A Method for Optimizing the Radiation Pattern of Metasurface Array Antennas Based on Semi-Deterministic Relaxation

Abstract

This invention discloses a method for optimizing the radiation pattern of a metasurface array antenna based on semidefinite relaxation. First, starting from the array factor formula of the antenna, an original optimization problem model is established according to the expected requirements. Then, each complex element of the array excitation vector is separated into real and virtual components, and the original problem is equivalently transformed. Next, a non-convex constraint in the original problem is discarded, resulting in a convex positive semidefinite relaxation problem with an optimal solution. Finally, cases are discussed: when the rank of the optimal solution is 1, the optimal solution is the optimal solution of the original problem; when the rank of the optimal solution is not 1, a feasible solution to the original problem is obtained through eigenvalue decomposition, thereby solving for the amplitude and phase information of the array that meets the expected requirements. This invention can quickly solve for the excitation vector of the array elements based on a specified array far-field radiation pattern. Subsequent full-wave simulations and experimental tests have proven the effectiveness of the proposed optimization method.

Description

Technical Field

[0001] This invention belongs to the field of metasurface antenna array synthesis and mathematical optimization, and particularly relates to a method for optimizing the radiation pattern of metasurface array antennas based on semidefinite relaxation. Background Technology

[0002] Array antennas, due to their outstanding beamforming capabilities, have wide applications in wireless communication fields such as communications, broadcasting, and radar. Electromagnetic metasurfaces can flexibly control electromagnetic waves and also have the advantages of low cost and simple architecture. Synthesis problems aim to find the optimal excitation vector of an array that satisfies specific far-field constraints. Pattern optimization is a type of synthesis problem, and rapidly solving for the amplitude and phase distribution of array elements in a metasurface antenna based on a predicted pattern is of significant research importance. Analytical algorithms, such as the Dolph-Chebyshev synthesis method, Taylor synthesis method, and Woodward-Lawson synthesis method, are only suitable for relatively simple pattern synthesis problems. Stochastic algorithms, such as particle swarm optimization and simulated annealing, while possessing high universality and flexibility, suffer from slow convergence, sensitivity to initial conditions, and susceptibility to local optima. Artificial intelligence algorithms, such as machine learning and neural networks, are not yet mature in the field of array synthesis. All the methods mentioned above have their own limitations and cannot effectively solve problems tailored to specific requirements. Summary of the Invention

[0003] The purpose of this invention is to provide a method for optimizing the radiation pattern of a metasurface array antenna based on semi-definite relaxation, so as to solve the technical problem that existing methods have their own defects and cannot effectively solve the problem in combination with specific requirements.

[0004] To solve the above-mentioned technical problems, the specific technical solution of the present invention includes the following three steps:

[0005] Step 1: First, consider a linear array with an element spacing of d and an element count of N. Write the far-field electric field expression for the array. Based on the specific requirements of the desired far-field radiation pattern, divide the main lobe region and the side lobe region. Determine the beam shape to be formed in the main lobe region and the far-field radiation pattern level value in the side lobe region. Set the complex excitation vector of the array as the initial optimization variable and establish the original optimization problem model, which is a non-convex problem model.

[0006] Step 2: Then, separate the real and imaginary parts of each element of the complex excitation vector to obtain a 2N-dimensional variable;

[0007] Step 3: Equivalently rewrite the original problem, discarding the non-convex constraints, thus obtaining a convex semidefinite relaxation problem. This problem has an optimal solution. Subsequently, the rank minimization theory is introduced to make the rank of the solution close to 1. Finally, we discuss the cases separately. If the rank of the aforementioned solution is 1, then it is the optimal solution of the original problem. If the rank is not 1, then the approximate solution of the original problem, which is also the feasible solution, can be obtained through eigenvalue decomposition.

[0008] Furthermore, step 1 specifically includes the following steps: Consider a linear array with element spacing of d and number of elements of N, whose array factor expression is written as:

[0009] G(θ)=v(θ) H w(1)

[0010] v(θ) is the steering vector in the θ direction:

[0011]

[0012] E i This is the element radiation pattern of the i-th element, where i = 1, 2, ..., N; k is the wave number, and j is the imaginary unit; x i Here, represents the coordinates of the i-th element, i = 1, 2, ..., N; H represents the conjugate transpose operation; w is the complex excitation vector of the array, containing the amplitude and phase information of each element, w = (w1, ..., w2). N ) T G(θ) refers to the far-field pattern of the array in the θ direction, where θ∈[-90°,90°]; the following pattern optimization problem (3) model is constructed:

[0013] find w (3a)

[0014] suchthatl(θ M ) <<|G(θ) M )|<<u(θ M ),θ M ∈ML (3b)

[0015] |G(θ S )|<<ρ(θ S ),θ S ∈SL (3c)

[0016] The optimization problem (3) consists of (3a)-(3c), where ML refers to the azimuth angle of the main lobe region, SL refers to the azimuth angle of the side lobe region, and the step size of the azimuth angle is 1°; (3b) indicates that the array should fluctuate within a certain range in the main lobe region, and (3c) indicates that the side lobe level should be less than an upper bound value, l(θ M ) represents the upper bound of the radiation pattern level in the main lobe region, u(θ) M) represents the lower bound of the radiation pattern level in the main lobe region, ρ(θ) S The upper bound of the radiation pattern level in the sidelobe region is l(θ). M ), u(θ) M ) and ρ(θ) S All of these are normal numbers.

[0017] Furthermore, step 2 specifically includes the following steps: performing virtual-real separation on each element in w, resulting in:

[0018] x=[R(w1),…,R(w N ),I(w1),…,I(w N )] T (4)

[0019] Where R(·) represents the operation of extracting the real part, I(·) represents the operation of extracting the imaginary part, and T represents the transpose operation; similarly, the far-field electric field value in the θ direction is separated into real and imaginary parts, resulting in a two-dimensional column vector:

[0020] d = [R(G(θ)) I(G(θ))] T (5)

[0021] Construct a new matrix U:

[0022]

[0023] At this point, the real and imaginary parts of the electric field value are obtained by multiplying the first and second rows of matrix U by the vector x, respectively. Therefore, the expression for the square of the electric field magnitude is written as:

[0024] |G(θ)| 2 =x T U T Ux (7)

[0025] Let P represent U T According to linear algebra theory, equation (7) is equivalent to:

[0026] |G(θ)| 2 =x T U T Ux = tr(Pxx) T )=tr(PX) (8)

[0027] Where X = xx T Here, X is a symmetric positive semi-definite matrix with a rank of 1.

[0028] Furthermore, step 3 specifically includes the following steps: transforming the pattern synthesis problem (3) into an equivalent problem (9):

[0029] find X (9a)

[0030] such that L(θ i )<tr(P i X) << U(θ) i ),θ i ∈ML (9b)

[0031] tr(P j X) << ρ(θ) j ),θ j ∈SL (9c)

[0032] rank(X) = 1, X >> 0 (9d)

[0033] (9a)-(9d) constitute problem (9), where U(θ) i ) refers to θ i The upper bound of the radiation pattern level, L(θ) i ) refers to θ i The lower bound of the radiation pattern level, ρ(θ) j ) is in θ j The upper bound of the radiation pattern level value is given, and all three are constants. At this point, problem (9) is equivalent to the original problem (3). The optimization variable of problem (9) is transformed into matrix X. In problem (9), only the rank 1 constraint is a non-convex constraint. Discarding it, we obtain a convex optimization problem (10).

[0034] find X (10a)

[0035] suchthatL(θ i ) << tr(P i X) << U(θ) i ),θ i ∈ML (10b)

[0036] X>>0,tr(P j X) << ρ(θ) j ),θ j ∈SL (10c)

[0037] Problem (10) has an optimal solution; when the solution X of problem (10) is... * When the rank of X is 1, * It is also the optimal solution to the original problem (3), when X * When the rank is not 1, we introduce the rank minimization theory and solve for the minimum rank solution iteratively:

[0038]

[0039] stL(θ i ) << tr(P i X) << U(θ)i ),θ i ∈ML (11b)

[0040] X>>0,tr(P j X) << ρ(θ) j ),θ j ∈SL (11c)

[0041] Where δ is a small positive real number, I is a 2N-order identity matrix, t represents the t-th iteration, and the solution to equation (11) is X′. When the rank of X′ is 1, X′ is the optimal solution to the original problem. When the rank is not 1, an approximate solution to the original problem can be obtained by performing eigenvalue decomposition on X′:

[0042] X=λaa T (12)

[0043]

[0044] Where λ is the largest eigenvalue of X′, and a is the corresponding eigenvector;

[0045] After obtaining x, the complex excitation vector w of the original problem is obtained by equation (4).

[0046] The method for optimizing the radiation pattern of a metasurface array antenna based on semi-definite relaxation, as proposed in this invention, has the following advantages:

[0047] 1. This invention separates the real and imaginary parts of each element of the complex excitation vector, transforming an optimization problem in the complex field into an optimization problem in the real field. Then, it solves the problem through semidefinite relaxed convex programming and eigenvalue decomposition, which can obtain the array element amplitude and phase required for a specified pattern in a very short time.

[0048] 2. The method of this invention is simple, and subsequent full-wave simulation and experimental testing have proven its effectiveness, which has practical engineering significance. Attached Figure Description

[0049] Figure 1 This is a one-dimensional linear array spatial diagram of an embodiment of the present invention;

[0050] Figure 2 A diagram of a metasurface unit structure with independently adjustable amplitude and phase used in an embodiment of the present invention;

[0051] Figure 3 This is a full-wave simulation 3D scene diagram of an embodiment of the present invention;

[0052] Figure 4 This is a comparison chart of numerical values, full-wavelength data, and test results for an embodiment of the present invention. Detailed Implementation

[0053] To better understand the purpose, structure, and function of this invention, the following detailed description of a metasurface array antenna pattern optimization method based on semi-definite relaxation is provided in conjunction with the accompanying drawings.

[0054] A method for optimizing the radiation pattern of a metasurface array antenna based on semi-definite relaxation is proposed. First, consider a one-dimensional N-element linear array placed at equal intervals along the x-axis. The array's excitation vector w(w=(w1,…,w)) is then used. N ) T Let w be the variable to be optimized, and establish an optimization problem according to the specified objective; separate the virtual and real elements for each element of w, and rewrite the original optimization problem equivalently, while discarding the non-convex constraints in the equivalent problem to obtain a semidefinite relaxed convex problem; finally, perform eigenvalue decomposition on the optimal solution of the convex problem to obtain the optimal solution or feasible solution of the original problem.

[0055] Step 1: Establish the original optimization problem model

[0056] The far-field electric field expression for a linear array is:

[0057] G(θ)=v(θ) H w(1)

[0058] Where v(θ) is the steering vector in the θ direction:

[0059]

[0060] Where E i This is the element radiation pattern of the i-th element, where i = 1, 2, ..., N; k is the wave number, and j is the imaginary unit; x i Here, represents the coordinates of the i-th element, i = 1, 2, ..., N; H represents the conjugate transpose operation; w is the complex excitation vector of the array, containing the amplitude and phase information of each element, w = (w1, ..., w2). N ) T G(θ) refers to the far-field pattern of the array in the θ direction, where θ∈[-90°,90°]. The following pattern optimization problem model is constructed:

[0061] find w (3a)

[0062] such thatl(θ M ) <<|G(θ) M )|<<u(θ M ),θ M ∈ML (3b)

[0063] |G(θ S )|<<ρ(θ S ),θ S ∈SL (3c)

[0064] The optimization problem (3) consists of (3a)-(3c), where (3b) and (3c) represent the desired radiation pattern where the voltage level fluctuates within a certain range in the main lobe region and the voltage level is less than a certain upper bound in the side lobe region. M ), u(θ) M ) represent the upper and lower bounds of the radiation pattern level in the main lobe region, respectively, ρ(θ) S The upper bound of the radiation pattern level in the sidelobe region is given by the formula, where l(θ) M ), u(θ) M ) and ρ(θ) S All of these are normal numbers.

[0065] Step 2: Rewrite the optimization variable w

[0066] Next, the complex excitation vector is separated into real and virtual elements to obtain a vector with 2N elements:

[0067] x=[R(w1),…,R(w N ),I(w1),…,I(w N )] T (4)

[0068] Simultaneously, the far-field electric field value is also separated into real and virtual values:

[0069] d = [R(G(θ)) I(G(θ))] T (5)

[0070] To facilitate an equivalent rewrite of the original problem, a new matrix U is constructed:

[0071]

[0072] Therefore, the expression for the square of the electric field modulus can be written as:

[0073] |G(θ)| 2 =x T U T Ux (7)

[0074] Let P represent U T U, (7) is equivalent to:

[0075] |G(θ)| 2 =x T U T Ux = tr(Pxx) T )=tr(PX) (8)

[0076] Step 3: Transform the original optimization problem

[0077] The equivalent transformation of equation (3) in the pattern optimization problem is:

[0078] find X (9a)

[0079] such that L(θ) i ) << tr(P i X) << U(θ) i ),θ i ∈ML (9b)

[0080] tr(P j X) << ρ(θ) j ),θ j ∈SL (9c)

[0081] rank(X) = 1, X >> 0 (9d)

[0082] Among them U(θ i ) and L(θ i ) respectively refer to θ i The upper and lower bounds of the radiation pattern level, ρ(θ) j ) is in θ j The upper bound of the radiation pattern level value, all three are constants. Problem (9) is equivalent to the original problem (3). In this case, the optimization variable of problem (9) is matrix X. In equation (9), only the rank 1 constraint is a non-convex constraint. After discarding it, we get a convex optimization problem (10):

[0083] find X (10a)

[0084] such that L(θ) i ) << tr(P i X) << U(θ) i ),θ i ∈ML (10b)

[0085] X>>0,tr(P j X) << ρ(θ) j ),θ j ∈SL (10c)

[0086] When equation (10) is solved, X * When the rank of X is 1, * It is the optimal solution to the original problem (3), when X * When the rank is not 1, we introduce the rank minimization theory to find the optimal or feasible solution to the original problem:

[0087]

[0088] stL(θ i ) << tr(P i X) << U(θ) i ),θi ∈ML (11b)

[0089] X>>0,tr(P j X) << ρ(θ) j ),θ j ∈SL (11c)

[0090] Where δ is a small positive real number, I is a 2N-order identity matrix, t represents the t-th iteration, and the solution to equation (11) is X′. When the rank of X′ is 1, X′ is the optimal solution to the original problem. When the rank is not 1, an approximate solution to the original problem can be obtained by performing eigenvalue decomposition on X′:

[0091] X=λaa T (12)

[0092]

[0093] Where λ is the largest eigenvalue of X′, and a is the eigenvector corresponding to the largest eigenvalue. After obtaining x, the complex excitation vector w of the original problem can be obtained from equation (4).

[0094] The following is a specific implementation case:

[0095] For a 22-element one-dimensional linear array with an element spacing of 0.005 mm, an operating frequency of 16 GHz, a main lobe direction cosecant beam, and an azimuth range of [5°, 36°], with the center direction at 5°, the level value of the main lobe region should fluctuate within ±0.5 dB of the cosecant function range; the maximum sidelobe level should be less than -7.8 dB, with the corresponding angular region being [-90°, 4°]∪[37°, 90°]. The array excitation vector corresponding to the desired radiation pattern can be obtained from equations (1) to (13). Similar radiation pattern optimization problems can be solved quickly using this invention method, with the fastest iteration time being 1.6 s.

[0096] Figure 2 This is a metasurface structure with independent amplitude-phase adjustment. The invented method was verified using full-wave simulation in CST with this structure. The aperture size α, characterizing this open-ring metasurface structure, is used to control the amplitude of the metasurface; the aperture direction β, characterizing this open-ring metasurface structure, is used to control the phase of the metasurface. The amplitude and phase distributions of 22 array elements were obtained using the invented method. Parameter scanning was performed in CST to obtain 22 open rings with different aperture directions and sizes, and then full-wave simulation was performed.

[0097] Figure 3 This is a 3D scene diagram used in CST for full-wave simulation.

[0098] Figure 4This chart compares the results of the invented pattern optimization method, full-wave simulation, and hardware testing. The full-wave simulation results agree well with the numerical optimization results, while the test results show some error due to the influence of the testing environment. Overall, this demonstrates the effectiveness of the numerical method.

[0099] In summary, the pattern optimization method of this invention is a very fast and effective method. The effectiveness of this optimization method has been strongly demonstrated by full-wave simulation and hardware testing, and it has great practical significance for the research of array synthesis.

[0100] It is understood that the present invention has been described through some embodiments, and those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of the present invention.

Claims

1. A method for metasurface array antenna pattern optimization based on semi-definite relaxation, characterized in that, The method comprises the following three steps: Step 1: considering a linear array with element spacing , the number of elements , write the far-field electric field expression of the array, combine the requirements of the far-field pattern to be formed, divide the main lobe region and the side lobe region, determine the beam shape to be formed in the main lobe region, and determine the far-field pattern level value of the side lobe region; set the complex excitation vector of the array as the initial optimization variable, and establish an original optimization problem model, which is a non-convex problem model; Step 2: Real and imaginary part separation is performed on each element of the complex excitation vector, resulting in one dimensional variable; Step 3: The original problem is equivalently rewritten and the non-convex constraints are removed, so that a convex semi-definite relaxation problem is obtained, which has an optimal solution, and the rank minimization theory is introduced to make the solution close to ; Finally, the cases are discussed. If the rank of the foregoing solution is , it is the optimal solution of the original problem. If the rank is not , the approximate solution of the original problem, i.e. the feasible solution, can be obtained by eigenvalue decomposition. Step 1 specifically comprises the following steps: considering a linear array with an element spacing of d and an element number of N, an array factor expression of the linear array is written as: ； is a direction of the guide vector: ； is the directional pattern of the unit , where ; is the wave number, is the imaginary unit; is the coordinate position of the unit , where ;H represents the conjugate transpose operation; is the complex excitation vector of the array, containing the amplitude and phase information of each unit, ; denotes the far-field directional pattern of the array in the direction , ; the directional pattern optimization problem (3) model is constructed as follows: (3a)； (3b)； (3c)； (3a) - (3c) consisting of an optimization problem (3), where denotes the azimuth angle of the main lobe region, denotes the azimuth angle of the side lobe region, the step size of the azimuth angle is taken as ; (3b) represents the requirement that the array fluctuates within a range in the main lobe region, (3c) represents that the side lobe level is less than an upper limit value, represents the upper limit of the main lobe region pattern level value, represents the lower limit of the main lobe region pattern level value, represents the upper limit of the side lobe region pattern level value, , and are all positive numbers.

2. The semidefinite relaxation based metasurface array antenna pattern optimization method of claim 1, wherein, The step 2 includes the following steps: separating the virtual and real of each element in to obtain: ； where R(·) represents the real part operation, I(·) represents the imaginary part operation, and T represents the transpose operation; similarly, the far-field electric field values in the direction are separated into real and imaginary parts to obtain a two-dimensional column vector: where R(·) represents the real part operation, I(·) represents the imaginary part operation, and T represents the transpose operation; similarly, the far-field electric field values in the direction are separated into real and imaginary parts to obtain a two-dimensional column vector: ； A new matrix U is constructed: ； The real and imaginary parts of the electric field value at this time are obtained by multiplying the first and second rows of the matrix by the vectors respectively, so the expression for the square of the electric field modulus is written as: ； with denotes According to linear algebra theory, (7) is equivalent to ； wherein here is a symmetric positive semi-definite matrix, and has a rank of .

3. The semidefinite-relaxation-based metasurface array antenna pattern optimization method of claim 2, wherein, Step 3 specifically comprises the following steps: the directivity pattern synthesis problem (3) is equivalently transformed into problem (9): (9a)； (9b)； (9c)； (9d)； (9a)-(9d) constitute Problem (9), where denote an upper bound on the directional pattern level at denote a lower bound on the directional pattern level at is an upper bound on the directional pattern level at are constants; in this case, Problem (9) is equivalent to the original Problem (3), and the optimization variable of Problem (9) is transformed into the matrix In Problem (9), only the rank constraint is non-convex, and it is removed to obtain a convex optimization problem (10): (10a)； (10b)； (10c)； Problem (10) has an optimal solution; when the rank of the solution of problem (10) is , the solution is also the optimal solution of the original problem (3), when the rank of the solution of problem (10) is not , the rank minimization theory is introduced, and the solution with the minimum rank is solved by iteration: ​ (11a)； (11b)； (11c)； where is a small positive real number, is the identity matrix of order represents the th iteration, the solution of (11) is when the rank of is the optimal solution of the original problem, and when the rank is not an approximate solution of the original problem is obtained by performing an eigenvalue decomposition of ​ ； ； wherein is the largest eigenvalue of is the corresponding eigenvector; obtained The complex excitation vector of the original problem is then obtained from equation (4) .

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