Optimization method, product and equipment for multi-collaborative monitoring underwater acoustic covert communication

Abstract

The application discloses a kind of transmission power optimization methods, products and equipment for multi-cooperative monitoring underwater acoustic covert communication, comprising: estimating when sending node sends preset symbol, each listening node listens to the received signal, according to received signal, the transmission power observed by each listening node is calculated, and then the decision statistics of each listening node is calculated;Select the fusion strategy corresponding to the underwater acoustic covert communication environment where it is located, fuse the decision statistics of each listening node to obtain joint decision statistics;Determine whether joint decision statistics meet preset condition, if yes, establish transmission power optimization model, the joint false alarm rate of multiple listening nodes cooperative monitoring is lowest as the goal of transmission power optimization model, with total communication throughput greater than or equal to preset threshold as constraint condition, transmission power is disturbed in each communication time slot Control;The optimal transmission power is obtained by solving.This application is applicable to underwater acoustic covert communication scene with multiple listening nodes.

Description

Technical Field

[0001] This invention relates to covert communication technology, and more particularly to a method, product, and equipment for optimizing the transmission power of underwater acoustic covert communication for multi-cooperative monitoring. Background Technology

[0002] In traditional wireless communication systems, covert communication refers to the process of transmitting information without being detected by the adversary. In the field of underwater communication, underwater acoustic communication has become the mainstream due to the ineffective propagation of electromagnetic waves. However, underwater acoustic communication faces many challenges, such as limited bandwidth, significant multipath effects, large propagation delays, and low channel reachability, making the design of covert communication more complex.

[0003] Current mainstream research on covert communication focuses on single-node (Willie) models in Gaussian noise backgrounds, such as adversarial optimization based on GLRT (Generalized Likelihood Ratio Detector) or energy detectors. In these scenarios, the transmitting node (Alice) is typically designed with a power scheduling strategy that is not detected by the eavesdropper, controlling the false detection probability or information rate to achieve the goal of covert communication.

[0004] However, in real-world underwater acoustic scenarios, multiple cooperative monitoring nodes (such as underwater acoustic sensor networks) are typically deployed. These Willie nodes can share observations, conduct joint statistical analysis, and coordinate threshold decisions, thereby significantly improving the ability to detect communication behavior. Furthermore, underwater background noise does not follow a Gaussian distribution but tends towards a non-Gaussian stable distribution, such as the stable α-stable distribution (SαS) noise model, which is sensitive to impact noise. In this environment, traditional single-point modeling methods, static threshold detection methods, and uniform power scheduling mechanisms will struggle to meet practical concealment requirements. Summary of the Invention

[0005] To address the problems existing in the prior art, the purpose of this invention is to provide a method, product, and device for optimizing the transmission power of underwater acoustic covert communication for multi-cooperative monitoring.

[0006] To achieve the above-mentioned objectives, the present invention provides the following technical solution:

[0007] A method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring includes the following steps:

[0008] (1) Estimate the received signal heard by each listening node when the transmitting node transmits the preset symbol. The received signal is estimated by considering the multipath interference between the transmitting node and the listening node, and the noise at the listening node follows an α-stable distribution.

[0009] (2) Based on the received signals monitored by each monitoring node, calculate the transmission power of the transmitting node when transmitting the preset symbol as observed by each monitoring node using the EM algorithm;

[0010] (3) Calculate the decision statistics of each listening node for the received signal based on the transmission power observed by each listening node;

[0011] (4) According to the correspondence between environment and fusion strategy, select the fusion strategy corresponding to the underwater acoustic covert communication environment, fuse the decision statistics of each monitoring node, and obtain the joint decision statistics of multi-monitoring node collaborative monitoring;

[0012] (5) Determine whether the joint decision statistic meets the preset conditions. If so, proceed to step (6).

[0013] (6) Establish a transmission power optimization model, wherein the transmission power optimization model aims to minimize the joint false detection rate of multi-monitoring node collaborative monitoring, with the total communication throughput being greater than or equal to a preset threshold as a constraint, and transmission power as a decision variable;

[0014] (7) Solve the transmission power optimization model to obtain the optimal transmission power.

[0015] Furthermore, the received signal monitored by the monitoring node in step (1) specifically includes:

[0016]

[0017] In the formula, This indicates that the listening node m is targeting the symbol The received signal that was monitored, This represents the main path gain. This represents the gain of the k-th multipath path. This represents the i-th symbol sent by the sending node. Let K represent the time delay of the k-th multipath path relative to the main path, where K represents the number of paths. This represents the noise at the monitoring node m. It follows an α-stable distribution.

[0018] Furthermore, step (2) specifically includes:

[0019] (2.1) For each listening node m, the initial transmit power estimate is set as follows: The number of iterations is s=1;

[0020] (2.2) In the s-th iteration, based on the previously estimated transmit power Calculate the posterior probability of the observed values ​​and construct the expectation function:

[0021]

[0022] Where z is a latent variable. , Let be the expected function. Indicates the transmit power variable. Represents a probability function. = This indicates the received signal of the monitoring node m; Represent the expected function;

[0023] (2.3) With the goal of maximizing the value of the constructed expectation function, solve for the transmit power in this iteration. Value:

[0024]

[0025] (2.4) If satisfied , If it is a threshold, then... The transmitted power observed by the listening node m Output; otherwise, s = s + 1, return to step (2.2).

[0026] Furthermore, the calculation expression for the decision statistic in step (3) is as follows:

[0027] ,

[0028]

[0029] In the formula, This is the decision statistic for monitoring node m. This refers to the detection statistics of the monitoring node m. This represents the noise power at monitoring node m. For multipath interference power, The transmit power observed by the listening node m, Main path gain, Indicates the underwater acoustic path loss factor. This represents the received signal detected by the monitoring node m, and N represents the total number of symbols counted. For the detection order, satisfying α is the characteristic index of the α-stable distribution that the noise at the monitoring node follows.

[0030] Furthermore, the calculation expression for the multipath interference power is as follows:

[0031]

[0032] In the formula, This represents the gain of the k-th multipath path, where K represents the number of multipath paths.

[0033] The formula for calculating the underwater acoustic path loss factor is as follows:

[0034]

[0035]

[0036] In the formula, The transmission energy loss is given when the communication distance is l and the signal frequency is f. This represents the underwater propagation attenuation coefficient. For the normalized reference distance, This represents the absorption loss coefficient.

[0037] Furthermore, step (4) specifically includes:

[0038] (4.1) Obtain the correspondence between the environment and the fusion strategy, wherein the correspondence between the environment and the fusion strategy is specifically as follows:

[0039] When the underwater acoustic covert communication environment is characterized by heterogeneous node channel states, a weighted fusion strategy is selected.

[0040] When the underwater acoustic covert communication environment has significant differences in local detection capabilities, the maximum value fusion strategy is selected.

[0041] When the underwater acoustic covert communication environment is a low-communication environment, the majority voting strategy is selected.

[0042] (4.2) Select a fusion strategy that corresponds to the underwater acoustic covert communication environment according to the correspondence between the environment and the fusion strategy;

[0043] (4.3) The decision statistics of each monitoring node are fused using a selected fusion strategy to obtain the joint decision statistics of multi-monitoring node collaborative monitoring:

[0044] When a weighted fusion strategy is selected, the joint decision statistic is:

[0045]

[0046] In the formula, For joint judgment statistics, To monitor the weight of node m, Here, M represents the decision statistics for monitoring node m, where M is the number of monitoring nodes.

[0047] When the maximum value fusion strategy is selected, the joint decision statistic is:

[0048]

[0049] When the majority voting strategy is chosen, the joint decision statistic is:

[0050]

[0051] In the formula, This indicates an indicator function; its value is 1 if the condition within the parentheses is true, and 0 otherwise. This is the decision threshold at the monitoring node m.

[0052] Furthermore, when the selected fusion strategy is a weighted fusion strategy or a maximum value fusion strategy, the preset condition is: , The threshold is used; when the selected fusion strategy is the majority voting strategy, the preset condition is: M represents the number of monitoring nodes.

[0053] Furthermore, the specific transmit power optimization model is as follows:

[0054]

[0055]

[0056]

[0057]

[0058]

[0059] In the formula, This represents the transmit power of the transmitting node in time slot t. This represents the joint false detection rate of all monitoring nodes within T time slots, where M represents the number of monitoring nodes. This represents the average false detection rate of monitoring node m within T time slots. It is the Gaussian Q-function. This represents the transmit power observed by the listening node m in time slot t. This represents the difference in statistical measures. Main path gain, Indicates the underwater acoustic path loss factor. Where N is the detection order, and N is the number of symbols counted. This represents the noise power at monitoring node m. The decision threshold for monitoring node m. Indicates multipath interference power. This represents the perturbation variable. This represents the power distribution density function after the perturbation. Indicates in Next Communication throughput per time slot.

[0060] A computer program product includes a computer program that, when executed by a processor, implements the above-described method.

[0061] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method described above.

[0062] Compared with existing technologies, the advantages of this invention are as follows: This invention achieves highly covert underwater acoustic communication in strong detection environments by combining a generalized likelihood ratio detector and a power perturbation mechanism under a multi-monitoring node collaborative detection underwater acoustic communication model. Compared with existing technologies, this invention enhances anti-detection capabilities while introducing a dynamic perturbation strategy (the communicating party actively introduces power interference), effectively weakening the accurate estimation of transmission power by the monitoring nodes, increasing the probability of false detection, and reducing the risk of detection. Furthermore, the joint optimization design of communication power and transmission rate further improves system throughput performance while ensuring covertness, possessing good practicality and robustness, and is suitable for various complex underwater communication scenarios. Attached Figure Description

[0063] Figure 1 This is a flowchart illustrating the underwater acoustic covert communication power optimization method for multi-cooperative monitoring environments provided in this embodiment of the invention.

[0064] Figure 2 This is a schematic diagram of the underwater acoustic covert communication environment provided in an embodiment of the present invention;

[0065] Figure 3 This is a schematic diagram of the transmission power calculation provided in an embodiment of the present invention;

[0066] Figure 4 This is a schematic diagram of the collaborative monitoring mechanism provided in an embodiment of the present invention;

[0067] Figure 5 This is a schematic diagram illustrating the effects of adding and not adding a disturbance mechanism provided in an embodiment of the present invention;

[0068] Figure 6 This is a schematic diagram of a computer device provided in an embodiment of the present invention. Detailed Implementation

[0069] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0070] Example 1

[0071] This invention proposes a power optimization method for underwater acoustic covert communication in a multi-cooperative monitoring environment, which combines key processes such as monitoring detection, perturbation modulation, and throughput optimization. Its process structure is as follows: Figure 1 As shown. The core idea of ​​this method is to perform perturbation control on the transmission power in each communication time slot while ensuring communication concealment, so as to break the eavesdropper's power estimation model and improve the system's false detection rate. At the same time, combined with throughput function modeling and constraint construction, the communication throughput is maximized while ensuring concealment.

[0072] This invention is applicable to underwater acoustic covert communication environments such as Figure 2 As shown, it includes the sending node Alice, the receiving node Bob, and multiple listening nodes Willie.

[0073] like Figure 1 As shown, the present invention specifically includes the following steps:

[0074] S101. Estimate the received signals heard by each listening node when the transmitting node transmits the preset symbol.

[0075] The received signal estimation takes into account multipath interference between the transmitting node and the listening node, and the noise at the listening node follows an α-stable distribution. The specific received signal is as follows:

[0076]

[0077] In the formula, This indicates that the listening node m is targeting the symbol The received signal that was monitored, This represents the main path gain. This represents the gain of the k-th multipath path. This represents the i-th symbol sent by the sending node. Let K represent the time delay of the k-th multipath path relative to the main path, where K represents the number of paths. This represents the noise at the monitoring node m. It follows an α-stable distribution, that is:

[0078]

[0079] In the formula, Let α represent the α-stable distribution, and α represent the characteristic index of the α-stable distribution. Indicates the scale parameter.

[0080] S102. Based on the received signals monitored by each monitoring node, calculate the transmission power of the transmitting node when transmitting the preset symbol as observed by each monitoring node using the EM algorithm.

[0081] like Figure 3 As shown, this step specifically includes:

[0082] (2.1) For each listening node m, the initial transmit power estimate is set as follows: The number of iterations is s=1;

[0083] (2.2) In the s-th iteration, based on the previously estimated transmit power Calculate the posterior probability of the observed values ​​and construct the expectation function:

[0084]

[0085] Where z is a latent variable. , Let be the expected function. Indicates the transmit power variable. Represents a probability function. ={ } indicates that the received signal of the listening node m; Represent the expected function;

[0086] (2.3) With the goal of maximizing the value of the constructed expectation function, solve for the transmit power in this iteration. Value:

[0087]

[0088] (2.4) If satisfied , If it is a threshold, then... The transmitted power observed by the listening node m Output; otherwise, s = s + 1, return to step (2.2).

[0089] S103. Based on the transmission power observed by each monitoring node, calculate the decision statistics for the received signal by each monitoring node.

[0090] In traditional Gaussian noise models, detectors typically use second-order moments (energy) as their statistical basis. However, in the context of this invention... In a stable distribution environment, the noise signal of the monitoring node has infinite variance, rendering the second-order moment statistical method inapplicable. To address the problem of non-Gaussian, heavy-tailed background noise, this invention employs the fractional lower-order moment (FLOM) method as the statistical basis for monitoring node detection. The FLOM statistic is defined as follows:

[0091] In the formula, This represents the received signal detected by the monitoring node m, and N represents the total number of symbols counted. For the detection order, satisfying α is the characteristic index of the α-stable distribution that the noise at the monitoring node follows.

[0092] In this invention, the listening node needs to determine whether communication activity exists based on the observed transmission power. To overcome the limitation of traditional energy detection relying on known power, this invention constructs a generalized likelihood ratio detector (GLRT) based on FLOM statistics for signal presence detection in covert communication scenarios.

[0093] The monitoring node collects data during each detection cycle. For each sample, the FLOM statistic is calculated, and then the likelihood ratio is constructed based on the following two hypothetical models:

[0094] 1) (Null hypothesis): There is no communication signal in the listening signal. ;

[0095] 2) (Alternative hypothesis): The monitored signal contains communication signals and noise. ,in This is a signal component.

[0096] Using the maximum likelihood estimation principle, the decision statistic of GLRT is constructed as follows:

[0097]

[0098] In the formula, This is the decision statistic for monitoring node m. This refers to the detection statistics of the monitoring node m. This represents the noise power at monitoring node m. For multipath interference power, The transmit power observed by the listening node m, Main path gain, This represents the underwater acoustic path loss factor.

[0099] The judgment rule is as follows:

[0100]

[0101] That is, when Greater than At that time, the judgment was That is, the listening signal contains a communication signal, when Less than At that time, the judgment was This means that the listening signal does not contain a communication signal.

[0102] In underwater acoustics, path loss has a crucial impact on the signal strength received by the monitoring node. Its physical characteristics determine the degree of signal attenuation during propagation. Therefore, this invention considers the propagation loss of the underwater acoustic path to establish a more accurate model. The underwater acoustic path loss model is as follows:

[0103]

[0104] In the formula, Transmission energy loss when the communication distance is l (in km) and the signal frequency is f (in kHz).

[0105] Considering the underwater acoustic environment, and taking into account both geometric spread loss and absorption loss, the transmission loss energy is modeled as follows:

[0106]

[0107] In the formula, This represents the underwater propagation attenuation coefficient. For the normalized reference distance, This represents the absorption loss coefficient.

[0108] The absorption loss coefficient is a frequency-dependent function that reflects the frequency-selective attenuation of signals in water due to heat dissipation, molecular vibrations, etc., and is defined as:

[0109]

[0110] Where f is in kHz.

[0111] The calculation expression for the multipath interference power is as follows:

[0112]

[0113] In the formula, This represents the gain of the k-th multipath path, where K represents the number of multipath paths.

[0114] S104. According to the correspondence between environment and fusion strategy, select the fusion strategy corresponding to the underwater acoustic covert communication environment, fuse the decision statistics of each monitoring node, and obtain the joint decision statistics of multi-monitoring node collaborative monitoring.

[0115] In this invention, to further improve the detection accuracy of the system in complex distributed eavesdropping environments and enhance the anti-detection capability of covert communication, a multi-eavesdropping node collaborative detection mechanism is proposed, such as... Figure 4 As shown, this mechanism achieves unified judgment and performance measurement of the transmitting end's communication behavior by fusing and judging the local detection statistics of each monitoring node and combining them with the system's false detection probability function.

[0116] The specific relationship between the environment and the fusion strategy is as follows: when the underwater acoustic covert communication environment is characterized by heterogeneous node channel states, a weighted fusion strategy is selected; when the underwater acoustic covert communication environment is characterized by significant differences in local detection capabilities, a maximum value fusion strategy is selected; and when the underwater acoustic covert communication environment is characterized by low communication conditions, a majority voting strategy is selected.

[0117] Based on the correspondence between environment and fusion strategy, a fusion strategy corresponding to the underwater acoustic covert communication environment is selected; the selected fusion strategy is used to fuse the decision statistics of each monitoring node to obtain the joint decision statistics of multi-monitoring node collaborative monitoring.

[0118] When a weighted fusion strategy is selected, the joint decision statistic is:

[0119]

[0120] In the formula, For joint judgment statistics, To monitor the weight of node m, Here, M represents the decision statistics for monitoring node m, where M is the number of monitoring nodes.

[0121] When the maximum value fusion strategy is selected, the joint decision statistic is:

[0122]

[0123] When the majority voting strategy is chosen, the joint decision statistic is:

[0124]

[0125] In the formula, This indicates an indicator function; its value is 1 if the condition within the parentheses is true, and 0 otherwise. This is the decision threshold at the monitoring node m.

[0126] S105. Determine whether the joint decision statistic meets the preset conditions. If so, proceed to step S106.

[0127] Wherein, when the selected fusion strategy is a weighted fusion strategy or a maximum value fusion strategy, the preset condition is: , The threshold is used; when the selected fusion strategy is the majority voting strategy, the preset condition is: M represents the number of monitoring nodes. If the joint decision statistics do not meet the preset conditions, steps (6) and (7) will not be executed, meaning that there is no need to optimize the transmission power.

[0128] S106. Establish a transmission power optimization model. The transmission power optimization model aims to minimize the joint false detection rate of multi-monitoring nodes, with the total communication throughput being greater than or equal to a preset threshold as a constraint, and transmission power as a decision variable. The transmission power is subjected to disturbance control in each communication time slot.

[0129] The specific transmit power optimization model is as follows:

[0130]

[0131]

[0132]

[0133]

[0134]

[0135] In the formula, This represents the transmit power of the transmitting node in time slot t. This represents the joint false detection rate of all monitoring nodes within T time slots, where M represents the number of monitoring nodes. This represents the average false detection rate of monitoring node m within T time slots. It is the Gaussian Q-function. This represents the transmit power observed by the listening node m in time slot t. This represents the difference in statistical measures. Main path gain, Indicates the underwater acoustic path loss factor. Where N is the detection order, and N is the number of symbols counted. This represents the noise power at monitoring node m. The decision threshold for monitoring node m. Indicates multipath interference power. This represents the perturbation variable. This represents the power distribution density function after the perturbation. Indicates in Next Communication throughput per time slot.

[0136] This invention introduces a power perturbation mechanism, which applies minute perturbations to the transmit power in each communication time slot, reducing the detection accuracy of the listening node's transmission behavior and thus improving the stealth of the communication system. Based on this, a power scheduling optimization problem is constructed to maximize the overall throughput of the system while satisfying the stealth constraint. The variable is a small, near-zero mean perturbation, the purpose of which is to disrupt the power convergence accuracy of the eavesdropper in the EM estimation algorithm and reduce the effectiveness of its detector construction.

[0137] Under the aforementioned interference conditions, the Fisher information constructed by the listening node in time slot t will decrease significantly, as expressed mathematically:

[0138] in To monitor the noise power at the listening node. The lower the information level, the worse the estimation accuracy, which ultimately leads to an increase in the overall false detection rate of the system, thus helping to achieve the goal of ensuring the system's communication concealment.

[0139] In the constraints of the model, in the first... The communication throughput per time slot is:

[0140]

[0141] Among them, the communication channel bandwidth is The monitoring noise power is .

[0142] S107. Solve the aforementioned transmission power optimization model to obtain the optimal transmission power.

[0143] To verify the detection and countermeasure performance of this invention in covert communication scenarios. Figure 5 The simulation results of the detection error probability comparison using MATLAB are presented. The horizontal axis represents the transmission power, and the vertical axis represents the joint detection false alarm probability. The data is displayed using a logarithmic coordinate system to facilitate observation of the changing trends in the low-probability region.

[0144] In the figure, the solid blue line represents the false alarm probability of the monitoring node (Willie) detecting communication signals without using the perturbation strategy proposed in this invention; while the dashed red line represents the false alarm probability that Willie can achieve under the power perturbation mechanism proposed in this invention. It is clear that, under the same transmit power conditions, the perturbation mechanism introduced in this invention significantly reduces Willie's detection performance, resulting in a higher joint false detection rate, and maintaining a significant interval with power variations.

[0145] The simulation results verify the effectiveness of the perturbation strategy proposed in this invention in interfering with the statistical distribution of the monitoring node detector. Even in the face of multi-node joint detection scenarios, it can still effectively improve the concealment of the communication system.

[0146] This invention provides a covert communication method suitable for underwater acoustic background noise conditions. It employs an adversarial analysis framework based on FLOM detection, combined with power perturbation design and channel uncertainty enhancement techniques, effectively disrupting the detector statistical characteristics of the eavesdropping party. Simulation results further verify that this strategy still possesses excellent covert communication countermeasure capabilities under multi-eavesdropping node collaborative detection, providing theoretical support and implementation methods for covert information transmission in underwater or near-ground communication scenarios.

[0147] Example 2

[0148] This invention also provides a computer product, such as an app on a mobile phone or tablet, or an installer on a computer. This product includes a computer program / instructions that, when executed by a processor, implement the method described in Embodiment 1. The code for a computer-executable program that performs the operations of this invention can be written in one or more programming languages ​​or a combination thereof. Programming languages ​​include object-oriented programming languages ​​such as Java, Smalltalk, and C++, as well as conventional procedural programming languages ​​such as C or similar languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computer (e.g., via the Internet using an Internet service provider).

[0149] Example 3

[0150] Figure 6 This is a schematic diagram of the structure of a computer device provided in an embodiment of the present invention. The embodiments of the present invention provide services for implementing the method of the first embodiment of the present invention described above. Figure 6 As shown, the device may include: a memory 301 storing a computer-executable program; a processor 302 coupled to the memory 301; the processor 302 calls the computer-executable program stored in the memory 301 to perform the steps in the method described in Embodiment 1.

[0151] Memory 301 may include computer system readable media in the form of volatile memory, such as random access memory (RAM) and / or cache memory. The device may further include other removable / non-removable, volatile / non-volatile computer system storage media. By way of example only, memory 301 may be used to read and write non-removable, non-volatile magnetic media (commonly referred to as a "hard disk drive"). A program / utility having a set (at least one) of program modules may be stored, for example, in memory 301. Such program modules include, but are not limited to, an operating system, one or more application programs, other program modules, and program data. Each or some combination of these examples may include an implementation of a network environment. The computer-executable program of the program modules typically performs the functions and / or methods described in the embodiments of the present invention.

[0152] The processor 302 executes various functional applications and data processing by running programs stored in the memory 301, such as implementing the method provided in Embodiment 1 of the present invention.

[0153] The code of a computer executable program can be written in one or more programming languages ​​or a combination thereof. Programming languages ​​include object-oriented programming languages ​​such as Java, Smalltalk, and C++, as well as conventional procedural programming languages ​​such as the "C" language or similar programming languages.

[0154] It should be understood that the embodiments and descriptions above are only the principles, main features and advantages of the present invention. Various changes and modifications can be made to the present invention without departing from the spirit and scope of the invention, and all such changes and modifications fall within the protection scope of the present invention.

Claims

1. A method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring, characterized in that, Includes the following steps: (1) Estimate the received signal heard by each listening node when the transmitting node transmits the preset symbol. The received signal is estimated by considering the multipath interference between the transmitting node and the listening node, and the noise at the listening node follows an α-stable distribution. (2) Based on the received signals monitored by each monitoring node, calculate the transmission power of the transmitting node when transmitting the preset symbol as observed by each monitoring node using the EM algorithm; (3) Calculate the decision statistics of each listening node for the received signal based on the transmission power observed by each listening node; (4) According to the correspondence between environment and fusion strategy, select the fusion strategy corresponding to the underwater acoustic covert communication environment, fuse the decision statistics of each monitoring node, and obtain the joint decision statistics of multi-monitoring node collaborative monitoring; (5) Determine whether the joint decision statistic meets the preset conditions. If so, proceed to step (6). (6) Establish a transmission power optimization model. The transmission power optimization model aims to minimize the joint false detection rate of multi-monitoring nodes, with the total communication throughput being greater than or equal to a preset threshold as a constraint, and transmission power as a decision variable. The transmission power is perturbed and controlled in each communication time slot. (7) Solve the aforementioned transmission power optimization model to obtain the optimal transmission power; The expression for calculating the decision statistic in step (3) is as follows: ， ， In the formula, This is the decision statistic for monitoring node m. This refers to the detection statistics of the monitoring node m. This represents the noise power at monitoring node m. For multipath interference power, The transmit power observed by the listening node m, Main path gain, Indicates the underwater acoustic path loss factor. This represents the received signal detected by the monitoring node m, and N represents the total number of symbols counted. For the detection order, satisfying α is the characteristic index of the α-stable distribution that the noise at the monitoring node follows; Step (4) specifically includes: (4.1) Obtain the correspondence between the environment and the fusion strategy, wherein the correspondence between the environment and the fusion strategy is specifically as follows: When the underwater acoustic covert communication environment is characterized by heterogeneous node channel states, a weighted fusion strategy is selected. When the underwater acoustic covert communication environment has significant differences in local detection capabilities, the maximum value fusion strategy is selected. When the underwater acoustic covert communication environment is a low-communication environment, the majority voting strategy is selected. (4.2) Select a fusion strategy that corresponds to the underwater acoustic covert communication environment according to the correspondence between the environment and the fusion strategy; (4.3) The decision statistics of each monitoring node are fused using a selected fusion strategy to obtain the joint decision statistics of multi-monitoring node collaborative monitoring: When a weighted fusion strategy is selected, the joint decision statistic is: ， In the formula, For joint judgment statistics, To monitor the weight of node m, Here, M represents the decision statistics for monitoring node m, where M is the number of monitoring nodes. When the maximum value fusion strategy is selected, the joint decision statistic is: ， When the majority voting strategy is chosen, the joint decision statistic is: ， In the formula, This indicates an indicator function; its value is 1 if the condition within the parentheses is true, and 0 otherwise. This is the decision threshold at the monitoring node m.

2. The method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring according to claim 1, characterized in that, The received signal monitored by the monitoring node in step (1) is specifically as follows: ， In the formula, This indicates that the listening node m is targeting the symbol The received signal that was monitored, Indicates the main path gain. This represents the gain of the k-th multipath path. This represents the i-th symbol sent by the sending node. Let K represent the time delay of the k-th multipath path relative to the main path, where K represents the number of paths. This represents the noise at the monitoring node m. obey Stable distribution.

3. The method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring according to claim 1, characterized in that: Step (2) specifically includes: (2.1) For each listening node m, the initial transmit power estimate is set as follows: The number of iterations is s=1; (2.2) In the s-th iteration, based on the previously estimated transmit power Calculate the posterior probability of the observed values ​​and construct the expectation function: ， Where z is a latent variable. , Let be the expected function. Indicates the transmit power variable. Represents a probability function. = This indicates the received signal of the monitoring node m; Represent the expected function; (2.3) With the goal of maximizing the value of the constructed expectation function, solve for the transmit power in this iteration. Value: ， (2.4) If satisfied , If it is a threshold, then... The transmitted power observed by the listening node m Output; otherwise, s = s + 1, return to step (2.2).

4. The method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring according to claim 1, characterized in that: The calculation expression for the multipath interference power is as follows: ， In the formula, This represents the gain of the k-th multipath path, where K represents the number of multipath paths. The formula for calculating the underwater acoustic path loss factor is as follows: ， ， In the formula, The transmission energy loss is given when the communication distance is l and the signal frequency is f. This represents the underwater propagation attenuation coefficient. For the normalized reference distance, This represents the absorption loss coefficient.

5. The method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring according to claim 1, characterized in that: When the selected fusion strategy is a weighted fusion strategy or a maximum value fusion strategy, the preset condition is: , The threshold is used; when the selected fusion strategy is the majority voting strategy, the preset condition is: M represents the number of monitoring nodes.

6. The method for optimizing transmit power in underwater acoustic covert communication for multi-cooperative monitoring according to claim 1, characterized in that: The specific transmit power optimization model is as follows: ， ， ， ， ， In the formula, This represents the transmit power of the transmitting node in time slot t. This represents the joint false detection rate of all monitoring nodes within T time slots, where M represents the number of monitoring nodes. This represents the average false detection rate of monitoring node m within T time slots. It is the Gaussian Q-function. This represents the transmit power observed by the listening node m in time slot t. This represents the difference in statistical values. Main path gain, Indicates the underwater acoustic path loss factor. Where N is the detection order, and N is the number of symbols counted. This represents the noise power at monitoring node m. The decision threshold for monitoring node m. Indicates multipath interference power. This represents the perturbation variable. This represents the power distribution density function after the perturbation. Indicates in Next Communication throughput per time slot.

7. A computer program product comprising a computer program / instructions, characterized in that: When the computer program / instructions are executed by the processor, they implement the method of any one of claims 1-6.

8. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: The processor executes the computer program to implement the method as described in any one of claims 1-6.

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