A joint optimization method and system for multipath aerial computing time synchronization transmission and reception

Abstract

This invention relates to the fields of wireless communication and edge intelligence, and discloses a joint optimization method and system for multipath over-the-air computing time synchronization transmission and reception. Existing multipath over-the-air computing suffers from multipath delay and gain mismatch problems, disrupting perfect signal superposition and reducing aggregation accuracy. This invention constructs a block delay matrix and a multipath channel matrix; introduces a precoding matrix set and a power vector at the transmitter, and solves for the minimum mean square error compensation operator using a fusion center; employs an alternating optimization strategy: when the precoding matrix set is fixed, the power vector is optimized using the gradient projection method; when the power vector is fixed, a weighted target channel is constructed, and the precoding matrices of each device are solved using norm-constrained least squares, Lagrange multipliers, and a bisection method. After alternating iterations, the aggregation estimation result is output. This invention is applicable to 6G edge intelligence and federated learning scenarios, effectively alleviating multipath synchronization mismatch and improving the aggregation accuracy of over-the-air computing.

Description

Technical Field

[0001] This invention relates to the fields of wireless communication and edge intelligence, and in particular to a method and system for joint optimization of multipath aerial computing time synchronization transmission and reception. Background Technology

[0002] As fifth-generation mobile communication systems evolve into sixth-generation systems, massive connectivity, ultra-low latency, and intelligent service scenarios place higher demands on the integration of communication and computing capabilities. Over-the-air computing utilizes the signal superposition characteristics of wireless multiple access channels to directly perform function aggregation during transmission, making it suitable for low-latency data fusion and edge intelligence scenarios.

[0003] In real-world wireless environments, multipath propagation and hardware deviations lead to inconsistent propagation delays, resulting in channel gain mismatch and time asynchrony issues. For over-the-air computing, time mismatch disrupts the signal synchronization and superposition characteristics, increasing aggregation estimation errors. Existing asynchronous over-the-air computing compensation methods primarily focus on single-path, flat fading, orthogonal frequency division multiplexing (OFDM) aggregation, or majority voting scenarios, failing to adequately consider the dual delay and gain mismatch issues caused by multipath superposition in simulated multipath over-the-air computing. Summary of the Invention

[0004] (a) Technical problems to be solved

[0005] The technical problem this invention aims to solve is that in simulated multipath aerial computing scenarios, multiple propagation paths from multiple edge devices arrive at the fusion center at different times and with different gains, leading to inconsistencies in effective channels and increased aggregation estimation errors.

[0006] (II) Technical Solution

[0007] To address the aforementioned issues, this invention establishes a multipath aerial computing system model considering integer multiples of sampling delay, fractional multiples of sampling delay, and channel complex gain. At the edge device side, a time-domain linear transmit precoding matrix set and a scalar transmit power vector are jointly designed. At the fusion center side, a minimum mean square error compensation operator is obtained based on the mean square error between the received signal and the ideal aggregated signal. Through an iterative optimization process of updating the power vector with a fixed precoding matrix set and updating the precoding matrix set with a fixed power vector, the multipath aerial computing aggregation estimation result is output.

[0008] When the precoding matrix group is fixed, the gradient projection method is used to update the transmit power vector under individual power constraints and total power constraints, and the step size is adjusted by Armijo backtracking search to ensure the improvement of the objective function. When the transmit power vector is fixed, a target channel matrix with power square root weighting is constructed, and a least squares problem with Frobenius norm equality constraints is solved for each edge device. The closed transmit precoding matrix containing Lagrange multipliers is obtained through the Lagrange function, and the Lagrange multipliers are determined by the monotonicity of the auxiliary function and the bisection method.

[0009] (III) Beneficial Effects

[0010] This invention simultaneously characterizes multipath delay mismatch and channel gain mismatch using a block delay matrix and a multipath channel matrix, enabling compensation and optimization to directly address the multipath in-flight computation aggregation process.

[0011] This invention improves the effective channel consistency of different edge devices at the fusion center by jointly and alternately optimizing the receiver minimum mean square error compensation operator, the transmit precoding matrix group, and the transmit power vector, thereby reducing the aggregation error caused by multipath propagation.

[0012] This invention decomposes the originally non-convex and highly coupled joint optimization problem into a power allocation subproblem and a precoding matrix subproblem, taking into account both aggregation accuracy and computational feasibility. Attached Figure Description

[0013] Figure 1 This is a flowchart of the transmit / receive joint optimization method and system of the present invention. The process includes: inputting system parameters, constructing a multipath channel matrix and the received signal; solving the receiver compensation operator based on the minimum mean square error criterion; entering the transmitter alternating optimization, sequentially updating the transmit power vector with a fixed transmit precoding matrix group, updating the transmit precoding matrix group with a fixed transmit power vector, and outputting the optimal aggregate estimation result after iteration.

[0014] Figure 2 The flowchart for the subproblem of optimizing the transmit power vector with a fixed transmit precoding matrix set is as follows: When the transmit precoding matrix set is fixed, the gradient projection method is used to iteratively optimize the transmit power vector. The process is as follows: calculate the power gradient of the objective function and update it along the ascending direction; sequentially perform individual power and total power constraint projections; combine Armijo line search to adjust the step size; and output the optimal transmit power vector.

[0015] Figure 3 The flowchart for optimizing the transmit precoding matrix set for a fixed transmit power vector is as follows: Design the transmit precoding matrix independently for each device, construct a power-weighted target channel, and transform the problem into a least-squares solution with norm constraints. The process is as follows: Introduce Lagrange multipliers to handle the constraints, solve for the multipliers using the bisection method, and then substitute back to obtain the optimal transmit precoding matrix set. Detailed Implementation

[0016] To make the objectives, technical solutions, and beneficial effects of the present invention clearer, the specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0017] Reference Figures 1 to 3 This invention provides a multipath aerial computing time synchronization transmission and reception joint optimization method and system, the specific formulas and symbols of which are explained below.

[0018]

[0019] in, Represents an ideal aggregation signal. Indicates the first Data symbols from edge devices participating in aggregation This indicates the number of edge devices. This formula defines the function aggregation result that the fusion center hopes to directly obtain.

[0020]

[0021] in, Indicates the first The first device The propagation delay of each path, Indicates an integer multiple of the sampling delay. This indicates a fractional multiple of the sampling delay. This represents the sampling interval. This formula is used to characterize multipath time mismatch.

[0022]

[0023] in, Represents the integer block delay matrix. Indicates the length of the symbol block. and This represents the matrix index. The formula expresses integer delays as a shift matrix of symbol blocks.

[0024]

[0025] in, This represents the block delay matrix corresponding to the fractional delay. The formula characterizes the fractional delay through a linear combination of adjacent integer delay matrices.

[0026]

[0027] in, Indicates the first Multipath channel matrix of each edge device Indicates the number of paths. This represents the path complex gain. This formula fully describes the linear transformation of the transmitted symbol block after passing through the multipath channel.

[0028]

[0029] in, Indicates sending a signal. Indicates scalar transmit power. This represents the transmit precoding matrix. This formula provides the degrees of freedom for transmitter alignment adjustment.

[0030]

[0031] in, This indicates the signal observed at the receiving end. This represents additive complex white Gaussian noise. This formula reflects superposition transmission in wireless multiple access channels.

[0032]

[0033] in, Represents the transmit power vector. Indicates the transmit precoding matrix group, This represents the minimum mean square error compensation operator. Let squared Euclidean norm of a vector be denoted as . This represents the mathematical expectation. The formula measures the mean square error of the compensated aggregation estimate relative to the ideal aggregation.

[0034]

[0035] in, and These represent the upper and lower bounds of individual power, This represents the total power budget. This formula ensures that power allocation meets the system's physical constraints.

[0036]

[0037] in, This represents the optimal minimum mean square error compensation operator. express and The cross-correlation matrix, express The autocorrelation matrix, This represents the inversion of a matrix. This formula is used to recover the aggregated signal at the fusion center.

[0038]

[0039] in, Indicates the noise variance. Represents the identity matrix. This represents the conjugate transpose. This formula incorporates channel and noise statistics into the receive compensation.

[0040]

[0041] in, and It is a matrix composed of equivalent channel and power. This formula represents finding the rank of a matrix. This formula allows the optimization objective to be further transformed by equivalent means.

[0042]

[0043] in, The sum of the weighted equivalent channel matrices is given by... This represents the covariance correlation matrix of the received signal. This formula connects the transmit power, precoding, and receive error.

[0044]

[0045] in, Let represent the objective function for minimizing the mean square error. This formula equivalently transforms the original problem of minimizing the mean square error into maximizing the objective function. This provides a unified optimization scalar for subsequent power optimization.

[0046]

[0047] in, This represents the intermediate transmit power vector after unconstrained update. Indicates the first The transmit power vector of the next iteration. Indicates the current iteration step size. This represents the gradient vector of the current objective function with respect to each power. This formula is used to obtain the unconstrained updated power based on the gradient vector.

[0048]

[0049] in, Indicates the first update under constraints The transmit power vector of the next iteration. This represents the projection onto the feasible region. This formula is used to obtain the transmit power vector that satisfies the power constraints.

[0050]

[0051] in, This represents the feasible power region that satisfies both individual power constraints and total power constraints. This formula ensures that the power vector updated by the projected gradient still satisfies the system constraints.

[0052]

[0053] in, Indicates the first The gradient of the transmit power, To take the real part of a complex number, This represents the square of the Frobenius norm. This formula provides the ascending direction for the gradient projection method.

[0054]

[0055] in, Indicates the first Intermediate power values ​​of the devices after unconstrained updates This represents the power value after individual power constraints. This formula rigidly truncates the updated power value into the individual power constraints.

[0056]

[0057] in, Indicates the first After the device undergoes individual power constraints and total power constraints, the [number]th [device] is [number]. The power value after the next iteration This represents a global penalty factor (uniform power reduction). This formula applies a global power constraint to the total power exceeding the limit.

[0058]

[0059] The left side of the equation represents the total power after applying a uniform reduction. This formula is used to solve... The core determination condition can be solved efficiently and uniquely using the bisection method, ensuring the accurate satisfaction of the total power constraint.

[0060]

[0061] in, It is a constant (usually taken as) ), This is the squared magnitude of the gradient vector. This formula is used to dynamically adjust the step size, ensuring that the objective function grows sufficiently in each iteration.

[0062]

[0063] in, Represents the target channel matrix. Indicates the first The power square root normalized weights of the edge devices. This formula is used to construct the weighted target channel matrix under fixed power conditions.

[0064]

[0065] in, This represents the precoding matrix variable to be solved. This means finding the matrix that minimizes the subsequent objective function. , This is a preset constraint value. The formula represents how to make the precoded equivalent channel approximate the target channel.

[0066]

[0067] in, Indicates about and The Lagrange function, This represents the Lagrange multiplier introduced by the Frobenius norm equality constraint. This formula is used to transform the constrained least squares problem into an unconstrained optimization form containing the Lagrange multiplier.

[0068]

[0069] This formula is used to obtain the first [unit / type] under constant power conditions. The transmit precoding matrix of each edge device.

[0070]

[0071] in, Indicates the solution The auxiliary function. This formula transforms the problem of solving norm constraints into the problem of finding the roots of a one-dimensional nonlinear equation.

[0072]

[0073] in, This indicates the Moore-Penrose pseudo-inverse. This formula demonstrates that the norm of the unconstrained solution does not exceed a predefined value. Directly take the unconstrained solution .

[0074]

[0075] The formula shows For the unique Lagrange multiplier that satisfies the norm constraint, since Strictly monotonically decreasing, when If the equation has a unique solution, it can be solved efficiently using the bisection method.

[0076]

[0077] in, for Left endpoint, for The right endpoint. This formula is used to determine solutions containing a unique solution. initial interval Guarantee through the doubling method This ensures the convergence of the bisection method.

[0078] Example

[0079] like Figure 1 As shown, this embodiment provides an implementation method and system for joint optimization of multipath over-the-air computation time synchronization and transmission. This implementation method is applicable to, including... A multipath aerial computing system consisting of a single-antenna edge device and a fusion center. Each edge device transmits local data symbol blocks to be aggregated, while the fusion center receives the observation signals superimposed through the multipath channels and outputs the multipath aerial computing aggregation estimation results.

[0080] In this embodiment, the fusion center first obtains model parameters, which include the number of edge devices. Symbol block length The parameters include the number of paths for each edge device, the propagation delay of each path, the complex gain of each path, the sampling interval, the signal variance, the noise variance, the upper and lower bounds of individual power, and the total power budget.

[0081] Subsequently, the propagation delay of each propagation path is decomposed into integer multiples and fractional multiples of sampling delay. Then, according to the propagation delay decomposition formula, the integer block delay matrix formula, and the fractional block delay matrix construction formula, the block delay matrix corresponding to each propagation path is obtained. For the ... For each edge device, the complex gain and corresponding block delay matrix of each path are superimposed according to the multipath channel matrix superposition formula to obtain the multipath channel matrix of that edge device. To make the channel amplitudes of different devices comparable, in one specific implementation, the multipath channel matrix can also be normalized according to the average channel power normalization formula.

[0082] At each edge device, a time-domain linear transmit precoding matrix and a scalar transmit power are configured. Each edge device generates a transmit signal according to the transmit signal generation formula, where... Indicates the actual transmission power. As the transmission amplitude weight. The transmission precoding matrix satisfies the Frobenius norm normalization constraint, thus ensuring that the actual transmission energy is mainly determined by the transmission amplitude weight. Control. The observation signals received by the fusion center are represented according to the formula for observation signals received at the receiver.

[0083] The fusion center aims to minimize the mean square error between the compensated aggregated estimate and the ideal aggregated signal. For a given transmit power vector and transmit precoding matrix set, the fusion center constructs the matrix required to obtain the minimum mean square error compensation operator based on the formulas for the power-weighted equivalent channel matrix and the autocorrelation matrix of the received signal. Finally, it calculates the receiver compensation operator using the closed-form formula of the minimum mean square error compensation operator. This compensation operator is recalculated after each round of outer variable updates, rather than being used as a fixed constant.

[0084] When the transmit precoding matrix group is fixed, the fusion center calculates the objective function value of the current joint scheme based on the theoretically compensated mean square error formula, and calculates the analytical gradient of the objective function with respect to the transmit power vector based on the actual transmit power gradient formula. Subsequently, a trial power vector is obtained along the negative gradient direction, and this trial power vector is projected onto the feasible region that satisfies both individual power constraints and total power constraints. During projection, device-by-device individual power constraint pruning is performed first; if the pruned total power exceeds the total power budget, the common offset is solved using a bisection method based on the common offset projection formula, thus completing the total power constraint projection. The step size is adjusted using an Armijo backtracking search to ensure that the updated power vector satisfies the sufficient descent condition.

[0085] With a fixed transmit power vector, the fusion center constructs the target channel matrix based on the power square root weighted target channel matrix construction formula. Subsequently, for each edge device, a precoding update subproblem is established based on the precoding least squares formula with Frobenius norm constraints, and the corresponding transmit precoding matrix is ​​determined by a bisection solution using the Lagrangian function, the closed precoding matrix formula, and auxiliary functions.

[0086] During the alternating iteration process, the transmit precoding matrix of each edge device is first initialized to an identity matrix, and the transmit power vector is initialized to a feasible point that satisfies the power constraint. Each round of outer iteration includes: recalculating the minimum mean square error compensation operator; updating the transmit precoding matrix set under the current power; updating the transmit power vector under the updated equivalent channel; and calculating the new theoretically compensated mean square error. If the new mean square error is reduced by more than the preset alternation tolerance relative to the current recorded value, the update for this round is accepted; otherwise, the iteration stops. After the iteration is completed, the optimized transmit power vector, transmit precoding matrix set, equivalent channel matrix, and minimum mean square error compensation operator are output, and the multipath aerial computation aggregation estimation result is obtained by the fusion center.

[0087] In a specific simulation implementation, the settings are as follows: , Each device has a signal variance of 1, a signal-to-noise ratio (SNR) range of -10dB to 20dB, and a step size of 1dB. 300 Monte Carlo repeats are performed at each SNR. The lower bound for individual power is 0.1, the upper bound for individual power is 1.5, and the total power budget is... The upper bound of individual power is set to 1 / 2. The initial step size for the projected gradient method is 0.5, the Armijo backtracking reduction factor is 0.5, and the minimum step size is set to 1 / 2. The maximum number of iterations for power optimization is set to 200, and the power vector convergence threshold is set to... The convergence threshold of the objective function is taken as Armijo fully descent parameters take In the precoding matrix update, the Lagrange multiplier binary search precision is taken as... The maximum number of iterations for the binary search method is set to 80, the maximum number of iterations for the outer layer alternating optimization is set to 6 to 8, and the outer layer target improvement threshold is set to... .

[0088] In the above simulation implementation, each of the four edge devices can be configured to have three multipath paths. For example, the propagation delays of the four edge devices are [0.0,2.4,5.7], [0.3,3.1,7.6], [1.2,4.5,8.3] and [0.0,2.8,6.4], respectively; the corresponding path complex gains are [1.00+0.00i,0.45-0.20i,0.22+0.10i], [0.92+0.08i,0.36+0.18i,0.20-0.12i], [1.05-0.05i,0.40-0.16i,0.18+0.09i] and [0.98+0.02i,0.42+0.15i,0.24-0.08i]. The parameters described above are only used to illustrate a reproducible experimental configuration and do not constitute a limitation on the scope of protection of this invention.

[0089] To verify the effectiveness of this embodiment, multiple sets of comparative experiments can be conducted under the same multipath channel, the same source signal, and the same noise samples. Specifically, four comparison schemes are set up: the first scheme only uses the minimum mean square error compensation operator at the receiver for compensation, without optimizing the transmit power vector and designing the transmit precoding matrix group; the second scheme further optimizes the transmit power vector based on the minimum mean square error compensation operator at the receiver; the third scheme adds a transmit precoding matrix group to the second scheme; and the fourth scheme adopts the joint optimization method of transmit power vector, transmit precoding matrix group, and minimum mean square error compensation operator at the receiver proposed in this embodiment.

[0090] Experimental results show that this embodiment can still reduce aggregation error under low signal-to-noise ratio (SNR) conditions. For example, when the SNR is -10dB, the mean square errors (MSEs) of the four schemes are 3.1536, 3.1536, 2.8533, and 2.8533, respectively; when the SNR is -5dB, the MSEs are 2.5651, 2.5000, 2.2612, and 1.9247, respectively; and when the SNR is 0dB, the MSEs are 2.0837, 1.8718, 1.4155, and 1.0066, respectively. It can be seen that when noise is strong, transmitter alignment and joint optimization can reduce the error, but the improvement is relatively limited because system performance is still dominated by noise.

[0091] As the signal-to-noise ratio (SNR) increases, the advantages of this embodiment become more apparent. For example, at an SNR of 15 dB, the mean square errors (MSEs) for the four schemes are 1.7302, 1.3518, 0.60568, and 0.076605, respectively. Compared to the receiver-only minimum MSE compensation scheme, the joint optimization method in this embodiment reduces the MSE by approximately 95.57%. At an SNR of 20 dB, the MSEs for the four schemes are 1.7248, 1.3393, 0.57268, and 0.040338, respectively. This embodiment reduces the MSE by approximately 97.66% compared to the receiver-only minimum MSE compensation scheme. Therefore, this embodiment not only suppresses the aggregation error caused by noise and multipath mismatch under low SNR conditions but also significantly reduces the structural error caused by multipath channel mismatch under medium-to-high SNR conditions, demonstrating the effectiveness of the collaborative optimization of transmit power, transmit precoding matrix, and receiver compensation operator.

Claims

1. A joint optimization method for multipath aerial computing time synchronization transmission and reception, characterized in that, include: S1. Obtain the parameters of the multipath aerial computation model, including the complex gain and propagation delay of each propagation path of each edge device; S2. Decompose the propagation delay into integer multiple sampling delay and fractional multiple sampling delay, construct a block delay matrix based on the integer multiple sampling delay and the fractional multiple sampling delay, and generate the multipath channel matrix of each edge device based on the complex gain of each propagation path and the block delay matrix; S3. Introduce a time-domain linear transmit precoding matrix and scalar transmit power at each edge device, and generate an equivalent channel matrix and the receiver observation signal based on the multipath channel matrix, the time-domain linear transmit precoding matrix, and the scalar transmit power; S4. Compare the aggregation estimate after linear compensation at the fusion center with the ideal aggregation signal. The mean square error is the optimization objective. Given the transmit power vector and transmit precoding matrix set, the minimum mean square error compensation operator is obtained. In S5, the transmit precoding matrix set is fixed, and the transmit power vector is updated under individual power constraints and total power constraints using the gradient projection method. In S6, the transmit power vector is fixed, a power-weighted target channel matrix is ​​constructed, and the transmit precoding matrix of each edge device is updated by least squares problem with Frobenius norm equality constraints, closed-form solution of Lagrange multipliers, and bisection solution. In S7, S5 and S6 are executed alternately until the mean square error convergence condition is met, and the optimized transmit power vector, transmit precoding matrix set, and minimum mean square error compensation operator are output to obtain the multipath aerial computation aggregation estimation result.

2. The method according to claim 1, characterized in that, In S2, the propagation delay of the l-th propagation path of the k-th edge device is represented as a combination of integer multiple sampling delay and fractional multiple sampling delay, and the block delay matrix corresponding to the fractional delay is constructed by linear interpolation of adjacent integer block delay matrices.

3. The method according to claim 1, characterized in that, In S2, the multipath channel matrix of the k-th edge device is represented as the superposition of the complex gain of each propagation path and the corresponding block delay matrix. The multipath channel matrix is ​​a Toeplitz convolution matrix describing the linear transformation of the signal transmitted by the edge device through the multipath channel.

4. The method according to claim 1, characterized in that, In S3, the transmission signal of each edge device is generated by processing the scalar transmit power and the time-domain linear transmit precoding matrix to aggregate the data symbol block, and the time-domain linear transmit precoding matrix satisfies the Frobenius norm normalization constraint.

5. The method according to claim 1, characterized in that, In S4, based on the orthogonality principle between the observed signal at the receiving end and the ideal aggregated signal, the closed Wiener solution of the minimum mean square error compensation operator is obtained according to the product form of the cross-correlation matrix and the autocorrelation matrix.

6. The method according to claim 1, characterized in that, In S5, the gradient projection method includes: obtaining the gradient of the objective function with respect to each transmit power based on matrix differentiation; obtaining an unconstrained update value along the gradient direction; projecting the unconstrained update value into the feasible region constrained by individual power and total power; and adjusting the step size through Armijo backtracking search.

7. The method according to claim 6, characterized in that, The projection includes: first, truncating each transmit power to an individual power constraint range; when the sum of the truncated transmit powers exceeds the total power budget, by solving the nonlinear equation with the truncated structure, projecting the transmit power vector onto a hyperplane that satisfies the total power budget.

8. The method according to claim 1, characterized in that, In S6, the equivalent channel matrix is ​​weighted and summed according to the square root normalized weight of the transmit power of each edge device to construct the target channel matrix; and for each edge device, the least squares problem that makes the equivalent channel approximate the target channel is solved under the constraints of the Frobenius norm equation, the Lagrange function is constructed, the closed-form solution of the transmit precoding matrix containing Lagrange multipliers is obtained, and the Lagrange multipliers are determined by the monotonicity of the auxiliary function and the bisection method.

9. The method according to claim 1, characterized in that, In S7, the transmit precoding matrix group is initialized as an identity matrix group, and the transmit power vector is initialized as a feasible point that satisfies the individual power constraint and the total power constraint. In each round of outer layer iteration, the transmit precoding matrix group and the transmit power vector are updated sequentially, and the current round of update is accepted when the new mean square error is less than the mean square error of the current record minus the alternation tolerance.

10. A multipath over-the-air computing time synchronization and transceiver joint optimization system, characterized in that, The method includes K single-antenna edge devices and a fusion center, wherein the edge devices are configured to generate a transmit signal according to any one of claims 1 to 9, and the fusion center is configured to obtain a minimum mean square error compensation operator and output a multipath aerial computation aggregation estimation result according to any one of claims 1 to 9.

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