A graph neural network-based downlink precoding design method in a cell-free network

Abstract

The application provides a kind of based on graph neural network's no-cell network downlink precoding design method, comprising: step 1, the total rate optimization problem of user terminal equipment in no-cell network is established;Step 2, the optimization problem in step 1 is reconverted;Step 3, the sub-problem is solved by the successive convex approximation SCA method;Step 4, the converted problem obtained in step 2 is processed using the block coordinate descent BCD method;Step 5, a learning framework based on graph neural network for wireless communication is established;Step 6, the antenna connection relationship between base station and user is predicted using a supervised learning method based on graph neural network;Step 7, the precoding matrix is designed according to the existing antenna connection relationship between base station and user.The advanced precoding algorithm, graph neural network algorithm is used, and the capacity constraint is considered, and the scheme provided by the application can provide a practical solution for realizing high-performance no-cell large-scale MIMO network.

Description

Technical Field

[0001] This invention relates to a downlink precoding design method for cellular-free networks based on graph neural networks. Background Technology

[0002] Typically, the performance of cellular networks is limited by inter-cell interference. In particular, users near cell boundaries experience severe interference (relative to their required signal power). Multi-Input Multi-Output (MIMO) wireless communication systems (as well as distributed MIMO, coordinated multipoint transmission, and distributed antenna systems) can reduce this inter-cell interference through cooperation between base stations. In these networks, base stations cooperate via fronthaul and backhaul links, jointly transmitting signals on the downlink and jointly detecting signals on the uplink. However, research shows that cooperation between base stations has fundamental limitations; even with full cooperation, spectral efficiency has only a finite upper bound as transmit power approaches infinity. Nevertheless, significantly higher spectral efficiency can still be achieved compared to ignoring interference. The main challenges in implementing distributed MIMO are the need for substantial backhaul overhead, high deployment costs, and a sufficiently powerful central processing unit. These issues need to be considered, especially when precoded signals and channel state information (CSI) need to be shared between base stations. Recently, cellular-free massive MIMO has been introduced as a practical and useful embodiment of the network MIMO concept.

[0003] In cellular massive MIMO, a large number of base stations (or radio access points) equipped with single or multiple antennas are distributed over a large area to provide services to many users coherently. Similar to (cellular) centralized massive MIMO, cellular massive MIMO utilizes favorable propagation and channel hardening characteristics to multiplex many users on the same frequency resources at the same time, resulting in low inter-user interference. Therefore, it can provide significant spectral efficiency and simplified signal processing. More importantly, in cellular massive MIMO configurations, the serving antennas are close to the users, resulting in lower path loss; thus, many users can simultaneously receive a uniformly good quality of service. However, since all base stations (BSs) are connected to the CPU via a front-end link (similar to the back-end in distributed MIMO), this can lead to very heavy front-end link loads. Once there are too many active antennas, the front-end capacity cannot meet communication demands. Many researchers have optimized the back-end link capacity problem in Coordinated Multipoint Transmission (CoMP) networks, and this remains one of the issues that must be considered in cellular networks. It is important to note that many scholars assume infinite forward capacity in beamforming designs using cellular-free models, which may deviate further from reality with the development of large-scale distributions. Therefore, a joint optimization scheme is needed to design the precoding matrix while satisfying the forward link capacity.

[0004] In current research on noncellular networks, many researchers have adopted precoding design schemes such as zero-forcing (ZF) and maximum ratio transmission. While these have extremely low computational complexity, they incur significant performance losses and limit optimization operations on upstream links.

[0005] It is worth noting that various mathematical optimization algorithms currently used in communication scenarios often have high time complexity, which can lead to high computation time for communication scenarios with a large number of antennas or users, and may not be suitable for actual production environments.

[0006] Graph Neural Networks (GNNs) are machine learning models used to process graph-structured data. Unlike traditional neural networks, which primarily focus on data in Euclidean space, GNNs can effectively capture and analyze non-Euclidean data with complex connections. They construct graph data by modeling antennas as nodes and connections between antennas as edges in a communication scenario, and utilize attributes such as location and channel state information as node and edge features for learning and optimization within the GNN model. GNNs can capture spatial relationships, interference, and cooperative effects between antennas, and generate optimal weight configurations to achieve better system performance. Recently, they have demonstrated excellent performance in the field of communications.

[0007] Graph neural network-based learning methods offer excellent computational speed. Although the training time is relatively long, the trained network can quickly output precoding matrices under different channel environments. Furthermore, for network structures with good generalization performance, small changes in the number of base stations (BS) and user units (UEs) do not cause significant performance loss. They are also more closely aligned with production environments in relatively stable communication settings. Summary of the Invention

[0008] Objective of the Invention: The technical problem to be solved by the present invention is to address the shortcomings of existing technologies by providing a downlink precoding design method for cellular-free networks based on graph neural networks, specifically including the following steps:

[0009] Step 1: Establish the overall rate optimization problem for user terminal devices in a non-cellular network;

[0010] Step 2: Establish an iterative strategy based on the fractional programming (FP) algorithm to jointly optimize the total rate and the upstream capacity, and re-transform the optimization problem in Step 1;

[0011] Step 3: Obtain the subproblems of the optimization problem, and solve the subproblems using the Successive Convex Approximation (SCA) method;

[0012] Step 4: Use the Block Coordinate Descent (BCD) method to process the transformed problem obtained in Step 2;

[0013] Step 5: Establish a graph neural network-based learning framework for wireless communication;

[0014] Step 6: Use a supervised learning method based on graph neural networks to predict the antenna connection relationship between the base station and the user;

[0015] Step 7: Use an unsupervised learning method based on graph neural networks to design a precoding matrix based on the existing antenna connection relationship between base stations and users.

[0016] Step 1 includes: setting up a cellular network consisting of B base stations (BS) and K user equipment (UE), wherein the B base stations simultaneously transmit data symbols to the K user equipment via the downlink. The signal is first transmitted from the central processing unit (CPU) to the base station via the frontlink, and then transmitted to the user equipment by the base station antenna; each base station is configured to have M antennas, and each user equipment has one antenna;

[0017] The data symbol s for all user terminal equipment (UE) is represented as: Where E(s·s) H ) = I K The channel from the i-th BS to all user terminal equipment (UE) is represented as follows: The channel from all base stations (BS) to all users is represented as follows:

[0018] The CPU determines the actual precoding vector for each base station, and transmits the data symbol s and the precoding matrix V respectively. i The precoding matrix at the i-th base station is represented as The symbol vector transmitted at the i-th base station is:

[0019] x i =V i s (1)

[0020] The channel from the i-th base station to the k-th user terminal device is denoted as... It is also H i The k-th row, then the signal y received by the k-th UE. k Represented as:

[0021]

[0022] The signal-to-interference-plus-noise ratio (SINR) received by the k-th user terminal device k Represented as:

[0023]

[0024] in, This represents the Gaussian white noise at the k-th user location;

[0025] The overall rate optimization problem for user terminal devices is:

[0026]

[0027] in s .t. represents the set of constraints imposed on the optimization problem, and i represents the indicator function, whose expression is:

[0028] ||·|| F The F-norm of a vector. This represents the maximum transmit power of the i-th BS. This represents the maximum forward link capacity of all BSs.

[0029] In this invention, uppercase bold letters represent matrices, lowercase bold letters represent vectors, and lowercase letters represent scalars. Hi H represents the i-th column of matrix H. i，j Let h represent the element in the i-th column and j-th row of matrix H. i Let represent the i-th element of vector h. E(·) represents the expectation. Let diag(A) represent a complex field of dimension A*B, and let diag(A) represent a vector consisting of the diagonal elements of A. K Represents a K-dimensional identity matrix, with superscript... T , H They represent the transpose and conjugate transpose of a matrix, respectively, with superscripts... * , t These represent the optimal solution and the value at iteration t, respectively. A backslash \ indicates exclusion from the set.

[0030] Step 2 includes: by using SINR k Replace with γ k And by adding equality constraints, the problem can be expressed as:

[0031]

[0032] Use the Lagrange multiplier method to handle the equality constraints in problem (5):

[0033]

[0034] According to Lagrange's theorem, the optimal closed-form solution for λ is:

[0035]

[0036] Substituting the optimal Lagrange multiplier into the Lagrange dual function, we obtain the following formula:

[0037]

[0038] After finding the common denominator of the expression, we obtain the following equation:

[0039]

[0040] The objective function now has the following form:

[0041]

[0042] According to the fractional programming method, the complex fraction in formula (9) is equivalently transformed to obtain the following objective function:

[0043]

[0044] Where y k These are introduced auxiliary variables;

[0045] The problem in step 1 is then rephrased as:

[0046]

[0047] Step 3 includes:

[0048] For the indicator function I in the forward link constraint in formula (12), a smooth indicator function is used for approximation. The smooth indicator function is:

[0049]

[0050] Although the function is already a smooth function, the constraint is still non-convex. To make it feasible in convex optimization, for any given point... Performing a first-order Taylor expansion yields:

[0051]

[0052] The superscript ' indicates the derivative function;

[0053] For fixed γ and y, we obtain subproblem (15):

[0054]

[0055] The SCA method is used to solve the subproblem (15).

[0056] Step 3, which involves using the SCA method to solve subproblem (15), specifically includes the following steps:

[0057] Inputs are: accuracy ε, feasible beamforming matrix V 0 Maximum number of iterations N;

[0058] The output is: the optimal beamforming matrix V * ;

[0059] Step a1, set t=1 and initialize f3(V) 0 );

[0060] Step a2: If t≤N, proceed to step a3; otherwise, proceed to step a7.

[0061] Step a3, using a fixed γ t and y t The problem is solved using a convex optimization algorithm (15) to obtain V. t ;

[0062] Step a4, update f3(V) t );

[0063] Step a5, if |f3(V t )-f3(Vt-1 If |ε| > ε|, then set t = t + 1 and proceed to step 2; otherwise, return to the optimal beamforming matrix V. * =V t ;

[0064] Step a6, End.

[0065] Step 4 includes:

[0066] Step 4-1, optimizing γ with fixed V and y, is a standard convex problem; its first derivative is set to 0.

[0067]

[0068] Then the optimal solution for γ is obtained:

[0069]

[0070] Step 4-2, optimizing y with fixed V and γ, is a standard convex problem; its first derivative is set to 0.

[0071]

[0072] Then we obtain the optimal solution for y:

[0073]

[0074] Step 4-3, optimize V with y and γ fixed, and use the BCD method to handle the problem (12).

[0075] In step 4-3, the BCD method is used to handle problem (12), which specifically includes the following steps:

[0076] Step b1: Initialize accuracy ε, feasible beamforming matrix V0, iteration number t = 1;

[0077] Step b2, calculate γ0 using formula (17);

[0078] Step b3, calculate y using formula (19) o ;

[0079] Step b 4, calculate the objective function value of problem (15) as f(t-1);

[0080] Step b5, given V t-1 γ t-1 y t-1 The problem is solved using the SCA method for backward link optimization described in step 3 (15);

[0081] Step b 6, update V t γt y t and f(t);

[0082] Step b7: If f(t) - f(t-1) > ε, then t = t + 1 and proceed to step b5; otherwise, return to V. * γ * and y * .

[0083] Step 5-1 includes: using graph neural networks to model various communication problems as follows:

[0084]

[0085] Where TX nodes correspond to the transmitting nodes in the heterogeneous graph, RX nodes correspond to the receiving nodes in the heterogeneous graph, and S... TX =[s TX，1 , ..., s TX，M ] T and S RX =[s RX，1 , ..., s RX，K ] T This represents the variables on the TX and RX nodes. and The feature matrices representing the TX and RX nodes; Describe the characteristics of the edges. d represents the variable on the edge; TX d RX and d E These represent the feature dimensions.

[0086] Step 5-2 includes: constructing a graph neural network model with the following structure:

[0087] The graph neural network model consists of a preprocessing layer, multiple update layers, and a postprocessing layer. The preprocessing layer structure first processes the complex features (F... TX F RX E) is converted to real number characteristics

[0088]

[0089] in Indicates the real part, The imaginary part is represented; the initial representations of the TX node, RX node, and edges are transformed using a single-layer MLP with Rectified Linear Unit (ReLU) as the activation function:

[0090]

[0091] in These are all learnable parameters;

[0092] In the update layer, the update strategy for the TX node features is as follows:

[0093]

[0094] in yes The m-th line, yes The kth row, Let AGG represent the set of neighboring RX nodes of the m-th TX node, and let AGG represent the pooling function. They are two different MLPs;

[0095] The update strategy for RX node features is as follows:

[0096]

[0097] The update strategy for edge features is as follows:

[0098]

[0099] in It is E (l-1) The (m, k)th element, They are three different MLPs;

[0100] The feature representation of the graph (F) is performed in the post-processing layer. TX F RX E) mapped to output variable Specifically:

[0101]

[0102] in These are trainable parameters; then the output variables are... Normalize to (S) TX S RX ,Ξ) makes it satisfy the constraints.

[0103] Step 6 includes:

[0104] Training data for the network is prepared using the proposed FP-SCA method. The beamforming matrix output by the FP-SCA method is converted into a 0-1 binary classification matrix, where elements with magnitudes less than a threshold are set to zero (indicating no connection between TX-RX antennas), and elements with magnitudes greater than a threshold are set to one (indicating a connection between TX-RX antennas). A large amount of sample data is obtained as the training and test sets. The inputs to the datasets are channel information, base station transmit power constraints, and base station maximum front-range capacity constraints, and the outputs are the connection relationships between the BS antennas and the UE. Y represents the known connection relationship matrix, and X represents the predicted connection matrix of the neural network. The binary cross-entropy loss function is used as the objective function. The specific problem is described as follows:

[0105]

[0106] Where Sigmoid is an S-shaped activation function, p is the maximum transmit power vector of the base station, σ is the user noise vector, c is the maximum front-end link capacity vector of the base station, and H is the channel matrix between the base station and the user; in step 6, the TX node features are set to the maximum front-end capacity and maximum transmit power of the BS, the RX node features are set to the user noise, and the edge features are set to the channel information between the TX and RX nodes.

[0107] Step 7 includes:

[0108] Applying this graph neural network framework to the problem of maximizing users and rate proposed above, we can obtain the following problem statement:

[0109]

[0110] In this problem (28), the TX node features are set as base station transmit power, the RX node features are set as user noise, and the edge features are set as channel information between antennas; the connection matrix output by the network in step 6 is applied to this problem (28) to output a suboptimal solution of the precoding matrix.

[0111] Cellular-free massive MIMO technology is considered a communication technology for the post-5G / 6G era. A large number of distributed wireless access points not only break down cell boundaries and eliminate edge interference, but also significantly improve spectral efficiency. However, connecting a large number of access points to the CPU brings a heavy front-end transmission load. In transmitter precoding design, front-end link capacity constraints become an unavoidable problem. This invention proposes a mathematical optimization scheme that combines Fractional Programming (FP) and SCA (hereinafter referred to as the FP-SCA method) to solve the precoding design problem in downlink transmission. Furthermore, the results of the above mathematical optimization methods are combined with graph neural networks to provide a suboptimal design scheme that guarantees approximate performance while significantly reducing computation time. In this invention, the FP-SCA method and a graph neural network-based machine learning method are proposed. Simulation verification shows that the proposed method has good performance and can maximize users and rates under limited front-end link capacity.

[0112] Beneficial Effects: Currently, the zero-forcing equalization scheme widely used in non-cellular networks cannot achieve satisfactory user and rate performance, and it cannot limit the capacity of the upstream link. Compared with the zero-forcing equalization scheme, the FP-SCA method and graph neural network method proposed in this invention can bring better performance. The graph neural network learning method proposed in this invention has similar performance, lower computation time, and good generalization performance compared with the above-mentioned FP-SCA method. The trained model can adapt well to changes in the number of BS and UE.

[0113] In summary, the contributions of this invention aim to address some key challenges in implementing cellular massive MIMO systems, such as front-end load and precoding design. By using advanced precoding algorithms and graph neural network algorithms, and considering front-end capacity constraints, the proposed solution can provide a practical solution for realizing high-performance cellular massive MIMO networks. Attached Figure Description

[0114] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the present invention in the above and / or other aspects will become clearer.

[0115] Figure 1 This is a simulation diagram of the application scenario of the method of the present invention.

[0116] Figure 2 This is a topological diagram of the network structure of the present invention.

[0117] Figure 3 This is a comparative diagram of the cumulative distribution function of the method of the present invention.

[0118] Figure 4This is a schematic diagram illustrating the generalization capability of the method of the present invention for the number of BS.

[0119] Figure 5 This is a schematic diagram comparing the average calculation time for the number of BS in the method of this invention.

[0120] Figure 6 This is a schematic diagram illustrating the generalization capability of the method of the present invention for the number of UEs.

[0121] Figure 7 This is a schematic diagram comparing the average calculation time for the number of UEs in the method of this invention.

[0122] Figure 8 This is a schematic diagram comparing the performance of different forward link capacities in the method of this invention.

[0123] Figure 9 This is a schematic diagram of the overall GNN architecture. Detailed Implementation

[0124] This invention provides a downlink precoding design method for cellular-free networks based on graph neural networks, comprising the following steps:

[0125] Step 1, Problem Description:

[0126] Consider as Figure 2 The diagram illustrates a cellular network in which B base stations (BS) simultaneously transmit data symbols to K user terminal equipment (UE) via the downlink. The signal is first transmitted to the base stations (BS) through the uplink, and then to the UEs. Each base station (BS) is assumed to have M antennas, and each UE has one antenna.

[0127] The data symbol s for all user terminal equipment (UE) is represented as: Where E(s·s) H ) = I K The channel from the i-th BS to all user terminal equipment (UE) is represented as follows: The channel from all base stations (BS) to all users is represented as follows:

[0128] The CPU determines the actual beamforming vector for each base station (BS) and aims to sparsify it. That is, the CPU transmits the data symbol s and the precoding matrix V separately. i (Assuming the channel changes are not rapid) the precoding matrix at the i-th base station is represented as follows: The symbol vector transmitted at the i-th base station is:

[0129] x i =V i s (1)

[0130] The channel from the i-th base station to the k-th user terminal device is denoted as... It is also H i The k-th row, then the signal y received by the k-th UE. k Represented as:

[0131]

[0132] The signal-to-noise ratio (SINR) received by the k-th user terminal device k Represented as:

[0133]

[0134] in, This represents the Gaussian white noise at the k-th user location.

[0135] In order to improve the overall user rate with as few precoded data symbols as possible, the method of the present invention addresses the following problem:

[0136]

[0137] Where st represents the set of constraints imposed on the optimization problem, and I represents the indicator function, whose expression is:

[0138] ||·|| F The F-norm of a vector. This represents the maximum transmit power of the i-th BS. V represents the maximum forward link capacity of the i-th BS. i Let represent the beamforming matrix of the i-th BS. The first constraint ensures that the transmit power of each UE does not exceed its maximum power limit. The second constraint limits the network's forward link capacity, meaning that each BS can only serve a limited number of users and is limited by the current receive rate of the users it serves.

[0139] Step 2, Fractional Programming Optimization Design: In this step, an iterative strategy is proposed to jointly optimize the total rate and backhaul capacity to solve the problem in Step 1.

[0140] Optimization of overall user rate: This section focuses on optimizing matrix V to improve the overall rate. To simplify the solution, a new variable γ is introduced. k By SINR k Replace with γ k The target problem is:

[0141]

[0142] First, the Lagrange multiplier method is used to handle the equality constraints in the equation:

[0143]

[0144] According to Lagrange's theorem, the optimal closed-form solution for λ is:

[0145]

[0146] Substituting the optimal Lagrange multiplier into the Lagrange dual function, we obtain the following formula:

[0147]

[0148] After finding the common denominator of the expression, we obtain the following equation:

[0149]

[0150] The objective function now has the following form:

[0151]

[0152] Note that the complex fractions in the objective function f2 are difficult to handle, so this invention uses fractional programming to simplify them.

[0153] Lemma 1: The following nonconvex fractional programming problems can be equivalently transformed:

[0154]

[0155] This form can be equivalently converted to:

[0156]

[0157] Where y is an auxiliary variable introduced in this invention, the closed-form solution of y can be obtained directly:

[0158]

[0159] Applying Lemma 1 to the problem of this invention, we obtain the following objective function:

[0160]

[0161] To date, the problem has been restated as follows:

[0162]

[0163] Step 3, SCA method for upstream link optimization

[0164] The indicator function in the above problem makes the optimization problem challenging. A potential solution is to use an exhaustive search method, where all possible combinations of selected links are examined and the specific set of links that produces the maximum total rate is chosen. However, the total number of links is B*M*K, which results in extremely high computational complexity due to the huge number of antennas in non-cellular systems, making it impractical for real-world applications.

[0165] Next, a low-complexity algorithm is proposed to handle non-smooth indicator functions by approximating them as a convex, continuous, smooth function i. Specifically, consider the following commonly used smooth convex functions:

[0166]

[0167] Although the function is already a smooth function, the constraint is still non-convex. To make it feasible in convex optimization, for any given point... Performing a first-order Taylor expansion yields:

[0168]

[0169] The superscript ' indicates the derivative function.

[0170] For fixed γ and y, we obtain the subproblem:

[0171]

[0172] The algorithm for the SCA method is summarized as follows (Algorithm 1):

[0173] Algorithm 1: The SCA method solves the above problem;

[0174] Inputs are: accuracy ε, feasible beamforming matrix V 0 Maximum number of iterations N;

[0175] The output is: the optimal beamforming matrix V * ;

[0176] Step a1, set t=1 and initialize f3(V) 0 ).

[0177] Step a2: If t≤N, proceed to step a3; otherwise, proceed to step a7.

[0178] Step a3, using a fixed γ t and y t The problem is solved using a convex optimization algorithm (18) to obtain V. t ;

[0179] Step a4, update f3(V) t );

[0180] Step a5, if |f3(V t )-f3(V t-1 If |ε| > ε|, then set t = t + 1 and proceed to step 2; otherwise, return to the optimal beamforming matrix V. * =V t ;

[0181] Step 4, use the BCD method for overall rate optimization: use the BCD method to handle the problems after conversion (15).

[0182] Phase 1: Optimize γ with fixed V and y. This is a standard convex problem, with its first derivative set to 0:

[0183]

[0184] Then the optimal solution for γ is obtained:

[0185]

[0186] Phase 2: Optimize y with fixed V and γ. This is a standard convex problem, and its first derivative is set to 0.

[0187]

[0188] Then we obtain the optimal solution for y:

[0189]

[0190] Phase 3: Optimize V with y and γ fixed. This problem has already been discussed in step 3 and can be solved efficiently using standard convex optimization algorithms. The global BCD approach to solving problem (15) is summarized in Algorithm 2.

[0191] Algorithm 2 provides a fractional programming approach to solve problem (15):

[0192] Step b1: Initialize accuracy ε, feasible beamforming matrix V0, iteration number t = 1;

[0193] Step b2, calculate γ0 using formula (20);

[0194] Step b 3, use formula (22) to calculate y0;

[0195] Step b4: Calculate the objective function value as f(t-1);

[0196] Step b5, given V t-1 γ t-1 y t-1 Solve the problem using the SCA method described in step 3 (18);

[0197] Step b 6, update V t γ t y t and f(t);

[0198] Step b7: If f(t) - f(t-1) > ε, then t = t + 1 and proceed to step b5; otherwise, return to V. * γ * and y * .

[0199] Step 5: Establish a machine learning framework based on graph neural networks.

[0200] The notation in heterogeneous graph neural networks is as follows: TX nodes correspond to transmitting nodes in the heterogeneous graph, and RX nodes correspond to receiving nodes in the heterogeneous graph. Let represent the feature vectors of the i-th TX node and the k-th RX node. Where, d TX and d RX These represent the corresponding feature dimensions. Therefore, the feature matrices of the TX and RX nodes can be represented as follows: Similarly, the characteristics of an edge can also be represented by a tensor. Where d E This is the edge feature dimension. On the other hand, the variables on the tx and rx nodes can be represented as S. TX =[s TX，1 , ..., s TX，M ] T and S RX =[s RX，1 , ..., s RX，K ] T in Let d′ represent the variables on the i-th TX node and the k-th RX node. TX ,d′ RX These represent the corresponding variable feature dimensions. Similarly, edge variables are represented as: d′ E Indicates the dimension of the edge variable.

[0201] For a heterogeneous graph problem, it can be modeled as follows:

[0202]

[0203] Where φ is the mapping function from features to variables.

[0204] The overall GNN architecture is as follows: Figure 9 As shown;

[0205] The preprocessing layer converts the input features into initial node and edge feature representations. These representations are updated according to the node and edge update mechanisms in L update layers. During the update process, the dimension of each feature remains unchanged. Finally, the postprocessing layer converts the feature representations into variables and outputs them.

[0206] In the preprocessing layer, since the current neural network is not suitable for handling complex numbers, the features (F) that may be complex numbers must first be processed. TX F RX E) is converted to real number characteristics Specifically, by extracting and concatenating the real and imaginary parts of a complex number, we obtain the following representation:

[0207]

[0208] in Indicates the real part, The imaginary part is represented. Then, a single-layer MLP using the Rectified Linear Unit (ReLU) as the activation function is used to transform the initial representations of the TX nodes, RX nodes, and edges:

[0209]

[0210] in These are all learnable parameters.

[0211] In the update layer, the update strategy for the TX node features is as follows:

[0212]

[0213] in yes The m-th line, yes The kth row, Let AGG represent the set of neighboring RX nodes of the m-th TX node, and let AGG represent the pooling function. They are two different MLPs.

[0214] The update strategy for RX node features is as follows:

[0215]

[0216] The update strategy for edge features is as follows:

[0217]

[0218] in It is E (l-1) The fm,k)th element These are three different MLPs. The update strategies in these three graph neural networks have been proven by existing work to have permutational isovariance, which can greatly reduce the number of training samples.

[0219] The feature representation of the graph (F) is performed in the post-processing layer. TX F RX E) mapped to output variable Specifically as follows:

[0220]

[0221] in These are trainable parameters. Then, the output variables... Normalize to (S) TX， S RX， Ξ) makes it satisfy the constraints.

[0222] Step 6: Use a supervised learning method based on graph neural networks to predict the antenna connection relationship between the base station and the user.

[0223] The beamforming design problem under the front-end link constraint is solved using two serial networks. Since the front-end link constraint limits the number of connections between the BS antenna and the UE, the connection relationship between TX and RX is first obtained in step 6 using the neural network framework shown in the figure above (hereinafter referred to as the connection prediction network). Then, in step 7, a heterogeneous graph is constructed based on this connection relationship to solve for the precoding matrix (hereinafter referred to as the precoding design network).

[0224] Training data for the network is prepared using the FP-SCA method proposed in this invention. The beamforming matrix output by the FP-SCA method is converted into a 0-1 binary classification matrix, where elements with magnitudes less than a threshold are set to zero (indicating no connection between TX-RX antennas), and elements with magnitudes greater than a threshold are set to one (indicating a connection between TX-RX antennas). A large amount of sample data can be obtained as training and testing sets. The inputs to the datasets are channel information, base station transmit power constraints, and base station maximum front-range capacity constraints, and the outputs are the connection relationships between the BS antennas and the UE. Y represents the known connection relationship matrix, X represents the predicted connection matrix of the neural network, and a binary cross-entropy loss function is used as the objective function. The specific problem is described as follows:

[0225]

[0226] Here, Sigmoid is an S-shaped activation function, p is the maximum transmit power vector of the base station, σ is the user-end noise vector, c is the maximum front-range capacity vector of the base station, and H is the channel matrix between the base station and the user. In this step, the TX node features are set to the maximum front-range capacity and maximum transmit power of the base station, the RX node features are set to user-end noise, and the edge features are set to the channel information between the TX and RX nodes. After training, this network can quickly obtain the connection matrix between antennas based on the base station's front-range constraints and channel information, and directly apply it to the construction of the heterogeneous graph in step 7, thereby outputting a precoding matrix that satisfies the front-range constraints.

[0227] Step 7: Use an unsupervised learning method based on graph neural networks to design a precoding matrix based on the existing antenna connection relationship between base stations and users.

[0228] Applying this graph neural network framework to the problem of maximizing users and rate proposed above, we can obtain the following problem statement:

[0229]

[0230] In this problem, the TX node characteristics are set to the base station transmit power, the RX node characteristics are set to user-end noise, and the edge characteristics are set to the channel information between antennas. After training, the network can produce approximate performance of the FP-SCA method proposed above, and the network has good generalization ability, adapting well to changes in the number of UEs and BSs, as well as changes in the connections between them. Applying the connection matrix output by the network in step 6 to this problem, a suboptimal solution for the precoding matrix is ​​output.

[0231] Performance Evaluation: This section presents numerical results to demonstrate the effectiveness of the proposed cellular-free MIMO communication system. Specific simulation parameters are as follows: Consider a 2×2km... 2 The region has no downlink cellular network, where BSs and UEs are uniformly distributed. The path loss is 30.5 + 36.7 log. 10 (d)dB, where d is the distance between the transmitting and receiving antennas. The maximum transmit power of each base station is 30dBm, the small-scale channel follows Rayleigh fading, and the noise power is -99dBm. Unless otherwise specified, we assume the following parameter settings for network training and simulation testing: 4 base stations, 2 antennas per base station, 4 users, and a single antenna per user.

[0232] For the graph neural network approach, all aggregation functions are implemented using the maximum aggregation function, and all MLPs consist of three linear layers, each followed by a ReLU activation function. The number of update layers is set to 2. During training, the number of epochs is set to 500, the batch size to 256, with a training set size of 10,000 for the connection prediction network and 25,600 for the precoding design network. The learning rate is set to 10%. -4 The Adam optimizer was used. All experiments were implemented using PyTorch on an Intel Core i7-11800H (2.30GHz) and an NVIDIA GeForce RTX 3060 GPU (12GB).

[0233] Training of the Connection Prediction Network: First, the dataset for the connection prediction network was obtained using the FP-SCA method. For 10200 different channel implementations, the maximum forward link capacity of the base station was set to [0, 10] bps / Hz. The connection matrix for each implementation was obtained and stored. Then, the connection prediction network was trained using a training set of size 10000 and a test set of size 200. The simulation results are shown in Table 1. The average prediction accuracy for the 200 implementations in the test set with 4 base stations and 4 users was 96.5%. To test the generalization performance of the network, tests were conducted with 200 random forward link capacity samples with 8 base stations and 4 users, and 4 base stations and 8 users, respectively. The average prediction accuracies obtained were 95.5% and 95.0%, respectively. It can be seen that the network has high prediction accuracy and good generalization ability, adapting to changes in the number of base stations and users. The output connection matrix can lay a good foundation for the subsequent design of precoding matrices that meet capacity constraints.

[0234] Table 1

[0235] Number of base stations Number of users Sample size (number of samples) <![CDATA[C max (bps / Hz)]]> Prediction accuracy 4 4 200 [0, 10] random values 96.5％ 8 4 200 [0, 10] random values 95.5％ 4 8 200 [0, 10] random values 95.0％

[0236] The overall structure of the two networks will be referred to as the GNN method in the following text.

[0237] Figure 3This paper presents a comparison of the overall performance of the FP-SCA and GNN methods. Using the zero-forcing equalization algorithm as a benchmark, cumulative distribution curves for the three methods across 200 random implementations are plotted, with the horizontal axis representing users and rate, and the vertical axis representing cumulative distribution probability. The graph shows that FP-SCA and GNN produce higher user and rate rates compared to ZF. Specifically, the average user and rate of the GNN algorithm is 95.45% of that of the FP-SCA algorithm, while the average user and rate of the ZF algorithm is 86.02% of that of the FP-SCA algorithm. The precoding matrix output by the GNN method is validated for its forward link constraint satisfaction rate. Across 200 channel implementations, the GNN method satisfies the forward link constraints with a probability of 97.5%. This accuracy is slightly higher than the connection prediction accuracy because even for channel implementations with incorrect predictions, there is still a relatively high probability of outputting a precoding solution that satisfies the constraints.

[0238] Figure 4 and Figure 5 This demonstrates the generalization performance of the GNN method with varying numbers of business bases (BSs). The method was trained on a training set with 4 BSs, and its performance was tested on test sets with 5, 6, 7, and 8 BSs. Figure 4 In the graph, the horizontal axis represents the number of business units (BS), and the vertical axis represents the number of users and the data rate. Figure 4 As can be seen, the GNN algorithm has good generalization performance and can produce approximate performance of the FP-SCA algorithm for different numbers of base pairs. Figure 5 In the graph, the horizontal axis represents the number of BS (Browser Components) and the vertical axis represents the average computation time. Figure 5 As can be seen, for different numbers of base stations (BS), the GNN algorithm has a faster computation speed than FP-SCA (ignoring training time). Taking 8 BS as an example, the GNN algorithm only used 0.63% of the computation time compared to FP-SCA, achieving 94.51% of the user and speed. It can be seen that the GNN method proposed in this invention can significantly reduce computation time while producing suboptimal solutions, making it more suitable for production environments in large-scale MIMO applications.

[0239] Figure 6 and Figure 7 This demonstrates the generalization performance of the GNN method with varying numbers of user units (UEs). The method was trained on a training set with 4 UEs and tested on test sets with 5, 6, 7, and 8 UEs. Figure 6 In the graph, the horizontal axis represents the number of UEs, and the vertical axis represents the number of users and the data rate. Figure 6 As can be seen, the GNN algorithm has good generalization performance and can produce approximate performance of the FP-SCA algorithm for different numbers of UEs. Figure 7 In the graph, the horizontal axis represents the number of UEs, and the vertical axis represents the average computation time. Figure 7As can be seen, for different numbers of UEs, the GNN algorithm has a faster computation speed than FP-SCA (ignoring training time). Taking a number of 8 UEs as an example, the GNN algorithm only used 0.55% of the computation time compared to FP-SCA, and achieved 94.34% of the users and the rate. It can be seen that the GNN method proposed in this invention can greatly reduce the computation time when producing suboptimal solutions, and is more in line with the needs of production environments in the practical application of large-scale MIMO.

[0240] Figure 8 This paper demonstrates the performance of the FP-SCA algorithm and GNN algorithm proposed in this invention in solving the upstream link capacity constraint problem. The horizontal axis represents upstream link capacity, and the vertical axis represents users and rate. It can be seen that as the upstream link capacity increases, the GNN algorithm can produce approximate performance of the FP-SCA algorithm, maximizing users and rate under the upstream link capacity constraint. When the upstream link capacity exceeds the capacity required for the BS to serve all users (corresponding to horizontal axis 8-10 in the figure), the number of users and rate no longer increases, which is consistent with expectations.

[0241] This invention provides a downlink precoding design method for cellular-free networks based on graph neural networks. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

Claims

1. A downlink precoding design method for cellular-free networks based on graph neural networks, characterized in that, Includes the following steps: Step 1: Establish the overall rate optimization problem for user terminal devices in a non-cellular network; Step 2: Establish an iterative strategy based on fractional programming (FP) algorithm to jointly optimize the total rate and the upstream capacity, and re-transform the optimization problem in Step 1. Step 3: Obtain the subproblems of the optimization problem, and solve the subproblems using the continuous convex approximation SCA method; Step 4: Use the Block Coordinate Descent (BCD) method to process the transformed problem obtained in Step 2; Step 5: Establish a graph neural network-based learning framework for wireless communication; Step 6: Use a supervised learning method based on graph neural networks to predict the antenna connection relationship between the base station and the user; Step 7: Using an unsupervised learning method based on graph neural networks, design a precoding matrix according to the existing antenna connection relationship between base stations and users; Step 6 includes: Training data for the network is prepared using the FP-SCA method. The beamforming matrix output by the FP-SCA method is converted into a 0-1 binary classification matrix, where elements with magnitudes less than a threshold are set to zero, and elements with magnitudes greater than a threshold are set to one. Sample data is acquired as the training and test sets. The inputs to the datasets are channel information, base station transmit power constraints, and base station maximum front-range capacity constraints, and the output is the connection relationship between the BS antenna and the UE. To represent a known connectivity matrix, use The problem is defined as follows: The prediction connection matrix of the neural network is represented by the binary cross-entropy loss function as the objective function. （27）， Where st represents the set of constraints imposed on the optimization problem, K is the number of user terminals (UEs), M is the number of antennas, and sigmoid is a sigmoid activation function. It is the maximum transmit power vector of the base station. It is the user-side noise vector. It is the maximum forward link capacity vector of the base station. It is the channel matrix between the base station and the user; in step 6, the TX node characteristics are set to the maximum front-end capacity and maximum transmit power of the BS, the RX node characteristics are set to user-end noise, and the side characteristics are set to the channel information between the TX and RX nodes. Step 7 includes: Applying the graph neural network framework to the problem of maximizing users and rate yields the following problem statement: （28）， B represents the signal-to-interference-plus-noise ratio received by the k-th user terminal device; B is the number of base stations. The F-norm of a vector. This represents the maximum transmit power of the i-th BS; In this problem (28), the TX node features are set as base station transmit power, the RX node features are set as user noise, and the edge features are set as channel information between antennas; the connection matrix output by the network in step 6 is applied to this problem (28) to output a suboptimal solution of the precoding matrix.

2. The method according to claim 1, characterized in that, Step 1 includes: setting up a non-cellular network consisting of B base stations (BS) and K user terminals (UE), wherein the B base stations simultaneously transmit data symbols to the K UE devices via the downlink. The signal is first transmitted from the central processing unit (CPU) to the base station via the frontlink, and then transmitted to the UE device by the base station antenna; setting each base station to have M antennas, and each UE device to have one antenna. Data symbols for all user terminal equipment (UE) Represented as ,in The channel from the i-th BS to all user terminal equipment (UE) is represented as follows: The channel from all base stations (BS) to all users is represented as follows: ; The CPU determines the actual precoding vector for each base station, and transmits the data symbol s and the precoding matrix respectively. The precoding matrix at the i-th base station is represented as Then the symbol vector transmitted at the i-th base station is: （1）， The channel from the i-th base station to the k-th user terminal device is denoted as... It is also The k-th row, then the signal received by the k-th UE. Represented as: （2）， The signal-to-interference-plus-noise ratio received by the kth user terminal device Represented as: （3）， in, This represents the Gaussian white noise at the k-th user location; The overall rate optimization problem for user terminal devices is: （4）， Where I1 represents the indicator function, and its expression is: , This represents the maximum forward link capacity of all BSs.

3. The method according to claim 2, characterized in that, Step 2 includes: by Replace with And by adding equality constraints, the problem can be expressed as: （5）， Use the Lagrange multiplier method to handle the equality constraints in problem (5): （6）， According to Lagrange's theorem, we get The optimal closed-form solution is: （7）， Substituting the optimal Lagrange multiplier into the Lagrange dual function, we obtain the following formula: （8）， After finding the common denominator of the expression, we obtain the following equation: （9）， The objective function now has the following form: （10）， According to the fractional programming method, the complex fraction in formula (9) is equivalently transformed to obtain the following objective function: （11）， in These are introduced auxiliary variables; The problem in step 1 is then rephrased as: （12）。 4. The method according to claim 3, characterized in that, Step 3 includes: For the indicator function in the forward link constraint in formula (12) An approximation is made using a smoothing indicator function, which is: （13）， For any given point Performing a first-order Taylor expansion yields: （14）， superscript Describe the derivative function; For fixed and This leads to subproblem (15): （15）， The SCA method is used to solve subproblem (15).

5. The method according to claim 4, characterized in that, Step 3, which involves using the SCA method to solve subproblem (15), specifically includes the following steps: Inputs are: accuracy Z1, feasible beamforming matrix. Maximum number of iterations N; The output is: the optimal beamforming matrix. ; Step a1, set t = 1 and initialize ; Step a2: If t ≤ N, proceed to step a3; otherwise, proceed to step a7. Step a3, using fixed and The problem is solved using a convex optimization algorithm (15) to obtain... ; Step a4, Update ; Step a5, if If so, set t = t + 1 and proceed to step 2; otherwise, return to the optimal beamforming matrix. ; Step a6, End.

6. The method according to claim 5, characterized in that, Step 4 includes: Step 4-1, optimize γ with V and y fixed by setting its first derivative to 0: （16）， Then the optimal solution for γ is obtained: （17）， Step 4-2, optimize y with V and γ fixed, and set its first derivative to 0: （18）， Then we obtain the optimal solution for y: （19）， Step 4-3, optimize V with y and γ fixed, and use the BCD method to handle the problem (12).

7. The method according to claim 6, characterized in that, In step 4-3, the BCD method is used to handle problem (12), which specifically includes the following steps: Step b1: Initialize accuracy Z2, feasible beamforming matrix V0, iteration number t = 1; Step b2, calculate using formula (17) ; Step b3, calculate using formula (19) ; Step b 4, calculate the objective function value of problem (15) as ; Step b5, given , , The problem is solved using the SCA method for backward link optimization in step 3 (15). Step b 6, Update , , and ; Step b7, if If t=t+1, then proceed to step b5; otherwise, return. , and .

8. The method according to claim 7, characterized in that, Step 5-1 includes: using graph neural networks to model various communication problems as follows: （20）， In this diagram, TX nodes correspond to transmitting nodes in the heterogeneous graph, and RX nodes correspond to receiving nodes in the heterogeneous graph. and This represents the variables on the TX and RX nodes. and The feature matrices representing the TX and RX nodes; Describe the characteristics of the edges. Represents the variables on the edge; , and These represent the feature dimensions.

9. The method according to claim 8, characterized in that, Step 5-2 includes: constructing a graph neural network model with the following structure: The graph neural network model consists of a preprocessing layer, multiple update layers, and a postprocessing layer. The preprocessing layer structure first processes the features of complex numbers... Convert to real number characteristics : （21）， in Indicates the real part, The imaginary part is represented; the initial representations of the TX nodes, RX nodes, and edges are transformed using a single-layer MLP with ReLU as the activation function: （22）， in These are all learnable parameters; In the update layer, the update strategy for the TX node features is as follows: （23）， in yes The m-th line, yes The kth row, Let represent the set of neighboring RX nodes of the m-th TX node. Represents the pooling function, They are two different MLPs; The update strategy for RX node features is as follows: （24）， The update strategy for edge features is as follows: （25）， in yes The (m,k)th element, They are three different MLPs; The features in the graph are represented in the post-processing layer. Mapped to output variables Specifically: （26）， in These are trainable parameters; then the output variables are... Normalize to This ensures that it satisfies the constraints.

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