Millimeter wave MIMO channel estimation method based on dual non-convex relaxation

Abstract

The invention discloses a millimeter wave large-scale MIMO channel estimation method based on dual non-convex non-smooth rank relaxation, and the method comprises the steps: constructing channel estimation into a low-rank matrix recovery model through employing the low-rank characteristic of a millimeter wave channel; a dual non-convex non-smooth rank relaxation norm is adopted, differential constraint is carried out on singular values through dynamic weights of the norm, and a matrix rank is accurately approached; mapping an alternating direction multiplier method iteration process into a trainable multilayer feedforward neural network, and adaptively learning key parameters of each layer in an end-to-end mode to obtain an optimal network model; received signals are input into the completely trained network, and a high-precision channel estimation result can be output through single forward propagation. According to the method, dependence on manual parameter adjustment is remarkably reduced, adaptive optimization solution is achieved in a data driving mode, and the convergence speed and robustness are greatly improved while high precision is guaranteed.

Description

Technical Field

[0001] This invention relates to the field of mobile communication technology, and specifically to a millimeter-wave MIMO channel estimation method based on dual non-convex relaxation. Background Technology

[0002] With 5G entering large-scale commercial use and 6G research launching globally, mobile traffic is growing exponentially, and network capacity demand is expected to increase more than a thousandfold over the next decade. To address this challenge and alleviate the increasing scarcity of spectrum resources below 6 GHz, developing wireless communication technologies utilizing higher frequency bands has become an inevitable choice. Millimeter wave bands, due to their abundant untapped spectrum resources, have become a key technology, capable of supporting extremely high speeds and ultra-large capacity transmission requirements in 5G evolution and 6G systems. Massive MIMO technology, by deploying large-scale antenna arrays at the base station side, can effectively utilize spatial dimensions to improve spectrum efficiency and anti-interference capabilities, and is a core means to fully realize the potential of millimeter waves.

[0003] Existing channel estimation methods require significantly increased pilot overhead and computational complexity, which is a bottleneck restricting the practical deployment of systems. This results in low channel estimation accuracy and poor convergence. Furthermore, relying on manual parameter tuning and using excessively large penalty parameters can lead to numerical instability and low adaptability to different channel environments. Summary of the Invention

[0004] This invention proposes a millimeter-wave MIMO channel estimation method based on dual non-convex relaxation. This method addresses the technical problems of existing channel estimation methods, which suffer from rapidly increasing pilot overhead and computational complexity, which are bottlenecks restricting the practical deployment of systems. These problems result in low channel estimation accuracy and poor convergence. Furthermore, the reliance on manual parameter tuning and the use of extremely large penalty parameters can lead to numerical instability and low adaptability to different channel environments.

[0005] To achieve the above objectives, the present invention employs the following technical solution: A millimeter-wave MIMO channel estimation method based on dual non-convex relaxation, comprising the following steps: S1. Configure the channel parameters for the base station and the user terminal; S2. The base station and user terminal set the array element spacing, and construct the channel matrix according to the channel parameters configured by the base station and user terminal, and obtain the low-rank characteristics of the channel matrix. S3. The base station sends a training sequence, and the user receives the signal and feeds it back to the base station via the uplink. The base station constructs a millimeter-wave large-scale MIMO channel estimation optimization model and observation equation based on low-rank constraints, according to the received signal, the training sequence and the low-rank characteristics of the channel matrix. S4. Introduce a double non-convex, non-smooth rank relaxation norm, use it to construct dynamic weights, construct an objective function that can approach the rank of the matrix, and then combine the objective function with the observation equation by introducing an auxiliary matrix and equality constraints to form the final optimization model. S5. Solve the final optimization model based on the alternating direction multiplier method framework, and map its iterative process to a multi-layer feedforward neural network. Each layer of the network corresponds to one iteration. By setting the loss function and jointly training the independent parameters of each layer of the network in an end-to-end manner, the network converges and the final network model is obtained. S6. The received signal to be processed is directly input into the final network model. After a complete forward propagation calculation, the final channel matrix estimate is output.

[0006] Preferably, the base station in S1 is equipped with root antenna and One radio frequency link, equipped with user terminal root antenna and One radio frequency link, to meet , The base station transmits a random number of training sequences through the downlink channel.

[0007] Preferably, in step S2, both the base station and the user terminal use uniform linear arrays and the set element spacing is [value missing]. The channel estimation model for millimeter-wave massive MIMO, i.e., the formula for calculating the channel matrix, is as follows: ; in, For path loss, The number of effective transmission paths, For the first Complex gain of the path, as well as These are the array response vectors for the base station and the user terminal; The formulas for calculating the array response vectors of the base station and the user terminal are as follows:

[0008] in, It is the wavelength of millimeter waves.

[0009] Preferably, the received signal in S3 is as follows: ; in, To train the symbol matrix, For the number of training symbols, The downlink channel matrix for the base station and the user terminal. This is the noise matrix; At the base station, channel state information is jointly restored, and the user directly transmits the signal. The feedback to the base station, the feedback uplink signal is represented as The formula for calculating the equivalent signal estimate of the channel is as follows: ; in, This is the conjugate transpose of the uplink Rayleigh fading channel matrix.

[0010] Preferably, the channel estimation model for the low-rank constrained optimization problem is as follows:

[0011] in, Let be the channel matrix to be estimated. The estimation is performed under the premise that some sampled elements are known, that is, there exists an index set. Make .

[0012] Preferably, the steps for constructing the doubly nonconvex, nonsmooth rank relaxation norm model are as follows: S41. Combine the Schatten-p norm and the truncated Schatten-p norm, perform a first-order Taylor expansion on the truncated Schatten-p norm, and then obtain the weight vector from the truncated Schatten-p norm. S42. Based on the weighted vector, construct a doubly nonconvex, nonsmooth rank relaxation norm model to update the channel estimation model for the low-rank constrained optimization problem. The truncated Schatten-p norm, or TSPN, is defined as follows:

[0013] in, The number of segments to be truncated indicates the number of segments to retain. The number of maximum singular values, Because the TSPN truncation operation directly discards the previous value. The value depends on the sorting of singular values. Directly optimizing TSPN requires sorting and truncating the singular values ​​in each iteration, which is computationally complex and prone to getting trapped in local optima. To solve this problem, two auxiliary matrices are introduced. , Its construction method is as follows: For matrix Perform singular value decomposition to obtain the left and right singular value matrices:

[0014] in, , Extract the previous one from it. A matrix is ​​formed by two singular vectors. , ,satisfy , .matrix The singular values ​​satisfy the following properties:

[0015] Based on this characteristic, a first-order Taylor expansion of TSPN is performed as follows:

[0016] The formula for calculating the weight vector is as follows: .

[0017] This weighted form no longer involves explicit sorting and truncation, but instead achieves soft truncation through a continuous weight function.

[0018] Preferably, the updated channel estimation model for the low-rank constrained optimization problem is obtained as follows:

[0019] The final channel estimation model is as follows:

[0020] in, For auxiliary matrix, It is a differentiable convex function.

[0021] Preferably, in S5, each layer of the feedforward neural network corresponds to one iteration, and the iteration process is as follows:

[0022] in, , and They represent the corresponding variables respectively. , and Network layer, It refers to a specific layer in the total number of layers in the network.

[0023] Preferably, the independent parameters are as follows: ; The loss function is as follows: The loss function is as follows:

[0024] in, The iteration step size, For the vehicle, , This represents the set of all trainable parameters in the network. This represents the estimated channel matrix output after the network passes through layers of forward propagation. Its value is determined by the network parameters. Decide, This represents the channel matrix generated during the training phase, which serves as the target label during training.

[0025] Preferably, the update rule for the final network model is as follows: ; in, The training learning rate.

[0026] As can be seen from the above technical solution, this invention provides a millimeter-wave MIMO channel estimation method based on dual non-convex relaxation. Compared with the prior art, this invention has the following advantages: by mapping the multi-round iterative process of the traditional alternating direction multiplier method to a trainable feedforward network structure, a fundamental shift from iterative optimization to one-time forward inference is achieved; the network learns a series of key parameters such as penalty parameters, update step size, and near-end threshold in each network layer end-to-end, thereby approaching or even surpassing the convergence accuracy of the traditional alternating direction multiplier method with a very small number of network layers; this method significantly reduces the dependence on manual parameter tuning, avoids the problem of relying on experience trial and error in traditional methods, and improves the universality and robustness of the algorithm in different channel environments. The feedforward neural network model proposed in this invention can effectively handle such non-convex optimization problems, not only achieving or surpassing the traditional method in estimation accuracy, but also showing a significant advantage in convergence speed, thus providing a practical solution for achieving low-complexity, high-real-time millimeter-wave large-scale MIMO channel estimation. Attached Figure Description

[0027] Figure 1 This is a flowchart illustrating a millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to the present invention. Figure 2 This is a schematic diagram comparing the N-DNNR and TSPN models in this invention; Figure 3 This is a schematic diagram illustrating the effect of the p-value on the N-DNNR model in this invention; Figure 4 This is a schematic diagram comparing the traditional alternating direction multiplier method and the alternating direction multiplier method network in this invention. Detailed Implementation

[0028] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments.

[0029] like Figure 1 As shown in the figure, a millimeter-wave MIMO channel estimation method based on dual non-convex relaxation in this embodiment includes the following steps: S1. Configure the number of antennas and radio frequency links for the base station and user terminals, and configure the precoding matrix for the base station and the merging matrix for the user terminals. S2. The spacing between array elements is set at the base station and the user terminal to construct the channel matrix, thereby obtaining the low-rank characteristics of the channel matrix. S3. The base station sends a training sequence, and the user receives the signal and feeds it back to the base station via the uplink. The base station constructs a millimeter-wave large-scale MIMO channel estimation optimization model based on low-rank constraints based on the received signal and the known training sequence. S4. Introduce a dual nonconvex nonsmooth rank relaxation norm and use the dynamic weights constructed by it to construct an objective function that can more accurately approximate the rank of the matrix. Then, by introducing an auxiliary matrix and equality constraints, combine the objective function with the observation equation to form a final optimization model that can be decomposed and solved. S5. The above model is solved based on the alternating direction multiplier method framework, and the deep unfolding technique is applied to map its iterative process into a trainable multi-layer feedforward neural network. Each layer of the network corresponds to one iteration. By setting the loss function and jointly training the independent parameters of each layer of the network in an end-to-end manner until convergence, a fully trained network model with optimal parameters is obtained. S6. In the channel estimation stage, the received signal to be processed is directly input into the trained network model. After one complete forward propagation calculation, the final channel matrix estimate can be directly output.

[0030] Furthermore, S1 employs a narrowband millimeter-wave massive MIMO system in frequency division duplex mode, and the base station is equipped with... root antenna and One radio frequency link, equipped with user terminal root antenna and One radio frequency link, to meet , The base station transmits a random number of training sequences through the downlink channel.

[0031] Furthermore, in S2, both the base station and the user terminal use uniform linear arrays with a set element spacing of [value missing]. The channel estimation model for millimeter-wave massive MIMO, i.e., the formula for calculating the channel matrix, is as follows: ; in, For path loss, The number of effective transmission paths, For the first Complex gain of the path, as well as These are the array response vectors for the base station and the user terminal; The millimeter-wave channel matrix is ​​obtained as a product of the array flow matrix and the path gain. ; in, The additive complex Gaussian matrix is ​​a diagonal gain matrix, satisfying the following conditions: , , These represent the transceiver array flow matrices, respectively. The formulas for calculating the array response vectors at the base station and the user end are as follows:

[0032] in, It is the wavelength of millimeter waves.

[0033] Due to the millimeter wave channel matrix The number of antennas at the base station and user end is far greater than the number of transceiver paths. From this, we can deduce that:

[0034] Therefore, the channel matrix It exhibits significant low-rank characteristics, providing theoretical support for subsequent channel estimation based on low-rank matrix recovery.

[0035] Furthermore, the received signal in S3 is as follows: ; in, To train the symbol matrix, For the number of training symbols, The downlink channel matrix for the base station and the user terminal. This is the noise matrix; Considering the joint recovery of channel state information at the base station, the user directly transmits the signal. The feedback to the base station, the feedback uplink signal is represented as The formula for calculating the equivalent signal estimate of the channel is as follows: ; in, This is the conjugate transpose of the uplink Rayleigh fading channel matrix.

[0036] The channel estimation problem can be transformed into the following low-rank constrained optimization problem, and the channel estimation model for the low-rank constrained optimization problem is as follows:

[0037] in, Let be the channel matrix to be estimated. The estimation is performed under the premise that some sampled elements are known, that is, there exists an index set. Make .

[0038] Furthermore, the steps for constructing a doubly nonconvex, nonsmooth rank relaxation norm model are as follows: S41. Combine the Schatten-p norm and the truncated Schatten-p norm, perform a first-order Taylor expansion on the truncated Schatten-p norm, and then obtain the weight vector from the truncated Schatten-p norm. S42. Based on the weighted vector, construct a doubly nonconvex, nonsmooth rank relaxation norm model to update the channel estimation model for the low-rank constrained optimization problem. The Schatten-p norm is a non-convex function between the nuclear norm and the rank function. A smaller p-value is closer to the rank function. However, the non-convexity of the Schatten-p norm is sensitive to initialization, requiring complex algorithm design. Smaller p-values ​​can slow down the convergence speed or even cause the algorithm to diverge. The TSPN, on the other hand, ignores the impact of large singular values ​​on the matrix rank. Its truncation strategy only penalizes smaller singular values, thus avoiding excessive shrinkage for larger singular values ​​and preserving key information, resulting in good recovery performance. Therefore, the D-DNNR model is constructed by combining the Schatten-p norm and TSPN:

[0039] in, The weight vector is TSPN, and , This represents the Schatten-p norm.

[0040] The doubly nonconvex and nonsmooth norm is transformed into a nonconvex and nonsmooth function problem by utilizing the weighted vector in the definition of the hypergradient property of nonconvex and nonsmooth functions. Since TSPN ignores the first k large singular values, the number of weights calculated from its gradient does not match the number of singular values ​​of the Schatten-p norm. Furthermore, its truncation operation directly discards the first k singular values, heavily relying on the instantaneous sorting of singular values. This forces any gradient-based optimization algorithm to repeatedly perform complete singular value decomposition and sorting, resulting in extremely high computational complexity. To address these issues, it is replaced with an equivalent norm. The first-order Taylor expansion of the truncated Schatten-p norm is as follows:

[0041] in, For rank r The channel matrix, To determine the number of cutoffs, , They are respectively The left and right singular values, , They are respectively and The former k List, , , ; set up ,but about The derivative is To calculate the weight vector, TSPN is placed in the th... Next iteration point The first-order Taylor expansion at point A is as follows:

[0042] Therefore, the formula for calculating the weight vector is as follows:

[0043] when hour, This achieves dynamic suppression of penalties, retaining the previous... Principal components, avoiding the instability of hard truncation; when hour, This design enhances the regularization penalty, compresses subsequent singular values, and thus implicitly achieves low-rank constraints. This significantly improves the model's generalization ability and convergence. By introducing the derived weights, a channel estimation model based on the doubly nonconvex and nonsmooth norm is obtained, which is the updated channel estimation model for the low-rank constrained optimization problem.

[0044] Furthermore, the channel estimation model for the low-rank constrained optimization problem is updated, resulting in the following updated channel estimation model for the low-rank constrained optimization problem:

[0045] Among them, the channel matrix The estimation is performed under the premise that some sampled elements are known, that is, there exists an index set. Make .

[0046] Because the non-convexity of the objective function and the strong coupling with the equality constraints can easily lead to getting trapped in local minima during direct optimization, and convergence is difficult to guarantee, an auxiliary matrix is ​​introduced. The final channel estimation model is as follows:

[0047] in, For auxiliary matrix, Let be a differentiable convex function. For ease of representation, let... Therefore, the above model satisfies the assumptions of the alternating direction multiplier algorithm. By relaxing the Lagrange multipliers into the objective function, its augmented Lagrange function is expressed as:

[0048] Regarding updates For the subproblem, gradient descent is used instead of closed-form solution for updating as follows:

[0049] in, , This is the iteration step size.

[0050] Regarding updates The subproblem is addressed by introducing the proximal gradient algorithm to update the variables. as follows:

[0051] in, for The proximal operator, It is the Lipschitz constant (which needs to be set). Otherwise, it may cause divergence. Let the step size of the proximal operator be denoted as . According to the definition of the proximal operator, the variable The iteration is simplified as follows:

[0052] The optimal solution to this problem is obtained using the weighted singular value function threshold operator. ,in, as well as It is a unitary matrix, by and If ,but ,for The solution formula is as follows:

[0053] For ease of subsequent symbol representation, let... , , for The optimal solution to the equation is obtained by calculating the closed-form expression as follows: 1. When hour, The solution can be simplified to:

[0054] Its closed-form solution is as follows:

[0055] in, ; 2. When hour, The solution can be simplified to:

[0056] Its closed-form solution is as follows:

[0057] in, ; therefore, The update rules are as follows: ; in, , ; Update dynamic weights Question, seeking left and right singular values and Take the projection matrix , ; therefore,

[0058] Update about For the subproblems, the following update rules are adopted: ; Update about For the subproblems, the following update rules are adopted: ; This embodiment presents a deep-expansion-based alternating direction multiplier method network. By solidifying the complete computational flow of the alternating direction multiplier iterative algorithm into a trainable multilayer feedforward network, a fundamental shift is achieved from online iterative optimization to offline training and online single forward propagation. Furthermore, in S5, each layer of the feedforward neural network corresponds to one iteration, and the iteration process is as follows:

[0059] in, , and They represent the corresponding variables respectively. , and Network layer, It refers to a specific layer in the total number of layers in the network.

[0060] Furthermore, the network parameters are set as follows: ; The loss function is set as follows: The loss function is as follows:

[0061] in, The iteration step size, For the vehicle, , This represents the set of all trainable parameters in the network. This represents the estimated channel matrix output after the network passes through layers of forward propagation. Its value is determined by the network parameters. Decide, This represents the channel matrix generated during the training phase, which serves as the target label during training.

[0062] Furthermore, the final network model update rules are as follows: ; in, The training learning rate.

[0063] In the offline phase, the feedforward neural network is trained using massive amounts of channel data and automatically learns the optimal set of internal parameters. This results in the final network model, thus completely eliminating the reliance on online parameter tuning. During the channel estimation stage, only the pre-trained optimal network hyperparameter set needs to be loaded. This allows for obtaining relatively accurate channel estimation results with a very small number of layers.

[0064] To verify that the depth-expansion-based alternating direction multiplier algorithm mentioned in this invention can improve the convergence capability of traditional algorithms, the experiment was designed to first approximate the channel matrix using N-DNNR and TSPN models respectively, comparing the average number of iterations required for convergence. Then, the impact of different values ​​of p on channel estimation performance was compared. Finally, under the same conditions, the channel state information estimation performance of the traditional alternating direction multiplier method and the alternating direction multiplier method network was compared. The training network was set to 100 layers, the required parameters for channel state information estimation were a millimeter-wave frequency band of 90Hz, and the antenna AoD angle was [missing information].

[0065] For an effective transmission path, the channel state information estimation error uses the normalized mean square error and Frobenuis norm error as follows:

[0066] like Figure 2 As shown, by different numbers of cutoffs k Compare the number of iterations required for convergence of the N-DNNR and TSPN models, and analyze the number of truncations. k Impact on convergence performance. Test matrix. Size is And it follows Gaussian Rayleigh decay, rank r =13, number of cutoffs k Iterate through 1-17. By comparing the average number of iterations required for the N-DNNR and TSPN models to converge, it can be seen that when... At that time, the N-DNNR model requires fewer iterations than the TSPN model. At that time, the two models had the same number of iterations.

[0067] like Figure 3 As shown, considering the number of transmit and receive antennas Channel length Signal-to-noise ratio The rank of the channel matrix was calculated, and the maximum number of network layers was set to 100 for comparison. Simulation results show that when... At that time, NMSE converged earlier than The NMSE at a given time converges relatively quickly, but when the number of iterations increases... , The NMSE at that time was higher than The NMSE, and approximately at the number of iterations. It gradually converges to a stable value, while when At that time, It gradually converges to a stable value. Based on the above analysis, it can be inferred that... The N-DNNR model reduces error faster in the early stages of iteration, but The N-DNNR model at that time lies between sparsity and smoothness, thus helping it to better approximate the real channel in later stages.

[0068] like Figure 4 As shown, the convergence performance of the traditional alternating direction multiplier method and the alternating direction multiplier method network is compared under the same initial conditions. Simulation results show that the alternating direction multiplier method network exhibits faster convergence speed and better steady-state performance during iterative computation. The fundamental reason for this performance advantage lies in the essential difference in their parameter update mechanisms: the traditional alternating direction multiplier method relies on only a fixed set of global hyperparameters throughout the iteration process. This parameter set cannot adaptively adjust with the optimization process, resulting in a relatively rigid convergence path and a tendency to get stuck in local optima or converge slowly in non-convex problems. In contrast, the alternating direction multiplier method network proposed in this invention, through deep unfolding structure and end-to-end training, can learn independent optimal parameters for each network layer that match the optimization state of that layer. Therefore, it can achieve lower convergence error while significantly reducing the number of iterations.

[0069] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. A computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the flow or function according to the embodiments of this application is generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., a solid-state disk (SSD)).

[0070] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes the element.

[0071] The various embodiments in this specification are described in a related manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the system embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0072] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to the present invention, characterized in that, Includes the following steps: S1. Configure the channel parameters for the base station and the user terminal; S2. The base station and user terminal set the array element spacing, and construct the channel matrix according to the channel parameters configured by the base station and user terminal, and obtain the low-rank characteristics of the channel matrix. S3. The base station sends a training sequence, and the user terminal receives the signal and feeds it back to the base station via the uplink. Based on the received signal, training sequence, and low-rank characteristics of the channel matrix, the base station constructs a low-rank constraint-based optimization model for millimeter-wave large-scale MIMO channel estimation and observation equations. S4. Introduce a double non-convex, non-smooth rank relaxation norm, use it to construct dynamic weights, construct an objective function that can approach the rank of the matrix, and then combine the objective function with the observation equation by introducing an auxiliary matrix and equality constraints to form the final optimization model. S5. Solve the final optimization model based on the alternating direction multiplier method framework, and map its iterative process to a multi-layer feedforward neural network. Each layer of the network corresponds to one iteration. By setting the loss function and jointly training the independent parameters of each layer of the network in an end-to-end manner, the network converges and the final network model is obtained. S6. The received signal to be processed is directly input into the final network model. After a complete forward propagation calculation, the final channel matrix estimate is output.

2. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 1, characterized in that: The base station in S1 is equipped with root antenna and One radio frequency link, equipped with user terminal root antenna and One radio frequency link, to meet , The base station transmits a random number of training sequences through the downlink channel.

3. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 2, characterized in that: In S2, both the base station and the user terminal use uniform linear arrays, and the set element spacing is... The channel estimation model for millimeter-wave massive MIMO, i.e., the formula for calculating the channel matrix, is as follows: ； in, For path loss, The number of effective transmission paths, For the first Complex gain of the path, as well as These are the array response vectors for the base station and the user terminal; The formulas for calculating the array response vectors of the base station and the user terminal are as follows: in, It is the wavelength of millimeter waves.

4. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 3, characterized in that: The signals received by the user terminal in S3 are as follows: ； in, To train the symbol matrix, For the number of training symbols, The downlink channel matrix for the base station and the user terminal. This is the noise matrix; At the base station, channel state information is jointly restored, and the user directly transmits the signal. The feedback to the base station, the feedback uplink signal is represented as The formula for calculating the equivalent signal estimate of the channel is as follows: ； in, This is the conjugate transpose of the uplink Rayleigh fading channel matrix.

5. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 4, characterized in that: The low-rank constrained millimeter-wave massive MIMO channel estimation optimization model is as follows: in, Let be the channel matrix to be estimated. The estimation is performed under the premise that some sampled elements are known, that is, there exists an index set. Make .

6. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 5, characterized in that: The steps for constructing the doubly nonconvex, nonsmooth rank relaxation norm are as follows: S41. Combine the Schatten-p norm and the truncated Schatten-p norm, perform a first-order Taylor expansion on the truncated Schatten-p norm, and then obtain the dynamic weight vector from the truncated Schatten-p norm. S42. Based on the weighted vector, construct a doubly nonconvex, nonsmooth rank relaxation norm model to update the channel estimation model for the low-rank constrained optimization problem. The truncated Schatten-p norm, or TSPN, is defined as follows: in, The number of segments to be truncated indicates the number of segments to retain. The number of maximum singular values, Introduce two auxiliary matrices , Its construction method is as follows: For the channel matrix Perform singular value decomposition to obtain the left and right singular value matrices: in, , Extract the previous A matrix is ​​formed by two singular vectors. , ,satisfy , ,matrix The singular values ​​satisfy the following properties: Based on this characteristic, a first-order Taylor expansion of TSPN is performed as follows: The formula for calculating the weight vector is as follows: 。 7. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 6, characterized in that: The final optimized model is as follows: in, For auxiliary matrix, It is a differentiable convex function.

8. The millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 7, characterized in that: In S5, each layer of the feedforward neural network corresponds to one iteration, and the iteration process is as follows: in, , and They represent the corresponding variables respectively. , and Network layer, It refers to a specific layer in the total number of layers in the network.

9. A millimeter-wave MIMO channel estimation method based on dual non-convex relaxation as described in claim 8, characterized in that: The independent parameters are as follows: ； The loss function is as follows: in, The iteration step size, For the vehicle, , This represents the set of all trainable parameters in the network. This represents the estimated channel matrix output after the network passes through layers of forward propagation. Its value is determined by the network parameters. Decide, This represents the channel matrix generated during the training phase, which serves as the target label during training.

10. A millimeter-wave MIMO channel estimation method based on dual non-convex relaxation according to claim 9, characterized in that: The update rules for the final network parameter model are as follows: ； in, The training learning rate.

Images