Air computing optimization method based on IRS, CF-mMIMO and digital twinning

Abstract

The invention belongs to the technical field of wireless communication and calculation, discloses an air calculation optimization method based on IRS, CF-mMIMO and digital twinning, and aims to solve the problems that in the prior art, a compensation mechanism for low-performance users is lacked in a large-scale, multi-user and complex channel environment, the digital twinning technology is not fully utilized, the algorithm expandability is insufficient and the like. Network and user states are perceived and predicted in real time by introducing a digital twinning technology, so that finer dynamic scheduling and compensation are carried out in multi-dimensional resources (including transmitting power, beam forming, IRS phase shift, computing power and the like), and finally, the communication efficiency is improved while efficient communication is ensured. According to the algorithm, the mean square error of data aggregation is further reduced, the overall stability and expandability of the system are improved, the algorithm is particularly suitable for a complex channel environment and a scene with an enlarged user scale, and in addition, the alternating optimization strategy of the algorithm ensures efficient convergence in a non-convex optimization problem.

Description

Technical Field

[0001] The present invention relates to the field of wireless communications and computing technologies, and in particular to an air computing optimization method based on IRS, CF-mMIMO and digital twins. Background Art

[0002] With the rapid adoption of IoT devices, the demand for real-time data aggregation, transmission, and computation has increased significantly. Traditional wireless communication and computing technologies struggle to efficiently handle such large-scale, data-intensive applications. To address this issue, over-the-air computing technology has emerged. By leveraging the superposition properties of wireless multiple-access channels, over-the-air computing enables direct computation of data during transmission, significantly reducing communication latency and computational overhead. Compared to traditional methods that require decoding user signals individually, over-the-air computing can directly compute functions such as summation and averaging during signal transmission, significantly improving data processing efficiency.

[0003] However, the effectiveness of over-the-air computing relies heavily on the quality of the wireless channel, which is often affected by fading, interference, and noise. In complex multi-user channel environments, the performance of over-the-air computing is limited, and the mean square error of data aggregation increases significantly. To overcome these challenges, CF-mMIMO has received widespread attention in recent years. By deploying a large number of distributed access points throughout the network, CF-mMIMO eliminates the boundary constraints of traditional cellular networks, reduces interference, and enhances the system's spatial diversity and beamforming gain, significantly improving the performance of over-the-air computing in large-scale user environments.

[0004] Furthermore, IRS, as an emerging wireless communication technology, optimizes the signal propagation environment, enhances the signal-to-noise ratio, reduces multi-user interference, and provides additional signal transmission paths by dynamically adjusting the phase and amplitude of reflective elements. This technology offers significant advantages, particularly in non-line-of-sight (NLOS) scenarios. The introduction of IRS further enhances the robustness and reliability of airborne computing systems in complex and large-scale IoT deployments.

[0005] However, as the number of IoT devices continues to grow, system complexity and computational load are also increasing. Digital twin technology provides an effective solution for wireless communication systems by creating virtual replicas of physical systems, enabling real-time monitoring, simulation, and optimization. Digital twin technology can optimize network deployment, dynamically allocate resources, and improve system performance and reliability, making it particularly valuable in multi-user and complex network environments.

[0006] While existing technologies have significantly improved data aggregation efficiency and communication quality in large-scale IoT systems by introducing over-the-air computing, CF-mMIMO, and IRS, many challenges remain in joint optimization and resource allocation. In particular, in complex channel environments and large-scale user scenarios, further reducing the mean squared error (MSE) of data aggregation and improving overall system performance remain pressing challenges. Therefore, a novel approach that comprehensively leverages these technologies is urgently needed to achieve more efficient and reliable data aggregation and resource optimization. Summary of the Invention

[0007] To solve the above problems, the present invention provides an air computing optimization method based on IRS, CF-mMIMO and digital twins. The present invention is implemented through the following technical solutions.

[0008] An air computing optimization method based on IRS, CF-mMIMO, and digital twins includes the following steps:

[0009] S1, build an IoT communication system, which includes:

[0010] There are L access points, denoted as Each access point is equipped with N antennas;

[0011] M IRSs, denoted as Each IRS consists of B reflection units;

[0012] And K users, denoted as Each user is equipped with T antennas;

[0013] Each user communicates with the access point via a wireless link, and the access point is connected to the central processing unit (CPU) via a high-speed optical fiber backhaul link;

[0014] The over-the-air computing process involves two hops of data aggregation: from the user to the access point and from the access point to the CPU. In the first hop, the user transmits their data to both the access point and the IRS. The IRS, a passive intelligent reflector, reflects the incoming data back to the access point. In the second hop, the access point transmits the data to the CPU for aggregation. Each data signal contains Q dimensions.

[0015] For users with insufficient computing power, a twin digital twin virtual body is generated at the access point. K1 represents the set of users without digital twin virtual bodies, and K2 represents the set of users with digital twin virtual bodies. K1 and K2 constitute the digital twin system.

[0016] S2, under the constraints of relay transmission power and IRS component physical conditions, obtain the optimal solution of the deviation matrix Δ in the digital twin system, and fix the phase shift matrix at each IRS Introducing the denoising factor η and the normalized aggregation factor matrix F, the original and variable user transmit beamforming matrix B k , receive beamforming matrix A at the access point l The related over-the-air computation optimization problem is transformed into an optimization problem about the normalized aggregation factor matrix F;

[0017] S3, under the constraints of relay transmission power and IRS component physical conditions, the transmit beamforming matrix B at each user is fixed. k and the receive beamforming matrix A at the access point l , transforming the air computing optimization problem into a phase shift matrix Optimization problem;

[0018] S4, fix the phase shift matrix at each IRS According to the optimized normalized aggregation factor matrix F obtained by S2, the deviation matrix Δ, the denoising factor η, and the transmit beamforming matrix B at each user in the digital twin system are calculated. k and the receive beamforming matrix A at the access point l ;

[0019] S5, fix the transmit beamforming matrix B at each user k and the receive beamforming matrix A at the access point l , determine the phase shift matrix obtained by S3 The optimal solution of

[0020] S6, repeat steps S4 and S5 until the mean square error iteration converges.

[0021] As a further solution of the present invention, the step S2 specifically includes:

[0022] S21, calculating the received signal at the CPU;

[0023] Let S kl represents the signal sent by the kth user to the lth access point, let S kml The ideal signal received by the CPU is expressed as follows:

[0024]

[0025] The signal received at the access point is the sum of all transmitted signals, which can be converted into any nomogram function. Therefore, the signal received by the lth access point is expressed as:

[0026]

[0027] in It is represented as the channel matrix from the kth user to the mth IRS; It is represented as the channel matrix from the mth IRS to the lth access point; A is the channel matrix from the kth user to the lth access point; l represents the receive beamforming matrix at the lth access point, B k represents the transmit beamforming matrix at the kth user; represents the phase shift matrix at the mth IRS, n~CN(0,σ 2 I) represents additive white Gaussian noise;

[0028] The signal received by the CPU can be expressed as the sum of the signals received by all access points, that is:

[0029]

[0030] S22 uses the transmit beamforming matrix as the representation of the user state, and the state mapping formula is:

[0031]

[0032] Where Δ is the deviation matrix; represents the transmit beamforming matrix of the user who has a virtual replica in the digital twin; B k Represents the transmit beamforming matrix for the remaining users who perform calculations locally and do not rely on the digital twin for calculations;

[0033] S23, calculate the mean square error;

[0034] Introducing formulas (5) and (6), the signal received by the CPU is re-expressed as:

[0035]

[0036] The calculation error of the system will be quantified by the mean square error between the received signal and the ideal signal, which is:

[0037]

[0038] in, represents the expectation operator;

[0039] S24, for each user, the power consumption can be expressed as:

[0040]

[0041] S25, construct the optimization problem;

[0042] With the minimum mean square error as the goal and combined with the constraints, the optimization problem is constructed as follows:

[0043] P1:

[0044]

[0045] S26, calculating the optimal solution of the deviation matrix;

[0046] Performing basic mathematical operations on the expression of MSE, we can get:

[0047]

[0048] in, and are all positive, so using the zero-forcing principle,

[0049]

[0050] Can make

[0051]

[0052] make Then M 1,kl Vectorized into m 1,kl =vec(M 1,kl )m 1,kl =vec(M 1,kl ), M 2,kl Vectorized into m 2,kl =vec(M 2,kl ), we can get:

[0053]

[0054] Let m 2,kl =x+jy,m 1,kl =u+jv, where x, y, u, j are all real numbers and j is a complex factor, then (14) can be rewritten as:

[0055] (x+u) T (x+u)+(y+v) T (y+v)=‖u‖ 2 +‖v‖ 2 . (15)

[0056] According to (15), on the real plane, the vector (x, y) must lie on the hypersphere, where the center is (-u, -v) and the radius is For a non-zero matrix A l For example, there must exist a complex scalar λ such that m 2,kl =λm1,kl ,λ≠0m 2,kl =λm 1,kl ,λ≠0, then (15) can be transformed into:

[0057] (Re(λ)+1) 2 +(Im(λ)) 2 =1. (16)

[0058] This means that λ must be on the unit circle in the complex plane, with the center of the unit circle at (-1,0) and a radius of 1. Otherwise, there is no multiplication relationship between the two vectors. When the multiplication relationship is not satisfied, it means that m can be 2,kl Decompose into and m 1,kl The linearly related part and m 1,kl There is no linearly related part, that is, m 2,kl =λm 1,kl +C, substituting formula (16) into the equation, we can obtain:

[0059]

[0060] Since λ lies on the unit circle in the complex plane, it means |λ| 2 +2Re(λ)=0, unless C=0, otherwise the equation cannot be established, which means that the multiplication relationship must be satisfied. At this time, the optimal solution of the deviation matrix can be solved as:

[0061]

[0062] |λ+1|=1,λ≠0. (18)

[0063] S27, transforming the air computing optimization problem into an optimization problem about the normalized aggregation factor matrix F;

[0064] After substituting the deviation matrix and applying the zero-forcing principle again, we can get:

[0065]

[0066] Now we can get the equation:

[0067]

[0068] Since the basic limitation of MIMO spatial multiplexing is well known, the number of antennas at the transmitter and receiver limits the maximum number of data streams, so the maximum number is limited to min{T,N}, and the number of antennas must be greater than the number of transmitted parameters, so With full row rank, there exists a right inverse matrix, and the optimal solution can be obtained:

[0069]

[0070] Introduce the denoising factor η, let In addition, the normalized aggregation factor matrix F is introduced to satisfy FF H =I, at this time You can get:

[0071] P2:

[0072]

[0073] F H F=I. (22)

[0074] By changing the constraints of formula (22), we can obtain:

[0075]

[0076] When the constraints meet the equality conditions, η reaches its minimum value, and the optimal value can be expressed as:

[0077]

[0078] The optimal value η * Substitute into the optimization problem P2 to transform it into a new single-variable optimization problem and express it as follows:

[0079] P3:

[0080] sttr({FF H )=1. (25)

[0081] By unevenly relaxing some constraints, we can approximate P3 as:

[0082]

[0083] Among them, λ min (·) represents the minimum eigenvalue of the matrix. Using the cyclic property of matrix multiplication, the objective function becomes:

[0084] P4:

[0085] sttr(FF H )=1. (27)

[0086] Problem P4 is a convex optimization problem, and its optimal solution can be determined.

[0087] As a further aspect of the present invention, in step S25, the first constraint states that the total power consumption of all users must be less than a fixed value. The second constraint states that the amplitude of the reflective element on each IRS is typically fixed at 1, and its phase can be adjusted continuously or discretely within the range [0, 2π]. The third constraint states that users are divided into two groups based on their conditions to determine whether to generate virtual replicas in the digital twin layer. These two groups of users constitute all users, and there is no duplication.

[0088] As a further solution of the present invention, in step S4:

[0089] Based on the optimal solution of the determined normalized aggregation factor matrix F, the denoising factor η can be calculated according to formula (24);

[0090] Based on the formula According to F and η, the receiving beamforming matrix A at the entry point can be calculated l ;

[0091] Based on formula (21), the transmit beamforming matrix B can be calculated k ;

[0092] Based on formula (18), the deviation matrix Δ can be calculated.

[0093] As a further solution of the present invention, step S3 specifically includes:

[0094] Phase shift matrix The optimization problem can be rewritten as:

[0095] P5:

[0096]

[0097] For each fixed IRS, define C1=G UI,1m B'1+G UI,2m B'2...+G UI,Km B' K , at this time B' k Representative B k and make At this time, the objective function is transformed into ||C m Θ m C1 m +P|| 2 ; then order is the phase shift matrix, that is, the original diagonal phase shift matrix is written as a row matrix, and |z mb |=1, ξ=diag{C m}C1 m, at this time the objective function is transformed into ||z H ξ+P|| 2 , the problem is further transformed into:

[0098] P6:min {z} {z H ξξ H z+z H ξP+P H ξ H z+||P|| 2}

[0099] st

[0100] Next, we introduce two intermediate variables The original problem is transformed into:

[0101] P7:

[0102]

[0103] Since both the feasible set and the objective function are non-convex, P7 is an NP problem. Through mathematical transformation, the objective function can be rewritten as

[0104] Then make Transform the original problem into:

[0105] P8:

[0106] Z b,b =1,b=1,2,...,B+1,

[0107] Z~0.(31)

[0108] Formula (31) is a convex optimization problem.

[0109] As a further aspect of the present invention, standard convex optimization techniques are used to determine the optimal solution of equation (31) to minimize the objective function under given constraints.

[0110] The beneficial effects of the present invention are as follows:

[0111] 1. Significantly Reduces the Mean Squared Error of Data Aggregation: By jointly optimizing the IRS phase shift matrix, user precoding matrix, and receive beamforming matrix, the present invention achieves more accurate data aggregation in large-scale user and complex channel environments. In simulation analysis, the DT-AOmin algorithm reduces the mean squared error (MSE) compared to traditional full digital zero forcing (FD-ZF), random IRS phase shift scheme (Random IRS), manifold optimized alternating minimization (MO-AltMin), and optimization schemes that do not use digital twins. The performance advantage is particularly evident when the number of users increases and the channel conditions are complex.

[0112] 2. Improved system resource utilization and fairness: Digital twin technology offloads and compensates computing tasks for users with limited performance, optimizing the dynamic allocation of system resources and reducing the negative impact of low-performance users on overall system performance. This not only improves resource utilization efficiency but also enhances system fairness, ensuring that all users, regardless of their hardware performance, receive stable and high-quality services during the data aggregation process.

[0113] 3. Enhanced system scalability and robustness: The method of the present invention is designed with large-scale users and multi-IRS deployment in mind. Through an alternating optimization algorithm, efficient iteration and convergence of the algorithm are achieved, ensuring that the system can maintain low-error and high-efficiency data aggregation performance even when the number of users, access points, and IRSs increases significantly.

[0114] 4. Promoting Economic Benefits and Social Applications: The optimization method proposed in this paper improves data aggregation efficiency and system reliability while reducing resource consumption and operating costs of communication infrastructure. Efficient data processing capabilities enable large-scale IoT applications (such as smart cities, industrial automation, and intelligent transportation) to operate more efficiently, promoting the development of related industries and enhancing the overall level of informatization and intelligent capabilities of society. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] In order to more clearly illustrate the technical solution of the present invention, the following is a brief introduction to the drawings required for use in the description of the specific implementation methods. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0116] Figure 1 : Schematic diagram of data transmission in the present invention. DETAILED DESCRIPTION

[0117] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.

[0118] like Figure 1 As shown in FIG, an air computing optimization method based on IRS, CF-mMIMO, and digital twins includes the following steps:

[0119] S1, build an IoT communication system, which includes:

[0120] There are L access points, denoted as Each access point is equipped with N antennas;

[0121] M IRSs, denoted as Each IRS consists of B reflection units;

[0122] And K users, denoted as Each user is equipped with T antennas;

[0123] Each user communicates with the access point via a wireless link, and the access point is connected to the central processing unit (CPU) via a high-speed optical fiber backhaul link;

[0124] The over-the-air computing process involves two hops of data aggregation: from the user to the access point and from the access point to the CPU. In the first hop, the user transmits their data to both the access point and the IRS. The IRS, a passive intelligent reflector, reflects the incoming data back to the access point. In the second hop, the access point transmits the data to the CPU for aggregation. Each data signal contains Q dimensions.

[0125] For users with insufficient computing power, a twin digital twin virtual body is generated at the access point. K1 represents the set of users without digital twin virtual bodies, and K2 represents the set of users with digital twin virtual bodies. K1 and K2 constitute the digital twin system.

[0126] S2, under the constraints of relay transmission power and IRS component physical conditions, obtain the optimal solution of the deviation matrix Δ in the digital twin system, and fix the phase shift matrix at each IRS Introducing the denoising factor η and the normalized aggregation factor matrix F, the original and variable user transmit beamforming matrix B k , receive beamforming matrix A at the access point l The related over-the-air computation optimization problem is transformed into an optimization problem about the normalized aggregation factor matrix F;

[0127] S3, under the constraints of relay transmission power and IRS component physical conditions, the transmit beamforming matrix B at each user is fixed. k and the receive beamforming matrix A at the access point l , transforming the air computing optimization problem into a phase shift matrix Optimization problem;

[0128] S4, fix the phase shift matrix at each IRS According to the optimized normalized aggregation factor matrix F obtained by S2, the deviation matrix Δ, the denoising factor η, and the transmit beamforming matrix B at each user in the digital twin system are calculated. k and the receive beamforming matrix A at the access point l ;

[0129] S5, fix the transmit beamforming matrix B at each user and the receive beamforming matrix A at the access point l , determine the phase shift matrix obtained by S3 The optimal solution of

[0130] S6, repeat steps S4 and S5 until the mean square error iteration converges.

[0131] Preferably, step S2 specifically includes:

[0132] S21, calculating the received signal at the CPU;

[0133] Let S kl represents the signal sent by the kth user to the lth access point, let S kml The ideal signal received by the CPU is expressed as follows:

[0134]

[0135] The signal received at the access point is the sum of all transmitted signals, which can be converted into any nomogram function. Therefore, the signal received by the lth access point is expressed as:

[0136]

[0137] in It is represented as the channel matrix from the kth user to the mth IRS; It is represented as the channel matrix from the mth IRS to the lth access point; A is the channel matrix from the kth user to the lth access point; l represents the receive beamforming matrix at the lth access point, B krepresents the transmit beamforming matrix at the kth user; represents the phase shift matrix at the mth IRS, n~CN(0,σ 2 I) represents additive white Gaussian noise;

[0138] The signal received by the CPU can be expressed as the sum of the signals received by all access points, that is:

[0139]

[0140] S22: Since the user state is one of the key factors affecting the design and performance of the transmit beamforming matrix, the transmit beamforming matrix is used as a representative of the user state in this application. The state mapping formula is:

[0141]

[0142] Where Δ is the deviation matrix, which is used to describe the difference between the state of the physical entity and the state of the replica model. The controllability of the deviation matrix refers to the ability to influence and manage the deviation matrix in the digital twin through human intervention or adjustment. For users with limited computing power, the digital twin can transfer part of these users' computing tasks to the access point for processing, thereby reducing the burden on the user device. By selecting an appropriate deviation matrix, the error caused by hardware performance differences can be offset, thereby optimizing the mean square error.

[0143] represents the transmit beamforming matrix of the user who has a virtual replica in the digital twin;

[0144] B k Represents the transmit beamforming matrix for the remaining users who perform calculations locally and do not rely on the digital twin for calculations;

[0145] S23, calculate the mean square error;

[0146] Introducing formulas (5) and (6), the signal received by the CPU is re-expressed as:

[0147]

[0148] Assuming that both the transmitter and receiver have complete and accurate knowledge of the channel state information, the system's computational error is quantified by the mean square error between the received signal and the ideal signal, which is:

[0149]

[0150] in, represents the expectation operator;

[0151] and, Used to simplify the expression of multi-level channel matrices;

[0152] S24, for each user, the power consumption can be expressed as:

[0153]

[0154] S25, construct the optimization problem;

[0155] With the minimum mean square error as the goal and combined with the constraints, the optimization problem is constructed as follows:

[0156] P1:

[0157]

[0158] S26, calculating the optimal solution of the deviation matrix;

[0159] Performing basic mathematical operations on the expression of MSE, we can get:

[0160]

[0161] in, and are all positive, so using the zero-forcing principle,

[0162]

[0163] Can make

[0164]

[0165] make Then M 1,kl Vectorized into m 1,kl =vec(M 1,kl )m 1,kl =vec(M 1,kl ), M 2,kl Vectorized into m 2,kl =vec(M 2,kl ), we can get:

[0166]

[0167] Let m 2,kl =x+jy,m 1,kl =u+jv, where x, y, u, j are all real numbers and j is a complex factor, then (14) can be rewritten as:

[0168] (x+u) T (x+u)+(y+v) T (y+v)=‖u‖ 2 +‖v‖ 2 .(15)

[0169] According to (15), on the real plane, the vector (x, y) must lie on the hypersphere, where the center is (-u, -v) and the radius is For a non-zero matrix A l For example, there must exist a complex scalar λ such that m 2,kl =λm 1,kl ,λ≠0m 2,kl =λm 1,kl ,λ≠0, then (15) can be transformed into:

[0170] (Re(λ)+1) 2 +(Im(λ)) 2 =1. (16)

[0171] This means that λ must be on the unit circle in the complex plane, with the center of the unit circle at (-1,0) and a radius of 1. Otherwise, there is no multiplication relationship between the two vectors. When the multiplication relationship is not satisfied, it means that m can be 2,kl Decompose into and m 1,kl The linearly related part and m 1,kl There is no linearly related part, that is, m 2,kl =λm 1,kl +C, substituting formula (16) into the equation, we can obtain:

[0172]

[0173] Since λ lies on the unit circle in the complex plane, it means |λ| 2 +2Re(λ)=0, unless C=0, otherwise the equation cannot be established, which means that the multiplication relationship must be satisfied. At this time, the optimal solution of the deviation matrix can be solved as:

[0174]

[0175] |λ+1|=1,λ≠0. (18)

[0176] S27, transforming the air computing optimization problem into an optimization problem about the normalized aggregation factor matrix F;

[0177] After substituting the deviation matrix and applying the zero-forcing principle again, we can get:

[0178]

[0179] Now we can get the equation:

[0180]

[0181] Since the basic limitation of MIMO spatial multiplexing is well known, the number of antennas at the transmitter and receiver limits the maximum number of data streams, so the maximum number is limited to min{T,N}, and the number of antennas must be greater than the number of transmitted parameters, so With full row rank, there exists a right inverse matrix, and the optimal solution can be obtained:

[0182]

[0183] In getting B k Then, the denoising factor η is introduced, and Its role is to reduce the reception noise of the access point while increasing the transmission power. Therefore, η must be optimized according to the objective function to achieve the best balance between noise reduction and power consumption;

[0184] In addition, the normalized aggregation factor matrix F is introduced to satisfy FF H =I, at this time You can get:

[0185] P2:

[0186]

[0187] F H F=I. (22)

[0188] By analyzing Equation (22), we find that the original variables are replaced by η and F, which produces a new non-convex optimization problem related to η. By transforming the constraints, we obtain:

[0189]

[0190] When the constraints meet the equality conditions, η reaches its minimum value, and the optimal value can be expressed as:

[0191]

[0192] The optimal value η * Substitute into the optimization problem P2, thus transforming it into a new single-variable optimization problem. This substitution effectively reduces the complexity of the problem, allowing us to focus only on optimizing a single variable. At this point, the optimization problem can be further simplified and expressed as follows:

[0193] P3:

[0194] sttr({FF H )=1. (25)

[0195] However, problem P3 is still a non-convex optimization problem characterized by linear constraints, which are difficult to solve using traditional optimization techniques. To solve this problem, we use unequal relaxation conditions to transform the original non-convex problem into a more solvable form. By unevenly relaxing some constraints, we can approximate P3 as:

[0196]

[0197] Among them, λ min (·) represents the smallest eigenvalue of the matrix. Since F is an identity matrix, its conjugate transpose is also its inverse. This means that multiplying the identity matrix by its conjugate transpose yields the same matrix. and Unitary invariance has the same eigenvalue spectrum. Specifically, the unit transformation preserves the eigenvalues of the matrix. This is because the unit matrix represents rotations and reflections, and does not change the inherent properties of vectors, such as the length and angle between them. Therefore, under favorable channel conditions, the equation holds. Utilizing the cyclic property of matrix multiplication, the objective function becomes:

[0198]

[0199] sttr(FF H )=1. (27)

[0200] Problem P4 is a convex optimization problem, and its optimal solution can be determined.

[0201] Preferably, in step S25, the first constraint states that the total power consumption of all users must be less than a fixed value. The second constraint states that the amplitude of the reflective element on each IRS is typically fixed at 1, and its phase can be adjusted continuously or discretely within the range [0, 2π]. The third constraint states that users are divided into two groups based on their conditions to determine whether to generate virtual replicas in the digital twin layer. These two groups of users constitute all users, and there will be no duplication.

[0202] Preferably, in step S4:

[0203] Based on the determined optimal solution of the normalized aggregation factor matrix F, the denoising factor η can be calculated according to formula (24);

[0204] Based on the formula According to F and η, the receiving beamforming matrix A at the entry point can be calculated l ;

[0205] Based on formula (21), the transmit beamforming matrix B can be calculated k ;

[0206] Based on formula (18), the deviation matrix Δ can be calculated.

[0207] Preferably, step S3 specifically includes:

[0208] Here we aim to optimize the IRS phase shift Θm using a known matrix. However, since the phase shift variable of each IRS element is located in a continuous closed interval, solving the IRS phase shift optimization subproblem will bring significant challenges. The phase shift matrix can be The optimization problem can be rewritten as:

[0209] P5:

[0210]

[0211] For each fixed IRS, define C1=G UI,1m B'1+G UI,2m B'2...+G UI,Km B' K , at this time B' k Representative B k and Since both are known matrices, there is no need to distinguish them; let At this time, the objective function is transformed into ||C m Θ m C1 m +P|| 2 ; then order is the phase shift matrix, that is, the original diagonal phase shift matrix is written as a row matrix, and |z mb |=1, ξ=diag{C m}C1 m , at this time the objective function is transformed into ||z H ξ+P|| 2 , the problem is further transformed into:

[0212] P6:min {z} {z H ξξ H z+z H ξP+P H ξ H z+||P|| 2}

[0213] st

[0214] Next, we introduce two intermediate variables The original problem is transformed into:

[0215] P7:

[0216]

[0217] Since both the feasible set and the objective function are non-convex, P7 is an NP problem. Through mathematical transformation, the objective function can be rewritten as

[0218] Then make Transform the original problem into:

[0219] P8:

[0220] Z b,b =1,b=1,2,...,B+1,

[0221] Z~0.(31)

[0222] Formula (31) is a convex optimization problem.

[0223] Preferably, standard convex optimization techniques are used to determine the optimal solution to Equation (31) so as to minimize the objective function under given constraints.

[0224] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to specific embodiments. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention.

Claims

1. An air computing optimization method based on IRS, CF-mMIMO and digital twin, characterized by: The following steps are involved: S1, build an IoT communication system, which includes: There are L access points, denoted as Each access point is equipped with N antennas; M IRSs, denoted as Each IRS consists of B reflection units; And K users, denoted as Each user is equipped with T antennas; Each user communicates with the access point via a wireless link, and the access point is connected to the central processing unit (CPU) via a high-speed optical fiber backhaul link; The over-the-air computing process involves two hops of data aggregation: from the user to the access point and from the access point to the CPU. In the first hop, the user transmits their data to both the access point and the IRS. The IRS, a passive intelligent reflector, reflects the incoming data back to the access point. In the second hop, the access point transmits the data to the CPU for aggregation. Each data signal contains Q dimensions. For users with insufficient computing power, a twin digital twin virtual body is generated at the access point. K1 represents the set of users without digital twin virtual bodies, and K2 represents the set of users with digital twin virtual bodies. K1 and K2 constitute the digital twin system. S2, under the constraints of relay transmission power and IRS component physical conditions, obtain the optimal solution of the deviation matrix Δ in the digital twin system, and fix the phase shift matrix at each IRS Introducing the denoising factor η and the normalized aggregation factor matrix F, the original and variable user transmit beamforming matrix B k , receive beamforming matrix A at the access point l The related over-the-air computation optimization problem is transformed into an optimization problem about the normalized aggregation factor matrix F; S3, under the constraints of relay transmission power and IRS component physical conditions, the transmit beamforming matrix B at each user is fixed. k and the receive beamforming matrix A at the access point l , transforming the air computing optimization problem into a phase shift matrix Optimization problem; S4, fix the phase shift matrix at each IRS According to the optimized normalized aggregation factor matrix F obtained by S2, the deviation matrix Δ, the denoising factor η, and the transmit beamforming matrix B at each user in the digital twin system are calculated. k and the receive beamforming matrix A at the access point l ; S5, fix the transmit beamforming matrix B at each user k and the receive beamforming matrix A at the access point l , determine the phase shift matrix obtained by S3 The optimal solution of S6, repeat steps S4 and S5 until the mean square error iteration converges.

2. The air computing optimization method based on IRS, CF-mMIMO and digital twin according to claim 1, characterized in that: The step S2 specifically includes: S21, calculating the received signal at the CPU; Let S kl represents the signal sent by the kth user to the lth access point, let S kml The ideal signal received by the CPU is expressed as follows: The signal received at the access point is the sum of all transmitted signals, which can be converted into any nomogram function. Therefore, the signal received by the lth access point is expressed as: in It is represented as the channel matrix from the kth user to the mth IRS; It is represented as the channel matrix from the mth IRS to the lth access point; A is the channel matrix from the kth user to the lth access point; l represents the receive beamforming matrix at the lth access point, B k represents the transmit beamforming matrix at the kth user; represents the phase shift matrix at the mth IRS, n~CN(0,σ 2 I) represents additive white Gaussian noise; The signal received by the CPU can be expressed as the sum of the signals received by all access points, that is: S22 uses the transmit beamforming matrix as the representation of the user state, and the state mapping formula is: Where Δ is the deviation matrix; represents the transmit beamforming matrix of the user who has a virtual replica in the digital twin; B k Represents the transmit beamforming matrix for the remaining users who perform calculations locally and do not rely on the digital twin for calculations; S23, calculate the mean square error; Introducing formulas (5) and (6), the signal received by the CPU is re-expressed as: The calculation error of the system will be quantified by the mean square error between the received signal and the ideal signal, which is: in, represents the expectation operator; S24, for each user, the power consumption can be expressed as: S25, construct the optimization problem; With the minimum mean square error as the goal and combined with the constraints, the optimization problem is constructed as follows: S26, calculating the optimal solution of the deviation matrix; Performing basic mathematical operations on the expression of MSE, we can get: in, and are all positive, so using the zero-forcing principle, Can make make Then M 1,kl Vectorized into m 1,kl =vec(M 1,kl )m 1,kl =vec(M 1,kl ), M 2,kl Vectorized into m 2,kl =vec(M 2,kl ), we can get: Let m 2,kl =x+jy,m 1,kl =u+jv, where x, y, u, j are all real numbers and j is a complex factor, then (14) can be rewritten as: (x+u) T (x+u)+(y+v) T (y+v)=‖u‖ 2 +‖v‖ 2 . (15) According to (15), on the real plane, the vector (x, y) must lie on the hypersphere, where the center is (-u, -v) and the radius is For a non-zero matrix A l For example, there must exist a complex scalar λ such that m 2,kl =λm 1,kl ,λ≠0m 2,kl =λm 1,kl ,λ≠0, then (15) can be transformed into: (Re(λ)+1) 2 +(Im(λ)) 2 =1. (16) This means that λ must be on the unit circle in the complex plane, with the center of the unit circle at (-1,0) and a radius of 1. Otherwise, there is no multiplication relationship between the two vectors. When the multiplication relationship is not satisfied, it means that m can be 2,kl Decompose into and m 1,kl The linearly related part and m 1,kl There is no linearly related part, that is, m 2,kl =λm 1,kl +C, substituting formula (16) into the equation, we can obtain: Since λ lies on the unit circle in the complex plane, it means |λ| 2 +2Re(λ)=0, unless C=0, otherwise the equation cannot be established, which means that the multiplication relationship must be satisfied. At this time, the optimal solution of the deviation matrix can be solved as: S27, transforming the air computing optimization problem into an optimization problem about the normalized aggregation factor matrix F; After substituting the deviation matrix and applying the zero-forcing principle again, we can get: Now we can get the equation: Since the basic limitation of MIMO spatial multiplexing is well known, the number of antennas at the transmitter and receiver limits the maximum number of data streams, so the maximum number is limited to min{T,N}, and the number of antennas must be greater than the number of transmitted parameters, so With full row rank, there exists a right inverse matrix, and the optimal solution can be obtained: Introduce the denoising factor η, let In addition, the normalized aggregation factor matrix F is introduced to satisfy FF H =I, at this time You can get: By changing the constraints of formula (22), we can obtain: When the constraints meet the equality conditions, η reaches its minimum value, and the optimal value can be expressed as: The optimal value η * Substitute into the optimization problem P2 to transform it into a new single-variable optimization problem and express it as follows: By unevenly relaxing some constraints, we can approximate P3 as: Among them, λ min (·) represents the minimum eigenvalue of the matrix. Using the cyclic property of matrix multiplication, the objective function becomes: Problem P4 is a convex optimization problem, and its optimal solution can be determined.

3. The air computing optimization method based on IRS, CF-mMIMO and digital twin according to claim 2, characterized in that: In step S25, the first constraint states that the total power consumption of all users must be less than a fixed value. The second constraint states that the amplitude of the reflective element on each IRS is typically fixed at 1, and its phase can be adjusted continuously or discretely within the range [0, 2π]. The third constraint states that users are divided into two groups based on their conditions to determine whether to generate virtual replicas in the digital twin layer. These two groups of users constitute all users, and there will be no duplication.

4. The air computing optimization method based on IRS, CF-mMIMO and digital twin according to claim 4, characterized in that: In the step S4: Based on the optimal solution of the determined normalized aggregation factor matrix F, the denoising factor η can be calculated according to formula (24); Based on the formula According to F and η, the receiving beamforming matrix A at the entry point can be calculated l ; Based on formula (21), the transmit beamforming matrix B can be calculated k ; Based on formula (18), the deviation matrix Δ can be calculated.

5. The air computing optimization method based on IRS, CF-mMIMO and digital twin according to claim 1, characterized in that: The step S3 specifically includes: Phase shift matrix The optimization problem can be rewritten as: For each fixed IRS, define C1=G UI,1m B'1+G UI,2m B'2...+G UI,Km B' K , at this time B' k Representative B k and make At this time, the objective function is transformed into ||C m Θ m C1 m +P|| 2 ; then order is the phase shift matrix, that is, the original diagonal phase shift matrix is written as a row matrix, and At this time, the objective function is transformed into ||z H ξ+P|| 2 , the problem is further transformed into: Next, we introduce two intermediate variables The original problem is transformed into: Since both the feasible set and the objective function are non-convex, P7 is an NP problem. Through mathematical transformation, the objective function can be rewritten as Then make Transform the original problem into: Formula (31) is a convex optimization problem.

6. The air computing optimization method based on IRS, CF-mMIMO and digital twin according to claim 5, characterized in that: Standard convex optimization techniques are used to determine the optimal solution of Equation (31) that minimizes the objective function under the given constraints.

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