Phase-only reconfigurable antenna array beam forming method based on positive semidefinite relaxation

Abstract

The invention relates to the technical field of array antennas, in particular to a phase-only reconfigurable antenna array beam forming method based on positive semidefinite relaxation. According to the method, the performance requirements of diversified radiation patterns are met, and the co-weighting design of different beam excitation amplitudes is effectively realized. The method comprises the following steps: converting a complex relation of excitation co-weighted amplitude into an expression form of a matrix trace, and further simplifying an original problem into an optimization problem of a positive semi-definite matrix containing solving excitation. A positive semidefinite relaxation principle is utilized to relax the constraint of a positive semidefinite matrix to be solved with the rank being 1, and the constraint is converted into a convex optimization problem which is more concise and easy to process. In the solving process, an eigenvalue decomposition method is combined, an original non-convex optimization problem can be accurately approached, and high accuracy of the solution is ensured. In addition, the method does not need complex parameter adjustment and setting, is easy and convenient to implement and efficient in calculation, and can be widely applied to various array forms.

Description

Technical Field

[0001] The present invention relates to the technical field of array antennas, and in particular to a phase-only reconfigurable antenna array beamforming method based on semi-definite relaxation. Background Art

[0002] Array antennas, core components in numerous key fields such as wireless communications, radar systems, remote sensing technology, and satellite communications, demonstrate exceptional beamforming flexibility. They can flexibly generate and adjust the shape of beam patterns based on the complexity and diverse requirements of specific application scenarios. Currently, array antenna beamforming technologies have been extensively researched and can be broadly categorized into three main types: complex weighting, which achieves beamforming by simultaneously adjusting the excitation amplitude and phase of antenna elements; amplitude weighting, which focuses on adjusting the excitation amplitude of antenna elements while maintaining the phase; and phase weighting, which achieves beamforming solely by precisely controlling the feed phase, eliminating the need to adjust the feed signal amplitude. This approach not only avoids the need for additional hardware and simplifies the feed system structure, but also significantly reduces system losses and manufacturing costs. Therefore, phase weighting, as an efficient and cost-effective solution, has attracted considerable attention in the field of active phased array antennas.

[0003] Existing phase-only beamforming methods primarily include iterative Fourier transforms, adaptive algorithms, and evolutionary algorithms (such as differential evolution, genetic algorithms, and particle swarm optimization). The iterative Fourier transform method is favored for its high efficiency, but it is susceptible to limitations in array structure. Adaptive algorithms leverage the antenna system's real-time adjustment capabilities to dynamically adjust array element excitations, enhancing the desired signal and suppressing interfering signals. While this method is flexible and efficient, practical applications still require addressing slow convergence and unstable algorithm accuracy. Evolutionary optimization algorithms are widely applicable due to their ability to flexibly address diverse scenarios. However, these algorithms also face challenges such as slow convergence, low computational efficiency, numerous algorithm parameters, and a tendency to fall into local optimality.

[0004] Chinese Patent 201710101241.9 discloses a phase-only weighted array antenna beamforming optimization method based on a hybrid MIFT and CP algorithm. This method first uses an iterative Fourier transform algorithm to obtain the complex excitations for each beam and fix their phase information. It then employs a convex optimization algorithm to optimize the excitation amplitude shared by these beams to meet the overall radiation performance requirements. However, this method is currently only applicable to equally spaced linear arrays and cannot be directly applied to beamforming for other array types. On the other hand, Chinese Patent 202311824304.5 discloses a reconfigurable sparse array beamforming method based on oscillatory fusion differential evolution. By introducing an oscillatory fusion differential evolution algorithm, this method significantly reduces computational complexity compared to traditional differential evolution algorithms. Furthermore, the paper "A Phase-Only Method for Reconfigurable Array Antenna Patterns" uses a real-coded genetic algorithm for phase-only pattern optimization, effectively avoiding the complex encoding and decoding processes of traditional genetic algorithms. However, these two methods are still evolutionary algorithms in essence, and will inevitably face the common shortcomings of evolutionary algorithms, such as computational efficiency and parameter adjustment.

[0005] Therefore, developing a phase-only reconfigurable array antenna beamforming method that can adapt to various array forms, has efficient computing capabilities, high stability, and is easy to implement is of great significance for practical engineering applications. Summary of the Invention

[0006] The purpose of the present invention is to overcome the challenges commonly faced by current phase-only reconfigurable beamforming technologies, such as lengthy processing time, insufficient accuracy, and limited antenna forms, and to propose a phase-only reconfigurable array antenna beamforming method based on semi-definite relaxation technology, thereby significantly improving the efficiency and flexibility of beamforming.

[0007] The present invention comprises the following steps:

[0008] 1) Construct the excitation relation expression of phase-only reconfigurable beamforming;

[0009] 2) Express the incentive relationship in the form of matrix trace Tr(·);

[0010] 3) Use mathematical models to describe the beam reconfigurability of array antennas under phase-only control;

[0011] 4) Use semi-positive definite relaxation and eigenvalue decomposition to solve the excitation vector.

[0012] In step 1), the excitation relationship expression of phase-only reconfigurable beamforming is constructed, and the specific steps are:

[0013] If the complex excitation of the nth element in two different beams of the array antenna in different application scenarios is expressed as and Where n=1,2...,N, then the excitation vectors corresponding to the two beams can be expressed as and Right now

[0014]

[0015] These complex excitations determine the radiation characteristics and directivity of each beam. Under the condition of phase control only, it is required and The amplitudes of the beams must remain consistent, and the beam reconfigurability is achieved only by adjusting their phases. Therefore, the expression of the excitation relationship of different beams is:

[0016]

[0017] In step 2), the incentive relationship is expressed in the form of a matrix trace Tr(·), and the specific steps are:

[0018] First, the excitations corresponding to the two different beams and Redefined and described through mathematical quadratic forms

[0019]

[0020]

[0021] Among them, Q n is the nth diagonal matrix of size 2N×2N, and

[0022]

[0023] Therefore, the excitation relationship of phase-only reconfigurable beamforming is Can be further written as

[0024]

[0025] Secondly, in order to and Put it in a vector to facilitate the solution, let but It can be equivalently converted into

[0026]

[0027] Among them, Q n1 and Q n2 are all 4N×4N diagonal matrices, and

[0028]

[0029] Finally, further Converted into the matrix trace form Tr(·), we have and Introducing a new variable at this time It is equivalent to W being a semi-positive symmetric matrix of rank 1, that is, W ≥ 0 and rank(W) = 1, then the incentive relationship can be finally expressed in the form of matrix trace as

[0030] Tr(Q n1 W)=Tr(Q n2 W) and W≥0, rank(W)=1

[0031] In step 3), the array antenna beam reconfigurability problem under phase-only control can be expressed as: find W so that it satisfies the following constraints:

[0032]

[0033] Among them, C1 and C2 respectively represent a series of constraint sets that need to be followed for two different beam designs. C1 and C2 can be flexibly set as specific constraints when designing a shaped beam, and can also be adjusted to the constraints required for focusing a beam, thus achieving wide adaptability and flexibility for beamforming problems.

[0034] In step 4), the excitation vector is solved by using semi-definite relaxation and eigenvalue decomposition. Specifically, the non-convex constraint rank(W)=1 in step 3) is relaxed, and an iterative method is used to gradually minimize the rank of the matrix W. This relaxation is called semi-definite relaxation, and the reconfigurable beamforming problem after relaxation can be effectively solved using a convex optimization toolkit. Mathematically, it can be expressed as

[0035] W k+1 =argmin Tr((W k +δI) -1 W k )

[0036]

[0037] Where δ>0 is a small regularization parameter introduced. Once the optimal solution of the matrix W is obtained through the convex optimization algorithm, the vector can be further recovered from W using the eigenvalue decomposition technique. That is, W≈σ max vv T , where σ max is the maximum eigenvalue of the matrix W, and v is the corresponding eigenvector. According to the eigenvalue decomposition,

[0038] Compared with the prior art methods, the present invention has the following beneficial effects:

[0039] 1) The phase-only reconfigurable array antenna beamforming method based on semi-positive definite relaxation proposed in the present invention realizes the co-weighted design of different beam excitation amplitudes while meeting the performance requirements of different radiation patterns. This method cleverly transforms the complex relationship of the co-weighted excitation amplitudes into a matrix trace expression, and then simplifies the original problem into an optimization problem of the semi-positive definite matrix containing the excitation solution. By applying the semi-positive definite relaxation principle, the strict constraint of the rank of 1 of the semi-positive definite matrix to be solved is relaxed and converted into a simpler and more efficient convex optimization problem. In the solution process, combined with the eigenvalue decomposition method, the present invention can accurately approximate the original non-convex optimization problem and ensure the high accuracy of the solution. At the same time, relying on the perfect theoretical system of convex optimization and with the help of a mature convex optimization toolkit, the computational efficiency is significantly improved. In addition, compared with evolutionary algorithms, this method effectively avoids tedious parameter adjustment and setting, and further simplifies the operation process.

[0040] 2) The present invention exhibits broad applicability. It is not limited to a specific array format and can be flexibly applied to linear arrays, planar arrays, uniform arrays, and non-uniform arrays. This universality makes the present invention highly practical in a variety of real-world scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 The figure is a flow chart of the phase-only reconfigurable antenna array beamforming method based on semi-definite relaxation of the present invention.

[0042] Figure 2 The far-field pattern of the reconfigurable focused beam and shaped beam of the present invention.

[0043] Figure 3 In order to meet the Figure 2 Array element excitation amplitude distribution diagram under the directivity pattern requirements.

[0044] Figure 4 In order to meet the Figure 2 Array element excitation phase distribution diagram under the directivity pattern requirements. DETAILED DESCRIPTION

[0045] The terms used in various embodiments of the present invention are only used to describe the purpose of specific embodiments and are not intended to limit the various embodiments of the present invention. As used herein, the singular form is intended to also include the plural form, unless the context clearly indicates otherwise. Unless otherwise limited, all terms used here (including technical terms and scientific terms) have the same meaning as those of ordinary skill in the art generally understood by the various embodiments of the present invention. The terms (such as those defined in generally used dictionaries) will be interpreted as having the same meaning as the contextual meaning in the relevant technical field and will not be interpreted as having idealized meaning or too formal meaning, unless clearly defined in various embodiments of the present invention.

[0046] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with examples and drawings. The exemplary embodiments of the present invention and their descriptions are only used to explain the present invention and are not intended to limit the present invention.

[0047] The implementation process of the present invention is as follows:

[0048] like Figure 1 As shown, a phase-only reconfigurable antenna array beamforming method based on semi-positive relaxation includes the following steps:

[0049] 1) Construct the excitation relationship expression of phase-only reconfigurable beamforming:

[0050] If the complex excitation of the nth element in two different beams of the array antenna in different application scenarios is expressed as and Where n=1,2...,N, then the excitation vectors corresponding to the two beams can be expressed as and Right now

[0051]

[0052] These complex excitations determine the radiation characteristics and directivity of each beam. Under the condition of phase control only, it is required and The amplitudes of the beams must remain consistent, and the beam reconfigurability is achieved only by adjusting their phases. Therefore, the expression of the excitation relationship of different beams is:

[0053]

[0054] 2) Incentive relationships It is expressed in the form of matrix trace Tr(·):

[0055] First, the excitations corresponding to the two different beams and Redefined and described through mathematical quadratic forms

[0056]

[0057] Among them, Q n is the nth diagonal matrix of size 2N×2N, and

[0058]

[0059] Therefore, the excitation relationship of phase-only reconfigurable beamforming is Can be further written as

[0060]

[0061] Secondly, in order to and Put it in a vector to facilitate the solution, let but It can be equivalently converted into

[0062]

[0063] Among them, Q n1 and Q n2 are all 4N×4N diagonal matrices, and

[0064]

[0065] Finally, further Converted into the matrix trace form Tr(·), we have

[0066]

[0067] Introducing a new variable at this time This is equivalent to W being a semi-positive symmetric matrix of rank 1, i.e. W ≥ 0 and rank(W) = 1. Then the incentive relationship can be finally expressed in the form of matrix trace as

[0068] Tr(Q n1 W)=Tr(Q n2 W) and W≥0, rank(W)=1

[0069] 3) Description of the problem of array antenna beam reconfigurability under phase-only control:

[0070] In the framework of pattern synthesis, if C1 and C2 represent a set of constraints that must be followed for two different beam designs, the core of the problem is transformed into solving the constant amplitude excitation vector and These two vectors must satisfy the condition that the beam generated by them can strictly follow the constraints defined by C1 and C2 respectively. Here, C1 and C2 can be flexibly set as specific constraints when designing a shaped beam, or they can be adjusted to the constraints required for focusing the beam, thus achieving wide adaptability and flexibility in beamforming problems. Specifically, mathematically, it can be expressed as finding W so that it satisfies the following constraints

[0071]

[0072] 4) Solve the optimization problem in step 3 using semi-definite relaxation and eigenvalue decomposition:

[0073] Since the optimization problem in step 3 contains the non-convex rank constraint rank(W)=1, this directly limits the possibility of using a standard convex optimization algorithm to solve it directly. To solve this problem, an alternative strategy is adopted, which is to relax the strict constraints on the rank and instead use an iterative method to gradually minimize the rank of the matrix W. Specifically, W is optimized through an iterative process so that while maintaining other optimization objectives, it gradually tends to a lower rank structure, in order to eventually approximately satisfy the key properties of the original problem. This relaxation is called semi-definite relaxation, which transforms the original non-convex optimization problem into a convex optimization problem, so that it can be efficiently solved using a convex optimization algorithm. Mathematically, it can be expressed as

[0074] W k+1 =argmin Tr((W k +δI) -1 W k )

[0075]

[0076] Where δ>0 is a small regularization parameter introduced to smooth the optimization problem and promote numerical stability. Once the optimal solution of the matrix W is obtained by the convex optimization algorithm, the vector can be further recovered from W using the eigenvalue decomposition technique. Right now

[0077] W≈σ max vv T

[0078] Among them, σ max is the maximum eigenvalue of the matrix W, and v is the corresponding eigenvector. According to the eigenvalue decomposition, Specific embodiment:

[0080] The beam of a uniform linear array composed of 20 ideal point sources is reconfigurable, and the array element spacing is set to half a wavelength. In this embodiment, the array radiation pattern is flexibly switched between a focused beam and a flat-top beam by adjusting only the excitation phase of each array element without changing its excitation amplitude. For the focused beam, the constraint conditions are: in the region of |sinθ|≥0.15, the sidelobe level is lower than -25dB; for the flat-top beam, the constraint conditions are: in the region of |sinθ|≥0.3, the sidelobe level is lower than -20dB, and in the region of |sinθ|≤0.2, a flat-top beam with an upper and lower fluctuation amplitude of ±0.25dB is formed. The integrated far-field pattern is shown in Figure 2 , where both the focused beam and the flat-top beam strictly meet their preset sidelobe level and shaping requirements. Figure 3 and Figure 4 The excitation amplitude distribution and phase distribution under the two beam states are given respectively. The results show that the two beams share the same excitation amplitude distribution, and the beam shape can be significantly changed only by changing the phase distribution, which verifies the effectiveness of the phase reconfiguration technology in the present invention and shows that the method proposed in the present invention can efficiently and flexibly meet the needs of phase-only reconfiguration of the beam in different application scenarios.

[0081] The embodiment described above is only a specific example of the present invention. Those skilled in the art will understand that various changes, modifications, substitutions and variations may be made to these embodiments without departing from the principles and purpose of the present invention. The scope of the present invention is defined by the claims and their equivalents.

Claims

1. A phase-only reconfigurable antenna array beamforming method based on semi-definite relaxation, characterized by The following steps are involved: 1) Construct the excitation relation expression of phase-only reconfigurable beamforming; 2) Express the incentive relationship in the form of matrix trace Tr(·); 3) Use mathematical models to describe the beam reconfigurability of array antennas under phase-only control; 4) Use semi-positive definite relaxation and eigenvalue decomposition to solve the excitation vector.

2. A phase-only reconfigurable antenna array beamforming method based on semi-positive relaxation according to claim 1, characterized in that In step 1), the excitation relationship expression of the phase-only reconfigurable beamforming is constructed, specifically: if the complex excitation of the nth array element in two different beams of the array antenna in different application scenarios is expressed as and Where n=1,2...,N, then the excitation vectors corresponding to the two beams can be expressed as and Right now These complex excitations determine the radiation characteristics and directivity of each beam. Under the condition of phase control only, it is required and The amplitudes of the beams must remain consistent, and the beam reconfigurability is achieved only by adjusting their phases. Therefore, the expression of the excitation relationship of different beams is:

3. The method for beamforming a phase-only reconfigurable antenna array based on semi-positive relaxation according to claim 1, characterized in that In step 2), the excitation relationship is expressed in the form of a matrix trace Tr(·), and the specific steps are: the excitation corresponding to two different beams and Redefined and described through mathematical quadratic forms Among them, Q n is the nth diagonal matrix of size 2N×2N, and Therefore, the excitation relationship of phase-only reconfigurable beamforming is Can be further written as In order to and Put it in a vector to facilitate the solution, let but It can be equivalently converted into Among them, Q n1 and Q n2 are all 4N×4N diagonal matrices, and Further Converted into the matrix trace form Tr(·), we have T and Introducing a new variable at this time It is equivalent to W being a semi-positive symmetric matrix of rank 1, that is, W ≥ 0 and rank(W) = 1, then the incentive relationship can be finally expressed in the form of matrix trace as Tr(Q n1 W) = Tr(Q n2 W) and W ≥ 0, rank(W) = 1.

4. The method for beamforming a phase-only reconfigurable antenna array based on semi-definite relaxation according to claim 1, characterized in that In step 3), the array antenna beam reconfigurability problem under phase-only control can be expressed as: find W so that it satisfies the following constraints: Among them, C1 and C2 respectively represent a series of constraint sets that need to be followed for two different beam designs. C1 and C2 can be flexibly set as specific constraints when designing a shaped beam, and can also be adjusted to the constraints required for focusing a beam, thus achieving wide adaptability and flexibility for beamforming problems.

5. The method for beamforming a phase-only reconfigurable antenna array based on semi-positive relaxation according to claim 1, characterized in that In step 4), the excitation vector is solved by using semi-positive definite relaxation and eigenvalue decomposition. Specifically, the non-convex constraint condition rank(W)=1 in step 3) is relaxed, and an iterative method is used to gradually minimize the rank of the matrix W. This relaxation is called semi-positive definite relaxation, and the reconfigurable beamforming problem after relaxation can be effectively solved using a convex optimization toolkit, which can be mathematically expressed as IN k+1 =argmin Tr((W k +δI) -1 IN k ) Among them, δ>0 is a small regularization parameter introduced. Once the optimal solution of the matrix W is obtained through the convex optimization algorithm, the vector can be further recovered from W using the eigenvalue decomposition technique. That is, W≈σ max vv T , where σ max is the maximum eigenvalue of the matrix W, v is the corresponding eigenvector, and according to the eigenvalue decomposition,

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