Double-hop covert communication system power distribution method assisted by full-duplex relay

Abstract

The invention discloses a power distribution method for a double-hop covert communication system assisted by a full-duplex relay. For a covert communication scene of a relay-based long-distance transmission and energy-limited system, the mode of equally distributing system energy to a sender and a relay is often not optimal. In order to solve the problem, a covert communication analysis framework based on a full duplex relay is constructed, and the minimum error detection probability of a detector and a covert constraint condition required to be met by the system are analyzed. Furthermore, the joint optimization problem of the transmission power distribution of the transmitter and the full-duplex relay is solved, and the closed expression of the transmission power of the transmitter and the full-duplex relay is solved. Furthermore, the correctness of theoretical analysis is verified through simulation of an MATLAB (Matrix Laboratory) platform. The result shows that the power distribution problem of the double-hop covert communication system with the limited total power is well solved, and the influence of transmission power distribution on the covert transmission rate is given so as to guide the design of a covert transmission strategy.

Description

Technical Field

[0001] The 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 increasing development of wireless communication technology, all walks of life in society are increasingly dependent on wireless networks. A large amount of private information is transmitted in wireless networks, and the security of its transmission has always attracted much attention. Although traditional encryption technology or increasingly mature physical layer security technology can effectively protect the legitimacy of communication users and the confidentiality of communication content, due to the broadcast characteristics of wireless transmission, the transmission of signals can still be detected, resulting in the exposure of communication behavior and even communication location, which brings security risks. Therefore, the covert wireless communication technology that uses a variety of signal processing methods to achieve low probability detection by the enemy has attracted the attention and research of many experts and scholars.

[0003] The purpose of covert wireless communication is to prevent communication behavior from being discovered, thereby achieving safer and more private communication. However, long-distance point-to-point transmission requires very high transmission power, making the implementation of covert transmission quite difficult. How to improve the concealment rate of long-distance transmission has attracted widespread attention from researchers. Related studies have shown that long-distance transmission assisted by relays is an effective way to achieve concealment. 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 power optimization problem of the transmitter when the relay power is fixed, and the optimization target is the system's concealed transmission rate. However, in this study, the relay is used as an interference node to send artificial noise. Although it 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 the transmitter power and relay power. In fact, in a double-hop covert communication scenario, the transmitter or relay is the sender of the communication signal. To achieve covert communication, the transmission power of the transmitter or relay needs to be reasonably 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 dual-hop covert communication system.

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

[0007] Step 1: describe and establish a mathematical model for the 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 concealed transmission rate of the system;

[0009] Step 3, analyze the joint optimization problem in step 2, and decompose the problem by analytical method and graphical method to obtain the optimization sub-problem that only considers the total power constraint, the optimization sub-problem that only considers the hidden constraint, and the optimization sub-problem that considers 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 hidden constraint to obtain the optimal transmitter transmit power and relay transmit power in this case;

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

[0013] Furthermore, in step 1, the research scenario includes a transmitter Alice, a legitimate receiver Bob, a detector Willie, and a full-duplex relay Relay, wherein 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;

[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 ar ,h rb ,h rr h rr ,h aw h aw and h rwrepresent 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 represents the corresponding channel gain; where |h rr | 2 represents the self-disturbance 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 share a codebook in advance and use a full-duplex relay to encrypt the communication through 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] Wherein, l=1,2,...,Ll=1,2,...,L represents the channel index; P a and P r They represent the transmission power of Alice and Relay respectively;

[0018]

[0019] 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 observed by the full-duplex relay when the llth channel is used. The complex Gaussian noise and the mean of the noise observed by Bob when using the lth channel are 0 and the variance is Complex Gaussian noise;

[0020] Analyzing the concealment performance of the two hops, the instantaneous signal to interference noise ratio at the relay and Bob is expressed as:

[0021]

[0022] in, The instantaneous signal to interference and noise ratio from Alice to Bob is expressed as:

[0023]

[0024] Since Alice uses finite block length coding, the concealed transmission rate of full-duplex relay in full-duplex mode is approximately:

[0025]

[0026] Among them, Q-1 (.) is the inverse Q function, L is the packet length, and δ is Bob’s packet error rate;

[0027] Since Willie does not know the key and codebook, he can only perform binary detection on whether Alice and Relay have sent data packets through environmental detection; specifically, it is expressed as:

[0028]

[0029] Among them, H0 is the null hypothesis, indicating that no data packets are sent in the current time slot of the system; 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;

[0030] P0 and P1 are used 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:

[0031]

[0032] in, It represents the false alarm probability of Willie, that is, the judgment is H1 under the condition of H0; represents the probability of Willie's missed alarm, that is, the judgment is H0 under the condition of H1; V(P0 L ,P1 L ) represents the total variation of Willie's observed signal under the conditions of H0 and H1. According to Pinsker's inequality:

[0033]

[0034] 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:

[0035] D(P0 L ||P1 L )=LD(P0||P1)(36)

[0036] 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:

[0037]

[0038] 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 the random guess of whether Alice and Bob have communicated, that is, the required minimum error detection probability satisfies:

[0039]

[0040] Among them, ε is the tolerance level of covert communication when the prior probability is equal;

[0041] Combined with formula (11), we get a strict hidden constraint, expressed as

[0042]

[0043] in,

[0044]

[0045] f0(x) and f1(x) represent the probability density functions of Willie received signals under the conditions of H0 and H1, respectively. Considering the AWGN channel, they are expressed as

[0046]

[0047] Substituting formulas (14)-(16) into formula (13), we can obtain the hidden constraint:

[0048]

[0049] Among them, when x>0, ln(1+x)<x, the hidden constraint can be simplified to

[0050]

[0051] Further, we analyze the hidden constraints; let ξ = P a γ aw +P r γ rw , the hidden constraint can be expressed as

[0052]

[0053] Taking the derivative of f(ξ) with respect to ξ, we get

[0054]

[0055] This shows that f(ξ) is a monotonically increasing function of ξ, let Since ξ>0, we can obtain

[0056]

[0057] Then formula (20) is valid, and ξ≤ξ * , that is, P a γ aw +P r γ rw ≤ξ * ; This shows that the hidden constraint (18) is equivalent to P a γ aw +P r γ rw ≤ξ * ,in

[0058] Furthermore, in step 2, a mathematical model for joint optimization of transmitter power and relay power is established to maximize the concealed transmission rate of the system. This problem includes the optimization objectives, optimization variables and constraints as follows:

[0059] 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; C FD SINR FD The partial derivative of :

[0060]

[0061] Let satisfy the equation The signal-to-interference-noise ratio is Then we have:

[0062]

[0063] Observe that the equation becomes 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

[0064] 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

[0065]

[0066] in, Constraint C1 is a hidden constraint, and constraint C2 is the total system power constraint. It, together with C3 and C4, constitutes Alice's transmit power constraint and the full-duplex relay Relay's transmit power constraint.

[0067] 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 The function of P r The value range is an 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, so that the system signal to interference and noise ratio can reach the maximum value.

[0068] Furthermore, in step 3, the optimization problem is decomposed into the following form using a graphical method for the coupling of the constraints in step 2:

[0069] 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. The feasible domain is divided into three categories using a graphical method:

[0070] 1) When When only the total power constraint is considered, the optimization problem can be rewritten as

[0071]

[0072] The optimal solution when only the total power constraint is considered is

[0073] 2) When When only the hidden constraints are considered, the optimization problem can be rewritten as

[0074]

[0075] in,

[0076] The optimal solution when only the total power constraint is considered is

[0077] 3) When When , the total power constraint and the hidden constraint need to be considered compatibly; at this time, the intersection of the constraints C1 and C2 is within the feasible domain, and the transmitter transmission power at the intersection is The full-duplex relay transmission power is When the total power constraint and the hidden constraint need to be compatible, 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 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.

[0078] 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 of is as follows:

[0079] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter transmission power is

[0080] 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 of is as follows:

[0081] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter transmission power is

[0082] Furthermore, in step 6, the solution to the joint optimization problem of transmitter power and relay power is obtained, thereby obtaining the power allocation method of the full-duplex relay-assisted two-hop covert communication system as follows:

[0083] Combining steps 2-5, we get the optimization problem The solution is

[0084]

[0085] Among them, the set {(P a ,P r )} means (P ap ,P rp ), and The set of points in the feasible domain 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; It 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.

[0086] Compared with the prior art, the significant progress of the present invention is that: 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 transmitter and full-duplex relay transmission power allocation, gives a closed-form expression for the transmitter and full-duplex relay transmission power, and provides a feasible power allocation idea.

[0087] In order to more clearly illustrate the functional characteristics and structural parameters of the present invention, further description is given below in conjunction with the accompanying drawings and specific implementation methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] 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:

[0089] Figure 1 It is a schematic diagram of a double-hop wireless full-duplex relay covert communication system model;

[0090] Figure 2 It is a diagram that divides the feasible domain of the joint optimization problem into three cases by graphical method;

[0091] Figure 3 It is a graph showing the relationship between the optimal transmitter transmission power and the optimal full-duplex relay transmission power and the maximum total power of the system;

[0092] Figure 4 It is a graph showing the relationship between the system's concealed transmission rate and minimum error detection probability as a function of the total power limit level. DETAILED DESCRIPTION

[0093] 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.

[0094] A power allocation method for a full-duplex relay-assisted dual-hop covert communication system, characterized in that it comprises the following steps:

[0095] Step 1, describe and establish a mathematical model for the full-duplex relay-assisted two-hop covert communication scenario, and analyze the covert performance in this scenario;

[0096] 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 concealed transmission rate of the system.

[0097] 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.

[0098] 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 transmission power and relay transmission power in this case.

[0099] Step 5, similar to step 4, solves optimization sub-problem 2 based on step 3 to obtain the optimal transmitter transmission power and relay transmission power in this case.

[0100] Step 6, combining steps 2, 3 and 4, obtains the solution to the joint optimization problem of transmitter power and relay power, thereby obtaining a power allocation method for a full-duplex relay-assisted two-hop covert communication system.

[0101] Specifically, step 1 is as follows:

[0102] 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, where 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 ar ,h rb ,h rr h rr ,h aw h aw and h rw 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-interfering channel gain of full-duplex relay. The channel gain can be obtained using the channel estimation method.

[0103] Consider a finite-length channel L. Alice and Bob share a codebook in advance and use a full-duplex relay to encrypt the communication through the available channel, while Bob 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:

[0104]

[0105] Wherein, l=1,2,...,Ll=1,2,...,L represents the channel index; P a and P r They represent the transmission 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 observed by the full-duplex relay when the llth channel is used. The complex Gaussian noise and the mean of the noise observed by Bob when using the lth channel are 0 and the variance is of complex Gaussian noise.

[0106] The concealment performance of two hops is analyzed below.

[0107] The instantaneous signal-to-interference-noise ratio at Relay and Bob can be expressed as:

[0108]

[0109] in, The instantaneous signal to noise ratio from Alice to Bob can be expressed as:

[0110]

[0111] Since Alice uses finite block length coding, the concealed transmission rate of full-duplex relay in full-duplex mode can be approximated as:

[0112]

[0113] Among them, Q -1 (.) is the inverse Q function, L is the packet length, and δ is Bob’s packet error rate.

[0114] 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. Specifically, it can be expressed as:

[0115]

[0116] Among them, H0 is the null hypothesis, indicating that no data packets are sent in the current time slot of the system; 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 of complex Gaussian noise.

[0117] P0 and P1 are used 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:

[0118]

[0119] in, It represents the false alarm probability of Willie, that is, the judgment is H1 under the condition of H0; represents the probability of Willie's missed alarm, that is, the judgment is H0 under the condition of H1; V(P0 L,P1 L ) represents the total variation of Willie's observed signal under the conditions of H0 and H1. According to Pinsker's inequality:

[0120] Among them, D(P0 L ||P1 L ) is P0 L To P1 L The relative entropy 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 the random guess of whether Alice and Bob have communicated, that is, the required minimum error detection probability satisfies:

[0125]

[0126] Among them, 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, expressed as

[0128]

[0129] in,

[0130]

[0131] f0(x) and f1(x) represent the probability density functions of Willie's 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, we 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 valid, and ξ≤ξ * , that is, P a γ aw +P r γ rw ≤ξ * . This 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 concealed transmission rate of the system. This problem includes the optimization objectives, optimization variables and constraints as follows:

[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 The partial derivative of :

[0146]

[0147] Let satisfy the equation The signal-to-interference-noise ratio is Then we have:

[0148]

[0149] Observe that the equation becomes 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 .

[0150] 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

[0151]

[0152] in, Constraint C1 is a hidden constraint, and constraint C2 is a system total power constraint. Constraint C1, C3, and C4 together constitute Alice's transmit power constraint and the full-duplex relay Relay's transmit power constraint.

[0153] By analyzing the optimization objective function expression, we can find that, in addition to environmental factors, the signal-to-interference-to-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 The function of P r The value range is an 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, so that the system signal to noise ratio reaches the maximum value.

[0154] Specifically, in step 3, the optimization problem is decomposed into the following form using a graphical method for the coupling of the constraints in step 2:

[0155] 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. The feasible domain is divided into three categories using a graphical method, as shown in the figure.

[0156] 1) When When Figure 2 As shown in (a), considering only the total power constraint, the optimization problem can be rewritten as

[0157]

[0158] The optimal solution when only the total power constraint is considered is

[0159] 2) When When Figure 2 As shown in (b), considering only the hidden constraints, the optimization problem can be rewritten as

[0160]

[0161] in,

[0162] The optimal solution when only the total power constraint is considered is

[0163] 3) When When Figure 2 As shown in (c), the total power constraint and the concealment constraint need to be considered compatible. At this time, the intersection of constraints C1 and C2 is within the feasible domain, 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 compatible, 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 within the feasible region, therefore, determine (P ap ,P rp ),

[0164] 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 of 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 transmission 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] P a =P total -P r Substituting into the objective function, the optimization problem is rewritten as:

[0170]

[0171] Derivative the objective function and find two extreme points:

[0172]

[0173] 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 the maximum value.

[0174] 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 of is as follows:

[0175] When satisfied When , the optimal full-duplex relay transmission power that maximizes the system's concealed transmission rate is Correspondingly, the optimal transmitter transmission power is

[0176] The proof is as follows: Analyze the objective function, when P rWhen 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:

[0177]

[0178] Will Substituting into the objective function, the optimization problem is rewritten as:

[0179]

[0180] Derivative the objective function and find two extreme points:

[0181]

[0182] 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 the maximum value.

[0183] Specifically, in step 6, the solution to the joint optimization problem of transmitter power and relay power is obtained, thereby obtaining the power allocation method of the full-duplex relay-assisted two-hop covert communication system as follows:

[0184] Combining steps 2, 3, and 4, we can get the optimization problem The solution is

[0185]

[0186] Among them, the set {(P a ,P r )} means (P ap ,P rp ), and The set of points in the feasible domain of the three. 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.

[0187] Based on this, the algorithm proposed in this application 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),

[0188] (600m, 0) and (-20m, 50m), and the noise equations at each location are the same, all of which are 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 transmission power of the transmitter and the optimal transmission power of the full-duplex relay are given in relation to the maximum total power of the system under different concealment tolerance levels. It can be observed from the figure that the values ​​of the algorithm proposed in this application and the exhaustive search method are exactly the same under different conditions, which shows the correctness of the theoretical derivation of this application. Figure 4 The relationship between the system's concealed transmission rate and Willie's minimum error detection probability and the total power limit level is given. Figure 4 (a) The concealed transmission rate of the system using the power allocation method of the present application is compared with the concealed transmission rate of the system using the power average allocation method under the concealed condition. 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) The minimum error detection under the KL divergence constraint derived in this application (theoretical value) is compared with the minimum error detection probability (simulated value) using likelihood ratio detection. 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.

[0189] It should be noted that, in this application, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device.

[0190] Although 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 the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A power allocation method for a full-duplex relay-assisted dual-hop covert communication system, characterized in that: The following steps are involved: Step 1: describe and establish a mathematical model for the 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 concealed transmission rate of the system; Step 3, analyze the joint optimization problem in step 2, and decompose the problem by analytical method and graphical method to obtain the optimization sub-problem that only considers the total power constraint, the optimization sub-problem that only considers the hidden constraint, and the optimization sub-problem that considers 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 hidden constraint to obtain the optimal transmitter transmit power and relay transmit power in this case; Step 6: Combining steps 2-5, the solution to the joint optimization problem of transmitter transmission power and relay transmission power is obtained, thereby obtaining a power allocation method for a full-duplex relay-assisted dual-hop covert communication system.

2. The power allocation method for a full-duplex relay-assisted dual-hop covert communication system according to claim 1, characterized in that: In step 1, the study scenario includes the transmitter Alice, the legitimate receiver Bob, the detector Willie, and the 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. 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 ar ,h rb ,h rr h rr ,h aw h aw and h rw 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 represents the corresponding channel gain; where |h rr | 2 represents the self-disturbance 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 share a codebook in advance and use a full-duplex relay to encrypt the communication through 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 transmission 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 observed by the full-duplex relay when the llth channel is used. The complex Gaussian noise and the mean of the noise observed by Bob when using the lth channel are 0 and the variance is Complex Gaussian noise; Analyzing the concealment performance of the two hops, the instantaneous signal to interference noise ratio at the relay and Bob is expressed as: in, The instantaneous signal to noise ratio from Alice to Bob is expressed as: Since Alice uses finite block length coding, the concealed 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 codebook, he can only perform binary detection on whether Alice and Relay have sent data packets through environmental detection; specifically, it is expressed as: Among them, H0 is the null hypothesis, indicating that no data packets are sent in the current time slot of the system; 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; P0 and P1 are used 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, It represents the false alarm probability of Willie, that is, the judgment is H1 under the condition of H0; represents the probability of Willie's missed alarm, that is, the judgment is H0 under the condition of H1; V(P0 L ,P1 L ) represents the total variation of Willie's observed 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 the random guess of whether Alice and Bob have communicated, that is, the required minimum error detection probability satisfies: Among them, ε is the tolerance level of covert communication when the prior probability is equal; Combined with formula (11), we get a strict hidden constraint, expressed as in, f0(x) and f1(x) represent the probability density functions of Willie received signals under the 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 valid, and ξ≤ξ * , that is, P a γ aw +P r γ rw ≤ξ * ; This shows that the hidden constraint (18) is equivalent to P a γ aw +P r γ rw ≤ξ * ,in 3. The power allocation method for a full-duplex relay-assisted dual-hop covert communication system according to claim 1, characterized in that: In step 2, a mathematical model for joint optimization of transmitter power and relay power is established to maximize the system's concealed transmission rate. This problem includes the optimization objectives, optimization variables, and constraints as follows: Assume that the packet length L and Bob's packet error rate δ are both fixed values, and prove that the effective concealed transmission rate C FD Can be considered as SINR FD A monotonically increasing function; C FD SINR FD The partial derivative of : Let satisfy the equation The signal-to-interference-noise ratio is Then we have: Observe that the equation becomes 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 hidden constraint, and constraint C2 is the total system power constraint. It, together with C3 and C4, constitutes Alice's transmit power constraint and the full-duplex relay 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 The function of P r The value range is an 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, so that the system signal to interference and noise ratio can reach the maximum value.

4. The power allocation method for a full-duplex relay-assisted dual-hop covert communication system according to claim 1, characterized in that: 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: 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. The feasible domain is divided into three categories using a graphical method: 1) When When only the total power constraint is considered, the optimization problem can be rewritten as The optimal solution when only the total power constraint is considered is 2) When When only the hidden constraints are considered, the optimization problem can be rewritten as in, The optimal solution when only the total power constraint is considered is 3) When When , the total power constraint and the hidden constraint need to be considered compatibly; at this time, the intersection of the constraints C1 and C2 is within the feasible domain, and the transmitter transmission power at the intersection is The full-duplex relay transmission power is When the total power constraint and the hidden constraint need to be compatible, 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 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.

5. The power allocation method for a full-duplex relay-assisted dual-hop covert communication system according to claim 1, characterized in that: 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 of 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 transmission power is 6. The power allocation method for a full-duplex relay-assisted dual-hop covert communication system according to claim 1, characterized in that: 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 of 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 transmission power is 7. The power allocation method for a full-duplex relay-assisted dual-hop covert communication system according to claim 1, characterized in that: In step 6, the solution to the joint optimization problem of transmitter power and relay power is obtained, so that the power allocation method of the full-duplex relay-assisted two-hop covert communication system is 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 feasible domain 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; It 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.

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