A covert wireless communication beamforming optimization method under limited character input

Abstract

The application discloses a covert wireless communication beamforming optimization method under limited character input and belongs to the technical field of wireless communication.The application constructs a joint optimization strategy of the transmission power and the beamforming vector optimization strategy of the transmission end under different modulation modes, designs a semi-positive definite relaxation algorithm on the basis of using the lower bound approximation of the objective function, equivalently converts the initial non-convex objective function and the covert constraint into a more easily handled convex form, solves by using the CVX tool package in MATLAB, and obtains the design of the optimal beamforming vector of the problem.Further, in order to reduce the calculation complexity, the application proposes a low-complexity optimization strategy, and simulation results show that the method can significantly reduce the calculation complexity while meeting the performance of covert communication.

Description

An Optimization Method for Covert Wireless Communication Beamforming under Limited Character Input Technical Field

[0001] This invention relates to the field of wireless communication technology, and in particular to an optimization method for covert wireless communication beamforming under limited character input. Background Technology

[0002] With the rapid development of wireless communication technology, the development of 6G demands extremely high data transmission rates, which brings new challenges to security-related traffic processing. Simultaneously, as communication scenarios become increasingly complex and diverse, individuals' reliance on wireless devices grows daily, the risks of malicious eavesdropping are constantly rising, and the sharing of private and sensitive information is becoming more common. The importance of security and privacy protection has risen to an unprecedented level, and ensuring privacy and security during communication has become a crucial research direction. To address the problem of monitoring nodes detecting communication behavior in covert communication, covert communication employs covert transmission behavior for protection, reducing the probability of unauthorized nodes detecting the transmission behavior. This is of great significance for scenarios requiring high levels of concealment.

[0003] Existing research on covert communication largely focuses on single-input single-output systems or ideal Gaussian input signal models. However, in practical wireless communication systems, the transmitter typically employs finite character modulation schemes, such as PSK and QAM, which are finite discrete signal forms. In these cases, the mutual information expression is more complex than that of a Gaussian signal model, posing challenges to performance analysis and algorithm design. Meanwhile, multiple-input single-output (MISO) systems, utilizing beamforming technology, can improve signal quality at legitimate receivers while suppressing the received power at monitoring nodes, offering significant advantages in covert transmission. However, jointly optimizing the transmit power and beamforming vector under covert constraints often results in a non-convex optimization problem, requiring complex processes such as semi-definite relaxation and matrix lifting, leading to high computational complexity and making them unsuitable for scenarios with limited computational resources.

[0004] Therefore, existing technologies still suffer from problems such as high algorithm complexity, difficulty in engineering implementation, and difficulty in balancing performance and computational efficiency in covert communication environments with limited character input and multiple antennas. There is an urgent need to propose an optimization method for covert wireless communication beamforming that can satisfy covertness constraints while taking into account communication performance and computational efficiency. Summary of the Invention

[0005] The purpose of this invention is to provide an optimization method for covert wireless communication beamforming under limited character input and a method to reduce complexity to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A method for optimizing beamforming in covert wireless communication with limited character input, specifically including the following steps:

[0008] S1. Construct a three-dimensional system model: Select the transmitter, receiver, and listener as the entities to be studied in the three-dimensional system model. The transmitter is equipped with multiple antennas, while the receiver and listener are equipped with single antennas. The spatial position information of each entity is represented by a Cartesian coordinate system.

[0009] S2. Solve for the minimum total detection error probability of the eavesdropper: The eavesdropper judges whether the transmitter sends hidden information to the receiver based on the signal samples it observes. It counts the signals received by the eavesdropper within the symbol period and calculates the minimum total detection error probability of the eavesdropper by combining the cases of no information and information sent by the transmitter in the received signals.

[0010] S3. Constructing the optimization problem: Selecting the total mutual information as the system performance metric, deriving the hidden constraint expression for finite character input, and constructing an optimization problem that maximizes the total mutual information;

[0011] S4. Simplify the optimization problem: Simplify the optimization problem constructed in S3, transform it into an easier optimization problem to handle, derive the lower bound of the total mutual information of the objective function, calculate the hidden constraint expression under finite character input, and transform it into an easier upper bound to handle;

[0012] S5. Solving the Optimization Problem: A semi-definite relaxation algorithm is used to approximate and convexize the objective function and hidden constraints using mathematical methods, transforming the original non-convex optimization problem into a standard convex optimization problem. This problem is then solved using the CVX toolkit in MATLAB to obtain the optimal beamforming vector design. Simultaneously, a low-complexity algorithm is proposed to handle matrix variables, decoupling the transmit power and beamforming vector optimization. In the first stage, the beamforming vector is designed based on the maximum signal leakage-to-noise ratio criterion, and the optimal beamforming vector is solved using a closed-form expression. In the second stage, with the beam vector fixed, the original joint optimization problem is degenerated into a one-dimensional transmit power optimization problem, which is then efficiently solved using the CVX toolkit in MATLAB.

[0013] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0014] 1. This invention couples the transmitter power (Alice) and the optimal beamforming vector into a single variable for optimization.

[0015] 2. Given that the objective function and constraint set of the constructed problem are non-convex, they are usually difficult to solve directly. This invention simplifies the objective function and hidden constraints through mathematical analysis and effectively solves the optimization problem using a semidefinite relaxation method.

[0016] 3. The low-complexity scheme proposed in this invention cleverly avoids processing the matrix variable W, degenerates the joint optimization problem into a one-dimensional transmit power optimization problem, and solves it efficiently using convex optimization methods. Attached Figure Description

[0017] Figure 1 is a schematic diagram of a system model of an optimization method for covert wireless communication beamforming under finite character input proposed in this invention.

[0018] Figure 2 is a schematic diagram of the algorithm flow of an optimization method for covert wireless communication beamforming under limited character input proposed in this invention.

[0019] Figure 3 shows the mutual information of the covert wireless communication beamforming optimization method proposed in this invention under finite character input as a function of the covertness level parameter. A schematic diagram of the change curve.

[0020] Figure 4 shows the transmit power of the covert wireless communication beamforming optimization method proposed in this invention under finite character input as a function of the covertness level parameter. A schematic diagram of the change curve.

[0021] Figure 5 is a schematic diagram showing the relationship between mutual information and the number of antennas at the transmitting end in an optimization method for covert wireless communication with limited character input proposed in this invention.

[0022] Figure 6 shows the mutual information of the covert wireless communication beamforming optimization method proposed in this invention under finite character input as a function of average maximum transmit power. A schematic diagram of the change curve. Detailed Implementation

[0023] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings.

[0024] Example 1:

[0025] Please refer to Figures 1-2. This invention proposes a method for optimizing beamforming in covert wireless communication under limited character input, specifically including the following steps:

[0026] S1. Constructing a 3D System Model: The transmitter (Alice), receiver (Bob), and listener (Willie) are selected as the entities in the 3D system model. The transmitter is equipped with multiple antennas, while the receiver and listener are equipped with single antennas. The spatial location information of each entity is represented using a Cartesian coordinate system, specifically including the following:

[0027] The coordinates of the transmitter (Alice), receiver (Bob), and listener (Willie) are set to (0,0,0), (70,0,0), and (110,0,0) respectively, meaning the listener (Willie) is deployed on the same straight line as the transmitter (Alice) and receiver (Bob); the channel vectors from the transmitter (Alice) to the receiver (Bob), from the transmitter (Alice) to the listener (Willie), and from the receiver (Bob) to the listener (Willie) are respectively... , , It is assumed that all channels are Rayleigh fading channels, and the channel loss coefficient between the transmitter and receiver is 2.3, while the channel loss coefficient between the transmitter and the receiver is 3.6.

[0028] The constellation modulation of the input signal constellation points considered in this invention is based on an equal probability distribution, meaning that all constellation points are taken from a set Ω containing M discrete constellation points. ;in It is the first discrete constellation adopted One modulation point.

[0029] S2. Solving for the minimum total detection error probability of the eavesdropper (Willie): The eavesdropper (Willie) determines whether the transmitter (Alice) has sent a hidden message to the receiver (Bob) based on the signal samples it observes. It counts the signals received by the eavesdropper (Willie) within the symbol period and, combining the received signals with and without information sent by the transmitter (Alice), calculates the minimum total detection error probability of the eavesdropper (Willie). The specific solution method is as follows:

[0030] S201. In covert communication, the eavesdropper (Willie) bases their observations on signal samples... To determine whether the transmitter (Alice) has sent covert information to the receiver (Bob), This indicates that the listener (Willie) was in the [number]th [period]. The signals received on each channel can be represented as:

[0031] (1)

[0032] in, This indicates that the transmitter is not transmitting information, and the listener only receives additive white Gaussian noise. This indicates that the transmitter is sending information, and the listener receives an observation signal that is a superposition of pure signal and noise.

[0033] in, Indicates the transmitter is at the 1st Signals transmitted during the use of each channel ,in The total number of times the channel was used.

[0034] in, Additive white Gaussian noise representing the listener Its mean is zero and its variance is ; The superscript represents the channel vector between the transmitter and the listener. H Indicates conjugate transpose; parameter Denotes the equivalent beamforming vector at the transmitting end, where For transmission power, For the unit norm beamforming vector; therefore, under the assumption Below, the pure signal component received by the eavesdropper is It is determined by the channel vector, the equivalent beamforming vector, and the transmitted signal.

[0035] S202, Assuming the false alarm error probability is used The probability of a missed alarm is expressed as... It means that, respectively by and Given, among which and Let represent the binary decision made by the eavesdropper (Willie) regarding whether the transmitter (Alice) should transmit the message, and let represent the total detection error probability of the eavesdropper (Willie). for:

[0036] (2)

[0037] S203. In covert wireless communication, the eavesdropper (Willie) wants to minimize its total probability of detection error. The optimal detector is the likelihood ratio detector, and the corresponding likelihood ratio function is as follows:

[0038]

[0039] in, and They represent in Assume and Under the assumption that, the observation is The likelihood function; and They represent in and Prior probabilities under the assumption; and These represent the decisions to send and receive data, respectively; when the likelihood ratio is greater than the decision threshold, the eavesdropper determines... If true, otherwise judge Established.

[0040] S204. Based on the likelihood ratio function in S203, analyze the optimal detection threshold and the corresponding minimum detection error probability for the eavesdropper (Willie). Considering The Gamma function is often incomplete, which is detrimental to subsequent covert transmission design; therefore, it is necessary to consider... The lower bound of can be represented as:

[0041] (4)

[0042] in, It is a limited character input from arrive Upper bound of Kullback-Leibler divergence;

[0043] This inequality shows that the smaller the difference in distributions measured by divergence under the two hypotheses, the closer the minimum detection error probability achievable by the eavesdropper is to 1, meaning the detection performance is worse, and the eavesdropper can hardly distinguish between the two hypotheses. Therefore, in the design of covert communication systems, by constraining the Kullback-Leibler divergence to not exceed a certain small positive number... ,Right now This ensures that the minimum total detection error probability of the eavesdropper is satisfied. To achieve covert communication.

[0044] S3. Constructing the Optimization Problem: Choosing the total mutual information as the system performance metric, derive the implicit constraint expression for finite character input, and construct an optimization problem that maximizes the total mutual information, specifically including the following:

[0045] In covert communication with finite character input, the objective is to maximize the total mutual information obtained at the receiver (Bob). For the th... The total mutual information of each channel at any given time. The expression is:

[0046]

[0047] in, Indicates the receiver at the Signals received on each channel ,symbol This represents additive white Gaussian noise at the receiving end, with a mean of zero and a variance of... ; Represents the first digit in a finite set of constellation points. The constellation point and the first Subtract the points from each constellation.

[0048] Using the total mutual information received at the receiver as a system performance metric, and under the premise of satisfying the concealment constraint and transmit power constraint, the transmit power and beamforming vector at the transmitter are jointly optimized and coupled into a single variable. , .

[0049] The optimization problem constructed is as follows:

[0050]

[0051] Here, it is assumed that the channel is stationary and discrete with no memory, and that the components of the input vector are independent of each other; at this time, after... Mutual information after secondary channel use Represented as: .

[0052] Where C1 is the concealment constraint, here express The upper boundary.

[0053] Where C2 is the power constraint of the transmitter under discrete input.

[0054] because For unit norm beamforming vectors, This represents the actual power, and its value must not exceed the maximum power allowed by the system. .

[0055] S4. Simplify the optimization problem constructed in S3, transforming it into a more manageable optimization problem. Derive the lower bound of the total mutual information of the objective function, calculate the hidden constraint expression under finite character input, and transform it into a more manageable upper bound. Specifically, this includes the following:

[0056] S401, For the objective function containing factors including about The expected operation involves summation of exponential terms and nested logarithms within the inner layer, making it impossible to directly obtain a closed-form expression, which brings inconvenience to both theoretical analysis and numerical calculation. Therefore, this invention derives its lower bound using Jensen's inequality:

[0057]

[0058] S402. The KL divergence within the hidden constraints requires integration over a high-dimensional probability density function, which is computationally complex and difficult to apply directly to optimization design. Therefore, this invention uses inequality relaxation and expectation expansion on the integral expression of the KL divergence to transform the original expression into a more manageable upper bound form. To this end, the upper bound is derived using the log-sum inequality:

[0059] (8)

[0060] So, after simplification, regarding The optimization problem is as follows:

[0061]

[0062] S5. Solve the optimization problem constructed in S4. For the objective function of optimization problem (9), since maximizing the mutual information of the objective function is equivalent to maximizing the lower bound of the mutual information, and since it contains a non-convex exponential function, the overall problem is a non-convex optimization problem, which is difficult to solve directly. Therefore, this invention adopts the positive semidefinite relaxation SDR method.

[0063] S501, all related to the original problem All related quadratic forms can be uniformly written as about The trace form, as can be seen from the convexity of Log-Sum-Exp, indicates that the objective function is about... Given a concave function, maximizing a concave function is a convex optimization problem, which can be solved using the CVX toolkit. .

[0064] Similarly, after applying similar processing to the hidden constraints, the following optimization problem is obtained:

[0065]

[0066] Problem (10) is equivalent to Problem (9) and has the same optimal solution.

[0067] S502. For the optimization problem (10), after semidefinite relaxation, the optimization problem is a convex SDP problem with a convex objective function and a convex constraint set. It can be solved using the CVX toolkit in MATLAB to find the optimal matrix variables. .

[0068] S503. For SDR joint optimization, matrix variables need to be handled. The resulting high computational complexity problem is addressed by proposing a two-stage low-complexity optimization strategy for the optimization problem (9), which optimizes the power and beamforming vector separately. .

[0069] In the first phase, the closed-form SLNR criterion is based on the maximum (Signal-to-Leakage-plus-Noise Ratio) criterion. The beamforming vector direction can be solved using a closed-form equation. In the second stage, based on the fixed beam vector, the original joint optimization problem is degenerated into a one-dimensional transmit power optimization problem and solved efficiently using convex optimization methods.

[0070] This strategy avoids the process of solving a semidefinite programming problem, significantly reducing the overall computational cost. The low-complexity optimization problem based on optimization problem (9) is as follows:

[0071]

[0072] The optimization problem is simplified into a convex optimization problem, namely, maximizing mutual information with respect to transmission power while satisfying concealment constraints and transmission power constraints. The problem is considered as a one-dimensional optimization problem with respect to transmit power when the beamforming vector is determined by a closed-form solution. This problem is solved numerically using the CVX toolbox in MATLAB to obtain the optimal transmit power. .

[0073] Example 2:

[0074] This invention proposes an optimization method for covert wireless communication beamforming under finite character input, aiming to effectively improve the covert transmission performance of the system. Simulation verification results are shown in Figures 3-6, respectively:

[0075] Figure 3 shows the concealment constraints under different modulation methods. The impact on mutual information. In this group's simulations... With the value fixed at 30, it can be observed from the figure that the mutual information increases with the hidden constraint. The increase is due to relaxation. The larger the order, the easier it is to satisfy the concealment constraint. At the same time, with the higher the order of the modulation scheme, the greater the mutual information obtained by the receiver (Bob). This is because higher-order modulation schemes carry more information, which is caused by the different constellation point arrangements of the two.

[0076] Figure 4 shows the concealment constraints under different modulation methods. Impact on transmitter power. With concealment constraints... With the relaxation of the transmission, the transmit power at the transmitting end increases accordingly, and increasing the transmit power can achieve higher communication performance. Overall, the optimal power corresponding to the SDR algorithm is generally higher than that of the low-complexity algorithm, while the low-complexity method, while maintaining lower complexity, exhibits a more conservative but stable power trend. In addition, 8PSK requires higher transmit power than QPSK to support reliable transmission of higher modulation orders. These are consistent with the mutual information performance improvement trend in Figure 3, which indirectly verifies the correspondence and consistency between the two.

[0077] Figure 5 shows the different numbers of antennas at the transmitter. Below, the mutual information variation trends of each algorithm in QPSK and 8PSK scenarios are analyzed. The results show that as the number of antennas increases, the overall system mutual information shows an increasing trend and remains relatively stable. Under these conditions, the system gradually approaches the maximum value of the objective function. This is because increasing the number of antennas at the transmitting end provides higher spatial degrees of freedom, giving the system stronger beam focusing capabilities, thereby effectively enhancing the received signal power at the Bob end while ensuring concealment constraints.

[0078] Figure 6 shows the different transmit power limits. Under the given conditions, the mutual information changes of the two algorithms in QPSK and 8PSK scenarios are analyzed. In this simulation, the concealment level is fixed at [value missing]. It can be observed that, in order to At that time, the mutual information results corresponding to each curve are the same as those in Figure 3 at the same concealment level ( The performance values ​​obtained under both perspectives are consistent, which indicates that the optimal solutions obtained from the two perspectives are consistent, thus further verifying the correctness of the simulation results.

[0079] The simulation results above show that the proposed optimal design can achieve significant performance gains while maintaining low time complexity.

[0080] In summary, this invention proposes an optimization method for covert wireless communication beamforming under finite character input. This method first simplifies the optimization problem by deriving the upper bound of the objective function and the lower bound of the covertness constraint. To address the non-convex nature of the simplified problem, this invention designs a positive semidefinite relaxation algorithm. Based on this, this invention innovatively proposes a two-stage, low-complexity optimization scheme that does not require changes to matrix variables. Direct computation significantly reduces the algorithm's time complexity. Simulation results demonstrate that, under strict concealment constraints, the proposed method can significantly improve system communication performance, reaching the theoretical limit of the system's objective function under finite character input.

Claims

1. A method for optimizing beamforming in covert wireless communication under limited character input, characterized in that, Specifically, the following steps are included: S1. Construct a 3D system model: Select the transmitter, receiver, and listener as the entities in the 3D system model. The transmitter is equipped with multiple antennas, while the receiver and listener are equipped with single antennas. A Cartesian coordinate system is used to represent the spatial location information of each entity. S2. Solve for the minimum total detection error probability of the listener: The listener determines whether the transmitter has sent concealed information to the receiver based on its observed signal samples. The signals received by the listener within the symbol period are statistically analyzed. Combining the received signals with and without transmitted information, the minimum total detection error probability of the listener is calculated. S3. Construct an optimization problem: Select the total mutual information as the system performance metric. Derive the concealment constraint expression for finite character input, constructing an optimization problem that maximizes the total mutual information. S4. Simplify the optimization problem: Simplify the optimization problem constructed in S3, transforming it into a more manageable optimization problem. The process involves several steps: First, deriving the lower bound of the total mutual information of the objective function, calculating the hidden constraint expression under finite character input, and transforming it into an easily manageable upper bound. Second, solving the optimization problem: A semi-definite relaxation algorithm is used to approximate and convexize the objective function and hidden constraints using mathematical methods, transforming the original non-convex optimization problem into a standard convex optimization problem. This problem is then solved using the CVX toolkit in MATLAB to obtain the design of the optimal beamforming vector. Simultaneously, a low-complexity algorithm is proposed to handle the matrix variables, decoupling the transmit power from the beamforming vector. In the first stage, the beamforming vector is designed based on the maximum signal leakage-to-noise ratio criterion, and the optimal beamforming vector is solved using a closed-form expression. In the second stage, with the beamform vector fixed, the original joint optimization problem is degenerated into a one-dimensional transmit power optimization problem, which is then efficiently solved using the CVX toolkit in MATLAB.

2. The method for optimizing covert wireless communication beamforming under finite character input as described in claim 1, characterized in that, The use of Cartesian coordinates to represent the spatial location information of the transmitter, receiver, and eavesdropper as described in S1 specifically includes the following: setting the position coordinates of the transmitter, receiver, and eavesdropper to (0,0,0), (70,0,0), and (110,0,0) respectively, that is, deploying the eavesdropper on the same straight line as the transmitter and receiver; The channel vectors from transmitter to receiver, transmitter to listener, and receiver to listener are respectively used as follows: 、 、 This indicates that all channels are Rayleigh fading channels, and the channel loss coefficient between the transmitter and receiver is 2.3, while the channel loss coefficient between the transmitter and the receiver is 3.

6. The input signal under consideration is constellation modulation with an equal probability distribution of constellation points, meaning that all constellation points are taken from a set Ω containing M discrete constellation points. ;in It is the first discrete constellation adopted One modulation point.

3. The method for optimizing covert wireless communication beamforming under finite character input as described in claim 2, characterized in that, The solution to the minimum total detection error probability of the eavesdropper described in S2 specifically includes the following: S201. In covert communication, the eavesdropper uses observed signal samples... To determine whether the transmitting end has sent covert information to the receiving end, among which, Indicates the listener is at the The signals received on each channel are represented as follows: (1) Among them, This indicates that the transmitter is not transmitting information, and the listener only receives additive white Gaussian noise. This indicates that the transmitter is sending information, and the listener receives an observation signal that is a superposition of pure signal and noise; among which, Indicates the transmitter is at the 1st Signals transmitted during the use of each channel ,in The total number of times the channel was used; where, Additive white Gaussian noise representing the listener Its mean is zero and its variance is ； The superscript represents the channel vector between the transmitter and the listener. H Indicates conjugate transpose; parameter Denotes the equivalent beamforming vector at the transmitting end, where For transmission power, For the unit norm beamforming vector; therefore, under the assumption Below, the pure signal component received by the eavesdropper is The false alarm probability is determined by the channel vector, the equivalent beamforming vector, and the transmitted signal; S202, the false alarm probability is... The probability of a missed alarm is expressed as... It means that, respectively by and Given, among which and Let represent the binary decision made by the eavesdropper regarding whether the transmitter should transmit the message, and let represent the total detection error probability of the eavesdropper. for: (2) S203. In covert wireless communication, the eavesdropper minimizes its total detection error probability. The optimal detector is the likelihood ratio detector, and the corresponding likelihood ratio function is as follows: in, and They represent in Hypotheses and Under the assumption that, the observation is The likelihood function; and They represent in and Prior probabilities under the assumption; and These represent the decisions to send and receive data, respectively; when the likelihood ratio is greater than the decision threshold, the eavesdropper determines... If true, otherwise judge Established; S204, Based on the likelihood ratio function in S203, analyze the optimal detection threshold of the eavesdropper and the corresponding minimum detection error probability. ;consider The lower bound of is expressed by the formula: (4) Among them, It is a limited character input from arrive The upper bound of the Kullback-Leibler divergence; this inequality shows that the smaller the difference in distributions measured by divergence under the two hypotheses, the closer the minimum detection error probability that the eavesdropper can achieve is to 1, that is, the worse the detection performance, and the eavesdropper can hardly distinguish between the two hypotheses; therefore, in the design of covert communication systems, by constraining the Kullback-Leibler divergence to not exceed a certain small positive number. ,Right now This ensures that the minimum total detection error probability of the eavesdropper is satisfied. To achieve covert communication.

4. The method for optimizing covert wireless communication beamforming under finite character input as described in claim 3, characterized in that, The construction optimization problem described in S3, specifically Includes the following: In covert communication with finite character input, the objective is to maximize the total mutual information obtained at the receiver, for the The total mutual information of each channel at any given time. The expression is: in, Indicates the receiver at the Signals received on each channel ,symbol This represents additive white Gaussian noise at the receiving end, with a mean of zero and a variance of... ； Represents the first digit in a finite set of constellation points. The constellation point and the first Subtract the points from each constellation; using the total mutual information received at the receiver as a system performance metric, and under the premise of satisfying the concealment constraint and transmit power constraint, jointly optimize the transmitter power and beamforming vector at the transmitter, and couple them into a single variable. ， The optimization problem constructed is: Here, it is assumed that the channel is stationary and discrete with no memory, and that the components of the input vector are independent of each other; at this time, after... Mutual information after secondary channel use Represented as: Where C1 is the concealment constraint, here... express The upper bound of; where C2 is the power constraint of the transmitter under discrete input; because For unit norm beamforming vectors, This represents the actual power, and its value must not exceed the maximum power allowed by the system. 。 5. The method for optimizing covert wireless communication beamforming under finite character input as described in claim 4, characterized in that, The simplification and optimization problem described in S4 specifically includes the following: S401, For the objective function containing factors including about The expectation operation involves nested summations of exponential terms and logarithms within the inner layer. Its lower bound is derived using Jensen's inequality: S402. Using the integral expression of KL divergence, we perform inequality relaxation and expectation expansion to transform the original expression into an upper bound form that is easier to process. To this end, we derive its upper bound through the log-sum inequality: (8) So, after simplification, regarding The optimization problem is as follows: 。 6. The method for optimizing covert wireless communication beamforming under finite character input as described in claim 5, characterized in that, The solution to the optimization problem described in S5 specifically includes the following: S501. For the objective function of the optimization problem (9), the semidefinite relaxation SDR method is adopted, and all functions in the original problem are related to the objective function. Related quadratic forms are uniformly written as about The trace form; the objective function with respect to Given a concave function, maximizing a concave function is a convex optimization problem, which can be solved using the CVX toolkit. After processing the hidden constraints as described above, the following optimization problem is obtained: Problem (10) is equivalent to problem (9) and has the same optimal solution; S502, for optimization problem (10), after semidefinite relaxation, the optimization problem is a convex SDP problem with a convex objective function and a convex constraint set. It is solved by using the CVX toolkit in MATLAB to find the optimal matrix variables. S503. A two-stage low-complexity optimization strategy is proposed for optimization problem (9), which optimizes the power and beamforming vector separately. In the first stage, based on the maximum SLNR criterion, closed-form... The beamforming vector direction can be solved using a closed-form equation. In the second stage, based on the fixed beam vector, the original joint optimization problem is degenerated into a one-dimensional transmit power optimization problem and solved efficiently using a convex optimization method; the low-complexity optimization problem based on optimization problem (9) is as follows: The optimization problem is simplified into a convex optimization problem, namely, maximizing mutual information with respect to transmission power while satisfying concealment constraints and transmission power constraints. The problem is considered as a one-dimensional optimization problem with respect to transmit power when the beamforming vector is determined by a closed-form solution. This problem is solved numerically using the CVX toolbox in MATLAB to obtain the optimal transmit power. 。

Images