A full-duplex relay-assisted two-hop covert communication system power allocation method

Abstract

The application discloses a kind of full duplex relay assisted double-hop covert communication system power distribution method.For a long-distance transmission based on relay and energy limited system's covert communication scene, the way of dividing system energy to sender and relay is often not optimal.In order to solve this problem, the application constructs the analysis framework of covert communication based on full duplex relay, analyzes the minimum false detection probability of detector and the covert constraint condition that system needs to meet.Further, the application solves the joint optimization problem of transmitter and full duplex relay transmission power distribution, and solves the closed expression of transmitter and full duplex relay transmission power.Further, the correctness of theoretical analysis is verified by using MATLAB platform simulation.The results show that the application solves the power distribution problem of double-hop covert communication system with total power limited well, and gives the influence of transmission power distribution on covert transmission rate, to guide the design of covert transmission strategy.

Description

Technical Field

[0001] The present invention belongs to the technical field of covert communication strategy design, and in particular relates to a power allocation method for a full-duplex relay-assisted double-hop covert communication system. Background Art

[0002] With the advancement of wireless communication technology, all walks of life are becoming increasingly reliant on wireless networks. Large amounts of private information are transmitted over these networks, and the security of this transmission has long been a major concern. While traditional encryption techniques and increasingly sophisticated physical layer security technologies can effectively safeguard the legitimacy of communication users and the confidentiality of communication content, the broadcast nature of wireless transmission allows for signal transmission to be detected, exposing communication activity and even location, posing security risks. Consequently, covert wireless communication technologies that utilize various signal processing techniques to minimize detection by the enemy have garnered significant attention and research from numerous experts and scholars.

[0003] The goal of covert wireless communications is to prevent detection, thereby achieving more secure and private communications. However, long-distance point-to-point transmission requires high transmit power, making covert transmission quite difficult to implement. Improving the stealthiness of long-distance transmission has garnered considerable research attention. Related research has shown that relay-assisted long-distance transmission is an effective approach to achieving covertness. However, existing research on two-hop covert communication often only considers the single optimization problem of the transmitter node or the relay node. For example, the literature [Ranran S, Bin Y, Siqi M, et al. Covert Rate Maximization in Wireless Full-Duplex Relaying Systems With Power Control[J]. IEEE Transactions on Communications, 2021, 69(9): 6198-6212.] studied the power optimization problem of the relay in a two-hop covert communication system. The optimization target is the covert transmission rate of the system. However, in this study, only the power of the relay was optimized to ensure concealment, and the power optimization problem of the transmitter was ignored; the literature [Shahzad K. Relaying via cooperative jamming in covert wireless communications[C] / / 201812thInternational Conference on Signal Processing and Communication Systems (ICSPCS). IEEE, 2018: 1-6.] studied the transmitter power optimization problem when the relay power is fixed. The optimization goal is the system's covert transmission rate. However, in this study, the relay is used as an interference node to send artificial noise. Although this can effectively improve the concealment effect of the transmitter, it ignores the concealment protection of the relay node and the optimizability of the relay power.

[0004] None of the above existing studies have jointly optimized transmitter and relay power. In fact, in a two-hop covert communication scenario, both the transmitter and the relay are senders of the communication signal. To achieve covert communication, the transmit power of both the transmitter and the relay must be properly set to ensure the covert performance of the entire system. Summary of the Invention

[0005] The present invention aims to solve the problems existing in the above-mentioned background technology and provide a power allocation method for a full-duplex relay-assisted two-hop covert communication system.

[0006] To achieve the purpose of the present invention, the present invention discloses a power allocation method for a full-duplex relay-assisted two-hop covert communication system, comprising the following steps:

[0007] Step 1: Describe and establish a mathematical model for a full-duplex relay-assisted two-hop covert communication scenario, and analyze the covert performance in this scenario.

[0008] Step 2: Based on the scenario mathematical model and concealment performance of step 1, a mathematical model for joint optimization of transmitter power and relay power is established with the goal of maximizing the system's concealed transmission rate.

[0009] Step 3: Analyze the joint optimization problem in step 2 and decompose it using analytical and graphical methods to obtain optimization subproblems that only consider the total power constraint, optimization subproblems that only consider the hidden constraint, and optimization subproblems that consider both the total power constraint and the hidden constraint.

[0010] Step 4: Based on step 3, solve the optimization sub-problem that only considers the total power constraint to obtain the optimal transmitter transmit power and relay transmit power in this case;

[0011] Step 5: Based on step 3, solve the optimization sub-problem that only considers the concealment constraint to obtain the optimal transmitter transmit power and relay transmit power in this case;

[0012] Step 6: Combining steps 2-5, a solution to the joint optimization problem of transmitter power and relay power established with the goal of maximizing the system's covert transmission rate is obtained, thereby obtaining a power allocation method for a full-duplex relay-assisted two-hop covert communication system.

[0013] Furthermore, in step 1, the scenario under consideration includes a transmitter named Alice, a legitimate receiver named Bob, a detector named Willie, and a full-duplex relay. The transmitter sends sensitive information to the legitimate receiver via a full-duplex relay, while the detector attempts to monitor whether the transmitter communicates with the full-duplex relay and whether the full-duplex relay communicates with the receiver. Because the relay operates in full-duplex mode, i.e., it can simultaneously receive and transmit information, when the system transmits a data packet, Alice first sends the data to the relay, which then forwards it to Willie.

[0014] Considering the AWGN channel, the channel parameter between node i and node j is defined as h ij , where ij can be ar, rb, rr, aw, or rw; therefore, h ar , h rb , h rr , h aw and h rwThey represent the channels from transmitter to full-duplex relay, full-duplex relay to legal receiver, full-duplex relay to full-duplex relay, transmitter to detector, and full-duplex relay to detector, respectively. ij | 2 Denotes the corresponding channel gain; where |h rr | 2 represents the self-interference channel gain of full-duplex relay, and the channel gain is obtained using the channel estimation method;

[0015] Consider a finite-length channel L. Alice and Bob pre-share a codebook and use a full-duplex relay to encrypt the communication over the available channel. Bob directly decodes the codebook. When a data packet is sent, the received signals at the full-duplex relay and Bob can be expressed as:

[0016]

[0017] Where, l=1,2,...,L l=1,2,...,L represents the channel index; P a and P r They represent the transmit power of Alice and Relay respectively; They represent the signal transmitted by Alice when the lth channel is used, the signal transmitted by the full-duplex relay when the lth channel is used, and the mean and variance of the full-duplex relay observed when the lth channel is used are 0 and 1, respectively. The complex Gaussian noise and the mean of the noise observed by Bob when he uses the lth channel are 0 and the variance is Complex Gaussian noise;

[0018] To analyze the two-hop stealth performance, the instantaneous signal-to-interference-and-noise ratios at the relay and Bob are expressed as:

[0019]

[0020] in, The instantaneous signal to interference and noise ratio from Alice to Bob is expressed as:

[0021]

[0022] Since Alice uses finite block length coding, the covert transmission rate of full-duplex relay in full-duplex mode is approximately:

[0023]

[0024] Among them, Q -1 (.) is the inverse Q function, L is the packet length, and δ is Bob’s packet error rate;

[0025] Since Willie does not know the key and password, he can only perform binary detection on whether Alice and Relay have sent data packets through environmental detection. The specific expression is:

[0026]

[0027] Among them, H0 is the null hypothesis, indicating that the system has no data packets sent in the current time slot; H1 is the alternative hypothesis, indicating that the system has data packets sent in the current time slot; It means that the mean value observed by Willie when using the lth channel is 0 and the variance is Complex Gaussian noise;

[0028] Use P0 and P1 to represent the probability distribution of Willie's observation channel under H0 and H1 respectively. Therefore, when the number of channels is L, the data received by Willie is The probability distribution is P0 L and P1 L When Alice has equal probability of being silent and sending data packets in each time slot, Willie’s minimum error detection probability is:

[0029]

[0030] in, represents the false alarm probability of Willie, that is, the judgment is H1 under the condition of H0; represents the probability of missed alarm of Willie, that is, the judgment is H0 under the condition of H1; V(P0 L ,P1 L ) represents the total variation of Willie's observation signal under the conditions of H0 and H1. According to Pinsker's inequality:

[0031]

[0032] Among them, D(P0 L ||P1 L ) is P0 L to P1 L The relative entropy of is also called KL divergence; according to the chain rule of relative entropy:

[0033] D(P0 L ||P1 L )=LD(P0||P1) (36)

[0034] Substituting formula (9) and formula (10) into formula (8), we can obtain the lower bound of Willie's minimum false detection probability based on KL divergence:

[0035]

[0036] In order to achieve covert communication between Alice and Bob, Willie needs to ensure that the judgment result based on the observed signal is close to randomly guessing whether Alice and Bob have communicated. That is, the required minimum false detection probability satisfies:

[0037]

[0038] Where ε is the tolerance level of covert communication when the prior probability is equal;

[0039] Combined with formula (11), we get a strict hidden constraint, which is expressed as

[0040]

[0041] in,

[0042]

[0043] f0(x) and f1(x) represent the probability density functions of the Willie received signal under conditions of H0 and H1, respectively. Considering the AWGN channel, they are expressed as

[0044]

[0045] Substituting formulas (14)-(16) into formula (13), we can obtain the hidden constraint:

[0046]

[0047] Among them, when x>0, ln(1+x)<x, the hidden constraint can be simplified to

[0048]

[0049] Further, we analyze the hidden constraints; let ξ=P a γ aw +P r γ rw , the hidden constraint can be expressed as

[0050]

[0051] Taking the derivative of f(ξ) with respect to ξ, we get

[0052]

[0053] This shows that f(ξ) is a monotonically increasing function of ξ, let Since ξ>0, we can obtain

[0054]

[0055] Then formula (20) is established, and ξ≤ξ * , that is, P a γ aw +P r γ rw ≤ξ * ; This shows that the implicit constraint (18) is equivalent to P a γ aw +P r γ rw ≤ξ * ,in

[0056] Furthermore, in step 2, a mathematical model for the joint optimization of transmitter power and relay power is established to maximize the system's covert transmission rate. This problem includes the following optimization objectives, optimization variables, and constraints:

[0057] Assuming that the packet length L and Bob's packet error rate δ are fixed values, for a reasonable packet length and packet error rate, it can be proved that the effective concealed transmission rate C FD Can be regarded as SINR FD A monotonically increasing function of FD SINR FD Partial derivatives of :

[0058]

[0059] Among them, FD refers to full-duplex relay, SINR FD C is the instantaneous signal-to-interference-and-noise ratio from Alice to Bob via full-duplex relay. FD It refers to the covert transmission rate from Alice to Bob via full-duplex relay;

[0060] Let satisfy the equation The signal-to-interference-noise ratio is Then we have:

[0061]

[0062] It can be observed that the equation is valid There is only one positive value and it can be ignored; hence, Established, that is, the effective transmission rate C FD With SINR FD increases with the increase of

[0063] Therefore, with the transmission power P of the transmitter Alice a and the transmit power P of the full-duplex relay r As variables, the optimization problem of maximizing the system's covert transmission rate is modeled as

[0064]

[0065] in, Constraint C1 is a concealment constraint, and constraint C2 is the total system power constraint. Together with C3 and C4, they constitute Alice's transmit power constraint and the full-duplex relay's transmit power constraint.

[0066] By analyzing the optimization objective function expression, we can find that, in addition to environmental factors, the signal-to-interference-noise ratio of the system is determined by the transmission power P of the transmitter Alice. a and the transmit power P of the full-duplex relay r Decide, and when P r After being fixed, the objective function changes with P a increases with the increase of; analyzing the constraints C1~C4, it is found that the two variables P a and P r is deeply coupled, when P r After fixation, P a =min{P total -P r ,(ξ * -P r γ rw ) / γ aw}; Therefore, the objective function can be regarded as a single variable P r Function, P r The value range is open interval (0, min{P total ,ξ * / γ rw At the endpoints of the interval, the objective function value is 0. According to Rolle's theorem, there must be an optimal transmitter transmit power and full-duplex relay transmit power in the open interval to maximize the system signal-to-interference-and-noise ratio.

[0067] Furthermore, in step 3, the optimization problem is decomposed into the following form using a graphical method based on the coupling of the constraints in step 2:

[0068] Optimization Problem Medium variable P a and P r There is a coupling relationship, which makes it difficult to solve directly. However, the constraints are all linear constraints. A graphical method is used to divide the feasible region into three categories:

[0069] 1) When When only the total power constraint is considered, the optimization problem can be rewritten as

[0070]

[0071] The optimal solution when only considering the total power constraint is set to

[0072] 2) When When only the implicit constraints are considered, the optimization problem can be rewritten as

[0073]

[0074] in,

[0075] The optimal solution when only considering the total power constraint is set to

[0076] 3) When When , it is necessary to consider the total power constraint and the concealment constraint compatibly; at this time, the intersection of the constraints C1 and C2 is in the feasible region, and the transmitter power at the intersection is The full-duplex relay transmission power is When the total power constraint and the hidden constraint need to be considered compatibly, the feasible region of the maximum point is: at the intersection of the constraints C1 and C2 (P ap ,P rp ) and only consider the optimal solution when the total power constraint is and the optimal solution when only the hidden constraints are considered Therefore, determine (P ap ,P rp ), and After finding the point set of the feasible domain of the three, the point that maximizes the objective function is the optimal solution in this case.

[0077] Furthermore, in step 4, for the optimization problem in step 3 It is further transformed into an equality constraint problem for analysis, and the optimization problem is obtained. The closed-form solution is as follows:

[0078] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter power is

[0079] Furthermore, in step 5, for the optimization problem in step 3 It is further transformed into an equality constraint problem for analysis, and the optimization problem is obtained. The closed-form solution is as follows:

[0080] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter power is

[0081] Furthermore, in step 6, the solution to the joint optimization problem of transmitter power and relay power established with the goal of maximizing the system's covert transmission rate is obtained, thereby obtaining the power allocation method for the full-duplex relay-assisted two-hop covert communication system as follows:

[0082] Combining steps 2-5, we get the optimization problem The solution is

[0083]

[0084] Among them, the set {(P a ,P r )} means (P ap ,P rp ), and The set of points in the inner feasible region of the three; (P ap ,P rp ) is the intersection of constraints C1 and C2; is the optimal solution when only the total power constraint is considered; is the optimal solution when only the concealment constraint is considered; this conclusion shows that the optimal transmitter power and relay power are given in the full-duplex relay-assisted two-hop covert communication system, which guides the configuration of the concealment power.

[0085] Compared with the existing technology, the significant progress of the present invention is: 1) The present invention derives and constructs a covert communication system model based on full-duplex relay, analyzes the minimum error detection probability of the detector based on divergence and the covert constraints that the system needs to meet, and constructs a joint optimization problem for the transmitter transmission power and the full-duplex relay transmission power with the goal of maximizing the system's covert transmission rate; 2) The present invention solves the joint optimization problem of the transmission power allocation of the transmitter and the full-duplex relay, gives a closed-form expression for the transmission power of the transmitter and the full-duplex relay, and provides a feasible power allocation idea.

[0086] In order to more clearly illustrate the functional characteristics and structural parameters of the present invention, further description is given below with reference to the accompanying drawings and specific embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0088] Figure 1 This is a schematic diagram of a dual-hop wireless full-duplex relay covert communication system model;

[0089] Figure 2 It is a diagram that divides the feasible domain of the joint optimization problem into three cases by graphical method;

[0090] Figure 3 This is a graph showing the relationship between the optimal transmitter transmit power and the optimal full-duplex relay transmit power and the maximum total system power.

[0091] Figure 4 This is a graph showing the relationship between the system's covert transmission rate and minimum error detection probability as a function of the total power limit level. DETAILED DESCRIPTION

[0092] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments; based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0093] A power allocation method for a full-duplex relay-assisted two-hop covert communication system, characterized by comprising the following steps:

[0094] Step 1: Describe and establish a mathematical model for a full-duplex relay-assisted two-hop covert communication scenario, and analyze the covert performance in this scenario.

[0095] Step 2: Based on the scenario mathematical model and concealment performance of step 1, a mathematical model for joint optimization of transmitter power and relay power is established with the goal of maximizing the system's concealed transmission rate.

[0096] Step 3: Analyze the joint optimization problem in step 2, decompose the problem using analytical and graphical methods, and obtain three sub-optimization sub-problems.

[0097] Step 4: Based on step 3, solve the optimization sub-problem 1, transform its inequality constraint problem into an equality constraint problem for analysis, and obtain the optimal transmitter transmit power and relay transmit power in this case.

[0098] Step 5 is similar to step 4. Based on step 3, the optimization sub-problem 2 is solved to obtain the optimal transmitter transmission power and relay transmission power in this case.

[0099] In step 6, combining steps 2, 3, and 4, the solution to the joint optimization problem of transmitter power and relay power is obtained with the goal of maximizing the system's covert transmission rate, thereby obtaining a power allocation method for a full-duplex relay-assisted two-hop covert communication system.

[0100] Specifically, step 1 is as follows:

[0101] Research scenarios such as Figure 1As shown, it includes a transmitter Alice, a legitimate receiver Bob, a detector Willie, and a full-duplex relay Relay. The transmitter sends sensitive information to the legitimate receiver through the full-duplex relay, and the detector attempts to monitor whether the transmitter communicates with the full-duplex relay and whether the full-duplex relay communicates with the receiver. Since the relay works in full-duplex mode, that is, it can receive and send information at the same time, when the system sends a data packet, Alice first sends the data to the Relay, and the Relay forwards it to Willie at the same time. This application considers the AWGN channel and defines the channel parameter between node i and node j as h ij , where ij can be ar, rb, rr, aw or rw. Therefore, h ar , h rb , h rr , h aw and h rw They represent the channels from transmitter to full-duplex relay, full-duplex relay to legal receiver, full-duplex relay to full-duplex relay, transmitter to detector, and full-duplex relay to detector, respectively. ij | 2 It represents the corresponding channel gain. rr | 2 represents the self-interference channel gain of full-duplex relay. The channel gain can be obtained using the channel estimation method.

[0102] Consider a finite-length channel L. Alice and Bob pre-share a codebook and use a full-duplex relay to encrypt the communication over the available channel. Bob then directly decodes the codebook. Specifically, when a data packet is sent, the received signals at the full-duplex relay and Bob can be expressed as:

[0103]

[0104] Where, l=1,2,...,L l=1,2,...,L represents the channel index; P a and P r They represent the transmit power of Alice and Relay respectively; They represent the signal transmitted by Alice when the lth channel is used, the signal transmitted by the full-duplex relay when the lth channel is used, and the mean and variance of the full-duplex relay observed when the lth channel is used are 0 and 1, respectively. The complex Gaussian noise and the mean of the noise observed by Bob when he uses the lth channel are 0 and the variance is complex Gaussian noise.

[0105] The concealment performance of two hops is analyzed below.

[0106] The instantaneous signal-to-interference-and-noise ratio at Relay and Bob can be expressed as:

[0107]

[0108] in, The instantaneous signal to interference and noise ratio from Alice to Bob can be expressed as:

[0109]

[0110] Since Alice uses finite block length coding, the concealed transmission rate in full-duplex relay mode can be approximated as:

[0111]

[0112] Among them, Q -1 (.) is the inverse Q function, L is the packet length, and δ is Bob's packet error rate.

[0113] Since Willie does not know the key and password, he can only perform binary detection by checking the environment to see if Alice and Relay have sent data packets. Specifically, it can be expressed as:

[0114]

[0115] Among them, H0 is the null hypothesis, indicating that the system has no data packets sent in the current time slot; H1 is the alternative hypothesis, indicating that the system has data packets sent in the current time slot; It means that the mean value observed by Willie when using the lth channel is 0 and the variance is complex Gaussian noise.

[0116] Use P0 and P1 to represent the probability distribution of Willie's observation channel under H0 and H1 respectively. Therefore, when the number of channels is L, the data received by Willie is The probability distribution is P0 L and P1 L When Alice has equal probability of being silent and sending data packets in each time slot, Willie's minimum error detection probability is:

[0117]

[0118] in, represents the false alarm probability of Willie, that is, the judgment is H1 under the condition of H0; represents the probability of missed alarm of Willie, that is, the judgment is H0 under the condition of H1; V(P0 L ,P1 L) represents the total variation of Willie's observation signal under the conditions of H0 and H1. According to Pinsker's inequality:

[0119]

[0120] Among them, D(P0 L ||P1 L ) is P0 L to P1 L The relative entropy of is also called KL divergence. According to the chain rule of relative entropy:

[0121] D(P0 L ||P1 L )=LD(P0||P1) (62)

[0122] Substituting formula (9) and formula (10) into formula (8), we can obtain the lower bound of Willie's minimum false detection probability based on KL divergence:

[0123]

[0124] In order to achieve covert communication between Alice and Bob, Willie needs to ensure that the judgment result based on the observed signal is close to randomly guessing whether Alice and Bob have communicated. That is, the required minimum false detection probability satisfies:

[0125]

[0126] Where ε is the tolerance level of covert communication when the prior probability is equal.

[0127] Combined with formula (11), we get a strict hidden constraint, which is expressed as

[0128]

[0129] in,

[0130]

[0131] f0(x) and f1(x) represent the probability density functions of the Willie received signal under the conditions of H0 and H1, respectively. Considering the AWGN channel, it can be expressed as

[0132]

[0133] Substituting formulas (14)-(16) into formula (13), we can obtain the hidden constraint:

[0134]

[0135] Among them, when x>0, ln(1+x)<x, the hidden constraint can be simplified to

[0136]

[0137] Further, let us analyze the hidden constraints. Let ξ=P a γ aw +P r γ rw , the hidden constraint can be expressed as

[0138]

[0139] Taking the derivative of f(ξ) with respect to ξ, we get

[0140]

[0141] This shows that f(ξ) is a monotonically increasing function of ξ. Let Since ξ>0, we can obtain

[0142]

[0143] Then formula (20) is established, and ξ≤ξ * , that is, P a γ aw +P r γ rw ≤ξ * . It shows that the hidden constraint (18) is equivalent to P a γ aw +P r γ rw ≤ξ * ,in

[0144] Specifically, the mathematical model for joint optimization of transmitter power and relay power described in step 2 is established to maximize the system's covert transmission rate. This problem includes the following optimization objectives, optimization variables, and constraints:

[0145] Assuming that the packet length L and Bob's packet error rate δ are fixed values, for a reasonable packet length and packet error rate, it can be proved that the effective concealed transmission rate C FD Can be regarded as SINR FD A monotonically increasing function of C. FD SINR FD Partial derivatives of :

[0146]

[0147] Among them, FD refers to full-duplex relay, SINR FD C is the instantaneous signal-to-interference-and-noise ratio from Alice to Bob via full-duplex relay.FD It refers to the covert transmission rate from Alice to Bob via full-duplex relay;

[0148] Let satisfy the equation The signal-to-interference-noise ratio is Then we have:

[0149]

[0150] It can be observed that the equation is valid There is only one positive value. When the packet error rate and data packet length take appropriate values, is a very small number and can be ignored. Therefore, Established, that is, the effective transmission rate C FD With SINR FD increases with the increase of .

[0151] Therefore, with the transmission power P of the transmitter Alice a and the transmit power P of the full-duplex relay r As variables, the optimization problem of maximizing the system's covert transmission rate can be modeled as

[0152]

[0153] in, Constraint C1 is a hidden constraint, and constraint C2 is the total system power constraint. Together with C3 and C4, they constitute Alice's transmit power constraint and the transmit power constraint of the full-duplex relay.

[0154] By analyzing the optimization objective function expression, we can find that, in addition to environmental factors, the signal-to-interference-and-noise ratio of the system is determined by the transmission power P of the transmitter Alice. a and the transmit power P of the full-duplex relay r Decide, and when P r After being fixed, the objective function changes with P a By analyzing the constraints C1 to C4, we can find that the two variables P a and P r is deeply coupled, when P r After fixation, P a =min{P total -P r ,(ξ * -P r γ rw ) / γ aw}. Therefore, the objective function can be regarded as a single variable P r Function, P r The value range is open interval (0, min{P total ,ξ* / γ rw At the endpoints of the interval, the objective function value is 0. According to Rolle's theorem, there must be an optimal transmitter transmit power and full-duplex relay transmit power within the open interval to maximize the system signal-to-interference-and-noise ratio.

[0155] Specifically, in step 3, based on the coupling of the constraints in step 2, the optimization problem is decomposed into the following parts using a graphical method:

[0156] Optimization Problem Medium variable P a and P r There is a coupling relationship, which makes it difficult to solve directly, but the constraints are all linear constraints. A graphical method is used to divide the feasible domain into three categories, as shown in the figure.

[0157] 1) When When, such as Figure 2 As shown in (a), considering only the total power constraint, the optimization problem can be rewritten as

[0158]

[0159] The optimal solution when only considering the total power constraint is set to

[0160] 2) When When, such as Figure 2 As shown in (b), considering only the implicit constraints, the optimization problem can be rewritten as

[0161]

[0162] in,

[0163] The optimal solution when only considering the total power constraint is set to

[0164] 3) When When, such as Figure 2 As shown in (c), it is necessary to consider the total power constraint and the concealment constraint. At this time, the intersection of the constraints C1 and C2 is within the feasible region, and the transmitter power at the intersection is The full-duplex relay transmission power is When the total power constraint and the hidden constraint need to be considered compatibly, the maximum point may be at the intersection of the constraints C1 and C2 (P ap ,P rp ) and only consider the optimal solution when the total power constraint is or the optimal solution when only the hidden constraints are considered The intersection of constraints C1 and C2 (P ap ,P rp) must be in the feasible region, and and is not necessarily in the feasible region, so determine (P ap ,P rp ), and After finding the point set of the feasible domain of the three, the point that maximizes the objective function is the optimal solution in this case.

[0165] Specifically, in step 4, for the optimization problem in step 3 It is further transformed into an equality constraint problem for analysis, and the optimization problem is obtained. The closed-form solution is as follows:

[0166] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter power is

[0167] The proof is as follows: Analyze the objective function, when P r When P is fixed, the objective function changes with a Therefore, the inequality constraint problem can be transformed into an equality constraint problem. The optimization problem can be rewritten as:

[0168]

[0169] sT a +P r =P total

[0170] P a >0

[0171] P r >0 (79)

[0172] P a =P total -P r Substituting into the objective function, the optimization problem is rewritten as:

[0173]

[0174] st0 <P r <P total (80)

[0175] Derivative the objective function and find two extreme points:

[0176]

[0177]

[0178] It can be verified that and Analyzing the objective function, we can see that It is the only maximum point in the interval and makes the objective function reach its maximum value.

[0179] Specifically, in step 5, for the optimization problem in step 3 It is further transformed into an equality constraint problem for analysis, and the optimization problem is obtained. The closed-form solution is as follows:

[0180] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter power is

[0181] The proof is as follows: Analyze the objective function, when P r When P is fixed, the objective function changes with a Therefore, the inequality constraint problem can be transformed into an equality constraint problem. The optimization problem can be rewritten as:

[0182]

[0183] Will Substituting into the objective function, the optimization problem is rewritten as:

[0184]

[0185] Derivative the objective function and find two extreme points:

[0186]

[0187]

[0188] It can be verified that and Analyzing the objective function, we can see that It is the only maximum point in the interval and makes the objective function reach its maximum value.

[0189] Specifically, in step 6, the solution to the joint optimization problem of transmitter power and relay power established with the goal of maximizing the system's covert transmission rate is obtained, thereby obtaining the power allocation method for the full-duplex relay-assisted two-hop covert communication system as follows:

[0190] Combining steps 2, 3, and 4, the solution to the optimization problem P0 is

[0191]

[0192] Among them, the set {(P a ,P r )} means (P ap ,P rp ), and The set of points in the inner feasible region of the three. (P ap ,P rp ) is the intersection of constraints C1 and C2; is the optimal solution when only the total power constraint is considered; is the optimal solution when only the concealment constraint is considered. This conclusion shows that the optimal transmitter power and relay power are given in the full-duplex relay-assisted two-hop covert communication system, which guides the configuration of the concealment power.

[0193] Based on this, the proposed algorithm is simulated and verified. Unless otherwise specified, some simulation parameters in the system are fixed as follows: the legitimate receiver Bob, the detector Willie, and the full-duplex relay Relay are distributed at (0, 0), (0, 150m),

[0194] (600m, 0) and (-20m, 50m), and the noise equations at each location are the same, both Wireless channel gain h ij =h0(d ij / d0) -α , d0 = 1m represents the reference distance, h0 = -40dB represents the channel gain at the reference distance, α ar =2.3α rb =2.5α aw =α rw =3.5 represents the loss index corresponding to each path. Figure 3 The optimal transmit power of the transmitter and the optimal transmit power of the full-duplex relay are plotted as a function of the system's maximum total power under different concealment tolerance levels. The figures show that the proposed algorithm and the exhaustive search method achieve identical results under different conditions, demonstrating the correctness of the theoretical derivation. Figure 4 The relationship between the system's covert transmission rate and Willie's minimum error detection probability and the total power limit level is given. Figure 4 (a) compares the system concealed transmission rate using the power allocation method of the present application and the system concealed transmission rate using the power average allocation method under the concealment conditions. It can be observed that the power allocation method of the present application can obtain a higher system concealed transmission rate, and the advantage is more obvious as the total power increases. Figure 4(b) compares the minimum error detection under the KL divergence constraint derived in this application (theoretical value) and the minimum error detection probability using likelihood ratio detection (simulated value). It can be observed that the minimum error detection under the KL divergence constraint derived in this application is the upper bound of the simulation value. As the total power increases, the difference between the simulation value and the theoretical value becomes larger, but It is the lower bound of the simulation value, which illustrates the feasibility of the hidden constraint based on KL divergence derived in this application.

[0195] It should be noted that, in this application, relational terms such as first and second, etc., are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprises," "comprising," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus comprising a series of elements includes not only those elements, but also other elements not explicitly listed, or elements inherent to such process, method, article, or apparatus.

[0196] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

Claims

1. A power allocation method for a full-duplex relay-assisted two-hop covert communication system, characterized in that: The following steps are involved: Step 1: Describe and establish a mathematical model for a full-duplex relay-assisted two-hop covert communication scenario, and analyze the covert performance in this scenario. Step 2: Based on the scenario mathematical model and concealment performance of step 1, a mathematical model for joint optimization of transmitter power and relay power is established with the goal of maximizing the system's concealed transmission rate. Step 3: Analyze the joint optimization problem in step 2 and decompose it using analytical and graphical methods to obtain optimization subproblems that only consider the total power constraint, optimization subproblems that only consider the hidden constraint, and optimization subproblems that consider both the total power constraint and the hidden constraint. Step 4: Based on step 3, solve the optimization sub-problem that only considers the total power constraint to obtain the optimal transmitter transmit power and relay transmit power in this case; Step 5: Based on step 3, solve the optimization sub-problem that only considers the concealment constraint to obtain the optimal transmitter transmit power and relay transmit power in this case; Step 6: Combining steps 2-5, a solution to the joint optimization problem of transmitter power and relay power established with the goal of maximizing the system's covert transmission rate is obtained, thereby obtaining a power allocation method for a full-duplex relay-assisted two-hop covert communication system. In step 1, the scenario involves a transmitter named Alice, a legitimate receiver named Bob, a detector named Willie, and a full-duplex relay. The transmitter sends sensitive information to the legitimate receiver via a full-duplex relay, while the detector attempts to monitor whether the transmitter communicates with the relay and vice versa. Because the relay operates in full-duplex mode, meaning it can simultaneously transmit and receive information, when the system sends a data packet, Alice first sends the data to the relay, which then forwards it to Willie. Considering the AWGN channel, the channel parameter between node i and node j is defined as h ij , where ij can be ar, rb, rr, aw, or rw; therefore, h ar , h rb , h rr , h aw and h rw They represent the channels from transmitter to full-duplex relay, full-duplex relay to legal receiver, full-duplex relay to full-duplex relay, transmitter to detector, and full-duplex relay to detector, respectively. ij | 2 Denotes the corresponding channel gain; where |h rr | 2 represents the self-interference channel gain of full-duplex relay, and the channel gain is obtained using the channel estimation method; Consider a finite-length channel L. Alice and Bob pre-share a codebook and use a full-duplex relay to encrypt the communication over the available channel. Bob directly decodes the codebook. When a data packet is sent, the received signals at the full-duplex relay and Bob can be expressed as: Wherein, l=1,2,...,Ll=1,2,...,L represents the channel index; P a and P r They represent the transmit power of Alice and Relay respectively; They represent the signal transmitted by Alice when the lth channel is used, the signal transmitted by the full-duplex relay when the lth channel is used, and the mean and variance of the full-duplex relay observed when the lth channel is used are 0 and 1, respectively. The complex Gaussian noise and the mean of the noise observed by Bob when he uses the lth channel are 0 and the variance is Complex Gaussian noise; To analyze the two-hop stealth performance, the instantaneous signal-to-interference-and-noise ratios at the relay and Bob are expressed as: in, The instantaneous signal to interference and noise ratio from Alice to Bob is expressed as: Since Alice uses finite block length coding, the covert transmission rate of full-duplex relay in full-duplex mode is approximately: Among them, Q -1 (.) is the inverse Q function, L is the packet length, and δ is Bob’s packet error rate; Since Willie does not know the key and password, he can only perform binary detection on whether Alice and Relay have sent data packets through environmental detection. The specific expression is: Among them, H0 is the null hypothesis, indicating that the system has no data packets sent in the current time slot; H1 is the alternative hypothesis, indicating that the system has data packets sent in the current time slot; It means that the mean value observed by Willie when using the lth channel is 0 and the variance is Complex Gaussian noise; Use P0 and P1 to represent the probability distribution of Willie's observation channel under H0 and H1 respectively. Therefore, when the number of channels is L, the data received by Willie is The probability distribution is P0 L and P1 L When Alice has equal probability of being silent and sending data packets in each time slot, Willie’s minimum error detection probability is: in, represents the false alarm probability of Willie, that is, the judgment is H1 under the condition of H0; represents the probability of missed alarm of Willie, that is, the judgment is H0 under the condition of H1; V(P0 L ,P1 L ) represents the total variation of Willie's observation signal under the conditions of H0 and H1. According to Pinsker's inequality: Among them, D(P0 L ||P1 L ) is P0 L to P1 L The relative entropy of is also called KL divergence; according to the chain rule of relative entropy: D(P0 L ||P1 L )=LD(P0||P1) (10) Substituting formula (9) and formula (10) into formula (8), we can obtain the lower bound of Willie's minimum false detection probability based on KL divergence: In order to achieve covert communication between Alice and Bob, Willie needs to ensure that the judgment result based on the observed signal is close to randomly guessing whether Alice and Bob have communicated. That is, the required minimum false detection probability satisfies: Where ε is the tolerance level of covert communication when the prior probability is equal; Combined with formula (11), we get a strict hidden constraint, which is expressed as in, f0(x) and f1(x) represent the probability density functions of the Willie received signal under conditions of H0 and H1, respectively. Considering the AWGN channel, they are expressed as Substituting formulas (14)-(16) into formula (13), we can obtain the hidden constraint: Among them, when x>0, ln(1+x)<x, the hidden constraint can be simplified to Further, we analyze the hidden constraints; let ξ=P a γ aw +P r γ rw , the hidden constraint can be expressed as Taking the derivative of f(ξ) with respect to ξ, we get This shows that f(ξ) is a monotonically increasing function of ξ, let Since ξ>0, we can obtain Then formula (20) is established, and ξ≤ξ * , that is, P a γ aw +P r γ rw ≤ξ * ; This shows that the implicit constraint (18) is equivalent to P a γ aw +P r γ rw ≤ξ * ,in In step 2, a mathematical model for the joint optimization of transmitter power and relay power is established to maximize the system's covert transmission rate. This problem includes the following optimization objectives, optimization variables, and constraints: Assume that the packet length L and Bob's packet error rate δ are fixed values, and prove that the effective concealed transmission rate C FD Can be considered as SINR FD A monotonically increasing function of FD SINR FD Partial derivatives of : Among them, FD refers to full-duplex relay, SINR FD C is the instantaneous signal-to-interference-and-noise ratio from Alice to Bob via full-duplex relay. FD It refers to the covert transmission rate from Alice to Bob via full-duplex relay; Let satisfy the equation The signal-to-interference-noise ratio is Then we have: It can be observed that the equation is valid There is only one positive value and it can be ignored; hence, Established, that is, the effective transmission rate C FD With SINR FD increases with the increase of Therefore, with the transmission power P of the transmitter Alice a and the transmit power P of the full-duplex relay r As variables, the optimization problem of maximizing the system's covert transmission rate is modeled as in, Constraint C1 is a concealment constraint, and constraint C2 is the total system power constraint. Together with C3 and C4, they constitute Alice's transmit power constraint and the full-duplex relay's transmit power constraint. By analyzing the optimization objective function expression, we can find that, in addition to environmental factors, the signal-to-interference-noise ratio of the system is determined by the transmission power P of the transmitter Alice. a and the transmit power P of the full-duplex relay r Decide, and when P r After being fixed, the objective function changes with P a increases with the increase of; analyzing the constraints C1~C4, it is found that the two variables P a and P r is deeply coupled, when P r After fixation, P a =min{P total -P r ,(ξ * -P r γ rw ) / γ aw }; Therefore, the objective function can be regarded as a single variable P r Function, P r The value range is open interval (0, min{P total ,ξ * / γ rw At the endpoints of the interval, the objective function value is 0. According to Rolle's theorem, there must be an optimal transmitter transmit power and full-duplex relay transmit power in the open interval to maximize the system signal-to-interference-and-noise ratio. In step 3, based on the coupling of the constraints in step 2, the optimization problem is decomposed into the following parts using a graphical method: Optimization Problem Medium variable P a and P r There is a coupling relationship, which makes it difficult to solve directly. However, the constraints are all linear constraints. A graphical method is used to divide the feasible region into three categories: 1) When When only the total power constraint is considered, the optimization problem can be rewritten as The optimal solution when only considering the total power constraint is set to 2) When When only the implicit constraints are considered, the optimization problem can be rewritten as in, The optimal solution when only considering the total power constraint is set to 3) When When , it is necessary to consider the total power constraint and the concealment constraint compatibly; at this time, the intersection of the constraints C1 and C2 is in the feasible region, and the transmitter power at the intersection is The full-duplex relay transmission power is When the total power constraint and the hidden constraint need to be considered compatibly, the feasible region of the maximum point is: at the intersection of the constraints C1 and C2 (P ap ,P rp ) and only consider the optimal solution when the total power constraint is and the optimal solution when only the hidden constraints are considered Therefore, determine (P ap ,P rp ), and After finding the set of points in the feasible region of the three, the point that maximizes the objective function is the optimal solution in this case; In step 4, for the optimization problem in step 3 It is further transformed into an equality constraint problem for analysis, and the optimization problem is obtained. The closed-form solution is as follows: When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter power is In step 5, for the optimization problem in step 3 It is further transformed into an equality constraint problem for analysis, and the optimization problem is obtained. The closed-form solution is as follows: When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter power is In step 6, the solution to the joint optimization problem of transmitter power and relay power, which aims to maximize the system's covert transmission rate, is obtained. The power allocation method for the full-duplex relay-assisted two-hop covert communication system is then obtained as follows: Combining steps 2-5, we get the optimization problem The solution is Among them, the set {(P a ,P r )} means (P ap ,P rp ), and The set of points in the inner feasible region of the three; (P ap ,P rp ) is the intersection of constraints C1 and C2; is the optimal solution when only the total power constraint is considered; is the optimal solution when only the hidden constraints are considered.

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