Graph Convolutional Neural Network-based Modeling of Ultra-Scale MIMO Channels in 6G Networks

With the rapid development of mobile communication technology, the next-generation wireless communication network 6G will meet the increasingly high demand for Internet services, such as ultra-low access delay, ultra-low bit error rate and ultra-high transmission data rate. Based on the demand and technology evolution of new applications and new scenarios in the future digital era, the ultra-large-scale MIMO technology is further proposed [1–3]. Through the deployment of ultra-large-scale antennas, not only can provide ultra-high-speed wireless data transmission, but also significantly improve the spectral efficiency of the communication system and network coverage, and realize further performance breakthroughs. Ultra-large-scale MIMO technology can play a key role in meeting the 6G performance indexes, especially the three major indexes of spectral efficiency, energy efficiency, and mobility speed [4–7]. Shannon’s information theory reveals two main ways to increase the system capacity, i.e., increasing the system bandwidth and improving the spectral efficiency, which will remain an important basis for the development of 6G, and the ultra-large-scale MIMO technology is precisely through the parallel data streams multiplexed on the same frequency channel to achieve the goal of significantly improving the spectral efficiency [8–11]. This technology is not only the expansion of the number of base station antennas, but also involves the innovation of antenna array arrangement, deployment scenarios and forms, as well as the innovation of technology applications. Therefore, ultra-large-scale MIMO technology will become one of the key technologies in the physical layer of future 6G mobile communications [12–15]. At present, the research on ultra-large-scale MIMO technology has achieved certain results, such as the study of ultra-large-scale MIMO channel characteristics and channel modeling, system performance analysis, etc. However, new application scenarios also bring new problems. The research on exascale MIMO is not comprehensive enough, some research scenarios are still different from the actual communication, and some of the assumptions in the research lack of reasonableness, therefore, the research on exascale MIMO technology is of great significance both in theory and in practical applications [16–20].

In this paper, we successfully realize the ultra-large-scale MIMO channel modeling in 6G networks based on graph convolutional neural network, which makes the relationship between channel characteristics and system parameters transparent and credible by establishing a graph convolutional neural network channel model, and adopts self-learning and self-optimization training methods to automatically adjust the model parameters, so as to make it highly accurate and adaptive in complex scenarios. In order to verify the reliability of the proposed model, this paper simulates and analyzes it, and proves that the simulated 6G MIMO channel has the long-term dependence property on the time series as well as the non-smooth property. Meanwhile, the channel model is validated for delay extension and analyzed for power delay characteristics.

2

Hyperscale MIMO channel characterization and modeling

2.1

Hyperscale MIMO Channel Characterization

Conventional antenna sizes are usually proportional to the wavelength, typically half or 1/4 wavelength, so that the antenna sizes for ultra-large-scale MIMO are generally small only in the UHF band. In the future 6G network, in the 1 THz band, for example, the wavelength is just 0.3 mm, so it is entirely possible to place hundreds of antennas within a distance of a few tens of centimeters, and by taking advantage of this, it is expected that the 6G system will be able to achieve a 10-fold increase in the spectral efficiency of 5G. In an exascale MIMO system, the terminal is most of the time in the near-field region of the base station antenna array, and the use of large-size antennas and the proximity of the antenna array to the user will bring new channel characteristics to exascale MIMO. In this section, the new channel characteristics of the hyperscale MIMO system in 6G networks - near-field characteristics and spatial non-stationary characteristics - are summarized.

2.1.1

Near-field characteristics

Electromagnetic radiation fields in wireless communication systems can be categorized into far field and near field, where different fields give rise to different channel models. The demarcation between the near and far fields is determined by the Rayleigh distance 2D²/λ, where D and λ denote the array aperture and wavelength, respectively. In traditional large-scale communication systems, due to the small number of antennas, the near-field region of the antenna array is very small, and the distance from the scatterer to the base station is often much larger than the Rayleigh distance, which satisfies the far-field propagation condition. With the significant increase in the number of antennas in the future 6G system, the Rayleigh distance between the antenna arrays will also increase, and the near-field region of the ultra-large-scale antenna array will be enlarged by several orders of magnitude.

The far-field plane wave to near-field spherical wave comparison is shown in Fig. 1, and in the near-field case, it is not possible to model all antenna paths using a single angle, i.e., a plane wave cannot be used to approximate a spherical wave. Therefore, for ultra-large-scale MIMO systems, where users and scatterers tend to be more in the near-field region, it is more reasonable to choose a spherical wave model for channel modeling.

2.1.2

Spatial non-stationary properties

In an ultra-large-scale MIMO system, due to the influence of the size, shape, location of the scatterer and the obstacles between the scatterer and the base station, different parts of the array antennas are said to be able to see different scatterers or terminals with different signal arrival angles and reception probabilities. The energy of the scatterers or terminals is concentrated on some of the antennas of the array, which is called the user’s VR.With the expansion of the antenna scale of the base station, the performance of individual terminals will be affected by the visible area, which leads to a more complex transmission environment for spatially non-smooth channels. Depending on the distribution of VRs, the VRs of different users may be completely non-overlapping, partially overlapping, or completely overlapping, and the smooth massive MIMO and spatially non-smooth mega MIMO are schematically shown in Fig. 2 and Fig. 3, respectively.

As can be seen from the figures, for smooth massive MIMO, the terminal is able to see all the antennas in the array through the scatterer. While for non-smooth massive MIMO, the signal energy from terminal 1 can be concentrated in VR-A1 and VR-A2 of the antenna array through scatterers C1 and C2, while the signal energy from terminal 1 cannot be received in VR-A3. Therefore, with a significant increase in the array dimension, the performance of each terminal is affected by its VR.

2.2

Hyperscale MIMO channel modeling

The design of ultra-large-scale MIMO wireless communication systems is based on channel modeling, which accurately characterizes the system performance that can be supported by the channel under the given channel model and the constraints satisfied by the system, and thus reveals the impact of various channel characteristics on the system performance, which can provide an important modeling basis for the optimal design of the wireless transmission system and performance evaluation. The current channel modeling approaches in wireless communication systems mainly include: geometric scatterer-based statistical stochastic model, geometric scatterer-based deterministic model, and correlation-based statistical stochastic model. Different scenarios consider different channel modeling approaches, and in the standardization of communication systems, the statistical stochastic model based on geometric scatterers is usually adopted in consideration of the universality and accuracy of channel modeling.

Compared with the traditional large-scale MIMO system, the antenna array of the ultra-large-scale MIMO system is larger, and the base station antennas may be distributed in a larger area in the future deployment, in which case, the scatterer is likely to be in the near-field area of the antenna array, and at this time, the plane wave assumption of the traditional large-scale MIMO is no longer valid, and the received electromagnetic wave can not be approximated as a plane wave, and it has a non-smooth characteristics. In this section, two types of channel modeling for ultra-large-scale MIMO systems are outlined: near-field channel modeling and spatially unsteady channel modeling.

2.2.1

Near-field channel modeling

To address the problem of the proliferation of user terminals, future wireless communications tend to employ mega-scale antenna arrays on base stations, a technique that will allow the near-field region of the base station to be extended to tens or even hundreds of meters, when wireless communications will still take place in that region.

Consider a hyperscale MIMO base station deployed with M antenna, communicating with K single-antenna users. When the distance between the base station and the users is less than the Rayleigh distance, the traditional plane-wave assumption will be invalid. In addition, when the wireless communication occurs in the near-field region, the arrival/departure angles of all antennas cannot be approximately equal, and the guidance vector of the array will involve the arrival/departure angles between the base station and the users as well as the distance between them. Therefore, under the spherical wave assumption, the near-field channel can be modeled as: 1 $h = \sqrt{\frac{M}{L}} \sum_{l = 1}^{L} α_{l} b (θ_{l}, r_{l})$ where L denotes the number of path components between the base station and the user, α_l and θ_l denote the gain and angle of the l th path, respectively, and b(θ_l,r_l) denotes the array-oriented vector of the near-field channel model, which can be expressed as: 2 $b (θ_{l}, r_{l}) = \frac{1}{\sqrt{M}} {[e^{- j \frac{2 π}{λ} (r_{l}^{(1)} - r_{l})}, \dots, e^{- j \frac{2 π}{λ} (r_{l}^{(M)} - r_{l})}]}^{H}$ where r_l denotes the distance from user i to the center of the antenna array, $r_{l}^{(n)} = \sqrt{r_{l}^{2} + δ_{n}^{2} d^{2} - 2 r_{l} δ_{n} d θ_{l}}$ denotes the distance from user i to the n th antenna of the base station, $δ_{n} = \frac{2 n - M - 1}{2}$ , n = 1,2,…,N.

2.2.2

Modeling spatially non-stationary channels

In a MIMO system with very large antenna arrays, a new property of the channel, spatial nonsmoothness, emerges, where the user will only see a portion of the base station antenna array, and in order to take advantage of the effects of spatial nonsmoothness, VR needs to be considered in channel modeling.

Assuming that the classical spatial correlation matrix for user k under a smooth channel is R_k ∈ ℂ ^M×M and D_k a diagonal matrix, if user k receives only signals transmitted from the D_k antenna, then D_k it has D_k non-zero diagonal terms. In addition, this D_k simultaneously models the VR of user k. For spatially non-smooth channels, a new correlation matrix Θ_k is introduced such that: 3 $Θ_{k} = D_{k}^{\frac{1}{2}} R_{k} D_{k}^{\frac{1}{2}}$

Therefore, the non-smooth channel between user k and the base station is modeled as: 4 $h_{k} = Θ_{k}^{\frac{1}{2}} g_{k}$ where g_k~ CN (0,1). In addition, for smooth channels D_k = 1, Θ_k = R_k.

3

Ultra-large-scale MIMO channel modeling based on graph convolutional neural networks

3.1

Diagram Definition and Diagram Representation

In the field of deep learning, graph neural networks provide an effective method for processing and analyzing graph-structured data. The core idea of graph neural network is to utilize the topology and node attributes of the graph to propagate and aggregate the feature information of the nodes on the graph, so as to obtain a richer and higher-level feature representation.

Graph is an abstract mathematical structure for describing relationships between objects. In graph theory, graphs are widely used to model and solve various practical problems. A graph is usually defined as a collection of nodes (V) and edges (E) representing complex relationships between entities.Vertex denotes the basic element of a graph and can represent an entity, an event, or an abstract concept.Edge is a line connecting two nodes and is used to represent the relationship between nodes. Edges can have direction (Directed Graph) or no direction (Undirected Graph). In Directed Graph, edges have an explicit direction, indicating a relationship that flows from one node to another. In Undirected Graph, edges have no direction and reflect bidirectional or peer-to-peer relationships between nodes.

In addition, edges have weights. It gives additional numerical attributes to the edges which are used to indicate the strength, correlation or other metrics of the edges. The weights of edges are important in solving various practical problems such as network analysis, social network modeling, path planning, etc.

3.1.1

Adjacency matrix

In representing the edge and node information of a graph, this can be achieved by means of the adjacency matrix (A) and the identity matrix (F). Among them, the adjacency matrix is mainly used to describe the connections between nodes in the graph and is a matrix of n × n, where n represents the total number of nodes in the graph [21]. A = (a_ij)_n×n and the element a_ij of the matrix is represented as shown in equation (5): 5 $a_{i j} = {\begin{array}{l} 1 & (v_{i}, v_{j}) \in E \\ 0 & (v_{i}, v_{j}) \notin E \end{array}$

For directed graphs, the element a_ij of the adjacency matrix indicates whether there is a directed edge from node i to node j. If it exists, a_ij is 1; otherwise, it is 0. For undirected graphs, the adjacency matrix is symmetric, i.e., a_ij is equal to a_ji. The identity matrix, on the other hand, represents the attributes of the nodes themselves, which can be of any data type, such as numeric, textual, or image data.

3.1.2

Degree matrix

The degree matrix is an important matrix used to represent the structure of a graph. In graph theory, the degree matrix reflects the degree of each node, i.e., the number of edges connected to that node. The degree matrix, usually denoted by the symbol D, is a matrix of n × n, where n is the number of nodes in the graph, and the expression is shown in equation (6): 6 $D = [\begin{matrix} d (v_{1}) & 0 & \dots & 0 \\ 0 & d (v_{2}) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & d (v_{n}) \end{matrix}]$

3.1.3

Laplace matrix

The Laplace matrix, usually denoted by the symbol L, is a key matrix obtained from the computation of the adjacency matrix A and the degree matrix D based on the graph [22]. The computation of the Laplace matrix incorporates the information of the adjacency and degree matrices, thus reflecting the structure and properties of the graph more comprehensively. In practice, there are two common Laplace matrix representations: the unnormalized Laplace matrix L (also known as the symmetric Laplace matrix) and the normalized Laplace matrix L_norm. The formula for the calculation of the unnormalized Laplace matrix L is given in Equation (7), and that for the normalized Laplace matrix L_norm is given in Equation (8): 7 $L = D - A$ 8 $L_{n o r m} = I - D^{- \frac{1}{2}} A D^{- \frac{1}{2}}$

3.2

Graph Convolutional Neural Networks

Graph Convolutional Neural Networks (GCNs) draw on the achievements of traditional convolutional neural networks in image processing and are specifically designed to analyze and process graph-structured data [23]. Unlike traditional fully-connected neural networks, GCNs make effective use of the structural information of the graph by effectively exploiting it so that the representation of each node depends not only on its own features, but also on the information of its neighboring nodes. This enables GCNs to capture the local structure and global patterns in the graph during the learning process, thus allowing them to better adapt to complex system data.

GCNs typically consist of multiple graph convolutional layers (GCLs), each with its own weight parameter, and by combining multiple layers, the network is able to gradually learn more abstract and higher-level feature representations. This layer-stacked structure allows GCNs to perform deep learning in graph structures, capturing complex graph relationships. The core graph convolutional layer updates a node’s representation to a weighted sum of its neighboring nodes by normalizing the adjacency matrix, where the weights are determined by the adjacency matrix and the degree matrix.

This design enables the GCN to make full use of the graph structure information while preserving the node’s own features to achieve effective feature propagation and learning.

The computational process of each graph convolution layer can be briefly described as follows: 1)

Node representation update: For each node in the graph, its own features are aggregated with those of its neighboring nodes to form a new node representation, the expression is shown in equation (9): 9 $Z^{l + 1} = A \cdot H^{(l)} \cdot W^{(l)}$

Where Z^l+1 is the set of nodes to be updated, obtained by the product of the adjacency matrix A and the node features H^(l). W^(l) is the weight matrix of layer l.

2)

Aggregation approach: usually, aggregation is done by weighted summation of the features of neighboring nodes, with weights determined by the adjacency matrix and degree matrix of the graph. This weighted approach is able to take into account the strength of the relationship between the nodes and the topology. Namely: 10 ${\hat{Z}}^{(l + 1)} = D^{- \frac{1}{2}} \cdot Z^{(l + 1)} \cdot D^{- \frac{1}{2}}$

Equation (10) is a symmetric normalization of the nodes, where D is the degree matrix.

3)

Nonlinear transformation: a nonlinear transformation is performed on the updated node representations, usually using an activation function, such as the Revised Linear Unit (ReLU), to introduce nonlinearity. The computational procedure is shown in (11): 11 $A^{(l + 1)} = {\hat{Z}}^{(l + 1)} \cdot W^{(l)}$

Equation (12) is a linear transformation of the normalized nodes: 12 $H^{(l + 1)} = Re L U (A^{(l + 1)})$

4)

Multi-layer stacking: the above process is stacked multiple times, with each layer using the output of the previous layer as input to learn more advanced node representations.

Such multi-layer stacking enables GCN to gradually expand the sensory field and capture feature information at different levels in the graph structure. This also allows the GCN to excel in handling tasks such as graph classification, node classification, and link prediction. In each layer, the network is able to utilize the information learned in the previous layer to better understand the structure and characteristics of the graph.

3.3

Neural network channel modeling

The schematic of neural network for channel modeling is shown in Fig. 4. The channel data can be generated by simulation or obtained by real measurements, and all the channel data are divided into a training set and a test set. The training set is fed into the neural network and the weight parameters are estimated using the back propagation algorithm or Newton’s algorithm to model the channel. Then, the test data is used to test the model and obtain the fit values for the channel model. Finally, the test set channel parameter values and model fit values are compared to evaluate the accuracy of wireless channel modeling based on metrics such as residuals and root mean square error.

However, the neural network channel model, as a black-box model, only cares about the inputs and outputs of the model, and does not care about the internal structure and principles of the model, which makes the channel modeling results only numerically approximated, and cannot explain the intrinsic connection between the channel characteristics and the system parameters, and at the same time, most of the researches in the field use the simulation data to train the model, and lack of the real measurement data. To address the above problems, this paper proposes a self-learning GCN-based method for modeling the ultra-large-scale MIMO channel. The neural network channel model outputs the results at the same time and can explain the reasons, so as to update the model parameters in time, optimize the model structure design, and improve the adaptive ability of different scenarios. At the same time, the channel model is trained using measured and simulated data, and the performance of the model is verified in laboratory scenarios to realize the prediction of channel characteristics at different locations in indoor three-dimensional space, including, but not limited to, the channel characteristics in the power domain, delay domain, and angle domain.

3.4

GCN-based self-learning channel modeling process

Channel modeling is the process of exploring and characterizing channel properties in a given scenario, the flow of GCN-based self-learning channel modeling to predict the characteristics of the ultra-large-scale MIMO channel is shown in Fig. 5. The flowchart includes three main parts: the first preprocessing part, which includes data acquisition and partitioning, building graph convolutional neural network models, and initialization. The second training part involves training the model using the training algorithm, updating the internal parameters of the model, obtaining the graph convolutional neural network ultra-large-scale MIMO channel model, and fine-tuning it afterwards. The third prediction section includes channel characterization prediction using the channel model and superimposing Gaussian noise to obtain the final prediction results.

3.4.1

Pre-processing module

The raw channel data includes system parameters such as the 3D coordinates of the base station, the 3D coordinates of the user, the system operating frequency, the distance between the base station and the user, and also channel characteristic parameters such as path loss, delay extension, and angle extension.

In the raw channel data used for neural network model training, the magnitude difference between the system parameters as input and the channel characteristic parameters as output is large and belongs to different dimensions. Directly using it for neural network training will make the model training slow, may not converge for a long time, or may deviate from the model effect. Therefore, it is necessary to complete the data preprocessing before inputting the neural network for training. Usually use the minimum-maximum data normalization method, the transformation formula is: 13 $x^{*} = \frac{x - x_{\min}}{x_{\max} - x_{\min}} (x_{\max}^{*} - x_{\min}^{*}) + x_{\min}^{*}$

Where x denotes the original data and x^* denotes the normalized data, by selecting the appropriate $x_{\min}^{*}$ and $x_{\max}^{*}$ , the data can be normalized and transformed to the same dimension, generally [–1,1] or [0,1] is selected as the mapping range for neural network training.

The inputs to the graph convolutional neural network model are the system parameters in the channel data, and the outputs are the channel characteristic parameters. Before training the neural network model, all the parameters in the model need to be initialized. Different methods for initializing the weight of neural networks can be used as needed, such as normal distribution initialization, random initialization, and others. When the parameter initialization is completed, the maximum number of iterations and the threshold error are set, and the training of the neural network model can be started.

3.4.2

Training modules

The first thing that needs to be made clear is that the expression for the graph convolutional neural network model is: 14 $\hat{y} = α + \sum_{j = 1}^{k} β_{j} {\tilde{h}}_{j} (w_{j}^{T} x)$

From the above equation it can be seen that the parameters of the learning update are: ${α, β_{j}, {\tilde{h}}_{j}, w_{j}}$ , j = 1,…,k. The network loss function is defined as the mean square error, while the L1 penalty term is added in order to control the model complexity and to reduce the overfitting: 15 $L = \frac{1}{n} \sum_{i = 1}^{n} {({\hat{y}}_{i} - y_{i})}^{2} + λ_{1} \sum_{j = 1}^{k} {‖ w_{j} ‖}_{L 1} + λ_{2} {‖ β ‖}_{L_{1}}$ where n is the batch size of the training process, k denotes the number of sub-networks, ${\hat{y}}_{i}$ denotes the predicted values output by the graph convolutional neural network, y_i denotes the true values in the dataset, w_j denotes the projection weight vector, βdenotes the sub-network weight vector, ${\tilde{h}}_{j}$ denotes the sub-network, and λ₁ and λ₂ denote the hyper-parameters, which are used to constrain the L1 penalty term in the regularization. For the projection weight matrix W, it is iteratively updated by the Kellet transform with orthogonal constraint restrictions: 16 $W^{(t + 1)} = W^{t} (s) = {(I + \frac{s}{2} A)}^{- 1} (I - \frac{s}{2} A) W^{(t)}$ where $A = \nabla_{W} W^{T} - W \nabla_{W}^{T}$ , ∇W are the network loss functions L for the projection weight matrix W for the gradient, and s is the step size. For parameters θ other than the projection weight matrix W that need to be learned to be updated, they are updated by the gradient descent algorithm: 17 $θ^{(t + 1)} = θ^{(t)} - η_{t} \nabla_{θ}^{(t)}$

Where, ∇_θ is the network loss function L for parameter θ to find the gradient, and η_t is the learning rate at each step of the update, which can be automatically adjusted during the training process by the Adam optimizer.

The validation error of the graph convolutional neural network model is the mean square error between the actual values in the validation set and the network predictions. After training the network model parameters are updated and optimized, the network validation error gradually converges, and if it meets the set threshold error, the training is stopped and the model is saved. The sub-networks of the model and the corresponding projection weights are visualized, and the feature importance analysis is performed based on the visualization results, and the importance ratio of the sub-networks in the model is calculated at the same time: 18 $I R_{j} = | β_{j} | / \sum_{j = 1}^{k} | β_{j} |, j = 1, \dots, k$

In addition, the noise source module is introduced in the training process, which simulates the actual environment by adding Gaussian noise to the training data, making the data more realistic. At the same time, when adding Gaussian noise to the model output data, it means drawing new samples from the neighborhood of the known sample data, which will expand the size of the training dataset, and the data space will become smoother, which is more conducive to the training and learning of the model. In addition, since the added noise is random, the training samples will be changing, which will result in a lower generalization error of the trained network.

3.4.3

Forecasting module

The channel model based on graph convolutional neural network is obtained after the training is completed, after which the system parameters are obtained and input into the channel model to obtain the channel characteristic prediction results. Considering that the actual measurement environment and the measurement system may generate errors, the Gaussian noise error is superimposed on the output of the neural network model as the final prediction result of the channel characteristics. The mean value of the Gaussian noise error obtained from the sample data is $μ = \frac{1}{T} \sum_{i = 1}^{T} ({\hat{y}}_{i} - y_{i})$ and the variance is $σ^{2} = \frac{1}{T - 1} \sum_{i = 1}^{T} {({\hat{y}}_{i} - y_{i} - μ)}^{2}$ . Therefore, the final channel characteristic prediction result ${\dot{y}}_{i}$ can be expressed as the sum of the neural network model output ${\hat{y}}_{i}$ and the Gaussian noise error ò_i: 19 ${\dot{y}}_{i} = {\hat{y}}_{i} + ò_{i}$ $${\dot y_i} = {\hat y_i} + {\unicode {x00F2} _i}$$

In order to evaluate the effectiveness of channel characteristics prediction, the root mean square error (RMSE) of the final prediction of channel characteristics ${\dot{y}}_{i}$ , and the actual value y_i is used as a measure with the expression: 20 $R M S E = \sqrt{\frac{1}{T} \sum_{i = 1}^{T} {({\dot{y}}_{i} - y_{i})}^{2}}$ where T is the size of the test set. By calculating the RMSE under different test sets, the accuracy of model prediction and the generalization effect of the model can be initially determined.

4

Model simulation experiments and analysis of results

4.1

Simulation Analysis of Ultra-Scale MIMO Channel Models

According to the modeling process, this paper selects the simulation scenario as urban micro-area propagation environment. Nowadays, 6G MIMO wireless communication system will be configured with a huge number of antennas at the transmitter and receiver ends, and the configuration of large-scale antenna arrays has developed a variety of different geometrical schemes under the continuous research, and the common antenna centralized deployment schemes are Uniform Linear Array (ULA), where the antenna elements are uniformly arranged in a straight line, which is a relatively simple deployment method. Uniform rectangular array (URA) antenna elements are uniformly arranged in a matrix. Uniform Circular Array (UCR), where the antenna elements are uniformly distributed on a circle with radius R. The antenna components are first arranged according to the uniform circular array, and then multiple circular arrays are placed in parallel to form a uniform cylindrical array. In this paper, a uniform rectangular array is chosen to deploy the antenna array elements. Equipped with 1 and 64 antennas at the transmitter and receiver ends, respectively, the geometric relationship of the antenna array is shown in Fig. 6.

A three-dimensional coordinate system is established with the transmitting end configured with one antenna as the origin, and the spatial relative positions of the 64 antennas at the receiving end as well as the scatterer clusters are determined using this as a reference, and the large-scale parameters are shown below:

1 antenna at the sending end, 64 antennas at the receiving end, the height of the antenna at the sending end is 1.5 m, the height of the antenna at the receiving end is 30 m, the carrier frequency is 2 GHz, the antenna polarization mode is XPOL, 12 LOS clusters, 19 NLOS clusters, and 20 in-cluster ray components, the two-dimensional distance is 200 m, the moving speed is 54 Km/h, and the antenna layout is URA.

The time correlation functions for antenna pair 1 and antenna pair 64 are shown in Fig. 7. Its horizontal coordinate is the time interval in milliseconds and the vertical coordinate is the correlation coefficient.

From Fig. 7, it can be seen that as the sampling time interval increases, the temporal correlation between each antenna pair in the subsequent decline, and the overall trend of the two antenna pairs tends to be the same, this is because, in the calculation of the temporal correlation, for each antenna pair before and after its own time for comparison, when the time interval increases, which is equivalent to the distance between the two sampling points in the temporal domain increases, does not involve the spatial domain of the change. Therefore, the trend of the time correlation is independent of the antenna pair spatial location. And it is observed that the correlation tends to stabilize when the time interval is greater than 30ms, but the correlation still remains around 0.1, which can be inferred that the 6G MIMO channel has a long-term dependence on the time series.

The simulation of the effect of different angular distributions on the spatial correlation is shown in Fig. 8. From Fig. 8, it can be seen that when the angular distribution obeys the von Mises distribution, the correlation between antenna pairs is low, and the spatial correlation coefficient is only 0.32 when the array element spacing is 2 wavelengths.When the angle obeys the Gaussian distribution or the Laplace distribution, the spatial correlation is high, but there is almost no difference between the two, and the curves are more overlapped.

The simulation of the evolution process of clusters in a 6G MIMO channel is shown in Fig. 9. Where the horizontal coordinate indicates the antenna array element index and the vertical coordinate indicates the cluster index.

From Fig. 9, it can be seen that since the antenna array is an 8 × 8 rectangular array, the scatterer clusters numbered 6 and 11 cannot be observed by the first antenna if none of the antennas in the same column can be observed. The scatterer clusters numbered 9 could not be observed by antennas 2, 10, 18, 19, 26, 27, 34, 35, 42, 43, 50, 51, and 52, indicating that the number of unobservable antennas increases according to the number of rows of the array as the cluster spacing antenna array increases. The same case for the cluster numbered 10 shows that each antenna array element has a separate set of effective scatterer clusters, which is manifested by the appearance of vanishing scatterer clusters on the array axis, verifying the non-smooth nature of the 6G MIMO channel.

The normalized channel strength simulation is shown in Fig. 10, where the horizontal coordinate is the antenna array element index and the vertical coordinate is the Eq. normalized array axis strength. Combining this with Fig. 9 shows that the channel strengths of the antennas are categorized into three echelons according to their size due to the non-smoothness of the channel space. Antennas 2, 10, 18, 19, 26, 27, 34, 35, 42, 43, 50, 51, 52 are unable to observe clusters 6, 9, 10, 11 due to distance and therefore have lower channel strength values than the other antenna array elements. Antennas 19 and 44 are unable to observe clusters 6, 9, and 11, and thus the channel strength is in the middle echelon. The remaining antennas are only unable to observe clusters 6, 11 and hence have higher channel strength. The simulation confirms that the large-scale antenna array causes the clusters to evolve along the array axis due to its spatial geometric layout.

4.2

6G network channel model validation and characterization

To ensure the correctness of the hyperscale MIMO channel model in the constructed 6G network, this paper carries out delay extension validation and power delay characterization of the channel model.

4.2.1

Delay Scaling Verification and Analysis

1)

Validation of time-delay expansion

The time-delay extension is usually used to study the time-dispersion characteristics of multipath fading channels, which is determined by the power-delay spectrum of multipath clusters, and is calculated as follows: 21 $σ_{D S} = \sqrt{μ_{τ} - μ_{τ}^{2}}$ 22 $μ_{τ} = \frac{\sum_{n = 1}^{N} τ_{n}^{2} P_{n}}{\sum_{n = 1}^{N} P_{n}}$ where τ_n represents the delay of the n nd cluster, P_n represents the power of the n th cluster, and N represents the number of multipath clusters.

Firstly, the delay modeling results are verified, and the input parameters for model simulation are configured as follows:

Satellite orbit altitude: 800km. scene: dense urban area. Band: S-band, Ka-band. Link state: LOS/NLOS. Satellite elevation angles: 20°, 40°, 60°, 80°. Satellite-side antenna configuration: a = 10c/f, orthogonal dual-polarized, number of antennas is 2. User-side antenna configuration: omnidirectional dual-polarized, number of antennas is 2.

Based on the delay matrix and power matrix under the corresponding propagation conditions in the LOS or NLOS structure output from the parameter generation module, the cluster delay and cluster power in the parameter generation process are obtained and substituted into Eq. (21) to calculate the simulated values of the delay extension and plot its cumulative distribution function (CDF) curve. Meanwhile, for the theoretical values of delay extension, the mean and variance of delay extension under the corresponding scene, band, link state and elevation angle are obtained, and the CDF curves of delay extension which theoretically obey the lognormal distribution are plotted.

The corresponding parameters when the satellite elevation angle is 60° are taken for simulation comparison, and the validation results of th e delay extension distribution under S-band and Ka-band are shown in Fig. 11, (a) and (b) represent the validation results under S-band and Ka-band, respectively, and the solid line in the figure is the simulated CDF curve of the delay extension, and the dashed line is the theoretical CDF curve of the delay extension.

From Fig. 11, it can be seen that under LOS and NLOS propagation conditions, the fit of the delay extensions generated using the simulation results and calculated using the theoretical data are both very high, and the simulation curves and the theoretical curves are almost completely overlapped after 400 ns, which fully verifies the accuracy of the delay modeling. It can also be seen that under the same elevation angle, the value of the delay extension for the NLOS propagation condition is significantly higher than that for the LOS propagation condition when it is below 200ns.

2)

Delay Expansion Analysis

To further support the validation results, the simulation results of delay extension under different satellite elevation angles are compared and analyzed under the same parameter configurations as in the previous section. Taking the NLOS propagation conditions as an example, the CDF curves of delay extension under different elevation angles are shown in Fig. 12.

From Fig. 12, it can be seen that the higher the elevation angle, the faster the delay extension converges, both in S-band and Ka-band. The reason is that when the elevation angle is lower, the transmission signal between the satellite and the ground terminal is more likely to be reflected and scattered by the obstacles in the receiving environment, and the multipath effect is stronger, so the corresponding delay extension is also larger. As the elevation angle increases, the probability of the signal in the star-ground link being blocked by obstacles decreases, the multipath effect is weakened, and the delay extension is also reduced.

4.2.2

Power delay characterization

The power delay spectrum is used to describe the time dispersion characteristics of the wireless channel, which can intuitively reflect the multipath fading effect of the channel, and the power delay characteristic (PDP) is generally used to describe the power delay spectrum in modeling, which needs to give the average power under different time delays. The satellite mobile channel has a more open signal reception environment compared with the terrestrial mobile channel, with obvious direct signal components and a small number of multipath signal components, so the number of channel multipath clusters is generally smaller, usually only 2~4 clusters, and there is a large difference between the power delay characteristics under LOS and NLOS propagation conditions. In one simulation, the PDP under S-band and Ka-band for the dense urban scenario is schematically shown in Fig. 13.

The number of multipath clusters is 3 in both S-band and Ka-band. Under LOS propagation conditions, the power of the first cluster containing the LOS direct path is much larger than that of the other two clusters, which is due to the fact that the longer propagation distance leads to a more severe attenuation of the multipath signal component in the star-ground link, and thus the signal power is mainly concentrated in the direct path, whereas the power difference between the clusters in NLOS propagation conditions is relatively more even due to the absence of the direct path.

5

Conclusion

In this paper, based on the neural network channel modeling method, a self-learning channel modeling method based on graph convolutional neural network is proposed, which is successfully applied to the modeling of the ultra-large-scale MIMO channel in 6G networks, and the model is simulated to verify and characterize the model.

The temporal correlation between each antenna pair decreases with the increase of the sampling time interval, and the overall trend of the two antenna pairs with serial numbers 1 and 64 converges to the same, and the correlation converges to 0.1 after the time interval is larger than 30ms, which illustrates the long term dependence property of the 6G MIMO channel on the time series. In addition, the number of unobservable antennas increases by the number of array rows as the cluster spacing antenna array increases, indicating that each antenna array element has an independent set of effective scatterer clusters, which verifies the non-stationary property of the 6G MIMO channel.

Under both LOS and NLOS propagation conditions, the delay extensions generated using the simulation results and calculated using the theoretical data fit very well and almost completely overlap after 400ns, verifying the accuracy of the delay modeling. Meanwhile, under the same elevation angle, the delay extension values of the NLOS propagation condition are significantly higher than those of the LOS propagation condition when they are below 200ns. And under different satellite elevation angles, the convergence speed of delay extension in both S-band and Ka-band accelerates with the increase of elevation angle.

When the number of multipath clusters under both S-band and Ka-band is 3, the power difference between the clusters under NLOS propagation condition is smaller than that under LOS propagation condition, i.e., the power distribution of the clusters under NLOS propagation condition is more even.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

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Graph Convolutional Neural Network-based Modeling of Ultra-Scale MIMO Channels in 6G Networks

Jinhui Chen

Haochen He

Zhan Xu

Published Online: Mar 19, 2025

Received: Nov 02, 2024

Accepted: Feb 16, 2025

DOI: https://doi.org/10.2478/amns-2025-0470

Keywords6G network, Hyperscale MIMO, Graph convolutional neural network, Channel modeling

© 2025 Jinhui Chen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
6G network, Hyperscale MIMO, Graph convolutional neural network, Channel modeling