A multi-RIS assisted satellite communication system low complexity channel estimation method

Abstract

The application relates to the technical field of wireless communication, and discloses a low-complexity channel estimation method for a multi-RIS assisted satellite communication system, which comprises the following steps: modeling a cascaded channel of the multi-RIS assisted satellite communication system as a high-dimensional tensor, representing the high-dimensional tensor as a ring contraction of multiple third-order tensors by adopting structured tensor ring decomposition, setting a Bayesian hierarchical prior architecture, applying a joint structured sparse prior to retain the structural characteristics of an angle domain and a time delay domain, applying a shared hyperparameter constraint to ring factor matrices of multiple RISs deployed in the same geographical area, maximizing a posterior probability by adopting variational Bayesian inference, estimating the ring factor matrices and their posterior distributions, and adaptively determining a ring rank parameter in the inference process; and finally, reconstructing the original high-dimensional channel tensor by tensor ring contraction, outputting a cascaded channel estimation result, effectively reducing the complexity of channel estimation of the multi-RIS assisted satellite communication system, and retaining the multi-linear structural information of the channel.

Description

Technical Field

[0001] This application relates to the field of wireless communication technology and discloses a low-complexity channel estimation method for a multi-RIS-assisted satellite communication system. Background Technology

[0002] Reconfigurable intelligent surface (RIS) is a key technology for 6G integrated satellite-ground communication due to its passive low power consumption and flexible beam control. By deploying multiple RIS at multiple ground locations to build a multi-level cascaded transmission link between satellites and users, it can avoid building obstruction and fill ground blind spots in satellite communication. However, existing channel estimation schemes still have many shortcomings. For example, the channel dimension is jointly determined by the satellite antenna, the reflective units of each level of RIS, and the user antenna. The channel parameters to be estimated are the product of the values ​​of each dimension, and the number of parameters increases exponentially with the number of RIS deployed. Traditional LS and compressed sensing estimation algorithms have exponentially increased pilot overhead and excessively high computing and storage costs, making them difficult to implement in engineering. Existing Tucker and CP tensor decomposition kernel tensor sizes expand exponentially with tensor order. General tensor ring decomposition is only designed for terrestrial millimeter-wave communication and lacks customized structures adapted to satellite chain cascade topologies. Dimensionality reduction methods such as random projection and simple tensor truncation destroy three physical characteristics of the channel: continuous distribution in the angle domain, clustered sparsity of multipath delays, and statistical correlation of neighboring RIS channels. Conventional Sparse Bayes (SBL) only achieves independent sparsity of matrix elements and does not utilize the spatial continuity constraints of satellite channel angles and delays. After compression, the angle spectrum and delay spectrum are discrete and fragmented, significantly degrading the channel estimation accuracy. RISs deployed in the same park, on the same building exterior wall, and at close range are in the same electromagnetic scattering environment and have similar channel statistical characteristics. However, existing solutions model the channel parameters of each RIS independently and cannot share prior information. Low signal-to-noise ratio RISs require a large number of pilots for support. At the same time, the ring rank of existing tensor rings relies on manual pre-setting. Fixed ring ranks cannot be adaptively matched in dense multipath scenarios in urban areas and open few-path scenarios in suburban areas, easily leading to underfitting or overfitting. Summary of the Invention

[0003] The purpose of this section is to outline some aspects of the embodiments of this application and to briefly describe some preferred embodiments. Simplifications or omissions may be made in this section, as well as in the abstract and title of this application, to avoid obscuring the purpose of these documents, and such simplifications or omissions should not be construed as limiting the scope of this application.

[0004] This application provides a low-complexity channel estimation method for satellite communication systems assisted by multiple RIS (Reference-Induced Rigs). It addresses the technical problems of excessive computational complexity, large pilot overhead, and easy loss of multilinear channel structure information caused by the dimensionality explosion of cascaded multi-RIS channels. The method achieves this by compressing the high-dimensional channel tensor into a ring-shaped contraction of multiple third-order core tensors through structured tensor ring decomposition, reducing the number of parameters to be estimated. Simultaneously, a joint structured sparse prior is applied to the ring factor matrix through a Bayesian hierarchical prior architecture, fully preserving the continuous structural features in the angle and time delay domains. Shared hyperparameter constraints are applied to multiple RIS within the same geographical region to maintain channel correlation between spatially adjacent RIS. Furthermore, variational Bayesian inference is used to adaptively determine the ring rank parameter, automatically balancing the degree of compression and expressive power during the estimation process. Finally, the high-dimensional channel is reconstructed through tensor ring contraction, outputting low-complexity, high-precision cascaded channel estimation results, reducing pilot overhead and computational complexity, and improving system spectrum utilization efficiency.

[0005] On the one hand, this application provides a low-complexity channel estimation method for satellite communication systems assisted by multiple RIS (Reference to RIS), including:

[0006] S1 models the cascaded channel of the multi-RIS assisted satellite communication system as a high-dimensional tensor, and uses structured tensor ring decomposition to represent the high-dimensional tensor as a ring-shaped contraction of multiple third-order tensors. Each third-order tensor corresponds to one RIS. The tensor ring decomposition has a configurable ring rank parameter.

[0007] S2 sets a Bayesian hierarchical prior architecture for each ring factor matrix obtained by the tensor ring decomposition, applies a joint structured sparse prior to each ring factor matrix, and applies shared hyperparameter constraints to the ring factor matrices of multiple RISs deployed in the same geographical region.

[0008] S3 establishes the likelihood function between the received signal and the ring factor matrix through linear observation, uses variational Bayesian inference to maximize the posterior probability, estimates each ring factor matrix and its posterior distribution, and determines the ring rank parameter of the tensor ring decomposition during the inference process.

[0009] S4 reconstructs the original high-dimensional channel tensor of the ring factor matrix by shrinking the tensor ring, and outputs the cascaded channel estimation results of the multi-RIS assisted satellite communication system.

[0010] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0011] Based on the number of RIS deployed in the satellite communication system and the number of reflection units contained in each RIS, the number of satellite-side antennas and the number of user-side antennas are also set, and the channels from the satellite to the first RIS, between adjacent RIS, and between the last RIS and the user are represented in matrix form.

[0012] The matrices are arranged in a concatenated order to construct a higher-order tensor. The order of the higher-order tensor is equal to the number of RIS plus two. The size of each dimension corresponds to the number of satellite antennas, the number of each RIS reflection unit, and the number of user antennas, respectively.

[0013] The structured tensor ring decomposition is configured as follows:

[0014] The higher-order tensor is decomposed into a ring-shaped connection structure of a set of third-order core tensors. The number of core tensors is equal to the order of the tensor. Each core tensor has three dimensions, two of which are associated with the corresponding channel dimension, and the other dimension serves as a link connecting adjacent core tensors. The link dimension between the first and last core tensors is fixed to a single dimension.

[0015] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0016] The value of each element in the constructed high-order channel tensor is expressed as the sum of the products of the corresponding elements in all core tensors;

[0017] Each core tensor contributes a factor. After multiplying the factors of all core tensors, the indices on all the connecting link dimensions are summed to complete one circumferential contraction operation.

[0018] The first core tensor is connected to the input of the second core tensor through its output link, the output of the second core tensor is connected to the input of the third core tensor, and so on, with the output of the last core tensor connected to the input of the first core tensor, forming a closed loop chain.

[0019] Each element value in a high-dimensional tensor is equal to the sum of the internal link indices after traversing all core tensors along the circular chain;

[0020] Through cyclic contraction, the number of parameters in the original high-dimensional tensor is compressed to the sum of the number of parameters in the core tensor, and the link dimension of the core tensor is smaller than the original channel dimension.

[0021] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0022] Assign an independent third-order core tensor to each deployed RIS;

[0023] The third-order core tensor includes three dimensions: the first dimension represents the signal direction or path index from the previous hop channel, the second dimension represents the index of each reflection unit on the current RIS, and the third dimension represents the signal direction or path index to the next hop channel.

[0024] For the link from the satellite to the first RIS and the link from the last RIS to the user, a core tensor is assigned, and the relationship between the antenna port and the signal direction is defined in the dimension of the assigned core tensor.

[0025] The dimension of the link connecting adjacent core tensors is called the ring rank, and the dimension of the link is a positive integer that can be dynamically adjusted during the estimation process.

[0026] The ring rank is used to represent the degree of compression and expressive power of the decomposition. If the ring rank is less than the standard ring rank, the ring rank achieves a high compression ratio. If the ring rank is greater than the standard ring rank, channel information is preserved. The maximum allowable value of the ring rank does not exceed the smaller of the channel dimensions corresponding to the two core tensors connected.

[0027] The optimal ring rank is automatically inferred from the received data, with practically acceptable computational complexity and expected channel estimation accuracy.

[0028] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0029] After completing the tensor ring decomposition, the ring factor matrix is ​​extracted from each third-order core tensor;

[0030] The method for extracting the ring factor matrix involves expanding or reshaping each core tensor along its connecting link dimension to obtain multiple two-dimensional matrices, which are the ring factor matrices.

[0031] Each ring factor matrix includes directional information of a link and the coupling relationship between RIS reflection units;

[0032] For each cyclic factor matrix, maintain an independent parameter structure, do not mix or superimpose, and record the topological connections between the cyclic factor matrices.

[0033] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, the method for setting up a Bayesian hierarchical prior architecture and applying a joint structured sparse prior to each ring factor matrix in S2 is as follows:

[0034] A probability distribution prior is set for each ring factor matrix. The probability distribution prior adopts a hierarchical structure: the first layer assumes that each row or column in the ring factor matrix follows a complex Gaussian distribution with a mean of zero. The variance of the Gaussian distribution is controlled by the hyperparameter of the second layer. The second layer sets a gamma distribution for the variance hyperparameter to form an automatic adjustment mechanism for sparsity.

[0035] The row indices of the ring factor matrix are mapped to spatial angular directions, and the column indices are mapped to signal propagation time delays. In the Bayesian prior, rows corresponding to adjacent angles share similar sparse patterns, and columns corresponding to adjacent time delays share similar sparse patterns, thus preserving the continuous structure of the angle domain and the time delay domain during compression.

[0036] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0037] Based on the actual deployment location information of each RIS, determine the geographical area of ​​each RIS; the geographical area includes building exterior walls, campus, or the distance between each RIS being less than a preset distance threshold;

[0038] For the ring factor matrices corresponding to multiple RIS belonging to the same geographical region, the same set of hyperparameters is set in the Bayesian hierarchical prior architecture, that is, the second-level variance hyperparameters of the ring factor matrices share the same prior distribution.

[0039] The ring factor matrices of different RIS are statistically drawn to similar sparse structures and energy distributions.

[0040] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0041] Pilot symbols are acquired, and the ring factor matrix is ​​reconstructed through tensor ring shrinkage to obtain the channel tensor. After linear operation with the pilot symbols, noise is superimposed to form the received signal.

[0042] The channel tensor is replaced with a tensor ring reconstruction expression of the ring factor matrix, and the likelihood function is obtained through the noise distribution.

[0043] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0044] Using the ring factor matrix as a latent variable, the posterior probability is constructed by combining the likelihood function and Bayesian hierarchical prior.

[0045] We introduce a decomposable variational distribution to approximate the true posterior. By iteratively optimizing the variational parameters, we minimize the difference between the two and update the posterior mean and covariance of each cyclic factor matrix in turn, while also updating the noise variance and hyperparameters.

[0046] As a preferred embodiment of the low-complexity channel estimation method for a multi-RIS-assisted satellite communication system proposed in this application, wherein:

[0047] Based on the ring connection topology established during tensor ring decomposition, each ring factor matrix is ​​sequentially condensed along the link dimension, that is, the link dimensions shared by adjacent ring factor matrices are multiplied and summed to finally recover the complete high-dimensional tensor. Each dimension of the high-dimensional tensor corresponds to the number of satellite antennas, the number of each RIS reflector unit, and the number of user antennas.

[0048] The beneficial effects of this application are as follows:

[0049] This application uses structured tensor ring decomposition to represent high-dimensional channel tensors as a ring-shaped contraction of multiple third-order tensors, and sets a configurable ring rank parameter to control the degree of compression, thereby reducing the number of parameters to be estimated from exponential to linear and solving the dimensionality explosion problem of multi-RIS cascaded channels.

[0050] This application establishes a Bayesian hierarchical prior architecture based on tensor ring decomposition. On the one hand, it applies a joint structured sparse prior to each ring factor matrix, mapping the matrix row index to spatial angles and column index to time delays. This allows matrix elements with adjacent angles and adjacent time delays to share sparse patterns, thus fully preserving the continuous structure in the angle and time delay domains during compression. On the other hand, it applies shared hyperparameter constraints to the ring factor matrices of multiple RISs deployed in the same geographical area, causing them to be statistically pulled to have similar sparse structures and energy distributions. This effectively preserves the inherent channel correlation between spatially adjacent RISs, reducing dimensionality while avoiding the destruction of the channel's multilinear structure and physical coupling relationship.

[0051] This application calculates the energy value of the ring factor matrix corresponding to each link dimension during the variational Bayes inference process, and sets the ring rank below the preset threshold to zero, thereby achieving adaptive determination of the ring rank parameter without the need for manual preset or cross-validation, and improving the adaptability to different channel environments.

[0052] This application compresses high-dimensional channel parameters into low-dimensional ring factor parameters through structured tensor ring decomposition, which greatly reduces the number of unknowns that need to be estimated. This enables channel estimation to be completed with only a small number of pilot symbols, effectively reducing pilot overhead and improving spectrum utilization efficiency. Attached Figure Description

[0053] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained through the drawings without creative effort. Wherein:

[0054] Figure 1 The overall flowchart of a low-complexity channel estimation method for a multi-RIS-assisted satellite communication system provided in this application;

[0055] Figure 2 This application provides a structured tensor ring decomposition and ring factor extraction sub-process for a low-complexity channel estimation method for multi-RIS assisted satellite communication systems;

[0056] Figure 3 The present application provides a variational Bayesian inference and ring-rank adaptive deterministic subprocess for a low-complexity channel estimation method for a multi-RIS-assisted satellite communication system.

[0057] Figure 4 A schematic diagram of tensor compression and recovery for a low-complexity channel estimation method for a multi-RIS-assisted satellite communication system provided in this application;

[0058] Figure 5 A schematic diagram of tensor accumulation for a low-complexity channel estimation method for a multi-RIS-assisted satellite communication system provided in this application. Detailed Implementation

[0059] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the specific embodiments of this application will be described in detail below with reference to the accompanying drawings.

[0060] Many specific details are set forth in the following description in order to provide a full understanding of this application. However, this application may also be implemented in other ways different from those described herein. Those skilled in the art can make similar extensions without departing from the spirit of this application. Therefore, this application is not limited to the specific embodiments disclosed below.

[0061] Secondly, the term "an embodiment" or "embodiment" as used herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of this application. The phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it a single embodiment or an embodiment selectively excluded from other embodiments.

[0062] Example 1

[0063] like Figure 1 and Figure 4 As shown, a low-complexity channel estimation method for a multi-RIS-assisted satellite communication system includes:

[0064] S1 models the cascaded channel of the multi-RIS assisted satellite communication system as a high-dimensional tensor and uses structured tensor ring decomposition to represent the high-dimensional tensor as a ring-shaped contraction of multiple third-order tensors. Each third-order tensor corresponds to one RIS. The tensor ring decomposition has a configurable ring rank parameter to control the degree of compression, so as to reduce the channel dimension and the number of parameters to be estimated.

[0065] Based on the number of RIS deployed in the satellite communication system and the number of reflection units contained in each RIS, the number of satellite-side antennas and the number of user-side antennas are also set, and the channels from the satellite to the first RIS, between adjacent RIS, and between the last RIS and the user are represented in matrix form.

[0066] This application first considers the number R (R≥1) of reconfigurable smart surfaces (RIS) actually deployed in the satellite communication system, and the number of reflective units contained in each RIS. , ,…, At the same time, set the number of antennas on the satellite side. And the number of antennas on the user side The cascaded channel is decomposed into several independent segments according to the physical order of signal propagation:

[0067] Channel segment 0: The channel between the satellite and the first RIS, represented as a complex matrix. ∈ ;

[0068] The r-th channel segment (r=1,…,R-1): The channel between the r-th RIS and the (r+1)-th RIS, represented as a complex matrix. ∈ ;

[0069] The R-th channel segment: the channel between the last RIS and the user, represented as a complex matrix. ∈ .

[0070] The number of rows and columns of each channel matrix segment is uniquely determined by the corresponding number of antenna ports or RIS reflector units. Meanwhile, since the dimension of each channel segment is much lower than the dimension of the overall channel after concatenation, the decomposition representation itself contains preliminary structured information, that is, the overall coupling is decomposed into local chain associations.

[0071] The channel matrices are arranged according to the order in which the signals propagate in the satellite-to-ground link. , … The users are arranged sequentially, and these matrices are used as different moduli or slices of a tensor, combined into a higher-order tensor. The order of the higher-order tensor is equal to the number of RIS R plus 2, corresponding to the satellite side, the 1st RIS, the 2nd RIS, ..., the Rth RIS, and the user side, for a total of R+2 dimensions. The size of the first dimension is equal to the number of satellite antennas. The size of the (r+1)th dimension (r=1,…,R) is equal to the number of reflection units in the r-th RIS. The size of the last dimension is equal to the number of user antennas. .

[0072] Specifically: The size of the first dimension = the number of satellite antennas ;

[0073] The size of the (r+1)th dimension (r=1,…,R) = the number of reflection units in the r-th RIS. ;

[0074] The size of the (R+2)th dimension = number of user antennas .

[0075] The structured tensor ring decomposition is configured as follows:

[0076] The higher-order tensor is decomposed into a ring-shaped structure of three-order core tensors, with the number of core tensors equal to the order of the tensor. Each core tensor has three dimensions, two of which are associated with the corresponding channel dimension, and the third dimension serves as a link connecting adjacent core tensors. The link dimension between the first and last core tensors is fixed as a single dimension to ensure the closure of the entire ring structure. Through this setup, the original complex higher-order tensor is represented as a series of lower-order core tensors that are mutually condensed along the link dimension.

[0077] This application employs structured tensor ring decomposition on the higher-order tensors constructed above, as follows:

[0078] The higher-order tensor is decomposed into a ring-connected structure of a set of third-order core tensors. The number of core tensors is equal to the order of the higher-order tensor, that is, R+2. Each core tensor has three dimensions, two of which are associated with the corresponding channel dimension, and the third dimension, which serves as the link connecting adjacent core tensors, is called the ring-rank dimension.

[0079] Specifically, for the k-th core tensor, its first correlation dimension corresponds to the dimension of the (k-1)-th channel segment, i.e., the number of ports in the previous hop; the second correlation dimension corresponds to the dimension of the k-th channel segment, i.e., the number of ports in the current hop; and the third dimension is used to connect with the (k-1)-th and (k+1)-th core tensors. The connection link dimension between the first and last two core tensors, i.e., the 1st and (R+2)th, is fixed to a single dimension, i.e., a dimension size of 1, to ensure that the entire ring structure can be closed without introducing additional degrees of freedom.

[0080] By decomposing a high-order tensor into a series of ring-connected third-order core tensors, this application achieves the first level of dimensionality reduction and compression of the original high-dimensional channel parameters, ensuring the closure and uniqueness of the decomposition, while avoiding parameter redundancy caused by endpoint divergence. Figure 2 As shown.

[0081] The matrices are arranged in a concatenated order to construct a higher-order tensor. The order of the higher-order tensor is equal to the number of RIS plus two. The size of each dimension corresponds to the number of satellite antennas, the number of each RIS reflection unit, and the number of user antennas, respectively.

[0082] The value of each element in the constructed high-order channel tensor is expressed as the sum of the products of the corresponding elements in all core tensors;

[0083] For the high-order channel tensor H constructed through the aforementioned steps, this application expresses the value of each element as the sum of the products of the corresponding elements in all core tensors.

[0084] Specifically, in a higher-order tensor, one is indexed ( , ,…, The element specified by ) is equal to the element at index () in the first core tensor. , The element of the second core tensor is indexed as ( , , The elements of the last core tensor are indexed as (...). , After multiplying the elements of ), then index all intermediate links. , ,…, Summation is the result obtained by traversing all possible combinations of bonds, where R is a positive integer representing the count. This is the index of the R-th link dimension.

[0085] By writing each element as the sum of the products of the corresponding factors of the core tensor, each degree of freedom of the higher-order tensor can be decoupled into the coupling of multiple lower-order degrees of freedom, thereby decomposing the global parameter estimation problem into local subproblems.

[0086] Each core tensor contributes a factor. After multiplying the factors of all core tensors, the indices on all the connecting link dimensions are summed to complete one circumferential contraction operation.

[0087] The process of summing the indices on all the link dimensions after multiplication is called a circumferential contraction operation. Each core tensor contributes a factor. After multiplying the factors of all core tensors, the indices on all the link dimensions are summed to complete the calculation of the element.

[0088] In this application, a preferred expression for tensor ring reconstruction calculation includes:

[0089]

[0090] Decompose H into R+2 third-order core tensors. , ,..., The number of core tensors is equal to the order of H, and each core tensor... It has three dimensions, and its size is denoted as ,in:

[0091] Associated with the dimension of the k-th channel segment;

[0092] and This is called the ring rank, which represents the dimension of the connection link between the current core tensor and the previous and next core tensors, respectively.

[0093] By defining a cyclic contraction operation, this application models complex multi-RIS channels as a computable tensor network structure, avoiding the exponential computation required to directly process high-dimensional tensors and simplifying the design of inference algorithms.

[0094] By combining cascaded multi-segment matrices into a single high-order tensor, the inherent high-order coupling relationship in multi-RIS cascaded channels is fully preserved. That is, the channel value of any transmit / receive combination depends on the combined effect of the satellite port, the reflection units of each RIS level, and the user port. This tensor representation is a prerequisite for subsequent structured tensor ring decomposition, avoiding the loss of cross-RIS correlation information due to processing each segment matrix separately.

[0095] The higher-order tensor is decomposed into a ring-shaped connection structure of a set of third-order core tensors. The number of core tensors is equal to the order of the tensor. Each core tensor has three dimensions, two of which are associated with the corresponding channel dimension, and the other dimension serves as a link connecting adjacent core tensors. The link dimension between the first and last core tensors is fixed to a single dimension.

[0096] This application employs structured tensor ring decomposition for the constructed higher-order tensors, as follows:

[0097] The higher-order tensor is decomposed into a ring-connected structure of a set of third-order core tensors. The number of core tensors is equal to the order of the higher-order tensor, i.e., R+2. Each core tensor has three dimensions, two of which are associated with the corresponding channel dimension, and the third dimension, which serves as the link between adjacent core tensors, can also be called the ring-rank dimension.

[0098] Specifically, for the k-th core tensor, the first correlation dimension corresponds to the dimension of the (k-1)-th channel segment, i.e., the number of ports in the previous hop; the second correlation dimension corresponds to the dimension of the k-th channel segment, i.e., the number of ports in the current hop; and the third dimension is used to connect with the (k-1)-th and (k+1)-th core tensors. The connection link dimension between the first and last two core tensors, i.e., the 1st and (R+2)th, is fixed to a single dimension, i.e., a dimension size of 1, to ensure that the entire ring structure can be closed without introducing additional degrees of freedom.

[0099] By decomposing a high-order tensor into a series of ring-connected third-order core tensors, this application achieves the first level of dimensionality reduction and compression of the original high-dimensional channel parameters. The number of parameters in the original high-order tensor is the product of the dimensions, while the total number of parameters after decomposition becomes the sum of the number of parameters in each core tensor. Each core tensor is associated with only two adjacent channel segments and one link dimension. The link dimension, i.e., the ring rank, is much smaller than the original channel dimension, thereby reducing the overall number of parameters. Fixing the first and last links as a single dimension ensures the closure and uniqueness of the decomposition, while avoiding parameter redundancy caused by endpoint divergence.

[0100] The value of each element in the constructed high-order channel tensor is expressed as the sum of the products of the corresponding elements in all core tensors;

[0101] Each core tensor contributes a factor. After multiplying the factors of all core tensors, the indices on all the connecting link dimensions are summed to complete one circumferential contraction operation.

[0102] The first core tensor is connected to the input of the second core tensor through its output link, the output of the second core tensor is connected to the input of the third core tensor, and so on, with the output of the last core tensor connected to the input of the first core tensor, forming a closed loop chain.

[0103] In tensor ring decomposition, the first core tensor is connected to the second core tensor's input link (its first dimension) via its output link (its third dimension); the second core tensor's output link is connected to the third core tensor's input link; and so on, until the last core tensor's output link is connected to the first core tensor's input link, thus forming a closed loop. This closed connection ensures that all core tensors form a topological ring with no free endpoints.

[0104] The closed-loop chain used in this application allows information to flow bidirectionally on the loop, enabling each core tensor to simultaneously integrate information from both directions. This achieves the same expressive power as T-decomposition with a smaller ring rank, increasing the flexibility of decomposition. The closed-loop chain can more accurately fit the actual physical propagation characteristics, and the closed structure also makes the uniqueness condition of the decomposition more relaxed, reducing the difficulty of parameter identification.

[0105] Each element value in a high-dimensional tensor is equal to the sum of the internal link indices after traversing all core tensors along the circular chain;

[0106] Each element value in a high-dimensional tensor is equal to the sum of the internal link indices after traversing all core tensors along a closed loop chain. The channel coefficients corresponding to any transmit / receive combination and each level of RIS reflection unit combination can be obtained by sequentially accessing the corresponding entries in each core tensor, multiplying the values ​​of all entries, and then summing over all possible intermediate connection states.

[0107] Through cyclic contraction, the number of parameters in the original high-dimensional tensor is compressed to the sum of the number of parameters in the core tensor. The link dimension of the core tensor is smaller than the original channel dimension, thus achieving dimensionality reduction and compression. Figure 2 As shown, the structured tensor ring decomposition and ring factor extraction sub-process of this application includes: first, arranging each segment of the channel matrix into a high-order tensor, then decomposing it into a ring-connected third-order core tensor, and finally expanding each core tensor into a ring factor matrix along the link dimension.

[0108] Although multi-RIS cascaded channels have extremely high dimensionality, their inherent degrees of freedom are limited by the finite number of scatterers and multipath propagation paths in the physical propagation environment. Therefore, they are essentially low-rank tensors. Tensor ring decomposition utilizes this low-rank characteristic to capture the main channel variation patterns with a small number of ring-rank parameters, while attributing unimportant disturbances to noise terms. Compared with traditional matrix decomposition or higher-order tensor decomposition, the parameter growth of tensor ring decomposition is linear, while the kernel tensor size of Tucker decomposition still grows exponentially with order. Therefore, this application has an overwhelming complexity advantage when the number of RIS, R, is large.

[0109] The compressed core tensors are fewer in number and have lower dimensionality, which allows subsequent Bayesian hierarchical priors and variational inference to be performed in a low-dimensional space, avoiding the huge computational and storage overhead caused by direct estimation in the high-dimensional channel space.

[0110] Assign an independent third-order core tensor to each deployed RIS;

[0111] For each reconfigurable smart surface deployed in a satellite communication scenario, this application assigns it an independent third-order core tensor. The core tensor is physically mapped one-to-one with the corresponding RIS: the reflection coefficient matrix, cell position layout, and directional characteristics of the incident and outgoing beams of the RIS are all encoded in the numerical structure of the core tensor.

[0112] By assigning a core tensor to each RIS independently, this application decouples the complex coupling relationship of the entire cascaded channel into a combination of several local modules.

[0113] The signal modulation effect of each RIS depends only on its own reflection coefficient and the direction of the incident signal, and its modulation behavior is conditionally independent of that of other RISs. The channel characteristics of each RIS are characterized by independent third-order tensors, conforming to the Markov property of physical propagation. (Satellite to...) The channel and arrive The channel in a given The states are independent of each other. This allocation method allows Bayesian inference to update factors one by one according to RIS, avoiding the curse of dimensionality caused by jointly estimating all RIS parameters.

[0114] The third-order core tensor includes three dimensions: the first dimension represents the signal direction or path index from the previous hop channel, the second dimension represents the index of each reflection unit on the current RIS, and the third dimension represents the signal direction or path index to the next hop channel.

[0115] In this application, the third-order core tensor corresponding to each RIS contains three dimensions, each assigned a clear physical meaning:

[0116] The first dimension represents the signal direction or path index from the previous hop channel. The size of the dimension is equal to the equivalent channel dimension of the previous hop, such as the number of satellite antenna ports or the number of the previous RIS reflector units. Each index in this dimension corresponds to an incident direction or incident path, representing the spatial distribution characteristics of the signal arriving at the current RIS.

[0117] The second dimension represents the index of each reflection unit on the current RIS. The size of the dimension is equal to the total number of reflection units on the current RIS. The index on the dimension directly corresponds to the physical unit position on the RIS panel. Each unit can independently adjust the reflection coefficient, i.e., the phase and amplitude.

[0118] The third dimension represents the signal direction or path index to the next hop channel. The size of the dimension is equal to the equivalent channel dimension of the next hop. The index on the dimension corresponds to the outgoing direction or outgoing path after reflection, which describes the radiation characteristics of the signal after RIS modulation.

[0119] This application explicitly divides the core tensor into three dimensions: the incident direction, the RIS element, and the exit direction. Each RIS element independently controls the reflection phase and amplitude of the incident wave, and the overall reflection field is the superposition of the contributions of each element. By treating the incident and exit directions as independent dimensions, the core tensor can naturally express the angular selectivity in multipath propagation.

[0120] For the link from the satellite to the first RIS and the link from the last RIS to the user, core tensors are assigned respectively. The assigned core tensor dimensions do not include RIS reflection element indices, but only reflect the relationship between the antenna port and the signal direction.

[0121] For the link from the satellite to the first RIS and the link from the last RIS to the user, core tensors are assigned respectively. The assigned core tensor dimensions do not include RIS reflection element indices, but only reflect the relationship between the antenna port and the signal direction.

[0122] Specifically, the satellite link core tensor contains only the satellite antenna port index and the signal direction index to the first RIS; the user link core tensor contains only two valid dimensions: the signal direction index from the last RIS and the user antenna port index. Without affecting the ring structure, the third or first dimension of the link core tensor is set to a single dimension to adapt to the ring connection requirement where the beginning and end are fixed to a single dimension.

[0123] The dimension of the link connecting adjacent core tensors is called the ring rank, and the dimension of the link is a positive integer that can be dynamically adjusted during the estimation process.

[0124] In this application, the ring rank is a positive integer that can be dynamically adjusted during the estimation process. Its value determines the number of channels for information flow between adjacent core tensors. The larger the ring rank, the more information patterns can be exchanged between adjacent core tensors, and the stronger the fitting ability of the decomposition. The smaller the ring rank, the tighter the parameter constraints of the decomposition and the higher the degree of compression.

[0125] The inherent degrees of freedom of a real wireless channel, namely the effective multipath number and the number of resolvable scatterers, are usually limited and vary with the environment. A fixed ring rank is either too large, leading to overfitting and wasted computation, or too small, failing to capture the changes in the real channel. This application allows the ring rank to be automatically adjusted based on the received data during inference, so that the complexity of the decomposition matches the complexity of the channel itself. In open scenarios with low degrees of freedom, a small ring rank is automatically selected, while in urban scenarios with abundant multipaths, a large ring rank is automatically selected, avoiding the tedious manual parameter tuning and improving the accuracy of the method for different deployment environments.

[0126] The ring rank is used to represent the degree of compression and expressive power of the decomposition. If the ring rank is less than the standard ring rank, the ring rank achieves a high compression ratio. If the ring rank is greater than the standard ring rank, channel information is preserved. The maximum allowable value of the ring rank does not exceed the smaller of the channel dimensions corresponding to the two core tensors connected.

[0127] The standard ring rank is the theoretical minimum ring rank that can completely represent a channel.

[0128] The specific value of the ring rank reflects the trade-off between the compression and expressive power of the ring tensor decomposition. A smaller ring rank achieves a higher compression ratio but may lose some channel detail information; a larger ring rank retains more channel information and improves model fitting accuracy, but also increases the number of parameters and computational burden. The maximum allowable value of the ring rank is limited by the smaller of the channel dimensions of the two core tensors it connects to, because the link dimension cannot exceed the minimum value of the two adjacent related dimensions, otherwise redundant zero parameters will appear.

[0129] When multiplying two matrices, the rank of the product matrix cannot exceed the minimum of the ranks of the two factor matrices. In tensor ring decomposition, the link dimension plays a role similar to an intermediate rank, so its maximum value is naturally limited by the lower of the two channel dimensions. Explicitly disclosing this upper limit allows those skilled in the art to reasonably set the search range for the ring rank and avoid unnecessary calculations in invalid intervals.

[0130] With practically acceptable computational complexity and desired channel estimation accuracy, the optimal ring rank is automatically inferred from the received data, so that the decomposition result can both fully fit the channel structure and avoid generating redundant parameters.

[0131] This application does not rely on manually preset fixed ring rank values. Instead, it automatically infers the optimal ring rank during the channel estimation process using received data, based on a practically acceptable upper limit of computational complexity and the desired channel estimation accuracy requirements. Specifically, in each iteration of the variational Bayesian inference, the energy value of the ring factor matrix corresponding to each link dimension is calculated, for example, the square of the Frobenius norm or the magnitude of the singular values.

[0132] With practically acceptable computational complexity and desired channel estimation accuracy, the optimal ring rank is automatically inferred from the received data, so that the decomposition result can both fully fit the channel structure and avoid generating redundant parameters.

[0133] S2 sets up a Bayesian hierarchical prior architecture for each ring factor matrix obtained by the tensor ring decomposition, applies a joint structured sparse prior to each ring factor matrix to preserve the structural features in the angle domain and the time delay domain, and applies shared hyperparameter constraints to the ring factor matrices of multiple RISs deployed in the same geographical area to maintain the correlation between different RIS channels.

[0134] After completing the tensor ring decomposition, the ring factor matrix is ​​extracted from each third-order core tensor;

[0135] After completing the structured tensor ring decomposition, this application extracts the corresponding ring factor matrix from each third-order core tensor. Specifically, each core tensor is expanded or reshaped along its connecting link dimension, i.e., the third or first dimension serving as the input / output connection channel, transforming the original third-order tensor into a set of two-dimensional matrices. These two-dimensional matrices are the ring factor matrices described in this application.

[0136] Converting the third-order core tensor into a ring factor matrix is ​​to unify the decomposed parameters into matrix form.

[0137] In tensor ring decomposition, each core tensor is connected to its adjacent core tensors only through a link dimension. Therefore, by taking this link dimension as the row or column direction of the matrix and merging the other two channel dimensions into one direction, the third-order tensor can be flattened into a two-dimensional matrix. This does not change the value of the parameters, but only changes the indexing method of the parameters. Therefore, it does not introduce any approximation or information loss. The flattened ring factor matrix can be directly used for reasoning using standard linear algebra and probability theory tools, which greatly reduces the complexity of the algorithm implementation.

[0138] The method for extracting the ring factor matrix involves expanding or reshaping each core tensor along its connecting link dimension to obtain multiple two-dimensional matrices, which are the ring factor matrices.

[0139] The specific operation for extracting the ring factor matrix in this application is as follows: For the k-th third-order core tensor G(k), its dimension is... When expanding along the link dimension (i.e., the ring-rank dimension connecting the preceding and following sections), there are two common patterns:

[0140] The first mode is to merge the first two dimensions into one dimension and use the third dimension as another dimension to obtain a new matrix.

[0141] The second method involves treating the first dimension as a row and merging the last two dimensions into one dimension to obtain a new matrix.

[0142] The mode adopted depends on the application direction of the sparse prior. For example, if the joint features of the incident direction-RIS unit are to be sparsified, then mode one is adopted; if the energy is to be estimated for the ring rank dimension, then mode two is adopted.

[0143] Each ring factor matrix includes directional information of a link and the coupling relationship between RIS reflection units;

[0144] Each ring factor matrix carries the directional information of a specific link segment and the coupling relationship between RIS reflection units.

[0145] Specifically:

[0146] For the ring factor matrix extracted from the core tensor of the corresponding RIS, its row (or column) direction reflects the combination pattern between the previous hop channel and the current RIS unit, and its column (or row) direction reflects the combination pattern between the current RIS unit and the next hop channel.

[0147] For the ring factor matrix extracted from the core tensor of the corresponding satellite link or user link, since one dimension is fixed as a single dimension, the ring factor matrix degenerates into a vector or a simple matrix to represent the mapping relationship between the satellite antenna port and the incident direction of the first RIS, or the mapping relationship between the exit direction of the last RIS and the user antenna port.

[0148] For each cyclic factor matrix, maintain an independent parameter structure, do not mix or superimpose, and record the topological connections between the cyclic factor matrices.

[0149] The method for setting up a Bayesian hierarchical prior architecture and applying a joint structured sparse prior to each cyclic factor matrix in S2 is as follows:

[0150] A probability distribution prior is set for each ring factor matrix. The probability distribution prior adopts a hierarchical structure: the first layer assumes that each row or column in the ring factor matrix follows a complex Gaussian distribution with a mean of zero. The variance of the Gaussian distribution is controlled by the hyperparameter of the second layer. The second layer sets a gamma distribution for the variance hyperparameter to form an automatic adjustment mechanism for sparsity.

[0151] For the ring factor matrix extracted from the core tensor of the corresponding RIS, the rows of the ring factor matrix are mapped to the incident direction, and the columns are mapped to the time delay; for the ring factor matrix corresponding to the satellite link, only one dimension has physical meaning, and the mapping method is analogous.

[0152] The first layer prior is a complex Gaussian distribution of rows / columns;

[0153] For any ring factor matrix denoted as matrix U, the row index corresponds to the spatial angular direction and the column index corresponds to the signal propagation delay. In the first-level prior, this application assumes that each row of the matrix follows a complex Gaussian distribution with a mean of zero and that different rows are independent of each other.

[0154] It should be noted that, taking rows as an example, assuming the ring factor matrix has P rows, each containing Q elements, the first-level prior stipulates that all elements in the p-th row (p=1,…,P) follow a common complex Gaussian distribution with a mean of zero and a positive variance. Mathematically, this can be expressed as: the probability density function of the p-th row is uniquely determined by the variance. It is worth noting that the variances corresponding to different rows can be different, which means that the signal energy in different angular directions can differ. The angular direction with stronger energy corresponds to a larger variance, while the angular direction with weaker energy corresponds to a smaller variance.

[0155] The first-level prior associates each row of the ring factor matrix with a variance parameter. The magnitude of the variance parameter determines the overall amplitude level of the elements in that row: the larger the variance, the more likely the elements in that row are to take larger values, indicating significant channel energy in the angular direction; when the variance approaches zero, all elements in that row are forced to be compressed to near zero, indicating that there is almost no effective signal in that angular direction.

[0156] The second-level prior is to set the variance parameter as a gamma distribution;

[0157] The variance parameter in the first-level prior is not a fixed constant, but is treated as a random variable and assigned to the second-level prior. This application uses the gamma distribution as the prior distribution for the variance parameter. The gamma distribution is a continuous probability distribution on positive real numbers, possessing shape and scale parameters. The shape and scale parameters are set to very small positive numbers (e.g., ...). The prior constructed from these priors is called the uninformative prior, which means that no subjective preference is added to the magnitude of the variance parameter before the observation data is obtained.

[0158] During Bayesian inference, the received pilot signal works in conjunction with prior information to update the posterior distribution of each variance parameter. If channel energy does exist in a certain angular direction, the data will support the variance parameter corresponding to that direction to take a large value. Conversely, if there is no effective signal energy in a certain angular direction, the data will make the posterior estimate of that variance parameter approach zero, causing the elements of the corresponding row in the first-level prior to be strongly compressed to near zero. There is no need to manually set a sparsity penalty coefficient, nor is it necessary to pre-specify how many angular directions to retain.

[0159] The row indices of the ring factor matrix are mapped to spatial angular directions, and the column indices are mapped to signal propagation time delays. In the Bayesian prior, rows corresponding to adjacent angles share similar sparse patterns, and columns corresponding to adjacent time delays share similar sparse patterns.

[0160] The specific method to achieve a smooth shared sparse pattern is as follows: set a first-order Markov chain prior for the logarithmic precision hyperparameter of each row or column. Taking the angle domain as an example, let the difference between the logarithmic precision of the p-th row and the logarithmic precision of the (p-1)-th row follow a normal distribution with zero mean and minimal variance. This variance value can be pre-fixed to a very small positive number, for example... This indicates that the channel energy changes continuously between adjacent angles, the processing in the time delay domain is completely symmetrical, and the difference in logarithmic precision between adjacent columns also follows the same microvariance normal distribution. In variational Bayesian iteration, when updating the precision hyperparameter of each row, if an adjacent row has higher precision, the precision of this row will also be pushed to a higher value; and vice versa. This coupling mechanism makes the estimated angle power spectrum and time delay power spectrum present a natural continuous cluster structure, avoiding isolated discrete peaks. Those skilled in the art do not need to manually set the sparsity threshold. All smoothing constraints are automatically driven by the received data through Bayesian inference, preserving the continuous structure of the angle domain and time delay domain during compression.

[0161] Based on the geometry of the satellite and RIS and the operating frequency of the system, the possible range of signal incident angles is determined.

[0162] For example, for a RIS deployed on the ground, the incident angles from the satellite are typically concentrated within a small angular sector. This angular range is uniformly divided into P smaller intervals, each corresponding to a central angle value. Then, the first row of the ring factor matrix is ​​defined to correspond to the first angular interval, the second row to the second angular interval, and so on, with the Pth row corresponding to the Pth angular interval.

[0163] Based on the typical distance of the satellite-to-ground link and the maximum possible delay difference in multipath propagation, the maximum delay spread is determined. Using the system's sampling interval as the delay resolution, the maximum delay range is divided into Q delay grid points, each corresponding to a specific delay value. Then, the first column of the ring factor matrix is ​​defined as corresponding to the smallest delay grid point (usually the line-of-sight path), the second smallest as the second smallest, and so on, with the Qth column corresponding to the largest delay grid point. Thus, a definite mapping relationship is established between column indices and delays.

[0164] The sparse state of each column in the ring factor matrix directly corresponds to the intensity of the multipath component at that delay position. The columns corresponding to important delay paths are retained, while the columns corresponding to delay positions without multipath are automatically compressed, so that the estimation results can clearly show the multipath delay distribution of the channel.

[0165] If only a hierarchical prior is used, the rows corresponding to different angles are statistically independent, leading to unreasonable phenomena in the estimation results: for example, the p-th row is retained, while the adjacent p+1-th row is completely compressed. Although physically, due to the angular expansion of the scatterer, the signal energy of adjacent angles usually changes continuously, there will not be a situation where an isolated discrete angle has a signal while its adjacent angle has no signal at all.

[0166] This application further introduces constraints in the Bayesian prior, so that rows corresponding to adjacent angles share similar sparse patterns.

[0167] As a preferred embodiment, the transfer variance τ² can be preset to a very small positive number, such as τ² = Alternatively, an inverse gamma prior can be introduced for τ² and updated in variational Bayesian inference.

[0168] Specifically, this includes establishing a correlation between adjacent angles for the hyperparameters that control the sparsity of each row, namely the variance parameter in the first-layer prior.

[0169] A preferred example includes assuming that the logarithmic form of the hyperparameters follows a first-order Gaussian Markov chain, that is, the difference between the logarithmic hyperparameters corresponding to adjacent angles follows a Gaussian distribution with a mean of zero. In this case, the hyperparameters will not change drastically between adjacent angles, but will change smoothly.

[0170] Let the precision hyperparameter of the p-th row be... ,make =ln ,set up:

[0171] ~N(0, ),

[0172] | ~N( , )

[0173] p=2,…,P

[0174] in As a smoothing coefficient, it can be preset to a fixed value (e.g. =0.01), or learn by giving inverse gamma priors.

[0175] In the time-delay domain, this application forces columns corresponding to adjacent time delays to share similar sparsity patterns and establishes smooth constraints between adjacent time delays by controlling the sparsity of each column hyperparameter. For example, it is assumed that the difference between the logarithmic sparse hyperparameters corresponding to adjacent time delays follows a Gaussian distribution with small variance.

[0176] A preferred example: Let the precision hyperparameter of the q-th column be... ,make ;

[0177]

[0178]

[0179] in, is the time-delay domain smoothing coefficient.

[0180] Multipath components typically appear in the form of clusters, i.e. multiple paths generated by the same reflector with very small time delay differences, forming a continuous time delay interval. This forces adjacent columns to share similar sparse patterns, ensuring that the estimated time delay power spectrum presents a reasonable continuous cluster structure rather than isolated, discrete single-point peaks.

[0181] This application does not require prior knowledge of the number of incoming signals. It automatically identifies which angles and directions contain valid signals from the data and compresses the parameters corresponding to other angles and directions to near zero. It also does not require prior knowledge of the number of multipath signals. It automatically identifies which time delay positions contain valid multipath components and compresses the parameters corresponding to other time delays. The sparse patterns between adjacent angles are smoothly correlated, and the estimated angular power spectrum is continuous and unbroken, conforming to physical reality. The sparse patterns between adjacent time delays are smoothly correlated, and the estimated time delay power spectrum exhibits a natural cluster structure. All sparse patterns are automatically determined by the received data through Bayesian inference, without the need for manual setting of sparsity, penalty coefficients, or truncation thresholds.

[0182] The method for applying shared hyperparameter constraints to the ring factor matrices of multiple RIS deployed in the same geographical region in S2 is as follows:

[0183] To implement the shared hyperparameter constraints in variational Bayesian updates, variables and update rules are defined as follows:

[0184] Suppose that the same geographical region group S contains L RIS, for each RIS... ∈S, whose ring factor matrix is ​​denoted as Where k corresponds to the k-th core tensor, for all ∈S, matrix The p-th row corresponds to the same physical angle interval and shares the same precision hyperparameter. .

[0185] Where, let prior knowledge be... ,Pick = = Contribution in likelihood: For each ,for The p-th row conforms to CN(0, );

[0186] definition for The number of columns;

[0187] Shared precision hyperparameters The variational posterior remains a Gamma distribution with the following parameters:

[0188]

[0189]

[0190] in, = and The first The posterior mean and covariance sub-block in the p-th row of the k-th ring factor matrix of a RIS.

[0191] Based on the actual deployment location information of each RIS, determine the geographical area of ​​each RIS; the geographical area includes building exterior walls, campus, or the distance between each RIS being less than a preset distance threshold;

[0192] For the ring factor matrices corresponding to multiple RIS belonging to the same geographical region, the same set of hyperparameters is set in the Bayesian hierarchical prior architecture, that is, the second-level variance hyperparameters of the ring factor matrices share the same prior distribution.

[0193] The ring factor matrices of different RIS are statistically drawn to similar sparse structures and energy distributions. In actual computation with shared hyperparameter constraints, it is necessary to extract the posterior second moment of a certain row in each ring factor matrix, that is, the expected square length of the row vector.

[0194] The specific extraction method is as follows: First, it is agreed that the ring factor matrix is ​​vectorized in column-major order, that is, all elements in the first column are arranged first, then the second column, and so on. Under this agreement, the position of the element in the p-th row and q-th column of the original matrix in the vectorized long vector is p+(q-1)×D, where D is the total number of rows in the matrix, and the indices of all Q elements in the p-th row after vectorization are p, p+D, p+2D, ..., p+(Q-1)×D. The submatrix formed by the intersection of rows and columns is extracted from the overall posterior covariance matrix, which is the row covariance sub-block. The expected square length of the row vector is equal to the square of the length of the current row's posterior mean vector plus the trace of the covariance sub-block (i.e., the sum of the diagonal elements). This calculation process is a standard multivariate statistical operation, which can be independently implemented by those skilled in the art based on Bayesian inference and the above rules, thereby completing the update of shared hyperparameters.

[0195] This preserves the channel correlation between RIS due to their spatial proximity; the reflection path angle distribution of each RIS is similar, the signal attenuation is similar, and the multipath delay structure is highly consistent; this constraint embeds the physical coupling information between RIS into the estimation model without increasing the number of parameters to be estimated, thus avoiding the loss of correlation due to independent estimation.

[0196] After completing the joint structured sparse prior setting for a single ring factor matrix, this application further considers the spatial correlation between multiple RIS. In actual satellite communication scenarios, multiple RIS are deployed in the same geographical area, such as different exterior walls of the same building, multiple light poles in the same park, or multiple RIS with small distances between them. The electromagnetic propagation environment of the RIS is similar, such as having similar building obstruction distribution, similar atmospheric attenuation characteristics, and similar scatterer layout. The channels of the RIS show statistical correlation, specifically: the reflection path angle distribution of each RIS is similar, the signal attenuation degree is similar, and the multipath delay structure is consistent.

[0197] This application links multiple RIS within the same geographical region in a Bayesian hierarchical prior architecture, so that the ring factor matrix is ​​statistically pulled toward similar sparse structures and energy distributions.

[0198] Specifically, the actual deployment location information of each RIS is first obtained. This information can be obtained through methods including, but not limited to: GPS coordinate positioning, reading location parameters from base station configuration files, or manual input of on-site survey data. Based on preset geographical region determination rules, these RIS are then grouped. These determination rules include one or more of the following:

[0199] If multiple RIS are installed on the exterior wall of the same building, such as on different facades or different floors, they are considered to belong to the same geographical area.

[0200] If multiple RIS are located in the same park, such as a campus, airport, stadium, or science park, they are considered to belong to the same geographical area.

[0201] If the spacing is less than a preset distance threshold, and the straight-line spatial distance between any two RIS is less than a preset distance threshold, such as 50 meters or 100 meters, and this threshold can be set according to the propagation characteristics of urban or suburban environments, then they are classified into the same geographical area. The distance threshold can be a configurable parameter of the system, allowing it to be adjusted according to different scenarios.

[0202] Furthermore, all RIS are divided into several geographical regions. RIS within each group are considered to share similar propagation environments, while RIS between different groups may face different propagation conditions. For example, one group may be located in a densely built-up city center, while another group may be located in an open suburb.

[0203] Transforming proximity in physical space into grouping criteria in statistical modeling ensures that shared constraints are only applied between RISs that truly have environmental similarities, avoiding the unreasonable forced association of RISs that are far apart and have vastly different propagation environments.

[0204] For multiple RISs that are determined to belong to the same geographical region, this application sets the same set of hyperparameters for the ring factor matrix in the Bayesian hierarchical prior architecture.

[0205] Specifically, each ring factor matrix is ​​equipped with a set of Bayesian hierarchical priors, where the second-level hyperparameters, such as gamma distribution parameters that control the sparsity of each row or column, or smoothing coefficients in the angular and temporal domains, determine the sparsity pattern and energy distribution. All RIS ring factor matrices in the same geographic region share the exact same set of second-level hyperparameters.

[0206] There are L RIS in a certain geographical region group, and the corresponding ring factor matrices are U(1), U(2), ..., U(L).

[0207] For each matrix, the row variance hyperparameters γ1, γ2, ... involved in its first-level a priori row sparse prior are, according to this application, common and shared among the L matrices. That is, regardless of which RIS (Relative Rig) it is, the ring factor matrices of all RISes follow a row sparse pattern controlled by the same set of hyperparameters. Similarly, the same sharing strategy is adopted for column-direction hyperparameters and smoothing constraint hyperparameters in the angle / delay domain.

[0208] In the absence of sharing, the first The precision hyperparameter of the p-th row of the RIS ring factor matrix is ​​denoted as γp( These are independent of each other. Under shared constraints, the same row index of all RISes shares the same hyperparameter. In the Bayesian model, the precision hyperparameter is replaced by the same random variable.

[0209] When updating the ring factor matrix of each RIS, its sparsity pattern is not determined solely by its own received signal, but also by borrowing information from other RISs in the same region. For example, if a RIS has poor pilot quality due to occlusion or noise, and its own data is insufficient to reliably determine which angles and directions are active, then the shared hyperparameters will borrow data from other RISs in the same region with higher signal-to-noise ratios to provide the RIS with a robust prior to the sparse structure.

[0210] By sharing hyperparameter constraints, the ring factor matrices of different RIS within the same geographic region will statistically be naturally drawn towards similar sparse structures and energy distributions. Specifically, this includes similar angular distributions, similar energy attenuation levels, and consistent multipath delay structures.

[0211] The similarity of angular distributions means that the main incident angles of each RIS tend to be consistent. For example, if the dominant signal in the region comes from a specific azimuth of the satellite, then the ring factor matrices of all RIS will retain significant energy in the corresponding angular row, while being compressed to near zero in other angular rows.

[0212] Similar energy attenuation means that the overall energy level of the signals received by each RIS is comparable. Shared hyperparameters keep the amplitude scale of the ring factor matrix of different RISs on the same order of magnitude, avoiding energy jumps caused by independent estimation. For example, if one RIS estimates the channel energy as 0.1, another neighboring RIS might estimate it as 10.

[0213] The multipath delay structure is consistent in that the delay power spectrum shapes of each RIS are similar. If there is a principal path delay extension caused by reflection from buildings in the region, then the ring factor matrices of all RIS will retain energy in the corresponding delay columns, forming a consistent multipath distribution.

[0214] The shared constraints embed the spatial correlation between RIS into the estimation model without increasing the number of parameters to be estimated, preserving the inherent channel correlation between RIS: in real propagation environments, spatially adjacent RIS will inevitably have similar statistical characteristics. This application uses shared hyperparameters to make the estimation results naturally exhibit this correlation, avoiding the problem of losing correlation due to noise interference in independent estimation; when the pilot overhead of a certain RIS is small or the instantaneous signal-to-noise ratio is low, its own observation data is insufficient to obtain a reliable sparse estimate; in independent estimation, shared hyperparameters bind RIS in the same region, reducing the overall model complexity and effectively suppressing overfitting, especially when the number of RIS is large and the total pilot resources are limited.

[0215] S3 establishes the likelihood function between the received signal and the ring factor matrix through linear observation, uses variational Bayesian inference to maximize the posterior probability, estimates each ring factor matrix and its posterior distribution, and determines the ring rank parameter of the tensor ring decomposition during the inference process.

[0216] For example: Calculate the ring factor matrix corresponding to the kth link dimension. The square of the Frobenius norm is denoted as .

[0217] The ring rank adaptive adjustment is performed after each iteration. For each link index m=1,2,…,R+1, the link index corresponds to the ring rank. Perform the following judgment and update steps.

[0218] Extract the posterior mean of the left and right ring factor matrices adjacent to the m-th link:

[0219] Left-hand ring factor matrix: When m=1, When m>1, ;

[0220] Right-hand ring factor matrix: First, reshape the right-hand ring factor matrix as follows: , recorded as .

[0221] is an m-order left-hand ring factor matrix, and L is the total number of pilot time slots, i.e., the number of columns in the pilot symbol matrix.

[0222] It is an m-order right-hand ring factor matrix.

[0223] Further, calculate the energy matrix: ;

[0224] Furthermore, regarding the energy matrix Perform singular value decomposition, i.e.: ,in ;

[0225] Calculate the cumulative energy ratio Choose the smallest Make ( )≥1- ,in This is a preset threshold. If... < If so, then the cropping will be performed.

[0226] The cutting method includes:

[0227] reserve The former Column, denoted as ;

[0228] Update the left-hand matrix: ,correspond Becoming a new dimension ;

[0229] Update the right-hand matrix: Later reshaped into ,correspond Becoming a new dimension Finally, the ring rank Updated to .

[0230] It should be noted that, to prevent underfitting of the model due to an excessively small initial ring rank, this application introduces an active expansion mechanism during the variational Bayes iteration process. Specifically:

[0231] Set a maximum number of iterations. When the number of iterations does not exceed half of the maximum number of iterations, after each iteration, determine whether the current ring rank needs to be increased. The judgment condition is: the increment of the variational lower bound calculated by three consecutive iterations is less than a minimum threshold, and the current ring rank has not reached its theoretical upper limit, i.e., it does not exceed the smaller of the two channel dimensions connected, and it also does not exceed the value of the ring rank of the adjacent link.

[0232] When the above conditions are met, an expansion operation is performed: the ring rank is increased by 1, but not exceeding the theoretical upper limit. During expansion, the dimensions of the two ring factor matrices adjacent to the link are increased by adding a column to the right side of the left matrix and a row to the bottom of the right matrix. The values ​​of the new elements are randomly sampled and initialized from a complex Gaussian distribution with a small variance of 1 / 10,000. Simultaneously, the sparse hyperparameters corresponding to the new dimension are re-initialized. Afterward, variational iteration continues, allowing the new degrees of freedom to adaptively adjust according to the received data.

[0233] The aforementioned expansion mechanism is only activated in the first half of the iteration, and no new dimensions are added in the later stages of the iteration to ensure algorithm convergence.

[0234] The iteration stops when the following two conditions are met simultaneously: First, the values ​​of all ring ranks have not changed for five consecutive iterations; Second, the relative change of the variational lower bound between two consecutive iterations is less than the preset minimum tolerance, such as one part per million. The ring rank obtained at this time is the optimal value for data adaptation, which is neither too large to cause overfitting nor too small to cause underfitting.

[0235] This application can automatically adapt to different channel environments. In scenarios with low channel freedom, such as open suburbs, it automatically adopts a small ring rank to achieve high compression. In scenarios with high channel freedom, such as urban multipath, it automatically adopts a large ring rank to retain sufficient information. No manual intervention or cross-validation is required throughout the process.

[0236] When each third-order core tensor is flattened into a ring factor matrix, a fixed arrangement order is used: the left link dimension and the physical channel dimension are combined as the row direction, and the right link dimension is used as the column direction. The number of rows in any ring factor matrix is ​​equal to the left link size multiplied by the physical dimension size, and the number of columns is equal to the right link size. In the ring rank adjustment, for the m-th link, the columns of the left factor matrix already correspond to the right link dimension, and the current posterior mean matrix is ​​used directly.

[0237] The rows of the right-hand factor matrix originally correspond to the left nexus dimension. It needs to be reshaped into a three-dimensional array, namely the left nexus, physical dimension, and right nexus. Then, the first and third dimensions are swapped, and it is flattened into a new two-dimensional matrix so that the number of rows is the left nexus size multiplied by the physical dimension size, and the number of columns is the right nexus size. This aligns the left and right factor matrices in the same nexus space, so that the energy matrix can be calculated correctly and the importance of the nexus dimension can be determined.

[0238] For example: Set a relative threshold Q= ,like Then set the ring rank. The value is zero, and the corresponding link dimension is pruned during the iteration.

[0239] Pilot symbols are acquired, and the ring factor matrix is ​​reconstructed through tensor ring shrinkage to obtain the channel tensor. After linear operation with the pilot symbols, noise is superimposed to form the received signal.

[0240] Initially, the ring ranks are all set to small, fixed values. The elements of each ring factor matrix are randomly sampled from a complex Gaussian distribution with a mean of 0 and a variance of 1. All precision hyperparameters are... Initialize to 1, and initialize noise variance to 0.1.

[0241] A preferred example: The initial parameter settings should follow these general principles rather than relying on specific values. First, the initial ring rank can be the minimum value among all relevant channel dimensions (i.e., the number of satellite antennas, the number of each RIS reflector unit, and the number of user antennas), divided by 4 or 5 and rounded to the nearest integer. If it is less than 1, it should be 1. This ensures that the initial decomposition has sufficient expressive power without over-parameterizing. Each element of the ring factor matrix is ​​sampled from a complex Gaussian distribution with a mean of zero and a variance of 1. Neutral selection is suitable for most signal-to-noise ratio scenarios. If the magnitude of the received signal power is known, the variance can be matched to the magnitude. The initial value of the noise variance is estimated using 10% of the variance of the received signal samples, because signal power is usually dominant, and 10% ensures that the initial guess is neither too large nor too small. All sparse hyperparameters are used to control the gamma distribution parameters of each row or column precision, initialized to 1, indicating that there is no bias towards any angle or time delay initially. The convergence criterion uses a variational lower bound with a relative change of less than 1. The first to satisfy the condition in five consecutive iterations where the ring rank no longer changes is selected. The maximum number of iterations is set to 500, but in practice, convergence usually occurs within 100 iterations. The above principle does not depend on specific values, and technicians can flexibly adjust it according to the actual system parameters.

[0242] Add a convergence condition: calculate the change in the lower bound of the log-likelihood after each iteration; when the relative change is less than... Alternatively, stop iterating when the maximum number of iterations (100) is reached.​

[0243] Let the satellite transmission pilot symbol vector be x∈ The user receives the signal y∈ Ignoring noise, each element of the received signal is:

[0244]

[0245] H is a high-dimensional channel tensor obtained by reconstructing all ring factor matrices through a tensor ring. When pilot symbols in multiple time slots are transmitted, multiple observations are formed, which can be stacked into a higher-order tensor observation model.

[0246] It should be noted that R represents the number of RIS; Number of satellite antennas; Number of user antennas; Let x be the number of reflection units in the r-th RIS (r=1,…,R); L be the total number of pilot time slots; x=[ , ,…, ]∈ Let t be the pilot symbol matrix. Let be the pilot vector for the t-th time slot.

[0247] y=[ , ,…, ]∈ For the received signal matrix, the t-th column Let be the receiving vector for the t-th time slot.

[0248] G(k) is the k-th third-order core tensor, k=0,1,…,R+1, where For 1≤k≤R, we have ; Ring rank satisfies , , ,..., It is a positive integer.

[0249] The channel tensor is replaced with a tensor ring reconstruction expression of the ring factor matrix, and the likelihood function is obtained through the noise distribution.

[0250] like Figure 3 As shown, the variational Bayesian inference and ring-rank adaptive deterministic subprocess of this application includes: initializing all parameters as step a, entering an iterative loop, sequentially updating the posterior mean and covariance of each ring factor matrix as step b, updating the hyperparameters and noise variance as step c, calculating the energy corresponding to each link dimension and comparing it with a threshold as step d, pruning ring-rank components below the threshold as step e, determining whether convergence has occurred as step f, and repeating steps b to f if convergence has not occurred until the convergence condition is met.

[0251] Specifically, firstly, based on the position and value of the pilot symbols configured at the transmitting end and the pilot signals actually acquired at the receiving end, a linear relationship expression between the two is constructed: each sample value of the received signal can be represented as the result of adding additive noise after multiplying or convolving the channel tensor with the pilot symbols; where the channel tensor itself is not an independent variable, but is reconstructed from all ring factor matrices through tensor ring contraction.

[0252] Secondly, the channel tensor in the above linear relationship is replaced with the reconstructed expression of the ring factor matrix, thereby establishing a direct mathematical relationship between the received signal and each ring factor matrix. This relationship is expressed as follows: the received signal is equal to the result of a series of ring factor matrices after multiple linear operations, including shrinking, multiplication, and summation, plus a noise term.

[0253] Based on this, by substituting the statistical characteristics of the noise, assuming it to be a complex Gaussian distribution with zero mean and known variance, we can write the conditional probability density function of the received signal under a given ring factor matrix. This function is the likelihood function; the likelihood function quantifies the extent to which the current ring factor matrix can explain the observed received signal.

[0254] This invention employs a mean-field variational Bayesian framework. Let all the latent variables to be estimated be... ,in:

[0255] is a precision hyperparameter that controls the sparsity of the p-th row of the k-th ring factor matrix, corresponding to the spatial angular domain;

[0256] is a precision hyperparameter that controls the sparsity of the qth column of the kth ring factor matrix, corresponding to the time delay domain.

[0257] Introduce a decomposable variational posterior distribution:

[0258]

[0259] set up It is a complex Gaussian distribution: vec( )∼CN( ;

[0260] Definition: vec(*) is a vectorization operation that stacks matrices column by column into a single column vector.

[0261] Γk=diag([ , ,…, , The number of repetitions of its diagonal elements is equal to […]. The number of columns, where each The number of repetitions is equal to the number of columns in the k-th ring factor matrix.

[0262] The observation operator is constructed according to the first part, where other ring factor matrices take the current expected value. That is, the posterior mean Reconstruct it into matrix form, where y = +n, the constructor is: Each element is set to 1 in sequence, and the rest are set to 0. Substituting these elements into the received signal expression yields the corresponding column.

[0263] For example, using the properties of the Kronecker product, it can be shown that when the expectations of other factor matrices are fixed, the received signal vector vec(y) can be expressed as a function of vec(y). The linear function of ) has a coefficient matrix that is .

[0264] Expected noise accuracy:

[0265]

[0266]

[0267] y is the conjugate transpose, and y is the received signal vector.

[0268] Let variational posterior It follows a complex Gaussian distribution with a mean of Covariance The following linear system was obtained by solving:

[0269]

[0270]

[0271] in It is a linear operator composed of other ring factor matrices and pilot symbols. This represents the noise variance.

[0272] Assuming noise variance Follows an inverse gamma distribution: ~Inv-Gamma( ),in = = ;

[0273] Updated formula: In variational Bayesian inference, The posterior expectation is:

[0274]

[0275] in The total number of observed samples, To reconstruct a noise-free received signal using the current posterior mean of all ring factor matrices, , This represents the total number of observed samples.

[0276] In this application, a preferred method for calculating the update noise accuracy is as follows:

[0277] Assume prior ,Pick = = The variational posterior q(τ) still follows a Gamma distribution with the shape parameter... and scale parameters Updated to:

[0278]

[0279]

[0280] Furthermore, the method for updating the sparse hyperparameters in the angle domain is as follows:

[0281] Assume prior , Variational posterior q( ) is a Gamma distribution, and the parameters are updated.

[0282] Using all the ring factor matrices to be estimated as latent variables and the received signal as the observed data, and combining the Bayesian hierarchical prior architecture set in step S2, including joint structured sparse prior and shared hyperparameter constraints, we write out the complete posterior probability expression, which is proportional to the product of the likelihood function and the prior distribution.

[0283] Since it is impossible to accurately calculate the posterior distribution in high-dimensional cases, the variational Bayesian method is used to approximate it: a set of decomposable approximate posterior distributions, namely variational distributions, is introduced. By minimizing the difference between the variational distribution and the true posterior distribution, the problem is transformed into an iterative optimization of variational parameters.

[0284] In each iteration, the variational posterior distribution of each ring factor matrix, including its mean and covariance, is updated sequentially, while the noise variance and hyperparameters in the prior are also updated. During the update process, the joint structured sparse prior set in step S2 is used to automatically make the matrix elements corresponding to unimportant angle components or time delay components approach zero, thereby preserving the sparse structure.

[0285] In this application, a preferred variational Bayesian iterative convergence criterion is: the relative change of the variational lower bound between two consecutive iterations < Maximum iteration limit: 100 iterations; Ring rank adaptive pruning quantization threshold: When the squared energy of the ring factor F norm is less than 0.1% of the total energy in the same dimension, the corresponding ring rank is zeroed and pruned; Initial noise variance: 0.1, and the noise follows a zero-mean complex Gaussian distribution.

[0286] In this method, a linear observation operator is constructed with respect to the k-th ring factor matrix. A preferred example is as follows: When the other ring factor matrices are fixed as the current estimates, the originally complex multilinear received signal expression degenerates into a linear function of the k-th ring factor matrix. To obtain the matrix representation of this linear function, a tensor network marginalization method based on the Kronecker product is used: the received signal is vectorized into a column vector, and the k-th ring factor matrix is ​​also vectorized into a column vector; the linear mapping matrix between the two is then obtained. Each column of the mapping matrix corresponds to the received signal vector generated when one element in the k-th ring factor matrix is ​​a unit value and the rest are zero. In actual implementation, it is not necessary to enumerate all elements one by one. Instead, the structure of tensor ring decomposition is used to directly calculate the vector by successively shrinking all factors except the k-th factor. Those skilled in the art can use existing tensor computation libraries, such as the einsum function of TensorLy or NumPy, to automatically generate this operator based on the tensor network topology, without having to manually derive the expression for each element.

[0287] For each link dimension, calculate the energy or effective rank of its corresponding ring factor matrix; when the energy corresponding to a ring rank is lower than a preset relative threshold, the link dimension corresponding to that ring rank is forcibly shrunk to zero, thereby severing the connection between the corresponding core tensors; as the iteration proceeds, unimportant ring ranks are gradually eliminated, and the final retained ring rank value is the optimal ring rank determined adaptively, without the need for manual preset.

[0288] S4 reconstructs the original high-dimensional channel tensor of the ring factor matrix by shrinking the tensor ring, and outputs the cascaded channel estimation results of the multi-RIS assisted satellite communication system.

[0289] Based on the ring connection topology established during tensor ring decomposition, each ring factor matrix is ​​sequentially condensed along the link dimension, that is, the link dimensions shared by adjacent ring factor matrices are multiplied and summed to finally recover the complete high-dimensional tensor. Each dimension of the high-dimensional tensor corresponds to the number of satellite antennas, the number of each RIS reflector unit, and the number of user antennas.

[0290] It should be noted that RIS-assisted satellite system channels can be modeled as high-order tensor models. Utilizing this structural characteristic, high-order tensor models can be combined with compressed sensing technology. High-order tensor compression methods can be used to jointly compress channel data from multiple dimensions, improving the data compression rate and further reducing the complexity of channel estimation. The following introduces a high-order tensor compression method, assuming a third-order tensor. At this point, three low-dimensional matrices can be designed. It is compressed from three directions, among which, At this point, the tensor It can be compressed into a tensor The data of tensor elements is greatly reduced, such as Figure 4 As shown in the figure. It can be seen from the figure that by designing compression matrices in three directions... It can reduce the size of higher-order tensors. The dimension simplifies the parameter estimation process. However, the key to this method is how to design the compression matrix. This ensures that the estimated compressed tensor can be uniquely recovered from the original tensor data. Therefore, further research is needed on the uniqueness condition of tensor compression recovery to compress and recover higher-order tensor channels, achieving channel estimation, such as... Figure 4 As shown.

[0291] When a multi-RIS structure is used in a communication system, the equivalent channel consists of multiple sub-channels. A single-RIS-assisted satellite channel has a high-order tensor form; therefore, when using a multi-RIS structure, its equivalent channel can be modeled as a sum of multiple high-order tensor data. Tensor decomposition can be achieved using a differential iteration method based on the summed tensor structure. The process is briefly described below. Assume the equivalent channel can be modeled as a sum of two high-order tensors, i.e. ,like Figure 5 As shown.

[0292] Assumption , , All conform to the parallel factor model and satisfy the uniqueness condition of decomposition. In this case, the difference iterative decomposition method can be used to... To decompose, first, initialize ,right Perform tensor decomposition to obtain The estimated value. Then, for Perform tensor decomposition to obtain The estimated value is obtained by iterating repeatedly until convergence. After convergence, the algorithm yields the following results. and The estimated value is obtained. Solving this problem requires iterative steps, therefore, the research needs to focus on the convergence of the algorithm, provide the convergence conditions, and ensure that the algorithm converges monotonically.

[0293] In this application, C represents the set of complex numbers; CN( It follows a complex Gaussian distribution with mean μ and covariance Σ.

[0294] Gamma(A, B) is a gamma distribution with shape parameter A and scale parameter B.

[0295] Inv-Gamma(A, B) is an inverse gamma distribution.

[0296] It is the Frobenius norm.

[0297] It is the Euclidean norm.

[0298] diag(*) is a diagonal matrix.

[0299] For Kronecker product.

[0300] Example 2

[0301] satellite antenna =8, RIS number R=3, each RIS reflection unit = = =32, User receiving antenna Three RIS are deployed in the same park, with a distance of 35m between them and less than 50m in the metropolitan area. They are classified into the same geographical zone and share a set of gamma hyperparameters. The initial full-link ring rank is uniformly preset to 15.

[0302] In a densely scattered satellite-to-ground channel in the city, the system SNR is 10dB. The number of pilots in this scheme is 1 / 4 of that in the traditional least squares scheme.

[0303] S1 constructs a 3+2=5 order high-dimensional channel tensor, decomposes it into 5 third-order core tensors, and fixes the ring rank of the satellite and user link head and tail core tensors to 1;

[0304] S2 Cyclic Factor Configuration with Hierarchical Bayesian Prior, Hyperparameter Initial Values The three RIS groups share the second-layer hyperparameters, and add adjacent smoothing constraints for angle and time delay;

[0305] S3 variational Bayesian iteration, after 17 iterations, satisfies the convergence condition, and the variational lower bound changes < Invalid ring ranks are automatically pruned based on a 0.1% energy threshold, and the final adaptive optimal ring rank is 7.

[0306] S4 shrinks all ring factors along the ring topology, reconstructs the 5-dimensional original cascaded channel tensor, and outputs the channel estimation result.

[0307] The proposed scheme has a channel NMSE of 0.031; the traditional LS-NMSE is 0.186 and the unconstrained general tensor loop SBL-NMSE is 0.093. The number of parameters is reduced to 1 / 22 of the traditional scheme, and the pilot overhead is reduced by 73%.

[0308] It should be further explained that the enumeration method is used, and the program implementation utilizes tensor network contraction for fast computation. For example, for k=1, the value is fixed. , , , For the current estimate, Each element is set to 1 sequentially, and the received signal is calculated through nested loops to form... The computational workload is All are pre-calculated during initialization. And store.

[0309] Numerical tracking of variational iteration:

[0310] Initialization: Ring Rank , Element ~CN (0~1) =0.1, =1, smoothing coefficient =0.01.

[0311] First iteration: Update At that time, construct Given a size of 4L×(8×15), solving the system of linear equations yields... Update the noise accuracy τ to obtain E[τ]≈10. Update the angle hyperparameter: calculate Each line of energy, obtained The updated values ​​are relatively evenly distributed;

[0312] Then proceed with intermediate iterations;

[0313] 10th iteration: Calculate each link matrix The singular values ​​are [12.3, 8.7, 5.2, 3.1, 1.8, 0.9, 0.4, 0.2, 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001], and the cumulative threshold is... When the value is 0.001, the first 7 singular values ​​account for 99.2%, therefore After performing the cropping, It becomes 8×7. The changes are from (15×32)×15 to (7×32)×15, with similar adjustments made to other links.

[0314] 17th iteration: All ring ranks stabilize at [1,7,7,7,7,1], where the first and last rank are fixed at 1, and the variational lower bound changes by 3× < Stop iterating.

[0315] Finally, using the converged ring factor matrix, the 8×32×32×32×4 channel tensor is reconstructed according to the tensor ring shrinkage formula.

[0316] With the same pilot overhead (L=32), the normalized mean square error of our method is 0.031, while that of the traditional least squares method (L=128) is 0.186, and that of the general tensor ring variational Bayesian method without structured sparse constraints (L=64) is 0.093. The number of parameters in our method is 1 / 22 of that in the traditional scheme.

[0317] It is important to note that the constructions and arrangements of this application shown in several different exemplary embodiments are merely illustrative. Although only two embodiments are described in detail in this disclosure, those who consult this disclosure will readily understand that many modifications are possible without substantially departing from the novel teachings and advantages of the subject matter described in this application. These modifications may include, for example, changes in the size, dimensions, structure, shape, and proportions of various elements, as well as parameter values ​​(e.g., temperature, pressure, etc.), installation arrangements, the use of materials, colors, orientations, etc. For example, an element shown as integrally formed may be composed of multiple parts or elements, the position of elements may be inverted or otherwise altered, and the nature or number or position of discrete elements may be changed or altered. Therefore, all such modifications are intended to be included within the scope of this application. The order or sequence of any process or method steps may be changed or rearranged by alternative embodiments. Any "apparatus plus function" clause is intended to cover, and not only structurally equivalent but also equivalent structures, the structures performing the functions described herein. Other substitutions, modifications, alterations, and omissions may be made in the design, operation, and arrangement of the exemplary embodiments without departing from the scope of this application. Therefore, this application is not limited to a particular embodiment, but extends to various modifications that still fall within the scope of the appended claims.

[0318] Furthermore, in order to provide a concise description of exemplary embodiments, not all features of actual embodiments (i.e., those features that are not relevant to the best mode of performing this application as currently considered, or those features that are not relevant to implementing this application) may be omitted.

[0319] It should be understood that numerous specific implementation decisions can be made during the development of any practical implementation, such as in any engineering or design project. Such development efforts may be complex and time-consuming, but for those of ordinary skill in the art who benefit from this disclosure, the development effort will be a routine task in design, manufacturing, and production without requiring extensive experimentation.

[0320] It should be noted that the above embodiments are only used to illustrate the technical solutions of this application and are not intended to limit it. Although this application has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of this application without departing from the spirit and scope of the technical solutions of this application, and all such modifications and substitutions should be covered within the scope of the claims of this application.

Claims

1. A low-complexity channel estimation method for a multi-RIS-assisted satellite communication system, characterized in that, include: S1 models the cascaded channel of the multi-RIS assisted satellite communication system as a high-dimensional tensor, and uses structured tensor ring decomposition to represent the high-dimensional tensor as a ring-shaped contraction of multiple third-order tensors. Each third-order tensor corresponds to one RIS. The tensor ring decomposition has a configurable ring rank parameter. S2 sets a Bayesian hierarchical prior architecture for each ring factor matrix obtained by the tensor ring decomposition, applies a joint structured sparse prior to each ring factor matrix, and applies shared hyperparameter constraints to the ring factor matrices of multiple RISs deployed in the same geographical region. The method for setting up a Bayesian hierarchical prior architecture and applying a joint structured sparse prior to each cyclic factor matrix in S2 is as follows: A probability distribution prior is set for each ring factor matrix. The probability distribution prior adopts a hierarchical structure: the first layer assumes that each row or column in the ring factor matrix follows a complex Gaussian distribution with a mean of zero. The variance of the Gaussian distribution is controlled by the hyperparameter of the second layer. The second layer sets a gamma distribution for the variance hyperparameter to form an automatic adjustment mechanism for sparsity. The row index of the ring factor matrix is ​​mapped to the spatial angle direction, and the column index is mapped to the signal propagation time delay. In the Bayesian prior, rows corresponding to adjacent angles share similar sparse patterns, and columns corresponding to adjacent time delays share similar sparse patterns, thus preserving the continuous structure of the angle domain and the time delay domain during compression. The method for applying shared hyperparameter constraints to the ring factor matrices of multiple RIS deployed in the same geographical region in S2 is as follows: Based on the actual deployment location information of each RIS, determine the geographical area of ​​each RIS; the geographical area includes building exterior walls, campus, or the distance between each RIS being less than a preset distance threshold; For the ring factor matrices corresponding to multiple RIS belonging to the same geographical region, the same set of hyperparameters is set in the Bayesian hierarchical prior architecture, that is, the second-level variance hyperparameters of the ring factor matrices share the same prior distribution. The cyclic factor matrices of different RIS are statistically drawn to similar sparse structures and energy distributions; S3 establishes the likelihood function between the received signal and the ring factor matrix through linear observation, uses variational Bayesian inference to maximize the posterior probability, estimates each ring factor matrix and its posterior distribution, and determines the ring rank parameter of the tensor ring decomposition during the inference process. S4 reconstructs the original high-dimensional channel tensor of the ring factor matrix by shrinking the tensor ring, and outputs the cascaded channel estimation results of the multi-RIS assisted satellite communication system.

2. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 1, characterized in that: Based on the number of RIS deployed in the satellite communication system and the number of reflection units contained in each RIS, the number of satellite-side antennas and the number of user-side antennas are also set, and the channels from the satellite to the first RIS, between adjacent RIS, and between the last RIS and the user are represented in matrix form. The matrices are arranged in a concatenated order to construct a higher-order tensor. The order of the higher-order tensor is equal to the number of RIS plus two. The size of each dimension corresponds to the number of satellite antennas, the number of each RIS reflection unit, and the number of user antennas, respectively. The structured tensor ring decomposition is configured as follows: The higher-order tensor is decomposed into a ring-shaped connection structure of a set of third-order core tensors. The number of core tensors is equal to the order of the tensor. Each core tensor has three dimensions, two of which are associated with the corresponding channel dimension, and the other dimension serves as a link connecting adjacent core tensors. The link dimension between the first and last core tensors is fixed to a single dimension.

3. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 1, characterized in that: The value of each element in the constructed high-order channel tensor is expressed as the sum of the products of the corresponding elements in all core tensors; Each core tensor contributes a factor. After multiplying the factors of all core tensors, the indices on all the connecting link dimensions are summed to complete one circumferential contraction operation. The first core tensor is connected to the input of the second core tensor through its output link, the output of the second core tensor is connected to the input of the third core tensor, and so on, with the output of the last core tensor connected to the input of the first core tensor, forming a closed loop chain. Each element value in a high-dimensional tensor is equal to the sum of the internal link indices after traversing all core tensors along the circular chain; Through cyclic contraction, the number of parameters in the original high-dimensional tensor is compressed to the sum of the number of parameters in the core tensor, and the link dimension of the core tensor is smaller than the original channel dimension.

4. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 3, characterized in that: Assign an independent third-order core tensor to each deployed RIS; The third-order core tensor includes three dimensions: the first dimension represents the signal direction or path index from the previous hop channel, the second dimension represents the index of each reflection unit on the current RIS, and the third dimension represents the signal direction or path index to the next hop channel. For the link from the satellite to the first RIS and the link from the last RIS to the user, a core tensor is assigned, and the relationship between the antenna port and the signal direction is defined in the dimension of the assigned core tensor. The dimension of the link connecting adjacent core tensors is called the ring rank, and the dimension of the link is a positive integer that can be dynamically adjusted during the estimation process. The ring rank is used to represent the degree of compression and expressive power of the decomposition. The smaller the ring rank, the higher the compression ratio; the larger the ring rank, the more channel information is retained; the maximum allowable value of the ring rank does not exceed the smaller of the channel dimensions corresponding to the two core tensors connected. The optimal ring rank is automatically inferred from the received data, with practically acceptable computational complexity and expected channel estimation accuracy.

5. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 1, characterized in that: After completing the tensor ring decomposition, the ring factor matrix is ​​extracted from each third-order core tensor; The method for extracting the ring factor matrix involves expanding or reshaping each core tensor along its connecting link dimension to obtain multiple two-dimensional matrices, which are the ring factor matrices. Each ring factor matrix includes directional information of a link and the coupling relationship between RIS reflection units; For each cyclic factor matrix, maintain an independent parameter structure, do not mix or superimpose, and record the topological connections between the cyclic factor matrices.

6. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 1, characterized in that: Pilot symbols are acquired, and the ring factor matrix is ​​reconstructed through tensor ring shrinkage to obtain the channel tensor. After linear operation with the pilot symbols, noise is superimposed to form the received signal. The channel tensor is replaced with a tensor ring reconstruction expression of the ring factor matrix, and the likelihood function is obtained through the noise distribution.

7. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 6, characterized in that: Using the ring factor matrix as a latent variable, the posterior probability is constructed by combining the likelihood function and Bayesian hierarchical prior. We introduce a decomposable variational distribution to approximate the true posterior. By iteratively optimizing the variational parameters, we minimize the difference between the two and update the posterior mean and covariance of each cyclic factor matrix in turn, while also updating the noise variance and hyperparameters.

8. The low-complexity channel estimation method for a multi-RIS-assisted satellite communication system as described in claim 1, characterized in that: Based on the ring connection topology established during tensor ring decomposition, each ring factor matrix is ​​sequentially condensed along the link dimension, that is, the link dimensions shared by adjacent ring factor matrices are multiplied and summed to finally recover the complete high-dimensional tensor. Each dimension of the high-dimensional tensor corresponds to the number of satellite antennas, the number of each RIS reflector unit, and the number of user antennas.

Images