An optimization method for heterogeneous over-the-air computing networks with imperfect channels

Abstract

This invention belongs to the field of the Internet of Things (IoT) and relates to an optimization method for heterogeneous over-the-air computing networks with imperfect channels. The method includes: constructing a RIS-assisted heterogeneous over-the-air computing network model MD for imperfect channels; calculating the mean square error of model MD; constructing an optimization problem P based on the channel error matrix Δ, device transmit power vector b, base station receive beamforming vector v, and smart reflector phase shift vector θ with the objective of minimizing and maximizing the mean square error; and solving P to obtain the optimal device transmit power vector b. * Base station received beamforming vector v * and the phase shift vector θ of the intelligent reflector * ; Set b, v, and θ of the heterogeneous network for aerial computing as b respectively * v * and θ * The optimized heterogeneous network for aerial computing was obtained. This invention constructs an objective optimization problem based on Δ, b, v and θ with the mean square error of the minimax network model MD as the objective, which significantly improves the performance of the heterogeneous network for aerial computing.

Description

Technical Field

[0001] This invention belongs to the field of the Internet of Things and relates to an optimization method for heterogeneous over-the-air computing networks with imperfect channels. Background Technology

[0002] In recent years, with the development of wireless communication technology, the Internet of Things (IoT) has gradually become an important part of modern society. By connecting various physical devices through sensors, software, and network technologies, and enabling real-time information transmission and processing, these devices can sense environmental changes and exchange data, thus providing various intelligent services such as smart homes, smart cities, and industrial automation. However, after decades of rapid development, the number of IoT nodes has increased dramatically. Due to this surge in the number of nodes, the IoT faces significant challenges in data aggregation and communication, such as high latency, high energy consumption, and limited bandwidth. One existing solution is Wireless Data Fusion (WDA), which aims to integrate, aggregate, and process data from multiple sensors and devices to simplify transmission, reduce data volume, and improve overall system performance. However, for large-scale WDA with a massive number of nodes, the traditional method of communicating first and then computing inevitably suffers from low bandwidth efficiency and high latency.

[0003] To address these challenges in the Internet of Things (IoT), over-the-air computing (AirComp), as an emerging technology, is considered an effective way to overcome the limitations of wireless data fusion. By utilizing the waveform superposition characteristics of multiple access channels, AirComp combines concurrent data from multiple IoT devices with nomographic functions (e.g., averages and geometric averages) calculations. Through concurrent transmission, AirComp achieves low transmission latency independent of the number of IoT devices, thus enabling fast WDA (Wireless Data Acquisition). This approach allows for parallel processing of computational tasks on the wireless link, significantly reducing data transmission latency and bandwidth pressure. Due to these advantages, AirComp has gained widespread attention in federated learning, signal processing, and unmanned aerial vehicles (UAVs). However, AirComp also faces some challenges. As the number of connected wireless devices increases, poor communication environments can significantly impact AirComp performance. To address this issue, researchers have proposed numerous methods for AirComp cellular applications, such as power control and massively multi-input multi-output (MIMO). However, these techniques are not suitable for multi-cellular scenarios, where AirComp performance is severely affected by inter-cell interference. Some studies have proposed cooperative interference management mechanisms suitable for multi-cellular scenarios and interference management mechanisms for AirComp coexisting networks. However, these methods offer limited performance improvements to AirComp by adjusting the transmit and receive coefficients at the transceiver ends. How to further suppress interference, thereby reducing the system mean square error and improving AirComp performance, remains a pressing issue.

[0004] Reconfigurable Intelligent Surface (RIS) is a novel technology that effectively and cost-effectively assists AirComp in suppressing interference. An RIS is a plane composed of multiple passive reflective elements that can adjust the phase and amplitude of the reflected signal. By controlling these reflective elements, this technology can reconfigure the phase and amplitude of the signal to optimize the propagation path of the wireless signal, improve signal quality, increase coverage, and reduce interference. Furthermore, RIS is characterized by low cost, low power consumption, and flexible deployment. Based on these excellent characteristics, RIS is considered an important technology for improving network capacity, coverage, and energy efficiency in 5G and future 6G communication systems. Similarly, applying RIS to heterogeneous over-the-air computing networks can also achieve interference suppression, thereby improving the performance of AirComp.

[0005] However, existing RIS-assisted heterogeneous networks for airborne computing fail to take channel errors into account, resulting in high latency and insufficient mean square error (MSE) for smart reflector-assisted heterogeneous networks for airborne computing in the face of imperfect channels, thus leading to poor performance. Summary of the Invention

[0006] To address the aforementioned problems in the prior art, this invention employs an optimization method for heterogeneous over-the-air computing networks with imperfect channels, comprising:

[0007] S1. Construct a heterogeneous network model MD for airborne computing assisted by RIS for imperfect channels; Model MD includes: airborne computing cell, traditional cell and RIS, wherein the airborne computing cell includes: base station and airborne computing device, and the traditional cell includes: base station and traditional device; wherein, RIS is a smart reflector.

[0008] S2. Calculate the mean square error of the model MD, and construct an objective optimization problem based on the channel error matrix, the device transmit power vector, the base station receive beamforming vector, and the smart reflector phase shift vector with the goal of minimizing and maximizing the mean square error.

[0009] S3. Construct a first-level objective optimization problem based on the channel error matrix according to the objective optimization problem, and solve the first-level objective optimization problem analytically to obtain the closed-form solution of the optimal channel error matrix;

[0010] S4. Based on the closed-form solution of the target optimization problem and the optimal channel error matrix, construct a secondary target optimization problem based on the device transmit power vector, the base station receive beamforming vector, and the intelligent reflector phase shift vector; solve the secondary target optimization problem using the alternating optimization algorithm to obtain the optimal device transmit power vector, the optimal base station receive beamforming vector, and the optimal intelligent reflector phase shift vector.

[0011] S5. Set the device transmit vector, base station receive beamforming vector, and RIS phase shift vector of the RIS-assisted heterogeneous network for imperfect channels to the optimal device transmit power vector, the optimal base station receive beamforming vector, and the optimal smart reflector phase shift vector, respectively, to obtain the optimized RIS-assisted heterogeneous network for imperfect channels.

[0012] A cellular over-the-air computing unit consists of one base station and K single-antenna over-the-air computing devices; a traditional cellular unit consists of one base station and one single-antenna traditional device; in the heterogeneous over-the-air computing network model, one cell is the target cell, and the rest are interfering cells. For calculating the set of cells l in the air, For the set of traditional cellular n, the target cellular device For the set of devices in a target cell, the airborne computing cell is the device. For aerial computation, the set of devices in cellular l is compared with the devices in traditional cellular n. It is a collection of devices in a traditional cellular n; the smart reflective surface has M reflective elements.

[0013] Target optimization problem for:

[0014]

[0015] Where C1 is the constraint of the phase shift matrix of the smart reflector, and C2 is the transmit power b of the target cell device. k The constraints are as follows: C3 and C4 are the channel error matrices of the direct link path between the device and the target base station and the path through the smart reflector, respectively; b is the device transmit power vector of the target cell; v is the receive beamforming vector of the base station of the target cell; and θ is the phase shift vector of the smart reflector. m For the phase shift of the reflective element m of the intelligent reflective surface, P max The maximum transmit power of the device is given by Δ, where MSE is the mean square error of the heterogeneous network model for intelligent reflector-assisted airborne computing in imperfect channels. p Let be the channel error matrix of the direct link path between device p and the base station of the target cell. Let ρ be the channel error matrix between device p and the base station of the target cell via the smart reflector path, and let ρ be the maximum error of the channel error matrix.

[0016] The mean square error (MSE) of the heterogeneous network model for smart reflector-assisted airborne computing in imperfect channels is:

[0017]

[0018] Among them, b i Let represent the transmission power of device i, and (·) H This indicates the conjugate transpose. and Let represent the measurement channel vectors of the direct link paths between the target cell's device k, the over-the-air computing cellular device i, the traditional cellular device j, and the target cell's base station, respectively. and Let Δ represent the cascaded channel error matrices between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station via the smart reflector path. k Δ i and Δ jLet σ represent the channel error matrices of the direct link paths between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station, respectively. 2 This represents the power of additive white Gaussian noise. and These represent the cascaded measurement channel matrices between the target cell's device k, the over-the-air computing cellular device i, the traditional cellular device j, and the target cell's base station via the smart reflector path, respectively.

[0019] The primary objective optimization problem is:

[0020]

[0021] Solving first-order objective optimization problems using analytical methods includes:

[0022] S31. Use the complex trigonometric inequality to calculate the inequalities for the first-order objective optimization problem;

[0023] S32. Based on the requirement that the inequalities of the first-level objective optimization problem satisfy the equation, we obtain the equation containing the optimal channel error matrix;

[0024] S33. Solve the equation containing the optimal channel error matrix based on the constraints C1 and C2 of the channel error matrix to obtain the closed-form solution Δ of the optimal channel error matrix. * .

[0025] Closed-form solution Δ of the optimal channel error matrix * for:

[0026]

[0027] Where, Δ k * Δ i * Δ j * Δ k Δ i and Δ j The closed-form solution, They are respectively and The closed-form solution.

[0028] The secondary objective optimization problem is:

[0029]

[0030] The optimization of secondary objective problems using the alternation optimization method includes:

[0031] S41. Fix v and θ of the second-level objective optimization problem to obtain a subproblem. Optimize the subproblem using the Lagrange multiplier method Get b k The optimal value b k * ; based on all devices k of b k * Obtain the optimal device transmission coefficient b * ;

[0032] S42, b k * Substituting these values ​​into the second-order objective optimization problem, we obtain subproblems based on v and θ. Set the initial value of θ to θ0 and the iteration stopping condition threshold ∈;

[0033] S43, Subproblem θ is fixed as θ from the previous iteration. t-1 This yields a subproblem based on v. Apply gradient descent to the subproblem Optimize to obtain the optimized v t ;

[0034] S44, Subproblems The value of v is fixed as the optimized v. t This yields a subproblem based on θ. Using the semidefinite relaxation method and Taylor series expansion method on the subproblem Processing the problem yields the subproblems. Using convex optimization tools on subproblems Optimize to obtain the optimized θ t ;

[0035] S45, Based on the optimized v t and θ t Calculate the ratio q t If q t If v is less than the iteration stopping condition threshold ∈, then t and θ t These are respectively used as the optimal base station receive beamforming vectors v * and the optimal intelligent reflector phase shift vector θ * Otherwise, return to step S43.

[0036] According to the optimized v t and θ t Calculate the ratio q t Includes: based on the optimized v t and θ t Calculate the mean squared error (MSE) of the current iteration t. Calculate the absolute value of the difference between the MSE of the current iteration t and the MSE of the previous iteration. Divide the absolute value by the MSE of the previous iteration to obtain the ratio q.t .

[0037] Beneficial effects:

[0038] 1. This invention constructs a RIS-assisted heterogeneous network model MD for imperfect channels. Considering channel errors, it constructs an objective optimization problem based on the channel error matrix, device transmit power vector, base station receive beamforming vector, and intelligent reflector phase shift vector with the objective of minimizing and maximizing the mean square error of the network model MD. The optimization problem is solved to obtain the optimal device transmit power vector, the optimal base station receive beamforming vector, and the optimal intelligent reflector phase shift vector, which significantly improves the performance of heterogeneous networks for imperfect channels. 2. This invention uses an alternating optimization method to solve the objective optimization function, which significantly reduces the complexity of the method. Attached Figure Description

[0039] Figure 1 A flowchart illustrating an optimization method for heterogeneous over-the-air computing networks with imperfect channels, provided as an embodiment of the present invention.

[0040] Figure 2 A schematic diagram of a system model for heterogeneous over-the-air computing networks oriented towards imperfect channels, provided in an embodiment of the present invention.

[0041] Figure 3 The flowchart illustrates the optimization of the objective optimization problem using the alternating optimization algorithm provided in this embodiment of the invention. Detailed Implementation

[0042] 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 only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0043] like Figure 1 As shown, this invention employs an optimization method for heterogeneous over-the-air computing networks with imperfect channels, comprising:

[0044] S1. Construct a RIS-assisted heterogeneous network model for airborne computing oriented towards imperfect channels (MD).

[0045] like Figure 2As shown, consider a RIS-assisted heterogeneous cellular network consisting of multiple AirComp cells and multiple traditional cells. In this network, the target cell is surrounded by multiple other cells, thus receiving interference from the uplink transmissions of other cells. Each AirComp cell consists of a base station with Nr antennas and K single-antenna devices, while each traditional cell consists of a single-antenna base station and a single-antenna device. The network is also equipped with a RIS with M reflectors. For model generality, the base station locations are determined using the Matern hard core point process (MHCPP). The AirComp cell density is λ. a The density of traditional cellular cells is λ. b Each cell has a coverage radius of R, and the locations of devices within the coverage area are uniformly distributed.

[0046] In one embodiment, the target cell is an AirComp cell; let Represents the collection of devices in the target cell, let Let L represent the set of L interfering AirComp cells, and let This represents a set of N conventional cells that generate interference. This represents the set of K devices in the AirComp cell that caused the interference. Represents a set of devices in a traditional cellular network. Represents the set of all devices in the system, that is

[0047] Let x p Representative equipment The measured data, where it is assumed that x p They are independent and identically distributed. For ease of power control, let... This represents the data transmitted by device p, where... Represents the normalization function, and s p The mean is 0 and the variance is 1, that is Therefore, the signal received by the base station of the target cell is represented as follows:

[0048]

[0049] Where n represents additive white Gaussian noise;

[0050] The post-processed signal recovered by the target base station can be represented as:

[0051]

[0052] In this case, it is assumed that the AirComp cell causing the interference uses an optimal wireless power control strategy, i.e., transmit power. Among them, f i Indicates equipment and base stations Channel vectors between Representing base stations The receiving beamforming vector.

[0053] The channel model in this embodiment of the invention is the Rician model (Rician fading channel model). The channel vector between device k and the base station T of the target cell, the channel vector between device k and RIS, and the channel matrix between RIS and the base station T of the target cell are defined as follows:

[0054]

[0055] in, Let represent the path loss of the links between device k and the target cell's base station T, between device k and RIS, and between RIS and the target cell's base station, respectively, denoted by . This represents the reference path loss and path loss exponent at d0 = 1 meter. k,T d k,R and d R,T Let represent the distances between device k and the target cell's base station T, between device k and RIS, and between RIS and the target cell's base station, respectively; β represents the Rician factor. Represents the real number field. Represents the complex field; This represents the non-line-of-sight (NLoS) component, whose elements are independent and identically distributed and follow a complex Gaussian distribution. The line-of-sight (LoS) component is represented by a uniform linear array (ULA), whose array response is: in, Let represent the antenna spacing and wavelength, respectively, and ω represent the angle of departure (AoD) or the angle of arrival (AoA). Where ω AOA ,ω AOD These represent AOA and AOD, respectively.

[0056] S2. Calculate the mean square error of the RIS-assisted heterogeneous network model for air computing in an imperfect channel. Construct an objective optimization problem based on the channel error matrix, device transmit power vector, base station receive beamforming vector, and smart reflector phase shift matrix with the goal of minimizing and maximizing the mean square error.

[0057] The mean square error (MSE) of a heterogeneous network model can be calculated using the function g and the corresponding estimation function. express:

[0058]

[0059] Denotes the diagonal phase shift matrix of RIS, where The phase shift of the reflecting element is represented by the function g, which represents the ideal received signal, i.e., the signal transmitted by the device.

[0060] However, the measurement of channel state information is subject to error; therefore, the mean square error under imperfect channels is:

[0061]

[0062] Among them, b k Let k represent the transmit coefficient of device k in the target cell, and v be the receive beamforming vector of the base station in the target cell. H This indicates the conjugate transpose. This represents the measurement channel matrix between the smart reflector and the base station of the target cell. and Let k represent the measurement channel matrix between the target cell device k, the over-the-air computing cell device i, the traditional cell device j, and the smart reflector, respectively. and Let represent the measurement channel vectors of the direct link paths between the target cell's device k, the over-the-air computing cellular device i, the traditional cellular device j, and the target cell's base station, respectively. and Let Δ represent the channel error matrices Δ between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station via the smart reflector path. k Δ i and Δ j Let represent the channel error vectors of the direct link paths between the target cell device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell base station, respectively; θ represents the phase shift vector of the smart reflector; and σ represents the phase shift vector of the smart reflector. 2 This represents the power of additive white Gaussian noise. and These represent the cascaded measurement channel matrices between the target cell's device k, the over-the-air computing cellular device i, the traditional cellular device j, and the target cell's base station via the smart reflector path, respectively.

[0063] To minimize the MSE under constraints of the channel error matrix, device power, and unity modulus of the phase shift of the RIS reflector, the optimization problem can be obtained. as follows:

[0064]

[0065] Where C1 is the constraint of the phase shift matrix of the smart reflector, and C2 is the transmit power b of the target cell device. k The constraints are as follows: C3 and C4 are the channel error matrices of the direct link path between the device and the target base station and the path through the smart reflector, respectively; b is the device transmit power vector of the target cell; θ m For the phase shift of the reflective element m of the intelligent reflective surface, P max Δ represents the maximum transmit power of the device. p Let be the channel error matrix of the direct link path between device p and the base station of the target cell. Let ρ be the channel error matrix between device p and the base station of the target cell via the smart reflector path, and let ρ be the maximum error of the channel error matrix.

[0066] S3. Construct a first-level objective optimization problem based on the channel error matrix according to the objective optimization problem, and solve the first-level objective optimization problem analytically to obtain the closed-form solution of the optimal channel error matrix;

[0067] First-order optimization problem based on channel error matrix Δ for:

[0068]

[0069] Solve the first-order optimization problem using analytical methods. include:

[0070] S31. Optimize the primary objective problem Expanding the terms within the second normal form, we obtain the inequalities for the objective optimization problem using the complex triangle inequality.

[0071]

[0072] S322. Based on the requirement that the inequalities of the first-level objective optimization problem satisfy the equation, we obtain the equation containing the optimal channel error matrix;

[0073] Specifically, the equation holds if and only if the following expression is satisfied, then the first-order optimization problem is valid. Reaching the maximum value:

[0074]

[0075] Where λ1, λ2, λ3, λ4, λ5, and λ6 are real numbers.

[0076] S33. Solve the equation containing the optimal channel error matrix based on the constraints C1 and C2 of the channel error matrix to obtain the optimal channel error matrix Δ. * Closed-form solution:

[0077]

[0078] S4. Based on the objective optimization problem and the closed-form solution of the optimal channel error matrix, construct a second-level objective optimization problem based on the device transmit power vector, the base station receive beamforming vector, and the smart reflector phase shift vector; solve the second-level objective optimization problem using the alternating optimization algorithm to obtain the optimal device transmit power vector b. * The optimal base station receive beamforming vector v * and the optimal intelligent reflector phase shift vector θ * ;

[0079] The secondary objective optimization problem based on the device transmit power vector, the base station receive beamforming vector, and the smart reflector phase shift vector is as follows:

[0080]

[0081] S5. Solve the second-level objective optimization problem using the alternating optimization algorithm to obtain the optimal device transmit power vector b for the target cell. * Base station received beamforming vector v * and the phase shift vector θ of the intelligent reflector * ;

[0082] like Figure 3 As shown, solving the second-order objective optimization problem using the alternating optimization algorithm includes:

[0083] S41. Fix v and θ of the second-level objective optimization problem to obtain a subproblem. Optimize the subproblem using the Lagrange multiplier method Get b k The closed solution b k * ; b of all devices k k * As the optimal device transmission coefficient b * ;

[0084] S42. The equipment transmit power vector b in the second-level objective optimization problem... k Fixed as b k *This yields a subproblem containing only v and θ. Set the initial values ​​of v and θ, v0 and θ0, and the iteration stopping condition threshold ∈;

[0085] S43, Subproblem With θ fixed at θ0, we obtain a subproblem based on v. Apply gradient descent to the subproblem Optimize to obtain the optimized v1;

[0086] S44, Subproblems With v fixed as the optimized v1, we obtain a subproblem based on θ. Using the semidefinite relaxation method and Taylor series expansion method on the subproblem Processing the problem yields the subproblems. Using convex optimization tools, CVX pairs of problems Optimization is performed to obtain the optimized θ1;

[0087] S45. Calculate the ratio q1 based on the optimized v1 and θ1. If q1 is less than the iteration stopping condition threshold ∈, stop the iteration and use v1 and θ1 as the optimal base station receiving beamforming vector v. * and the phase shift vector θ of the intelligent reflector * Otherwise, proceed to step S46;

[0088] S46, Subproblem θ is fixed as θ from the previous iteration. t-1 This yields a subproblem based on v. Apply gradient descent to the subproblem Optimize to obtain the optimized v t ;

[0089] S47, Subproblem The value of v is fixed as the optimized v. t This yields a subproblem based on θ. Using the semidefinite relaxation method and Taylor series expansion method on the subproblem Processing the problem yields the subproblems. Using convex optimization tools on subproblems Optimize to obtain the optimized θ t ;

[0090] S48. Based on the optimized v t and θ t Calculate the ratio q t If q t If the value is less than the iteration stopping condition threshold ∈, then the iteration stops, and v is set to... t and θ t As the optimal base station receive beamforming vector v* and the phase shift vector θ of the intelligent reflector * Otherwise, return to step S46.

[0091] Subproblems for:

[0092]

[0093] in,

[0094]

[0095] Among them, A k B and C are intermediate parameters. and b represents the concatenated equivalent channel vectors from the target cellular device, the over-the-air computing cellular device, and the traditional cellular device to the base station of the target cellular device, respectively. k Let represent the transmit coefficient of device k, and v represent the receive beamforming vector of the base station of the target cell. H P represents the conjugate transpose. max Let θ represent the maximum transmit power of device k, θ represent the phase shift vector of RIS, and σ represent the maximum transmit power of device k. 2 This represents the power of additive white Gaussian noise. and Let represent the cascaded measurement channel matrices between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station via the smart reflector path, respectively. and b represents the measurement channel matrix of the direct link path between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station, respectively. i This represents the emission coefficient of the over-the-air cellular device i. It is a complex field.

[0096] Optimize the subproblem using the Lagrange multiplier method. include:

[0097] Constructing the Lagrange function: Where, λ k It is a Lagrange multiplier.

[0098] For each variable x k Find the first derivative and the point where the derivative is zero:

[0099]

[0100] Summarized as follows:

[0101]

[0102] in, Indicates conjugate;

[0103] When λ k =0, that is When λ > 0, there is an optimal solution; when λ > 0, there is an optimal solution. k When >0, that is When considering the constraints of the modulus:

[0104]

[0105] After sorting, we get information about λ. k The quadratic equation:

[0106]

[0107] Due to λ k ≥0, therefore:

[0108]

[0109] Will Substitute into b k From the expression, we can obtain:

[0110]

[0111] Solving for the Lagrange function yields the optimal device transmission power.

[0112]

[0113] Substitute b k * subproblems As shown below:

[0114]

[0115] By fixing θ, we can obtain subproblems. as follows:

[0116]

[0117] Where, θ=θ t-1 , Let represent the cascaded equivalent measurement channel matrix between device k in the target cell, device i in the AirComp cell of the interfering cell, device j in the conventional cell of the interfering cell, and the BS of the target cell, respectively.

[0118] This is an unconstrained quadratic programming (QP) problem. Since the gradient of the objective function is solvable, its first derivative with respect to v can be obtained as follows:

[0119]

[0120] let The optimal solution is as follows:

[0121]

[0122] in, I is the identity matrix.

[0123] Let v be a fixed value. t Subproblems can be obtained as follows:

[0124]

[0125] in,

[0126]

[0127] v = v t D, E, and F are intermediate parameters, Tr is the trace of the matrix, and rank is the rank of the matrix. For matrix The elements in the m rows and m columns are positive semidefinite if they are greater than or equal to 1.

[0128] Use semidefinite relaxation (SDR) to relax. The constraints are determined, and the subproblems are obtained using Taylor series expansion. for:

[0129]

[0130] in, for First-order Taylor expansion, c0, t E t F c E c F These are all intermediate parameters. It represents The value of the Taylor expansion point, It represents The value of the Taylor expansion point, the first-order Taylor expansion of f(x) is: This corresponds to f(x0).

[0131] Through alternating optimization and The question is transformed into a question about Semidefinite programming (SDP) problems can be solved using existing convex optimization tools such as CVX or MOSEK.

[0132] According to the optimized v t and θ t Calculate the ratio q t Includes: based on the optimized v t and θ t Calculate the mean squared error (MSE) of the current iteration t. Calculate the absolute value of the difference between the MSE of the current iteration t and the MSE of the previous iteration. Divide the absolute value by the MSE of the previous iteration to obtain the ratio q. t .

[0133] S5. Set the device transmit vector, base station receive beamforming vector, and RIS phase shift vector of model MD to b respectively. * v * and θ * This yields an optimized RIS-assisted heterogeneous network for airborne computing oriented towards imperfect channels.

[0134] The above-described embodiments further illustrate the purpose, technical solution, and advantages of the present invention. It should be understood that the above-described embodiments are merely preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made to the present invention within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An optimization method for heterogeneous over-the-air computing networks with imperfect channels, characterized in that, include: S1. Construct a RIS-assisted heterogeneous network model for airborne computing oriented towards imperfect channels (MD). The model MD includes: over-the-air computing cell, traditional cell, and RIS. The over-the-air computing cell includes: base station and over-the-air computing device. The traditional cell includes: base station and traditional device. Among them, RIS is a smart reflector. S2. Calculate the mean square error of the model MD, and construct an objective optimization problem based on the channel error matrix, the device transmit power vector, the base station receive beamforming vector, and the smart reflector phase shift vector with the goal of minimizing and maximizing the mean square error. S3. Construct a first-level objective optimization problem based on the channel error matrix according to the objective optimization problem, and solve the first-level objective optimization problem analytically to obtain the closed-form solution of the optimal channel error matrix; S4. Based on the closed-form solution of the target optimization problem and the optimal channel error matrix, construct a secondary target optimization problem based on the device transmit power vector, the base station receive beamforming vector, and the intelligent reflector phase shift vector; solve the secondary target optimization problem using the alternating optimization algorithm to obtain the optimal device transmit power vector, the optimal base station receive beamforming vector, and the optimal intelligent reflector phase shift vector. S5. Set the device transmit vector, base station receive beamforming vector, and RIS phase shift vector of the RIS-assisted heterogeneous network for imperfect channels to the optimal device transmit power vector, the optimal base station receive beamforming vector, and the optimal smart reflector phase shift vector, respectively, to obtain the optimized RIS-assisted heterogeneous network for imperfect channels.

2. The optimization method for heterogeneous over-the-air computing networks oriented towards imperfect channels according to claim 1, characterized in that, A cellular over-the-air computing unit consists of one base station and K single-antenna over-the-air computing devices; a traditional cellular unit consists of one base station and one single-antenna traditional device; in the heterogeneous over-the-air computing network model, one cell is the target cell, and the rest are interfering cells. The set of cells l to be calculated in the air. For the set of traditional cellular n, the target cellular device For the set of devices in a target cell, the airborne computing cell is the device. For aerial computation, the set of devices in cellular l is compared with the devices in traditional cellular n. It is a collection of devices in a traditional cellular network. The intelligent reflective surface has M reflective elements.

3. The optimization method for heterogeneous over-the-air computing networks oriented towards imperfect channels according to claim 2, characterized in that, Target optimization problem for: Where C1 is the constraint of the phase shift matrix of the smart reflector, and C2 is the transmit power b of the target cell device. k The constraints are as follows: C3 and C4 are the channel error matrices of the direct link path between the device and the target base station and the path through the smart reflector, respectively; b is the device transmit power vector of the target cell; v is the receive beamforming vector of the base station of the target cell; and θ is the phase shift vector of the smart reflector. m For the phase shift of the reflective element m of the intelligent reflective surface, P max The maximum transmit power of the device is given by Δ, where MSE is the mean square error of the heterogeneous network model for intelligent reflector-assisted airborne computing in imperfect channels. p Let be the channel error matrix of the direct link path between device p and the base station of the target cell. Let ρ be the channel error matrix between device p and the base station of the target cell via the smart reflector path, and let ρ be the maximum error of the channel error matrix.

4. The optimization method for heterogeneous over-the-air computing networks with imperfect channels according to claim 3, characterized in that, The mean square error (MSE) of the heterogeneous network model for smart reflector-assisted airborne computing in imperfect channels is: Among them, b i Let represent the transmission power of device i, and (·) H This indicates the conjugate transpose. and Let represent the measurement channel vectors of the direct link paths between the target cell's device k, the over-the-air computing cellular device i, the traditional cellular device j, and the target cell's base station, respectively. and Let Δ represent the cascaded channel error matrices between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station via the smart reflector path. k Δ i and Δ j Let σ represent the channel error matrices of the direct link paths between the target cell's device k, the over-the-air computing cell device i, the traditional cell device j, and the target cell's base station, respectively. 2 This represents the power of additive white Gaussian noise. and These represent the cascaded measurement channel matrices between the target cell's device k, the over-the-air computing cellular device i, the traditional cellular device j, and the target cell's base station via the smart reflector path, respectively.

5. An optimization method for heterogeneous over-the-air computing networks with imperfect channels according to claim 4, characterized in that, The primary objective optimization problem is:

6. The optimization method for heterogeneous over-the-air computing networks oriented towards imperfect channels according to claim 5, characterized in that, Solving first-order objective optimization problems using analytical methods includes: S31. Use the complex trigonometric inequality to calculate the inequalities for the first-order objective optimization problem; S32. Based on the requirement that the inequalities of the first-level objective optimization problem satisfy the equation, we obtain the equation containing the optimal channel error matrix; S33. Solve the equation containing the optimal channel error matrix according to the constraints C1 and C2 of the channel error matrix to obtain the closed-form solution Δ* of the optimal channel error matrix.

7. An optimization method for heterogeneous over-the-air computing networks with imperfect channels according to claim 6, characterized in that, The closed-form solution Δ* of the optimal channel error matrix is: Where, Δ k * Δ i * Δ j * Δ k Δ i and Δ j The closed-form solution, They are respectively and The closed-form solution.

8. An optimization method for heterogeneous over-the-air computing networks with imperfect channels according to claim 3, characterized in that, The secondary objective optimization problem is:

9. An optimization method for heterogeneous over-the-air computing networks with imperfect channels according to claim 8, characterized in that, The optimization of secondary objective problems using the alternation optimization method includes: S41. Fix v and θ of the second-level objective optimization problem to obtain a subproblem. Optimize the subproblem using the Lagrange multiplier method Get b k The optimal value b k * ; based on all devices k of b k * Obtain the optimal device transmission coefficient b * ; S42, b k * Substituting these values ​​into the second-order objective optimization problem, we obtain subproblems based on v and θ. Set the initial value of θ to θ0 and the iteration stopping condition threshold ∈; S43, Subproblems θ is fixed as θ from the previous iteration. t-1 This yields a subproblem based on v. Apply gradient descent to the subproblem Optimize to obtain the optimized v t ; S44, Subproblems The value of v is fixed as the optimized v. t This yields a subproblem based on θ. Using the semidefinite relaxation method and Taylor series expansion method on the subproblem Processing the problem yields the subproblems. Using convex optimization tools on subproblems Optimize to obtain the optimized θ t ; S45, Based on the optimized v t and θ t Calculate the ratio q t If q t If v is less than the iteration stopping condition threshold ∈, then t and θ t These are respectively used as the optimal base station receive beamforming vectors v * and the optimal intelligent reflector phase shift vector θ * Otherwise, return to step S43.

10. An optimization method for heterogeneous over-the-air computing networks with imperfect channels according to claim 9, characterized in that, According to the optimized v t and θ t Calculate the ratio q t Includes: based on the optimized v t and θ t Calculate the mean squared error (MSE) of the current iteration t. Calculate the absolute value of the difference between the MSE of the current iteration t and the MSE of the previous iteration. Divide the absolute value by the MSE of the previous iteration to obtain the ratio q. t .

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