Optimization method for lower planet-ground integrated cell-free network

Abstract

The invention discloses an optimization method of a lower planet-earth integrated cell-free network, and belongs to the field of wireless communication. According to the method, a lower planet-ground integrated cell-free network optimization problem is constructed by taking a maximum sum rate (MSR) and a maximum-minimum fairness (MMF) as optimization targets. By adopting the maximum sum rate as an optimization target, the spectrum efficiency is maximized on the premise of meeting the TU / ES service quality requirement of each terminal user. Beam forming is designed by adopting maximum and minimum fairness, so that the fairness of a ground cell-free network is ensured, and the total rate of a satellite downlink multicast network is maximized. The stability of beam forming is ensured by introducing a constant modulus constraint CR1, and the service fairness among users is ensured through a fairness constraint CR2. According to the method, a joint precoding problem based on the MSR is converted into an equivalent quadratic constraint quadratic programming QCQP problem through fractional programming, alternate optimization and successive convex approximation methods, the planet-earth integrated cell-free network is optimized by adopting an algorithm based on the ADMM, and the optimization calculation complexity is remarkably reduced.

Description

Technical Field

[0001] The invention relates to an optimization method for a planet-ground integrated non-cell network, belonging to the field of wireless communications. Background Art

[0002] With the development of the sixth generation (6G) mobile communication system, comprehensive coverage and universal connectivity have become important goals in the design of communication systems. Due to the rapid development of the Internet of Things (IoT) technology, the number of wireless access nodes and the demand for communication coverage have shown explosive growth. In order to meet this demand, the integrated satellite-terrestrial network (ISGN) is considered to be a key technology. By combining the wide-area coverage capability of the satellite network and the efficient transmission performance of the terrestrial network, it can provide reliable communication support for remote areas, maritime communications, emergency communications and other scenarios, and has broad application prospects. Existing research mainly focuses on the beamforming design and resource allocation of the ISGN system, and improves system performance by optimizing these technologies. For example, satellite networks can provide efficient multicast services with a wide coverage range by utilizing the broadband characteristics of the millimeter wave frequency band and miniaturized antennas. The terrestrial network effectively eliminates inter-cell interference and improves spectrum efficiency and communication capacity by introducing the cell-free multiple-input multiple-output (CFmMIMO) technology. In addition, combined with the time division duplex (TDD) mode, the terrestrial network can obtain channel state information (CSI) through uplink training and perform precoding optimization in the downlink transmission stage. However, most existing methods assume that the system has an ideal CSI, which is difficult to establish in actual environments. At the same time, the joint optimization of resource allocation and beamforming strategies also faces problems such as high computational complexity and spectrum sharing conflicts, which restricts the further improvement of ISGN performance. To address the above problems, this study innovatively proposed a new downlink transmission optimization framework. By combining the characteristics of CFmMIMO technology and millimeter wave frequency bands, a joint beamforming and resource allocation algorithm was designed to significantly improve the system's spectrum efficiency and service quality without relying on ideal CSI. The proposed method further optimizes the collaborative communication of satellite-ground links, reduces computational complexity, and effectively solves the problem of spectrum sharing conflicts, providing a new solution for the practical application of ISGN systems in 6G communications. Summary of the invention

[0003] Aiming at the following technical defects of the existing ISGN system based on CFmMIMO: (i) the channel state information (CSI) of the ground communication system cannot be ideally obtained, resulting in the limitation of the optimization design; (ii) the existing optimization method does not fully consider the fairness of joint beamforming and resource allocation. The main purpose of the present invention is to provide an optimization method for a lower planet-ground integrated cell-free network, which optimizes the lower planet-ground integrated network without cells by using non-ideal channel state information, can achieve the fairness of joint beamforming and resource allocation, improve the maximum sum rate (MSR) of users, and has the advantage of high optimization efficiency.

[0004] The purpose of the present invention is achieved through the following technical solutions.

[0005] The present invention discloses an optimization method for a lower planet-ground integrated cell-free network, which takes the maximum sum rate MSR and the maximum minimum fairness MMF as optimization targets, and the constraints include the quality of service (QoS) of ground stations and ground users, the signal transmission restrictions between satellites and ground stations, the base station power constraints and the satellite power constraints, to construct the optimization problem of the lower planet-ground integrated cell-free network. By adopting the maximum sum rate MSR as the optimization target, the spectrum efficiency SE can be maximized under the premise of meeting the TU / ES service quality requirements of each terminal user. By adopting the maximum minimum fairness MMF to design beamforming, the fairness of the ground cell-free network is guaranteed, and the total rate of the satellite downlink multicast network is maximized. In addition, the stability of the beamforming is guaranteed by introducing the constant modulus constraint CR1, and the fairness of the service between each user is guaranteed by the fairness constraint CR2. The present invention converts the complex sum logarithmic objective function into a polynomial form by utilizing fractional programming FP, and decomposes the optimization problem into two simpler sub-problems by alternating optimization AO strategy. The continuous convex approximation (SCA) method is used to transform non-convex constraints into convex format, and the difficult-to-solve planet-ground integrated no-cell network optimization problem is transformed into a quadratic constrained quadratic programming (QCQP) problem. The ADMM-based algorithm is used to optimize the planet-ground integrated no-cell network optimization problem, thereby significantly reducing the computational complexity.

[0006] The present invention discloses a method for optimizing a planet-ground integrated cell-free network, comprising the following steps:

[0007] Step 1: Build a ground-to-planet integrated cell-free network consisting of satellite and ground-based non-cellular MIMO networks. Satellites use multicast communication to provide services to a group of ground stations ES, while the ground network collaborates to provide services to multiple ground users TU, ensuring that each user gets enough bandwidth and signal quality. The ground-based non-cellular MIMO network operates in the millimeter wave frequency band.

[0008] The signals received by the kth ground user TU are respectively

[0009]

[0010] Among them: signal x m Indicates that ω is the beamforming data that meets the transmission power constraint requirements. M is the number of ground base stations without cells, h mk represents the channel from the mth base station to the kth ground user, f k represents the satellite channel to the kth ground user, w s represents the precoding vector of the satellite downlink, s s is the multicast signal sent by the satellite, z k represents the additive Gaussian noise at the kth ground user; s k ,s j is an independent and identically distributed complex Gaussian random variable, represented by the received data of the kth / jth ground user; w mk ,w mj represents the downlink precoding vector of the kth / jth terrestrial user in the downlink data transmission phase in the terrestrial cell-free network; v s is the downlink beamforming vector applied to the satellite.

[0011] The signal received at the l ground station ES is

[0012]

[0013] Where: h ml represents the channel from the mth base station to the lth ground station, f l represents the channel of the satellite to the lth ground station, z l is the noise received by the user.

[0014] The channel estimation results of the mth base station for the kth ground user and the lth ground station are as follows:

[0015]

[0016] Where: K represents the number of ground users without cells, L is the number of ground stations, τ p is the number of pilot sequences, and satisfies L≤K≤τ p ≤L+K; is the pilot vector τ received by the mth base station p ×1 received signal expression. is the set of terrestrial users that reuse the same pilot as the kth terrestrial user, ρ TU represents the pilot transmission signal-to-noise ratio of the terrestrial user in the uplink pilot estimation stage, ρ sat represents the pilot transmission signal-to-noise ratio of the ground station in the uplink pilot estimation stage, Z mExpressed as a normalized additive Gaussian noise matrix, As the pilot sequence allocated to the kth ground user and the lth ground station.

[0017] The signal-to-interference-plus-noise ratio (SINR) γ between the kth ground user and the lth ground station obtained at the central processing unit (CPU) and the gateway (GW) k , γ l It is expressed as:

[0018]

[0019] in, represents the noise energy of the kth terrestrial user in the downlink data transmission phase, Represents the noise energy of the lth ground station in the downlink data transmission phase.

[0020] Step 2: The optimization target is determined by choosing between the maximum sum rate MSR and the maximum minimum fairness MMF. The constraints include the service quality QoS of ground stations and ground users, the signal transmission limit between satellites and ground stations, the base station power constraint and the satellite power constraint. The optimization problem of the planet-ground integrated cell-free network is constructed. By adopting the maximum sum rate MSR as the optimization target, the spectrum efficiency SE can be maximized under the premise of meeting the service quality requirements of each terminal user TU / ES. By adopting the maximum minimum fairness MMF to design beamforming, the fairness of the ground cell-free network is guaranteed, while the total rate of the satellite downlink multicast network is maximized. In addition, the stability of beamforming is guaranteed by introducing the constant modulus constraint CR1, and the fairness of service between users is guaranteed by the fairness constraint CR2. By using fractional programming FP, the complex sum logarithmic objective function is transformed into a polynomial form, and the optimization problem is decomposed into two sub-problems through the alternating optimization strategy AO.

[0021] The construction of the planet-ground integrated no-cell network optimization problem includes two sub-problems: the planet-ground integrated no-cell network optimization problem with maximum sum rate MSR as the optimization target and the planet-ground integrated no-cell network optimization problem with maximum minimum fairness MMF as the optimization target.

[0022] The optimization problem of the planet-ground integrated non-cell network with the maximum sum rate MSR as the optimization target is shown in the following formula:

[0023]

[0024] in: represents the minimum signal-to-noise ratio requirement of the ground station, represents the minimum signal-to-noise ratio requirement for ground users, P m represents the maximum power constraint of the base station in the terrestrial cell-free network, P satIndicates the maximum power constrained carrier frequency of the satellite.

[0025] The optimization problem of the planet-ground integrated non-cell network with maximum and minimum fairness MMF as the optimization target is as follows:

[0026]

[0027] Among them, constraints C1 and C2 represent the service quality QoS requirements of the ground station and ground users respectively, and C3 and C4 are the actual power constraints of the base station and satellite in downlink transmission.

[0028] Step 3: For the optimization problem of the integrated planet-ground cell-free network with the maximum sum rate MSR as the optimization target, the complex sum logarithmic objective function is converted into a polynomial form by using fractional programming FP, and the optimization problem is decomposed into two simpler sub-problems by using the alternating optimization AO strategy. The continuous convex approximation SCA method is used to convert the non-convex constraints into a convex format, and the difficult-to-solve optimization problem of the integrated planet-ground cell-free network is converted into a quadratic constrained quadratic programming QCQP problem. The ADMM-based algorithm is used to optimize the optimization problem of the integrated planet-ground cell-free network, and the optimization result of the integrated planet-ground cell-free network is obtained. According to the optimization result of the integrated planet-ground cell-free network, the maximum sum rate MSR of users is improved.

[0029] Step 3.1: Input SAT-ES, SAT-UT channel f l and f k ; BS-TU, BS-ES channel h mk and h ml ; Maximum power P of BS and SAT m and P sat ; Penalize variables and errors; Find the feasible point by the following formula and

[0030] Where: N t represents the number of antennas of the ground base station without cell, and the channel set is expressed as The set of precoding vectors is represented as and represents the covariance of the channel.

[0031] To express the problem as an SDP problem, define

[0032] In order to express the BS power constraint as a quadratic constrained SDP problem, a new variable is introduced: in The optimal solution of the SDP problem is obtained by solving the optimization problem through convex optimization tools. Then, SVD decomposition and randomization methods are used to generate and

[0033] Step 3.2: If If there is no convergence, repeat steps 3.3 to 3.6.

[0034] Step 3.3: According to Update x l and ψ k .

[0035] Step 3.4: According to Update q l and p k .

[0036] Step 3.5: Optimize w mk .

[0037] At a given χ l , ψ k ,q l , p k With v s , the optimization problem is rewritten as follows

[0038]

[0039] in:

[0040] Use the SCA algorithm to transform the non-convex problem into a convex optimization problem, and then use the ADMM algorithm to find the optimal solution to the convex problem. The specific iterative method after using the ADMM algorithm is as follows:

[0041] Step 3.5.1: Set the SCA iteration number t←0 and initialize

[0042] Step 3.5.2: If the convergence criterion of the SCA algorithm is not met, Then repeat steps 3.5.3 to 3.5.9.

[0043] Step 3.5.3: Initialize the ADMM algorithm and set the number of ADMM iterations j←0. Where: r l,i ,u k,i ,e i are the auxiliary residual variables of C1, C2, and C3 in step 3.1.

[0044] Step 3.5.4: Update auxiliary variables According to the following formula.

[0045]

[0046] Where: y l,i ,x k,i ,z i is the scaling dual variable of C1, C2, C3 in step 3.1, and π is the Lager dual variable, which is found exhaustively using the bisection method through the KKT conditions.

[0047] Step 3.5.5: Update the optimization variables according to the following formula

[0048]

[0049] Step 3.5.6: Update the auxiliary residual variable according to the following formula

[0050]

[0051] Step 3.5.7: j←j+1.

[0052] Step 3.5.8: If the convergence criterion of the ADMM algorithm is not met, Then repeat steps 3.5.4 to 3.5.8.

[0053] Step 3.5.9: t←t+1,

[0054] Step 3.5.10: Obtain the variable w with lower computational complexity by iterating from step 3.5.1 to step 3.5.9 mk The optimal solution

[0055] Step 3.6: Optimize v s .

[0056] At a given χ l , ψ k ,q l , p k With w i , the optimization problem is rewritten as follows

[0057]

[0058] To facilitate the writing of expressions, auxiliary variables are introduced:

[0059]

[0060] The optimal solution to this problem is found through the ADMM algorithm. The specific iterative method is as follows:

[0061] Step 3.6.1: Initialize the ADMM algorithm, set the number of ADMM iterations j←0, and initialize the variables Initialize residual variables κ 0 ←0. Among them: ι k ,κ is the auxiliary residual variable of C1, C2, C4 in step 3.1.

[0062] Step 3.6.2: Update auxiliary variables according to the following formula Δ j+1 .

[0063]

[0064] Where: k ,Λ l ,Δ is the scaling dual variable of C1, C2, C4 in step 3.1.

[0065] Step 3.6.3: Update the optimization variables according to the following formula

[0066]

[0067] Step 3.6.4: Update the auxiliary residual variable according to the following formula κ j+1 .

[0068]

[0069] Step 3.6.5: j←j+1.

[0070] Step 3.6.6: If the convergence criterion of the ADMM algorithm is not met, Through the iterations from step 3.6.2 to step 3.6.5, the optimization variables are reduced. The complexity of the optimal solution.

[0071] Step 3.7: Precoding vector optimized according to step 3.5 and step 3.6 Satisfy the overall constraints to obtain the optimized maximum sum rate

[0072] Step 4: For the optimization problem of the integrated planet-ground no-cell network with maximum and minimum fairness MMF as the optimization target, the complex sum logarithmic objective function is transformed into a polynomial form by using fractional programming FP, and the optimization problem is decomposed into two simpler sub-problems by using the alternating optimization AO strategy. The continuous convex approximation SCA method is used to transform the non-convex constraints into a convex format, and the difficult-to-solve optimization problem of the integrated planet-ground no-cell network is transformed into a quadratic constrained quadratic programming QCQP problem. The ADMM-based algorithm is used to optimize the optimization problem of the integrated planet-ground no-cell network, and the optimization result of the integrated planet-ground no-cell network is obtained. The fairness of joint beamforming and resource allocation is achieved based on the optimization result of the integrated planet-ground no-cell network.

[0073] Step 4.1: Enter SAT-TU channel f i and f c ; BS-TU, BS-ES channel b mk and h ml ; Maximum power P of BS and SAT m and P sat ; Since the constraints of the two sub-problems are the same, the feasible point found in step 3.1 is also applicable to step 4.1, and then the feasible point is found and

[0074] Step 4.2: If If there is no convergence, repeat steps 4.3 to 4.5.

[0075] Step 4.3: According to Update x l and q l .

[0076] Step 4.4: Obtain a new solution w by solving the following problem mk and α.

[0077]

[0078] Introducing auxiliary variables To simplify the expression, the problem is also a non-convex QCQP problem. The SCA algorithm is used to transform the problem into a convex problem and then the ADMM algorithm is applied to find the optimal solution of the problem. The specific implementation steps are as follows:

[0079] Step 4.4.1: Set the SCA iteration number t←0 and initialize

[0080] Step 4.4.2: If the convergence criterion of the SCA algorithm is not met, and Then repeat steps 4.4.3 to 4.4.9.

[0081] Step 4.4.3: Initialize the ADMM algorithm and set the number of ADMM iterations j←0. α 0 ←α (t) , Where: r l,i ,u k,i ,e i are the auxiliary residual variables of C1, C2, and C3 in step 3.1, and β k is the auxiliary residual variable of variable α.

[0082] Step 4.4.4: Update auxiliary variables according to the following formula

[0083]

[0084] Where: y l,i ,x k,i ,z i is the scaling dual variable of C1, C2, C3 in step 3.1, α k is the scaled dual variable of variable α, where π is the Lager dual variable, which is found exhaustively using the bisection method through the KKT conditions.

[0085] Step 4.4.5: Update the optimization variables according to the following formula α j+1 .

[0086]

[0087] Step 4.4.6: Update the auxiliary residual variable according to the following formula

[0088]

[0089] Step 4.4.7: j←j+1.

[0090] Step 4.4.8: If the convergence criterion of the ADMM algorithm is not met, and Then repeat steps 4.4.4 to 4.4.7.

[0091] Step 4.4.9: t←t+1, α t+1 ←α opt .

[0092] Step 4.4.10: Reduce the optimization variables by iterating from step 4.4.1 to step 4.4.9 α opt The complexity of the solution.

[0093] Step 4.5: Solve the following problem to obtain a new solution v s

[0094]

[0095] Introducing auxiliary variables to simplify expressions This problem is the same as the problem in step 3.6 in form, and only differs in the specific auxiliary variables, so the problem is solved by directly using the method in step 3.6. The specific implementation steps of step 4.5 are as follows:

[0096] Step 4.5.1: Initialize the ADMM algorithm, set the number of ADMM iterations j←0, and initialize the variables Initialize residual variables κ 0 ←0. Among them: ι k ,κ is the auxiliary residual variable of C1, C2, C4 in step 3.1.

[0097] Step 4.5.2: Update auxiliary variables according to the following formula Δ j+1 .

[0098]

[0099]

[0100] Where: k ,Λ l ,Δ is the scaling dual variable of C1, C2, C4 in step 3.1.

[0101] Step 4.5.3: Update the optimization variables according to the following formula

[0102]

[0103] Step 4.5.4: Update the auxiliary residual variable according to the following formula κ j+1 .

[0104]

[0105] Step 4.5.5: j←j+1.

[0106] Step 4.5.6: If the convergence criterion of the ADMM algorithm is not met, Then, by iterating from step 4.5.2 to step 4.5.5, the complexity of solving the optimal solution of the optimization variable v is reduced.

[0107] Step 4.6: Optimal precoding vector according to step 4.4 and step 4.5 and Obtain the maximum total rate of the satellite-to-ground station link after optimization under the constraints And the maximum minimum achievable rate α of the cell-free network based on the MMF fairness strategy opt .

[0108] Beneficial effects:

[0109] 1. The present invention discloses an optimization method for a lower planet-ground integrated cell-free network, with maximum sum rate MSR and maximum minimum fairness MMF as optimization targets, and constraints including the quality of service QoS of ground stations and ground users, signal transmission restrictions between satellites and ground stations, base station power constraints and satellite power constraints, to construct an optimization problem for a lower planet-ground integrated cell-free network; non-ideal channel state information is used to optimize the performance of the lower planet-ground integrated network without cells. This method ensures that the ISGN system based on cell-free multiple-input multiple-output (CFmMIMO) can achieve efficient downlink transmission, while meeting the fairness requirements of joint beamforming and resource allocation, and effectively improving the maximum sum rate MSR of users.

[0110] 2. The present invention discloses an optimization method for a ground-to-planet integrated cell-free network. By adopting the maximum sum rate MSR as the optimization target, the spectrum efficiency SE can be maximized while meeting the TU / ES service quality requirements of each terminal user. By adopting the maximum minimum fairness MMF to design beamforming, the fairness of the ground cell-free network is guaranteed, while the total rate of the satellite downlink multicast network is maximized. In addition, the stability of the beamforming is guaranteed by introducing the constant modulus constraint CR1, and the fairness of the service among users is guaranteed by the fairness constraint CR2. While maintaining the quality of network service, this method ensures the improvement of the efficiency of spectrum utilization, and through fairness design, ensures that all users can obtain a reasonable service share, thereby achieving the fairness of the overall network and improving user satisfaction.

[0111] 3. The present invention discloses an optimization method for a planet-ground integrated no-cell network. By utilizing fractional programming FP, the complex summed logarithmic objective function is converted into a polynomial form, and the optimization problem is decomposed into two simpler sub-problems through the alternating optimization AO strategy. The continuous convex approximation SCA method is used to convert non-convex constraints into a convex format, and the difficult-to-solve planet-ground integrated no-cell network optimization problem is converted into a quadratic constrained quadratic programming QCQP problem, and an ADMM-based algorithm is used to separately optimize the planet-ground integrated no-cell network optimization problem. This method significantly reduces the computational complexity, making the optimization problem of the planet-ground integrated no-cell network easier to solve, improving the practicality and efficiency of the algorithm, and providing an efficient and robust solution for network design and resource management. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] Figure 1 It is a structural block diagram of a satellite-terrestrial integrated network (ISGN) system based on cell-free multiple-input multiple-output (CFmMIMO) according to the present invention;

[0113] Figure 2 This is a topological structure diagram of an ISGN system based on CFmMIMO according to the present invention;

[0114] Figure 3 It is a comparison chart of SE performance of the method described in the present invention with the change of BSs maximum power constraint and the performance curve of the traditional method;

[0115] Figure 4 It is a comparison chart of SE performance of the method described in the present invention with the change of satellite maximum power constraint and the performance curve of the traditional method;

[0116] Figure 5 A flow chart of an optimization method for a planet-ground integrated cell-free network according to the present invention. DETAILED DESCRIPTION

[0117] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in combination with specific implementation examples and with reference to the accompanying drawings.

[0118] The system topology of the ISGN system based on CFmMIMO is established, where BSs, TUs, and ESs are randomly distributed in an area of ​​1 square kilometer. A total of 5 BSSs serve 4 TUs together, while 3 ESs in the area receive satellite multicast signals. In addition, the altitude of GEO is 35786 kilometers, and the altitudes of BSs, TUs, and ESs are set to 15 meters, 1.5 meters, and 6 meters, respectively.

[0119] This embodiment discloses an optimization method for a planet-to-ground integrated cell-free network, and its example optimization effect is as follows: Figure 3, 4 As shown in the figure, under different base station maximum power or satellite power constraints, the performance of the invented algorithm is always very close to the optimal algorithm, especially under P m / P sat In the case of higher power, the two are almost the same. And compared with the two comparison algorithms "Consensus ADMM" and "SDR-GAN", the invented algorithm provides a higher total rate in the entire power range. Figure 5 As shown, the present embodiment discloses a method for optimizing a planet-ground integrated cell-free network, and the specific implementation steps are as follows:

[0120] Step 1: Build a ground-to-planet integrated cell-free network consisting of satellite and ground-based non-cellular MIMO networks. Satellites use multicast communication to provide services to a group of ground stations ES, while the ground network collaborates to provide services to multiple ground users TU, ensuring that each user gets enough bandwidth and signal quality. The ground-based non-cellular MIMO network operates in the millimeter wave frequency band.

[0121] The signals received by the kth ground user TU are respectively

[0122]

[0123] Among them: signal x m Indicates that ω is the beamforming data that meets the transmission power constraint requirements. M is the number of ground base stations without cells, h mk represents the channel from the mth base station to the kth ground user, f k represents the satellite channel to the kth ground user, w s represents the precoding vector of the satellite downlink, s s is the multicast signal sent by the satellite, z k represents the additive Gaussian noise at the kth ground user; s k ,s j is an independent and identically distributed complex Gaussian random variable, represented by the received data of the kth / jth ground user; w mk ,w mj represents the downlink precoding vector of the kth / jth terrestrial user in the downlink data transmission phase in the terrestrial cell-free network; v s is the downlink beamforming vector applied to the satellite. N s Indicates the number of antennas of the GEO satellite (single antenna corresponds to single beam working mode), Indicates the noise energy of the mth ground base station in the uplink pilot estimation stage, which is 20 GHz, the data bandwidth is 20 MHz, and the number of satellite beams is N s =7, the maximum multi-beam gain from satellite to ground station b max=52dB, 3dB angle of satellite beam amplitude θ 3dB =0.4°, the antenna gain of the ground base station is G a =25dB, link noise factor NF = 9dB, satellite millimeter wave link rain attenuation parameters μ = -3.125, σ = 1.591, terrestrial millimeter wave link direct path shadow fading parameters ε = 1.9, ζ = 1.1, terrestrial millimeter wave link indirect path shadow fading parameters ε = 3.4, ζ = 9.1

[0124] The signal received at the l ground station ES is

[0125]

[0126] Where: h ml represents the channel from the mth base station to the lth ground station, f l represents the channel of the satellite to the lth ground station, z l is the noise received by the user.

[0127] The channel estimation results of the mth base station for the kth ground user and the lth ground station are as follows:

[0128]

[0129] Where: K represents the number of ground users without cells, L is the number of ground stations, τ p is the number of pilot sequences, and satisfies L≤K≤τ p ≤L+K;Y m P is the pilot vector τ received by the mth base station p ×1 received signal expression. is the set of terrestrial users that reuse the same pilot as the kth terrestrial user, ρ TU represents the pilot transmission signal-to-noise ratio of the terrestrial user in the uplink pilot estimation stage, ρ sat represents the pilot transmission signal-to-noise ratio of the ground station in the uplink pilot estimation stage, Z m Expressed as a normalized additive Gaussian noise matrix, As the pilot sequence allocated to the kth ground user and the lth ground station.

[0130] The signal-to-interference-plus-noise ratio (SINR) γ between the kth ground user and the lth ground station obtained at the central processing unit (CPU) and the gateway (GW) k , γ l It is expressed as:

[0131]

[0132] in, represents the noise energy of the kth terrestrial user in the downlink data transmission phase, Represents the noise energy of the lth ground station in the downlink data transmission phase.

[0133] Step 2: The optimization target is determined by choosing between the maximum sum rate MSR and the maximum minimum fairness MMF. The constraints include the service quality QoS of ground stations and ground users, the signal transmission limit between satellites and ground stations, the base station power constraint and the satellite power constraint. The optimization problem of the planet-ground integrated cell-free network is constructed. By adopting the maximum sum rate MSR as the optimization target, the spectrum efficiency SE can be maximized under the premise of meeting the service quality requirements of each terminal user TU / ES. By adopting the maximum minimum fairness MMF to design beamforming, the fairness of the ground cell-free network is guaranteed, while the total rate of the satellite downlink multicast network is maximized. In addition, the stability of beamforming is guaranteed by introducing the constant modulus constraint CR1, and the fairness of service between users is guaranteed by the fairness constraint CR2. By using fractional programming FP, the complex sum logarithmic objective function is transformed into a polynomial form, and the optimization problem is decomposed into two sub-problems through the alternating optimization strategy AO.

[0134] The construction of the planet-ground integrated no-cell network optimization problem includes two sub-problems: the planet-ground integrated no-cell network optimization problem with maximum sum rate MSR as the optimization target and the planet-ground integrated no-cell network optimization problem with maximum minimum fairness MMF as the optimization target.

[0135] The optimization problem of the planet-ground integrated non-cell network with the maximum sum rate MSR as the optimization target is as follows:

[0136]

[0137] in: represents the minimum signal-to-noise ratio requirement of the ground station, represents the minimum signal-to-noise ratio requirement for ground users, P m represents the maximum power constraint of the base station in the terrestrial cell-free network, P sat Indicates the maximum power constrained carrier frequency of the satellite.

[0138] The optimization problem of the planet-ground integrated non-cell network with maximum and minimum fairness MMF as the optimization target is as follows:

[0139]

[0140] Among them, constraints C1 and C2 represent the service quality QoS requirements of ES and TU respectively, and C3 and C4 are the actual power constraints of the base station and satellite in downlink transmission.

[0141] Step 3: For the optimization problem of the integrated planet-ground cell-free network with the maximum sum rate MSR as the optimization target, the complex sum logarithmic objective function is converted into a polynomial form by using fractional programming FP, and the optimization problem is decomposed into two simpler sub-problems by using the alternating optimization AO strategy. The continuous convex approximation SCA method is used to convert the non-convex constraints into a convex format, and the difficult-to-solve optimization problem of the integrated planet-ground cell-free network is converted into a quadratic constrained quadratic programming QCQP problem. The ADMM-based algorithm is used to optimize the optimization problem of the integrated planet-ground cell-free network, and the optimization result of the integrated planet-ground cell-free network is obtained. According to the optimization result of the integrated planet-ground cell-free network, the maximum sum rate MSR of users is improved.

[0142] Step 3.1: Input SAT-ES, SAT-UT channel f l and f k ; BS-TU, BS-ES channel h mk and h ml ; Maximum power P of BS and SAT m and P sat ; Penalize variables and errors; Find the feasible point by the following formula and

[0143]

[0144] Where: N t represents the number of antennas of the ground base station without cell, and the channel set is expressed as The set of precoding vectors is represented as and represents the covariance of the channel.

[0145] To express the problem as an SDP problem, define

[0146] In order to express the BS power constraint as a quadratic constrained SDP problem, a new variable is introduced: in The optimal solution of the SDP problem is obtained by solving the optimization problem through convex optimization tools. Then, SVD decomposition and randomization methods are used to generate and

[0147] Step 3.2: If If there is no convergence, repeat steps 3.3 to 3.6.

[0148] Step 3.3: According to Update x l and ψ k.

[0149] Step 3.4: According to Update q l and p k .

[0150] Step 3.5: Optimize w mk .

[0151] At a given χ l , ψ k ,q l , p k With v s , the optimization problem is rewritten as follows

[0152]

[0153] in:

[0154] Use the SCA algorithm to transform the non-convex problem into a convex optimization problem, and then use the ADMM algorithm to find the optimal solution to the convex problem. The specific iterative method after using the ADMM algorithm is as follows:

[0155] Step 3.5.1: Set the SCA iteration number t←0 and initialize

[0156] Step 3.5.2: If the convergence criterion of the SCA algorithm is not met, Then repeat steps 3.5.3 to 3.5.9.

[0157] Step 3.5.3: Initialize the ADMM algorithm and set the number of ADMM iterations j←0. Where: r l,i ,u k,i ,e i are the auxiliary residual variables of C1, C2, and C3 in step 3.1, and β k is the auxiliary residual variable of variable α.

[0158] Step 3.5.4: Update auxiliary variables According to the following formula.

[0159]

[0160] Where: y l,i ,x k,i ,z i is the scaling dual variable of C1, C2, C3 in step 3.1, and π is the Lager dual variable, which is found exhaustively using the bisection method through the KKT conditions.

[0161] Step 3.5.5: Update the optimization variables according to the following formula

[0162]

[0163] Step 3.5.6: Update the auxiliary residual variable according to the following formula

[0164]

[0165] Step 3.5.7: j←j+1.

[0166] Step 3.5.8: If the convergence criterion of the ADMM algorithm is not met, Then repeat steps 3.5.4 to 3.5.8.

[0167] Step 3.5.9: t←t+1,

[0168] Step 3.5.10: Reduce the optimization variable w by iterating from step 3.5.1 to step 3.5.9 mk The complexity of the optimal solution.

[0169] Step 3.6: Optimize v s .

[0170] At a given χ l , ψ k ,q l , p k With w i , the optimization problem is rewritten as follows:

[0171]

[0172] To facilitate the writing of expressions, auxiliary variables are introduced:

[0173]

[0174] The optimal solution to this problem is found through the ADMM algorithm. The specific iterative method is as follows:

[0175] Step 3.6.1: Initialize the ADMM algorithm, set the ADMM iteration number j←0, and initialize the variables Initialize residual variables κ 0 ←0. Among them: ι k ,κ is the auxiliary residual variable of C1, C2, C4 in step 3.1.

[0176] Step 3.6.2: Update auxiliary variables according to the following formula Δ j+1 .

[0177]

[0178] Where: k ,Λ l ,Δ is the scaling dual variable of C1, C2, C4 in step 3.1.

[0179] Step 3.6.3: Update the optimization variables according to the following formula

[0180]

[0181] Step 3.6.4: Update the auxiliary residual variable according to the following formula κ j+1 .

[0182]

[0183] Step 3.6.5: j←j+1.

[0184] Step 3.6.6: If the convergence criterion of the ADMM algorithm is not met, By iterating from step 3.6.2 to step 3.6.5, the optimization variable v is reduced. s The complexity of the optimal solution.

[0185] Step 3.7: Precoding vector optimized according to step 3.5 and step 3.6 Satisfy the overall constraints and obtain the optimized maximum sum rate

[0186] Step 4: For the optimization problem of the integrated planet-ground no-cell network with maximum and minimum fairness MMF as the optimization target, the complex sum logarithmic objective function is transformed into a polynomial form by using fractional programming FP, and the optimization problem is decomposed into two simpler sub-problems by using the alternating optimization AO strategy. The continuous convex approximation SCA method is used to transform the non-convex constraints into a convex format, and the difficult-to-solve optimization problem of the integrated planet-ground no-cell network is transformed into a quadratic constrained quadratic programming QCQP problem. The ADMM-based algorithm is used to optimize the optimization problem of the integrated planet-ground no-cell network, and the optimization result of the integrated planet-ground no-cell network is obtained. The fairness of joint beamforming and resource allocation is achieved based on the optimization result of the integrated planet-ground no-cell network.

[0187] Step 4.1: Enter SAT-TU channel f i and f c ; BS-TU, BS-ES channel b mk and h ml ; Maximum power P of BS and SAT m and P sat; Since the constraints of the two sub-problems are the same, the feasible point found in step 3.1 is also applicable to step 4.1, and then the feasible point is found and

[0188] Step 4.2: If If there is no convergence, repeat steps 4.3 to 4.5.

[0189] Step 4.3: According to Update x l and q l .

[0190] Step 4.4: Obtain a new solution w by solving the following problem mk and α.

[0191]

[0192] Introducing auxiliary variables To simplify the expression, the problem is also a non-convex QCQP problem. The SCA algorithm is used to transform the problem into a convex problem and then the ADMM algorithm is applied to find the optimal solution of the problem. The specific implementation steps are as follows:

[0193] Step 4.4.1: Set the SCA iteration number t←0 and initialize

[0194] Step 4.4.2: If the convergence criterion of the SCA algorithm is not met, and Then repeat steps 4.4.3 to 4.4.9.

[0195] Step 4.4.3: Initialize the ADMM algorithm and set the number of ADMM iterations j←0. α 0 ←α (t) , Where: r l,i ,u k,i ,,e i are the auxiliary residual variables of C1, C2, and C3 in step 3.1, and β k is the auxiliary residual variable of variable α

[0196] Step 4.4.4: Update auxiliary variables according to the following formula

[0197]

[0198] Where: y l,i ,x k,i ,z i is the scaling dual variable of C1, C2, C3 in step 3.1, ak is the scaled dual variable of variable α, where π is the Lager dual variable, which is found exhaustively using the bisection method through the KKT conditions.

[0199] Step 4.4.5: Update the optimization variables according to the following formula α j+1 .

[0200]

[0201] Step 4.4.6: Update the auxiliary residual variable according to the following formula

[0202]

[0203] Step 4.4.7: j←j+1.

[0204] Step 4.4.8: If the convergence criterion of the ADMM algorithm is not met, and Then repeat steps 4.4.4 to 4.4.7.

[0205] Step 4.4.9: t←t+1, α t+1 ←α opt .

[0206] Step 4.4.10: Reduce the optimal optimization variable by iterating from step 4.4.1 to step 4.4.9 α opt The complexity of solution.

[0207] Step 4.5: Solve the following problem to obtain a new solution v

[0208]

[0209] Introducing auxiliary variables to simplify expressions This problem is the same as the problem in step 3.6 in form, and only differs in the specific auxiliary variables, so I solve this problem by directly using the method in step 3.6. The specific implementation steps of step 4.5 are as follows:

[0210] Step 4.5.1: Initialize the ADMM algorithm, set the number of ADMM iterations j←0, and initialize the variables Initialize residual variables κ 0 ←0. Among them: ι k ,κ is the auxiliary residual variable of C1, C2, C4 in step 3.1

[0211] Step 4.5.2: Update auxiliary variables according to the following formula Δ j+1 .

[0212]

[0213]

[0214] Where: k ,Λ l ,Δ is the scaling dual variable of C1, C2, C4 in step 3.1.

[0215] Step 4.5.3: Update the optimization variables according to the following formula

[0216]

[0217] Step 4.5.4: Update the auxiliary residual variable according to the following formula κ j+1 .

[0218]

[0219] Step 4.5.5: j←j+1.

[0220] Step 4.5.6: If the convergence criterion of the ADMM algorithm is not met, Then, by iterating from step 4.5.2 to step 4.5.5, the optimization variables are reduced. The computational complexity of .

[0221] Step 4.6: Optimal precoding vector according to step 4.4 and step 4.5 and Satisfy the overall constraints and obtain the optimized maximum total rate of the satellite-to-ground station link And the maximum minimum achievable rate α of the cell-free network based on the MMF fairness strategy opt .

[0222] The specific description above further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for optimizing a planet-to-ground integrated cell-free network, characterized in that: The following steps are included: Step 1: Build a ground-to-planet integrated cell-free network consisting of satellite and ground-based non-cellular MIMO network; the satellite uses multicast communication to provide services to a group of ground stations ES, while the ground network collaboratively provides services to multiple ground users TU, ensuring that each user has sufficient bandwidth and signal quality; the ground-based non-cellular MIMO network operates in the millimeter wave frequency band; Step 2: The optimization target is determined by selecting either the maximum sum rate MSR or the maximum minimum fairness MMF. The constraints include the service quality QoS of ground stations and ground users, the signal transmission limit between satellites and ground stations, the base station power constraint and the satellite power constraint, and the optimization problem of the planet-ground integrated cell-free network is constructed; by adopting the maximum sum rate MSR as the optimization target, the spectrum efficiency SE can be maximized under the premise of meeting the service quality requirements of each terminal user TU / ES; by adopting the maximum minimum fairness MMF to design beamforming, the fairness of the ground cell-free network is guaranteed, and the total rate of the satellite downlink multicast network is maximized; in addition, the stability of the beamforming is guaranteed by introducing the constant modulus constraint CR1, and the fairness of the service between each user is guaranteed by the fairness constraint CR2; by using the fractional programming FP, the complex sum logarithmic objective function is converted into a polynomial form, and the optimization problem is decomposed into two sub-problems through the alternating optimization strategy AO; Step 3: For the optimization problem of the planet-ground integrated cell-free network with the maximum sum rate MSR as the optimization target, the complex sum logarithmic objective function is transformed into a polynomial form by using fractional programming FP, and the optimization problem is decomposed into two simpler sub-problems by using the alternating optimization AO strategy; The continuous convex approximation (SCA) method is used to transform non-convex constraints into convex format, and the difficult-to-solve planet-ground integrated no-cell network optimization problem is transformed into a quadratic constrained quadratic programming (QCQP) problem. The ADMM-based algorithm is used to optimize the planet-ground integrated no-cell network optimization problem, and the planet-ground integrated no-cell network optimization result is obtained. According to the planet-ground integrated no-cell network optimization result, the maximum sum rate (MSR) of users is improved. Step 4: For the optimization problem of the planet-ground integrated non-cell network with maximum and minimum fairness MMF as the optimization objective, the complex summation logarithmic objective function is transformed into a polynomial form by using fractional programming FP, and the optimization problem is decomposed into two simpler sub-problems by using the alternating optimization AO strategy; The continuous convex approximation (SCA) method is used to transform non-convex constraints into convex format, and the difficult-to-solve planet-ground integrated no-cell network optimization problem is transformed into a quadratic constrained quadratic programming (QCQP) problem. The ADMM-based algorithm is used to optimize the planet-ground integrated no-cell network optimization problem, and the optimization result of the planet-ground integrated no-cell network is obtained. According to the optimization result of the planet-ground integrated no-cell network, the fairness of joint beamforming and resource allocation is achieved.

2. The optimization method of the lower planet-ground integrated cell-free network according to claim 1, characterized in that: In step one, The signals received by the kth ground user TU are Among them: signal x m Indicates that ω is the beamforming data that meets the transmission power constraint requirements; M is the number of ground base stations without cells, and h mk represents the channel from the mth base station to the kth ground user, f k represents the satellite channel to the kth ground user, w s represents the precoding vector of the satellite downlink, s s is the multicast signal sent by the satellite, z k represents the additive Gaussian noise at the kth ground user; s k ,s j is an independent and identically distributed complex Gaussian random variable, represented by the received data of the kth / jth ground user; w mk ,w mj represents the downlink precoding vector of the kth / jth terrestrial user in the downlink data transmission phase in the terrestrial cell-free network; v s is the downlink beamforming vector applied to the satellite; The signal received at the lth ground station ES is Where: h ml represents the channel from the mth base station to the lth ground station, f l represents the channel of the satellite to the lth ground station, z l is the noise received by the user; The channel estimation results of the mth base station for the kth ground user and the lth ground station are as follows: Where: K represents the number of ground users without cells, L is the number of ground stations, τ p is the number of pilot sequences, and satisfies L≤K≤τ p ≤L+K; is the pilot vector τ received by the mth base station p ×1 when the received signal is expressed; is the set of terrestrial users that reuse the same pilot as the kth terrestrial user, ρ TU represents the pilot transmission signal-to-noise ratio of the terrestrial user in the uplink pilot estimation stage, ρ sat represents the pilot transmission signal-to-noise ratio of the ground station in the uplink pilot estimation stage, Z m Expressed as a normalized additive Gaussian noise matrix, As the pilot sequence allocated to the kth ground user and the lth ground station; The signal-to-interference-to-noise ratio (SINRγ) between the kth ground user and the lth ground station obtained at the central processor CPU and the gateway GW k , γ l It is expressed as: in, represents the noise energy of the kth terrestrial user in the downlink data transmission phase, Represents the noise energy of the lth ground station in the downlink data transmission phase.

3. The optimization method of the lower planet-ground integrated cell-free network as claimed in claim 2, characterized in that: The implementation method of step 2 is: The optimization problem of the planet-ground integrated no-cell network is constructed, which includes two sub-problems: the optimization problem of the planet-ground integrated no-cell network with the maximum sum rate MSR as the optimization target and the optimization problem of the planet-ground integrated no-cell network with the maximum minimum fairness MMF as the optimization target. The optimization problem of the planet-ground integrated non-cell network with the maximum sum rate MSR as the optimization target is shown in the following formula: C4:‖v s ‖ 2 ≤P sat . in: represents the minimum signal-to-noise ratio requirement of the ground station, represents the minimum signal-to-noise ratio requirement for ground users, P m represents the maximum power constraint of the base station in the terrestrial cell-free network, P sat Indicates the maximum power constrained carrier frequency of the satellite; The optimization problem of the planet-ground integrated non-cell network with maximum and minimum fairness MMF as the optimization target is as follows: stC1-C4. Among them, constraints C1 and C2 represent the service quality QoS requirements of the ground station and ground users respectively, and C3 and C4 are the actual power constraints of the base station and satellite in downlink transmission.

4. The optimization method of the lower planet-ground integrated cell-free network as claimed in claim 3, characterized in that: The implementation method of step three is: Step 3.1: Input SAT-ES, SAT-UT channel f l and f k ; BS-TU, BS-ES channel h mk and h ml ; Maximum power P of BS and SAT m and P sat ; Penalize variables and errors; Find the feasible point by the following formula and C5:Rank(X i )=1,i∈{0,1,...,K}. Where: N t represents the number of antennas of the ground base station without cell, and the channel set is expressed as , the set of precoding vectors is expressed as and represents the covariance of the channel; To express the problem as an SDP problem, define In order to express the BS power constraint as a quadratic constrained SDP problem, a new variable is introduced: in The optimal solution of the SDP problem is obtained by solving the optimization problem through convex optimization tools. Then, SVD decomposition and randomization methods are used to generate and Step 3.2: If If there is no convergence, repeat steps 3.3 to 3.6; Step 3.3: According to Update x l and ψ k ; Step 3.4: According to Update q l and p k ; Step 3.5: Optimize w mk ; At a given χ l , ψ k ,q l , p k With v s , the optimization problem is rewritten as follows in: Use the SCA algorithm to transform the non-convex problem into a convex optimization problem, and then use the ADMM algorithm to find the optimal solution to the convex problem; the specific iterative method after using the ADMM algorithm is as follows: Step 3.5.1: Set the SCA iteration number t←0 and initialize Step 3.5.2: If the convergence criterion of the SCA algorithm is not met, Then repeat steps 3.5.3 to 3.5.9; Step 3.5.3: Initialize the ADMM algorithm and set the number of ADMM iterations j←0. Where: r l,i ,u k,i ,e i are the auxiliary residual variables of C1, C2, and C3 in step 3.

1. Step 3.5.4: Update auxiliary variables According to the following formula; Where: y l,i ,x k,i ,z i is the scaling dual variable of C1, C2, C3 in step 3.1, π is the Lager dual variable, which is found by exhaustive bisection through KKT conditions; Step 3.5.5: Update the optimization variables according to the following formula Step 3.5.6: Update the auxiliary residual variable according to the following formula Step 3.5.7: j←j+1; Step 3.5.8: If the convergence criterion of the ADMM algorithm is not met, Then repeat steps 3.5.4 to 3.5.8; Step 3.5.9: t←t+1, Step 3.5.10: Obtain the variable w with lower computational complexity by iterating from step 3.5.1 to step 3.5.9 mk The optimal solution Step 3.6: Optimize v s ; At a given χ l , ψ k ,q l , p k With w i , the optimization problem is rewritten as follows To facilitate the writing of expressions, auxiliary variables are introduced: The optimal solution to this problem is found through the ADMM algorithm. The specific iterative method is as follows: Step 3.6.1: Initialize the ADMM algorithm, set the number of ADMM iterations j←0, and initialize the variables Initialize residual variables κ 0 ←0; where: ι k ,κ is the auxiliary residual variable of C1, C2, C4 in step 3.1; Step 3.6.2: Update auxiliary variables according to the following formula Δ j+1 ; Where: k ,Λ l ,Δ is the scaling dual variable of C1, C2, C4 in step 3.1; Step 3.6.3: Update the optimization variables according to the following formula Step 3.6.4: Update the auxiliary residual variable according to the following formula Step 3.6.5: j←j+1; Step 3.6.6: If the convergence criterion of the ADMM algorithm is not met, Through the iterations from step 3.6.2 to step 3.6.5, the optimization variables are reduced. The complexity of the optimal solution; Step 3.7: Precoding vector optimized according to step 3.5 and step 3.6 Satisfy the overall constraints to obtain the optimized maximum sum rate 5. The optimization method of the lower planet-ground integrated cell-free network as claimed in claim 4, characterized in that: The implementation method of step 4 is: Step 4.1: Enter SAT-TU channel f i and f c ; BS-TU, BS-ES channel b mk and h ml ; Maximum power P of BS and SAT m and P sat ; Since the constraints of the two sub-problems are the same, the feasible point found in step 3.1 is also applicable to step 4.1, and then the feasible point is found and Step 4.2: If If there is no convergence, repeat steps 4.3 to 4.5; Step 4.3: According to Update x l and q l ; Step 4.4: Obtain a new solution w by solving the following problem mk and α; Introducing auxiliary variables To simplify the expression, the problem is also a non-convex QCQP problem. The SCA algorithm is used to transform the problem into a convex problem and then the ADMM algorithm is applied to find the optimal solution of the problem. The specific implementation steps are as follows: Step 4.4.1: Set the SCA iteration number t←0 and initialize Step 4.4.2: If the convergence criterion of the SCA algorithm is not met, and Then repeat steps 4.4.3 to 4.4.9; Step 4.4.3: Initialize the ADMM algorithm and set the number of ADMM iterations j←0. α 0 ←α (t) , Where: r l,i ,u k,i ,e i are the auxiliary residual variables of C1, C2, and C3 in step 3.1, and β k is the auxiliary residual variable of variable α; Step 4.4.4: Update auxiliary variables according to the following formula Where: y l,i ,x k,i ,z i is the scaling dual variable of C1, C2, C3 in step 3.1, α k is the scaled dual variable of variable α, where π is the Lager dual variable, which is found exhaustively using the bisection method through the KKT conditions; Step 4.4.5: Update the optimization variables according to the following formula α j+1 ; Step 4.4.6: Update the auxiliary residual variable according to the following formula Step 4.4.7: j←j+1; Step 4.4.8: If the convergence criterion of the ADMM algorithm is not met, and Then repeat steps 4.4.4 to 4.4.7; Step 4.4.9: t←t+1, α t+1 ←α opt ; Step 4.4.10: Reduce the optimization variables by iterating from step 4.4.1 to step 4.4.9 α opt The complexity of the solution; Step 4.5: Solve the following problem to obtain a new solution v s Introducing auxiliary variables to simplify expressions This problem is the same as the problem in step 3.6 in form, and only differs in the specific auxiliary variables, so the problem is solved by directly using the method in step 3.6; the specific implementation steps of step 4.5 are as follows: Step 4.5.1: Initialize the ADMM algorithm, set the number of ADMM iterations j←0, and initialize the variables Initialize residual variables κ 0 ←0; where: ι k ,κ is the auxiliary residual variable of C1, C2, C4 in step 3.1; Step 4.5.2: Update auxiliary variables according to the following formula Δ j+1 ; Where: k ,Λ l ,Δ is the scaling dual variable of C1, C2, C4 in step 3.1; Step 4.5.3: Update the optimization variables according to the following formula Step 4.5.4: Update the auxiliary residual variable according to the following formula Step 4.5.5: j←j+1; Step 4.5.6: If the convergence criterion of the ADMM algorithm is not met, Then, by iterating from step 4.5.2 to step 4.5.5, the complexity of solving the optimal solution of the optimization variable v is reduced; Step 4.6: Optimal precoding vector according to step 4.4 and step 4.5 and Obtain the maximum total rate of the satellite-to-ground station link after optimization under the constraints And the maximum minimum achievable rate α of the cell-free network based on the MMF fairness strategy opt .

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