An optimization method for downlink and ground integrated cell-free network

Abstract

This invention discloses an optimization method for a terrestrial integrated planetary-terrestrial cellless network, belonging to the field of wireless communication. The invention constructs an optimization problem for a terrestrial integrated planetary-terrestrial cellless network with maximum sum rate (MSR) and maximum minimum fairness (MMF) as optimization objectives. By adopting maximum sum rate as the optimization objective, spectral efficiency is maximized while meeting the TU / ES service quality requirements of each terminal user. By employing maximum minimum fairness in beamforming design, the fairness of the terrestrial cellless network is guaranteed, maximizing the total rate of the satellite downlink multicast network. The stability of beamforming is ensured by introducing constant modulus constraint CR1, and service fairness among users is guaranteed by fairness constraint CR2. This invention transforms the MSR-based joint precoding problem into an equivalent quadratic constrained quadratic programming (QCQP) problem through fractional programming, alternating optimization, and successive convex approximation methods. An ADMM-based algorithm is used to optimize the terrestrial integrated planetary-terrestrial cellless network, significantly reducing the computational complexity of the optimization.

Description

Technical Field

[0001] This invention relates to an optimization method for a planetary-ground integrated cellless network, belonging to the field of wireless communication. Background Technology

[0002] With the development of sixth-generation (6G) mobile communication systems, comprehensive coverage and ubiquitous connectivity have become important goals in communication system design. Due to the rapid development of Internet of Things (IoT) technology, the number of wireless access nodes and the demand for communication coverage are experiencing explosive growth. To meet this demand, Integrated Satellite-Ground Network (ISGN) is considered a key technology. By combining the wide-area coverage of satellite networks with the efficient transmission performance of terrestrial networks, it can provide reliable communication support for scenarios such as remote areas, maritime communication, and emergency communication, and has broad application prospects. Existing research mainly focuses on beamforming design and resource allocation of ISGN systems, aiming to improve system performance by optimizing these technologies. For example, satellite networks utilize the broadband characteristics of the millimeter-wave band and miniaturized antennas to provide efficient multicast services with wide coverage. Terrestrial networks, by introducing cell-free multiple-input multiple-output (CFmMIMO) technology, effectively eliminate inter-cell interference, improving spectral efficiency and communication capacity. Furthermore, combined with Time Division Duplex (TDD) mode, terrestrial networks can acquire Channel State Information (CSI) through uplink training and perform precoding optimization during downlink transmission. However, existing methods often assume that the system has ideal CSI, an assumption that is difficult to hold in real-world environments. Furthermore, the joint optimization of resource allocation and beamforming strategies faces challenges such as high computational complexity and spectrum sharing conflicts, hindering further improvements in ISGN performance. To address these issues, this study innovatively proposes a novel downlink transmission optimization framework. By combining CFmMIMO technology with the characteristics of the millimeter-wave band, a joint beamforming and resource allocation algorithm is designed, significantly improving the system's spectral efficiency and quality of service without relying on ideal CSI. The proposed method further optimizes satellite-to-ground link cooperative communication, reduces computational complexity, and effectively resolves the spectrum sharing conflict problem, providing a new solution for the practical application of ISGN systems in 6G communication. Summary of the Invention

[0003] The present invention addresses the following technical shortcomings of existing CFmMIMO-based ISGN systems: (i) the Channel State Information (CSI) of terrestrial communication systems cannot be ideally obtained, limiting optimization design; and (ii) existing optimization methods do not fully consider the fairness of joint beamforming and resource allocation. The main objective of this invention is to provide an optimization method for cellless planetary-terrestrial integrated networks. This method utilizes non-ideal channel state information to optimize cellless planetary-terrestrial integrated networks, achieving fairness in joint beamforming and resource allocation, improving the maximum sum rate (MSR) for users, and offering high optimization efficiency.

[0004] The objective of this invention is achieved through the following technical solution.

[0005] This invention discloses an optimization method for a ground-to-satellite integrated cellless network. The optimization objectives are maximum sum rate (MSR) and maximum minimum fairness (MMF). Constraints include Quality of Service (QoS) for ground stations and users, signal transmission limitations between satellites and ground stations, base station power constraints, and satellite power constraints. This constructs a ground-to-satellite integrated cellless network optimization problem. By using maximum sum rate (MSR) as the optimization objective, the spectral efficiency (SE) can be maximized while meeting the TU / ES QoS requirements of each terminal user. Beamforming is designed using maximum minimum fairness (MMF) to ensure fairness in the ground-to-satellite cellless network while maximizing the total rate of the satellite downlink multicast network. Furthermore, constant modulus constraint (CR1) ensures the stability of beamforming, and fairness constraint (CR2) ensures service fairness among users. This invention utilizes fractional programming (FP) to transform the complex summation logarithmic objective function into a polynomial form and decomposes the optimization problem into two simpler subproblems through an alternating optimization (AO) strategy. The continuous convex approximation SCA method is used to transform non-convex constraints into convex schemes, and the difficult-to-solve lower planetary-ground integrated cellless network optimization problem is transformed into a quadratic constrained quadratic programming QCQP problem. An ADMM-based algorithm is then used to optimize the lower planetary-ground integrated cellless network optimization problem, thereby significantly reducing computational complexity.

[0006] This invention discloses an optimization method for a planetary-ground integrated cellless network, comprising the following steps:

[0007] Step 1: Construct a next-generation integrated planetary-to-ground cellless network consisting of satellite and terrestrial cellless MIMO networks. The satellites use multicast communication to provide services to a group of ground stations (ES), while the terrestrial network collaboratively provides services to multiple ground user units (TU), ensuring sufficient bandwidth and signal quality for each user. The terrestrial cellless MIMO network operates in the millimeter-wave band.

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

[0009]

[0010] Where: the signal is represented by x m This indicates that ω represents the beamforming data that meets the transmit power constraint requirements. M represents the number of ground-based cell-free base stations, and h... mk f represents the channel from the m-th base station to the k-th ground user. k w represents the satellite channel to the k-th ground user. s s represents the precoding vector of the satellite downlink. s It is a multicast signal transmitted by a satellite, z k s represents the additive Gaussian noise at the k-th ground user location; k ,s j It is an independent and identically distributed complex Gaussian random variable, represented as the received data of the k-th / j-th ground user; w mk ,w mj In a terrestrial cellless network, v represents the downlink precoding vector of the k-th / j-th terrestrial user during the downlink data transmission phase. s It is the downlink beamforming vector applied to satellites.

[0011] The signals received at the l ground stations ES are

[0012]

[0013] Where: h ml f represents the channel from the m-th base station to the l-th ground station. l This represents the channel between the satellite and the l-th ground station, z. l This is the noise received by the user.

[0014] The channel estimation results of the m-th base station for the k-th ground user and for the l-th ground station are as follows:

[0015]

[0016] Where: K represents the number of users without a ground cell, L is the number of ground stations, and τ p Let L be the number of pilot sequences, and let L ≤ K ≤ τ. p ≤L+K; It is the pilot vector τ received by the m-th base station. p The received signal representation when ×1. For the set of ground users who reuse the same pilot signal as the k-th ground user, ρ TU ρ represents the signal-to-noise ratio of pilot transmission for ground users during the uplink pilot estimation phase. sat Z represents the pilot transmission signal-to-noise ratio of the ground station during the uplink pilot estimation phase. mRepresented as a normalized additive Gaussian noise matrix, This serves as the pilot sequence allocated to the k-th ground user and the l-th ground station.

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

[0018]

[0019] in, This represents the noise energy of the k-th ground user during the downlink data transmission phase. This represents the noise energy of the l-th ground station during the downlink data transmission phase.

[0020] Step 2: The optimization objective is determined by choosing between Maximum Sum Rate (MSR) and Maximum Minimum Fairness (MMF). Constraints include Quality of Service (QoS) for ground stations and ground users, signal transmission limitations between satellites and ground stations, base station power constraints, and satellite power constraints, thus constructing a planetary-ground integrated cellless network optimization problem. By adopting Maximum Sum Rate (MSR) as the optimization objective, the spectral efficiency (SE) can be maximized while meeting the TU / ES QoS requirements of each terminal user. Beamforming is designed using Maximum Minimum Fairness (MMF) to ensure fairness in the terrestrial cellless network while maximizing the total rate of the satellite downlink multicast network. Furthermore, constant modulus constraint (CR1) ensures the stability of beamforming, and fairness constraint (CR2) ensures service fairness among users. Fractional programming (FP) transforms the complex summation logarithmic objective function into a polynomial form, and the optimization problem is decomposed into two subproblems using an alternating optimization strategy (AO).

[0021] The optimization problem of planetary-ground integrated cellless networks includes two subproblems: the optimization problem of planetary-ground integrated cellless networks with the maximum sum rate (MSR) as the optimization objective and the optimization problem of maximum minimum fairness (MMF) with the optimization objective.

[0022] The optimization problem of a planetary-ground integrated cellless network with the maximum sum rate (MSR) as the optimization objective is shown in the following equation.

[0023]

[0024] in: This indicates the minimum signal-to-noise ratio requirement for the ground station. P represents the minimum signal-to-noise ratio requirement for ground users. m P represents the maximum power constraint for base stations in a terrestrial uncelled network. satThis indicates the maximum power constraint carrier frequency for the satellite.

[0025] The optimization problem of a planetary-ground integrated cellless network with the optimization objective of maximum minimum fairness (MMF) is as follows:

[0026]

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

[0028] Step 3: For the planetary-ground integrated cellless network optimization problem with the maximum sum rate (MSR) as the optimization objective, the complex logarithmic summation objective function is transformed into a polynomial form using fractional programming (FP). The optimization problem is then decomposed into two simpler subproblems using an alternating optimization (AO) strategy. The continuous convex approximation (SCA) method is employed to transform the non-convex constraints into a convex scheme, converting the difficult-to-solve planetary-ground integrated cellless network optimization problem into a quadratic constraint quadratic programming (QCQP) problem. An ADMM-based algorithm is then used to optimize the planetary-ground integrated cellless network optimization problem, yielding the optimization results. Based on these results, the maximum sum rate (MSR) of users is improved.

[0029] Step 3.1: Input SAT-ES and SAT-UT channels f l and f k ; BS-TU, BS-ES channels mk and h ml The maximum power P of BS and SAT m and P sat Penalty variables and errors; finding the feasible point using the following formula. and

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

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

[0032] To express the BS power constraint as a quadratic constrained SDP problem, new variables are introduced. in The optimal solution to the SDP problem is obtained by solving this optimization problem using convex optimization tools. Then, SVD decomposition and randomization methods are used to generate results that satisfy the constraints. 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 χ l and ψ k .

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

[0036] Step 3.5: Optimize w mk .

[0037] In a given χ l , ψ k q l p k With v s The optimization problem can be rewritten as follows:

[0038]

[0039] in:

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

[0041] Step 3.5.1: Set the SCA iteration step 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, set the ADMM iteration step number j←0, Where: r l,i ,u k,i ,e i These are the auxiliary residual variables C1, C2, and C3 from step 3.1.

[0044] Step 3.5.4: Update auxiliary variables Based on the following formula.

[0045]

[0046] Where: y l,i ,x k,i ,z i In step 3.1, C1, C2, and C3 are scaling dual variables, and π is a Lagrange dual variable. These are found through exhaustive search using the bisection method based on 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: Through iterations from Steps 3.5.1 to 3.5.9, obtain the variable w with low computational complexity. mk optimal solution

[0055] Step 3.6: Optimize v s .

[0056] In a given χ l , ψ k q l p k With w i The optimization problem can be rewritten as follows:

[0057]

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

[0059]

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

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

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

[0063]

[0064] Wherein: Γ k ,Λ l Δ is the scaling dual variable of C1, C2, and 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 iterations from steps 3.6.2 to 3.6.5, the number of optimization variables is reduced. The complexity of the optimal solution.

[0071] Step 3.7: Optimize the precoding vector based on Steps 3.5 and 3.6 By satisfying the overall constraints, the optimized maximum sum rate can be obtained.

[0072] Step 4: For the planetary-ground integrated cellless network optimization problem with the maximum-minimum fairness (MMF) as the optimization objective, the complex summation logarithmic objective function is transformed into a polynomial form using fractional programming (FP). The optimization problem is then decomposed into two simpler subproblems using an alternating optimization (AO) strategy. The continuous convex approximation (SCA) method is employed to transform the non-convex constraints into a convex scheme, converting the difficult-to-solve planetary-ground integrated cellless network optimization problem into a quadratic constraint quadratic programming (QCQP) problem. An ADMM-based algorithm is then used to optimize the planetary-ground integrated cellless network optimization problem, yielding the optimization results. Based on these results, fairness in joint beamforming and resource allocation is achieved.

[0073] Step 4.1: Input SAT-TU channel f i and f c ; BS-TU, BS-ES channel b mk and h ml The maximum power P of BS and SAT m and P sat Since the constraints of the two subproblems are the same, the feasible point found in step 3.1 also applies to step 4.1, thus finding a feasible point. 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 χ 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, since this problem is 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. The specific implementation steps are as follows:

[0079] Step 4.4.1: Set the SCA iteration step 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, set the ADMM iteration step number j←0, α 0 ←α (t) , Where: r l,i ,u k,i ,e i These are the auxiliary residual variables C1, C2, and C3 from step 3.1, while β... k It is an auxiliary residual variable of variable α.

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

[0083]

[0084] Where: y l,i ,x k,i ,z i It refers to the scaling dual variables C1, C2, and C3 in step 3.1, α. k It is the scaling dual variable of variable α, where π is the Lagrange dual variable, which is found exhaustively using the bisection method through 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: Through iterations from Step 4.4.1 to Step 4.4.9, reduce the number of variables to be solved for the optimal solution. 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 similar in form to the problem in step 3.6, differing only in the specific auxiliary variables. Therefore, it can be solved directly using the method from step 3.6. The specific implementation steps for step 4.5 are as follows:

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

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

[0098]

[0099]

[0100] Wherein: Γ k ,Λ l Δ is the scaling dual variable of C1, C2, and 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, through the iterations from step 4.5.2 to step 4.5.5, the complexity of finding the optimal solution for the optimization variable v is reduced.

[0107] Step 4.6: Based on the optimal precoding vector from Steps 4.4 and 4.5 and The maximum total rate of the optimized satellite-to-ground link, satisfying the constraints, is obtained. And the maximum minimum achievable rate α for cellless networks based on the MMF fairness strategy. opt .

[0108] Beneficial effects:

[0109] 1. This invention discloses an optimization method for a cellless ISGN system based on a planetary-ground integrated network. The optimization objectives are maximum sum rate (MSR) and maximum minimum fairness (MMF). Constraints include Quality of Service (QoS) for ground stations and ground users, signal transmission limitations between satellites and ground stations, base station power constraints, and satellite power constraints. This constructs a cellless ISGN optimization problem. Non-ideal channel state information is used to optimize the performance of the cellless ISGN system. This method ensures efficient downlink transmission in the cellless multiple-input multiple-output (CFmMIMO) based ISGN system, while simultaneously meeting the fairness requirements of joint beamforming and resource allocation, effectively improving the maximum sum rate (MSR) for users.

[0110] 2. This invention discloses an optimization method for a terrestrial integrated cellless network. By adopting the maximum sum rate (MSR) as the optimization objective, it maximizes the spectrum efficiency (SE) while meeting the TU / ES service quality requirements of each terminal user. Through beamforming design using maximum minimum fairness (MMF), it ensures the fairness of the terrestrial cellless network while maximizing the total rate of the satellite downlink multicast network. Furthermore, it introduces a constant mode constraint (CR1) to ensure the stability of beamforming and a fairness constraint (CR2) to ensure service fairness among users. This method maintains network service quality while improving spectrum utilization efficiency, and through fairness design, ensures that all users receive a reasonable service share, thereby achieving overall network fairness and improving user satisfaction.

[0111] 3. This invention discloses an optimization method for a planetary-ground integrated cellless network. By utilizing fractional programming (FP), the complex summation logarithmic objective function is transformed into a polynomial form. Furthermore, an alternating optimization (AO) strategy decomposes the optimization problem into two simpler subproblems. The continuous convex approximation (SCA) method is employed to transform non-convex constraints into convex schemes, converting the difficult-to-solve planetary-ground integrated cellless network optimization problem into a quadratic constraint quadratic programming (QCQP) problem. An ADMM-based algorithm is then used to optimize the planetary-ground integrated cellless network optimization problem separately. This method significantly reduces computational complexity, making the optimization problem of planetary-ground integrated cellless networks easier to solve, improving the algorithm's practicality and efficiency, and providing an efficient and robust solution for network design and resource management. Attached Figure Description

[0112] Figure 1 This is a block diagram of a satellite-terrestrial integrated network (ISGN) system structure based on cell-free multiple-input multiple-output (CFmMIMO) as described in this invention;

[0113] Figure 2 This is a topology diagram of an ISGN system based on CFmMIMO as described in this invention;

[0114] Figure 3 This is a comparison graph of the SE performance of the method described in this invention as a function of the maximum power constraint of BSs and the performance curves of traditional methods.

[0115] Figure 4 This is a comparison chart of the SE performance of the method described in this invention as a function of satellite maximum power constraint and the performance curves of traditional methods;

[0116] Figure 5 A flowchart of an optimization method for a planetary-ground integrated cellless network according to the present invention. Detailed Implementation

[0117] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0118] A system topology for an ISGN system based on CFmMIMO was established, in which BSs, TUs, and ESs are randomly distributed within an area of ​​1 square kilometer. A total of 5 BSSs jointly serve 4 TUs, while the 3 ESs within this area receive satellite multicast signals. Furthermore, the GEO altitude is 35,786 km, and the altitudes of the 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 planetary-ground integrated cell-free network, and its optimization effect is as follows: Figure 3, 4 As shown, under different base station maximum power or satellite power limitations, the performance of the invented algorithm is always very close to the optimal algorithm, especially in P. m / P sat At higher power levels, the two almost overlap. Furthermore, compared to the "Consensus ADMM" and "SDR-GAN" algorithms, the invented algorithm provides a higher summing rate across the entire power range. Figure 5 As shown in the figure, the optimization method for a planetary-ground integrated cellless network disclosed in this embodiment has the following specific implementation steps:

[0120] Step 1: Construct a next-generation integrated planetary-to-ground cellless network consisting of satellite and terrestrial cellless MIMO networks. The satellites use multicast communication to provide services to a group of ground stations (ES), while the terrestrial network collaboratively provides services to multiple ground user units (TU), ensuring sufficient bandwidth and signal quality for each user. The terrestrial cellless MIMO network operates in the millimeter-wave band.

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

[0122]

[0123] Where: the signal is represented by x m This indicates that ω represents the beamforming data that meets the transmit power constraint requirements. M represents the number of ground-based cell-free base stations, and h... mk f represents the channel from the m-th base station to the k-th ground user. k w represents the satellite channel to the k-th ground user. s s represents the precoding vector of the satellite downlink. s It is a multicast signal transmitted by a satellite, z k s represents the additive Gaussian noise at the k-th ground user location; k ,s j It is an independent and identically distributed complex Gaussian random variable, represented as the received data of the k-th / j-th ground user; w mk ,w mj In a terrestrial cellless network, v represents the downlink precoding vector of the k-th / j-th terrestrial user during the downlink data transmission phase. s It is the downlink beamforming vector applied to satellites. N s This indicates the number of antennas on a GEO satellite (a single antenna corresponds to a single-beam operating mode). This represents the noise energy of the m-th ground base station during the uplink pilot estimation phase, with a GHz frequency of 20 GHz, a data bandwidth of 20 MHz, and N satellite beams. s =7, maximum multibeam gain b from satellite to ground station max=52dB, 3dB angle θ of satellite beam amplitude 3dB =0.4°, antenna gain G of ground base station a =25dB, link noise figure NF=9dB, rain attenuation parameters of satellite millimeter-wave link μ=-3.125, σ=1.591, shadow attenuation parameters of direct path in ground millimeter-wave link ε=1.9, ζ=1.1, shadow attenuation parameters of indirect path in ground millimeter-wave link ε=3.4, ζ=9.1

[0124] The signals received at the l ground stations ES are

[0125]

[0126] Where: h ml f represents the channel from the m-th base station to the l-th ground station. l This represents the channel between the satellite and the l-th ground station, z. l This is the noise received by the user.

[0127] The channel estimation results of the m-th base station for the k-th ground user and for the l-th ground station are as follows:

[0128]

[0129] Where: K represents the number of users without a ground cell, L is the number of ground stations, and τ p Let L be the number of pilot sequences, and let L ≤ K ≤ τ. p ≤L+K;Y m P It is the pilot vector τ received by the m-th base station. p The received signal representation when ×1. For the set of ground users who reuse the same pilot signal as the k-th ground user, ρ TU ρ represents the signal-to-noise ratio of pilot transmission for ground users during the uplink pilot estimation phase. sat Z represents the pilot transmission signal-to-noise ratio of the ground station during the uplink pilot estimation phase. m Represented as a normalized additive Gaussian noise matrix, This serves as the pilot sequence allocated to the k-th ground user and the l-th ground station.

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

[0131]

[0132] in, This represents the noise energy of the k-th ground user during the downlink data transmission phase. This represents the noise energy of the l-th ground station during the downlink data transmission phase.

[0133] Step 2: The optimization objective is determined by choosing between Maximum Sum Rate (MSR) and Maximum Minimum Fairness (MMF). Constraints include Quality of Service (QoS) for ground stations and ground users, signal transmission limitations between satellites and ground stations, base station power constraints, and satellite power constraints, thus constructing a planetary-ground integrated cellless network optimization problem. By adopting Maximum Sum Rate (MSR) as the optimization objective, the spectral efficiency (SE) can be maximized while meeting the TU / ES QoS requirements of each terminal user. Beamforming is designed using Maximum Minimum Fairness (MMF) to ensure fairness in the terrestrial cellless network while maximizing the total rate of the satellite downlink multicast network. Furthermore, constant modulus constraint (CR1) ensures the stability of beamforming, and fairness constraint (CR2) ensures service fairness among users. Fractional programming (FP) transforms the complex summation logarithmic objective function into a polynomial form, and the optimization problem is decomposed into two subproblems using an alternating optimization strategy (AO).

[0134] The optimization problem of planetary-ground integrated cellless networks includes two subproblems: the optimization problem of planetary-ground integrated cellless networks with the maximum sum rate (MSR) as the optimization objective and the optimization problem of maximum minimum fairness (MMF) with the optimization objective.

[0135] The optimization problem of a planetary-ground integrated cellless network with the maximum sum rate (MSR) as the optimization objective is shown in the following equation.

[0136]

[0137] in: This indicates the minimum signal-to-noise ratio requirement for the ground station. P represents the minimum signal-to-noise ratio requirement for ground users. m P represents the maximum power constraint for base stations in a terrestrial uncelled network. sat This indicates the maximum power constraint carrier frequency for the satellite.

[0138] The optimization problem of a planetary-ground integrated cellless network with the optimization objective of maximum minimum fairness (MMF) is as follows:

[0139]

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

[0141] Step 3: For the planetary-ground integrated cellless network optimization problem with the maximum sum rate (MSR) as the optimization objective, the complex logarithmic summation objective function is transformed into a polynomial form using fractional programming (FP). The optimization problem is then decomposed into two simpler subproblems using an alternating optimization (AO) strategy. The continuous convex approximation (SCA) method is employed to transform the non-convex constraints into a convex scheme, converting the difficult-to-solve planetary-ground integrated cellless network optimization problem into a quadratic constraint quadratic programming (QCQP) problem. An ADMM-based algorithm is then used to optimize the planetary-ground integrated cellless network optimization problem, yielding the optimization results. Based on these results, the maximum sum rate (MSR) of users is improved.

[0142] Step 3.1: Input SAT-ES and SAT-UT channels f l and f k ; BS-TU, BS-ES channels mk and h ml The maximum power P of BS and SAT m and P sat Penalty variables and errors; finding the feasible point using the following formula. and

[0143]

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

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

[0146] To express the BS power constraint as a quadratic constrained SDP problem, new variables are introduced. in The optimal solution to the SDP problem is obtained by solving this optimization problem using convex optimization tools. Then, SVD decomposition and randomization methods are used to generate results that satisfy the constraints. 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 χ l and ψ k.

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

[0150] Step 3.5: Optimize w mk .

[0151] In a given χ l , ψ k q l p k With v s The optimization problem can be rewritten as follows:

[0152]

[0153] in:

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

[0155] Step 3.5.1: Set the SCA iteration step 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, set the ADMM iteration step number j←0, Where: r l,i ,u k,i ,e i These are the auxiliary residual variables C1, C2, and C3 from step 3.1, while β... k It is an auxiliary residual variable of variable α.

[0158] Step 3.5.4: Update auxiliary variables Based on the following formula.

[0159]

[0160] Where: y l,i ,x k,i ,z i In step 3.1, C1, C2, and C3 are scaling dual variables, and π is a Lagrange dual variable. These are found through exhaustive search using the bisection method based on 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: Through iterations from Step 3.5.1 to Step 3.5.9, reduce the optimization variable w. mk The complexity of the optimal solution.

[0169] Step 3.6: Optimize v s .

[0170] In a given χ l , ψ k q l p k With w i The optimization problem can be rewritten as follows:

[0171]

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

[0173]

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

[0175] Step 3.6.1: Initialize the ADMM algorithm, set the ADMM iteration step number j←0, and initialize variables. Initialize residual variables κ 0 ←0. Wherein: ι k κ represents the auxiliary residual variables C1, C2, and C4 from step 3.1.

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

[0177]

[0178] Wherein: Γ k ,Λ l Δ is the scaling dual variable of C1, C2, and 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... Through iterations from steps 3.6.2 to 3.6.5, the solution for the optimization variable v is reduced. s The complexity of the optimal solution.

[0185] Step 3.7: Optimize the precoding vector based on Steps 3.5 and 3.6 By satisfying the overall constraints, the optimized maximum sum rate can be obtained.

[0186] Step 4: For the planetary-ground integrated cellless network optimization problem with the maximum-minimum fairness (MMF) as the optimization objective, the complex summation logarithmic objective function is transformed into a polynomial form using fractional programming (FP). The optimization problem is then decomposed into two simpler subproblems using an alternating optimization (AO) strategy. The continuous convex approximation (SCA) method is employed to transform the non-convex constraints into a convex scheme, converting the difficult-to-solve planetary-ground integrated cellless network optimization problem into a quadratic constraint quadratic programming (QCQP) problem. An ADMM-based algorithm is then used to optimize the planetary-ground integrated cellless network optimization problem, yielding the optimization results. Based on these results, fairness in joint beamforming and resource allocation is achieved.

[0187] Step 4.1: Input SAT-TU channel f i and f c ; BS-TU, BS-ES channel b mk and h ml The maximum power P of BS and SAT m and P satSince the constraints of the two subproblems are the same, the feasible point found in step 3.1 also applies to step 4.1, thus finding a feasible point. 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 χ 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, since this problem is 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. The specific implementation steps are as follows:

[0193] Step 4.4.1: Set the SCA iteration step 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, set the ADMM iteration step number j←0, α 0 ←α (t) , Where: r l,i ,u k,i ,,e i These are the auxiliary residual variables C1, C2, and C3 from step 3.1, while β... k It is the auxiliary residual variable of variable α

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

[0197]

[0198] Where: y l,i ,x k,i ,z i It refers to scaling the dual variables C1, C2, and C3 in step 3.1, ak It is the scaling dual variable of variable α, where π is the Lagrange dual variable, which is found exhaustively using the bisection method through 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: Through iterations from Step 4.4.1 to Step 4.4.9, reduce the optimal variables. α opt The solution complexity is...

[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 similar in form to the problem in step 3.6, differing only in the specific auxiliary variables. Therefore, I solved this problem by directly using the method from step 3.6. The specific implementation steps for step 4.5 are as follows:

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

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

[0212]

[0213]

[0214] Wherein: Γ k ,Λ l Δ is the scaling dual variable of C1, C2, and 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, through the iterations from steps 4.5.2 to 4.5.5, the number of optimization variables to be solved is reduced. The computational complexity is...

[0221] Step 4.6: Based on the optimal precoding vector from Steps 4.4 and 4.5 and By satisfying the overall constraints, the optimized maximum total rate of the satellite-to-ground station link is obtained. And the maximum minimum achievable rate α for cellless networks based on the MMF fairness strategy. opt .

[0222] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description 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 within the scope of protection of the present invention.

Claims

1. An optimization method for a planetary-ground integrated cellless network, characterized in that: Includes the following steps, Step 1: Construct a next-generation integrated planetary-to-ground cellless network consisting of satellite and terrestrial non-cellular MIMO networks; the satellite uses multicast communication to provide services to a group of ground stations (ES), while the terrestrial network coordinates to provide services to multiple ground user units (TU), ensuring that each user receives sufficient bandwidth and signal quality; the terrestrial non-cellular MIMO network operates in the millimeter-wave band; Step 2: The optimization objective is determined by choosing between Maximum Sum Rate (MSR) and Maximum Minimum Fairness (MMF). Constraints include Quality of Service (QoS) for ground stations and ground users, signal transmission limitations between satellites and ground stations, base station power constraints, and satellite power constraints. This constructs a planetary-ground integrated cellless network optimization problem. By adopting Maximum Sum Rate (MSR) as the optimization objective, the spectral efficiency (SE) can be maximized while meeting the TU / ES QoS requirements of each terminal user. By using Maximum Minimum Fairness (MMF) to design beamforming, the fairness of the terrestrial cellless network is guaranteed, while maximizing the total rate of the satellite downlink multicast network. In addition, the stability of beamforming is guaranteed by introducing constant modulus constraint (CR1), and the service fairness among users is guaranteed by fairness constraint (CR2). Fractional programming (FP) is used to transform the complex summation logarithmic objective function into a polynomial form, and the optimization problem is decomposed into two subproblems by using an alternating optimization strategy (AO). Step 3: For the optimization problem of planetary-ground integrated cellless network with maximum sum rate (MSR) as the optimization objective, the complex summation logarithm objective function is transformed into a polynomial form by using fractional programming (FP), and the optimization problem is decomposed into two simpler subproblems by using the alternating optimization (AO) strategy. The continuous convex approximation SCA method is used to transform non-convex constraints into convex schemes, and the difficult-to-solve lower planetary-ground integrated cellless network optimization problem is transformed into a quadratic constraint quadratic programming QCQP problem. An ADMM-based algorithm is used to optimize the lower planetary-ground integrated cellless network optimization problem, and the optimization results of the lower planetary-ground integrated cellless network are obtained. Based on the optimization results of the lower planetary-ground integrated cellless network, the maximum sum rate (MSR) of users is improved. Step 4: For the optimization problem of the planetary-ground integrated cellless network with the optimization objective of maximum minimum fairness (MMF), the complex summation logarithm objective function is transformed into a polynomial form by using fractional programming (FP), and the optimization problem is decomposed into two simpler subproblems by using the alternating optimization (AO) strategy. The continuous convex approximation SCA method is used to transform non-convex constraints into convex schemes, and the difficult-to-solve lower planetary-ground integrated cellless network optimization problem is transformed into a quadratic constrained quadratic programming (QCQP) problem. An ADMM-based algorithm is used to optimize the lower planetary-ground integrated cellless network optimization problem, and the optimization results are obtained. Based on the optimization results, the fairness of joint beamforming and resource allocation is achieved.

2. The optimization method for a planetary-ground integrated cellless network as described in claim 1, characterized in that: In step one, The signals received by the k-th ground user TU are respectively Where: the signal is represented by x m Indicates: M represents the number of ground-based cell-free base stations, h mk f represents the channel from the m-th base station to the k-th ground user. k w represents the satellite channel to the k-th ground user. s s represents the precoding vector of the satellite downlink. s It is a multicast signal transmitted by a satellite, z k s represents the additive Gaussian noise at the k-th ground user location; k ,s j It is an independent and identically distributed complex Gaussian random variable, represented as the received data of the k-th / j-th ground user; w mk ,w mj In a terrestrial cellless network, v represents the downlink precoding vector of the k-th / j-th terrestrial user during the downlink data transmission phase. s It is the downlink beamforming vector applied to satellites; The signal received at the l-th ground station ES is Where: h ml f represents the channel from the m-th base station to the l-th ground station. l This represents the channel between the satellite and the l-th ground station, z. l This is the noise received by the user; The channel estimation results of the m-th base station for the k-th ground user and for the l-th ground station are as follows: Where: τ p Let L be the number of pilot sequences, and let L ≤ K ≤ τ. p ≤L+K, where K represents the number of users without ground cells and L is the number of ground stations; It is the pilot vector τ received by the m-th base station. p The received signal representation when ×1; For the set of ground users who reuse the same pilot signal as the k-th ground user, ρ TU ρ represents the signal-to-noise ratio of pilot transmission for ground users during the uplink pilot estimation phase. sat Z represents the pilot transmission signal-to-noise ratio of the ground station during the uplink pilot estimation phase. m Represented as a normalized additive Gaussian noise matrix, As the pilot sequence allocated to the k-th ground user and the l-th ground station; The signal-to-interference-plus-noise ratio (SINR) γ obtained at the central processing unit (CPU) and gateway (GW) for the k-th ground user and the l-th ground station. k γ l Represented as: in, This represents the noise energy of the k-th ground user during the downlink data transmission phase. This represents the noise energy of the l-th ground station during the downlink data transmission phase.

3. The optimization method for a planetary-ground integrated cellless network as described in claim 2, characterized in that: The second step is implemented as follows: The optimization problem of planetary-ground integrated cellless network includes two sub-problems: the optimization problem of planetary-ground integrated cellless network with the maximum sum rate (MSR) as the optimization objective and the optimization problem of maximum minimum fairness (MMF) of planetary-ground integrated cellless network with the optimization objective. The optimization problem of a planetary-ground integrated cellless network with the maximum sum rate (MSR) as the optimization objective is shown in the following equation. in: This indicates the minimum signal-to-noise ratio requirement for the ground station. P represents the minimum signal-to-noise ratio requirement for ground users. m P represents the maximum power constraint for base stations in a terrestrial uncelled network. sat Indicates the maximum power constraint carrier frequency of the satellite; The optimization problem of a planetary-ground integrated cellless network with the optimization objective of maximum-minimum fairness (MMF) is as follows: Among them, constraints C1 and C2 represent the QoS requirements of the ground station and ground users, respectively, while C3 and C4 are the actual power constraints of the base station and satellite in downlink transmission.

4. The optimization method for a planetary-ground integrated cellless network as described in claim 3, characterized in that: The method for implementing step three is as follows: Step 3.1: Input SAT-ES and SAT-UT channels f l and f k ; BS-TU, BS-ES channels mk and h ml The maximum power P of BS and SAT m and P sat The feasible point is found by the following formula. and Where: N t This represents the number of antennas without a cell base station on the ground, and the channel set is represented as... The set of precoding vectors is represented as and Represents the covariance of the channel; To express the problem as an SDP problem, define To express the BS power constraint as a quadratic constrained SDP problem, new variables are introduced. in The optimal solution to the SDP problem is obtained by solving this optimization problem using convex optimization tools. Then, SVD decomposition and randomization methods are used to generate results that satisfy the constraints. and Step 3.2: If If there is no convergence, repeat steps 3.3 to 3.6; Step 3.3: According to Update χ l and ψ k ; Step 3.4: According to Update q l and p k ; Step 3.5: Optimize w mk ; In a given χ l , ψ k q l p k With v s The optimization problem can be rewritten as follows: in: The SCA algorithm is used to transform the non-convex problem into a convex optimization problem, and then the ADMM algorithm is used to find the optimal solution to this convex problem. The specific iterative method after using the ADMM algorithm is as follows: Step 3.5.1: Set the SCA iteration step 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, set the ADMM iteration step number j←0, Where: r l,i ,u k,i ,e i These are the auxiliary residual variables C1, C2, and C3 from step 3.

1. Step 3.5.4: Update auxiliary variables Based on the following formula; Where: y l,i ,x k,i ,z i In step 3.1, C1, C2, and C3 are scaling dual variables, and π is a Lagrange dual variable. These are found through exhaustive search using the bisection method based on the 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: Through iterations from Steps 3.5.1 to 3.5.9, obtain the variable w with low computational complexity. mk optimal solution Step 3.6: Optimize v s ; In a given χ l , ψ k q l p k With w i The optimization problem can be rewritten as follows: To simplify the writing of expressions, auxiliary variables are introduced: The optimal solution to this problem is found using the ADMM algorithm, with the specific iterative method as follows: Step 3.6.1: Initialize the ADMM algorithm, set the ADMM iteration step number j←0, and initialize variables. Initialize residual variables κ 0 ←0; where: ι k κ is the auxiliary residual variable C1, C2, C4 in step 3.1; Step 3.6.2: Update the auxiliary variable according to the following formula. Δ j+1 ; Wherein: Γ k ,Λ l Δ is the scaling dual variable of C1, C2, and 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. κ j+1 ; Step 3.6.5: j←j+1; Step 3.6.6: If the convergence criterion of the ADMM algorithm is not met... Through iterations from steps 3.6.2 to 3.6.5, the number of optimization variables is reduced. The complexity of the optimal solution; Step 3.7: Optimize the precoding vector based on Steps 3.5 and 3.6 By satisfying the overall constraints, the optimized maximum sum rate can be obtained.

5. The optimization method for a planetary-ground integrated cellless network as described in claim 4, characterized in that: Step four is implemented as follows: Step 4.1: Input SAT-TU channel f i and f c ; BS-TU, BS-ES channel b mk and h ml The maximum power P of BS and SAT m and P sat Since the constraints of the two subproblems are the same, the feasible point found in step 3.1 also applies to step 4.1, thus finding a feasible point. and Step 4.2: If If there is no convergence, repeat steps 4.3 to 4.5; Step 4.3: According to Update χ 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, since this problem is 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. The specific implementation steps are as follows: Step 4.4.1: Set the SCA iteration step 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, set the ADMM iteration step number j←0, α 0 ←α (t) , Where: r l,i ,u k,i ,e i These are the auxiliary residual variables C1, C2, and C3 from step 3.1, while β... k It is an auxiliary residual variable of variable α; Step 4.4.4: Update the auxiliary variable according to the following formula. Where: y l,i ,x k,i ,z i It refers to the scaling dual variables C1, C2, and C3 in step 3.1, α. k It is the scaling dual variable of variable α, where π is the Lag dual variable, which is found exhaustively using the bisection method through 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: Through iterations from Step 4.4.1 to Step 4.4.9, reduce the number of variables to be solved for the optimal solution. 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 similar in form to the problem in step 3.6, differing only in the specific auxiliary variables. Therefore, it can be solved directly using the method in step 3.

6. The specific implementation steps for step 4.5 are as follows: Step 4.5.1: Initialize the ADMM algorithm, set the ADMM iteration step number j←0, and initialize variables. Initialize residual variables κ 0 ←0; where: ι k κ is the auxiliary residual variable C1, C2, C4 in step 3.1; Step 4.5.2: Update the auxiliary variable according to the following formula. Δ j+1 ; Wherein: Γ k ,Λ l Δ is the scaling dual variable of C1, C2, and 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. κ j+1 ; Step 4.5.5: j←j+1; Step 4.5.6: If the convergence criterion of the ADMM algorithm is not met... Then, through the iterations from step 4.5.2 to step 4.5.5, the complexity of finding the optimal solution for the optimization variable v is reduced; Step 4.6: Based on the optimal precoding vector from Steps 4.4 and 4.5 and The maximum total rate of the optimized satellite-to-ground link, satisfying the constraints, is obtained. And the maximum minimum achievable rate α for cellless networks based on the MMF fairness strategy. opt .

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