Robust hybrid beam forming method for cooperative communication of multiple low-orbit satellites

Abstract

The invention provides a robust hybrid beam forming method for cooperative communication of multiple low-orbit satellites, which relates to the technical field of satellite communication and comprises the following steps of: S1, constructing a joint optimization problem for joint beam forming and user scheduling; s2, the minimum channel gain of all users is maximized, when the number of satellite antennas is larger than a preset threshold value, the step S3 is executed, and otherwise, the step S5 is executed; s3, constructing a minimum signal to interference plus noise ratio maximization problem, and solving to obtain an optimal all-digital precoding matrix; s4, based on the optimal all-digital precoding matrix, constructing a digital-analog hybrid precoding matrix design problem to obtain a digital-analog hybrid precoding matrix; and S5, constructing and solving an analog pre-coding matrix design problem based on maximization of the signal leakage and noise ratio to obtain an optimal analog pre-coding matrix, and then solving an optimal digital pre-coding matrix. According to the invention, the overall performance and the service quality of the multi-low-orbit satellite communication network can be obviously enhanced.

Description

Technical Field

[0001] This invention relates to the field of satellite communication technology, and in particular to a robust hybrid beamforming method for collaborative communication among multiple low-Earth orbit satellites. Background Technology

[0002] Low Earth Orbit (LEO) satellite communication systems are typically deployed in orbits below 2000 kilometers, offering significant advantages in latency and energy consumption compared to medium Earth Orbit (MEO) and geostationary orbit (GEO) satellites. Due to their lower orbital altitude, LEO satellites can achieve high-quality communication with lower power, making them suitable for providing wide-area coverage and high-capacity services to remote, maritime, and underdeveloped regions. Multiple LEO satellites can work collaboratively to achieve dynamic sharing of user resources, seamless network coverage, and improved communication link stability. This collaborative mechanism generally includes joint beamforming among multiple satellites, user scheduling strategies, and interference suppression techniques under asynchronous propagation delays, demonstrating good robustness and flexibility in addressing communication challenges in highly mobile and complex environments.

[0003] Multiple-input multiple-output (MIMO) technology significantly improves the communication quality and spectral efficiency of satellite links by constructing highly directional and high-gain beams. However, traditional all-digital beamforming architectures require an RF link for each antenna, leading to a sharp increase in system hardware costs and energy consumption, making them unsuitable for resource-constrained low-Earth orbit (LEO) satellite platforms. Furthermore, the high-speed motion of LEO satellites and the rapidly changing transmission environment make it difficult to accurately obtain channel state information, which is subject to measurement errors and feedback delays, severely impacting system performance. Summary of the Invention

[0004] The main objective of this invention is to propose a robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication, which can ensure the communication reliability and fairness of the system in complex dynamic channel environments, and significantly enhance the overall performance and service quality of multi-low-Earth orbit satellite communication networks.

[0005] This invention is achieved through the following technical solution:

[0006] A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication is characterized by the following steps:

[0007] Step S1: Based on a communication system with multiple low-Earth orbit satellites and ground users, construct a joint optimization problem for joint beamforming and user scheduling. The optimization objective of this joint optimization problem is to maximize the minimum signal-to-interference-plus-noise ratio (SINR) for all users. This joint optimization problem is related to the scheduling parameters between satellites and ground users and the hybrid precoding matrix.

[0008] Step S2: Use the bilinear penalty method to design the scheduling parameters between the satellite and the ground user to maximize the minimum channel gain of all users. Determine whether the number of satellite antennas is greater than a predetermined threshold. If it is less than the threshold, proceed to step S3; otherwise, proceed to step S5.

[0009] Step S3: Construct the minimum signal-to-interference-plus-noise ratio maximization problem under the communication interruption probability constraint of statistical channel state information error. Use Bernstein-type inequalities to relax the problem, introduce relaxation variables to transform the problem into a deterministic constraint, and solve the transformed problem to obtain the optimal all-digital precoding matrix.

[0010] Step S4: Based on the all-digital precoding matrix obtained in step S3, construct the design problem of the mixed digital-analog precoding matrix. Solve the design problem of the mixed digital-analog precoding matrix using an alternating optimization method to obtain the mixed digital-analog precoding matrix.

[0011] Step S5: Based on maximizing signal leakage and noise ratio, construct and solve the analog precoding matrix design problem to obtain the optimal analog precoding matrix. Then, construct and solve the digital precoding matrix design problem to obtain the optimal digital precoding matrix.

[0012] Furthermore, in step S1, the joint optimization problem is expressed as: Where, γ k w represents the signal-to-interference-plus-noise ratio (SIR) for user k. c,k =F RF,c f BB,c,k F represents the hybrid precoding matrix. RF,c f represents the analog precoding matrix of low-Earth orbit satellite c. BB,c,k Let represent the digital precoding matrix of low-Earth orbit satellite c, t represent the minimum signal-to-interference-plus-noise ratio for any user, Pr{.} represent the probability, τ represent the minimum allowable confidence probability, and a c,k P represents the scheduling parameters between low-Earth orbit satellite c and ground user k. T This indicates the maximum permissible launch power for low-Earth orbit satellites. This represents a set of low-Earth orbit (LEO) satellites, where C represents the number of LEO satellites. Let K represent the set of ground users, and K represent the number of ground users. c =N RF This indicates the maximum number of users a low-Earth orbit satellite can serve, [F RF,c ] n,m F represents RF,c The m-th element in the n-th row.

[0013] Furthermore, in step S2, a system is constructed for scheduling parameter a. c,k First sub-optimization problem To overcome the nonconvexity of the minimization function, the sub-optimization problem is transformed into a second sub-optimization problem using a smooth approximation of the minimization function. Introducing auxiliary matrices A and B to solve the binary constraint problem transforms the second sub-optimization problem into a third sub-optimization problem. By combining the bilinear penalty method and introducing the bilinear constraint into the objective function of the third sub-optimization problem through a penalty term, the standard convex problem form of the first sub-optimization problem is obtained. The block coordinate descent method is used to alternately optimize A and B to determine the scheduling parameters. Then, it is determined whether the number of satellite antennas is greater than a predetermined threshold. If it is less, steps S3 and S4 are executed; otherwise, step S5 is executed. Where h... c,k Let μ be the downlink channel vector from low-Earth orbit satellite c to ground user k, and μ be the control parameter for smoothing approximation. Tr{.} represents the trace function, 1 C×K This represents a C×K matrix of all 1s, where vec(·) denotes the vectorization operation, ||·|| ∞ Represents the infinite norm, Let Frobenius norm be denoted, and ρ be the penalty factor, which is gradually increased during iteration to satisfy the constraints.

[0014] Furthermore, in step S2, the method of alternately optimizing A and B using block coordinate descent specifically involves: when B is fixed, the optimization problem for A is... With A fixed, the optimization problem for B is: The penalty factor ρ gradually increases according to the rule ρ = min(ξρ, 2L), where L is the Lipschitz constant and ξ represents the multiplication factor.

[0015] Furthermore, the minimum signal-to-interference-plus-noise ratio (SINR) maximization problem in step S3 is expressed as: Introducing the auxiliary variable χ s The minimum signal-to-interference-plus-noise ratio maximization problem then becomes Introducing auxiliary parameters probability constraint formula Transform into And introduce auxiliary parameters The channel estimation error is expressed as... Thus, the probability constraint formula is reformulated as Among them, t (i) Let represent the value of t in the i-th iteration, Pr{.} denotes the probability calculation, τ denotes the minimum allowable confidence probability, and A S,k A represents the useful signal strength. I,k Indicates the intensity of interference between users. Indicates noise intensity.

[0016] F BB,c,-k =[a c,1 f BB,c,1 ,…a c,k-1 f BB,c,k-1 ,a c,k+1 f BB,c,k+1 …a c,K f BB,c,K ], h k ξ represents the channel vector. k =[ξ k,1 ,…ξ k,C ] T Let ξ represent the channel error vector. k,C N represents the variance of the channel state information error between ground user k and low-Earth orbit satellite C. t This indicates the number of antennas in a uniform planar array facing Earth that a low-Earth orbit satellite is equipped with. N represents t 3D identity matrix Indicates the real part.

[0017] Furthermore, in step S3, the problem transformed from the minimum signal-to-interference-plus-noise ratio (SINR) maximization problem is relaxed using Bernstein-type inequalities to obtain a relaxed expression. Introducing an auxiliary matrix and Then the constraints in the relaxation expression are re-expressed as Using continuous convex approximation techniques for non-convex constraints Perform a convex approximation as The relaxation expression is then transformed into Solving this transformed problem yields the optimal all-digital precoding matrix. Where ρ2 represents the penalty coefficient, x k and y k It is an auxiliary variable. express The value of the i-th iteration.

[0018] Furthermore, in step S4, the mixed-signal precoding matrix design problem is expressed as: First, the mixed-signal precoding matrix is ​​transformed into... To update the simulated precoding matrix, a relaxation constraint method is used, incorporating the constant modulus constraint as a penalty term into the objective function, resulting in... Using continuous approximation algorithms to handle non-convex penalty terms to obtain equivalent problems The analog precoding matrix is ​​obtained by solving this equivalent problem using the standard convex optimization algorithm. Then, the digital precoding matrix optimization problem is addressed by fixing the analog precoding matrix. Solving this problem yields the optimal digital precoding matrix, which is the hybrid precoding matrix for this iteration. In the i-th iteration, we obtain the result with respect to χ. s The problem Solving this problem yields χ s and according to Update parameter t, where, This represents the value of the i-th iteration.

[0019] Furthermore, in step S5, the number of ground users that the low-orbit satellite can serve is equal to the number of its radio frequency links N. RF The N of low-orbit satellite c RF Each radio frequency chain is assigned to its scheduled N RF A ground user, represented as in Let c be the set of users served by the c-th satellite, and the analog precoding matrix design problem is expressed as: The problem of designing the simulation precoding matrix can be transformed into a first subproblem using fractional programming. definition b c,k =α c,k h c,k Rewrite the first subproblem as the second subproblem Using an element-wise optimization method, for f RF,c,k With the other elements fixed, the second subproblem simplifies to a third subproblem for the nth element. The closed-form solution to the third subproblem is: Among them, f RF,c,k Let α represent the simulated beamforming vector of ground user k. c,k As an auxiliary variable in the fractional programming method, the optimal α c,k Represented as (f RF,c,k ) m f RF,c,k The m-th unit, (.) * This indicates the conjugate operation.

[0020] Furthermore, in step S5, the digital precoding matrix design problem is expressed as: Transform the probability constraint into Then, using the Bernstein-Type inequality, the probability constraint is reformulated as follows: The problem of designing a digital precoding matrix is ​​then transformed into Solving this transformed problem yields the optimal digital precoding matrix, where,

[0021] As can be seen from the above description of the present invention, compared with the prior art, the present invention has the following beneficial effects:

[0022] 1. This invention first constructs a joint optimization problem for joint beamforming and user scheduling. Then, it uses a bilinear penalty method to design scheduling parameters between satellites and ground users to maximize the minimum channel gain for all users. When the number of satellite antennas is less than a set threshold, it constructs a minimum signal-to-interference-plus-noise ratio (SINR) maximization problem under the constraint of communication interruption probability under statistical channel state information error. Bernstein-type inequalities are used to relax this problem, and relaxation variables are introduced to transform it into a deterministic constraint. Solving the transformed problem yields the optimal all-digital precoding matrix. Based on this all-digital precoding matrix, a hybrid digital-analog precoding matrix design problem is constructed, and an alternating optimization approach is used to solve the hybrid digital-analog precoding matrix design problem. The design problem is to obtain a hybrid analog-digital precoding matrix. When the number of satellite antennas exceeds a set threshold, the design problem of the analog precoding matrix is ​​constructed and solved based on maximizing the signal leakage and noise ratio to obtain the optimal analog precoding matrix. Then, the design problem of the digital precoding matrix is ​​constructed and solved to obtain the optimal digital precoding matrix. In this way, through the joint design of digital baseband and analog radio frequency, the number of radio frequency links is significantly reduced while maintaining the performance of a near-all-digital scheme. This ensures that the system can still achieve stable and efficient data transmission under imperfect channel state information conditions, thereby guaranteeing the communication reliability and fairness of the system in complex dynamic channel environments and significantly enhancing the overall performance and service quality of multi-low orbit satellite communication networks. Attached Figure Description

[0023] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0024] Figure 1 This is a flowchart of the present invention.

[0025] Figure 2 This is a schematic diagram of multi-low-orbit satellite cooperative communication according to the present invention.

[0026] Figure 3 This is a graph showing the minimum user channel gain of the present invention and the comparative scheme as a function of the ratio of the number of transmit antennas.

[0027] Figure 4 This is a simulation diagram comparing the minimum channel interference-to-noise ratio performance of the present invention and the comparative scheme under varying signal-to-noise ratio.

[0028] Figure 5This is a simulation comparison of the minimum channel interference-to-noise ratio performance of the present invention and the comparative scheme under the imperfect channel state information scaling factor.

[0029] Figure 6 This is a simulation diagram comparing the minimum channel interference-to-noise ratio performance of the present invention and the comparative scheme under varying antenna numbers. Detailed Implementation

[0030] The present invention will be further described below through specific embodiments.

[0031] like Figure 1 As shown, the robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication includes the following steps:

[0032] Step S1: Based on a communication system with multiple low-Earth orbit satellites and ground users, construct a joint optimization problem for joint beamforming and user scheduling. The optimization objective of this joint optimization problem is to maximize the minimum signal-to-interference-plus-noise ratio (SINR) for all users. This joint optimization problem is related to the scheduling parameters between satellites and ground users and the hybrid precoding matrix.

[0033] Specifically, the communication system includes C low-Earth orbit satellites, numbered as follows: Serving K single-antenna ground users, user number is Each satellite is equipped with an N-type satellite facing Earth. t A uniform planar antenna array, consisting of N antenna elements along the x-axis and y-axis, respectively. x and N y It consists of several components. Low-Earth orbit satellites are interconnected via inter-satellite links and connected to the ground via relay communication links. Therefore, each satellite can serve multiple ground users simultaneously, and each ground user can receive signals from multiple satellites.

[0034] Due to hardware limitations, the number of radio frequency links for each satellite is limited, denoted as N. RF Therefore, each satellite can have a maximum of N simultaneous operations. RF Each ground user processes and transmits signals. The scheduling relationship between the satellite and the ground user is represented by parameter a. c,k This means that when a c,k When a = 1, it means that the c-th satellite provides service to the k-th user; otherwise, a = 1. c,k =0 indicates no scheduling. The channel considered between the satellite and the ground user is a line-of-sight channel due to the lack of a scatterer. The downlink channel vector from the c-th satellite to the k-th ground user is denoted as... Where, ρ c,k For small-scale complex gain, it follows a standard complex Gaussian distribution; θ c,k and These are the azimuth and elevation angles of the satellite pointing towards the user; large-scale fading coefficient. Includes satellite antenna gain G S,c,k User receiving antenna gain G GU,k Path loss PL c,k Rain attenuation RA c,k The satellite antenna gain is approximately... in, J1(·) and J3(·) are first-order and third-order Bessel functions of the first kind, respectively, ξ c,k ξ is the satellite beam principal axis deviation angle. 3dB For a satellite transmitting antenna with a 3dB beamwidth, G max This represents the maximum antenna gain. The path loss is the free-space propagation path loss, expressed as: Where, d c,k Let f be the distance from satellite c to user k, f be the carrier frequency, and c be the speed of light. The rain attenuation power gain (in dB) follows a log-normal distribution, i.e. Array steering vector is Due to estimation errors, the actual channel model is expressed as follows: in For the channel estimation between the c-th satellite and the k-th ground user, Δh c,k The additive error of the channel state information is represented by Δh. c,k It follows a statistical error model, namely a complex Gaussian distribution, and satisfies... Where ξ c,k The variance represents the error of the channel state information. It is N t 3D identity matrix.

[0035] Based on the above model, the joint optimization problem is expressed as follows: Where, γ k This represents the signal-to-interference-plus-noise ratio (SIR) for user k. Indicates the strength of the useful signal. Indicates the intensity of interference between users. Indicates noise intensity, w c,k =F RF, c f BB,c,k F represents the hybrid precoding matrix. RF,c f represents the analog precoding matrix of low-Earth orbit satellite c. BB,c,k Let represent the digital precoding matrix of the low-Earth orbit satellite c, t represent the minimum signal-to-interference-plus-noise ratio (SINNR) for any user, Pr{.} denotes the probability calculation, τ represents the minimum allowable confidence probability, and a c,k P represents the scheduling parameters between low-Earth orbit satellite c and ground user k. T This indicates the maximum permissible launch power for low-Earth orbit satellites. This represents a set of low-Earth orbit (LEO) satellites, where C represents the number of LEO satellites. Let K represent the set of ground users, and K represent the number of ground users. c =N RF This indicates the maximum number of users a low-Earth orbit satellite can serve, [F RF,c ] n,m F represents RF,c The m-th element in the n-th row.

[0036] Step S2: Use the bilinear penalty method to design the scheduling parameters between the satellite and the ground user to maximize the minimum channel gain of all users. Determine whether the number of satellite antennas is greater than a predetermined threshold. If it is less than the predetermined threshold, proceed to steps S3 and S4; otherwise, proceed to step S5. The predetermined threshold is set using the application of satellite computing power and is related to the payload capacity.

[0037] Specifically, construct a system for scheduling parameter a c,k First sub-optimization problem To overcome the nonconvexity of the minimization function, the sub-optimization problem is transformed into a second sub-optimization problem using a smooth approximation of the minimization function. The smooth approximation of the minimum function used is expressed as follows: h c,k Let be the downlink channel vector from low-Earth orbit satellite c to ground user k, and μ be the control parameter for a smooth approximation, wherein the approximation error is less than 1.

[0038] Define the bilinear penalty set Introduce auxiliary matrices A and B to satisfy the condition in the bilinear penalty set. Solving the binary constraint problem involves transforming the second sub-optimization problem into a third sub-optimization problem. By combining the bilinear penalty method and introducing the bilinear constraint into the objective function of the third sub-optimization problem through a penalty term, the standard convex problem form of the first sub-optimization problem is obtained. Where Tr{.} represents the trace function, 1 C×K This represents a C×K matrix of all ones, vec(·) represents the vectorization operation, and ||·|| ∞ Represents the infinite norm, Let Frobenius norm be denoted, and ρ be the penalty factor, which is gradually increased during algorithm iterations to satisfy the constraints.

[0039] The block coordinate descent method is used to alternately optimize A and B to determine the scheduling parameters. Specifically, when B is fixed, the optimization problem of A is: With A fixed, the optimization problem for B is: The penalty factor ρ gradually increases according to the rule ρ = min(ξρ, 2L), where L is the Lipschitz constant of the objective function and ξ represents the multiplication coefficient.

[0040] Then determine whether the number of satellite antennas is greater than a predetermined threshold. If it is less than the threshold, proceed to step S3; otherwise, proceed to step S5.

[0041] Step S3: Construct the minimum signal-to-interference-plus-noise ratio maximization problem under the communication interruption probability constraint of statistical channel state information error. Use Bernstein-type inequalities to relax the problem, introduce relaxation variables to transform the problem into a deterministic constraint, and solve the transformed problem to obtain the optimal all-digital precoding matrix.

[0042] Specifically, considering the statistical channel state information error model, a robust minimum signal-to-interference-plus-noise ratio (SINR) maximization framework is adopted. This framework addresses the minimum SINR maximization problem constrained by the communication interruption probability under statistical channel state information error. Under the premise of ensuring a certain channel communication interruption probability, the goal is to maximize the minimum SINR for all users. In other words, the minimum SINR maximization problem is expressed as: Introducing the auxiliary variable χ s The minimum signal-to-interference-plus-noise ratio maximization problem then becomes Introducing auxiliary parameters probability constraint formula Transform into And introduce auxiliary parameters The channel estimation error is re-expressed as Thus, the probability constraint formula is reformulated as Among them, t (i) Let represent the value of t in the i-th iteration, Pr{.} denotes the probability calculation, τ denotes the minimum allowable confidence probability, and A S,k A represents the useful signal strength. I,k Indicates the intensity of interference between users. Indicates noise intensity. w c,k =F RF,c f BB,c,k , F BB,c,-k =[a c,1 f BB,c,1 ,…a c,k-1 f BB,c,k-1 ,a c,k+1 f BB,c,k+1 …a c,K f BB,c,K "-k" indicates taking any value other than k, h k ξ represents the channel vector. k =[ξ k,1 ,…ξ k,C ]T Let ξ represent the channel error vector. k,C N represents the variance of the channel state information error between ground user k and low-Earth orbit satellite C. t This indicates the number of antennas in a uniform planar array facing Earth that a low-Earth orbit satellite is equipped with. N represents t 3D identity matrix Indicates the real part, (.) H This indicates the conjugate transpose, (.). T This indicates transpose.

[0043] We apply Bernstein-type inequalities to relax the problem transformed from the minimum signal-to-interference-plus-noise ratio (SINR) maximization problem, resulting in the relaxed expression. Introducing an auxiliary matrix and Then the constraints in the relaxation expression are re-expressed as Using continuous convex approximation techniques for non-convex constraints Perform a convex approximation as The relaxation expression is then transformed into Solving this transformed problem yields the optimal all-digital precoding matrix. Where ρ2 represents the penalty coefficient, x k and y k It is an auxiliary variable. express The value of the i-th iteration.

[0044] Step S4: Based on the all-digital precoding matrix obtained in step S3, construct the design problem of the mixed digital-analog precoding matrix. Solve the design problem of the mixed digital-analog precoding matrix using an alternating optimization method to obtain the mixed digital-analog precoding matrix.

[0045] The problem of designing a mixed-signal precoding matrix is ​​expressed as follows: The design problem of this hybrid precoding matrix is ​​solved by using an alternating optimization approach.

[0046] First, the problem of designing a mixed-signal precoding matrix is ​​transformed into... To update the analog precoding matrix, a relaxation constraint method is used to apply the constant modulus constraint |[F RF,c ] n,m |=1, It is added to the objective function as a penalty term, resulting in Using continuous approximation algorithms to handle non-convex penalty terms to obtain equivalent problems The simulated precoding matrix is ​​obtained by solving this equivalent problem using the standard convex optimization algorithm.

[0047] Then, with the analog precoding matrix fixed, the problem of optimizing the digital precoding matrix is ​​addressed. Solving this problem yields the optimal digital precoding matrix, which is the hybrid precoding matrix for this iteration. In the i-th iteration, we obtain the matrix with respect to χ. s The problem Solving this problem yields χ s and according to Update parameter t, where, This represents the value of the i-th iteration.

[0048] Step S5: Based on maximizing the signal leakage and noise ratio, construct and solve the analog precoding matrix design problem to obtain the optimal analog precoding matrix, and then construct and solve the digital precoding matrix design problem to obtain the optimal digital precoding matrix;

[0049] The number of ground users that a low-Earth orbit satellite can serve is equal to the number of its radio frequency links, N. RF The N of low-orbit satellite c RF Each radio frequency chain is assigned to its scheduled N RF A ground user, represented as in Let c be the set of users served by the c-th satellite. The problem of simulating the design of the precoding matrix is ​​expressed as follows: The simulation precoding matrix design problem can be transformed into a first subproblem using fractional programming. definition b c,k =α c,k h c,k Rewrite the first subproblem as the second subproblem Using an element-wise optimization method, for f RF,c,k With the other elements fixed, the second subproblem simplifies to a third subproblem for the nth element. The closed-form solution to the third subproblem is: Among them, f RF,c,k Let α represent the simulated beamforming vector of ground user k. c,k As an auxiliary variable in the fractional programming method, the optimal α c,k Represented as (f RF,c,k ) m f RF,c,k The m-th unit, (.) * This indicates the conjugate operation.

[0050] Define the effective channel as in, Let c be the prewhitening matrix, satisfying The effective noise is Its distribution is The digital precoding matrix design problem is then expressed as: Transform the probability constraint into Among them are definition Its obedience

[0051]

[0052] make Transform the probability constraint into in,

[0053] Then, using the Bernstein-Type inequality, the probability constraint is reformulated as follows: The problem of designing a digital precoding matrix is ​​then transformed into Solving this transformed problem yields the optimal digital precoding matrix.

[0054] In this embodiment, the algorithm proposed in this invention is implemented using Matlab numerical calculation software, and its performance is revealed.

[0055] from Figure 3 It can be seen that the bilinear penalty-based allocation method proposed in this invention consistently outperforms the comparative scheme (greedy method). This is because the greedy method, with its heuristic nature, may fall into suboptimal solutions, while this invention approximates the optimal solution by relaxing the binary constraints. Furthermore, it can be observed that the minimum channel gain increases with the number of transmit antennas. This is because more antennas generate a higher-dimensional channel matrix, which typically leads to enhanced channel gain.

[0056] Figure 4 The minimum channel interference-to-noise ratio (MIN) performance of different schemes under varying signal-to-noise ratios was compared, taking into account errors in ideal channel state information and statistical channel state information. Figure 4 It can be seen that the robust algorithm of this invention when the satellite antenna is below the set threshold is naturally superior to the non-robust algorithm because it takes into account the uncertainty of channel state information. Furthermore, the low-complexity scheme adopted by this invention when the satellite antenna is above the set threshold achieves overall rate performance close to the alternating optimization scheme, while significantly reducing computational complexity.

[0057] Figure 5The impact of the imperfection ratio of channel state information on the minimum channel interference-to-noise ratio (MNR) of different schemes is demonstrated. It can be clearly observed that as the imperfection ratio increases, the MNR performance of each scheme considering channel state information errors monotonically decreases. Furthermore, the performance advantage of the robust scheme proposed in this invention compared to the non-robust scheme is particularly significant when the imperfection ratio is large.

[0058] Figure 6 This paper demonstrates the performance of the proposed low-complexity algorithm for large-scale arrays as the number of transmit antennas increases. It is evident that the minimum channel interference-to-noise ratio (MIC) of each scheme monotonically increases with the number of antennas, proving the effectiveness of the proposed low-complexity algorithm. Furthermore, with the increase in the number of transmit antennas, the analog precoder design based on MIC is more effective in suppressing inter-user interference, while the digital precoder enhances robustness to channel uncertainties.

[0059] In this invention, the terms "first," "second," and "third," etc., are used only to distinguish similar objects and are not necessarily used to describe a specific order or sequence, nor should they be construed as indicating or implying relative importance. The use of terms such as "upper," "lower," "left," "right," "front," and "rear" to indicate orientation or positional relationships is based on the orientation or positional relationships shown in the accompanying drawings and is only for the convenience of describing the invention, not to indicate or imply that the device referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, it should not be construed as a limitation on the scope of protection of this invention. Those skilled in the art can understand the specific meaning of the above terms in this application according to the specific circumstances.

[0060] Furthermore, in the description of this application, unless otherwise stated, "multiple" means two or more. "And / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone. The character " / " generally indicates that the preceding and following related objects have an "or" relationship.

[0061] The above are merely specific embodiments of the present invention, but the design concept of the present invention is not limited thereto. Any non-substantial modifications made to the present invention using this concept shall be considered as infringing upon the protection scope of the present invention.

Claims

1. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication, characterized in that: Includes the following steps: Step S1: Based on a communication system with multiple low-Earth orbit satellites and ground users, construct a joint optimization problem for joint beamforming and user scheduling. The optimization objective of this joint optimization problem is to maximize the minimum signal-to-interference-plus-noise ratio (SINR) for all users. This joint optimization problem is related to the scheduling parameters between satellites and ground users and the hybrid precoding matrix. Step S2: Use the bilinear penalty method to design the scheduling parameters between the satellite and the ground user to maximize the minimum channel gain of all users. Determine whether the number of satellite antennas is greater than a predetermined threshold. If it is less than the threshold, proceed to step S3; otherwise, proceed to step S5. Step S3: Construct the minimum signal-to-interference-plus-noise ratio maximization problem under the communication interruption probability constraint of statistical channel state information error. Use Bernstein-type inequalities to relax the problem, introduce relaxation variables to transform the problem into a deterministic constraint, and solve the transformed problem to obtain the optimal all-digital precoding matrix. Step S4: Based on the all-digital precoding matrix obtained in step S3, construct the design problem of the mixed digital-analog precoding matrix. Solve the design problem of the mixed digital-analog precoding matrix using an alternating optimization method to obtain the mixed digital-analog precoding matrix. Step S5: Based on maximizing signal leakage and noise ratio, construct and solve the analog precoding matrix design problem to obtain the optimal analog precoding matrix. Then, construct and solve the digital precoding matrix design problem to obtain the optimal digital precoding matrix.

2. The robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 1, characterized in that: In step S1, the joint optimization problem is expressed as: Where, γ k w represents the signal-to-interference-plus-noise ratio (SIR) for user k. c,k =F RF,c f BB,c,k F represents the hybrid precoding matrix. RF,c f represents the analog precoding matrix of low-Earth orbit satellite c. BB,c,k Let represent the digital precoding matrix of low-Earth orbit satellite c, t represent the minimum signal-to-interference-plus-noise ratio for any user, Pr{.} represent the probability, τ represent the minimum allowable confidence probability, and a c,k P represents the scheduling parameters between low-Earth orbit satellite c and ground user k. T This indicates the maximum permissible launch power for low-Earth orbit satellites. This represents a set of low-Earth orbit (LEO) satellites, where C represents the number of LEO satellites. Let K represent the set of ground users, and K represent the number of ground users. c =N RF This indicates the maximum number of users a low-Earth orbit satellite can serve, [F RF,c ] n,m F represents RF,c The m-th element in the n-th row.

3. The robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 2, characterized in that: In step S2, a system is constructed for scheduling parameter a. c,k First sub-optimization problem To overcome the nonconvexity of the minimization function, the sub-optimization problem is transformed into a second sub-optimization problem using a smooth approximation of the minimization function. Introducing auxiliary matrices A and B to solve the binary constraint problem transforms the second sub-optimization problem into a third sub-optimization problem. By combining the bilinear penalty method and introducing the bilinear constraint into the objective function of the third sub-optimization problem through a penalty term, the standard convex problem form of the first sub-optimization problem is obtained. The block coordinate descent method is used to alternately optimize A and B to determine the scheduling parameters. Then, it is determined whether the number of satellite antennas is greater than a predetermined threshold. If it is less, steps S3 and S4 are executed; otherwise, step S5 is executed. Where h... c,k Let μ be the downlink channel vector from low-Earth orbit satellite c to ground user k, and μ be the control parameter for smoothing approximation. Tr{.} represents the trace function, 1 C×K This represents a C×K matrix of all 1s, where vec(·) denotes the vectorization operation, ||·|| ∞ Represents the infinite norm, Let Frobenius norm be denoted, and ρ be the penalty factor, which is gradually increased during iteration to satisfy the constraints.

4. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 3, characterized in that: In step S2, the block coordinate descent method is used to alternately optimize A and B. Specifically, when B is fixed, the optimization problem of A is: With A fixed, the optimization problem for B is: The penalty factor ρ gradually increases according to the rule ρ = min(ξρ, 2L), where L is the Lipschitz constant and ξ represents the multiplication factor.

5. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 4, characterized in that: The minimum signal-to-interference-plus-noise ratio (SINR) maximization problem in step S3 is expressed as follows: Introducing the auxiliary variable χ s The minimum signal-to-interference-plus-noise ratio maximization problem then becomes Introducing auxiliary parameters probability constraint formula Transform into And introduce auxiliary parameters The channel estimation error is expressed as... Thus, the probability constraint formula is reformulated as Among them, t (i) Let represent the value of t in the i-th iteration, Pr{.} denotes the probability calculation, τ denotes the minimum allowable confidence probability, and A S,k A represents the useful signal strength. I,k Indicates the intensity of interference between users. Indicates noise intensity. F BB,c,-k =[a c,1 f BB,c,1 ,…a c,k-1 f BB,c,k-1 ,a c,k+1 f BB,c,k+1 …a c,K f BB,c,K ], h k ξ represents the channel vector. k =[ξ k,1 ,…ξ k,C ] T Let ξ represent the channel error vector. k,C N represents the variance of the channel state information error between ground user k and low-Earth orbit satellite C. t This indicates the number of antennas in a uniform planar array facing Earth that a low-Earth orbit satellite is equipped with. N represents t 3D identity matrix Indicates the real part.

6. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 5, characterized in that: In step S3, the problem transformed from the minimum signal-to-interference-plus-noise ratio (SINR) maximization problem is relaxed using Bernstein-type inequalities to obtain the relaxation expression. Introducing an auxiliary matrix and Then the constraints in the relaxation expression are re-expressed as Using continuous convex approximation techniques for non-convex constraints Perform a convex approximation as The relaxation expression is then transformed into Solving this transformed problem yields the optimal all-digital precoding matrix. Where ρ2 represents the penalty coefficient, x k and y k It is an auxiliary variable. express The value of the i-th iteration.

7. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 6, characterized in that: In step S4, the mixed-signal precoding matrix design problem is expressed as: First, the mixed-signal precoding matrix is ​​transformed into... To update the simulated precoding matrix, a relaxation constraint method is used, incorporating the constant modulus constraint as a penalty term into the objective function, resulting in... Using continuous approximation algorithms to handle non-convex penalty terms to obtain equivalent problems The analog precoding matrix is ​​obtained by solving this equivalent problem using the standard convex optimization algorithm. Then, the digital precoding matrix optimization problem is addressed by fixing the analog precoding matrix. Solving this problem yields the optimal digital precoding matrix, which is the hybrid precoding matrix for this iteration. In the i-th iteration, we obtain the result with respect to χ. s The problem Solving this problem yields χ s and according to Update parameter t, where, This represents the value of the i-th iteration.

8. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 7, characterized in that: In step S5, the number of ground users that the low-orbit satellite can serve is equal to the number of its radio frequency links N. RF The N of low-orbit satellite c RF Each radio frequency chain is assigned to its scheduled N RF A ground user, represented as in Let c be the set of users served by the c-th satellite, and the analog precoding matrix design problem is expressed as: The problem of designing the simulation precoding matrix can be transformed into a first subproblem using fractional programming. definition b c,k =α c,k h c,k Rewrite the first subproblem as the second subproblem Using an element-wise optimization method, for f RF,c,k With the other elements fixed, the second subproblem simplifies to a third subproblem for the nth element. The closed-form solution to the third subproblem is: Among them, f RF,c,k Let α represent the simulated beamforming vector of ground user k. c,k As an auxiliary variable in the fractional programming method, the optimal α c,k Represented as (f RF,c,k ) m f RF,c,k The m-th unit, (.) * This indicates the conjugate operation.

9. A robust hybrid beamforming method for multi-low-Earth orbit satellite cooperative communication according to claim 8, characterized in that: In step S5, the digital precoding matrix design problem is expressed as: Transform the probability constraint into Then, using the Bernstein-Type inequality, the probability constraint is reformulated as follows: The problem of designing a digital precoding matrix is ​​then transformed into Solving this transformed problem yields the optimal digital precoding matrix, where,

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