A frequency division covert communication method and system based on random frequency diversity array

Abstract

The application provides a frequency division concealment communication method and system based on a random frequency diversity array, and belongs to the technical field of concealment communication, and comprises the following steps: a random frequency diversity array concealment communication system model is established, the concealment communication system model adopts a frequency division strategy to divide a limited total bandwidth of a concealment user into a plurality of frequency subbands; a signal received by a receiving user in a frequency subband is constructed based on the concealment communication system model, and a concealment rate from the concealment user to the receiving user is calculated; noise uncertainty is used to analyze the distribution of noise power received by a monitoring user, and an overall false detection probability is obtained; under the condition of meeting a system concealment constraint, an optimal false detection probability of the monitoring user is obtained according to an optimal detection threshold; and a double-layer optimization method is adopted to optimize the frequency division strategy and power distribution, so that the maximum concealment rate is realized. The application effectively reduces the detection performance of a listener.

Description

Technical Field

[0001] This invention belongs to the field of covert communication technology, and particularly relates to a frequency division covert communication method and system based on a random frequency diversity array. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] In recent years, the demand for secure and covert data transmission has been growing across various fields, leading to a shift in communication security from traditional physical layer protection to covert communication. Covert communication embeds secret signals into environmental noise or legitimate traffic, making them statistically indistinguishable from noise. Random Frequency Diversity Array (RFDA) is an advanced technology applied to covert communication to enhance its performance in complex environments.

[0004] RFDA (Rapid Field Testing) is used in covert communications to accurately analyze and characterize user detection behavior (such as false alarm and false negative probabilities) and determine the fundamental limits of covertness. Using RFDA to achieve sharp beamforming and good range-angle decoupling characteristics requires a wider variety of selectable transmission frequencies, thus consuming significant bandwidth. Furthermore, when facing different communication or covert needs, directly using RFDA within limited bandwidth may negatively impact its performance due to the limited bandwidth resources, resulting in low covert efficiency. Summary of the Invention

[0005] To overcome the shortcomings of the prior art, this invention proposes a frequency division covert communication method and system based on random frequency diversity array, in order to solve the problem that bandwidth resources affect random frequency diversity array, leading to a decrease in covert efficiency.

[0006] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions: In a first aspect, the present invention discloses a frequency division multiplexing (FDM) covert communication method based on a random frequency diversity array, comprising: A model of a covert communication system using a random frequency diversity array is established. The covert communication system model uses a frequency division strategy to divide the covert user into several frequency sub-bands within a limited total bandwidth. Based on the aforementioned covert communication system model, the signal received by the receiving user in the frequency sub-band is constructed, and the covertness rate from the covert user to the receiving user is calculated. The distribution of noise power received by the monitoring user is analyzed by utilizing noise uncertainty to obtain the overall false detection probability. Under the condition of satisfying the system concealment constraint, the optimal false detection probability of the monitoring user is obtained according to the optimal detection threshold. A two-layer optimization method is used to optimize the frequency division strategy and power allocation to achieve maximum concealment rate.

[0007] A further technical solution is that the signal received by the user in the frequency sub-band is:

[0008]

[0009]

[0010]

[0011] in, This represents the Hermitian transpose operation; This represents the turning vector from the hidden user to the receiving user; For the first Covert users transmitting signals in each frequency sub-band; Indicates the first The transmission power of concealed users in the sub-band; Indicates the beamforming vector; It is the first one that satisfies the average power constraint. Information signals in each frequency sub-band; To reduce the path loss between receiving users and hidden users; This represents additive white Gaussian noise.

[0012] A further technical solution involves calculating the concealment rate from the concealed user to the receiving user, specifically as follows:

[0013] in, The limited bandwidth occupied by the transmitting end for transmitting signals; The number of frequency divisions; Let be the signal-to-noise ratio of the communication between the receiving user and the covert user in the i-th subband; Indicates the first The transmission power of concealed users in the sub-band; To reduce the path loss between receiving users and hidden users; It is the power spectral density at the receiving user.

[0014] A further technical solution is that the overall error detection probability is used to measure the detection performance of the monitoring user, and the expression is: The optimal error detection probability for monitoring users is:

[0015] in, It is a complete gamma function; , These are the lower and upper limits of the noise power range; The shape parameter of the incomplete gamma function; It is an incomplete gamma function; The scaling parameter of the incomplete gamma function.

[0016] A further technical solution involves employing a two-layer optimization method to optimize the frequency division strategy and power allocation, jointly optimizing the number of frequency divisions and the power allocation of each frequency band within the frequency band where the covert user is located, thereby maximizing the achievable covertness rate while satisfying the covertness constraint condition, which is that the optimal error detection probability of the eavesdropping user is not less than a specified threshold.

[0017] A further technical solution involves a two-layer structure for joint optimization. In the outer layer, the number of frequency sub-bands is traversed. In the inner layer, a composite optimization structure including total transmit power and power allocation weight vectors is employed. For a given total transmit power, the interior-point method is used to obtain the optimal power allocation vector to maximize the concealment rate while satisfying the concealment constraint. By comparing the upper and lower limits of the total transmit power values ​​for the maximum concealment rate, a binary search method is used to narrow down the feasible range of the total transmit power. After obtaining the new range of total transmit power, the above operations are performed alternately, and the maximum acceptable transmission power that satisfies the concealment constraint is gradually determined. Finally, by comparing the concealment rates obtained under all frequency sub-bands, the optimal parameter triplet is selected.

[0018] A further technical solution, the joint optimization, is expressed as:

[0019] in, Concealment rate; To determine the optimal error detection probability for monitoring users; The number of frequency divisions; Assign a weight vector to the power; Indicates the required level of concealment; This limits the maximum number of frequency dividers. Allocate power ratios for each sub-band; Maximum transmission power; This represents the total transmission power.

[0020] Secondly, this invention discloses a frequency division multiplexing (FDM) covert communication system based on a random frequency diversity array, comprising: The model building module is configured to: establish a model of a covert communication system using a random frequency diversity array, wherein the covert communication system model uses a frequency division strategy to divide the covert user into several frequency sub-bands within a limited total bandwidth; The covert calculation module is configured to: construct the signal received by the receiving user in the frequency subband based on the covert communication system model and calculate the covert rate from the covert user to the receiving user. The noise analysis module is configured to: analyze the distribution of noise power received by the monitoring user using noise uncertainty, obtain the overall false detection probability, and obtain the optimal false detection probability of the monitoring user based on the optimal detection threshold under the condition of satisfying the system concealment constraint; The joint optimization module is configured to optimize the frequency division strategy and power allocation using a two-layer optimization method to achieve maximum concealment rate.

[0021] Thirdly, the present invention discloses an electronic device, including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when run by the processor, complete the steps of the frequency division covert communication method based on random frequency diversity array described above.

[0022] Fourthly, the present invention discloses a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the frequency division covert communication method based on a random frequency diversity array described above.

[0023] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention divides the total bandwidth into multiple subbands, each employing RFDA technology independently. Analysis reveals that the false detection probability of a listening user varies significantly with the number of subbands and their location, verifying a non-monotonic relationship between the false detection probability and the number of subbands. By jointly optimizing the number of subbands and power allocation, the proposed scheme achieves higher concealment and, compared to traditional RFDA methods, enables a more adaptive system. This performance improvement stems from finer frequency control among antenna elements to reduce the received signal strength at the listening user's location, thereby simultaneously increasing transmission power and maintaining concealment.

[0024] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0025] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0026] Figure 1 This is a schematic diagram of the random frequency diversity array covert communication system model described in Embodiment 1 of the present invention.

[0027] Figure 2 This is a graph showing the variation of the optimal error detection probability at different positions as described in Embodiment 1 of the present invention.

[0028] Figure 3 This is a graph showing the variation of the optimal error detection probability for different frequency division numbers as described in Embodiment 1 of the present invention.

[0029] Figure 4 This is a diagram showing the results of the fixed total bandwidth joint optimization described in Embodiment 1 of the present invention. Detailed Implementation

[0030] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0031] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.

[0032] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0033] Terminology Explanation: Covert communication: By embedding secret signals into ambient noise or legitimate traffic, making them statistically indistinguishable from noise. Unlike traditional methods that only protect the message content, it hides the existence of the transmission itself, thus providing greater security.

[0034] Random Frequency Diversity Array (RFDA): This method is promising because it can jointly focus signal energy in the angular and range domains through random frequency offsets between antennas. By randomly assigning carrier frequencies to the transmit antennas, RFDA generates a frequency-dependent channel between the transmitter and receiver.

[0035] Unlike traditional methods that only protect the message content, covert communication hides the existence of the transmission itself, thus providing greater security. Due to this advantage, it has attracted widespread attention, and various advanced technologies, such as Random Frequency Diversity Array (RFDA), have been developed to enhance its performance in complex environments.

[0036] RFDA (Radio Frequency Adaptor) is a promising technique for achieving focused beamforming through random frequency offsets between antennas in both the angular and range domains. By randomly assigning carrier frequencies to the transmit antennas, RFDA generates a frequency-dependent channel between the transmitter and receiver. When combined with directional modulation (DM), desired receivers experience constructive signal enhancement, while unintentional receivers observe noise-like interference. This characteristic significantly reduces the probability of detection by the user, thereby improving the stealth of communication.

[0037] Over time, RFDA has expanded from physical layer security to the field of covert communications. For example, several studies have combined RFDA with directional modulation and artificial noise to improve confidential throughput and enhance the robustness of physical layer security schemes.

[0038] However, as mentioned in the background section, the performance of RFDA is affected by limited bandwidth. Therefore, the existing technology of directly applying RFDA to covert communication suffers from low covert efficiency.

[0039] Example 1 In one or more embodiments, a frequency division multiplexing (FDM) covert communication method based on a random frequency diversity array is disclosed. This method combines the effective beamforming effect of RFDA's angle-range decoupling with the high spectral efficiency of FDM to further reduce the detection performance of eavesdroppers while improving the transmission rate of covert information. The method includes the following steps: Step S1: Establish a model of a covert communication system using a random frequency diversity array. The covert communication system model uses a frequency division strategy to divide the covert user into several frequency sub-bands within a limited total bandwidth.

[0040] The covert communication system model proposed in this embodiment, such as Figure 1 As shown, the sender uses Random Frequency Diversity Array (RFDA) technology to conceal the user. This system employs a frequency division strategy, allowing the concealed user to operate within a limited total bandwidth. Internal transmission signal. This total bandwidth is divided equally into... Each frequency sub-band. Within each frequency sub-band, RFDA technology independently provides covert services for users. Each antenna is assigned a different transmission frequency. This shows that the same antenna can simultaneously transmit multiple orthogonal signals from different frequency sub-bands.

[0041] Below, we will first analyze the signal transmission within a single frequency sub-band, and then expand the discussion to all sub-bands. The covert user is in the... The first frequency sub-band The signal frequency used by each antenna is: (1) in, It is the first The center carrier frequency of each frequency sub-band It is the frequency increment. Let be the random mapping rule on the i-th subband, a random variable that determines the transmission frequency of different antennas and is different in each frequency subband. This work considers a uniform linear array (ULA) with at the concealed user location. The center of the ULA is set as a reference element. It is assumed that the location of the receiving user is known, and the location of the listening user is also known. In this invention, it is assumed that the receiving user is in the far field of the transmitter, therefore the first... The distance between the antenna and the receiver is used as... This can be approximated as: (2) in, Indicates the angle from the center of the ULA to the receiver. This indicates the distance from the center of the ULA to the receiver. It is the spacing between two adjacent ULA elements. The antenna position sequence can be represented as .

[0042] Considering path loss patterns, from concealed user equipment with RFDA functionality to locations at coordinates The channel vector of the receiver is: (3) in, For path loss, and ,in It is a constant that depends on the antenna characteristics and the attenuation at a reference distance of 1 meter. It is the path loss index. This indicates the transpose operation. The phase offset of the nth antenna element is defined as:

[0043] in, The phase offset of the nth antenna element; Phase offset for reference antenna element; The random mapping rule for the nth antenna; The center frequency.

[0044] because ,in, This represents the maximum value that can be taken in a random mapping. The limited bandwidth occupied by the transmitting end for transmitting signals. Therefore, It can be approximated as: (4) in, It is the speed of light.

[0045] The signal transmitted from the covert user to the receiving user is: (5) Since signals transmitted on the same antenna are orthogonal, these signals are divided into... Group, and determine the channel configuration of each sub-band, so the first The signals transmitted by covert users in each frequency sub-band are: (6) in, It is the first one that satisfies the average power constraint. Information signals in each frequency sub-band Indicates the first The transmit power of concealed users in the sub-band, and This represents the beamforming vector. To maximize the signal-to-noise ratio (SNR) of the receiving user, The expression is: (7) in, This represents the turning vector from the hidden user to the receiving user.

[0046] Step S2: Based on the covert communication system model, construct the signal received by the receiving user in the frequency sub-band and calculate the covert rate from the covert user to the receiving user.

[0047] Formula (8) is derived from formulas (3), (6) and (7), which receives the user's message in the first... The signals received in each frequency sub-band are: (8)

[0048]

[0049]

[0050] in, This represents the Hermitian transpose operation, therefore according to equation (3), ,in, To receive the angle from which the user is located, which is relatively concealed; To receive the user's relatively concealed distance; To reduce the path loss between receiving users and hidden users; This represents additive white Gaussian noise (AWGN), i.e. According to equation (8), the receiving user can recover the information signal from the concealed user. without needing to know the random mapping rules (i.e. This is because the covert user used a channel reversal strategy called Maximum Ratio Transmission (MRT), i.e. It cancels out the random mapping rule of the sender (i.e. The random channel caused by ). According to equation (9), the receiving user in the first The signal-to-noise ratio (SNR) in each frequency sub-band is: (9) in, It is the power spectral density at the receiving user, and This represents the noise power within a single frequency subband. Assuming RFDA is applied across all frequency bands, the concealment rate from the concealed user to the receiving user can be expressed as: (10) in, Let be the signal-to-noise ratio of the communication between the receiving user and the covert user in the i-th subband.

[0051] Step S3: Analyze the distribution of noise power received by the monitoring user using noise uncertainty to obtain the overall error detection probability. Under the condition of satisfying the system concealment constraint, obtain the optimal error detection probability of the monitoring user according to the optimal detection threshold.

[0052] This embodiment utilizes noise uncertainty and the randomness introduced to the channel by frequency division and RFDA to conceal the existence of communication, specifically: The signals received by the user are represented in the following composite model.

[0053] (11) in, This indicates the assumption that the covert user does not communicate with the receiving user, i.e., an invalid assumption; This indicates the scenario where a hidden user sends a signal, i.e., the alternative hypothesis; The additive white Gaussian noise representing the user being monitored has the following distribution: ; From hidden users to eavesdropping users in the first stage The steering vector for each sub-band is obtained by... Replace with This is derived from equation (3), where, To monitor the user's location from a relatively concealed angle; To monitor users at a relatively concealed distance.

[0054] In the assumption The signal received by the user can be represented as follows: (12) Therefore, the received power of the monitored user is given as follows: (13) in, This indicates the signal power received at the user's location. This represents the noise power in the channel from the covert user to the eavesdropping user.

[0055] Subsequently, the distribution of noise power received by the monitoring user was analyzed. In reality, the noise power observed at the monitoring user's location may fluctuate due to environmental changes and random interference. To reflect this impact, a bounded noise uncertainty model was adopted for the monitoring user, where the noise power... In the interval It is uniformly distributed within. Therefore, The probability density function (pdf) is: (14) in, , These are the lower and upper limits of the noise power range, and , ; Indicates nominal noise power. This indicates the degree of noise uncertainty.

[0056] In this embodiment, the monitoring user uses a radiometer as a detector to determine whether the covert user is transmitting a signal. This corresponds to the hypothetical scenario. and According to the Niemann-Pearson criterion, the likelihood ratio test (LRT) is used to minimize the false detection probability of the eavesdropping user, and its expression can be written as: (15) in, For the detection threshold, and These represent the listening user's decision regarding whether the hidden user sent or did not send the message.

[0057] The overall error detection probability is used to measure the detection performance of the monitoring user, and it is defined as follows: (16) in, and These represent the false alarm probability and the false negative probability, respectively. In this embodiment, it is assumed that the prior probabilities are equal, i.e. .

[0058] To ensure stealth, let's assume the eavesdropping user is a powerful probe who can choose the optimal detection threshold. Let its false detection probability (denoted as) This is reduced to a minimum. Therefore, the concealment constraint can be expressed as: (17) in, This indicates the required level of concealment. This condition ensures that, even under optimal detection conditions, the optimal false detection probability of the eavesdropping user will not be less than [a certain value]. .

[0059] To further improve the calculation of user error detection probability. The expression for the false alarm probability is first given by... Describe it. Under the assumption... Under these conditions, regardless of changes in the transmission strategy of the covert user, the noise power remains unaffected; therefore, for a given detection threshold... Monitor the probability of false alarms for users Remain unchanged: (18) Subsequently, it was deduced that The expression requires analysis under the assumptions Down The probability density function (pdf) is: (19) Based on the decision-making rules mentioned above, utilizing... Assuming The probability density function can be derived from this. The value is: (20) so, as follows: (twenty one) (twenty two) (twenty three) In order to find a way to Reaching the minimum value Value, pair Differentiate: (twenty four) according to The derivative shows that the false detection probability Firstly, with It decreases as it increases, then increases again, and... It reaches its minimum value at that time. Therefore, it will... Substitution In this study, the optimal error detection probability for monitoring users was obtained. : (25) in, It is a complete gamma function; The shape parameter of the incomplete gamma function; It is an incomplete gamma function; The scaling parameter of the incomplete gamma function.

[0060] This invention first divides the total available bandwidth into multiple sub-bands, each supporting independent RFDA signal transmission. A comprehensive analysis is conducted, assuming the eavesdropping user is located in a known position near the sender's concealed user, to determine the optimal detection threshold for the eavesdropping user and the corresponding minimum false detection probability.

[0061] Step S4: Optimize the frequency division strategy and power allocation using a two-layer optimization method to achieve maximum concealment rate.

[0062] For a fixed number of frequency divisions ( (Value), which can increase the concealment rate Power allocation strategies that achieve maximum power are typically associated with maximizing the probability of false detection. Different strategies lead to different maximum values. Therefore, this invention proposes an optimal power allocation strategy that appropriately balances these conflicting objectives, thereby promoting joint optimization of frequency division and power allocation to achieve maximum stealth rate. .

[0063] This embodiment constructs an optimization problem aimed at jointly optimizing the number of frequency partitions. This involves power allocation across frequency bands within the frequency band where the covert user is located, with the goal of maximizing the achievable covertness while satisfying covertness constraints. Specifically, this constraint requires the optimal false detection probability for the eavesdropping user. It is not less than a specified threshold. Therefore, the optimization problem is formulated as: (26) because and power allocation vector Optimal error detection probability for monitoring users It appears as a parameter of the gamma function, therefore With optimal concealment rate The relationships between them become so complex that they cannot be described by closed-form analytical expressions. Therefore, numerical methods are used to jointly optimize them. and The goal is to maximize while satisfying the concealment constraint. .

[0064] The optimization procedure proposed in this embodiment adopts a two-layer structure. In the outer layer, the number of frequency divisions is determined. For a given The power allocation vector is directly optimized in the inner layer. It is computationally inefficient because and or There is no obvious monotonic relationship between them. To address this issue, the power allocation vector... Reparameterized to total transmit power and the normalized power allocation weight vector for each frequency sub-band .

[0065] It is worth noting that for fixed and In terms of achievable concealment rate along with The probability increases monotonically with the increase of the optimal error detection probability. Then with The value increases and then monotonically decreases. Utilizing this monotonicity, a dichotomy can be used to effectively reduce the value. This greatly accelerates the convergence speed of the optimization process. Therefore, the original problem can be reformulated into an equivalent but more tractable form, expressed as: (27) in, This limits the maximum number of frequency dividers. Allocate power ratios for each sub-band; This is the maximum limited transmission power.

[0066] To solve the problem This embodiment proposes an iterative algorithm that combines traversal and binary search methods to determine the number of frequency divisions. Total transmission power Normalized power allocation weight vector of frequency sub-bands Perform a search.

[0067] Furthermore, the proposed optimization scheme follows a two-layer process. In the outer layer, the number of frequency partitions is traversed. In the inner layer, a method is employed that incorporates the total transmit power. and power allocation weight vector The composite optimized structure. For a given The optimal power allocation vector can be obtained using the interior-point method or other suitable optimization algorithms. Its goal is to maximize concealment. Simultaneously satisfying the concealment constraint By comparing the maximum concealment rate The upper and lower limits The value is narrowed down using a binary search method. The feasible range. In obtaining new... After defining the range, perform the above operations alternately, and gradually determine whether the conditions are met. Maximum acceptable transmission power Finally, by comparing all frequency subbands... The concealment rate obtained below Select the optimal parameter triplet .

[0068] In the system model, achieving the maximum concealment rate under confidentiality constraints requires optimizing the frequency division strategy, i.e., determining the optimal number of frequency bands. Furthermore, using frequency division introduces more degrees of freedom in power allocation across different frequency band sub-bands, providing significant optimization potential. This power allocation plays a crucial role in determining the achievable concealment rate.

[0069] Through the above technical solution, this invention discovers that the false detection probability of a listening user varies with the number of sub-bands and their spatial location. The change in the number of sub-bands significantly impacts the detection performance of the listening user in different locations, and the frequency division strategy design also provides room for optimization of the transmit power allocation in each sub-band. Therefore, by precisely optimizing the number of frequency bands and the power allocation of each sub-band based on the location of the listening user, high concealment is achieved while maximizing the transmission rate with legitimate receivers. Ultimately, the design proposed in this invention achieves a higher concealment rate than the traditional non-frequency-division RFDA scheme, providing a more flexible and spectrally efficient framework for next-generation covert communication systems.

[0070] The joint optimization framework proposed in this embodiment explicitly captures the inherent coupling relationship between the number of sub-bands, total transmit power, and power allocation weight vector. Furthermore, the outer traversal of the number of sub-bands ensures the completeness of the optimization process, enabling the selection of parameter configurations that maximize the achievable stealth rate. The inner composite structure further reduces search complexity by leveraging the interdependencies between some parameters, thereby significantly improving convergence speed and solution accuracy. Therefore, this joint optimization algorithm provides a complete and effective solution to the proposed optimization problem.

[0071] As one implementation method, further elaborating on the details of the invention, several practical factors must be considered during the simulation parameter design process, including spectrum allocation, antenna spacing, and transmission frequency distribution among antenna elements. The corresponding design choices are summarized below: First, when dividing the available system bandwidth into multiple subbands, a guard band accounting for 5% of the total bandwidth is inserted between adjacent subbands. This design can mitigate interference between adjacent subbands caused by spectral leakage and sidelobe effects, thereby ensuring a certain degree of independence for signal transmission in different frequency bands.

[0072] Secondly, the spacing between adjacent elements of the antenna is determined based on the lowest transmission frequency within the operating frequency band, i.e. Designing the array geometry under these most stringent conditions ensures that spacing requirements are met at all transmission frequencies, thereby avoiding spatial aliasing at higher frequencies and ensuring stable radiation and beamforming performance across the entire frequency band.

[0073] Furthermore, the transmit frequencies of the antenna elements are selected from a predefined set of discrete frequencies and follow a uniform distribution. A constraint is also imposed to ensure that each antenna is assigned a unique frequency. This design not only achieves balanced spectrum utilization but also frequency diversity, thereby providing performance improvements in terms of concealment and anti-jamming capabilities.

[0074] Because each antenna element has a unique transmission frequency, the number of available discrete frequencies is... Must meet The condition. In discrete array implementations, frequency allocation is inherently limited by the size of the available frequency set. Increasing It can increase the randomness of frequency selection, making it more difficult for eavesdropping users to deduce the transmission frequency used by the covert user.

[0075] However, when the number of transmitting antennas is large enough, further increasing the number of antennas... This will only result in a minor performance improvement. This indicates that for large antenna arrays, the randomness introduced by expanding the frequency set has reached saturation in terms of its impact on the detection capability of listening users.

[0076] Therefore, when the number of transmitting antennas is large enough, it is usually only necessary to... Set as This allows for a good balance between randomness, concealment, and spectral efficiency.

[0077] Through these designs, the proposed system model accurately captures key practical constraints and provides a solid foundation for reliable covert communication analysis.

[0078] Next, numerical results will be provided to evaluate system performance, with particular attention to the number of frequency divisions. The impact on the probability of false detection of monitored users and the achievable stealth rate. For example... Figure 2 As shown, this illustrates the number of divisions for different frequencies. The optimal error detection probabilities for listening users at different locations (with varying angles and distances relative to the concealed user) are 1, 10, and 100. The changes can be observed. It can be seen that the number of frequency divisions... It has a significant impact on the probability of false detection, and it has a significant impact on... The impact is closely related to the spatial location of the user being monitored.

[0079] In order to separate the number of frequency divisions The impact will be investigated by fixing the location of the monitoring user. Follow The extent of the change. For example... Figure 3 As shown, It exhibits an oscillating behavior, as The increase gradually converges to a stable value, rather than following a monotonic trend. This observation suggests that the increase... This does not necessarily improve system performance. This further emphasizes the need for careful optimization of the number of frequency allocations. Instead of arbitrarily increasing its importance.

[0080] Through joint optimization and To maximize the probability of false detection, higher transmission power can be used under the same concealment constraints, thereby improving the concealment rate. For example... Figure 4 As shown, with a fixed total bandwidth, two representative eavesdropping user locations are considered: one closer to the covert user than the receiving user, and the other farther away. The performance of the proposed optimized frequency-division RFDA covert communication scheme is compared with that of the traditional non-frequency-division RFDA covert communication scheme. The performance improvement mainly stems from the frequency division design, which allows for finer control of the transmit frequency on the antenna array and effectively suppresses the received signal power at the eavesdropping user location. These numerical results confirm that the optimized frequency-division RFDA covert communication scheme outperforms the traditional non-frequency-division RFDA covert communication scheme in covert communication scenarios.

[0081] Example 2 In one or more embodiments, a frequency division multiplexing (FDM) covert communication system based on a random frequency diversity array is disclosed, specifically including: The model building module is configured to: establish a model of a covert communication system using a random frequency diversity array, wherein the covert communication system model uses a frequency division strategy to divide the covert user into several frequency sub-bands within a limited total bandwidth; The covert calculation module is configured to: construct the signal received by the receiving user in the frequency subband based on the covert communication system model and calculate the covert rate from the covert user to the receiving user. The noise analysis module is configured to: analyze the distribution of noise power received by the monitoring user using noise uncertainty, obtain the overall false detection probability, and obtain the optimal false detection probability of the monitoring user based on the optimal detection threshold under the condition of satisfying the system concealment constraint; The joint optimization module is configured to optimize the frequency division strategy and power allocation using a two-layer optimization method to achieve maximum concealment rate.

[0082] Example 3 This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the computer instructions are executed by the processor, they complete the steps of the frequency division covert communication method based on random frequency diversity array described above.

[0083] Example 4 This embodiment provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the frequency division covert communication method based on a random frequency diversity array described above.

[0084] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0085] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1The function specified in one or more boxes.

[0086] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment, whereby a series of operational steps are performed to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0087] The descriptions of each embodiment in the above embodiments have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0088] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A frequency division multiplexing (FDM) covert communication method based on a random frequency diversity array, characterized in that, include: A model of a covert communication system using a random frequency diversity array is established. The covert communication system model uses a frequency division strategy to divide the covert user into several frequency sub-bands within a limited total bandwidth. Based on the aforementioned covert communication system model, the signal received by the receiving user in the frequency sub-band is constructed, and the covertness rate from the covert user to the receiving user is calculated. The distribution of noise power received by the monitoring user is analyzed by utilizing noise uncertainty to obtain the overall false detection probability. Under the condition of satisfying the system concealment constraint, the optimal false detection probability of the monitoring user is obtained according to the optimal detection threshold. A two-layer optimization method is used to optimize the frequency division strategy and power allocation to achieve maximum concealment rate.

2. The frequency division covert communication method based on a random frequency diversity array as described in claim 1, characterized in that, The signal received by the receiving user in the frequency subband is: in, This represents the Hermitian transpose operation; This represents the turning vector from the hidden user to the receiving user; For the first Covert users transmitting signals in each frequency sub-band; Indicates the first The transmit power of concealed users in the sub-band; Indicates the beamforming vector; It is the first one that satisfies the average power constraint. Information signals in each frequency sub-band; To reduce the path loss between receiving users and hidden users; This represents additive white Gaussian noise.

3. The frequency division covert communication method based on a random frequency diversity array as described in claim 1, characterized in that, The calculation of the concealment rate from the concealed user to the receiving user is specifically as follows: in, The limited bandwidth occupied by the transmitting end for transmitting signals; The number of frequency divisions; Let be the signal-to-noise ratio of the communication between the receiving user and the covert user in the i-th subband; Indicates the first The transmit power of concealed users in the sub-band; To reduce the path loss between receiving users and hidden users; It is the power spectral density at the receiving user.

4. The frequency division covert communication method based on a random frequency diversity array as described in claim 1, characterized in that, The overall error detection probability is used to measure the detection performance of the monitoring user, and its expression is: The optimal error detection probability for monitoring users is: in, It is a complete gamma function; , These are the lower and upper limits of the noise power range; The shape parameter of the incomplete gamma function; It is an incomplete gamma function; is the scaling parameter of the incomplete gamma function.

5. The frequency division multiplexing (FDM) covert communication method based on a random frequency diversity array as described in claim 1, characterized in that, The method employs a two-layer optimization approach to optimize the frequency division strategy and power allocation, jointly optimizing the number of frequency divisions and the power allocation of each frequency band within the frequency band where the covert user is located, thereby maximizing the achievable covertness rate while satisfying the covertness constraint condition, which is that the optimal false detection probability of the eavesdropping user is not less than a specified threshold.

6. The frequency division covert communication method based on a random frequency diversity array as described in claim 5, characterized in that, The joint optimization employs a two-layer structure. In the outer layer, the number of frequency sub-bands is traversed. In the inner layer, a composite optimization structure including total transmit power and power allocation weight vectors is used. For a given total transmit power, the interior-point method is used to obtain the optimal power allocation vector to maximize the concealment rate while satisfying the concealment constraint. By comparing the upper and lower limits of the total transmit power values ​​for the maximum concealment rate, a binary search method is used to narrow down the feasible range of the total transmit power. After obtaining the new range of total transmit power, the above operations are performed alternately, and the maximum acceptable transmission power that satisfies the concealment constraint is determined step by step. Finally, by comparing the concealment rates obtained under all frequency sub-bands, the optimal parameter triplet is selected.

7. The frequency division multiplexing covert communication method based on a random frequency diversity array as described in claim 5, characterized in that, The joint optimization is expressed as: in, Concealment rate; To determine the optimal error detection probability for monitoring users; The number of frequency divisions; Assign a weight vector to the power; Indicates the required level of concealment; This is the maximum number of frequency dividers. Allocate power ratios for each sub-band; Maximum transmission power; This represents the total transmission power.

8. A frequency division multiplexing covert communication system based on a random frequency diversity array, characterized in that, include: The model building module is configured to: establish a model of a covert communication system using a random frequency diversity array, wherein the covert communication system model uses a frequency division strategy to divide the covert user into several frequency sub-bands within a limited total bandwidth; The covert calculation module is configured to: construct the signal received by the receiving user in the frequency subband based on the covert communication system model and calculate the covert rate from the covert user to the receiving user; The noise analysis module is configured to: analyze the distribution of noise power received by the monitoring user using noise uncertainty, obtain the overall false detection probability, and obtain the optimal false detection probability of the monitoring user based on the optimal detection threshold under the condition of satisfying the system concealment constraint; The joint optimization module is configured to optimize the frequency division strategy and power allocation using a two-layer optimization method to achieve maximum concealment rate.

9. An electronic device, characterized in that, It includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, which, when executed by the processor, perform the frequency division covert communication method based on a random frequency diversity array as described in any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, Used to store computer instructions, which, when executed by a processor, complete the frequency division covert communication method based on a random frequency diversity array as described in any one of claims 1-7.

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