A First-Order Optimization Method for the Uplink Total Rate of De-Cellularized Massive MIMO

Abstract

The present invention provides a first-order optimization method for the uplink total rate of cellular massive MIMO, including the following steps: Step 1, establish an uplink pilot training model and derive the MMSE channel estimation; Step 2, derive the closed-form expression of the UE total rate and establish the UE total rate optimization problem; Step 3, design the UE total rate optimization method: adopt the total rate optimization algorithm based on weighted MMSE or the total rate optimization algorithm based on APG. Under the premise of satisfying the UE data transmission power constraint, the UE total rate is greatly improved, thereby compensating for the performance loss caused by the malicious attack of the jammer.

Description

Technical Field:

[0001] The present invention relates to a mobile communication system, and particularly to a first-order optimization method for the uplink total rate of a cellular-free massive MIMO. Background Art:

[0002] Cellular-free massive multiple-input multiple-output (MIMO) is a promising physical layer technology for future beyond 5G and 6G mobile communication networks. In a cellular-free massive MIMO system, a large number of access points (APs) equipped with multiple antennas are scattered at various positions within the service area. The APs are connected to a central processing unit (CPU) through a fronthaul network and can serve multiple users (UEs) on the same time-frequency resources, having extremely high macro-diversity gain and coverage rate. Therefore, ultra-high rates and energy efficiency can be achieved. Since the number of UEs in future beyond 5G and 6G networks will grow explosively, more and more APs need to be deployed to meet the high traffic demand of UEs. To solve the problems of increasing deployment cost and hardware power consumption, a feasible technical solution is to adopt low-resolution analog-to-digital converters (ADCs) at the AP side, but this inevitably brings performance losses, such as a decrease in the total rate.

[0003] Cellular-free massive MIMO technology benefits from strong array gain and spatial degrees of freedom, and can better resist passive eavesdropping and achieve better secrecy performance. However, since cellular-free massive MIMO allows UEs to transmit signals at a lower power, when there are high-power jammers in the system for malicious attacks, the accuracy of the system's minimum mean square error (MMSE) channel estimation and data decoding will be severely affected, thereby reducing the total system rate. To address this problem, a feasible solution is to optimize the transmit power of UEs to compensate for the rate loss caused by jammers by providing additional rate gain. However, most of the existing power optimization algorithms are based on successive convex approximation (SCA) technology, and SCA needs to use the interior point method to solve the numerical solution of the optimization problem, which will lead to a long running time and poor real-time performance of the optimization algorithm, and is not suitable for use in real-time cellular-free massive MIMO systems.

[0004] Therefore, aiming at the problems of long running time and poor real-time performance of the existing optimization algorithms based on SCA technology, how to optimize the UE transmission power more quickly in order to improve the total rate of the de-cellularized large-scale MIMO system with a low-precision ADC structure attacked by a jammer in real time is an urgent problem to be solved in this field. Summary of the Invention:

[0005] To solve the problems in the prior art that the total rate of the uplink de-cellularized large-scale MIMO system is vulnerable to malicious attacks by jammers and the existing total rate optimization algorithms based on SCA technology have long running time and poor real-time performance, etc., the present invention provides a first-order optimization method for the uplink total rate of a de-cellularized large-scale MIMO, which can reduce the impact of malicious attacks by jammers on the system total rate by quickly optimizing the transmission power of the UE.

[0006] To solve the above-mentioned existing technical problems, the present invention adopts the following scheme:

[0007] A first-order optimization method for the uplink total rate of a de-cellularized large-scale MIMO is as follows:

[0008] Step 1: Establish an uplink pilot training model and derive the MMSE channel estimation:

[0009] Under a general Rayleigh block fading channel, establish an uplink pilot training model of a de-cellularized large-scale MIMO system with a low-resolution ADC structure under malicious attacks by jammers, model the low-resolution ADC quantization output at the AP end during the pilot transmission stage based on AQNM (additive quantization noise model), and derive the MMSE channel estimation;

[0010] Step 2: Derive the closed-form expression of the UE total rate and establish the UE total rate optimization problem:

[0011] Establish an uplink data transmission model of a de-cellularized large-scale MIMO system with a low-resolution ADC structure under malicious attacks by jammers, and obtain the received data signal expression containing quantization noise at the AP end based on AQNM; the AP uses MRC (maximal ratio combining) technology to process the received signal and sends the processed signal to the CPU, and the CPU uses UatF (use-and-then-forget) technology to derive the closed-form expression of the UE total rate; with the UE data transmission power limit as the constraint condition, establish the UE total rate optimization problem;

[0012] Step 3: Design the UE total rate optimization method: adopt the total rate optimization algorithm based on weighted MMSE or the total rate optimization algorithm based on APG:

[0013] Weighted MMSE-based Total Rate Optimization Algorithm:

[0014] To solve the UE total rate optimization problem established in step 2, first, the weighted MMSE strategy is used to equivalently represent the non-convex total rate optimization problem as a convex minimization problem. Then, the Lagrange multiplier method is applied to solve the newly established convex minimization problem and derive a sub-optimal closed-form solution to the original optimization problem, ultimately achieving the goal of optimizing the system total rate.

[0015] APG (Accelerated Projected Gradient)-based Total Rate Optimization Algorithm:

[0016] To solve the UE total rate optimization problem established in step 2, the total rate optimization problem to be solved is differentiable with respect to the UE transmit power coefficient. First, the gradient of the objective function and the projection of the UE transmit power coefficient on the feasible set are calculated respectively. On this basis, the APG algorithm is used to solve a sub-optimal closed-form solution to this optimization problem, ultimately achieving the goal of optimizing the UE total rate.

[0017] Furthermore, the establishment of the uplink pilot training model and the derivation of the MMSE channel estimation in step 1 are specifically as follows:

[0018] Assume that when K UEs and an interferer with serial number j simultaneously transmit pilots to the AP, without considering ADC quantization, the pilot vector received by the AP l has the following expression:

[0019]

[0020] In the above formula, ρ p is the pilot transmission power of the UE, represents the pilot sequence of the UE k with length τ. The superscript H in the formula represents conjugate transpose. q p is the pilot transmission power of the interferer. Assume that the interferer does not know the pilot sequence of the UE and selects a random sequence uniformly distributed on the unit sphere as the pilot signal on each of its antennas, which satisfies where represents the expectation operator. In addition, l represents the additive white Gaussian noise at the AP is the channel vector between the AP l and the UE k , is the channel vector between the AP l and the m-th antenna of the interferer. Using AQNM to model the low-resolution ADC quantization output, the quantized y l has the following expression:

[0021]

[0022] Among them, represents the quantization distortion factor related to the ADC resolution b of the AP l When b l takes values of 1, 2, 3, 4, 5, the corresponding α l is equal to 0.6366, 0.8825, 0.9655, 0.9905, 0.9975; when b l > 5, l The matrix represents the quantization noise uncorrelated with y whose covariance matrix is l where diag(A) represents the diagonal matrix composed of the diagonal elements of matrix A. Using the MMSE criterion to estimate the channel, the MMSE estimation expression of the channel g is obtained as: lk

[0023]

[0024] where β lk is the large-scale fading coefficient between the AP l and the UE k β lj is the large-scale fading coefficient between the AP l and the jammer, δ 2 represents the Gaussian white noise power. In addition, The second moment of is equal to

[0025] Furthermore, the specific derivation of the closed-form expression of the total UE rate in step 2 and the establishment process of the total UE rate optimization problem are as follows:

[0026] Still assume that all UEs and jammers simultaneously transmit data signals to the AP. After the data signals received by the AP l are quantized by the ADC, its expression is:

[0027]

[0028] where x k and s j are the data signals transmitted by the UE k and the jammer respectively, ρ u and q u are the corresponding data transmission powers respectively. In addition, 0 ≤ η k ≤ 1 is the power control coefficient of the UE k , and n lis additive white Gaussian noise, is quantization noise, and its covariance matrix is where,

[0029] AP l adopts the MRC vector to decode and forward the decoded signal to the CPU. The expression of the total signal received by the CPU is:

[0030]

[0031] Using the UatF technology, the uplink rate expression of the UE k can be derived as:

[0032]

[0033] Based on the above formula, the total rate of the UE can be expressed as

[0034] Therefore, the UE total rate optimization problem can be expressed as:

[0035]

[0036]

[0037] where, η = [η 1 ,..., η K T represents the vector composed of the power control coefficients of all UEs, and Constraint 1 represents the condition that the UE data transmission power should satisfy.

[0038] Furthermore, the weighted MMSE technology is adopted in Step 3 to solve the optimization problem The specific steps include:

[0039] Step 1. Define Introduce the slack variables μ k and ν k , and using the weighted MMSE technology, the optimization problem established in Step 2 is equivalently transformed into:

[0040]

[0041]

[0042] where,

[0043] ​Step 2. The Lagrangian function related to the problem at the $(t + 1)$-th iteration is expressed as: :

[0044]

[0045] where $\lambda$ is the Lagrange multiplier. In addition, $\lambda$ (t+1) and respectively represent the values of the corresponding parameters at the $(t + 1)$-th iteration;

[0046] Step 3. The KKT conditions of the above Lagrangian function are:

[0047]

[0048] Step 4. Solving the above KKT condition equations, we can obtain:

[0049]

[0050]

[0051]

[0052]

[0053] Step 5. Judge whether holds, where is the maximum number of iterations. If it does not hold, let $t=t + 1$, and repeat Steps 2 to 5 until the termination condition is satisfied;

[0054] Step 6. Let to obtain a sub-optimal solution of the original optimization problem .

[0055] Furthermore, in Step 3, the APG technology is used to solve the optimization problem The specific steps are as follows:

[0056] Step 1. Define The total UE rate optimization problem in Step 2 can be equivalently written as:

[0057]

[0058]

[0059] where represents the feasible set of $\xi$,

[0060] Step 2. The objective function Differentiable with respect to the variable ξ. When using the APG algorithm to solve the optimization problem it is necessary to calculate the gradient of the function and the projection of the variable ξ onto the feasible set First, the gradient of the objective function can be calculated as:

[0061]

[0062] And the partial derivative of the function with respect to the variable can be written as:

[0063]

[0064] where, when k' = k, when k' ≠ k,

[0065] Next, since the projection of the variable ξ onto the feasible set is the solution of the optimization problem it can be obtained that:

[0066]

[0067] where, [x] + represents the projection of the vector x onto the first quadrant;

[0068] Step 3. At the (i + 1)-th iteration, the update function of the variable ξ (i+1) is:

[0069]

[0070] where, represents the gradient operator. In addition, υ > 0 represents the gradient descent step size and υ should be less than the reciprocal of the Lipschitz constant of to ensure the convergence of the algorithm,

[0071] Step 4. Determine whether holds, where is the maximum number of iterations. If it does not hold, let i = i + 1 and repeat Step 3 to Step 4 until the termination condition is satisfied;

[0072] Step 5. Let to obtain a sub-optimal solution of the original optimization problem

[0073] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0074] ​Compared with existing research, the present invention has the following remarkable advantages: The present invention studies the total rate optimization problem of a decellularized massive MIMO system with a low-resolution ADC structure under the malicious attack of a jammer. First, a closed-form expression for the total rate of the UE affected by the jammer attack and low-resolution ADC is derived, and the total rate optimization problem of the UE is proposed with the UE transmit power as the constraint condition; for this optimization problem, first-order optimization algorithms based on weighted MMSE technology and APG technology are designed respectively, which greatly improve the total rate of the UE within a short operation time and effectively compensate for the performance loss caused by the malicious attack of the jammer. BRIEF DESCRIPTION OF THE DRAWINGS:

[0075] Figure 1 It is a system model diagram of a decellularized massive MIMO system with a low-resolution ADC structure under the malicious attack of the jammer of the present invention;

[0076] Figure 2 It is a relationship diagram between the total rate of the UE of the present invention and the transmit power of the jammer;

[0077] Figure 3 It is a relationship diagram between the total rate of the UE of the present invention and the number of APs;

[0078] Figure 4 It is a comparison relationship diagram between the operation time-consuming of the weighted MMSE, APG and SCA optimization algorithms of the present invention. DETAILED DESCRIPTION OF THE INVENTION:

[0079] Next, in combination with the drawings and specific embodiments, the present invention will be further described. It should be noted that, on the premise of no conflict, the following-described embodiments or technical features can be combined arbitrarily to form new embodiments.

[0080] The present invention proposes a first-order optimization method for the uplink total rate of a decellularized massive MIMO, which greatly improves the total rate of the UE on the premise of satisfying the UE data transmit power constraint, thereby compensating for the performance loss caused by the malicious attack of the jammer. The following is a further detailed description of the present invention in combination with the drawings of the specification.

[0081] Embodiment 1:

[0082] A first-order optimization method for the uplink total rate of a decellularized massive MIMO includes the following steps:

[0083] Step 1, establish an uplink pilot training model and derive MMSE channel estimation:

[0084] As Figure 1As shown, the present invention studies an uplink de-cellular massive MIMO system with a jammer, and the AP side uses low-resolution ADCs to reduce the deployment cost and hardware power consumption. The considered system has L multi-antenna APs, K single-antenna UEs, and 1 multi-antenna jammer. Each AP is equipped with L antennas and the jammer is equipped with M antennas. The present invention assumes that the entire system operates in the time-division duplex mode, and the channel state information of the uplink is estimated by the UEs sending specific pilot sequences to the APs. Let denote the pilot sequence of length τ assigned to UE k , satisfying In addition, it is assumed that the jammer does not know the specific pilot sequences of the UEs and selects a random sequence uniformly distributed on the unit sphere as the pilot signal on each of its antennas, satisfying When all UEs and the jammer simultaneously transmit pilots to the AP, without considering ADC quantization, the received pilot vector expression at the AP l is

[0085]

[0086] where ρ p and q p are the pilot transmission powers of the UE and the jammer respectively, is the additive white Gaussian noise at the AP l , and represent the channel vectors between the m-th antenna of the AP l and the UE k and the jammer respectively. Using AQNM to model the quantization output of the low-resolution ADC, the expression of the quantized y l is

[0087]

[0088] where represents the linear quantization distortion factor related to the ADC resolution b l of the AP. When b l takes values of 1, 2, 3, 4, 5, the corresponding α l is equal to 0.6366, 0.8825, 0.9655, 0.9905, 0.9975; when b l > 5, l Moreover, the matrix represents the quantization noise uncorrelated with y , and its covariance matrix is l Based on the MMSE estimation expression of the channel g can be derived as lk ​

[0089]

[0090] Among them, β lk and β lj are the large-scale fading coefficients between AP l and UE k and the jammer respectively, and δ 2 represents the Gaussian white noise power. In addition, the second moment of the channel estimation is equal to

[0091] Step 2: Derive the closed-form expression of the total UE rate and establish the total UE rate optimization problem:

[0092] After the UE sends the pilot signal, all UEs still need to send valid data signals. Assume that the data signal transmitted by UE k is x k , which satisfies In addition, to reduce the system performance, the jammer also sends interference signals to the AP during this stage, s j , which satisfies Let ρ u and q u represent the transmission powers of the UE and the jammer during the data transmission stage respectively. When all UEs and the jammer transmit simultaneously, the expression of the data signal received by AP l after being quantized by the low-resolution ADC is

[0093]

[0094] Among them, n l represents the additive Gaussian white noise, represents the ADC quantization noise uncorrelated with the actual received signal, and its covariance matrix is where The present invention adopts an MRC receiver. AP l adopts an MRC vector to decode and forward the decoded signal to the CPU. The expression of the total signal received by the CPU is

[0095]

[0096] In the process of deriving the closed-form expression of the rate of the cell-free massive MIMO system, by applying the UatF strategy, r k can be equivalently rewritten in the form of a specific known signal plus uncorrelated interference noise. Based on the above analysis, a closed-form expression of a lower bound on the achievable rate of UE k is

[0097]

[0098] Therefore, the total rate of the UE can be expressed as

[0099] Next, the optimization problem of the total rate of the UE with the UE data transmission power limit as the constraint condition is expressed as

[0100]

[0101]

[0102] where η = [η 1 ,..., η K T represents the vector composed of the power control coefficients of all UEs, and Constraint 1 represents the condition that the UE data transmission power should satisfy. Obviously, this problem is non-convex with respect to the variable η, so it is difficult to obtain its optimal solution in polynomial time.

[0103] Step 3: Design a total rate optimization algorithm based on weighted MMSE

[0104] To effectively solve the non-convex problem in Step 2 , the present invention uses the weighted MMSE technique to equivalently transform the non-convex problem into a convex problem

[0105] Step 1: Define and introduce slack variables μ k and ν k , and the expression of the convex problem is

[0106]

[0107]

[0108] where

[0109] When solving the problem , first set the maximum number of iterations and the minimum iteration error. When the maximum number of iterations is reached or the minimum iteration error is satisfied, the solution obtained by solving the problem is a sub-optimal solution of the original problem . The steps for solving the problem include:

[0110] Step 2: If it is the first iteration, an initial power factor that satisfies the power constraint can be arbitrarily selected. Assume that the iteration progresses to the (t + 1)-th time. At this time, regarding the problem​ The Lagrangian function can be expressed as

[0111]

[0112] where λ (t+1) and respectively represent the values of the corresponding parameters at the (t + 1)-th iteration;

[0113] Step 3. By taking the partial derivatives of with respect to and λ (t+1) and setting the partial derivatives to 0, the KKT conditions of the above Lagrangian function can be written as

[0114]

[0115] Step 4. Solving the above KKT conditions, we can obtain

[0116]

[0117]

[0118]

[0119]

[0120] Step 5. Determine whether holds, where is the maximum number of iterations. If it does not hold, let t = t + 1, and repeat Steps 2 to 5 until the termination condition is satisfied;

[0121] Step 6. Let to obtain a sub-optimal solution of the original optimization problem

[0122] Example 2:

[0123] A first-order optimization method for the uplink total rate of cellular massive MIMO. Compared with Example 1, in Step 3, a total rate optimization algorithm based on APG is designed to quickly solve the non-convex problem in Step 2

[0124] Step 1. Define ξ = [ξ 1 , ξ 2 ,..., ξ K T The problem in Step 2 can be equivalently written as

[0125] ​​

[0126]

[0127] Among them, represents the feasible set of ξ,

[0128] Step 2: Carefully observe the optimization problem It can be seen that in the objective function is differentiable with respect to the variable ξ. Therefore, the APG algorithm given in the article ("Utility maximization for large-scale cell-free massive MIMO downlink, IEEE Transactions on Communications, vol. 69, no. 10, pp. 7050 - 7062, Oct. 2021) can be used to solve the problem It should be noted that when using the APG algorithm to solve the optimization problem it is necessary to calculate the gradient of the function and the projection of the variable ξ on the feasible set which is related to The projection is related to and The specific calculation process is described as follows:

[0129] Since the objective function is the sum of a series of functions therefore can be calculated as

[0130]

[0131] And the partial derivative of the function with respect to the variable can be written as

[0132]

[0133] Among them, when k' = k, when k' ≠ k,

[0134] Furthermore, since is the solution of the optimization problem it can be obtained that

[0135]

[0136] Among them, [x] +Denotes the projection of the vector x onto the first quadrant.

[0137] Step 3: With the above calculation basis, the APG algorithm can be used to solve the problem next. When solving , first set the maximum number of iterations and the minimum iteration error. When the maximum number of iterations is reached or the minimum iteration error is satisfied, solve the problem The obtained solution is a sub-optimal solution to the original problem . The steps of the APG algorithm for solving the problem include: If it is the first iteration, an initial power factor that satisfies the power constraint can be arbitrarily selected. Assume that the iteration proceeds to the (i + 1)-th time. At this time, the update function of the variable ξ (i+1) can be expressed as

[0138]

[0139] where represents the gradient descent step size and υ should be less than the reciprocal of the Lipschitz constant of where ρ is the auxiliary update constant;

[0140] Step 4: Judge whether holds, where is the maximum number of iterations. If it does not hold, let i = i + 1, and repeat steps 3 to 4 until the termination condition is satisfied;

[0141] Step 5: Let to obtain a sub-optimal solution to the original optimization problem .

[0142] The performance of the technical solution of the present invention is further described below in conjunction with simulation experiments.

[0143] Figure 2 is a graph showing the relationship between the total UE rate and the transmitter transmit power under five different optimization schemes. The abscissa is the transmitter transmit power, that is, the power of the transmitter to transmit pilots and data, and the ordinate is the total UE rate. The simulation parameters are taken as L = 120, K = 30, N = 4, M = 2, δ 2 = -126 dBw, ρ p = ρ u = 0.2 W, the maximum number of iterations and are respectively set to 150 and 3000, and the step size υ of the APG algorithm is set to 0.05. Figure 2 The FPC (full power control) scheme in As shown in the figure, as the transmitter power increases, the impact of the jammer on the total UE rate becomes more and more serious. Compared with the FPC scheme, the optimization algorithms based on WMMSE and APG proposed in the present invention can still effectively improve the total UE rate under strong interference power, and the rate gains provided by the above two optimization algorithms are roughly the same, both better than the rate gain provided by the optimization algorithm based on SCA. Therefore, it can better compensate for the impact of the jammer on the rate. In addition, it can also be found that when the AP side adopts a low-resolution ADC, the total UE rate will be reduced. From another perspective, the two rate optimization algorithms proposed in the present invention also play a role in reducing quantization distortion. Similar conclusions can also be observed from Figure 3 which is Figure 3 Figure 5 shows the relationship between the total UE rate and the number of APs under five different optimization schemes. Except for q p =q u =0.4W, the simulation parameters in Figure 3 are the same as those in Figure 2 . It can be found that when the number of APs is large, the two optimization algorithms proposed in the present invention can exceed the total UE rate without a jammer, which further confirms the superiority of the optimization algorithms designed in the present invention in improving the system rate.

[0144] Figure 4 Figure 6 compares the running times of the optimization algorithms based on MMSE and APG designed in the present invention and the traditional optimization algorithm based on SCA. Except for , the selected simulation parameters are the same as those in Figure 3 . It is not difficult to find that compared with the optimization algorithm based on SCA, the two optimization algorithms designed in the present invention greatly reduce the operation time-consuming, and as the number of APs increases, the advantage of the optimization algorithm based on APG in running time-consuming becomes more prominent. It is worth mentioning that although the present invention only considers the total rate optimization problem in the uplink cellular massive MIMO system, the present invention also provides a reference for other optimization problems in the same field, and can be extended based on this and applied to the technical solutions of other algorithms in the same field, with a very broad application prospect.

[0145] The above embodiments are only the preferred embodiments of the present invention, and the scope of protection of the present invention cannot be limited by this. Any non-substantive changes and substitutions made by those skilled in the art on the basis of the present invention belong to the scope of protection required by the present invention.

Claims

1. A first-order optimization method for the uplink total rate of cellular massive MIMO, characterized in that, it includes the following steps: Step 1, establish an uplink pilot training model and derive the MMSE channel estimation: Under the general Rayleigh block fading channel, an uplink pilot training model of a cell-free massive MIMO system with a low-resolution ADC structure under malicious attacks by a jammer is established. Based on AQNM, the low-resolution ADC quantization output at the AP side during the pilot transmission phase is modeled, and the MMSE channel estimation is derived. Specifically, it includes: considering a cell-free massive MIMO system with L multi-antenna APs, K single-antenna UEs, and 1 multi-antenna jammer, each AP is equipped with N antennas and the jammer is equipped with M antennas. When the K UEs simultaneously transmit pilots to the APs, the jammer with serial number j also transmits pilot signals to the APs. The expression of the pilot vector received by the l-th AP in the AP l is as follows: In the above formula, ρ p is the pilot transmission power of the UE, denotes the pilot sequence of the UE k with length τ, where the sequence number k represents the k-th UE; the superscript H in the formula represents conjugate transpose, q p is the pilot transmission power of the jammer. It is assumed that the jammer does not know the pilot sequence of the UE and selects a random sequence uniformly distributed on the unit sphere as the pilot signal on each of its antennas, which satisfies where denotes the expectation operator. In addition, represents the additive white Gaussian noise at the AP l location, is the AP l and the UE k channel vector between them, is the channel vector between the AP l and the m-th antenna of the jammer. The AQNM is used to model the quantization output of the low-resolution ADC. The quantized Y l expression is: Among them, represents the quantization distortion factor related to the ADC resolution b l of AP l . When b l takes values of 1, 2, 3, 4, 5, the corresponding α l is equal to 0.6366, 0.8825, 0.9655, 0.9905, 0.9975; when b l > 5, matrix represents the quantization noise uncorrelated with Y l , and its covariance matrix is where diag(A) represents the diagonal matrix composed of the diagonal elements of matrix A. Using the MMSE criterion to estimate the channel, the MMSE estimation expression of the channel g lk is: where, β lk is the large-scale fading coefficient between the AP l and the UE k ; β lj is the large-scale fading coefficient between the AP l and the jammer; δ 2 represents the Gaussian white noise power. In addition, the second moment of Step 2, derive the closed-form expression of the UE total rate and establish the UE total rate optimization problem: Establish an uplink data transmission model of a cell-free massive MIMO system with a low-resolution ADC structure under malicious attacks by jammers. Based on AQNM, obtain the expression of the received data signal containing quantization noise at the AP side; the AP uses MRC technology to process the received signal and sends the processed signal to the CPU, and the CPU uses the UatF technology to derive a closed-form expression of the total UE rate; with the UE data transmission power limit as a constraint condition, establish an optimization problem for the total UE rate; specifically including: based on the description in step 1 above, the AP l After the received data signal is quantized by the ADC, its expression is where x k and s j are the data signals transmitted by the UE k and the jammer respectively, and ρ u and q u represent the corresponding data transmission powers respectively; in addition, 0 ≤ η k ≤ 1 is the power control coefficient of the UE k , n l is the additive white Gaussian noise, is the quantization noise, and its covariance matrix is where AP l Adopt the MRC vector For Decode and forward the decoded signal to the CPU. The total signal expression received by the CPU is: Using the UatF technology, the uplink rate expression of the UE k can be derived as follows: Based on the above formula, the total rate of the UE can be expressed as Therefore, the UE total rate optimization problem can be expressed as: where η = [η 1 ,..., η K T represents the vector composed of the power control coefficients of all UEs, and Constraint 1 represents the condition that the UE data transmission power should satisfy;​ Step 3. Design the UE total rate optimization method: Adopt the total rate optimization algorithm based on APG: Solve the UE total rate optimization problem established in Step 2. The total rate optimization problem to be solved is differentiable with respect to the UE transmit power coefficient. First, calculate the gradient of the objective function and the projection of the UE transmit power coefficient on the feasible set respectively. On this basis, use the APG algorithm to solve a sub-optimal closed-form solution of this optimization problem, and finally achieve the purpose of optimizing the UE total rate. Use the APG technique to solve the optimization problem The specific steps are as follows: Step 1. Define ξ = [ξ 1 , ξ 2 ,..., ξ K T , and the UE total rate optimization problem in Step 2 can be equivalently rewritten as:​ Among them, represents the feasible set of ξ, Step 2. Objective function is differentiable with respect to the variable ξ. When using the APG algorithm to solve the optimization problem , it is necessary to calculate the gradient of the function and the projection of the variable ξ onto the feasible set . First, the gradient of the objective function can be calculated as follows: In the above formula, the function with respect to the variable The partial derivative can be written as: where, when k' = k, when k' ≠ k, Immediately, since the projection of the variable ξ onto the feasible set is the solution to the optimization problem we have: where, [x] + represents the projection of the vector x on the first quadrant; Step 3. At the (i + 1)-th iteration, the update function of variable ξ (i+1) is as follows: Among them, denotes the gradient operator. In addition, υ > 0 represents the gradient descent step size and υ should be less than the reciprocal of the Lipschitz constant of to ensure the convergence of the algorithm, where ρ is the auxiliary update constant; Step 4. Determine whether holds, where is the maximum number of iterations. If it does not hold, let i = i + 1, and repeat Steps 3 to 4 until the termination condition is satisfied; Step 5. Let to obtain a sub-optimal solution of the original optimization problem .

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