Three-dimensional spectrum map generation method

Abstract

The invention relates to the technical field of electromagnetic spectrum maps, and particularly discloses a three-dimensional spectrum map generation method, which comprises the following steps: dividing a three-dimensional geographic space of an electromagnetic environment to be measured into grids, and constructing a three-dimensional spectrum map tensor according to multiple pieces of spectrum data in the grids; the frequency spectrum data is the average received signal strength of one time slot in the grid; slicing the spectrum map tensor to obtain tensor slices, and calculating a local factor of the tensor slices to obtain a tensor local factor; applying smooth constraint to the tensor local factor to obtain a local smooth tensor; modeling according to the local smooth tensor to obtain a completion model; and solving the completion model by adopting an ADMM algorithm to obtain a completed three-dimensional electromagnetic spectrum map. According to the three-dimensional spectrum map generation method provided by the invention, the three-dimensional electromagnetic spectrum map can be accurately and efficiently constructed by using limited sampling data in a multi-radiation source scene.

Description

Technical Field

[0001] The present invention relates to the technical field of electromagnetic spectrum maps, and in particular to a method for generating a three-dimensional spectrum map. Background Art

[0002] With the rapid development of wireless communication technology, the number of wireless frequency devices has exploded, leading to an increasing shortage of electromagnetic spectrum resources. In this environment, cognitive radio technology has emerged, with the core goal of achieving efficient management and rational allocation of spectrum resources. Spectrum mapping is a critical task in radio systems. It visualizes the invisible electromagnetic spectrum space, providing an intuitive basis for spectrum resource management and allocation. Spectrum mapping provides a clear understanding of important information such as the spectrum environment, received signal strength, spectrum busy / idle status, and spectrum access protocols. This enables functions such as opportunistic spectrum access, spectrum management, and interference coordination, playing a vital role in improving the performance of wireless communication systems.

[0003] In recent years, numerous spectrum map reconstruction methods have emerged, primarily categorized as data-driven and model-driven. Data-driven methods typically treat spectrum map reconstruction as a matrix completion problem, employing interpolation algorithms. For example, the inverse distance weighted interpolation method (IDWI) performs weighted calculations based on the distance between sampling points and the interpolation points. However, this method fails to take into account any physical realities. In urban environments, where severe shadow fading exists, the performance of IDWI is significantly affected, making it difficult to accurately reflect the true spectrum. Kriging interpolation methods are also available, as well as the optimal data-driven spectrum mapping method based on distributed kriging. Furthermore, by extending the low-rank properties of matrices to tensors, tensor completion methods combined with predictive models have emerged. By imposing various constraints on the spectrum map tensor, map reconstruction can be achieved in various scenarios. However, current tensor completion algorithms mostly consider low rank and smoothness globally, insufficiently considering local features and failing to fully exploit known data characteristics. Furthermore, most current spectrum map generation methods only consider two-dimensional surfaces and fail to incorporate height information, making it difficult to fully represent the electromagnetic environment of a given area.

[0004] Model-driven methods primarily reconstruct spectrum maps based on radiation source information and electromagnetic propagation models. For example, algorithms based on location estimation estimate the location of primary users using received signal strength values, thereby reconstructing spectrum maps. However, in practical applications, the power and path loss models of primary users are often difficult to obtain, which limits the scope of this method's application. There are also indirect model-driven reconstruction methods, as well as schemes that utilize the least absolute shrinkage and selection algorithm (LASSO) combined with a path loss model to reconstruct spectrum maps. However, these model-driven methods have poor adaptability in scenarios with multiple radiation sources and are unable to accurately complete the task of constructing spectrum maps. Summary of the Invention

[0005] In response to the above problems, the purpose of the present invention is to provide a three-dimensional spectrum map generation method that can accurately and efficiently construct a three-dimensional electromagnetic spectrum map using limited sampling data in a multi-radiation source scenario.

[0006] The present invention provides a method for generating a three-dimensional spectrum map, comprising: Divide the three-dimensional geographic space of the electromagnetic environment to be measured into grids, and construct a three-dimensional spectrum map tensor based on multiple spectrum data in the grids; the spectrum data is the average received signal strength of a time slot in the grid; Slicing the spectrum map tensor to obtain tensor slices, and calculating local factors of the tensor slices to obtain tensor local factors; Applying a smoothness constraint to the local factor of the tensor to obtain a local smooth tensor; Modeling is performed based on the local smooth tensor, and combining the low-rank constraint of the tensor truncated nuclear norm to obtain a completion model; The ADMM algorithm is used to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map.

[0007] In a possible implementation, constructing a three-dimensional spectrum map tensor according to the plurality of spectrum data in the grid includes: The spectrum map tensor is determined according to the following formula : ; in, is the true value of the spectrum map tensor, is a tensor element, is a known grid index, is the horizontal coordinate of the grid, is the vertical coordinate of the grid, is the vertical coordinate of the grid, For The average received signal strength of the grid at the location, is the set of all grids and the average received signal strength.

[0008] In a possible implementation, calculating the local factor of the tensor slice to obtain the tensor local factor includes: The local factor of a tensor slice is calculated according to the following formula: ; ; ; ; ; in, For the tensor slices, is an orthogonal tensor, is a diagonal tensor, is the singular value decomposition operation, For the The front local factors of the tensor slices, For the The post-local factor of the tensor slice, For the orthogonal tensor Cut off the first R columns, To intercept the first R rows and R columns of the diagonal tensor, For the orthogonal tensor Cut off the first R columns, is the pre-local factor of the tensor, is the post-local factor of the tensor, represents the inverse process of Fourier transform, Indicates the number of the tensor slice, is the estimated tensor rank, represents the conjugate transpose.

[0009] In a possible implementation, applying a smoothness constraint to the local factor of the tensor to obtain a local smooth tensor includes: The smoothness constraint of the local factor is determined according to the following formula: ; ; ; ; in, is the pre-local factor of the tensor, is the post-local factor of the tensor, Indicates the number of the tensor slice, represents the three-modal total variation operator, is a first-order matrix difference operator, is the spectrum map tensor.

[0010] In one possible implementation, the difference between consecutive rows of the matrix is determined according to the following formula: and : ; ; in, is a first-order matrix difference operator, is the first local factor of the tensor slices, is the post-local factor of the tensor slices, The matrix represents the OK, The number representing the slice of the tensor.

[0011] In a possible implementation, the modeling based on the local smooth tensor and the low-rank constraint of the tensor truncated nuclear norm are combined to obtain the completion model, which includes: The completion model is obtained according to the following formula: ; in, , is the tensor local factor, is the observed value of the spectrum map tensor, Indicates the number of the tensor slice, is a first-order matrix difference operator, is the first local factor of the tensor slices, is the post-local factor of the tensor slices, and is the weighting coefficient, represents the mode of the tensor, is an auxiliary variable representing the tensor modulo-n expansion, is the cutoff coefficient, is the Frobenius norm, To get the value of a known grid index, is the true value of the spectrum map tensor, is the index of a known grid.

[0012] In one possible implementation, solving the completion model using the ADMM algorithm to obtain a completed three-dimensional electromagnetic spectrum map includes: Update according to the following formula : ; ; Update according to the following formula : ; ; Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; in, Represents the number of the tensor slice, 、 Indicates the number of cycles, The transpose of the first-order matrix difference operator, Indicates transposition, and is an additional slack variable used for algorithm convergence, 、 、 、 The coefficient variables introduced for the ADMM algorithm, Constraint coefficient, and Used to replace and 、 According to the tensor modulus The inverse process of unfolding, folding the matrix back into tensor form, is the Frobenius norm, .

[0013] In one possible implementation, solving the completion model using the ADMM algorithm to obtain a completed three-dimensional electromagnetic spectrum map includes: Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; Update the three-dimensional electromagnetic spectrum map according to the following formula : ; ; Update according to the following formula : ; in, represents the conjugate transpose, 、 、 is the constraint coefficient, Auxiliary variables representing the expansion of the tensor modulo-n, is the cutoff coefficient, is the multiplier of the constraint condition, is the set of all grids and the average received signal strength, is a wide-area singular value threshold operator related to minimization of the truncated nuclear norm.

[0014] In one possible implementation, solving the completion model using the ADMM algorithm to obtain a completed three-dimensional electromagnetic spectrum map includes: Update the multiplier for the constraint condition according to the following formula : ; ; ; ; ; in, , , , , is the multiplier of the constraint condition, Indicates the number of the tensor slice, Indicates the number of cycles, , is the tensor local factor; Update the constraint coefficient according to the following formula : ; ; ; in, is the increment multiple of each cycle coefficient, , , is the constraint coefficient.

[0015] In one possible implementation, solving the completion model using the ADMM algorithm to obtain a completed three-dimensional electromagnetic spectrum map includes: Determine whether the convergence condition is met or the preset number of iterations is reached; When the convergence condition is met or the preset number of iterations is reached, the completion model outputs a completed three-dimensional electromagnetic spectrum map; When the convergence condition is not met and the preset number of iterations is not reached, the parameters are updated iteratively.

[0016] The three-dimensional spectrum map generation method provided by this invention addresses the problem that existing spectrum map reconstruction techniques only consider global low-rank properties, ignoring the role of local smoothness in improving map reconstruction capabilities. In scenarios with multiple radiation sources, using limited sampled data, it can accurately and efficiently construct three-dimensional electromagnetic spectrum maps, meeting the growing demand for spectrum monitoring and management, and improving the performance and reliability of cognitive radio systems in complex electromagnetic environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 A schematic diagram of a flow chart of a method for generating a three-dimensional spectrum map provided by an embodiment of the present invention; Figure 2 This is a spectrum map tensor model provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0018] The following detailed description of the embodiments of the present invention is provided in conjunction with the accompanying drawings and examples. The following detailed description of the embodiments and the accompanying drawings are intended to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention. That is, the present invention is not limited to the preferred embodiments described, and the scope of the present invention is defined by the claims.

[0019] In the description of the present invention, it should be noted that, unless otherwise specified, “plurality” means two or more; the terms “first”, “second”, etc. are used for descriptive purposes only and cannot be understood as indicating or implying relative importance; for ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances.

[0020] To facilitate understanding of the present invention, the technical terms involved are first explained.

[0021] The Alternating Multiplier Algorithm (ADMM) is a classic algorithm for solving constrained minimization problems. Consider the following constrained minimization problem: ; in, and are all convex functions. The ADMM algorithm considers an augmented Lagrangian function as follows: ; Where y is the Lagrangian multiplier vector. The ADMM iteration mechanism is: ; in, is a positive penalty parameter.

[0022] As an extension of matrices to higher orders, tensor completion algorithms mainly include methods based on tensor decomposition and low-rank tensor completion based on rank minimization without decomposition structure.

[0023] The method based on tensor decomposition can be transformed into the following optimization problem: ; in, is the observation tensor, is the recovery tensor, represents the index set of observation items, represents the rank of the tensor, Is a given low-rank tensor The differences between different tensor decomposition methods mainly focus on the definition of rank, using the ranks obtained by different decomposition methods to replace the rank of tensor completion, such as CP decomposition, Tucker decomposition, tensor chain decomposition, and tensor ring decomposition.

[0024] In practice, tensor rank bounds may not be applicable in some applications. When there are few observations, choosing a high rank bound may lead to overfitting. To avoid this, another approach is to directly minimize the rank of the tensor, drawing on the idea of nuclear norm minimization from the matrix completion model. For the low-rank tensor completion problem, the general model for directly minimizing the tensor rank can be expressed as: ; There are many optimization models for tensor completion problems based on the definition of different tensor ranks. is a non-convex function, so the optimization problem becomes an NP-hard problem. In different methods, the tensor rank used can be selected from CP rank, Tucker rank, TT rank, TR rank, etc.

[0025] Suppose there is a third-order tensor Then, the tensor t-svd is decomposed into: ; in, and are all orthogonal tensors, is a diagonal tensor.

[0026] Figure 1 A flow chart of a method for generating a three-dimensional spectrum map according to an embodiment of the present invention is shown as follows: Figure 1 As shown, the present invention provides a method for generating a three-dimensional spectrum map, comprising: Step S1, dividing the three-dimensional geographic space of the electromagnetic environment to be measured into grids, and constructing a three-dimensional spectrum map tensor based on multiple spectrum data in the grids; The spectrum data is the average received signal strength of a time slot in the grid; the three-dimensional geographic space is divided into a group A cube grid, where Represents the x, y, z axis directions in space, spectrum map tensor .

[0027] In one possible implementation, the spectrum map tensor is determined according to the following formula: : ; in, is the true value of the spectrum map tensor, is a tensor element, is a known grid index, is the horizontal coordinate of the grid, is the vertical coordinate of the grid, is the vertical coordinate of the grid, For The average received signal strength of the grid at the location, is the set of all grids and the average received signal strength.

[0028] Step S2, slicing the spectrum map tensor to obtain tensor slices, and calculating local factors of the tensor slices to obtain tensor local factors; In one possible implementation, calculating the local factor of a tensor slice to obtain the tensor local factor includes: The local factor of a tensor slice is calculated according to the following formula: ; ; ; ; ; in, For the tensor slices, is an orthogonal tensor, is a diagonal tensor, is the singular value decomposition operation, For the The front local factors of the tensor slices, For the The post-local factor of the tensor slice, For the orthogonal tensor Cut off the first R columns, To intercept the first R rows and R columns of the diagonal tensor, For the orthogonal tensor Cut off the first R columns, is the pre-local factor of the tensor, is the post-local factor of the tensor, represents the inverse process of Fourier transform, Indicates the number of the tensor slice, is the estimated tensor rank, represents the conjugate transpose.

[0029] Step S3, applying a smoothness constraint to the local factor of the tensor to obtain a local smooth tensor; In one possible implementation, to impose smoothness constraints on the tensor local factors, Expand along three dimensions to get the matrix and .

[0030] Apply the difference operator to the expanded matrix : ; This can be achieved through and Get the differences between consecutive rows of the expanded matrix.

[0031] Determine the difference between consecutive rows of a matrix according to the following formula and : ; ; in, is a first-order matrix difference operator, is the first local factor of the tensor slices, is the post-local factor of the tensor slices, The matrix represents the OK, The number representing the slice of the tensor.

[0032] The smoothness constraint of the local factor is determined according to the following formula: ; ; ; ; in, is the pre-local factor of the tensor, is the post-local factor of the tensor, Indicates the number of the tensor slice, represents the three-modal total variation operator, is a first-order matrix difference operator, is the spectrum map tensor.

[0033] Step S4, modeling based on the local smooth tensor, combined with the low-rank constraint of the tensor truncated nuclear norm, to obtain the completed model; In one possible implementation, the general spectrum map generation problem can be modeled as a multidimensional tensor completion problem: ; in Represents the truncated nuclear norm of the tensor, which is approximately equal to the tensor rank.

[0034] Combined with the tensor local factor smoothness constraint, the spectrum map generation problem based on observations can be expressed as: ; is equivalent to: .

[0035] Since the truncated nuclear norm of any given tensor is equal to the weighted sum of the truncated nuclear norms on the tensor expansion matrix, It can be written as follows: ; in, express The modulo n expansion matrix of .

[0036] The problem to be solved can be expressed as the following formula, and the completion model is obtained according to the following formula: ; in, , is the tensor local factor, is the observed value of the spectrum map tensor, Indicates the number of the tensor slice, is a first-order matrix difference operator, is the first local factor of the tensor slices, is the post-local factor of the tensor slices, and is the weighting coefficient, represents the mode of the tensor, is an auxiliary variable representing the tensor modulo-n expansion, is the cutoff coefficient, is the Frobenius norm, To get the value of a known grid index, is the true value of the spectrum map tensor, is the index of a known grid.

[0037] Step S5: Use the ADMM algorithm to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map.

[0038] In one possible implementation, an auxiliary variable is introduced And order , then the tensor completion problem can be written as: ; The augmented Lagrangian function of the above problem can be written as: ; in, is the multiplier of the constraint condition.

[0039] According to the obtained augmented Lagrangian function, the solution of unknown variables is decomposed into several sub-problems for solution.

[0040] Update according to the following formula : ; ; Update according to the following formula : ; ; Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; in, Represents the number of the tensor slice, 、 Indicates the number of cycles, The transpose of the first-order matrix difference operator, Indicates transposition, and is an additional slack variable used for algorithm convergence, 、 、 、 The coefficient variables introduced for the ADMM algorithm, Constraint coefficient, and Used to replace and 、 According to the tensor modulus The inverse process of unfolding, folding the matrix back into tensor form, is the Frobenius norm, .

[0041] In one possible implementation, the ADMM algorithm is used to solve the completion model, and the completed three-dimensional electromagnetic spectrum map is obtained, including: Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; Update the auxiliary variable according to the following formula : ; Update the three-dimensional electromagnetic spectrum map according to the following formula : ; ; Update according to the following formula : ; in, represents the conjugate transpose, 、 、 is the constraint coefficient, Auxiliary variables representing the expansion of the tensor modulo-n, is the cutoff coefficient, is the multiplier of the constraint condition, is the set of all grids and the average received signal strength, is a wide-area singular value threshold operator related to minimization of the truncated nuclear norm.

[0042] In one possible implementation, the ADMM algorithm is used to solve the completion model, and the completed three-dimensional electromagnetic spectrum map is obtained, including: Update the multiplier for the constraint condition according to the following formula : ; ; ; ; ; in, , , , , is the multiplier of the constraint condition, Indicates the number of the tensor slice, Indicates the number of cycles, , is the tensor local factor; Update the constraint coefficient according to the following formula : ; ; ; in, is the increment multiple of each cycle coefficient, , , is the constraint coefficient. .

[0043] In one possible implementation, the ADMM algorithm is used to solve the completion model, and the completed three-dimensional electromagnetic spectrum map is obtained, including: Determine whether the convergence condition is met or the preset number of iterations is reached; When the convergence condition is met or the preset number of iterations is reached, the completion model outputs a completed three-dimensional electromagnetic spectrum map; When the convergence condition is not met and the preset number of iterations is not reached, the parameters are updated iteratively.

[0044] Compared with existing spectrum map reconstruction technologies, the three-dimensional spectrum map generation method provided by the present invention has the following significant benefits: (1) Multi-dimensional data feature fusion to improve reconstruction accuracy. By innovatively combining global low-rank and local smoothness constraints, multi-dimensional feature fusion is achieved in the tensor completion framework. This not only preserves the overall spatial correlation of spectral data but also enhances the refined expression of local electromagnetic features. This effectively solves the problem of detail loss caused by traditional methods that only focus on global characteristics, significantly improving the accuracy and physical authenticity of map reconstruction.

[0045] (2) Enhanced three-dimensional electromagnetic environment modeling capabilities. The height dimension is incorporated into the spectrum map construction system, and a three-dimensional tensor model is established to fully represent the spatial electromagnetic field distribution. Compared with traditional two-dimensional plane mapping, it can more realistically reflect the propagation and attenuation characteristics of electromagnetic waves in urban three-dimensional space, providing accurate three-dimensional visualization support for spectrum resource management in areas with dense high-rise buildings.

[0046] (3) High robustness under limited sampling. The developed local feature enhancement mechanism can extract deep spatial correlations from sparsely sampled data. Simulations have shown that at the same sampling rate, this scheme significantly reduces reconstruction error compared to the traditional Kriging interpolation method.

[0047] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A method for generating a three-dimensional spectrum map, characterized in that: include: The three-dimensional geographic space of the electromagnetic environment to be measured is divided into grids, and a three-dimensional spectrum map tensor is constructed based on multiple spectrum data in the grids; The spectrum data is the average received signal strength of a time slot in the grid; Slicing the spectrum map tensor to obtain tensor slices, and calculating local factors of the tensor slices to obtain tensor local factors; Applying a smoothness constraint to the local factor of the tensor to obtain a local smooth tensor; Modeling is performed based on the local smooth tensor, and combining the low-rank constraint of the tensor truncated nuclear norm to obtain a completion model; The ADMM algorithm is used to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map.

2. The method for generating a three-dimensional spectrum map according to claim 1, wherein: The step of constructing a three-dimensional spectrum map tensor based on the plurality of spectrum data in the grid includes: The spectrum map tensor is determined according to the following formula : ； in, is the true value of the spectrum map tensor, is a tensor element, is a known grid index, is the horizontal coordinate of the grid, is the vertical coordinate of the grid, is the vertical coordinate of the grid, For The average received signal strength of the grid at the location, is the set of all grids and the average received signal strength.

3. The method for generating a three-dimensional spectrum map according to claim 1, wherein: Calculating the local factor of the tensor slice to obtain the tensor local factor includes: The local factor of a tensor slice is calculated according to the following formula: ； ； ； ； ； in, For the tensor slices, is an orthogonal tensor, is a diagonal tensor, is the singular value decomposition operation, For the The front local factors of the tensor slices, For the The post-local factor of the tensor slice, For the orthogonal tensor Cut off the first R columns, To intercept the first R rows and R columns of the diagonal tensor, For the orthogonal tensor Cut off the first R columns, is the pre-local factor of the tensor, is the post-local factor of the tensor, represents the inverse process of Fourier transform, Indicates the number of the tensor slice, is the estimated tensor rank, represents the conjugate transpose.

4. The method for generating a three-dimensional spectrum map according to claim 1, wherein: Applying a smooth constraint on the local factor of the tensor to obtain a local smooth tensor includes: The smoothness constraint of the local factor is determined according to the following formula: ； ； ； ； in, is the pre-local factor of the tensor, is the post-local factor of the tensor, Indicates the number of the tensor slice, represents the three-modal total variation operator, is a first-order matrix difference operator, is the spectrum map tensor.

5. The method for generating a three-dimensional spectrum map according to claim 4, wherein: Determine the difference between consecutive rows of a matrix according to the following formula and : ； ； in, is a first-order matrix difference operator, is the first local factor of the tensor slices, is the post-local factor of the tensor slices, The matrix represents the OK, The number representing the slice of the tensor.

6. The method for generating a three-dimensional spectrum map according to claim 1, wherein: The modeling based on the local smooth tensor and the low-rank constraint of the tensor truncation nuclear norm are combined to obtain the completion model, which includes: The completion model is obtained according to the following formula: ； in, , is the tensor local factor, is the observed value of the spectrum map tensor, Indicates the number of the tensor slice, is a first-order matrix difference operator, is the first local factor of the tensor slices, is the post-local factor of the tensor slices, and is the weighting coefficient, represents the mode of the tensor, is an auxiliary variable representing the tensor modulo-n expansion, is the cutoff coefficient, is the Frobenius norm, To get the value of a known grid index, is the true value of the spectrum map tensor, is the index of a known grid.

7. The method for generating a three-dimensional spectrum map according to claim 1, wherein: The ADMM algorithm is used to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map, including: Update according to the following formula : ； ； Update according to the following formula : ； ； Update the auxiliary variable according to the following formula : ； Update the auxiliary variable according to the following formula : ； in, Represents the number of the tensor slice, 、 Indicates the number of cycles, The transpose of the first-order matrix difference operator, Indicates transposition, and is an additional slack variable used for algorithm convergence, 、 、 、 The coefficient variables introduced for the ADMM algorithm, Constraint coefficient, and Used to replace and 、 According to the tensor modulus The inverse process of unfolding, folding the matrix back into tensor form, is the Frobenius norm, .

8. The method for generating a three-dimensional spectrum map according to claim 7, wherein: The ADMM algorithm is used to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map, including: Update the auxiliary variable according to the following formula : ； Update the auxiliary variable according to the following formula : ； Update the auxiliary variable according to the following formula : ； Update the auxiliary variable according to the following formula : ； Update the three-dimensional electromagnetic spectrum map according to the following formula : ； ； Update according to the following formula : ； in, represents the conjugate transpose, 、 、 is the constraint coefficient, Auxiliary variables representing the expansion of the tensor modulo-n, is the cutoff coefficient, is the multiplier of the constraint condition, is the set of all grids and the average received signal strength, is a wide-area singular value threshold operator related to minimization of the truncated nuclear norm.

9. The method for generating a three-dimensional spectrum map according to claim 8, wherein: The ADMM algorithm is used to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map, including: Update the multiplier for the constraint condition according to the following formula : ； ； ； ； ； in, , , , , is the multiplier of the constraint condition, Indicates the number of the tensor slice, Indicates the number of cycles, , is the tensor local factor; Update the constraint coefficient according to the following formula : ； ； ； in, is the increment multiple of each cycle coefficient, , , is the constraint coefficient.

10. The method for generating a three-dimensional spectrum map according to claim 1, wherein: The ADMM algorithm is used to solve the completion model to obtain a completed three-dimensional electromagnetic spectrum map, including: Determine whether the convergence condition is met or the preset number of iterations is reached; When the convergence condition is met or the preset number of iterations is reached, the completion model outputs a completed three-dimensional electromagnetic spectrum map; When the convergence condition is not met and the preset number of iterations is not reached, the parameters are updated iteratively.

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