Low earth orbit satellite multi-group multicast robust beam forming method

Abstract

The invention particularly relates to a multi-group multicast robust beam forming method for a low earth orbit satellite. The method comprises the following steps: establishing an LEO satellite communication forward link propagation model under an SFPB reflector antenna structure; establishing a receiving signal model of the ground user terminal by considering a channel phase estimation error; deriving a reachable rate of each multicast group in a multi-group multicast scene based on a received signal model of the ground user terminal; establishing an optimization problem of multi-group multicast robust beam forming under beam priority and rate demand difference; approximately representing an expected rate, and converting an original optimization problem by using an SDR method; and solving the optimization problem under the relaxation unit rank constraint by using an MM algorithm. According to the method, errors existing in channel estimation are considered, and multi-group multicast robust beam forming of the low-orbit satellite under the difference of beam priorities and rate requirements is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of wireless communication, and particularly relates to a method for robust beamforming of multiple groups of multicast in low-earth orbit satellites. Background Art

[0002] With the continuous development of low-earth orbit (LEO) satellite communication systems, multi-beam and multi-group multicast communication has become an important technical architecture. Such satellite communication systems cover ground users through multiple beams simultaneously, and use beamforming technology to optimize resource allocation and improve communication efficiency. The existing technical solutions mainly involve the following methods:

[0003] Beamforming based on channel state information (CSI): Existing satellite communication systems dynamically adjust the direction and power of beams based on CSI to match the channel conditions of ground users. By obtaining CSI in real time, this method significantly improves spectral efficiency and system capacity, and performs well especially in environments with less interference. However, in practical applications, CSI usually has errors, which has a certain negative impact on the effect of beamforming.

[0004] Adaptive power allocation: This method dynamically adjusts the power allocation of each beam according to user requirements and channel conditions. Through this adaptive strategy, the system can minimize the total power consumption while meeting the user rate requirements. Although this method is relatively flexible, in an environment with large multi-beam interference, it may face the situation of uneven resource allocation.

[0005] Fixed beam allocation strategy: Some relatively traditional beamforming methods use a fixed beam allocation strategy, which is suitable for scenarios where user requirements are relatively uniform. This method is simple and easy to implement, but it lacks the ability to adapt to the complex user requirement differences in multi-beam systems and has poor performance in complex communication scenarios.

[0006] Although the existing technologies can effectively improve system performance in basic scenarios, they still face certain challenges when dealing with practical problems such as complex communication requirements of multi-beam and multi-group multicast, imperfect CSI, and non-uniform rate requirements; specifically as follows:

[0007] 1. Unreasonable beam resource allocation: Existing beam resource allocation methods usually ignore the priority differences between beams, resulting in the rate requirements of high-priority beams not being fully met, which affects the quality of service of key beams.

[0008] 2. CSI Imperfection Problem: Most of the existing technologies assume perfect CSI. However, in reality, due to channel estimation errors or time-varying channels, CSI is usually imperfect. This can lead to phase errors, thereby affecting the beamforming effect and increasing the system power consumption.

[0009] 3. Insufficient Consideration of Non-uniform Rate Requirements: The existing methods do not handle the differences in user rate requirements adequately. In the scenarios of multi-beam and multi-group multicast, the rate requirements of users vary greatly, while the existing technologies often assume uniform user rate requirements, resulting in some beams unable to meet the requirements of high-demand users and causing resource waste.

[0010] It should be noted that the information disclosed in the above background technology section is only used to enhance the understanding of the background of the present invention. Therefore, it may include information that does not constitute the prior art known to those of ordinary skill in the art. Summary of the Invention

[0011] The present invention provides a robust beamforming method for multi-group multicast in low-earth orbit satellites to solve the following technical problems:

[0012] 1. Resource Allocation Based on Beam Priority Differences: Ensure that in a multi-beam system, beams with higher priorities can obtain more resources to meet the requirements of critical beams and high-priority users.

[0013] 2. Consider the Impact of Imperfect CSI: Solve the problem of reduced beamforming accuracy caused by imperfect CSI and ensure that the system can still operate efficiently in the presence of channel estimation errors.

[0014] 3. Handling of Non-uniform Rate Requirements: Effectively address the differences in user rate requirements in a multi-group multicast system, avoid resource waste, and improve the system throughput and spectral efficiency.

[0015] Other features and advantages of the present invention will become apparent through the following detailed description, or be learned in part through the practice of the present invention.

[0016] According to a first aspect of the present invention, there is provided a robust beamforming method for multi-group multicast in low-earth orbit satellites, the method comprising:

[0017] Step 1: Establish a forward link propagation model for LEO satellite communication under the SFPB reflector antenna structure;

[0018] Step 2: Considering the channel phase estimation error, establish a received signal model for ground user terminals;

[0019] Step 3: Derive the achievable rates of each multicast group in a multi-group multicast scenario based on the received signal model of ground user terminals;

[0020] Step 4: Establish an optimization problem for multi-group multicast robust beamforming under beam priority and rate requirement differences;

[0021] Step 5: Approximately represent the desired rate and transform the original optimization problem using the SDR method;

[0022] Step 6: Solve the optimization problem under the relaxed unit rank constraint using the MM algorithm.

[0023] In some exemplary embodiments, the LEO satellite communication forward link propagation model is specifically:

[0024]

[0025] where G R is the receiving antenna gain of the UTs; ζ T = κBT represents the receiving end noise; κ is the Boltzmann constant; B is the link bandwidth; T is the receiving end noise temperature; C L is the free space loss coefficient; b is the beam gain; r is the rain fade coefficient; θ is the channel phase vector; h is the satellite-to-terminal forward link channel vector.

[0026] In some exemplary embodiments, the receiving signal model of the ground user terminal is specifically:

[0027]

[0028] where y m,i represents the received signal of the i-th UT in the m-th multicast group, h m,i represents the channel vector from the satellite to the ground UT. The first term on the right side of the equation represents the target received signal, the second term is the interference caused by the transmission signals of other multicast groups, η m,i is a circularly symmetric complex Gaussian distribution with a mean of 0 and a variance of N0, is the beamforming weight vector of all multicast groups, is the original transmission signal transmitted by all beams, w m is the beamforming weight vector of the m-th multicast group, s m is the original transmission signal with unit power sent to the multicast group Γ m .

[0029] In some exemplary embodiments, the achievable rate of each multicast group in the multi-group multicast scenario is specifically:

[0030] Q m = η m R m

[0031]

[0032] where \(w\) l is the multi - cast multi - group beamforming weight vector after removing the \(m\) - th beam, \(\eta\) m \(\in[0,1]\) is the beam priority factor, and \(Q\) m is the achievable rate of the \(m\) - th multi - cast group.

[0033] In some exemplary embodiments, the optimization problem of the multi - group multi - cast robust beamforming is specifically as follows:

[0034] Considering the influence of phase errors, that is, in the case of non - perfect CSI, design beamformers for each group to meet the differentiated beam priorities and non - uniform rate requirements. The goal is to minimize the total system power consumption, and this problem can be described as

[0035]

[0036] where \(C1\) is the rate requirement constraint for each beam, and the rate requirements for each beam are \(D=\{d1,d2,\cdots,d\) m ,\cdots,d M \}; \(C2\) is the range constraint for the beam priority factor; \(C3\) is the range constraint for the multi - cast group rate requirement factor. When the achievable rate of each beam exactly matches the required rate, \(\beta\) m \( = 1\); \(C4\) is the single - feed power constraint of the antenna.

[0037] In some exemplary embodiments, the transformation of the original optimization problem using the SDR method is specifically as follows:

[0038]

[0039] where \(H'\) m,i is the long - term channel correlation matrix, and \(C5\) is the new constraint.

[0040] In some exemplary embodiments, step 6 is specifically as follows:

[0041] By replacing the \(\cdots\) in the problem with the first - order Taylor expansion Then solve this convex optimization problem with an initial feasible point and continue to the next iteration; in the \(\lambda\) - th iteration, the optimization problem in step 5 is transformed into a convex optimization problem:

[0042]

[0043] Obtain the solution of the \(\lambda\) - th iteration. In the \((\lambda + 1)\) - th iteration, update using the solution of the \(\lambda\) - th iteration to obtain a new convex problem Finally, by solving the sub - convex problems updated in each iteration, the total system power consumption will converge to a stable minimum value;

[0044] If all the final solutions are of unit rank, then the solution is also optimal for the problem In this case, the beamformer can be obtained from W through eigenvalue decomposition opt The beamforming weights for each group can be expressed as where ν and z m are m the corresponding eigenvalue and eigenvector respectively

[0045] In some exemplary embodiments, the method further includes:

[0046] If what is obtained in step 6 is not of unit rank, the Gaussian randomization method needs to be used to further process the solution obtained by the MM algorithm

[0047] According to a second aspect of the present invention, there is provided a storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the robust beamforming method for multiple groups of multicast of low-earth orbit satellites described in the first aspect above is implemented.

[0048] According to a third aspect of the present invention, there is provided a computer program product, on which a computer program is stored, and when the computer program is executed by a processor, the robust beamforming method for multiple groups of multicast of low-earth orbit satellites described in the first aspect above is implemented.

[0049] According to a fourth aspect of the present invention, there is provided an electronic device, including:

[0050] a processor; and

[0051] a memory for storing executable instructions of the processor;

[0052] wherein the processor is configured to implement the robust beamforming method for multiple groups of multicast of low-earth orbit satellites described in the first aspect above when executing the executable instructions.

[0053] The robust beamforming method for multiple groups of multicast of low-earth orbit satellites provided by the embodiments of the present invention has the following beneficial effects compared with the prior art:

[0054] 1. Resource allocation based on differentiated beam priorities: The prior art usually ignores the priority differences between beams. By introducing a beam priority factor, the present invention dynamically adjusts resource allocation according to the importance of beams, ensuring that high-priority beams (such as beams covering key areas or high-priority users) receive more resource guarantees, and improving the overall service quality of the LEO satellite system.

[0055] ​2. Robust Beamforming Design: Most of the existing technologies assume perfect CSI. This invention takes into account the impact of channel estimation errors and time-varying channels, reduces the influence of phase errors on beamforming accuracy, and realizes effective power consumption reduction and rate requirement satisfaction under imperfect CSI.

[0056] 3. Resource Allocation Based on Non-uniform Multicast Group Rate Requirements: In existing technologies, most methods assume uniform user rate requirements and fail to fully address rate requirement differences. This invention considers the actual application scenario where the rate requirements of each multicast group are different. By introducing a rate matching factor, it adjusts the rate matching degree of each beam, improves spectral efficiency, and avoids resource waste.

[0057] It should be understood that the above general description and the following detailed description are only exemplary and explanatory, and cannot limit the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] The drawings herein are incorporated into the specification and constitute a part of this specification, showing embodiments consistent with the present invention, and are used together with the specification to explain the principles of the present invention. Obviously, the drawings in the following description are only some embodiments of the present invention, and those of ordinary skill in the art can obtain other drawings without creative efforts based on these drawings.

[0059] Figure 1 is a multi-user multi-beam satellite forward link communication network system;

[0060] Figure 2 is a flowchart of a low-earth orbit satellite multi-group multicast robust beamforming method according to an exemplary embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0061] Example embodiments will now be described more fully with reference to the accompanying drawings. However, the example embodiments can be implemented in various forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this invention will be more complete and comprehensive, and will fully convey the concept of the example embodiments to those skilled in the art. The features, structures, or characteristics described can be combined in any suitable manner in one or more embodiments.

[0062] In addition, the drawings are only schematic illustrations of the present invention and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and thus repeated descriptions thereof will be omitted. Some of the block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software form, or in one or more hardware modules or integrated circuits, or in different networks and / or processor devices and / or microcontroller devices.

[0063] The present invention is applicable to a full-frequency reuse multi-beam LEO satellite forward link communication system, such as Figure 1 shown. Among them, the LEO satellite is equipped with a reflector antenna with a single-feed-per-beam (SFPB) architecture, including a feed array with N s feeds, a beamforming network, and a reflector antenna. K fixed single-antenna ground UTs are randomly distributed within the coverage areas of each beam. Define to represent the index set of all UTs. The ground user terminals obtain the CSI of the downlink through channel estimation and send it to the gateway, and the gateway transmits the CSI to the satellite through the feedback link. Then, the array reflector antenna at the satellite end uses the CSI to generate N s adjacent beams and broadcasts the processed signal to each user within the beam through the downlink channel. Assume that one beam serves one multicast group, that is, M = N s , where represents the index set of all multicast groups. In addition, define the set of users in the m-th multicast group as Γ m , where each user belongs to only one group, that is,

[0064] The present invention considers the error in channel estimation and realizes robust beamforming for multi-group multicast of LEO satellites under beam priority and rate requirement differences. The process is as Figure 2 shown, and each step will be elaborated in detail below.

[0065] Step 1: Establish a forward link propagation model for LEO satellite communication under the SFPB reflector antenna structure.

[0066] Consider the effects of free space loss, rain attenuation, beam gain, and phase effect. The satellite-terminal forward link channel vector can be expressed as:

[0067]

[0068] In the formula, G R is the receiving antenna gain of the UTs; ζ T = κBT represents the receiving end noise; κ is the Boltzmann constant; B is the link bandwidth; T is the receiving end noise temperature. In addition, is the free space loss coefficient and can be written as:

[0069]

[0070] In the formula, c is the speed of light; f represents the carrier frequency; d h is the orbital altitude of the LEO satellite; dc is the distance from the UTs to the center of the ground beam.

[0071] In a multi-beam satellite communication system, the beam gain depends on the radiation pattern of the satellite antenna and the position of the UTs. The beam gain of the nth feed can be expressed as:

[0072]

[0073] where G t,max represents the maximum antenna gain at the satellite end; is the angle between the UTs and the line of sight of the nth beam; is the 3dB angle of the nth beam; J1(·) and J3(·) are the first-order and third-order Bessel functions of the first kind.

[0074] In addition, is the rain fade coefficient. According to ITU-R P.1853, rain fade is a slow fading process. Therefore, it is assumed that the rain fade within the same beam is the same, and the fades between different beams are independent of each other. The rain fade vector can be expressed as:

[0075]

[0076] where the power gain ξ 2 (dB) usually satisfies a lognormal distribution with a mean of μ dB and a standard deviation of σ dB , that is

[0077] Finally, is the channel phase vector. Due to the long propagation distance between the satellite and the ground UTs and the small spacing between the feeds, the signals at the satellite are highly correlated. Therefore, the phases of all feeds can be considered the same. The channel phase vector can be expressed as:

[0078]

[0079] where θ is a random value uniformly distributed between 0 and 2π.

[0080] Step 2: Considering the channel phase estimation error, establish the received signal model of the ground user terminal.

[0081] Due to factors such as atmospheric absorption, propagation delay, and the rapid relative motion between the satellite and the ground, the satellite channel will experience severe phase changes. Therefore, there may be channel phase estimation errors at the UTs. Assume that the i-th ground user in the m-th multicast group obtains the estimated channel vector at time t0 and feeds it back to the gateway station. After propagation delay and processing delay, the satellite uses the CSI at time t1. The channel phase at time t1 can be modeled as:

[0082]

[0083] where θ m,i (t1) is the channel vector of the i-th ground user in the m-th multicast group at time t1, is the estimated channel vector obtained by the i-th ground user in the m-th multicast group at time t0, is the channel phase error and follows a Gaussian distribution, i.e., where is the variance of the phase error, and C m,i is the normalized covariance matrix. Define to represent the estimated channel at time t0, then h k represents the actual channel at time t1. Therefore, the actual CSI can be expressed as:

[0084]

[0085] where q m,i = exp(je m,i ).

[0086] Define s m (t) as the original transmission signal with unit power sent to the multicast group Γ m , i.e., The retransmitted signal after satellite processing can be expressed as:

[0087]

[0088] where is the beamforming weight vector of the m-th multicast group. Then the satellite broadcasts the total transmission signal s to the ground UTs through the downlink channel. Therefore, the received signal of the i-th UT in the m-th multicast group can be expressed as:

[0089]

[0090] where represents the channel vector from the satellite to the ground UT. The first term on the right side of the equation represents the target received signal, the second term is the interference caused by the transmission signals of other multicast groups, and η m,i follows a circularly symmetric complex Gaussian distribution with a mean of 0 and a variance of N0, i.e.,

[0091] Step 3: Derive the achievable rates of each multicast group in multiple multicast scenarios.

[0092] According to the received signal model of UTs described by formula (9), the received signal-to-interference-plus-noise ratio γ of the l-th UT in the m-th multicast group is m,l as follows:

[0093]

[0094] Considering that accurate CSI cannot be obtained at the satellite side, the present invention mainly focuses on the beamforming design based on the desired rate. The desired rate of the i-th user in the multicast group m can be expressed as:

[0095]

[0096] In the multiple multicast transmission scenario, the rate achieved within the group is determined by the user with the lowest rate, that is At the same time, considering the priority of each beam, the achievable rate of the m-th multicast group is Q m = η m R m , η m ∈[0,1] is the beam priority factor. The larger η m , the higher the priority of the m-th beam, and more resources should be obtained during resource allocation to meet the higher rate requirements of the multicast group.

[0097] Step 4: Establish an optimization problem for multi-group multicast robust beamforming under beam priority and rate requirement differences.

[0098] Specifically, considering the influence of phase errors, that is, in the case of non-perfect CSI, design beamformers for each group to meet the differentiated beam priorities and non-uniform rate requirements. The goal is to minimize the total system power consumption. This problem can be described as

[0099]

[0100] In the formula, C1 is the rate requirement constraint of each beam, where the rate requirements of each beam are Dβ{d1, d2,..., d m ,..., d M}; C2 is the range constraint of the beam priority factor; C3 is the range constraint of the multicast group rate requirement factor. When the achievable rate of each beam exactly matches the required rate, β m = 1; C4 is the single-feed power constraint of the antenna.

[0101] Since the expected rate expression and the non-convex constraint C1 exist in formula (12), the above optimization problem (12) is non-convex and NP-Hard, and directly solving the problem is very difficult. To address the above challenges, the present invention first approximately represents the expected rate and uses semidefinite relaxation (SDR) to transform the original optimization problem into an easily tractable form. Then, the majorization-minimization (MM) algorithm is used to solve the convex problem under relaxed unit rank. Finally, for the case where the solution is not of unit rank, a Gaussian randomization method is used to obtain a feasible solution.

[0102] Step 5: Approximately represent the expected rate and use the SDR method to transform the original optimization problem.

[0103] First, the expected rate is approximately represented as follows:

[0104]

[0105] Then, the average signal-to-interference-plus-noise ratio can be expressed as:

[0106]

[0107] Although formula (13) uses an approximate function of the expected rate, the non-convex constraint C1 makes problem (12) still difficult to solve directly. Since is a classical QCQP problem, the present invention uses the SDR method for approximation, and the optimization variables of problem (12) can be transformed into This will add two new constraints and Rank(W m ) = 1. At the same time, the approximate rate in formula (13) can be rewritten as:

[0108]

[0109] where is the instantaneous channel correlation matrix and can be expressed according to formula (7) as:

[0110]

[0111] where In addition, the long-term channel correlation matrix H′ m,i can be expressed as:

[0112]

[0113] where is the correlation matrix of q m,i The matrix Q′ m,iThe position elements of are as follows:

[0114]

[0115] After the above conversion and definition Problem (12) can be converted to:

[0116]

[0117] In the formula,

[0118]

[0119] In problem (19), the rank constraint Rank(W m ) = 1 is non-convex. Furthermore, problem (19) can be relaxed to:

[0120]

[0121] Step 6: Use the MM algorithm to solve the optimization problem under the relaxed unit rank constraint.

[0122] Since and are both concave functions with respect to W , this may cause the constraint C1 in problem (22) to be non-convex. The present invention uses the MM algorithm to solve this problem.

[0123] First, use the first-order Taylor expansion to replace in problem (22). Then, solve this convex optimization problem with an initial feasible point and continue to the next iteration. Specifically, in the λ-th iteration, C1 in problem (22) can be re-expressed as:

[0124]

[0125] In the formula,

[0126]

[0127] In the formula, is the solution obtained for each variable in the λ-th iteration, and λ is the iteration index. In addition, The gradient with respect to W l is:

[0128]

[0129] Using formulas (23), (24), and (25), problem (22) can be transformed into the following problem:

[0130]

[0131] In problem (26), both the objective function and the constraints are convex. Therefore, the convex problem can be solved by existing tools such as CVX to obtain the solution at the λ-th iteration

[0132] In the (λ + 1)-th iteration, the formula (23) is updated using the solution of the λ-th iteration, thereby obtaining a new convex problem Finally, by solving the sub-convex problems updated at each iteration, the total system power consumption will converge to a stable minimum value.

[0133] If all the final solutions are of unit rank, then the solution is also optimal for the problem At this time, the beamformer can be obtained from W through eigenvalue decomposition opt The beamforming weights for each group can be expressed as where ν and z m and m are the corresponding eigenvalue and eigenvector respectively.

[0134] Step 7: Solve the optimization problem under the relaxed non-unit rank constraint by combining the Gaussian randomization method and the MM algorithm.

[0135] If what is obtained in step 6 is not of unit rank, then the Gaussian randomization method needs to be used to further process the solution obtained by the MM algorithm, as follows.

[0136] First, obtain the candidate Gaussian vectors: Perform eigenvalue decomposition on as follows:

[0137]

[0138] In addition, the candidate Gaussian vectors can be expressed as:

[0139]

[0140] where U and Σ can be calculated by formula (27), is a complex Gaussian random vector. Therefore, by repeating formula (27) and formula (28), G candidate Gaussian vectors can be generated for each multicast group.

[0141] Then, solve the power allocation problem: To ensure the feasibility of problem (26), it is necessary to re-allocate power to the beamforming vectors of each group. Specifically, for a specific set of candidate Gaussian vectors Combining formula (10) and formula (11), the power allocation problem among the candidate beamforming vectors can be expressed as:

[0142]

[0143] where p m is the power ratio factor of multicast m. Further, using the approximate function of the average rate in formula (13) and defining problem can be transformed into:

[0144]

[0145] where

[0146]

[0147] Similar to problem (22), in problem among them, and are both concave functions with respect to p, so the MM algorithm is still used to solve problem Then, in the t-th iteration, C6 in problem (30) can be rewritten as:

[0148]

[0149] where

[0150]

[0151] where is the solution obtained by each power ratio factor in the t-th iteration, and t is the number of iterations. In addition, in formula (34) with respect to p l the gradient is:

[0152]

[0153] Similarly, problem can be transformed into:

[0154]

[0155] Problem can also be effectively solved by iteration to obtain the corresponding beamforming weight vector

[0156] Finally, select the optimal candidate target: among all feasible candidate solutions, select the candidate object with the optimal objective value, that is, the one that minimizes the total system power consumption, as the beamforming vector for each multicast group.

[0157] In addition, the above-mentioned drawings are only schematic illustrations of the processes included in the method according to the exemplary embodiments of the present invention, rather than for the purpose of limitation. It is easy to understand that the processes shown in the above-mentioned drawings do not indicate or limit the chronological order of these processes. Additionally, it is also easy to understand that these processes can be executed synchronously or asynchronously in, for example, multiple modules.

[0158] Other embodiments of the present invention will be readily apparent to those skilled in the art upon consideration of the specification and practice of the invention herein. This application is intended to cover any variations, uses, or adaptations of the present invention that follow the general principles of the present invention and include known common knowledge or conventional technical means in the technical field not disclosed by the present invention. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of the present invention are pointed out by the claims.

[0159] It should be understood that the present invention is not limited to the exact structures described above and shown in the drawings, and various modifications and changes can be made without departing from its scope. The scope of the present invention is defined only by the appended claims.

Claims

1. A multi-group multicast robust beamforming method for low-earth orbit satellites, characterized in that The method includes the following: Step 1: Establish a propagation model for the forward link of LEO satellite communication under the SFPB reflector antenna structure; Step 2: Considering the channel phase estimation error, establish a received signal model for the ground user terminal; Step 3: Based on the received signal model of the ground user terminal, derive the achievable rates of each multicast group in multiple multicast scenarios; Step 4: Establish an optimization problem for multi-group multicast robust beamforming considering beam priority and rate requirement differences; Step 5: Approximately represent the desired rate and transform the original optimization problem using the SDR method; Step 6: Solve the optimization problem under the relaxed unit rank constraint using the MM algorithm.

2. The method according to claim 1, wherein The propagation model for the forward link of LEO satellite communication is specifically as follows: Where, G R is the receiving antenna gain of the UTs; ζ T = κBT represents the receiving-end noise; κ is the Boltzmann constant; B is the link bandwidth; T is the receiving-end noise temperature; C L is the free space loss coefficient; b is the beam gain; r is the rain fade coefficient; θ is the channel phase vector; h is the satellite-terminal forward link channel vector.

3. The method according to claim 2, characterized in that, The received signal model of the ground user terminal is specifically as follows: where y m,i represents the received signal of the \(i\)-th UT in the \(m\)-th multicast group, \(h m,i represents the channel vector from the satellite to the ground UT. The first term on the right side of the equation represents the target received signal, and the second term is the interference caused by the transmission signals of other multicast groups. \(\eta m,i is a circularly symmetric complex Gaussian distribution with a mean of 0 and a variance of \(N_0\). is the beamforming weight vector of all multicast groups. is the original transmission signal transmitted by all beams, \(w m is the beamforming weight vector of the \(m\)-th multicast group, \(s m is the original transmission signal with unit power sent to the multicast group \(\Gamma m .

4. The method according to claim 3, wherein The achievable rates of each multicast group in multiple multicast scenarios are specifically as follows: Q m = η m R m where \(w\) l is the multi - cast multi - group beamforming weight vector after removing the \(m\) - th beam, \(\eta\) m \(\in[0,1]\) is the beam priority factor, and \(Q\) m is the achievable rate of the \(m\) - th multi - cast group.

5. The method according to claim 1, wherein The optimization problem for multi-group multicast robust beamforming is specifically as follows: Considering the influence of phase errors, that is, in the case of non-perfect CSI, design beamformers for each group to meet the differentiated beam priorities and non-uniform rate requirements. The goal is to minimize the total system power consumption, and this problem can be described as s.t.C1: C2: C3: C4: where C1 is the rate requirement constraint for each beam, and the rate requirements for each beam are D = {d1, d2,..., d m ,..., d M}; C2 is the range constraint for the beam priority factor; C3 is the range constraint for the multicast group rate requirement factor. When the achievable rate of each beam exactly matches the required rate, β m = 1; C4 is the single-feed power constraint of the antenna.

6. The method according to claim 5, wherein The transformation of the original optimization problem using the SDR method is specifically as follows: s.t.C1: C2: C3: C4: C5:W m ≥0 where H′ m,i is the long-term channel correlation matrix, and C5 is the newly added constraint.

7. The method according to claim 6, characterized in that, Step 6 is specifically as follows: By replacing the in the problem with a first-order Taylor expansion, then solving this convex optimization problem with an initial feasible point and proceeding to the next iteration; in the $\lambda$-th iteration, the optimization problem in step 5 is transformed into a convex optimization problem: s.t.C1: C2: C3: C4: C5:W m ≥0 Obtain the solution of the λ-th iteration. In the (λ + 1)-th iteration, update using the solution of the λ-th iteration to obtain a new convex problem. Finally, by solving the sub-convex problems updated in each iteration, the total system power consumption will converge to a stable minimum value. If all the final solutions are of unit rank, then the solution is also optimal for the problem and the beamformer can be obtained from W through eigenvalue decomposition at this time opt The beamforming weights for each group can be expressed as where ν and z m are m the corresponding eigenvalue and eigenvector respectively ​ 8. The method according to claim 7, wherein The method further includes: If what is obtained in step 6 is not of unit rank, the solution obtained by the MM algorithm needs to be further processed using the Gaussian randomization method.

9. A storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the low-earth orbit satellite multi-group multicast robust beamforming method described in any one of claims 1 to 8.

10. A computer program product comprising a computer program, characterized in that, When the computer program is executed by a processor, it implements the low-earth orbit satellite multi-group multicast robust beamforming method described in any one of claims 1 to 8.

Images