High-directivity phased array synthesis method based on circular polarization axial ratio control

Abstract

The invention discloses a high-directivity phased array synthesis method based on circular polarization axial ratio control, and belongs to the technical field of satellite communication. The method comprises the following steps: firstly, establishing a one-to-one correspondence relationship between an axial ratio and an electric field; then, an array directivity coefficient maximization problem model with strict axial ratio constraint is established; a non-convex fractional programming problem is converted into a Rayleigh entropy form, axial ratio constraint is converted into linear constraint, the axial ratio constraint is replaced with first-dimension excitation through excitation dimension reduction, an original problem is converted into an unconstrained optimization problem, and the converted unconstrained optimization problem is further converted into the Rayleigh entropy form; and finally, carrying out eigendecomposition on the converted standard Rayleigh entropy form to obtain an eigenvector corresponding to the maximum eigenvalue, and recovering the excitation of the original problem according to the conversion form. According to the method, accurate axial ratio control can be achieved, meanwhile, the maximum directivity coefficient can be obtained under the condition that the current axial ratio is limited, and the iteration process is avoided.

Description

Technical Field

[0001] This invention relates to the field of satellite communication technology, and in particular to a high-directivity phased array synthesis method based on circular polarization axial ratio control. Background Technology

[0002] In satellite communication systems, communication quality depends on the transmission power and polarization mode of electromagnetic waves in the communication link. Antennas, as transmitters and receivers in satellite communication systems, play a crucial role in electromagnetic wave transmission. In satellite communication links, polarization mismatch caused by the Faraday rotation effect and a significant decrease in antenna directivity over ultra-wide scanning angles severely impact communication quality. To address these issues, most airborne and spaceborne phased array systems employ circularly polarized phased arrays to combat multipath effects. However, in ultra-wide scanning angles (elevation angles greater than ±60°), circularly polarized phased arrays experience polarization purity degradation, leading to a significant decrease in gain and severely affecting communication quality.

[0003] In satellite communication links, improving the polarization purity and directivity coefficient of the transmitting and receiving antennas are two important means to ensure communication quality. Since the polarization degradation and directivity reduction caused by the ultra-wide-angle scanning of phased arrays are highly coupled, simply improving polarization purity or directivity coefficient alone cannot solve this problem. Existing methods mainly address this issue by simultaneously controlling the directivity coefficient and cross-polarization. The first type of method aims to maximize the directivity coefficient in the main lobe region, with constraints of sidelobe suppression and cross-polarization suppression. The objective function of the optimization problem is simplified using Taylor expansion, and iterative convex optimization is employed for solution. The second type of method is a multi-objective optimization problem, simultaneously minimizing the difference between the subarray gain and the full array gain, as well as sidelobes and cross-polarization, and is solved using a particle swarm optimization algorithm.

[0004] by Taking a planar phased array antenna as an example, along Directional unit spacing is The number of array elements is ,along Directional unit spacing is The number of array elements is Then the far-field radiated electric field of the phased array antenna is:

[0005] (1)

[0006] In the formula, Indicates pitch angle, Indicates azimuth. represent Unit radiation pattern, Representing the The excitation amplitude and phase of each unit, , They represent Unit direction and Directional coordinates, defined , Represents spherical coordinate system to Coordinate system mapping, where , It represents the propagation constant of electromagnetic waves in free space.

[0007] The far-field directivity coefficient can be defined as:

[0008] (2)

[0009] Equations (1) and (2) can both be simplified to matrix form.

[0010] (3)

[0011] (4)

[0012] in, Indicates the scanning direction. Represents the activation vector. An auxiliary matrix representing the radiated power in the scanning direction. , , It is a Hermitian symmetric matrix, which can be obtained through eigenvalue decomposition. , An auxiliary matrix representing total power.

[0013] In the scheme of maximizing the directivity coefficient of the main lobe region, the optimization problem constrained by sidelobe suppression and cross-polarization suppression can be described as follows:

[0014] (5)

[0015] in, , electric field Quantity in Auxiliary matrix of radiated power in the direction, , An auxiliary matrix representing the total power. Indicates the first Excitation of each array element Represents the main lobe region. Represents the sidelobe region. The relaxation variable representing the main lobe constraint. The relaxation variable representing the sidelobe constraint, The relaxation variable representing the cross-polarization constraint, This represents the slack variable representing the dynamic range constraint of the excitation amplitude. Due to the non-convexity of this problem, an optimal solution cannot be obtained. Therefore, it is simplified through Taylor expansion to obtain...

[0016] (6)

[0017] in, The slack variable representing the directional coefficient constraint, Indicates the first The incentive for the next iteration This indicates taking the real part of the matrix. Indicates the first The incentive for the next iteration Indicates the first The iteration step size of each unit, express, This represents the maximum iteration step size. , Indicates the first The maximum value of the excitation amplitude in the next iteration , ,question For convex problems, the existing convex optimization toolkit CVX is used for solution, and iterative updates are performed. This continues until the maximum number of iterations is reached.

[0018] The main drawbacks of maximizing the directional coefficient of the main lobe region are: the relaxation control of cross-polarization cannot achieve good directional coefficient control while ensuring the axial ratio, and it does not take into account the practical application of ultra-wide-angle scanning. Furthermore, the formula... The problem described is a difficult second-order non-convex problem, which can only find suboptimal solutions. Furthermore, the algorithm depends on the selection of initial values ​​and is prone to getting trapped in local optima.

[0019] In schemes based on multi-objective optimization problems, with Taking a planar phased array antenna as an example, its element radiation pattern can be described as follows:

[0020] (7)

[0021] in Radiation pattern representing a unit cell. For array cell amplitude, The phase of the array unit.

[0022] The fitness function that minimizes the difference between the subarray gain and the full array gain, and the sidelobes, can be described as follows:

[0023] (8)

[0024] in, For the peak gain of a uniform array, To optimize the peak gain of the array, ~ The corresponding weighting coefficients are given, PSLL represents the peak sidelobe level, and the PSO algorithm is used to solve for the array element positions.

[0025] The scheme based on the multi-objective optimization problem only constrains the sidelobe and gain, and the axis ratio is controlled only by rotating the array elements. Without constraining according to a certain criterion, the axis ratio will deteriorate and the polarization performance will drop sharply. Summary of the Invention

[0026] This invention proposes a high-directivity phased array synthesis method based on circular polarization axial ratio control, which can achieve high directivity coefficient and accurate axial ratio control under ultra-wide angle scanning coverage.

[0027] The technical solution adopted in this invention is: a high-directivity phased array synthesis method based on circular polarization axial ratio control, which includes the following steps:

[0028] Step 1: Obtain the ellipticity based on the desired array axis ratio according to the axis ratio mapping relationship. And based on the set ellipse tilt angle Obtaining the amplitude coefficient and phase coefficient ;

[0029] Step 2: Construct an optimization model with axis ratio constraints and maximizing the directional coefficient in the scanning direction:

[0030] (9)

[0031] in, For array excitation, Let L represent the complex field, L represent the total number of array elements of the phased array, and the auxiliary matrix represent the total number of array elements of the phased array. Based on pitch angle Azimuth And guide vector calculation, ,in, , Characterizing the spherical coordinate system to Coordinate system mapping, where , Regarding the scanning direction The guide vector on; Represents a circularly polarized electric field Directional components and Directional components in the scanning direction The ratio on; , To correspond to the scanning direction The amplitude ratio and phase difference; the auxiliary matrix for total power. , For about , The guide vector, is a Hermitian symmetric matrix, where M and N are the number of array elements in the phased array in the horizontal and vertical directions, respectively;

[0032] Step 3, constrain the shaft ratio Convert to: ,in, Indicates the first Excitation of each array element Indicates the first Axial ratio auxiliary steering vector for each array element;

[0033] Step 4: Based on the transformed axis ratio constraint, solve the constructed optimization model, and obtain the optimal array excitation of the phased array based on the solution results. To excite based on this optimal array Achieve beam scanning.

[0034] Furthermore, a one-dimensional excitation is used to represent the axis ratio constraint: .

[0035] Furthermore, based on the constructed auxiliary matrix The optimization model is transformed into:

[0036] (10)

[0037] in, The activation function for dimensionality reduction, and the auxiliary matrix. for:

[0038] (11)

[0039] in, Indicates the first The axial ratio of each array element is used as an auxiliary guide vector.

[0040] Furthermore, auxiliary quantities are introduced. and auxiliary variables The optimization model is transformed into:

[0041] (12)

[0042] Among them, based on the matrix The eigenvalue decomposition yields the matrix ,in .

[0043] Furthermore, in step 4, when solving the constructed optimization model, the... Eigenvalue decomposition is performed, and the solution is obtained based on the eigenvector corresponding to the largest eigenvalue. Finally, the optimal array excitation is obtained. .

[0044] In this invention, a one-to-one correspondence between the axial ratio and the electric field is first established; then, a model for maximizing the array directivity coefficient with strict axial ratio constraints is established; next, the non-convex fractional programming problem is transformed into Rayleigh entropy form (convex problem form). First, the axial ratio constraint is transformed into a linear constraint, and then the original problem is transformed into an unconstrained optimization problem by replacing the axial ratio constraint with the first-dimensional stimulus through excitation dimensionality reduction. Second, the transformed unconstrained optimization problem is further transformed into Rayleigh entropy form (convex problem form); finally, the transformed standard Rayleigh entropy form is subjected to eigenvalue decomposition to obtain the eigenvector corresponding to its largest eigenvalue, and the excitation of the original problem is recovered according to the transformed form.

[0045] The technical solution provided by this invention brings at least the following beneficial effects:

[0046] This invention enables precise axis ratio control and achieves the maximum directivity coefficient within a given axis ratio. Compared to similar algorithms, the algorithm provided by this invention can obtain an upper limit for the directivity coefficient under a given axis ratio. Furthermore, because the method proposed in this invention has an analytical solution, it avoids iterative processes and has a faster computation speed, exhibiting significant computational advantages, especially when applied to large-scale arrays. Attached Figure Description

[0047] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0048] Figure 1 This is a schematic diagram of the coordinate system and circular polarization;

[0049] Figure 2 For the axis ratio AR and and A schematic diagram illustrating the specific mapping relationship;

[0050] Figure 3 A schematic diagram showing the change of the directionality coefficient with scanning angle under different axis ratios;

[0051] Figure 4 A schematic diagram showing the change of the directionality coefficient with scanning angle for different optimization methods;

[0052] Figure 5 This is a schematic diagram showing the change of axis ratio as a function of scanning angle for different optimization methods.

[0053] Figure 6 This is a schematic diagram of a measured scenario for a 2×6 planar phased array.

[0054] Figure 7 To provide 2×6 planar phased array beam coverage to A comparison chart of the measured data and the proposed method;

[0055] Figure 8 To provide 2×6 planar phased array beam coverage to The measured data is compared with the proposed method. Detailed Implementation

[0056] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be described in detail and completely below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. Generally, the components of the embodiments of the present invention described and shown in the accompanying drawings can be arranged and designed using different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely represents selected embodiments of the present invention.

[0057] This invention provides a high-directivity phased array synthesis method based on circular polarization axial ratio control. It is achieved by establishing an optimization problem of maximizing the directivity coefficient with axial ratio constraints. Due to the non-convexity of the problem, the axial ratio constraint is simplified to a first-order constraint by fractional programming. At the same time, the relationship between the axial ratio constraint and the excitation is constructed by using an excitation dimensionality reduction method, thereby simplifying the problem into an unconstrained optimization problem. This problem is in the standard Rayleigh entropy form. The analytical solution of the problem can be derived through formula derivation, so as to achieve high directivity coefficient and accurate axial ratio control under ultra-wide angle scanning coverage (pitch angle ±85° scanning).

[0058] First, establish the mapping relationship of axis ratio, such as Figure 1 The axial ratio shown is defined as follows:

[0059] (13)

[0060] in, Indicates the desired array axis ratio, The ellipticity can be calculated based on the desired axis ratio input. .and The relationship can be further expressed as electric field components and The ratio:

[0061] (14)

[0062] in, Represents array excitation, Indicates about The guide vector, amplitude coefficient (i.e., the amplitude ratio of the two electric field components) and phase coefficient (i.e., the phase difference between the two electric field components) and and elliptic inclination angle There is a one-to-one correspondence, as detailed below:

[0063] (15)

[0064] Therefore, it can be calculated according to formula (15):

[0065] (16)

[0066] For linearly polarized waves Directional polarization can be set to ,and Directional polarization can be set to axial ratio and and The specific mapping relationship is as follows Figure 2 As shown.

[0067] In this embodiment of the invention, an optimization problem is proposed that involves maximizing the directional coefficient in the scanning direction while constrained by the axis ratio. This problem can be described as follows:

[0068] (17)

[0069] in, , Represents a circularly polarized electric field Directional components and The ratio of the directional components, since the axial ratio constraint is a fractional programming problem, is a non-convex problem and cannot be solved directly. Therefore, the axial ratio constraint is further simplified as follows:

[0070] (18)

[0071] Let auxiliary variables be defined. ,Mode It can be expanded as follows:

[0072] (19)

[0073] in, Indicates the first Array excitation of each array element Indicates the first The axial ratio auxiliary steering vector of each array element Represents the total number of array cells, when the following conditions are met. When the condition is met, the constraint can be represented by a one-dimensional excitation as follows:

[0074] (20)

[0075] Therefore, the excitation with shaft ratio constraint can be expressed as:

[0076] (twenty one)

[0077] in, The activation function for dimensionality reduction, and the auxiliary matrix. for:

[0078] (twenty two)

[0079] in, Indicates the first The axial ratio of each array element is used as an auxiliary guide vector.

[0080] According to equation (21), the problem It can be transformed into:

[0081] (twenty three)

[0082] Since the problem is a difficult fractional programming problem, most existing methods for solving fractional programming problems use Taylor expansion for approximate simplification and then employ iterative convex optimization to solve the simplified problem. Although this approximation process can avoid the fractional nature of the problem and find a near-optimal solution, the iterative process significantly increases computational complexity, which is very costly when solving large-scale problems. Therefore, this invention proposes a low-complexity solution method, considering the establishment of an analytical solution form for this problem. It also takes into account the standard Rayleigh entropy form, specifically, since the denominator... The matrix is ​​a positive semi-definite Hermitian matrix, and the denominator is a positive semi-definite quadratic form. Further... Rewritten as ,at this time The matrix remains a positive semi-definite Hermitian matrix. Since symmetric matrices can be decomposed into vector products, the denominator can be simplified to the product of activation vectors. The method in this embodiment of the invention primarily focuses on the properties of positive semi-definite symmetric matrices, establishing the Rayleigh entropy canonical form. Based on Rayleigh's theorem, the global maximum value of the canonical Rayleigh entropy is equal to the largest eigenvalue of its corresponding Hermitian matrix, and the optimal solution is the eigenvector corresponding to that eigenvalue. Therefore, this invention completely transforms the originally complex fractional programming problem into a directly solvable matrix eigenvalue problem, thus obtaining an analytical solution. The specific steps are as follows:

[0083] The fractional form of this problem can be further simplified:

[0084] (twenty four)

[0085] In the formula, , All are Hermitian matrices, where Since it is a positive semi-definite symmetric matrix, it can be decomposed into... , Introducing auxiliary quantities and auxiliary variables Problem (23) can be further simplified to:

[0086] (twenty four)

[0087] Problem (24) is in the standard form of Ruili entropy, satisfying the following criteria:

[0088] (25)

[0089] in, Represents the smallest eigenvalue. The maximum upper bound of this problem is the maximum eigenvalue. The analytical solution to this problem This is the eigenvector corresponding to the largest eigenvalue. Therefore, the original problem... The analytical solution can be expressed as:

[0090] (25)

[0091] To further verify the performance of the method in this embodiment, Taking a planar phased array as an example, the radiation pattern of the active element is obtained through full-wave simulation software and optimized using the method of this embodiment. The method of this embodiment can achieve gain improvement under different axial ratios. At the same time, compared with existing methods, it can achieve a lower axial ratio and a higher directivity coefficient. The effectiveness of the proposed method can be verified by measured data.

[0092] like As shown in the figure, the method of this embodiment can achieve high directivity coefficient synthesis under different axis ratios. It can be seen from the figure that the directivity coefficient is the highest when the axis ratio is 2 dB in the ±60° scanning range; but the directivity coefficient is the highest when the axis ratio is 0 dB in the -85° to -75° scanning range; the directivity coefficient is the highest when the axis ratio is 3 dB in the -75° to -60° scanning range; the directivity coefficient is the highest when the axis ratio is 3 dB in the 60° to 70° scanning range; and the directivity coefficient is the highest when the axis ratio is 1 dB in the 75° to 85° scanning range.

[0093] like and As shown, compared with the two existing methods, the method in this embodiment can achieve a relatively flat change in the directional coefficient within a scanning range of ±85°, while ensuring good circular polarization performance. Compared with the iterative convex optimization method, it can obtain a better directional coefficient when scanning at a small angle, but the axis ratio is severely degraded, and it cannot guarantee a high directional coefficient under the condition of high circular polarization purity. Compared with the particle swarm optimization algorithm, the method in this embodiment can achieve a higher directional coefficient under the same axis ratio.

[0094] To verify the effectiveness of the method in this embodiment, a 2×6 planar phased array fabrication experiment was conducted to validate the proposed method. The specific test scenario is as follows: Figure 6 As shown, scan to and The normalized radiation pattern measured data and the simulation data of the proposed method are compared, for example... and As shown, the simulation data and the measured data of the method in this embodiment have a good matching degree.

[0095] 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 the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0096] The above descriptions are merely some embodiments of the present invention. Those skilled in the art can make various modifications and improvements without departing from the inventive concept of the present invention, and these all fall within the scope of protection of the present invention.

Claims

1. A high-directivity phased array synthesis method based on circular polarization axial ratio control, characterized in that, Includes the following steps: Step 1: Obtain the ellipticity based on the desired array axis ratio according to the axis ratio mapping relationship. And based on the set ellipse tilt angle Obtaining the amplitude coefficient and phase coefficient ; Step 2: Construct an optimization model with axis ratio constraints and maximizing the directional coefficient in the scanning direction: ； in, For array excitation, L represents the total number of array elements in the phased array, based on the scanning direction. Auxiliary matrix for calculating the guiding vector on ,in, , Characterizing the spherical coordinate system to Coordinate system mapping, where , For pitch angle, It is the azimuth angle. Regarding the scanning direction The guide vector on; Represents a circularly polarized electric field Directional components and Directional components in the scanning direction The ratio on; , To correspond to the scanning direction The amplitude ratio and phase difference; the auxiliary matrix for total power. , For about , The guide vector, is a Hermitian symmetric matrix, where M and N are the number of array elements in the phased array in the horizontal and vertical directions, respectively; Step 3, constrain the shaft ratio Convert to: ,in, Indicates the first Excitation of each array element Indicates the first Axial ratio auxiliary steering vector for each array element; Step 4: Based on the transformed axis ratio constraint, solve the constructed optimization model, and obtain the optimal array excitation of the phased array based on the solution results. .

2. The method as described in claim 1, characterized in that, The axis ratio mapping relationship is as follows: ； in, This indicates the desired array axis ratio.

3. The method as described in claim 2, characterized in that, According to the formula Obtaining the amplitude coefficient and phase coefficient .

4. The method as described in claim 1, characterized in that, One-dimensional excitation is used to represent the axis ratio constraint: .

5. The method as described in claim 1, characterized in that, Based on the constructed auxiliary matrix The optimization model is transformed into: ； in, The activation function for dimensionality reduction, and the auxiliary matrix. for: ； in, Indicates the first The axial ratio of each array element is used as an auxiliary guide vector.

6. The method as described in claim 1, characterized in that, Introducing auxiliary quantities and auxiliary variables The optimization model is transformed into: ； Based on the matrix The eigenvalue decomposition yields the matrix ,in .

7. The method as described in claim 1, characterized in that, In step 4, when solving the constructed optimization model, the following steps are performed: Eigenvalue decomposition is performed, and the solution is obtained based on the eigenvector corresponding to the largest eigenvalue. Finally, the optimal array excitation is obtained. .

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