Concealment communication method based on optimal subcarrier detection under frequency selective fading

Abstract

The application discloses a frequency selective fading based hidden communication method based on optimal subcarrier detection, and steps are as follows: step 1, according to a hidden communication scene, a multi-carrier hidden communication system model is constructed, and a probability density function of subcarrier channel gain is constructed; step 2, a probability distribution function of optimal subcarrier channel gain, a probability density function of optimal subcarrier channel gain are constructed, and a maximum transmission power upper limit meeting a hidden constraint condition is acquired; step 3, an effective transmission rate model is established, the model is solved, the maximum effective transmission rate is obtained, and communication work is carried out according to the maximum effective transmission rate. The method can not only guarantee system hidden performance under the most unfavorable detection condition, reduce communication behavior exposure risk, but also effectively balance complexity and performance.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication and information security technology, specifically relating to a covert communication method based on optimal subcarrier detection under frequency selective fading. Background Technology

[0002] Due to the openness and broadcasting nature of wireless channels, wireless communication systems are vulnerable to detection, interception, interference, and even malicious sabotage during information transmission. Existing communication security technologies, such as traditional encryption and Physical Layer Security (PLS) technologies, primarily focus on ensuring the confidentiality and integrity of communication content, but are insufficient to effectively conceal the communication activity itself. Once the communication activity is detected by the monitoring party, the adversary can further infer the location of communication nodes, network topology, and potential operational intentions, thus posing a serious threat to system security. Therefore, covert communication technology, which can hide the existence of communication at the physical layer, has gradually become an important research direction in the field of wireless security and has attracted widespread attention.

[0003] With the rapid development of multi-carrier wireless communication technology, multi-carrier transmission mechanisms such as Orthogonal Frequency Division Multiplexing (OFDM) have been widely applied in next-generation wireless communication systems. In multi-carrier communication systems, due to the multipath effect in the wireless propagation environment, different subcarriers typically experience varying degrees of fading, thus exhibiting significant frequency-selective fading characteristics. In such scenarios, the stealth performance of the communication system is not only related to the overall transmit power but also closely related to the channel state on different subcarriers.

[0004] Existing research on multi-carrier covert communication typically employs Kullback-Leibler divergence (KL divergence) of the received signal to evaluate the system's covert performance, using the statistical average of the KL divergences of all subcarriers as the basis for covert constraints, and then designing corresponding transmit power allocation strategies. However, in real-world frequency-selective fading channel environments, due to significant differences in channel gain among different subcarriers, monitoring parties can obtain Channel State Information (CSI) of the monitoring link through long-term listening, pilot leakage, channel estimation, or node location inference. Based on this, monitoring parties can perform subcarrier filtering on the received signal, prioritizing the detection of the subcarrier with the highest channel gain in the monitoring link (the optimal subcarrier), thereby significantly improving their detection capability. Under the most unfavorable detection conditions where the monitoring party knows the CSI of the monitoring link, traditional covert performance evaluation methods based on average subcarrier statistical characteristics will fail to accurately reflect the actual detection risks faced by the system.

[0005] Furthermore, when some subcarriers are under strong monitoring channel conditions, even if the system as a whole meets the global concealment constraint in an average sense, the monitoring party may still obtain a low detection error probability by detecting the optimal subcarrier, thus exposing the communication behavior. In other words, existing concealment design methods based on average KL divergence or average detection error probability are difficult to effectively address the problem of concealment performance degradation caused by "optimal subcarrier detection" in frequency-selective fading environments. Therefore, it is urgent to propose a multi-carrier concealed communication method that is suitable for frequency-selective fading channel environments and can still meet the concealment constraint under the most unfavorable detection condition where the monitoring party knows the monitoring link CSI, in order to improve the concealment performance of the system in complex wireless environments. Summary of the Invention

[0006] The purpose of this invention is to provide a covert communication method based on optimal subcarrier detection under frequency selective fading, and to solve the problem that existing methods are difficult to effectively deal with the deterioration of covert performance caused by optimal subcarrier detection under frequency selective fading environment.

[0007] The technical solution adopted in this invention is a covert communication method based on optimal subcarrier detection under frequency-selective fading, the steps of which are as follows: Step 1: Based on the covert communication scenario, construct a multi-carrier covert communication system model and construct the probability density function of the subcarrier channel gain; Step 2: Construct the probability distribution function and probability density function of the optimal subcarrier channel gain, and obtain the maximum transmit power upper limit that satisfies the concealment constraint. Step 3: Establish an effective transmission rate model, solve the model to obtain the maximum effective transmission rate, and conduct communication at the maximum effective transmission rate.

[0008] The invention is further characterized by: In step 1, the multi-carrier covert communication system model consists of Alice as the transmitter, Bob as the legitimate receiver, and Willie as the monitor. Alice sends multi-carrier signals to Bob as the legitimate receiver, while hoping that her communication behavior will not be detected by Willie.

[0009] In step 1, it is assumed that the transmitter Alice and the legitimate receiver Bob use a method with... A multi-carrier covert communication system model with consecutive subcarriers transmits information, with each subcarrier transmitting within a single transport block. Given three independent complex Gaussian signals, and the signals of different subcarriers are also independent of each other, then the total transmit power at a certain moment is: ,in, Indicates the first To suppress inter-symbol interference caused by multipath propagation, the transmitter Alice introduces a cyclic prefix before each transmission block. When the cyclic prefix length is not less than the maximum channel delay spread, removing the cyclic prefix transforms the linear convolution relationship between the transmitter Alice and the receiver m into a cyclic convolution, thus giving the channel matrix a cyclic structure. This indicates the distance from the transmitting end Alice to the receiving user. The distance, using This indicates the distance from the transmitting end Alice to the receiving user. The channel memory length, used This indicates the distance from the transmitting end Alice to the receiving user. The equivalent channel, where 'b' represents the legitimate receiver, Bob. Indicates Willie, the monitor; but The expression is: (1) In equation (1), The channel gain on each path is represented by a probability distribution that follows a complex Gaussian distribution with a mean of 0 and a variance of 1. , ; Then receive user The received time-domain signal can be represented as: (2) In equation (2), Indicates receiving user The received time-domain signal; Indicates receiving user The received additive white Gaussian noise signal, and Each element in the dataset follows a mean of 0 and a variance of 0. The complex Gaussian distribution; This represents the time-domain signal transmitted by the transmitter Alice within a transmission block, and Each element in the matrix follows a complex Gaussian distribution with mean 0 and variance 1. Indicates the spatial fading coefficient; make Indicates signal The frequency domain form, that is, satisfying ,in express The energy-normalized discrete Fourier transform matrix is ​​then... The Line number A column can be represented as: (3) In equation (3), ; ; In addition, due to for The circular matrix, therefore ,in Represent a a diagonal matrix; (4) In equation (4), Indicates receiving user No. Frequency domain channel state information corresponding to each subcarrier; (5) but obey Distribution, subcarrier channel gain The parameters are satisfied exponential distribution, subcarrier channel gain The probability density function is: (6) In equation (6), x is a variable in the probability density function.

[0010] The specific process of step 2 is as follows: To determine whether Alice, the transmitter, transmitted a signal, Willie, the monitor, faces a binary hypothesis testing problem: (8) In equation (8), This represents the assumption that Alice, the transmitter, did not transmit any signals; This represents the assumption that Alice, the transmitter, transmits signals. Monitor Willie uses an energy detection method to make decisions in latency-constrained covert communication scenarios, when the transmitter Alice transmits on each subcarrier. When there are several independent signals, its detection performance can be expressed as the detection error probability. Let represent , and obtain the following expression from Pinsker's inequality: (9) In equation (9), This represents the false alarm probability for monitor Willie; This indicates the probability of a missed detection by monitor Willie; Indicates that monitor Willie was in The probability density function under; Indicates that monitor Willie was in The probability density function under; This indicates that the monitor Willie was monitoring the transmitter Alice at a certain moment. When detecting the combination of signals transmitted on each subcarrier, and KL divergence in two states; in, (10) In equation (10), It indicates that monitor Willie was on the Signal received on each subcarrier and KL divergence under two assumptions; Since the signals on each subcarrier follow independent and identical distributions at different times, the detection error probability of the monitor Willie on each subcarrier can be estimated based on formula (9), and combined with formula (10), formula (11) is obtained, that is: (11) in, (12) In equation (12), Indicates the first Power allocated to each subcarrier; This represents the variance of noise on a single subcarrier; This indicates that when the receiving user is the monitor Willie, the first... Frequency domain channel gain corresponding to each subcarrier; The average error probability of the monitor Willie's detection can be calculated using formula (11): (13) In equation (13), This represents the expectation of a random variable; When the hidden constraint is Sometimes, (14) in, (15) In equation (15), Represents an exponential integral function. x Represents the variables in the exponential integral function; Indicates the memory length of the channel on the monitoring link; make The subcarrier index represents the optimal subcarrier, then the... The KL divergence corresponding to the subcarrier is: (16) In equation (16), Indicates the optimal subcarrier channel gain; Indicates the first Power allocated on subcarriers; Indicates the optimal subcarrier channel gain at The probability density function of a state; Indicates the optimal subcarrier channel gain at The probability density function of a state; make (17) In equation (17), Indicates about Expectation calculation of probability distributions; In multi-carrier covert communication system models, there are usually This leads to different subcarrier frequency domain channel gains. Although they are identically distributed, they are not independent; only when... hour, Only when they are independent of each other can we know from formula (1) that they are independent of each other. The channel only contains It consists of several independent channel parameters, equivalent to having One degree of freedom, when At this time If the elements on the diagonal are independent, then the probability distribution function of the optimal subcarrier channel gain is: (18) In equation (18), Represents an exponential function; x Represents the variables in an exponential function; Introducing correction factors , Formula (18) is modified to satisfy the following conditions: The probability distribution function of the optimal subcarrier channel gain is given by the following modified expression: (20) The maximum frequency domain channel gain probability density function approximation method based on MMSE is used to solve different... and Correction factor below ; The corrected factor after the solution Substituting into formula (20), we obtain the probability distribution function of the modified optimal subcarrier channel gain. ; The probability distribution function of the corrected optimal subcarrier channel gain Taking the derivative, we obtain the probability density function of the optimal subcarrier channel gain. ; make ,have: (twenty four) Combining formula (17), we obtain the result when the hidden constraint is... At that time, the transmission power The relationship that satisfies this is: (25) From formula (12), we can see that It is about If it is a monotonically increasing function, then by solving the equation To obtain the maximum transmit power upper limit that satisfies the concealment constraint on each subcarrier. , Alice was unaware of this at the transmitting end. In the case of, As a unified transmit power upper limit for all subcarriers, it can ensure that all subcarriers meet the concealment constraint conditions.

[0011] The maximum frequency domain channel gain probability density function approximation method based on MMSE is used to solve different... and Correction factor below The specific process is as follows: (1) Given and ,in and It is an integer greater than 0; (2) Simulate using Monte Carlo simulation. The probability distribution, setting the number of loops. and order Used to store the maximum subcarrier gain generated in each loop. ; (3) Setting The statistical range of values ​​is , express Upper bound of statistical values, statistics The elements in the middle are less than or equal to Number of ,use get Simulation probability distribution ; (4) Let and They represent The upper and lower bounds of the value range, the initial value is... , ,Pick In steps from arrive Changes, for ease of representation, let Indicates from Towards Forward The value of the step will Substituting into formula (20), we get: (twenty one) Calculate the corresponding and The one with the smallest mean square error value The correction after solving is because ; The expression is: (twenty two) In equation (22), Indicates correspondence and The one with the smallest mean square error value.

[0012] right Differentiation yields the probability density function of the optimal subcarrier channel gain. The expression is: (twenty three).

[0013] In step 3, the effective transmission rate model is: (26) In equation (26), This represents the vector of injected power on the subcarrier; Indicates the acceptable probability of interruption; express inverse function, , Describes the variable in the inverse function. .

[0014] Step 3 involves solving the model as follows: Order No. The effective transmission rate corresponding to each subcarrier is: (27) The maximum effective transmission rate can then be obtained by solving the following optimization problem; (28) Subject to, (29) (30) (31) In equation (30), This indicates the maximum transmission power of the transmitting end Alice; When Alice at the transmitting end only has When the probability distribution information is not available, it is impossible to accurately determine the specific subcarrier position corresponding to the optimal subcarrier in real time. In order to ensure that the information transmitted by all subcarriers satisfies the concealment constraint, formulas (29) and (31) are combined, and the optimization problem can be reformulated as: (32) Subject to, (33) (34) When the maximum transmit power is determined by concealment constraints When the power is lower than the optimal power allocated to each subcarrier by the water-filling algorithm, the optimal solution can be achieved by allocating equal power to all subcarriers at the maximum transmit power, that is: (35).

[0015] The beneficial effects of this invention are: (1) The covert communication method based on optimal subcarrier detection under frequency selective fading of the present invention can more accurately characterize the detection capability of the monitoring party under the most unfavorable detection conditions, can ensure the covert performance of the system under the most unfavorable detection conditions, reduce the risk of exposure of communication behavior, and achieve an effective balance between complexity and performance, thereby avoiding the problem of covert performance failure caused by subcarrier channel differences. (2) The covert communication method based on optimal subcarrier detection under frequency selective fading of the present invention constructs an optimization problem with the goal of maximizing the effective transmission rate under the premise of satisfying the covert constraints and reliability constraints, and proves that in the low power region dominated by the covert constraints, the equal power allocation strategy can achieve performance similar to the successive convex approximation (SCA) algorithm while significantly reducing the computational complexity. (3) The covert communication method based on optimal subcarrier detection under frequency selective fading of the present invention can provide a more rigorous and reliable assessment of covert performance and effectively reveal the true covert risks of the system under frequency selective fading conditions. Attached Figure Description

[0016] Figure 1 This is the multi-carrier covert communication system model used in the method of this invention; Figure 2 This is a curve showing the relationship between the KL divergence of MSCGD and ASCGD and the subcarrier transmit power under different numbers of multipaths in the method of this invention. Figure 3 This is the curve showing the relationship between the KL divergence of MSCGD and ASCGD and the subcarrier transmit power under the known subcarrier selection condition in the method of this invention. Figure 4 This is a curve showing the relationship between the average achievable covert rate and communication distance under different power allocation schemes in the method of this invention under MSCGD and ASCGD detection. Figure 5 This is a curve showing the relationship between the average achievable concealment rate and the receiver noise power under MSCGD and ASCGD detection in the method of this invention. Detailed Implementation

[0017] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0018] Example 1 The present invention provides a covert communication method based on optimal subcarrier detection under frequency-selective fading, comprising the following steps: Step 1: Based on the covert communication scenario, construct a multi-carrier covert communication system model and construct the probability density function of the subcarrier channel gain; Among them, such as Figure 1As shown, the multi-carrier covert communication system model consists of Alice as the transmitter, Bob as the legitimate receiver, and Willie as the monitor. Alice sends multi-carrier signals to Bob as the legitimate receiver, while hoping that her communication behavior will not be detected by Willie. The multicarrier covert communication system model is a multicarrier covert communication system model with time delay constraints under a quasi-static multipath fading channel. Assume that the transmitter Alice and the legitimate receiver Bob use a method with A multi-carrier covert communication system model with consecutive subcarriers transmits information, with each subcarrier transmitting within a single transport block. There are three independent complex Gaussian signals, and the signals on different subcarriers are also independent. Let the transmit power vector of the transmitter Alice on each subcarrier be expressed as: Then the total transmit power at a certain moment is ,in, Indicates the first To suppress inter-symbol interference (ISI) caused by multipath propagation, the transmitter Alice introduces a cyclic prefix (CP) before each transmission block. When the CP length is not less than the maximum channel delay spread, removing the CP transforms the linear convolution relationship between the transmitter Alice and the receiver m into a cyclic convolution, thus giving the channel matrix a cyclic structure. This indicates the distance from the transmitting end Alice to the receiving user. The distance, using This indicates the distance from the transmitting end Alice to the receiving user. The channel memory length, used This indicates the distance from the transmitting end Alice to the receiving user. The equivalent channel, where 'b' represents the legitimate receiver, Bob. Indicates Willie, the monitor; but The expression is: (1) In equation (1), The channel gain on each path is represented by a probability distribution that follows a complex Gaussian distribution with a mean of 0 and a variance of 1. , ; Then receive user The received time-domain signal can be represented as: (2) In equation (2), Indicates receiving user The received time-domain signal; Indicates receiving user The received additive white Gaussian noise signal, and Each element in the dataset follows a mean of 0 and a variance of 0. The complex Gaussian distribution; This represents the time-domain signal transmitted by the transmitter Alice within a transmission block, and Each element in the matrix follows a complex Gaussian distribution with mean 0 and variance 1. Indicates the spatial fading coefficient; make Indicates signal The frequency domain form, that is, satisfying ,in express The energy-normalized discrete Fourier transform matrix is ​​then... The Line number A column can be represented as: (3) In equation (3), ; ; In addition, due to for The circular matrix, therefore ,in Represent a a diagonal matrix; (4) In equation (4), Indicates receiving user No. Frequency domain channel state information corresponding to each subcarrier; (5) but obey Distribution, subcarrier channel gain The parameters are satisfied exponential distribution, subcarrier channel gain The probability density function is: (6) In equation (6), x These are variables in the probability density function; make Indicates signal The frequency domain form, i.e. ,make Indicates noise In its frequency domain form, we have: (7) because Since it is a unitary matrix, the time-domain and frequency-domain transformations do not change the statistical distribution characteristics of the signal and noise, thus the detection performance in the time domain and frequency domain is equivalent; at the same time, the multipath channel matrix is ​​transformed into a diagonal matrix form in the frequency domain, which can significantly reduce the complexity of subsequent covert detection performance and transmission performance analysis. In the design of multi-carrier covert communication system models, the average detection error probability of the monitor Willie is typically used as the core indicator to measure the system's covert performance. To ensure that the designed system can still meet the preset covert constraints under the most unfavorable detection conditions, a conservative modeling approach is usually adopted, assuming that the monitor Willie has strong detection capabilities: it can not only obtain relevant parameters of the eavesdropping channel by analyzing potential leaked pilot signals, but also obtain key parameters such as the location information and transmission power of the transmitter Alice using prior information or advanced detection technologies. Under these assumptions, a functional relationship between the monitor Willie's average detection error probability and the transmission power can be established, and the transmission power can be adjusted to ensure that the system meets the covert constraints.

[0019] Therefore, accurately estimating the minimum average detection error probability of the monitor Willie is a crucial aspect of covert communication system design. However, existing methods for estimating the covert performance of multi-carrier covert communication systems under time-delay constraints typically use the average detection error probability of the monitor Willie on the entire received signal as the evaluation criterion, failing to adequately consider the frequency-selective fading effect caused by multipath propagation. Under this effect, the channel gain corresponding to different subcarriers varies significantly, resulting in inconsistent detection performance of the monitor Willie on different subcarriers.

[0020] In actual detection, the monitor Willie can filter subcarriers and select the received signal corresponding to the optimal subcarrier for detection, thereby obtaining a lower average detection error probability and achieving a more effective judgment on communication behavior. Therefore, the stealth performance evaluation method based on the global average detection error probability may underestimate the detection capability of the monitor Willie, leading to the risk of insufficient stealth on some subcarriers. Based on the above analysis, this invention designs step 2 to achieve a more accurate characterization of stealth performance under the most unfavorable detection scenario, thereby improving the reliability and security of the system design.

[0021] Step 2: Construct the probability distribution function and probability density function of the optimal subcarrier channel gain, and obtain the maximum transmit power upper limit that satisfies the concealment constraint. The specific process is as follows: To determine whether Alice, the transmitter, transmitted a signal, Willie, the monitor, faces a binary hypothesis testing problem: (8) In equation (8), This represents the assumption that Alice, the transmitter, did not transmit any signals; This represents the assumption that Alice, the transmitter, transmits signals. Monitor Willie uses energy detection (power meter detection) to make decisions in latency-constrained covert communication scenarios, when the transmitter Alice transmits on each subcarrier When there are several independent signals, its detection performance can be expressed as the detection error probability. Let represent , and obtain the following expression from Pinsker's inequality: (9) In equation (9), This represents the false alarm probability for monitor Willie; This indicates the probability of a missed detection by monitor Willie; Indicates that monitor Willie was in The probability density function under; Indicates that monitor Willie was in The probability density function under; This indicates that the monitor Willie was monitoring the transmitter Alice at a certain moment. Detection is performed by combining the transmitted signals on each subcarrier (which can be equivalent to detecting the received signals). When detecting a specific column of a matrix signal, in and KL divergence in two states; In a multi-carrier covert communication system model, the KL divergence of the overall signal is equal to the sum of the KL divergences of each subcarrier signal. Therefore, the KL divergence of the overall signal can be estimated by calculating the sum of the KL divergences of each subcarrier signal at a given moment, i.e.: (10) In equation (10), It indicates that monitor Willie was on the Signal received on each subcarrier and KL divergence under two assumptions; Since the signals on each subcarrier follow independent and identical distributions at different times, the detection error probability of the monitor Willie on each subcarrier can be estimated based on formula (9), and combined with formula (10), formula (11) is obtained, that is: (11) in, (12) In equation (12), Indicates the first Power allocated to each subcarrier; This represents the variance of noise on a single subcarrier; This indicates that when the receiving user is the monitor Willie, the first... Frequency domain channel gain corresponding to each subcarrier; The average error probability of the monitor Willie's detection can be calculated using formula (11): (13) In equation (13), This represents the expectation of a random variable; When the hidden constraint is Sometimes, (14) in, (15) In equation (15), Represents an exponential integral function. x Represents the variables in the exponential integral function; Indicates the memory length of the channel on the monitoring link; It should be noted that the concealment performance estimation results obtained based on formula (13) The impact of frequency selective fading on the detection capability of the monitor Willie was not considered. In fact, the monitor Willie can use the frequency selectivity introduced by the multipath channel to decompose the received signal into multiple subcarrier components through a multi-band filter, and perform energy detection on each subcarrier signal separately, and then select the detection result corresponding to the optimal subcarrier as the final decision basis. In this case, if the method based on the average KL divergence of the subcarrier signal in formula (13) is still used to evaluate the system's concealment performance, it may cause some high channel gain subcarriers to fail to meet the concealment constraint conditions, thereby increasing the risk of communication behavior being detected. The root cause of the above problems is that the existing methods do not use the optimal detection performance of the monitor Willie as the basis for evaluating the concealment performance. Specifically, in a multipath channel environment, due to the existence of frequency selective fading, there are significant differences in the detection performance of different subcarriers. In the worst case where the monitor Willie can obtain the channel state information of the monitoring link, it can obtain the minimum detection error probability by selecting the optimal subcarrier for energy detection. make The subcarrier index represents the optimal subcarrier, then the... The KL divergence corresponding to the subcarrier is: (16) In equation (16), Indicates the optimal subcarrier channel gain; Indicates the first Power allocated on subcarriers; Indicates the optimal subcarrier channel gain at The probability density function of a state; Indicates the optimal subcarrier channel gain at The probability density function of a state; make (17) In equation (17), Indicates about Expectation calculation of probability distributions; Under this concealment constraint, the upper limit of the maximum permissible transmission power can be determined. When the transmit power on each subcarrier All constraints are satisfied In this way, the transmit power of all subcarriers can be guaranteed to meet the concealment constraint conditions, so that the system still has reliable concealment performance under the most unfavorable detection conditions. Therefore, this method can provide a more conservative and strict concealment performance guarantee. In multi-carrier covert communication system models, there are usually This leads to different subcarrier frequency domain channel gains. Although they are identically distributed, they are not independent; only when... hour, They are independent of each other, therefore, they cannot be directly obtained. Given the probability density function, the following steps were performed to solve the above problem; From formula (1), we know that The channel only contains It consists of several independent channel parameters, equivalent to having One degree of freedom, when At this time If the elements on the diagonal are independent, then the probability distribution function of the optimal subcarrier channel gain is: (18) In equation (18), Represents an exponential function; x Represents the variables in an exponential function; It should be noted that, as can be seen from formula (6), the exponential function in formula (18) within The probability distribution parameters corresponding to the exponential distribution (equivalent to) (average energy), while the outermost exponent term Corresponding to from The statistical property of selecting the maximum value from a number of independent samples; From formula (5), we know that Can be viewed as a vector In the truncated Fourier matrix Projection on; in, (19) exist Increase under unchanged conditions This is equivalent to converting a vector Projected onto a vector space matrix with smaller spacing This increases the expected value of the maximum channel gain, but... The number of independent samples did not change as a result, therefore a correction factor was introduced. , Formula (18) is modified to satisfy the following conditions: The probability distribution function of the optimal subcarrier channel gain is given by the following modified expression: (20) The maximum frequency domain channel gain probability density function approximation method based on MMSE is used to solve different... and Correction factor below The specific process is as follows: (1) Given and ,in and It is an integer greater than 0; (2) Simulate using Monte Carlo simulation. The probability distribution, setting the number of loops. and order Used to store the maximum subcarrier gain generated in each loop. ; for t=1:

[0022] (1) Generate obedience Complex random variables , ; (2) Calculate using formula (5) , ; (3) Record the maximum value: ; end (3) Setting The statistical range of values ​​is , express Upper bound of statistical values, statistics The elements in the middle are less than or equal to Number of ,use get Simulation probability distribution ; (4) Let and They represent The upper and lower bounds of the value range, the initial value is... , ,Pick In steps from arrive Changes, for ease of representation, let Indicates from Towards Forward The value of the step will Substituting into formula (20), we get: (twenty one) Calculate the corresponding and The one with the smallest mean square error value The correction after solving is because ; The expression is: (twenty two) In equation (22), Indicates correspondence and The one with the smallest mean square error value; The corrected factor after the solution Substituting into formula (20), we obtain the probability distribution function of the modified optimal subcarrier channel gain. ; This approximation algorithm can be used to obtain different... and Below Because the subsequent derivation of concealment performance usually takes place after determining... and Under the premise of..., for ease of expression, let... ; The probability distribution function of the corrected optimal subcarrier channel gain Taking the derivative, we obtain the probability density function of the optimal subcarrier channel gain. The expression is: (twenty three) make ,have: (twenty four) Combining formula (17), we obtain the result when the hidden constraint is... At that time, the transmission power The relationship that satisfies this is: (25) From formula (12), we can see that It is about If it is a monotonically increasing function, then by solving the equation To obtain the maximum transmit power upper limit that satisfies the concealment constraint on each subcarrier. , For the first subcarrier with the maximum channel gain Each subcarrier has a maximum transmit power that satisfies the concealment constraint. If it is the smallest of all subcarriers, Alice cannot know this at the transmitting end. In the case of, As a unified transmit power limit for all subcarriers, it can ensure that all subcarriers meet the concealment constraint conditions. Step 3: Establish an effective transmission rate model, solve the model to obtain the maximum effective transmission rate, and conduct communication at the maximum effective transmission rate. Since the signals of each subcarrier are independent, when the number of subcarriers participating in the transmission is... The signal time slot length is At that time, the effective transmission rate model is: (26) In equation (26), This represents the vector of injected power on the subcarrier; Indicates the acceptable probability of interruption; express inverse function, , Describes the variable in the inverse function. ; Order No. The effective transmission rate corresponding to each subcarrier is: (27) It is important to note that Regarding parameters It exhibits a monotonically increasing relationship; moreover, in Under the same range of change, Larger values ​​correspond to More significant changes; The maximum effective transmission rate can then be obtained by solving the following optimization problem; (28) Subject to, (29) (30) (31) In equation (30), This indicates the maximum transmission power of the transmitting end Alice; When Alice at the transmitting end only has When the probability distribution information is not available, it is impossible to accurately determine the specific subcarrier position corresponding to the optimal subcarrier in real time. In order to ensure that the information transmitted by all subcarriers satisfies the concealment constraint, the maximum transmit power upper limit that satisfies the concealment constraint can be obtained from formula (29). (See step 2 for the specific calculation process), then formulas (29) and (31) can be expressed as a unified power constraint on all subcarriers, i.e. Combining formulas (29) and (31), the optimization problem can be reformulated as: (32) Subject to, (33) (34) Existing literature typically employs the Successive Convex Approximation (SCA) algorithm to solve the aforementioned optimization problem. However, in practical covert communication scenarios, the noise power is usually low, resulting in a small maximum transmit power under covert constraints. In this case, the optimal power allocation strategy corresponding to the traditional water-filling theorem is no longer applicable.

[0023] Specifically, when the maximum transmit power is determined by the concealment constraint When the power is lower than the optimal power allocated to each subcarrier by the water-filling algorithm, the optimal solution can be achieved by allocating equal power to all subcarriers at the maximum transmit power, that is: (35).

[0024] This conclusion can be verified by numerical results from the SCA algorithm: except for a few subcarriers with extremely poor channel conditions that are allocated near-zero power, the optimal power allocation for most subcarriers approaches zero. This indicates that in low-power regions dominated by hidden constraints, complex power optimization algorithms offer limited improvement to system performance. Therefore, in the aforementioned scenario, adopting an equal-power allocation strategy not only achieves near-optimal effective transmission rates with almost no loss in system performance but also significantly reduces computational complexity, making it more suitable for practical resource-constrained wireless communication systems.

[0025] Example 2 Simulation verification: Simulation conditions: The simulation parameters are set as follows: The multi-carrier covert communication system model adopts a continuous spectrum multi-carrier communication structure, with a total number of subcarriers of [value missing]. The number of symbols transmitted by a single subcarrier within each transport block is The number of multipaths is The channel coefficients on each path are independent and all follow a complex Gaussian distribution with a mean of 0 and a variance of 1. Noise power dBm, the distance from the transmitter Alice to the monitor Willie is Spatial decay factor Hidden constraints are set as follows: The decoding error probability is set to .

[0026] Example 3 Based on Example 2, Figure 2 This shows the number of multipaths at the location of monitor Willie. The relationship between KL divergence and subcarrier transmit power is shown for values ​​of 8 and 16, respectively, where the number of cycles in the Monte Carlo simulation is set to 10. 5 As can be seen from the figure, the theoretical derivation results are highly consistent with the numerical simulation results, verifying the correctness of the proposed theoretical analysis. Furthermore, it can be observed that, under the same conditions, the detection performance of the optimal subcarrier-based detection method (MSCGD) is significantly better than that of the average subcarrier channel gain-based detection method (ASCGD). Specifically, when the number of multipath paths is fixed... At the same KL divergence level, the subcarrier transmit power of MSCGD is approximately 6 dB lower than that of ASCGD, indicating that MSCGD can achieve the same detection capability under lower transmit power conditions, thus imposing stricter detection constraints on covert communication systems. Furthermore, with the increase in the number of multipath paths... With the increase in the number of subcarriers, the average gain of each subcarrier channel increases, thereby enhancing Willie's overall detection capability. In other words, the detection performance of both MSCGD and ASCGD is improved. However, it is evident that the performance improvement of MSCGD is significantly greater than that of ASCGD. This further illustrates that under frequency-selective fading channel conditions, MSCGD can more effectively characterize the most unfavorable detection scenarios, thus providing a more rigorous and reliable assessment of the system's concealment performance.

[0027] Example 4 Based on Example 2, Figure 3 This paper describes the relationship between the KL divergence and the subcarrier transmit power when the subcarriers selected by Alice are no longer contiguous, given that the monitor Willie knows the set of communication subcarriers chosen by Alice. The simulation first presents the relationship when the total number of subcarriers is... The theoretical KL divergence results for MSCGD and ASCGD when there are 128 subcarriers and all subcarriers participate in communication; subsequently, through a cycle number of... Monte Carlo simulations were performed, and results were given for different selections of the number of subcarriers. The KL divergence values ​​are 96, 64, and 32, where the subcarrier set is generated randomly. Simulation results show that, given that the monitor Willie knows the selected subcarrier set, the KL divergence of ASCGD is essentially independent of the KL divergence. The KL divergence of MSCGD changes with the change; and the KL divergence of MSCGD changes with the change. The decrease is slight. Specifically, when the number of subcarriers selected is only half of the total number of subcarriers, that is... When the power range under consideration is 1, the average decrease in KL divergence is only about 0.08 compared to the case of full subcarrier communication. This phenomenon occurs because the selected subcarrier set may not include the subcarrier with the maximum channel gain from the original full subcarrier set. Therefore, in practical communication scenarios where spectrum utilization needs to be considered, the estimation error of concealment performance introduced by subcarrier selection is usually negligible.

[0028] Example 5 Based on Example 2, Figure 4 The comparison was made when the power of the additive white Gaussian noise (AWGN) received by the monitors Willie and Bob was the same, i.e. = Under dBm conditions, the average achievable concealment rate of the power allocation scheme based on successive convex approximation (SCA) and the equal power allocation scheme under MSCGD and ASCGD detection. The communication distance between the transmitting end Alice and the legitimate receiving end Bob varies. The relationship between changes. The Monte Carlo loop count in the simulation was set to 100. As shown in the figure, regardless of whether the monitor Willie uses the MSCGD or ASCGD detection method, the average achievable concealment rate obtained by the two power allocation strategies is basically the same. This indicates that under concealment constraints, the subcarrier transmit power is limited to a low level, thus rendering the traditional optimal power allocation strategy based on the water-filling theorem inapplicable. In this case, the computationally less complex equal power allocation scheme can achieve system performance similar to or even the same as the SCA method. Furthermore, the figure also shows that although MSCGD provides stricter concealment performance constraints, its corresponding transmission performance suffers a certain loss compared to ASCGD, and this performance gap widens further with increasing communication distance. This result reflects the trade-off between concealment and transmission performance. Further considering practical application scenarios, such as squad collaborative communication or internal communication within a secure area (in such scenarios, the communication distance is usually less than the potential monitoring distance), given the possible location of the monitoring party, the design method proposed in this invention can be used to evaluate communication security in different areas or along different paths, thus providing a valid reference for system deployment.

[0029] Example 6 Based on Example 2, Figure 5 The distances between Alice–Bob and Alice–Willie are equal, i.e. Under these conditions, the average concealment rate achievable under MSCGD and ASCGD detection is [missing information]. With Bob's AWGN power at the legitimate receiver The relationship of change. The Monte Carlo loop count was set to 1000 in the simulation. Simulation results show that adjusting the noise power at the legitimate receiver Bob end... With regulation This has an equivalent impact on the system's covert performance. The system can achieve a positive covert transmission rate when the difference between the noise power at the legitimate receiver (Bob) and the noise power at the monitor (Willie) is less than approximately 10 dB. From an engineering perspective, this result demonstrates that by constructing a favorable communication environment (e.g., deploying interference sources near the monitor (Willie) to enhance its received noise, or suppressing interference at the legitimate receiver (Bob) through signal processing techniques), the system's covert communication capabilities can be effectively improved, thereby achieving stable positive-rate covert transmission.

Claims

1. A method for covert communication based on optimal subcarrier detection under frequency selective fading, characterized in that, The steps are as follows: Step 1: Based on the covert communication scenario, construct a multi-carrier covert communication system model and construct the probability density function of the subcarrier channel gain; Step 2: Construct the probability distribution function and probability density function of the optimal subcarrier channel gain, and obtain the maximum transmit power upper limit that satisfies the concealment constraint. Step 3: Establish an effective transmission rate model, solve the model to obtain the maximum effective transmission rate, and conduct communication at the maximum effective transmission rate.

2. The method according to claim 1, wherein, In step 1, the multi-carrier covert communication system model consists of Alice as the transmitter, Bob as the legitimate receiver, and Willie as the monitor. Alice sends multi-carrier signals to Bob as the legitimate receiver, while hoping that her communication behavior will not be detected by Willie.

3. The covert communication method based on optimal subcarrier detection under frequency selective fading according to claim 2, characterized in that, In step 1, it is assumed that the transmitter Alice and the legitimate receiver Bob use a method with... A multi-carrier covert communication system model with consecutive subcarriers transmits information, with each subcarrier transmitting within a single transport block. Given three independent complex Gaussian signals, and the signals of different subcarriers are also independent of each other, then the total transmit power at a certain moment is: ,in, Indicates the first To suppress inter-symbol interference caused by multipath propagation, the transmitter Alice introduces a cyclic prefix before each transmission block. When the cyclic prefix length is not less than the maximum channel delay spread, removing the cyclic prefix transforms the linear convolution relationship between the transmitter Alice and the receiver m into a cyclic convolution, thus giving the channel matrix a cyclic structure. This indicates the distance from the transmitting end Alice to the receiving user. The distance, using This indicates the distance from the transmitting end Alice to the receiving user. The channel memory length, used This indicates the distance from the transmitting end Alice to the receiving user. The equivalent channel, where 'b' represents the legitimate receiver, Bob. Indicates Willie, the monitor; Then , the expression is: （1） In formula (1), denotes the channel gain on each path, whose probability distribution satisfies a complex Gaussian distribution with a mean of 0 and a variance of 1, i.e. , ; The receiving user The received time-domain signal can be expressed as: （2） In equation (2), Indicates receiving user The received time-domain signal; Indicates receiving user The received additive white Gaussian noise signal, and Each element in the dataset follows a mean of 0 and a variance of 0. The complex Gaussian distribution; This represents the time-domain signal transmitted by the transmitter Alice within a transmission block, and Each element in the matrix follows a complex Gaussian distribution with mean 0 and variance 1. Indicates the spatial fading coefficient; make Indicates signal The frequency domain form, that is, satisfying ,in express The energy-normalized discrete Fourier transform matrix is ​​then... The Line number A column can be represented as: （3） In equation (3), ; ; In addition, due to for The circular matrix, therefore ,in Represent a a diagonal matrix; （4） In equation (4), Indicates receiving user No. Frequency domain channel state information corresponding to each subcarrier; （5） but obey Distribution, subcarrier channel gain The parameters are satisfied exponential distribution, subcarrier channel gain The probability density function is: （6） In equation (6), x is a variable in the probability density function.

4. The covert communication method based on optimal subcarrier detection under frequency selective fading according to claim 3, characterized in that, The specific process of step 2 is as follows: To determine whether Alice, the transmitter, transmitted a signal, Willie, the monitor, faces a binary hypothesis testing problem: （8） In equation (8), This represents the assumption that Alice, the transmitter, did not transmit any signals; This represents the assumption that Alice, the transmitter, transmits signals. Monitor Willie uses an energy detection method to make decisions in latency-constrained covert communication scenarios, when the transmitter Alice transmits on each subcarrier. When there are several independent signals, its detection performance can be expressed as the detection error probability. Let represent , and obtain the following expression from Pinsker's inequality: （9） In equation (9), This represents the false alarm probability for monitor Willie; This indicates the probability of a missed detection by monitor Willie; Indicates that monitor Willie was in The probability density function under; Indicates that monitor Willie was in The probability density function under; This indicates that the monitor Willie was monitoring the transmitter Alice at a certain moment. When detecting the combination of signals transmitted on each subcarrier, and KL divergence in two states; in, （10） In equation (10), It indicates that monitor Willie was on the Signal received on each subcarrier and KL divergence under two assumptions; Since the signals on each subcarrier follow independent and identical distributions at different times, the detection error probability of the monitor Willie on each subcarrier can be estimated based on formula (9), and combined with formula (10), formula (11) is obtained, that is: （11） in, （12） In equation (12), Indicates the first Power allocated to each subcarrier; This represents the variance of noise on a single subcarrier; This indicates that when the receiving user is the monitor Willie, the first... Frequency domain channel gain corresponding to each subcarrier; The average error probability of the monitor Willie's detection can be calculated using formula (11): （13） In equation (13), This represents the expectation of a random variable; When the hidden constraint is Sometimes, （14） in, （15） In equation (15), Represents an exponential integral function. x Represents the variables in the exponential integral function; Indicates the memory length of the channel on the monitoring link; make The subcarrier index represents the optimal subcarrier, then the... The KL divergence corresponding to the subcarrier is: （16） In equation (16), Indicates the optimal subcarrier channel gain; Indicates the first Power allocated on subcarriers; Indicates the optimal subcarrier channel gain at The probability density function of a state; Indicates the optimal subcarrier channel gain at The probability density function of a state; make （17） In equation (17), Indicates about Expectation calculation of probability distributions; In multi-carrier covert communication system models, there are usually This leads to different subcarrier frequency domain channel gains. Although they are identically distributed, they are not independent; only when... hour, Only when they are independent of each other can we know from formula (1) that they are independent of each other. The channel only contains It consists of several independent channel parameters, equivalent to having One degree of freedom, when At this time If the elements on the diagonal are independent, then the probability distribution function of the optimal subcarrier channel gain is: （18） In equation (18), Represents an exponential function; x Represents the variables in an exponential function; Introducing correction factors , Formula (18) is modified to satisfy the following conditions: The probability distribution function of the optimal subcarrier channel gain is given by the following modified expression: （20） The maximum frequency domain channel gain probability density function approximation method based on MMSE is used to solve different... and Correction factor below ; The corrected factor after the solution Substituting into formula (20), we obtain the probability distribution function of the modified optimal subcarrier channel gain. ; The probability distribution function of the corrected optimal subcarrier channel gain Taking the derivative, we obtain the probability density function of the optimal subcarrier channel gain. ; make ,have: （24） Combining formula (17), we obtain the result when the hidden constraint is... At that time, the transmission power The relationship that satisfies this is: （25） From formula (12), we can see that It is about If it is a monotonically increasing function, then by solving the equation To obtain the maximum transmit power upper limit that satisfies the concealment constraint on each subcarrier. , Alice was unaware of this at the transmitting end. In the case of, As a unified transmit power upper limit for all subcarriers, it can ensure that all subcarriers meet the concealment constraint conditions.

5. The covert communication method based on optimal subcarrier detection under frequency-selective fading according to claim 4, characterized in that, The maximum frequency domain channel gain probability density function approximation method based on MMSE is used to solve different... and Correction factor below The specific process is as follows: (1) Given and ,in and It is an integer greater than 0; (2) Simulate using Monte Carlo simulation. The probability distribution, setting the number of loops. and order Used to store the maximum subcarrier gain generated in each loop. ; (3) Setting The statistical range of values ​​is , express Upper bound of statistical values, statistics The elements in the middle are less than or equal to Number of ,use get Simulation probability distribution ; (4) Let and They represent The upper and lower bounds of the value range, the initial value is... , ,Pick In steps from arrive Changes, for ease of representation, let Indicates from Towards Forward The value of the step will Substituting into formula (20), we get: （21） Calculate the corresponding and The one with the smallest mean square error value The correction after solving is because ; The expression is: （22） In equation (22), Indicates correspondence and The one with the smallest mean square error value.

6. The covert communication method based on optimal subcarrier detection under frequency selective fading according to claim 5, characterized in that, Probability density function of optimal subcarrier channel gain The expression is: （23）。 7. The covert communication method based on optimal subcarrier detection under frequency-selective fading according to claim 5, characterized in that, In step 3, the effective transmission rate model is: （26） In equation (26), This represents the vector of injected power on the subcarrier; Indicates the acceptable probability of interruption; express inverse function, x represents Describes the variable in the inverse function. .

8. The covert communication method based on optimal subcarrier detection under frequency selective fading according to claim 7, characterized in that, Step 3 involves solving the model as follows: Order No. The effective transmission rate corresponding to each subcarrier is: （27） The maximum effective transmission rate can then be obtained by solving the following optimization problem; （28） Subject to, (29) （30） （31） In equation (30), This indicates the maximum transmission power of the transmitting end Alice; When Alice at the transmitting end only has When the probability distribution information is not available, it is impossible to accurately determine the specific subcarrier position corresponding to the optimal subcarrier in real time. In order to ensure that the information transmitted by all subcarriers satisfies the concealment constraint, formulas (29) and (31) are combined, and the optimization problem can be reformulated as: （32） Subject to, (33) （34） When the maximum transmit power is determined by concealment constraints When the power is lower than the optimal power allocated to each subcarrier by the water-filling algorithm, the optimal solution can be achieved by allocating equal power to all subcarriers at the maximum transmit power, that is: （35）。

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