MA joint beam forming optimization method for low-orbit satellite communication ground station

Abstract

The invention provides an MA joint beam forming optimization method for a low-orbit satellite communication ground station, belongs to the technical field of satellite communication physical layers, and aims to solve the problems that in MA architecture related algorithms, antenna space layout and beam forming strategies cannot be effectively coordinated, but are limited by power consumption, size and mechanical structure constraints of a satellite-borne platform, the implementation difficulty is large, and the cost is high. The method comprises the following steps: establishing a three-dimensional geocentric spherical coordinate system and a three-dimensional geocentric Cartesian coordinate system; establishing a ground station center Cartesian coordinate system; acquiring an average reachable rate, a signal to interference plus noise ratio of a ground station and interference signal gains of other satellites; constructing an optimization problem P1 by taking the antenna position, the antenna spacing and the signal to interference plus noise ratio of the ground station and the interference signal gains of other satellites as constraint conditions; introducing an auxiliary variable, and converting the optimization problem P1 into a quadratic constraint problem p2; and decomposing the secondary constraint problem p2 into a sub-problem P3 to optimize AWV and a sub-problem P4 to optimize APV, and carrying out iterative solution.

Description

Technical Field

[0001] The present invention relates to an MA joint beamforming optimization method for a low-orbit satellite communication ground station, and belongs to the technical field of satellite communication physical layer. Background Art

[0002] In recent years, low-Earth Orbit (LEO) satellite communication systems, owing to their wide coverage, low latency, and high capacity, have become a key technology for achieving global communications coverage. This has driven the deep integration of sixth-generation mobile communications (6G) and non-terrestrial networks (NTN). In the context of ultra-dense constellation deployments, LEO satellite networks have been widely used in scenarios such as navigation and positioning, environmental monitoring, remote sensing, emergency communications, and large-scale Internet of Things (IoT) access. To achieve highly reliable and wide-coverage data transmission, ground receiving stations typically employ multi-antenna arrays and beamforming with antenna weight vectors (AWVs) to enhance target satellite signals and mitigate interference. However, existing solutions are mostly based on fixed-position antenna arrays (FPAs). Their array structure cannot be adjusted after initialization, resulting in limited spatial freedom and poor adaptability to highly dynamic interference environments. This is particularly true in ultra-dense constellations where multiple satellites simultaneously cover ground stations, which can lead to system performance degradation.

[0003] To enhance array spatial flexibility and interference resilience, the MA architecture has been proposed in recent years. This architecture allows each antenna element to move locally within a two-dimensional region. By optimizing the antenna position vector (APV) and AWV, the array beam is designed to enhance the serving satellite signal and suppress interfering satellite signals. Previous studies have demonstrated that MA technology offers significant performance advantages in scenarios such as terrestrial wireless communications, physical layer security, and aerial platform communications. However, research on the application of MA in LEO satellite communication systems is still in its infancy, and a complete and implementable system model and optimization strategy have yet to be established. In particular, considering the highly dynamic topology of LEO constellations and the limited hardware requirements of ground station receivers, effectively coordinating antenna spatial layout with beamforming strategies remains a key unresolved issue. Furthermore, some studies have attempted to deploy MA arrays on satellites, but these approaches are difficult to implement and have limited practicality due to the power consumption, volume, and mechanical constraints of the onboard platforms. Therefore, a ground-station-specific, operationally feasible joint optimization method is urgently needed to fully tap the potential of MA in LEO communication networks. Summary of the Invention

[0004] The present invention aims to solve the problems that the MA architecture-related algorithms cannot effectively coordinate the antenna spatial layout and beamforming strategy, and are limited by the power consumption, volume and mechanical structure constraints of the onboard platform, making their implementation difficult and practical. Therefore, an MA joint beamforming optimization method for low-orbit satellite communication ground stations is proposed.

[0005] The technical solution adopted by the present invention to solve the above problems is: the present invention comprises the following steps: Step 1: Establish a three-dimensional geocentric spherical coordinate system with the center of the earth as the center, convert the three-dimensional geocentric spherical coordinates into Cartesian coordinates, establish a three-dimensional geocentric Cartesian coordinate system, and obtain the ground station position based on the three-dimensional geocentric Cartesian coordinate system; Step 2: With the ground station as the center, establish the Cartesian coordinate system of the ground station center and place the satellite S jk As a service satellite, obtain the received signal of the ground station at time t, the signal-to-interference-and-noise ratio of the ground station at time t, the interference channel gain of other satellites, and t Achievable rate per unit bandwidth of time ; Step 3: Based on the signal-to-interference-and-noise ratio of the ground station at time t, the interference channel gain of other satellites and t Achievable rate per unit bandwidth of time Get the time interval Average achievable rate, signal-to-interference-noise ratio of ground stations and the interference signal gain of other satellites ; Step 4: Set the antenna position, antenna spacing, and signal-to-interference-noise ratio of the ground station and the interference signal gain of other satellites As constraints, construct the optimization problem P1; Step 5: Introduce auxiliary variables and transform the optimization problem P1 into a quadratic constraint problem P2 through the Lagrangian function; Step 6: Decompose the quadratic constraint problem p2 into subproblem P3 to optimize AWV and subproblem P4 to optimize APV. Iterate subproblems 1 and 2 to obtain the solution of optimization problem P1. Optimize antenna weight vector AWV and antenna position vector APV through subproblems P3 and P4 respectively. The maximum average achievable rate of the ground station movable antenna array is , completing the MA joint beamforming optimization of the ground station.

[0006] Furthermore, in the three-dimensional geocentric spherical coordinate system in step 1, the LEO constellation is represented by J orbital surfaces, each orbit contains K Satellites, S jk For the jOn the track k satellites, and , is the orbital inclination, that is, the angle between the orbital plane and the equatorial plane. Assuming the earth is a perfect sphere, the earth radius is R , the satellite altitude is H , the Earth's rotation period is T E , the orbital period is , G e and M e are the gravitational constant and the mass of the Earth respectively; is the pitch angle, which represents the angle between the reference plane and the direction of the given point, where a positive value represents the direction from south to north; is the bearing angle, which represents the angle between the reference azimuth direction and the projection of the point on the reference plane, where positive values ​​are measured from west to east; for t The ascending node and S jk The geocentric angle between for The initial angle at Convert the three-dimensional geocentric spherical coordinates into Cartesian coordinates, establish a three-dimensional geocentric Cartesian coordinate system, and obtain the satellite S jk Coordinates in a three-dimensional geocentric Cartesian coordinate system and ground station locations; Satellite S jk The position in the three-dimensional geocentric spherical coordinate system is: (1); In formula (1), is the distance from the satellite to the center of the Earth, The reference plane to S jk The elevation angle measured from the direction, is the reference azimuth direction to S jk Project the measured azimuth on the reference plane; Satellite S jk The coordinates in the three-dimensional geocentric Cartesian coordinate system are: (2); In formula (2), , which represents the time required for the next orbital plane to reach the same relative position of the ground station. The observation interval is ; The expression for the ground station position is: (3); In formula (3), is the ground station pitch angle, is the ground station azimuth.

[0007] Furthermore, in step 2, the coordinate transformation matrix Perform coordinate transformation on the three-dimensional geocentric Cartesian coordinate system to establish the ground station center Cartesian coordinate system. In the established ground station center Cartesian coordinate system, the ground station to satellite S jk The wave vector is ,in , is the carrier wavelength, y The axis is the tangent to the latitude line of the ground station, from west to east. z The axis points radially outward from the center of the Earth, and x Axis and y and z The planes formed by the axes are orthogonal, and the wave vector in the Cartesian coordinate system at the center of the ground station is ,in, is the coordinate transformation matrix; Coordinate transformation matrix The expression is: (4).

[0008] Furthermore, in step 2, obtaining the received signal of the ground station at time t, the signal-to-interference-and-noise ratio of the ground station at time t, and the interference channel gain of other satellites specifically includes: Step 2.1: Set each element of the ground station MA to the Cartesian coordinate system at the center of the ground station. x - y Two-dimensional region of the plane Moving within, based on the Cartesian coordinate system of the ground station center n The position of the element Get with Satellite S jk Associated MA array steering vector , where MA is the movable antenna array, and the first n The position of an element is represented by APV, and its expression is , , satellite S jk The path loss between the ground station and ; Step 2.2: At time t, select satellite S jk As a service satellite, the signals of other satellites are treated as co-channel interference, and the AWV at the ground station is combined to obtain the time of the ground station. t The received signal , where AWV is the antenna weight vector, and its expression is ; Step 2.3: Based on satellite S jkPath loss between the ground station and satellite S jk Associated MA array steering vector Get channel vector ; Step 2.4: Based on channel vector and the antenna weight vector at the ground station Get the moment t Signal-to-interference-and-noise ratio (SIR) of ground stations , interference channel gain of other satellites and in t Achievable rate per unit bandwidth of time ; With satellite S jk Associated MA array steering vector The calculation formula is: (5); In formula (5), Indicates all c n A collection of Satellite S jk Path loss between the ground station The calculation formula is: (6); In formula (6), and The reference distances are m Path loss exponent and path gain at meters, m is a constant; Ground station at the moment t The received signal The calculation formula is: (7); In formula (7), P s is the satellite transmission power, Indicates t The number of visible satellites in the jth orbital plane at time, where K j ( t ) is the number of satellites that the ground station can receive signals from. If the square of the Euclidean distance between the ground station and the satellite satisfies , then it is considered that the satellite can be received by the ground station, and the vector For AWV at the ground station, is independent and identically distributed additive Gaussian white noise, and ; Channel Vector The calculation formula is: (8); In formula (8), To set each satellite beam to point directly downward, the satellite S jk The directional antenna gain, x jk ( t ) is the satellite S jk The data transmitted, is the wave vector The angle between the direction opposite to the satellite position vector Satellite S jk Directional antenna gain The calculation formula is: (9); In formula (9), is the first-order Bessel function of the first kind, r is the antenna circular aperture radius; time t Signal-to-interference-and-noise ratio (SIR) of ground stations The calculation formula is: (10); In formula (10), is the interference channel gain of other satellites, and its calculation formula is: (11); t Achievable rate per unit bandwidth of time The calculation formula is: (12).

[0009] Furthermore, step 3 specifically includes: Step 3.1: Based on t Achievable rate per unit bandwidth of time Calculate in time interval The average achievable rate ; Step 3.2: Set the time interval Discrete M equal-length time slots ,in , No. m The time slot is ,in, t m As the midpoint, obtain the antenna weight vector of the discrete ground station ; Step 3.3: The discretized ground station antenna weight vector Substitute into formula (10) to obtain the signal-to-interference-noise ratio of the ground station and the interference signal gain of other satellites ; In time interval The average achievable rate The expression is: (13); Signal-to-interference-and-noise ratio (SIR) of ground stations The expression is: (14); In formula (14), , and , is the interference signal gain of other satellites, and its calculation formula is: (15).

[0010] Furthermore, the optimization problem P1 constructed in step 4 specifically includes: Position the ground station antenna As constraint C1, the antenna spacing As constraint C2, the antenna weight vector of the discretized ground station is in As constraint C3, the optimization problem P1 is constructed based on constraints C1, C2 and C3; The expression of the optimization problem P1 is: (16).

[0011] 7. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, wherein step 5 specifically comprises: Step 5.1: Use Lagrange dual transformation to introduce time slots m Corresponding auxiliary variables Solve the weighted sum of multiple logarithmic functions in the optimization problem P1; Step 5.2: Based on step 5.1, use secondary transformation to introduce auxiliary variables Convert the optimization problem P1 into the optimization problem P2; Introducing time slots m Corresponding auxiliary variables The optimization problem P1 is: (17); Introducing auxiliary variables The subsequent optimization problem P2 is: (18); (19); (20); The expression of the optimization problem P2 is: (twenty one); (twenty two); Formula (21)-Formula (22), , , 、 、 and All are parameters.

[0012] Furthermore, the steps of iteratively solving subproblems 1 and 2 in step 6 include: Step 6.1: Set the initial number of iterations , initialization parameters and ; Step 6.2: Set the number of iterations i = i +1, solution and , get the optimal auxiliary variable and ; Step 6.3: Fix the parameters in the optimization problem P2 , ,and , transform the optimization problem P2 into the optimization problem P3, solve the problem P3, and update the parameters ; Step 6.4: Fix all variables except parameter c in optimization problem P2, transform optimization problem P2 into optimization problem P4, solve optimization problem P4, and update parameters ; Step 6.5: Repeat steps 6.2 to 6.4 to iterate the calculation

[0013] Step 6.6: Determine if there is If yes, proceed to step 6.7, if no, return to step 6.2; Step 6.7: Output Parameters and , the maximum average achievable rate of the ground station movable antenna array is obtained ; Optimal auxiliary variables and The calculation formula is: (twenty three); The expression of the optimization problem P3 is: (twenty four); In formula (24), and are all equivalent variables, and their calculation formula is: (25); (26); parameter The calculation formula is: (27); The expression of the optimization problem P4 is: (28); (29); (30).

[0014] The beneficial effects of the present invention are: This paper constructs a LEO ground receiving system model based on a MA architecture. This system improves the service satellite connection quality and interference suppression capabilities through array reconfiguration. Simulation results demonstrate that the proposed method significantly outperforms traditional fixed array solutions under varying constellation densities, transmit powers, and antenna numbers. In particular, in high-density constellations, the proposed method demonstrates excellent communication reliability and interference robustness, making it suitable for ground receiving designs in future LEO satellite networks.

[0015] 2. This paper proposes an optimization framework based on Lagrangian duality and quadratic transformation, and uses the block coordinate descent (BCD) method to iteratively solve the APV and AWV. This effectively improves the average achievable rate of the system and reduces the computational complexity, verifying its practicality and robustness in ultra-dense LEO constellations. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 A flow chart of the MA joint beamforming optimization method for a low-orbit satellite communication ground station provided by the present invention; Figure 2 A flowchart for solving the optimization problem P1 provided by the present invention; Figure 3 Schematic diagram of the MA-assisted LEO satellite communication network established by the present invention; Figure 4 A schematic diagram of the geocentric coordinate system and related angles established by the present invention; Figure 5A schematic diagram showing the relationship between the number of iterations and the average achievable rate provided by the present invention; Figure 6 A schematic diagram showing the relationship between the number of iterations and the average power gain of the signal provided by the present invention; Figure 7 A schematic diagram of the change in achievable rate of the MA scheme and two fixed array schemes (SFPA and DFPA) under different time slots provided by the present invention; Figure 8 A schematic diagram of the antenna position distribution of the ground station provided by the present invention under different parameter settings; Figure 9 A schematic diagram showing the relationship between the average achievable speed rate and the latitude of the ground station under different ground station elevation conditions provided by the present invention; Figure 10 A schematic diagram showing the relationship between the number of ground station antennas and the average achievable rate provided by the present invention; Figure 11 This is a schematic diagram of the relationship between satellite transmission power and average achievable rate provided by the present invention. DETAILED DESCRIPTION

[0017] Combine Figure 1-4 This embodiment is described as follows. Figure 1 As shown, the steps of the MA joint beamforming optimization method for a low-orbit satellite communication ground station described in this embodiment include: S1: Establish a three-dimensional geocentric spherical coordinate system and a three-dimensional geocentric Cartesian coordinate system; The MA-assisted LEO satellite communication network established in this embodiment is as follows: Figure 3 As shown, in this system, the ground station is equipped with N Each antenna element can be moved in two dimensions and initialized before communication. Multiple satellites in the LEO constellation, originating from different orbital planes, simultaneously transmit signals to ground stations, generating strong interference. This implementation aims to maximize the average achievable rate by optimizing the APV during system initialization and dynamically adjusting the AWV during communication.

[0018] S101: In the three-dimensional geocentric spherical coordinate system, the three-dimensional geocentric spherical coordinate system and the related angles are as follows: Figure 4 As shown, the LEO constellation consists of J orbital surfaces, each orbit contains K Satellites. Let S jk Indicates the j On the track k satellites, and .make is the orbital inclination, that is, the angle between the orbital plane and the equatorial plane. Assuming that the earth is a perfect sphere with a radius ofR , the satellite altitude is H , the Earth's rotation period is T E , then the orbital period is , where G e and M e Denote the gravitational constant and the mass of the Earth respectively. A three-dimensional geocentric spherical coordinate system (GSCS) is established with the equatorial plane as the reference plane. It is the angle between the reference plane and the direction of the given point. A positive value indicates the direction from south to north. At the same time, the ascending node and the descending node are defined as the intersection of the satellite orbit and the equatorial plane, where the ascending node corresponds to the movement from south to north and the descending node corresponds to the movement from north to south. The direction of the ascending node of the orbit that intersects the equator at the Greenwich meridian is defined as the azimuth reference direction, that is, x Axis direction. Then, the azimuth It represents the angle between the reference azimuth direction and the projection of the point on the reference plane, with positive values ​​measured from west to east. Figure 2 Without loss of generality, consider the satellites in the orbit segment from the South Pole to the North Pole, assuming that the satellites in the opposite orbit segment operate in different frequency bands, and similar optimization strategies can be adopted. In the established GSCS, let (radians) indicates t The ascending node and S jk The geocentric angle between. express t =0, the initial angle can be obtained by Based on this, S jk In GSCS, position is represented as: (1); In formula (1), is the distance from the satellite to the center of the Earth, The reference plane to S jk The elevation angle measured from the direction, is the reference azimuth direction to S jk Project the measured azimuth on the reference plane; Therefore, S jk The coordinates in the three-dimensional Geocentric Cartesian Coordinate System (GCCS) are: (2); In formula (2), , which represents the time required for the next orbital plane to reach the same relative position of the ground station. The observation interval is ; S102: This embodiment considers the total number of tracks J and the Earth's rotation period T E , define the representative observation interval as ,in represents the time required for the next orbital plane to reach the same relative position of the ground station, and further assumes The interval period of satellites in orbit T / K Then, the ground station position can be expressed as: (3); In formula (3), is the ground station pitch angle, is the ground station azimuth.

[0019] S2: Establish the Cartesian coordinate system of the ground station center, obtain the received signal of the ground station at time t, the signal-to-interference-and-noise ratio of the ground station at time t, the interference channel gain of other satellites, and t The achievable rate per unit bandwidth of time; GCCS ground station to S jk The wave vector is ,in and Represents the carrier wavelength. Through the coordinate transformation matrix Perform coordinate transformation on the three-dimensional geocentric Cartesian coordinate system and establish a Cartesian coordinate system (Station-Centric Cartesian Coordinate System, SCCS) with the ground station as the center. In SCCS, y The axis is defined as the tangent along the latitude line of the ground station, from west to east; z The axis points radially outward from the center of the Earth; and x Axis and y and z The wave vector in SCCS is , where the coordinate transformation matrix is: (4).

[0020] Assume that each element of the ground station MA is x - y Two-dimensional region of the plane If the SCCS moves n The position of the element is available Indicates that , this position is called APV.jk The associated MA array steering vector can be expressed as: (5); In formula (5), Indicates all c n A collection of make and They represent the path loss index and path gain when the reference distance is 1 meter, respectively. Then S jk The path loss between the ground station and (6), assuming that at time t , the ground station selects S jk As a service satellite, the signals of other satellites are treated as co-channel interference. Therefore, the ground station t The received signal is: (7); In formula (7), P s is the satellite transmission power, Indicates t The number of visible satellites in the jth orbital plane at time, where K j ( t ) is the number of satellites that the ground station can receive signals from. If the square of the Euclidean distance between the ground station and the satellite satisfies , then it is considered that the satellite can be received by the ground station, and the vector For AWV at the ground station, is independent and identically distributed additive Gaussian white noise, and ; Based on this, the channel vector can be expressed as: (8); In formula (8), To set each satellite beam to point directly downward, the satellite S jk The directional antenna gain, x jk ( t ) is the satellite S jk The data transmitted, is the wave vector The angle between the directional antenna and the direction opposite to the satellite position vector. According to the 3GPP technical report TR 38.811, the directional antenna gain can be modeled as: (9); In formula (9), is the first-order Bessel function of the first kind, r is the antenna circular aperture radius; Based on channel vector and the antenna weight vector at the ground station Get the moment t Signal-to-interference-and-noise ratio (SIR) of ground stations , interference channel gain of other satellites and in t Achievable rate per unit bandwidth of time ; time t The Signal-to-Interference-Plus-Noise Ratio (SINR) of the ground station is expressed as: (10); In formula (10), is the interference channel gain of other satellites, and its calculation formula is: (11); t Achievable rate per unit bandwidth of time The calculation formula is: (12).

[0021] S3: Obtain the average achievable rate, the signal-to-interference-and-noise ratio of the ground station, and the interference signal gain of other satellites; The goal of this embodiment is to maximize the time interval The average achievable rate is: (13); S301: Set the interval Discrete M equal-length time slots ,in Time slot M The number is chosen to be large enough so that the satellite angle and can be considered approximately constant within each time slot. Using the midpoint To express the m time slots. ,definition: (14); S302: Since adjusting the APV in each time slot will result in a large amount of mobile overhead and energy consumption, the ground station MA is only configured once during initialization. The average achievable rate is: (15); Among them, the SINR of the ground station is Expressed as: (16); In formula (16), , and , is the interference signal gain of other satellites, and its calculation formula is: (17).

[0022] S4: Construct the optimization problem P1 by taking the ground station antenna position, antenna spacing, signal-to-interference-and-noise ratio, and interference signal gain of other satellites as constraints. Position the ground station antenna As constraint C1, the antenna spacing As constraint C2, the antenna weight vector of the discretized ground station is in As constraint C3, the optimization problem P1 is constructed based on constraints C1, C2 and C3; The expression of the optimization problem P1 is: (18).

[0023] S5: Introduce auxiliary variables and transform the optimization problem P1 into a quadratic constraint problem p2 through Lagrangian function; In order to more conveniently solve the optimization problem P1, this embodiment first uses the Lagrange dual transformation to process the weighted sum of multiple logarithmic functions, which is specifically expressed as: (19); In formula (17), Time slot m Corresponding auxiliary variables. Using secondary transformation, another auxiliary variable is introduced : (20); In formula (18), (twenty one); In formula (19), (twenty two); Therefore, the original problem (P1) can be transformed into the optimization problem P2: (twenty three); (twenty four); Formula (21)-Formula (22), , , 、 、 and All are parameters.

[0024] S6: Decompose the quadratic constraint problem p2 into subproblem P3 to optimize AWV and subproblem P4 to optimize APV, and perform iterative solutions to complete the MA joint beamforming optimization of the ground station; The present invention uses the Block Coordinate Descent (BCD) method to iteratively solve the APV and AWV, effectively improving the average achievable rate of the system and reducing the computational complexity, verifying its practicality and robustness in ultra-dense LEO constellations.

[0025] In this embodiment, the steps for iteratively solving subproblems P3 and P4 are as follows: Figure 2 Shown, including: S601: Set the initial number of iterations , initialization parameters and ; S602: Solve subproblem 1, optimize AWV, and set the number of iterations i = i +1, solution and , get the optimal auxiliary variable and : (25); S603: Fix the parameters in the optimization problem P2 , ,and , transform the optimization problem P2 into the optimization problem P3, solve the problem P3, and update the parameters : (26); In formula (26), and are all equivalent variables, and their calculation formula is: (27); (28); Then the optimal solution is: (29); S604: Fix the variables except parameter c in the optimization problem P2, transform the optimization problem P2 into the optimization problem P4, solve the optimization problem P4, and update the parameters : (30); (31); (32).

[0026] S605: Repeat S602-S604, iterative calculation

[0027] S606: Determine whether there is If yes, proceed to step 6.7, if no, return to step 6.2; S607: Output parameters and , the maximum average achievable rate of the ground station movable antenna array is obtained ; In summary, the optimization problem P1 can be solved by alternately solving formula (25), subproblem (P3) and subproblem (P4), and then the maximum average achievable rate of the ground station movable antenna array is obtained. , completing the MA joint beamforming optimization of the ground station.

[0028] In summary, this paper constructs a LEO ground receiving system model based on the MA architecture, improving the service satellite connection quality and interference suppression capabilities through array reconfiguration. Simulation results show that the proposed method significantly outperforms traditional fixed array solutions under various constellation densities, transmit powers, and antenna numbers. In particular, in high-density constellations, the proposed method demonstrates excellent communication reliability and interference robustness, making it suitable for ground receiving designs in future LEO satellite networks.

[0029] Example Combine Figure 5-Figure 11 To illustrate this embodiment, in order to verify the performance of the MA optimization solution proposed in the present invention, this embodiment compares and analyzes the communication performance and simulation process of three antenna array configuration solutions under different system parameters: 1) MA scheme (MA): The scheme proposed in this paper dynamically adjusts the antenna array structure by jointly optimizing AW and APV, so that the ground station can align the direction of the serving satellite at different times and suppress interfering satellite signals.

[0030] 2) Sparse Fixed-Position Antenna (SFPA): Distribute antenna units evenly across the area with large spacing. Inside, the position is fixed and only AWV is optimized.

[0031] 3) Dense Fixed-Position Antenna (DFPA): The antennas are placed at a minimum distance ( ) densely arranged in the region Inside, the position is fixed and only AWV is optimized.

[0032] In order to jointly optimize the AWV and APV of the ground station to maximize the average achievable rate, the simulation parameters are as follows: the satellite altitude is H =550 km, the radius of the Earth is R = 6371 km, and the orbital inclination is = 65 degrees, the Earth's rotation period is T E =15 T , the total number of time slots is M =500, the carrier frequency is f c =14 GHz, the minimum antenna spacing is d min = , antenna moving range = , noise power =-120 dBm, number of ground station antennas N =16, satellite transmission power P s =30 dBW, satellite directional antenna aperture radius r = , aperture efficiency =0.5.

[0033] like Figure 5 As shown, Figure 5 The convergence performance of the MA optimization algorithm proposed in the present invention in the LEO satellite network is demonstrated. The simulation results show that under all parameter configurations, the proposed MA optimization scheme significantly improves the average achievable rate of the ground station. With the increase of satellite transmission power, the system rate is further improved, and the optimization of antenna position plays a key role in this improvement. In addition, the figure also shows that when the number of orbits and satellites increases, the average achievable rate actually decreases. This is because the increase in constellation density brings stronger interference. However, the performance advantage of the proposed MA scheme is more obvious when the constellation density increases, indicating that the scheme has strong adaptability in interference suppression. Furthermore, in scenarios with a large number of orbits and satellites, after applying the MA scheme, its achievable rate even exceeds that of scenarios with a low satellite density. This is because while suppressing interference, MA also enhances the reception of useful signals, enabling ground stations to select service satellites with better link gain at different times.

[0034] pass Figure 6 This further supports the above conclusions. Figure 6The figure shows the average power gain of the serving signal and the interfering signal. It can be seen that the gain of the desired signal gradually increases with the number of orbits and satellites, as a denser constellation provides more connection options for ground stations. Applying the MA scheme further enhances the gain of the desired signal while effectively suppressing the interfering signal. This dual gain contributes to a significant improvement in the achievable rate of the system in a dense constellation environment.

[0035] Figure 7 The variation of the achievable rate of the MA scheme and two fixed array schemes (SFPA and DFPA) at different time slots is shown. The Earth's rotation period is set to T E =85952 seconds, and the corresponding optimization periods are Seconds and seconds. As can be seen from the figure, the achievable rate of SFPA is always lower than that of DFPA under the same parameter settings. On this basis, the MA scheme significantly improves the overall achievable rate of the system by further optimizing the antenna position. For the MA scheme, it can be observed that at the beginning of the time series, the service satellite is closer to the ground station, so the achievable rate is higher. Then, as the satellite moves away, its rate decreases, but as the next satellite approaches and takes over as the service satellite, the rate rises again, showing an approximately periodic fluctuation with a period of about 1194 seconds. In addition, although increasing the number of satellites and orbits will introduce more interference, it also expands the set of optional service satellites, giving the ground station a greater probability of connecting to satellites with better link quality. Therefore, in some time slots, the rate obtained when the constellation is denser is even higher than when the constellation is sparser. Overall, even considering the factor of the earth's rotation, the achievable rate under the MA scheme still shows a periodic law, indicating that the scheme proposed in the present invention has good stability and robustness during continuous operation cycles.

[0036] Figure 8 The antenna position distribution of the ground station under different parameter settings is shown. In the SFPA scheme, the antenna unit is based on the wavelength. For spacing, evenly distributed in a In the square area of ​​​​the DFPA scheme, the antenna unit is The minimum spacing is densely arranged in a In the area where the array structure remains unchanged during the communication process. In contrast, in the MA scheme, the antenna position is determined by an optimization algorithm based on the different constellation densities. As the number of satellites and orbits increases, the optimized antenna position tends to expand toward the edge of the array, thereby improving the angular resolution capability of the array and achieving more effective interference suppression and signal enhancement. The optimized antenna layout not only adapts to the spatial distribution of currently visible satellites, but also enhances the ability to distinguish between the main lobe directivity and the interference interval, enabling the ground station to connect to a service satellite with better link gain. Therefore, the MA scheme proposed in the present invention can adaptively configure the initial antenna position according to the changes in the low-orbit constellation structure, provide a more flexible and efficient array initialization strategy than a fixed array, and provide strong support for system deployment in complex dynamic interference environments.

[0037] Figure 9 The average achievable speed under different ground station elevation conditions is compared. The elevation angle of the ground station can be regarded as the corresponding parameter of its geographical latitude, so the horizontal axis in the figure reflects the geographical location change of the ground station. The simulation results show that the average achievable speed fluctuates with the change of the ground station location. The main reason is the dynamic change of the number of serving satellites and interfering satellites. Under the MA scheme proposed in this invention, the configuration K =96, J =72, the system achievable rate reaches the highest value when the elevation angle is 10 degrees. At this time, the ground station can select a service satellite with better link gain and less interference, and the system performance is the best. However, as the elevation angle increases further (that is, the latitude of the ground station increases), the achievable rate gradually decreases. This is because the number of visible satellites in higher latitudes increases significantly, resulting in increased interference, which offsets the link optimization benefits brought by the MA scheme. Although as shown in the previous figure, increasing the number of satellites and the number of orbits can improve the system achievable rate as a whole, in high-latitude areas, this improvement effect is limited by the dominant influence of interference, and the performance advantage gradually weakens. In summary, the MA scheme proposed in the present invention shows more significant performance gains in mid- and low-latitude areas, indicating that it has a better deployment effect in geographical areas where interference is controllable.

[0038] Figure 10 The influence of the number of receiving antennas at the ground station on the achievable rate performance of the system is demonstrated. The results show that under all schemes and parameter configurations, the average achievable rate shows a steady upward trend with the increase in the number of receiving antennas. This is because more antenna units provide the array with higher spatial degrees of freedom, thereby enhancing the signal gain and improving the interference suppression capability. Under the SFPA and DFPA schemes, the achievable rates are significantly lower than the MA scheme proposed in this invention. The MA scheme not only effectively suppresses interference signals by jointly optimizing the antenna position and beam direction, but also improves the connection quality of the service satellite, thereby further enhancing the overall communication performance. In addition, with the increase in constellation density (i.e. K andJ When the number of satellites and orbits increases, the performance gain of the MA solution is more significant. Comparing the MA solutions under different parameter configurations, it can be seen that the performance improvement of the MA solution is more prominent when the number of satellites and orbits increases. However, a phenomenon is also observed: when the number of receiving antennas is too large, the marginal rate improvement brought by the optimization of antenna positions gradually weakens. For example, when the parameter configuration is K =96, J =72, the performance gap between MA and DFPA becomes smaller as the number of antennas increases. This is because the antenna movement range is limited by the preset area. Due to the limitations of the array, as the array scales up, the minimum spacing between antennas limits the potential for improvement through position optimization. In summary, an appropriate number of antennas combined with a movable array structure can achieve optimal rate gain and interference control capabilities.

[0039] Figure 11 This study demonstrates the impact of satellite transmit power on the average achievable rate of the system. For fixed antenna position schemes (DFPA and SFPA), when optimizing only beamforming weights, the average achievable rate shows a gradual increase with increasing transmit power. However, when transmit power reaches a high level, the rate gain gradually saturates. This is because higher transmit power also leads to stronger multi-satellite interference, which dominates the system and limits the further performance improvement that can be achieved through beam optimization. In contrast, the MA scheme proposed in this paper achieves additional rate improvements at higher transmit power by jointly optimizing antenna position and beam direction. The MA scheme's interference suppression advantage is particularly significant in scenarios with high constellation density, further expanding the performance improvement in high-power configurations. Notably, even in the MA scheme, the system achievable rate does not always increase monotonically with transmit power. This is because system performance is also constrained by other parameters, such as the number of receiving antennas and antenna range. Therefore, in some configurations, the marginal benefit of increasing transmit power is reduced due to physical layer constraints. In summary, the MA scheme proposed in this invention demonstrates stronger robustness and scalability in medium-to-high transmit power and dense constellation scenarios, providing a practical engineering solution for power control and interference management in future large-scale low-orbit satellite systems.

[0040] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical content disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement of the above embodiments made according to the technical essence of the present invention, within the spirit and principles of the present invention, without departing from the content of the technical solution of the present invention, shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A MA joint beamforming optimization method for a low-orbit satellite communication ground station, characterized in that: include: Step 1: Establish a three-dimensional geocentric spherical coordinate system with the center of the earth as the center, convert the three-dimensional geocentric spherical coordinates into Cartesian coordinates, establish a three-dimensional geocentric Cartesian coordinate system, and obtain the ground station position based on the three-dimensional geocentric Cartesian coordinate system; Step 2: With the ground station as the center, establish the Cartesian coordinate system of the ground station center and place the satellite S jk As a service satellite, obtain the received signal of the ground station at time t, the signal-to-interference-and-noise ratio of the ground station at time t, the interference channel gain of other satellites, and t Achievable rate per unit bandwidth of time ; Step 3: Based on the signal-to-interference-and-noise ratio of the ground station at time t, the interference channel gain of other satellites and t Achievable rate per unit bandwidth of time Get the time interval Average achievable rate, signal-to-interference-noise ratio of ground stations and the interference signal gain of other satellites ; Step 4: Set the antenna position, antenna spacing, and signal-to-interference-noise ratio of the ground station and the interference signal gain of other satellites As constraints, construct the optimization problem P1; Step 5: Introduce auxiliary variables and transform the optimization problem P1 into a quadratic constraint problem P2 through the Lagrangian function; Step 6: Decompose the quadratic constraint problem p2 into subproblem P3 to optimize AWV and subproblem P4 to optimize APV. Iterate subproblems 1 and 2 to obtain the solution of optimization problem P1. Optimize antenna weight vector AWV and antenna position vector APV through subproblems P3 and P4 respectively. The maximum average achievable rate of the ground station movable antenna array is , completing the MA joint beamforming optimization of the ground station.

2. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: In the three-dimensional geocentric spherical coordinate system in step 1, the LEO constellation is composed of J orbital surfaces, each orbit contains K Satellites, S jk For the j On the track k satellites, and , is the orbital inclination, that is, the angle between the orbital plane and the equatorial plane. Assuming the earth is a perfect sphere, the radius of the earth is R , the satellite altitude is H , the Earth's rotation period is T E , the orbital period is , G e and M e are the gravitational constant and the mass of the Earth respectively; is the pitch angle, which represents the angle between the reference plane and the direction of the given point, where a positive value represents the direction from south to north; is the bearing angle, which represents the angle between the reference azimuth direction and the projection of the point on the reference plane, where positive values ​​are measured from west to east; for t The ascending node and S jk The geocentric angle between for The initial angle at Convert the three-dimensional geocentric spherical coordinates into Cartesian coordinates, establish a three-dimensional geocentric Cartesian coordinate system, and obtain the satellite S jk Coordinates in a three-dimensional geocentric Cartesian coordinate system and ground station locations; Satellite S jk The position in the three-dimensional geocentric spherical coordinate system is: (1)； In formula (1), is the distance from the satellite to the center of the Earth, The reference plane to S jk The elevation angle measured from the direction, is the reference azimuth direction to S jk Project the measured azimuth on the reference plane; Satellite S jk The coordinates in the three-dimensional geocentric Cartesian coordinate system are: (2)； In formula (2), , which represents the time required for the next orbital plane to reach the same relative position of the ground station. The observation interval is ; The expression for the ground station position is: (3)； In formula (3), is the ground station pitch angle, is the ground station azimuth.

3. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: In step 2, the coordinate transformation matrix Perform coordinate transformation on the three-dimensional geocentric Cartesian coordinate system to establish the ground station center Cartesian coordinate system. In the established ground station center Cartesian coordinate system, the ground station to satellite S jk The wave vector is ,in , is the carrier wavelength, y The axis is the tangent to the latitude line of the ground station, from west to east. z The axis points radially outward from the center of the Earth, and x Axis and y and z The planes formed by the axes are orthogonal, and the wave vector in the Cartesian coordinate system at the center of the ground station is ,in, is the coordinate transformation matrix; Coordinate transformation matrix The expression is: (4)。 4. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: In step 2, the received signal of the ground station at time t, the signal-to-interference-and-noise ratio of the ground station at time t, and the interference channel gain of other satellites are obtained specifically including: Step 2.1: Set each element of the ground station MA to the Cartesian coordinate system at the center of the ground station. x - y Two-dimensional region of the plane Moving within, based on the Cartesian coordinate system of the ground station center n The position of the element Get with Satellite S jk Associated MA array steering vector , where MA is the movable antenna array, and the first n The position of an element is represented by APV, and its expression is , , satellite S jk The path loss between the ground station and ; Step 2.2: At time t, select satellite S jk As a service satellite, the signals of other satellites are treated as co-channel interference, and the AWV at the ground station is combined to obtain the time of the ground station. t The received signal , where AWV is the antenna weight vector, and its expression is ; Step 2.3: Based on satellite S jk Path loss between the ground station and satellite S jk Associated MA array steering vector Get channel vector ; Step 2.4: Based on channel vector and the antenna weight vector at the ground station Get the moment t Signal-to-interference-and-noise ratio (SIR) of ground stations , interference channel gain of other satellites and in t Achievable rate per unit bandwidth of time ; With satellite S jk Associated MA array steering vector The calculation formula is: (5)； In formula (5), Indicates all c n A collection of Satellite S jk Path loss between the ground station The calculation formula is: (6)； In formula (6), and The reference distances are m Path loss exponent and path gain at meters, m is a constant; Ground station at the moment t The received signal The calculation formula is: (7)； In formula (7), P s is the satellite transmission power, Indicates t The number of visible satellites in the jth orbital plane at time, where K j ( t ) is the number of satellites that the ground station can receive signals from. If the square of the Euclidean distance between the ground station and the satellite satisfies , then it is considered that the satellite can be received by the ground station, and the vector For AWV at the ground station, is independent and identically distributed additive Gaussian white noise, and ; Channel Vector The calculation formula is: (8)； In formula (8), To set each satellite beam to point directly downward, the satellite S jk The directional antenna gain, x jk ( t ) is the satellite S jk The data transmitted, is the wave vector The angle between the direction opposite to the satellite position vector Satellite S jk Directional antenna gain The calculation formula is: (9)； In formula (9), is the first-order Bessel function of the first kind, r is the antenna circular aperture radius; time t Signal-to-interference-and-noise ratio (SIR) of ground stations The calculation formula is: (10)； In formula (10), is the interference channel gain of other satellites, and its calculation formula is: (11)； t Achievable rate per unit bandwidth of time The calculation formula is: (12)。 5. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: Step 3 specifically includes: Step 3.1: Based on t Achievable rate per unit bandwidth of time Calculate in time interval The average achievable rate ; Step 3.2: Set the time interval Discrete M equal-length time slots ,in , No. m The time slot is ,in, t m As the midpoint, obtain the antenna weight vector of the discrete ground station ; Step 3.3: The discretized antenna weight vector of the ground station Substitute into formula (10) to obtain the signal-to-interference-noise ratio of the ground station and the interference signal gain of other satellites ; In time interval The average achievable rate The expression is: (13)； Signal-to-interference-and-noise ratio (SIR) of ground stations The expression is: (14)； In formula (14), , and , is the interference signal gain of other satellites, and its calculation formula is: (15)。 6. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: The optimization problem P1 constructed in step 4 specifically includes: Position the ground station antenna As constraint C1, the antenna spacing As constraint C2, the antenna weight vector of the discretized ground station is in As constraint C3, the optimization problem P1 is constructed based on constraints C1, C2 and C3; The expression of the optimization problem P1 is: (16)。 7. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: Step 5 specifically includes: Step 5.1: Use Lagrange dual transformation to introduce time slots m Corresponding auxiliary variables Solve the weighted sum of multiple logarithmic functions in the optimization problem P1; Step 5.2: Based on step 5.1, use secondary transformation to introduce auxiliary variables Convert the optimization problem P1 into the optimization problem P2; Introducing time slots m Corresponding auxiliary variables The optimization problem P1 is: (17)； Introducing auxiliary variables The subsequent optimization problem P2 is: (18)； (19)； (20)； The expression of the optimization problem P2 is: (21)； (22)； Formula (21)-Formula (22), , , 、 、 and All are parameters.

8. The MA joint beamforming optimization method for a low-orbit satellite communication ground station according to claim 1, characterized in that: The steps for iteratively solving subproblems 1 and 2 in step 6 include: Step 6.1: Set the initial number of iterations , initialization parameters and ; Step 6.2: Set the number of iterations i = i +1, solution and , get the optimal auxiliary variable and ; Step 6.3: Fix the parameters in the optimization problem P2 , ,and , transform the optimization problem P2 into the optimization problem P3, solve the problem P3, and update the parameters ; Step 6.4: Fix all variables except parameter c in optimization problem P2, transform optimization problem P2 into optimization problem P4, solve optimization problem P4, and update parameters ; Step 6.5: Repeat steps 6.2 to 6.4 to iterate the calculation Step 6.6: Determine if there is If yes, proceed to step 6.7, if no, return to step 6.2; Step 6.7: Output Parameters and , the maximum average achievable rate of the ground station movable antenna array is obtained ; Optimal auxiliary variables and The calculation formula is: (23)； The expression of the optimization problem P3 is: (24)； In formula (24), and are all equivalent variables, and their calculation formula is: (25)； (26)； parameter The calculation formula is: (27)； The expression of the optimization problem P4 is: (28)； (29)； (30)。

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