MA-assisted multi-cell distributed air computing optimization method

Abstract

The invention discloses an MA-assisted multi-cell distributed air calculation optimization method, which comprises the following steps of: firstly, constructing an optimization problem based on minimum MSE (Mean Square Error), and then, providing a corresponding alternating iteration (AO) algorithm to solve the problem, specifically, firstly, providing a closed-form solution for updating a receiving beam vector of each base station, then, providing an optimal KKT solution of a transmitting coefficient of each user, and finally, providing an optimal KKT solution for updating a receiving beam vector of each base station. And finally, providing a gradient descent method for updating the MA position. A suboptimal solution of a mean square error (MSE) is obtained through iteration, and the solution can be quickly converged.

Description

TECHNICAL FIELD

[0001] The present application belongs to the network technology field in mobile communication system, and particularly relates to a multi-cell uplink computation optimization method assisted by a movable antenna (MA). BACKGROUND

[0002] The movable antenna (MA) technology can make the base station antenna move freely in one-dimensional, two-dimensional and three-dimensional space under certain constraints, thereby more fully utilizing the spatial freedom and improving the spatial resolution, so that the current wireless environment can be more effectively adapted and the communication performance of the system is improved. At present, no literature has studied the performance of the multi-cell uplink computation system assisted by the MA. Therefore, the present application constructs a corresponding optimization problem for this scheme, and proposes a low-complexity alternating iteration (AO) algorithm to solve this problem. SUMMARY

[0003] The present application aims at the deficiencies in the prior art, and provides a multi-cell uplink computation optimization method assisted by the MA, which optimizes the base station receiving beam matrix, the user transmitting coefficient and the MA antenna position respectively while fixing other optimization variables, has a low complexity, and can converge to the optimal value in a short time.

[0004] Technical scheme: To achieve the above-mentioned application purposes, the technical scheme adopted by the present application comprises the following steps:

[0005] A multi-cell uplink computation mean square error minimization method assisted by the MA comprises the following steps:

[0006] Step one: first, an AO-based optimization iteration algorithm is given to minimize the mean square error of the system. Specifically, the base station receiving precoding matrix, the user transmitting coefficient and the antenna position are optimized respectively, and each update scheme is given. The present application gives a low-complexity AO iteration algorithm, which can quickly obtain the minimum mean square error.

[0007] Step two: the present application gives the scheme for updating the MA position in step one. On the basis of limiting the moving range of each antenna and the interval between the antennas, the gradient descent algorithm is used to update the antenna position, so that the system mean square error is minimized.

[0008] The step one comprises:

[0009] The present application considers a multi-cell uplink computation system, each base station in each cell is equipped with M antennas, and each antenna can move in the area D, each base station simultaneously communicates with K far-field users in the cell, and the antenna set and the user set of each base station are respectively and There are L signal transmission paths between each user in each cell and the corresponding base station, and there are also L transmission paths and reception paths between base stations;

[0010] is the channel vector of the kth user in cell j to the base station, where is the reception field response matrix of user k in cell i to the base station, u j,k ∈ C L×1 is the channel fading vector of user k to base station j. It is assumed that the channel between the user and the base station is a Rayleigh channel, i.e. where d j，k is the distance from base station j to user k, and a is the channel fading coefficient. is the channel matrix of base station j to base station i, where is the reception field response matrix between base stations is the transmission field response matrix between base stations. ∑ i，j ∈ C L×L is the channel fading matrix between base stations i and j, u j，k ∈ C L×1 is the channel fading vector of user k to base station j. is the transmission direction vector, where is the transmission elevation angle of the lth path from base station j to base station i; is the transmission azimuth of the lth path from base station j to base station i;

[0011] is the reception direction vector, where is the reception elevation angle of the lth path from the base station in cell j to the base station in cell i; is the reception azimuth angle of the lth path from the base station in cell j to the base station in cell i;

[0012] where is the reception elevation angle of the lth path of user k to base station i; is the reception azimuth angle of the lth path of user k to base station i.

[0013] Each user is a single-antenna user, s i,k denotes the transmission symbol of the kth user under base station i, a i,k denotes the transmission coefficient of the kth user under base station i, and thus the transmission signal of the user in the jth cell received by the base station in the cell is After each cell base station receives the signal of the user in the current cell, the base stations share the signal and perform reception filtering on the received signal, w i is the reception filtering vector of base station i, and thus the combined reception signal of base station i is The sum of the mean square error (MSE) of each base station is as follows:

[0014] where the noise vector σ 2 is the noise power, B is the number of cells, and K is the number of users in a single cell. The mean square error (MSE) expression can be transformed into the following form by mathematical transformation:

[0015]

[0016] First, the optimization problem corresponding to the system is given:

[0017]

[0018] where P c is the maximum transmit power of the user, is the antenna moving area, and D0 is the minimum interval between antennas. The steps of initialization and variable exchange are omitted, and the specific steps for solving the equivalent problem are as follows:

[0019] First, fix the user transmit coefficients and the antenna positions, and optimize the optimal beam receiving vectors of each base station. After fixing other variables, the original optimization problem is simplified as follows:

[0020]

[0021] This is an unconstrained optimization problem, and the derivative of the receiving beam vector of each base station is obtained as the optimal receiving beam vector:

[0022]

[0023] Second, fix the receiving beam vectors of each base station and the antenna positions, and optimize the transmit coefficients of each user in each cell. First, fix other optimization variables, and the original problem is equivalent to the following:

[0024]

[0025] This problem is obviously a convex problem, and its KKT solution is its optimal solution. In order to facilitate the solution, first, the equivalent form is given as follows:

[0026] is the stacked vector of the transmit coefficients of each user, is the transmit coefficient of the i-th user in the i-th cell, b n is the n-th element of the vector b, is the equivalent coefficient from the k-th user in the j-th cell to the i-th base station, e j,k is a selection vector of the same dimension as a, and the elements at the same positions as a j,k are 1, and the rest are 0, Pc is the maximum transmit power of the user. Define Then R is a diagonal matrix, r n is the nth diagonal element of the matrix.

[0027] The KKT conditions of this equivalent problem are:

[0028]

[0029] λ n ≥0

[0030] ||a n || 2 ≤P c

[0031] λ n (||a n || 2 -P c )=0

[0032] where, K is the total number of users of all base stations; introduce E n After that the constraints can be converted to λ n is the Lagrange multiplier corresponding to the nth inequality. Solve the above KKT conditions, then the optimal solution of the user transmit coefficient is:

[0033]

[0034] where a n is the nth element of the vector a, r n is the nth diagonal element of R.

[0035] Thirdly, as claimed in claim 2, first give the optimization problem corresponding to the update of the antenna position:

[0036]

[0037] The specific steps for solving this problem are:

[0038] (i) initialize the antenna position where x i,m and y i,m are the horizontal and vertical coordinates of the mth antenna of the ith base station, the step size of antenna position update η, the current iteration number s = 0 and the maximum iteration number s max , M is the number of base station antennas, is the feasible region of the optimization problem;

[0039] (ii) Update the position of each antenna according to the gradient of the system mean square error (MSE) with respect to the antenna position as follows:

[0040]

[0041] where the expression of the mean square error is:

[0042]

[0043] (iii) If then reduce the step size to 0.5η and re-execute step (ii); otherwise let s = s + 1;

[0044] (iv) If s < s max then return to step (ii); otherwise output the final antenna position

[0045] Beneficial effects: Compared with the prior art, the application solves the problem of MA-assisted multi-cell uplink on-the-fly computation mean square error minimization, and proposes an efficient AO iterative algorithm. It has been verified that the algorithm has low complexity and accurate results.

[0046] Specifically:

[0047] 1. The application first solves the joint optimization problem of MA-assisted multi-cell uplink on-the-fly computation system. In the prior art, MA-assisted single base station on-the-fly computation system is mostly used, and no literature has involved MA-assisted distributed base station on-the-fly computation system. The application first constructs the MSE minimization problem of this complex system and provides a complete solution, filling the technical gap.

[0048] 2. In view of the pain point that the original non-convex optimization problem (P1) is difficult to solve directly, the application innovatively designs an iterative algorithm based on alternating optimization. The algorithm optimizes the precoding matrix, user transmission coefficient and antenna position by fixing other variables, decomposes the complex problem into a series of solvable sub-problems, and significantly reduces the computational complexity.

[0049] 3. By jointly optimizing the antenna position, precoding, and user transmission coefficient, the application makes full use of the spatial degrees of freedom provided by MA to significantly reduce the computation error of on-the-fly computation in a multi-cell system. Simulation results show that the scheme can effectively minimize the system MSE and outperforms the traditional fixed antenna on-the-fly computation system.

[0050] 4. The alternating optimization algorithm proposed in the application has a clear structure, and each step update has a closed-form solution or an efficient solving method (such as gradient descent), ensuring that the entire iterative process can quickly converge to the optimal solution or a high-quality suboptimal solution, meeting the real-time requirements of actual communication systems for algorithms. Attached Figure Description

[0051] Figure 1 This is a network structure diagram of MA-assisted multi-cell uplink over-the-air computing. Detailed Implementation

[0052] The technical solution of the present invention will be described in detail below, but the scope of protection of the present invention is not limited to the embodiments described.

[0053] This invention proposes a multi-cell uplink over-the-air calculation mean square error optimization method assisted by MA.

[0054] The following is an example:

[0055] This embodiment considers a multi-cell uplink over-the-air computing system assisted by a multi-mass communication (MA). Taking a three-base station system as an example, the three base stations are located at (-30m, 80m), (-200m, -80m), and (-140m, -160m) respectively, and conduct far-field communication with users in their respective cells. It is assumed that each cell contains 3 users. Each base station is equipped with M movable antennas, which can move freely within the range of [-6λ, 6λ], and the spacing between adjacent antennas must be no less than D0 = 0.5λ. Each user in a cell first transmits a signal to the corresponding cell base station. After receiving the user signal, each base station shares the signal among itself. The carrier frequency used for communication is f = 2.4GHz, corresponding to a wavelength of λ = c / f = 0.125m. The maximum transmit power P of each user is... C =10dBm, user noise power In addition, the number of transmit paths and the number of receive paths L t =L r =4, and the elevation angles of each signal propagation path are uniformly distributed within [0, π].

[0056] To execute the method of this invention, initialization is first performed:

[0057] 1) Initialize the transmission coefficients of each user Make it satisfy the power constraint For example, it can be simply set as

[0058] 2) Initialize the antenna positions of each base station Randomly generate within the movement region A, ensuring that the minimum spacing constraint is met.

[0059] 3) Based on the initial antenna position, calculate the initial channel state using the following channel model. and

[0060] 4) Set the convergence threshold of the alternating optimization algorithm ∈ = 10 -4 and the maximum number of iterations T = 100. max .

[0061] The method comprises the following steps:

[0062] Step one, first give the optimization iteration algorithm based on AO, to minimize the mean square error of the system. Specifically, the base station receiving precoding matrix, user transmitting coefficient and antenna position are optimized respectively, and each gives an update scheme. The present application gives a low complexity AO iteration algorithm, which can quickly obtain the minimum mean square error.

[0063] In the embodiment, the above iteration algorithm is executed according to the following flow:

[0064] Step A (initial iteration): let the iteration number t = 0. Based on the initialization setting of step one, calculate the initial system mean square error MSE (0) .

[0065] Step B (receiving beam update): fix the current user transmitting coefficient and antenna position update the receiving beam vector of each base station

[0066] Step C (transmitting coefficient update): fix the current receiving beam and antenna position update the user transmitting coefficient

[0067] Step D (antenna position update): fix the current receiving beam and user transmitting coefficient update the antenna position to

[0068] Step E (channel update): update the inter-base station channel matrix and the user-to-base station channel vector based on the updated antenna position

[0069] Step F (convergence judgment): calculate the system mean square error MSE (t+1) after this iteration.

[0070] If | MSE (t+1) - MSE (t) | / MSE (t) < ∈ or t + 1 ≥ T max , the iteration is terminated, and the optimization result is output Otherwise, let t = t + 1, return to step B to continue iteration.

[0071] Step two, the application gives the scheme of updating MA position in step one. On the basis of limiting the moving range of each antenna and the interval between antennas, the antenna position is updated by using gradient descent algorithm to minimize the system mean square error.

[0072] Wherein step one comprises:

[0073] The application considers a multi-cell uplink air computing system, each cell base station is equipped with M antennas, and each antenna can move within the area D, each base station simultaneously communicates with K users in the far field of the cell, and the antenna set and user set of each cell base station are respectively And There are L signal transmission paths between each user in each cell and the corresponding base station, and there are also L transmission paths and reception paths between base stations;

[0074] is the channel vector of the kth user in the cell j to the base station, wherein is the reception field response matrix of the user k in the cell i to the base station, u j，k ∈C L×1 is the channel fading vector of the user k to the base station j. It is assumed that the channel between the user and the base station is a Rayleigh channel, that is, where d j,k is the distance from the base station j to the user k, and a is the channel fading coefficient. is the channel matrix of the base station j to the base station i, wherein, is the reception field response matrix between base stations is the transmission field response matrix between base stations. Σ i,j ∈C L×L is the channel fading matrix between the base station i and the base station j, u j,k ∈C L×1 is the channel fading vector of the user k to the base station j. is the transmission direction vector, wherein is the transmission elevation angle of the lth path from the base station j to the base station i; is the transmission azimuth of the lth path from the base station j to the base station i;

[0075] is the reception direction vector, wherein is the reception elevation angle of the lth path from the base station in the cell j to the base station in the cell i; is the reception azimuth angle of the lth path from the base station in the cell j to the base station in the cell i;

[0076] wherein is the reception elevation angle of the lth path from the user k to the base station i; The received azimuth angle of the kth user to the ith base station along the lth path.

[0077] Each user is a single antenna user, s i，k aik represents the transmit symbol of the kth user under the ith base station, a i，k aik represents the transmit coefficient of the kth user under the ith base station, thus, the transmit signal of the user in the jth cell received by the base station in the cell is After the base station in each cell receives the signal of the user in the current cell, the base station shares the signal among the base stations and performs receive filtering on the received signal, w i wi represents the receive filtering vector of the ith base station, thus, the combined received signal of the ith base station is Thus, the sum of the mean square errors of each base station is as follows:

[0078] where the noise vector is σ 2 is the noise power, B is the number of cells, and K is the number of users in a single cell. The MSE expression can be transformed into the following expression through mathematical transformation:

[0079]

[0080] First, the optimization problem corresponding to the system is given:

[0081]

[0082] where P c is the maximum transmit power of the user, is the antenna moving area, and D0 is the minimum interval between the antennas.

[0083] The steps of initialization and variable exchange are omitted, and the specific steps for solving the equivalent problem are as follows:

[0084] First, fix the user transmit coefficient and the antenna position, and optimize the optimal beam receive vector of each base station. After fixing other variables, the original optimization problem is simplified as follows:

[0085]

[0086] This is an unconstrained optimization problem, and the derivative of the receive beam vector of each base station is obtained as the optimal receive beam vector:

[0087]

[0088] Second, fix the receive beam vector of each base station and the antenna position, and optimize the transmit coefficient of each user in each cell. First, fix other optimization variables, and the original problem is equivalent to the following:

[0089]

[0090] The problem is obviously a convex problem, so its KKT solution is its optimal solution. In order to facilitate the solution, first give its equivalent form:

[0091] Stack the vector of transmission coefficients for each user, is the transmission coefficient of each user in the i-th cell, b n is the n-th element of the vector b, is the equivalent coefficient of the k-th user in the j-th cell to the i-th base station, e j，k is a selection vector of the same dimension as a, which is 1 with the same position as a j,k and the rest are all 0, P c is the maximum transmission power of the user. Define Then R is a diagonal matrix, r n is the n-th diagonal element of the matrix.

[0092] The KKT conditions of this equivalent problem are:

[0093]

[0094] λ n ≥0

[0095] ||a n || 2 ≤P c

[0096] λ n (||a n || 2 -P c )=0

[0097] Where, K is the total number of base station users; Introduce E n After the constraint can be converted to λ n is the Lagrange multiplier corresponding to the n-th inequality. Solve the above KKT conditions, and the optimal solution of the user transmission coefficient is:

[0098]

[0099] Where, a n is the n-th element of the vector a, r n is the n-th diagonal element of R.

[0100] Third, as claimed in claim 2, first give the optimization problem corresponding to the update antenna position:

[0101]

[0102] The specific steps for solving this problem are as follows:

[0103] (i) Initialization of antenna positions wherein x i，m and y i，m are the horizontal and vertical coordinates of the mth antenna of the ith base station, the step size antenna position update η, the current iteration number s = 0 and the maximum iteration number s max , M is the number of base station antennas, is the feasible region of the optimization problem;

[0104] (ii) Update the position of each antenna according to the gradient of the system mean square error (MSE) with respect to the antenna position as follows:

[0105]

[0106] wherein the expression of the mean square error is:

[0107]

[0108] (iii) If , then the step size is reduced to 0.5η, and step (ii) is re-executed; otherwise, s = s + 1;

[0109] (iv) If s < s max , then return to step (ii); otherwise, output the final antenna positions

[0110] The above-described application is only a preferred embodiment of the present application, and it should be noted that for those skilled in the art, without departing from the principles of the present application, a number of predictable improvements and refinements can be made, and these improvements and refinements should also be considered as the protection scope of the present application.

Claims

1. A multi-cell distributed over-the-air computing optimization method assisted by MA, characterized in that, Includes the following steps: (1) Establish the original optimization problem (P1) of the multi-cell distributed air computing system with uplink MA assistance, with the goal of minimizing the mean square error of the received signal of each base station. Its constraints include user transmit power constraints, MA movement range constraints and MA antenna spacing constraints. (2) The optimization problem (P1) is solved iteratively using an alternating optimization framework. By fixing other variables, the user transmission coefficient, the base station receiving beam matrix and the antenna position are optimized in turn until the result converges.

2. The method according to claim 1, characterized in that, The alternating optimization iterative process in step (2) specifically includes the following steps: (a) Initialize system parameters and optimization variables; (b) Update the base station receive beam matrix using the closed-loop solution based on the current user transmit coefficients and antenna positions; (c) Based on the updated receive beam matrix in step (b), update the user transmit coefficients using the closed-loop solution; (d) Using the updated user transmission coefficients from step (c), update the antenna positions of each base station using the gradient descent method; (e) Update the channels between base stations and from the user to the base station based on the updated antenna locations; (f) Repeat steps (b) to (e) to calculate the MSE obtained in each iteration. If the change of the value relative to the previous iteration is less than the threshold and the maximum number of iterations has not been reached, return to step (b); otherwise, output the optimization result.

3. The method according to claim 2, characterized in that, The closed-form solution for the received beam vector of each base station in step (b) is: Where K is the total number of users in all communities, K j Let B be the number of users in cell j, B be the total number of cells, and a be the number of users in cell j. j,k Let σ be the emission coefficient of the k-th user within cell j. 2 h is the noise power. j,k H represents the channel vector from the k-th user in cell j to the corresponding base station. It is an M-dimensional column vector, where M is the number of base station antennas, and H is the number of antennas. i,j It is the channel matrix from base station j to base station i, an M*M matrix, where I is the identity matrix, and w i Let be the received beam vector of base station i, and let be an M-dimensional column vector. H Represents the conjugate transpose of the original vector or matrix, (·) * This represents the optimal solution for the current step.

4. The method according to claim 2, characterized in that, The optimal transmission coefficient update formula for each user in each cell in step (c) is: Among them, a n It is the nth element of vector a. The vector formed by stacking the emission coefficients of each user. It is the transmission coefficient of each user in the i-th cell. b n It is the nth element of vector b, where K is the total number of users in all cells. j Let B be the number of users in cell j, and let B be the total number of cells. Let e ​​be the equivalent coefficient from the k-th user in cell j to the i-th base station. j,k It is a selection vector of the same dimension as a, and it is related to a. j,k Elements at the same position are 1, and all others are 0. P c R is the user's maximum transmit power; R is a diagonal matrix. r n It is the nth diagonal element of the matrix.

5. The method according to claim 2, characterized in that, The method for updating the antenna position in step (d) is the gradient descent algorithm, which specifically includes: (i) Initialize antenna position in x i,m and y i,m The x and y coordinates of the m-th antenna of the i-th base station, the step size antenna position update η, the current iteration count s = 0, and the maximum iteration count s are respectively. max M is the number of base station antennas. It is the feasible region of the optimization problem; (ii) Based on the gradient of the system mean square error (MSE) with respect to the antenna position Update the position of each antenna as follows: The expression for the mean square error is: (iii) If If the step size is reduced to 0.5η, step (ii) is executed again; otherwise, let s = s + 1. (iv) If s < s max If the result is positive, return to step (ii); otherwise, output the final antenna position.

6. The method according to claim 2, characterized in that, The channel update expression in step (e) is: Let J be the channel matrix from base station j to base station i. Represents the coordinates of all antenna positions of base station i; The channel vector from the k-th user in cell j to this base station is: in, This is the received field response matrix between base stations; This is the transmit field response matrix between base stations; Let ∑ be the received field response matrix from user k in cell i to this base station. i,j ∈C L×L Let u be the channel fading matrix between base stations i and j. j,k ∈C L×1 Let L be the channel fading vector from user k to base station j; L is the number of paths, and λ is the carrier wavelength. Let be the launch direction vector, where Let be the transmit elevation angle of the l-th path from base station j to base station i; Let be the transmission azimuth angle of the l-th path from base station j to base station i; To receive the direction vector, where The elevation angle received along the l-th path from base station j to base station i; The receiving azimuth angle for the l-th path from base station j to base station i; in Let be the received elevation angle of the l-th path from user k to base station i; Let be the receiving azimuth angle of the l-th path from user k to base station i.

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