. Half of the diagonal elements are 1 and the other half are -1, which realizes the inverse control of the left and right half-plane array units. The differential channel excitation is represented as: ; in, Differential channel excitation.
[0023] make , , The original multi-constraint pattern optimization model is equivalent to: ; in, The minimum unit excitation magnitude, For the maximum unit excitation magnitude, To incentivize dynamic range ratio, To select a vector; make By introducing auxiliary variables, the equivalent multi-constraint pattern optimization model is represented as follows: ; in, The number of sampling points for the radiation pattern. For the first m The pitch angle of each sampling point For the first m The azimuth angle of each sampling point The first auxiliary variable for motivation n One element, For the auxiliary variables of the pattern m One element, The first auxiliary variable of the difference direction diagram m One element, For normalized incentives.
[0024] Step S4: The alternating direction multiplier method is used to solve the multi-constraint pattern optimization model. The constraints are gradually satisfied through iterative updates, so as to achieve simultaneous integration of the sum pattern and the difference pattern by sharing the power supply network.
[0025] The alternating direction multiplier method is used to solve the multi-constraint pattern optimization model. The specific steps are as follows: Step S41, for auxiliary variables , , Introduce the corresponding Lagrange multipliers respectively , , A quadratic penalty term is added to construct the augmented Lagrange function. The minimum value of the augmented Lagrange function is then calculated, as shown in the following formula: ; Step S42: Update variables and The remaining variables are treated as constant terms and ignored; let ,if It is known that the optimal solution is obtained by point-by-point projection. : ; in, For the first k +1 iterations of incentive auxiliary variables The n One element, for The nOne element, For the first k +1 iterations , For the first k The next iteration , For the first k Lagrange multipliers in the next iteration , This is the iteration step size; choose Elements greater than 1 are used as the endpoints of the interval, and the endpoints are sorted in ascending order to obtain... ,variable The domain is divided into S+1 adjacent subintervals; within the s-th subinterval, the following conditions will be met. The elements constitute the set of valid elements. And the set remains unchanged within the interval; where, For the first k The iteration's incentive auxiliary variable is... N The amplitude of each element, For the first k Maximum endpoint value in the next iteration; Substituting the updated values into the augmented Lagrange function, each subinterval contains values related to the variable. Related problems involving finding the minimum value of quadratic functions: ; ; in, The coefficient of the quadratic term, The coefficient of the linear term, For constant terms; Compare the candidate solutions obtained in each small interval The solution that minimizes the objective function value is selected as the optimal solution for the current iteration. And complete the variable projection method. Update; Step S43: Update variables The remaining variables are treated as constant terms and ignored; variables The update is transformed into a least squares problem, as shown in the following formula: ; Set its first derivative to 0 to complete the update. The formula is as follows: ; in, For the first k The sum of auxiliary variables for the pattern in the next iteration, For the first kLagrange multipliers in the next iteration , This is the short-circuit active radiation pattern matrix. For the first k Auxiliary variables for the difference pattern in the next iteration For the first k Lagrange multipliers in the next iteration H denotes the matrix conjugate transpose; Step S44: Update variables and The remaining variables are treated as constant terms and ignored; let , Perform point-by-point projection update, using the following formula: ; ; in, for The m One element, for The m One element, For the mask on the direction map m One element, for The m One element, for The m One element, The first mask on the difference direction map m One element; Step S45: Update variables The remaining variables are treated as constants and ignored; the value is updated by solving for the minimum of the quadratic function. The formula is as follows: ; ; Where R is the quadratic coefficient matrix and b is the linear coefficient vector. It is an N-order identity matrix; Step S46: Update the Lagrange multipliers , , The formula is as follows: ; Step S47: Repeat steps S43 to S46 until the iteration converges or the maximum number of iterations is reached, and output the final array excitation to achieve simultaneous synthesis of the array and the pattern and the difference pattern.
[0026] This embodiment is applicable to arrays with different dome shapes and polarizations. As long as the array short-circuit active radiation pattern data is obtained, the radiation pattern of arbitrary excitation can be synthesized, and the balance between various performance indicators can be achieved by adjusting the constraints.
[0027] The invention will be further illustrated below through specific implementation examples.
[0028] like Figure 2 As shown, consider a 1×10 equidistant dipole array operating at a frequency of 10 GHz, with a dipole array element length of... The center-to-center distance between adjacent array elements is .
[0029] Array and differential shared feed networks such as Figure 3 As shown, the system can be divided into a left half-plane and a right half-plane along the axis of symmetry. The left half-plane maintains the original feed phase, while the right half-plane introduces an inverter to achieve a 180° phase reversal. This allows for the formation of a sum graph after weighted superposition, or a difference graph after reverse superposition. In this scheme, the sum and difference channels share a single feed network. The switching of the sum and difference channels is controlled by the on / off state of the inverter, eliminating the need for additional hardware and significantly reducing the complexity of the feed network and manufacturing costs.
[0030] exist Optimize the orientation pattern in a plane. The range of variation is Sampling is performed at 1° intervals. The requirements are: maximum beam pointing in the sum pattern, null position in the difference pattern at 0°, expected sidelobe level of -20dB, and expected null depth of -25dB. After 150 iterations, the time taken is only 17.01s. A comparison of the sum and difference patterns before and after optimization is shown below. Figures 4-5 As shown, the optimized sidelobe level is -18.7dB and the zero depth is -50.2dB, which basically meets the requirements.
[0031] It is worth noting that all contents not described in detail in this invention are existing technologies and are well known to those skilled in the art.
[0032] Therefore, this invention provides a convex optimization method for the integrated radiation pattern of a phased array-radome. By acquiring the short-circuited active radiation pattern, constructing a multi-constraint optimization model, and using the alternating direction multiplier method for efficient solution, it achieves accurate synchronous synthesis of the sum and difference radiation patterns under multiple constraints. This not only significantly improves the data acquisition efficiency and the control accuracy of the radiation pattern, but also has a wide range of applications and strong engineering feasibility, reducing system design and manufacturing costs.
[0033] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.
Claims
1. A convex optimization method for the radiation pattern of an integrated phased array-radome, characterized in that, The specific steps are as follows: Step S1: Treat the phased array and radome as an N-port network, and obtain the overall short-circuit active radiation pattern of the array based on a fast electromagnetic algorithm; Step S2: Construct the upper and lower mask functions for the sum and difference patterns, and synthesize the overall pattern of the array based on the short-circuit active pattern; Step S3: Construct a multi-constraint pattern optimization model to minimize the excitation amplitude range while satisfying the upper and lower mask constraints of the sum and difference patterns. The multi-constraint pattern optimization model specifically adopts a shared feed network architecture for the sum and difference patterns. Step S4: The multi-constraint pattern optimization model is solved by using the alternating direction multiplier method. The constraint conditions are gradually satisfied through iterative updates, so as to achieve simultaneous integration of the sum pattern and the difference pattern by sharing the power supply network.
2. The convex optimization method for the integrated radiation pattern of a phased array-radome as described in claim 1, characterized in that, In step S1, the short-circuit active radiation pattern is: the radiation pattern when a certain cell of the array is fed with a unit voltage and all other cell ports are short-circuited, which is the short-circuit radiation pattern of that cell.
3. The convex optimization method for the integrated radiation pattern of a phased array-radome as described in claim 2, characterized in that, In step S2, the upper and lower mask functions of the sum pattern are constructed based on three core indicators: maximum beam pointing, 3dB beamwidth, and sidelobe level; the upper and lower mask functions of the difference pattern are constructed based on two core indicators: zero point position and zero depth.
4. The convex optimization method for the integrated radiation pattern of a phased array-radome as described in claim 3, characterized in that, In step S3, for simultaneous optimization of the sum and difference of the shared power supply network, the array is divided into a left half-plane and a right half-plane, the radiation patterns of the left half-plane and the right half-plane are calculated respectively, and an inverter is installed in the right half-plane. The switching between the sum channel and the difference channel is realized by controlling the on and off of the inverter.
5. The convex optimization method for the integrated radiation pattern of a phased array-radome according to claim 4, characterized in that, Construct a multi-constraint orientation pattern optimization model, as shown in the following formula: ； in, For the first n The incentive of each unit, For the mask function under the direction pattern, For mask functions on the direction pattern, This is a mask function for the difference direction pattern. For mask functions on the difference direction map, This is the short-circuit active pattern vector. Let them be the channel excitation vectors. It is a diagonal matrix. The pitch angle, It is the azimuth angle. n For unit numbering, N The number of units.
6. The convex optimization method for the integrated radiation pattern of a phased array-radome according to claim 5, characterized in that, diagonal matrix This achieves the conversion of sum-channel excitation to difference-channel excitation, the diagonal matrix Half of the diagonal elements are 1 and the other half are -1, which realizes the inverse control of the left and right half-plane array units. The differential channel excitation is represented as: ； in, Differential channel excitation.
7. The convex optimization method for the integrated radiation pattern of a phased array-radome as described in claim 5, characterized in that, make , , The original multi-constraint pattern optimization model is equivalent to: ； in, The minimum unit excitation magnitude, For the maximum unit excitation magnitude, To incentivize dynamic range ratio, To select a vector; make By introducing auxiliary variables, the equivalent multi-constraint pattern optimization model is represented as follows: ； in, The number of sampling points for the radiation pattern. For the first m The pitch angle of each sampling point For the first m The azimuth angle of each sampling point The first auxiliary variable for motivation n One element, For normalized incentives, For the auxiliary variables of the pattern m One element, The first auxiliary variable of the difference direction pattern m Each element.
8. The convex optimization method for the integrated radiation pattern of a phased array-radome according to claim 7, characterized in that, In step S4, the alternating direction multiplier method is used to solve the multi-constraint pattern optimization model. The specific steps are as follows: Step S41, for auxiliary variables , , Introduce the corresponding Lagrange multipliers respectively , , A quadratic penalty term is added to construct the augmented Lagrange function. The minimum value of the augmented Lagrange function is then calculated, as shown in the following formula: ； Step S42: Update variables and The remaining variables are treated as constant terms and ignored; let ,if It is known that the optimal solution is obtained by point-by-point projection. : ； in, For the first k +1 iterations of the incentive auxiliary variable The n One element, for The n One element, For the first k +1 iterations , For the first k The next iteration , For the first k Lagrange multipliers in the next iteration , This is the iteration step size; choose Elements greater than 1 are used as the endpoints of the interval, and the endpoints are sorted in ascending order to obtain... ,variable The domain is divided into S+1 adjacent subintervals; within the s-th subinterval, the following conditions will be met. The elements constitute the set of valid elements. And the set remains unchanged within the interval; where, For the first k The iteration's incentive auxiliary variable is... N The amplitude of each element, For the first k Maximum endpoint value in the next iteration; Substituting the updated values into the augmented Lagrange function, each subinterval contains values related to the variable. Related problems involving finding the minimum value of quadratic functions: ； ； in, The coefficient of the quadratic term, The coefficient of the linear term, For constant terms; Compare the candidate solutions obtained in each small interval The solution that minimizes the objective function value is selected as the optimal solution for the current iteration. And complete the variable projection method. Update; Step S43: Update variables The remaining variables are treated as constant terms and ignored; variables The update is transformed into a least squares problem, as shown in the following formula: ； Set its first derivative to 0 to complete the update. The formula is as follows: ； in, For the first k The sum of auxiliary variables for the pattern in the next iteration, For the first k Lagrange multipliers in the next iteration , This is the short-circuit active radiation pattern matrix. For the first k Auxiliary variables for the difference pattern in the next iteration For the first k Lagrange multipliers in the next iteration H denotes the matrix conjugate transpose; Step S44: Update variables and The remaining variables are treated as constant terms and ignored; let , Perform point-by-point projection update, using the following formula: ； in, for The m One element, for The m One element, For the mask on the direction map m One element, for The m One element, for The m One element, The first mask on the difference direction map m One element; Step S45: Update variables The remaining variables are treated as constants and ignored; the value is updated by solving for the minimum of the quadratic function. The formula is as follows: ； ； Where R is the quadratic coefficient matrix and b is the linear coefficient vector. It is an N-order identity matrix; Step S46: Update the Lagrange multipliers , , The formula is as follows: ； Step S47: Repeat steps S43 to S46 until the iteration converges or the maximum number of iterations is reached, and output the final array excitation to achieve simultaneous synthesis of the array and the pattern and the difference pattern.
9. The convex optimization method for the integrated radiation pattern of a phased array-radome according to claim 8, characterized in that, The method is applicable to arrays with different dome shapes and polarizations. As long as the array short-circuit active radiation pattern data is obtained, it is possible to synthesize radiation patterns of arbitrary excitations and achieve a balance between various performance indicators by adjusting constraints.