CN113726472A - Simultaneous interference and eavesdropping method based on Bayesian Stackelberg game - Google Patents

Simultaneous interference and eavesdropping method based on Bayesian Stackelberg game Download PDF

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CN113726472A
CN113726472A CN202111053174.0A CN202111053174A CN113726472A CN 113726472 A CN113726472 A CN 113726472A CN 202111053174 A CN202111053174 A CN 202111053174A CN 113726472 A CN113726472 A CN 113726472A
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CN113726472B (en
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王伟
刘一甲
戚楠
黄叶婷
王可为
苏悦悦
黄赞奇
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Nanjing University of Aeronautics and Astronautics
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04K—SECRET COMMUNICATION; JAMMING OF COMMUNICATION
    • H04K3/00—Jamming of communication; Counter-measures
    • H04K3/40—Jamming having variable characteristics
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04K—SECRET COMMUNICATION; JAMMING OF COMMUNICATION
    • H04K3/00—Jamming of communication; Counter-measures
    • H04K3/40—Jamming having variable characteristics
    • H04K3/45—Jamming having variable characteristics characterized by including monitoring of the target or target signal, e.g. in reactive jammers or follower jammers for example by means of an alternation of jamming phases and monitoring phases, called "look-through mode"
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04K—SECRET COMMUNICATION; JAMMING OF COMMUNICATION
    • H04K3/00—Jamming of communication; Counter-measures
    • H04K3/60—Jamming involving special techniques
    • H04K3/62—Jamming involving special techniques by exposing communication, processing or storing systems to electromagnetic wave radiation, e.g. causing disturbance, disruption or damage of electronic circuits, or causing external injection of faults in the information
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    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
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    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

The invention discloses a simultaneous interference and eavesdropping method based on a Bayesian Stackelberg game, which comprises the following steps: scene modeling: establishing an confrontation scene model based on the intelligent jammer of the party and the communication user pair of the enemy; game modeling: by utilizing a full-duplex technology, the communication countermeasure of the users of the enemy and the my parties under the incomplete information condition is modeled into a Bayesian Stackelberg game model, and the problem of simultaneously implementing interference and eavesdropping is converted into a game optimization problem; and (3) optimizing and solving: and transforming a non-convex optimization problem of the leader and the follower by adopting a continuous convex approximation SCA, and solving a Bayesian Stackelberg game equilibrium solution by using a KKT condition. Compared with half-duplex, single interference and single interception schemes, the simultaneous interference and interception simulation provided by the invention shows that the method has good accuracy and convergence and is superior to other schemes.

Description

Simultaneous interference and eavesdropping method based on Bayesian Stackelberg game
Technical Field
The invention belongs to the technical field of electronic battlefield wireless communication countermeasure, and particularly relates to a simultaneous interference and eavesdropping method based on a Bayesian Stackelberg game.
Background
In recent years, in the field of military wireless communication, there is an urgent need to monitor tactical information sent by an enemy transmitter to a target receiver in time and immediately interrupt transmission when necessary. The information acquisition advantage is an important factor for determining the battlefield advantage, and in the face of an opponent with intelligent anti-interference capability, the simple electromagnetic interference is not enough to exert effective lethality, and meanwhile, the effective information of the opposite user is difficult to acquire by simple eavesdropping.
Full duplex technology is of great advantage in meeting the above requirements because it facilitates simultaneous jamming and eavesdropping. Therefore, the development of power strategies for simultaneous interference and eavesdropping is also a current research hotspot. Based on the full-duplex simultaneous interference and eavesdropping technology, as an emerging hot problem, the method still has an insufficiently explored research direction at present. Some existing studies can be divided into two study directions, namely theoretical and experimental.
In theory, t.riihonen introduced the aggressive application of simultaneous transmit and receive capabilities in 2017, and this capability enabled joint interference and perception in hostile situations. Mietzner, in 2012, studied responsive attack applications to protect vehicles from radio controlled explosives. L.kong studied in 2016 the physical security problem in the presence of an active eavesdropper that could eavesdrop on user data transmissions while releasing interfering signals, and derived therein the privacy disruption probability of the victim node.
Experimentally, several laboratory experimental works were performed on a generic software radio defined radio, which verified the feasibility of full duplex simultaneous jamming and eavesdropping techniques. However, the currently published works focus on eavesdropping on the signal to interference and noise ratio on the link, i.e. the listening effect, ignoring the interference effect and the fact that the opposite user may be intelligent. Yang innovatively modeled the power control problem as the Stackelberg game in 2013. Firstly, the optimal response strategy of the jammer (as a follower) is estimated, and the optimal strategy of the leader is determined on the basis of the optimal response strategy. In 2017, tang studies the problem that energy-saving transmission with security has a full-duplex active eavesdropper under the Stackelberg game framework, which improves the defense against simultaneous eavesdropping and interference, but none of these works stands on the ground of an attacker to study the formulation of simultaneous interference and eavesdropping strategies.
In electronic countermeasures, electromagnetic interference alone is not sufficient to exert an effective killing force. Meanwhile, it is difficult to obtain effective information of the opposite user by simple eavesdropping. The traditional electromagnetic countermeasure technology is difficult to meet the requirements of communication battlefields. In summary, the existing wireless communication countermeasure model and angle mainly have the following problems: 1) incompleteness of channel information in a hostile environment is not considered. 2) Without considering the intelligence of the user, the intelligent user can change own policy dynamically according to the policy of the other party. 3) And the method considers how efficiently the simultaneous interference and interception strategies are formulated from the perspective of an attacker.
Disclosure of Invention
The invention aims to solve the technical problem of providing a simultaneous interference and interception method based on a Bayesian Stackelberg game, which utilizes a full duplex technology to model the communication countermeasure of the users of both the enemy and the my into the Bayesian Stackelberg game under the incomplete information condition, realizes simultaneous interference and interception, converts the non-convex optimization problem of a leader and a follower through continuous convex approximation, and solves the Stackelberg game balance through the KKT condition.
In order to achieve the technical purpose, the technical scheme adopted by the invention is as follows:
a simultaneous interference and eavesdropping method based on a Bayesian Stackelberg game comprises the following steps:
step 1: scene modeling: establishing an confrontation scene model based on the intelligent jammer of the party and the communication user pair of the enemy;
step 2: game modeling: based on the confrontation scene model in the step 1, by utilizing a full-duplex technology, the communication confrontation of the users of the two enemies and the two parties under the incomplete information condition is modeled into a Bayesian Stackelberg game model, a leader and a follower are determined, the problem of simultaneously implementing interference and eavesdropping is converted into a game optimization problem, and the game optimization problem is a non-convex optimization problem of the leader and the follower;
and step 3: and (3) optimizing and solving: and transforming a non-convex optimization problem of the leader and the follower by adopting a continuous convex approximation SCA, and solving a Bayesian Stackelberg game equilibrium solution by using a KKT condition.
In order to optimize the technical scheme, the specific measures adopted further comprise:
in the confrontation scene model in step 1, the two game parties are a pair of the intelligent jammer of my party and the communication user of the enemy party, wherein the working mode of the intelligent jammer of my party is full duplex, the interference is released, the eavesdropping is carried out on the communication user of the enemy party, the communication user of the enemy party has a pair of communication transceiving pairs, and the information is transmitted in real time, specifically:
r represents the number of the intelligent jammer of our party, S-D represents the number of the enemy communication transceiving pair, in the countermeasure scene, the signal transmission between any two nodes has path loss and small-scale fading at the same time, the two users hardly know the exact channel state information of the opposite party, and the two users have incomplete cognition on the small-channel fading;
for the same channel, two different probabilities are needed to represent the uncertainty of the channel power gain, b and e respectively represent the R and the S-D communication pair of the enemy, and a belongs to the group of b and e;
a is the cognitive set of small-scale fading gains from node x to node y:
Figure BDA0003253573840000021
Figure BDA0003253573840000031
the Z small scale fading gain value under certain probability is shown, Z belongs to {1,2, …, Z } represents the index number of the set,
Figure BDA0003253573840000032
the nodes S, d and R respectively represent a transmitter S, a receiver R and a full-duplex node R; "-" is an exclude operator;
therefore, based on the knowledge of a, the channel gains from node x to node y are:
Figure BDA0003253573840000033
wherein the content of the first and second substances,
Figure BDA0003253573840000034
for free space path loss, α is the path loss coefficient, dx，yIs the distance from node x to node y;
because intelligent jammer R adopts full-duplex communication mode, can have certain self-interference, and the cognitive set of a to R's self-interference channel gain is:
Figure BDA0003253573840000035
where N ∈ {1, 2., N } represents a set index number,
Figure BDA0003253573840000036
taking the value of the nth self-interference fading gain;
thus, based on the knowledge of a, the self-interference channel gain is defined as:
Figure BDA0003253573840000037
wherein k is0Is a self-interference cancellation factor.
In the countermeasure scene model in step 1, a cognitive signal-to-interference-and-noise ratio in the countermeasure scene where interference and eavesdropping exist is further defined, specifically:
in the S-D tactical communication process, the node D can be interfered and intercepted by a full-duplex jammer R, and based on the cognition of a, the signals received by the node D are as follows:
Figure BDA0003253573840000038
wherein
Figure BDA0003253573840000039
Is the signal received from S;
Figure BDA00032535738400000310
is an interference signal of node R; i belongs to {1,2, …, I }, I is a discrete sample of the small fading gain between S and D, J belongs to {1,2, …, J }, and J is a discrete sample of the small fading gain between R and D; p is a radical ofsAnd prRespectively, the transmission power of the node S and the interference power of the node R; x is the number ofsAnd xrTransmit signals of S and R, respectively; n is1Is additive white gaussian noise;
thus, a 'S knowledge of the received SINR at node D is a two-dimensional random variable, depending on a' S knowledge of the S-D and R-D channel gains, respectively denoted as
Figure BDA0003253573840000041
And
Figure BDA0003253573840000042
therefore, a considers the received signal-to-interference-and-noise ratio at node D to be:
Figure BDA0003253573840000043
wherein the content of the first and second substances,
Figure BDA0003253573840000044
N1is the unilateral power spectral density of the gaussian noise at node D; b issIs the S-D channel bandwidth;
to monitor what tactical data is sent by S to its target recipient D, the full-duplex node R eavesdrops on the S-D transmission signal, so the signal received at R consists of an eavesdropping signal and a self-interference signal, and the knowledge of a on the signal received at R is expressed as:
Figure BDA0003253573840000045
wherein
Figure BDA0003253573840000046
Is an eavesdropping signal;
Figure BDA0003253573840000047
is a self-interference signal; m is belonged to {1,2, …, M }, and M is
The S-R channel gains one discrete sample; n is2Is additive white gaussian noise;
since the knowledge of a on S-R and self-interference channel gains are respectively
Figure BDA0003253573840000048
And
Figure BDA0003253573840000049
thus, the eavesdropping signal-to-interference-and-noise ratio of node R is:
Figure BDA00032535738400000410
in the formula
Figure BDA00032535738400000411
N2A single-sided power spectral density that is gaussian noise;
assuming that the R-D channel bandwidth is the same as the S-D channel, denoted BsIn addition, with
Figure BDA00032535738400000412
In a similar manner to that described above,
Figure BDA00032535738400000413
is a two-dimensional random variable that depends on the knowledge of a on the S-R and self-interference channel gains.
The communication countermeasure of the users of the two enemies and the me under the incomplete information condition is a layered countermeasure process of the following power domain:
the behavior of S-D is taken as a leader, action is taken firstly, and the intelligent jammer R of the client is a follower, and action is taken after S-D, specifically:
the leader S executes tactical communications and adjusts its power policy, with the goal of ensuring secure communications by reducing the data rate overheard by jammers, and facing interference threats by increasing tactical communications capacity;
after observing the power policy of the leader S, the follower R releases the interfering signal to force S to increase its transmit power, allowing R to eavesdrop on the S-D transmission and also force D to receive its intended signal at a lower rate;
the intelligent jammer R learns the transmitting power of the S and adaptively adjusts the interference power of the intelligent jammer R so as to maximize the utility;
in addition, both enemy and my parties have only incomplete channel condition information, including interference and eavesdropping on the link.
In the step 2, based on channel cognition of the user and the opponent, a communication countermeasure process of the users of the two enemies and the me under the incomplete information condition is described through a Bayesian Stackelberg game, and a Bayesian Stackelberg game model is constructed, specifically:
the goal of node S is to increase the S-D tactical communication rate while reducing the data rate eavesdropped by R at a lower data transmission power cost, so the utility of transmitting node S is related to the S-D tactical communication rate, the data transmission power and the data rate eavesdropped by R;
the utility of S is defined as:
Figure BDA0003253573840000051
wherein D issIs a normal number, guaranteed UsIs positive, can be independently and autonomously determined by STaking a value;
Figure BDA0003253573840000052
expected S-D channel capacity for an adversary;
Figure BDA0003253573840000053
is the expected S-R eavesdropping channel rate of enemy e; thetasIs an interception factor which represents the attention degree of an enemy to the data rate intercepted by the R; etaspsIs the power cost at S; etasIs the cost per unit power at S;
compared with S, the full-duplex node R aims to realize higher S-R eavesdropping rate and reduce S-D transmission with lower power cost;
the utility of the interfering node R is defined as:
Figure BDA0003253573840000054
wherein D isrIs a normal number to ensure UrIs positive;
Figure BDA0003253573840000055
is the expected eavesdropping rate of R;
Figure BDA0003253573840000056
is the expected channel capacity of R versus S-D; thetarIs a rate suppression factor for R concerned about the extent to which the communication quality of its opponent is reduced, larger thetarIt is stated that R focuses more on suppressing S-D communication; etarprIs the power cost of R; etarA cost per unit power of R;
based on Bayes formula, gives
Figure BDA0003253573840000057
And
Figure BDA0003253573840000058
the specific definition of (1):
is provided with
Figure BDA0003253573840000061
Denotes the cognitive acquisition of a for the received signal to interference and noise ratio at R
Figure BDA0003253573840000062
Of value of (a), wherein
Figure BDA0003253573840000063
Thus, definition a for
Figure BDA0003253573840000064
The expected values of (c) are:
Figure BDA0003253573840000065
wherein the content of the first and second substances,
Figure BDA0003253573840000066
and
Figure BDA0003253573840000067
fractional S-D and R-D small channel fading gain acquisition
Figure BDA0003253573840000068
And
Figure BDA0003253573840000069
the probability of (d);
suppose that
Figure BDA00032535738400000610
And
Figure BDA00032535738400000611
is independent, therefore
Figure BDA00032535738400000612
According to
Figure BDA00032535738400000613
Defining eavesdropping rate
Figure BDA00032535738400000614
Comprises the following steps:
Figure BDA00032535738400000615
wherein the content of the first and second substances,
Figure BDA00032535738400000616
for eavesdropping signal-to-interference-and-noise ratio acquisition
Figure BDA00032535738400000617
Probability of value, and
Figure BDA00032535738400000618
suppose that
Figure BDA00032535738400000619
And
Figure BDA00032535738400000620
is independent, therefore
Figure BDA00032535738400000621
And
Figure BDA00032535738400000622
respectively S-R and self-interference channel gain taking
Figure BDA00032535738400000623
And
Figure BDA00032535738400000624
the probability of (c).
The step 3 specifically includes the following steps:
step 3-1: constructing an optimization problem model: respectively establishing an optimization problem model of a follower intelligent jammer R and an optimization problem model of a leader S based on the interference power of the R and the data transmission power of the S;
step 3-2: establishing a Stackelberg balance based on an optimization problem model, and proving the existence of the Stackelberg balance;
step 3-3: transforming a non-convex problem of the optimization problem model by using continuous convex approximation, decomposing the non-convex problem into a series of sub-convex functions, and solving the sub-convex functions through a KKT condition;
step 3-4: and solving the Stackelberg balance by adopting an inverse induction method based on the solution of the sub-convex function.
Constructing an optimization problem model in the step 3-1: respectively establishing an optimization problem model of a follower intelligent jammer R and an optimization problem model of a leader S based on the interference power of the follower intelligent jammer R and the data transmission power of the leader S, and specifically comprising the following steps of:
for the follower my intelligent jammer R, the optimization problem is defined as follows:
P1：
Figure BDA0003253573840000071
s.t.0＜pr≤pr，max
wherein p isr，maxIs the maximum interference power;
optimizing problem for leader S due to channel expected capacity of enemy communication user
Figure BDA0003253573840000072
Requiring more than a threshold value gamma0Namely:
Figure BDA0003253573840000073
due to the fact that
Figure BDA0003253573840000074
Relative to psIs monotonically increasing, so that p is presents，minWhen p iss≥ps，minWhen the temperature of the water is higher than the set temperature,
Figure BDA0003253573840000075
furthermore, psLess than maximum transmission power ps，max；
Thus, the optimization problem defining the leader S is:
P2：
Figure BDA0003253573840000076
s.t.ps，min＜ps≤ps，max。
the step 3-2 of constructing a Stackelberg balance based on the optimization problem model, and proving the existence of the Stackelberg balance specifically includes:
by using
Figure BDA0003253573840000077
And
Figure BDA0003253573840000078
represent the solutions of the optimization problems P1 and P2, respectively
Figure BDA0003253573840000079
And
Figure BDA00032535738400000710
satisfies the following conditions:
Figure BDA00032535738400000711
Figure BDA00032535738400000712
Figure BDA00032535738400000713
forming a Stackelberg balance;
thus, the existence of the above Stackelberg game balance is demonstrated as follows:
the follower optimization problem can be approximated by continuous convexity to convex form, thusIt has an asymptotic optimal solution, denoted as
Figure BDA00032535738400000714
Thus, given any S-strategy psThe following is true:
Figure BDA00032535738400000715
by using
Figure BDA00032535738400000716
Substituting to obtain
Figure BDA00032535738400000717
Similarly, the leader optimization problem can be approximated to be convex by continuous convex, and the progressive optimal solution of the leader optimization problem is recorded as
Figure BDA0003253573840000081
Thus, for any given R-policy prExistence of
Figure BDA0003253573840000082
By using
Figure BDA0003253573840000083
Is substituted to obtain
Figure BDA0003253573840000084
The non-convex problem of the optimization problem model is transformed by using the continuous convex approximation in the step 3-3, the non-convex problem is decomposed into a series of sub-convex functions, and the sub-convex functions are solved through the KKT condition, which specifically comprises the following steps:
1) and (3) problem decomposition: and (3) performing first-order Taylor expansion on the utility function of P1 to iteratively approximate a convex function of the utility function:
in each iteration, Ur(pr，ps) The approximate value of (d) is expressed as:
Figure BDA0003253573840000085
wherein, thetarIs that
Figure BDA0003253573840000086
In that
Figure BDA0003253573840000087
The first-order Taylor expansion form is as follows:
Figure BDA0003253573840000088
wherein phi1Is that
Figure BDA0003253573840000089
In that
Figure BDA00032535738400000810
Function value of (phi)2Is that
Figure BDA00032535738400000811
In that
Figure BDA00032535738400000812
A first derivative value of (d);
therefore, one further approximation problem to get P1 is:
SCP1:
Figure BDA00032535738400000813
s.t.pr≤pr,max
by solving for SCP1, the newly obtained
Figure BDA00032535738400000814
Taking the point as the next Taylor expansion point, and starting new iteration until the maximum iteration number is reached, or keeping the point unchanged;
when iteration stops, p of the last iteration roundrIs assigned to
Figure BDA00032535738400000815
2) Solving the sub-convex problem: the spread of any concave function at any point is the global upper bound, hence, U'r(pr，ps) As an original objective function Ur(pr，ps) By iteratively solving SCP1, the original objective function in P1 is approximated, thereby approximating P1:
for the convex optimization problem SCP1, the lagrange function is introduced as follows:
Lr(pr，ps)＝U′r(pr，ps)+λr(pr，max-pr)-μrpr
wherein λrAnd murIs the Lagrangian multiplier;
due to Lr(pr，ps) Is a concave function, so the dual gap between the lagrange dual problem and the original problem is zero;
then, under KKT conditions, a solution of SCP1 is obtained
Figure BDA0003253573840000091
Likewise, the same steps are taken to solve for P2.
The solution based on the sub-convex function in the step 3-4 is solved by adopting an inverse induction method to solve the Stackelberg equilibrium, and the specific solution is carried out in an iterative process according to the following steps:
a) setting S initial iteration power, an initial Taylor expansion point of S and an initial Taylor expansion point of R;
b) solving a follower sub-convex function according to the S initial iteration power and the R initial Taylor expansion point to obtain a solution of the follower sub-convex function of the current round;
c) taking the solution of the previous follower sub-convex function as a new Taylor expansion point, and iteratively solving the follower sub-convex function until convergence;
d) taking the solution of the follower sub-convex function of the last round as input, and solving the leader sub-convex function through combining the initial Taylor expansion point of the sum S to obtain the solution of the leader sub-convex function of the round;
e) taking the solution of the previous leader sub-convex function as a new round of Taylor expansion point, iteratively solving the leader sub-convex function until convergence, and replacing the S initial iteration power with the solution of the leader sub-convex function of the last round;
f) repeating the steps b) and e) until convergence, and obtaining a Stackelberg equilibrium solution
Figure BDA0003253573840000092
The invention has the following beneficial effects:
the invention researches an anti-game between an intelligent jammer with full duplex technology and an opposite user, and the specific form of the simultaneous interference and eavesdropping strategy of the invention is as follows: the intelligent jammer releases interference to reduce user data transmission of an enemy and eavesdrop data transmission of the other party. In order to describe the confrontation relationship of two parties under the incomplete information condition, the invention provides a Bayesian Stackelberg game frame model of a power domain, a Stackelberg game equilibrium solution is solved by adopting a continuous convex approximation SCA (successive convex approximation) to convert a non-convex problem and a KKT (Karush-Kuhn-Tucker) condition, namely how to utilize a full duplex technology in communication confrontation, the problem of efficiently and simultaneously implementing interference and eavesdropping is constructed into a Bayesian Stackelberg game model, the Bayesian Stackelberg game model is further converted into a game optimization problem, the non-convex optimization problem of a leader and a follower is converted by continuous convex approximation, and the Stackelberg game equilibrium is solved by the KKT condition.
Meanwhile, the invention proves that the game equilibrium solution is superior to Nash equilibrium, and the invention researches the influence of the rate suppression weight and the power cost coefficient of the full-duplex jammer on the power strategy and the utility. Compared with half-duplex, single interference and single interception schemes, the simultaneous interference and interception simulation provided by the invention shows that the method has good accuracy and convergence and is superior to other schemes.
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FIG. 1 is a schematic diagram of the present invention;
FIG. 2 is a diagram of a communication scenario between two parties;
FIG. 3 is a continuous convex approximation method convergence diagram;
FIG. 4 is a diagram of Stackelberg equalization iteration convergence;
FIG. 5 is a graph of performance analysis;
figure 6 is a graph comparing the utility of several reference protocols.
Detailed Description
Embodiments of the present invention are described in further detail below with reference to the accompanying drawings.
The embodiment of the invention is established under the condition of the communication countermeasure distribution of the users of the two enemies and the my people as shown in figure 2.
Fig. 1 is a flowchart showing the present invention, and the present invention provides a simultaneous jamming and eavesdropping method based on a bayesian Stackelberg game, which includes the following steps:
step 1: scene modeling: establishing an confrontation scene model based on the intelligent jammer of the party and the communication user pair of the enemy;
step 2: game modeling: based on the confrontation scene model in the step 1, by utilizing a full-duplex technology, the communication confrontation of the users of the two enemies and the two parties under the incomplete information condition is modeled into a Bayesian Stackelberg game model, a leader and a follower are determined, the problem of simultaneously implementing interference and eavesdropping is converted into a game optimization problem, and the game optimization problem is a non-convex optimization problem of the leader and the follower;
and step 3: and (3) optimizing and solving: and transforming a non-convex optimization problem of the leader and the follower by adopting a continuous convex approximation SCA, and solving a Bayesian Stackelberg game equilibrium solution by using a KKT condition.
In the embodiment, in the confrontation scene model described in step 1, the two game parties are a pair of an intelligent jammer of my party and a communication user of an enemy party, where the working mode of the intelligent jammer of my party is full duplex, so that the interference can be released and the communication user of the enemy party can be intercepted, the communication user of the enemy party has a pair of communication transceiver pairs to transmit information in real time, specifically:
r represents the number of the intelligent jammer of our party, S-D represents the number of the enemy communication transceiving pair, in the countermeasure scene, the signal transmission between any two nodes has path loss and small-scale fading at the same time, the two users hardly know the exact channel state information of the opposite party, and the two users have incomplete cognition on the small-channel fading;
for the same channel, two different probabilities are needed to represent the uncertainty of the channel power gain, b, e represent my full-duplex jammer R and enemy S-D communication pair, respectively, let a e (b, e),
wherein the cognitive set of the small-scale fading gains from node x to node y of a is:
Figure BDA0003253573840000111
Figure BDA0003253573840000112
the Z small scale fading gain value under certain probability is shown, Z belongs to {1,2, …, Z } represents the index number of the set,
Figure BDA0003253573840000113
the nodes S, d and R respectively represent a transmitter S, a receiver R and a full-duplex node R; "-" is an exclude operator;
therefore, based on the knowledge of a, the channel gains from node x to node y are:
Figure BDA0003253573840000114
wherein the content of the first and second substances,
Figure BDA0003253573840000115
for free space path loss, alpha is path loss coefficient, and the invention is set as 2, dx，yIs the distance from node x to node y;
because intelligent jammer R adopts full-duplex communication mode, can have certain self-interference, and the cognitive set of a to R's self-interference channel gain is:
Figure BDA0003253573840000116
where N ∈ {1, 2., N } represents a set index number,
Figure BDA0003253573840000117
taking the value of the nth self-interference fading gain;
thus, based on the knowledge of a, the self-interference channel gain is defined as:
Figure BDA0003253573840000118
wherein k is0Is a self-interference cancellation factor.
In the embodiment, in the countermeasure scenario model in step 1, a cognitive signal to interference plus noise ratio in the presence of an interference and eavesdropping is further defined, specifically:
in the S-D tactical communication process, the node D can be interfered and intercepted by a full-duplex jammer R, and based on the cognition of a, the signals received by the node D are as follows:
Figure BDA0003253573840000119
wherein
Figure BDA00032535738400001110
Is the signal received from S;
Figure BDA00032535738400001111
is an interference signal of node R; i belongs to {1,2, …, I }, I is a discrete sample of the small fading gain between S and D, J belongs to {1,2, …, J }, and J is a discrete sample of the small fading gain between R and D; p is a radical ofsAnd prRespectively, the transmission power of the node S and the interference power of the node R; x is the number ofsAnd xrTransmit signals of S and R, respectively; n is1Is additive white gaussian noise;
thus, a 'S knowledge of the received SINR at node D is a two-dimensional random variable, depending on a' S knowledge of the S-D and R-D channel gains, respectively denoted as
Figure BDA0003253573840000121
And
Figure BDA0003253573840000122
therefore, a considers the received signal-to-interference-and-noise ratio at node D to be:
Figure BDA0003253573840000123
wherein the content of the first and second substances,
Figure BDA0003253573840000124
N1is the unilateral power spectral density of the gaussian noise at node D; b issIs the S-D channel bandwidth;
in order to monitor what tactical data is sent by S to its target recipient D, the full-duplex node R eavesdrops on the S-D transmission signal, and therefore the signal received at R consists of an eavesdropping signal and a self-interference signal;
thus, the knowledge of a for the signal received at R is represented as:
Figure BDA0003253573840000125
wherein
Figure BDA0003253573840000126
Is an eavesdropping signal;
Figure BDA0003253573840000127
is a self-interference signal; m belongs to {1,2, …, M }, and M is a discrete sample of the S-R channel gain; n is2Is additive white gaussian noise;
since the knowledge of a on S-R and self-interference channel gains are respectively
Figure BDA0003253573840000128
And
Figure BDA0003253573840000129
thus, the eavesdropping signal-to-interference-and-noise ratio of node R is:
Figure BDA00032535738400001210
in the formula
Figure BDA00032535738400001211
N2A single-sided power spectral density that is gaussian noise;
without loss of generality, the R-D channel bandwidth is assumed to be the same as the S-D channel, denoted as BsIn addition, with
Figure BDA0003253573840000131
In a similar manner to that described above,
Figure BDA0003253573840000132
and is also a two-dimensional random variable that depends on the knowledge of a on the S-R and self-interference channel gains.
In an embodiment, the communication countermeasure of the users of the two enemies and the my party under the incomplete information condition in the step 2 is a layered countermeasure process of the following power domain:
the behavior of S-D is taken as a leader, action is taken firstly, and the intelligent jammer R of the client is a follower, and action is taken after S-D, specifically:
the leader S executes tactical communications and adjusts its power policy, with the goal of ensuring secure communications by reducing the data rate overheard by jammers, and facing interference threats by increasing tactical communications capacity;
after observing the power policy of the leader S, the follower R releases the interfering signal to force S to increase its transmit power, which helps R eavesdrop on S-D transmissions, also forcing D to receive its intended signal at a lower rate;
the intelligent jammer R can rapidly learn the transmitting power of the S and adaptively adjust the interference power of the S, so that the utility is maximized;
the above process can be expressed as a hierarchical countermeasure process for one power domain.
In addition, both enemy and my parties have only incomplete channel condition information, including interference and eavesdropping on the link.
In the embodiment, in the step 2, based on channel cognition on the user and the opponent, the invention describes the communication countermeasure process of the users of the two enemies and the me under the incomplete information condition through the bayesian Stackelberg game, and constructs a bayesian Stackelberg game model, specifically:
the goal of node S is to increase the S-D tactical communication rate while reducing the data rate eavesdropped by R at a lower data transmission power cost, so the utility of transmitting node S is related to the S-D tactical communication rate, the data transmission power and the data rate eavesdropped by R;
the utility of S is defined as:
Figure BDA0003253573840000133
wherein D issIs a normal number, guaranteed UsIf the S value is positive, the S value can be independently and autonomously determined;
Figure BDA0003253573840000134
expected S-D channel capacity for an adversary;
Figure BDA0003253573840000135
is the expected S-R eavesdropping channel rate of enemy e; thetasIs an interception factor which represents the attention degree of an enemy to the data rate intercepted by the R; etaspsIs the power cost at S; etasIs the cost per unit power at S;
compared with S, the full-duplex node R aims to realize higher S-R eavesdropping rate and reduce S-D transmission with lower power cost;
the utility of the interfering node R is defined as:
Figure BDA0003253573840000141
wherein D isrIs a normal number to ensure UrIs positive;
Figure BDA0003253573840000142
is the expected eavesdropping rate of R;
Figure BDA0003253573840000143
is the expected channel capacity of R versus S-D; thetarIs a rate suppression factor for R concerned about the extent to which the communication quality of its opponent is reduced, larger thetarIt is stated that R focuses more on suppressing S-D communication; etarprIs the power cost of R; etarIs the cost per unit power of R.
In the embodiment, based on Bayesian formula, the method is provided
Figure BDA0003253573840000144
And
Figure BDA0003253573840000145
the specific definition of (1):
is provided with
Figure BDA0003253573840000146
Denotes the cognitive acquisition of a for the received signal to interference and noise ratio at R
Figure BDA0003253573840000147
Probability of the value of (c). Note that, therein
Figure BDA0003253573840000148
Thus, definition a for
Figure BDA0003253573840000149
The expected values of (c) are:
Figure BDA00032535738400001410
wherein the content of the first and second substances,
Figure BDA00032535738400001411
and
Figure BDA00032535738400001412
fractional S-D and R-D small channel fading gain acquisition
Figure BDA00032535738400001413
And
Figure BDA00032535738400001414
the probability of (d);
suppose that
Figure BDA00032535738400001415
And
Figure BDA00032535738400001416
is independent, therefore
Figure BDA00032535738400001417
According to
Figure BDA00032535738400001418
Defining eavesdropping rate
Figure BDA00032535738400001419
Comprises the following steps:
Figure BDA00032535738400001420
wherein the content of the first and second substances,
Figure BDA00032535738400001421
for eavesdropping signal-to-interference-and-noise ratio acquisition
Figure BDA00032535738400001422
Of valueProbability of, and
Figure BDA00032535738400001423
suppose that
Figure BDA00032535738400001424
And
Figure BDA00032535738400001425
is independent, therefore
Figure BDA00032535738400001426
And
Figure BDA00032535738400001427
respectively S-R and self-interference channel gain taking
Figure BDA00032535738400001428
And
Figure BDA00032535738400001429
the probability of (c).
The step 3 specifically comprises the following steps:
step 3-1: constructing an optimization problem model: respectively establishing an optimization problem model of a follower intelligent jammer R and an optimization problem model of a leader S based on the interference power of the R and the data transmission power of the S;
specifically, the method comprises the following steps:
and constructing an optimization problem model. In the game, the interference power of R and the data transmission power of S need to be carefully designed. In particular, the present invention relates to a method for producing,
for R, blindly increasing the interference power may result in severe self-interference, resulting in a drop in the eavesdropping rate. Therefore, R needs to adjust its power to achieve maximum utility.
In addition, for S, to combat interference from R, blindly increasing the transmission power increases the risk of more data being overheard, increasing power costs.
Therefore, for the follower my intelligent jammer R, the optimization problem is defined as:
P1：
Figure BDA0003253573840000151
s.t.0＜pr≤pr，max
wherein p isr，maxIs the maximum interference power;
optimizing problem for leader S due to channel expected capacity of enemy communication user
Figure BDA0003253573840000152
Requiring more than a threshold value gamma0Namely:
Figure BDA0003253573840000153
due to the fact that
Figure BDA0003253573840000154
Relative to psIs monotonically increasing, so that p is presents，minWhen p iss≥ps，minWhen the temperature of the water is higher than the set temperature,
Figure BDA0003253573840000155
furthermore, psLess than maximum transmission power ps，max；
Thus, the optimization problem defining the leader S is:
P2：
Figure BDA0003253573840000156
s.t.ps，min＜ps≤ps，max
step 3-2: constructing a Stackelberg balance based on an optimization problem model, and proving the existence of the Stackelberg balance:
in gaming, it is intelligent to both parties, and therefore both party power strategies are interacting.
R as a follower, can quickly observe opponent's strategy and adjust its power to maximize its utility using smart sensors and positioning devices.
And S is used as a leader, the power strategy of the follower R can be predicted, and a decision is made according to the prediction.
By using
Figure BDA0003253573840000161
And
Figure BDA0003253573840000162
represent the solutions of the optimization problems P1 and P2, respectively
Figure BDA0003253573840000163
And
Figure BDA0003253573840000164
satisfies the following conditions:
Figure BDA0003253573840000165
Figure BDA0003253573840000166
this means that R and S cannot unilaterally change their power, otherwise their utility will decrease, at which point,
Figure BDA0003253573840000167
forming a Stackelberg balance;
thus, the existence of the above Stackelberg game balance is demonstrated as follows:
the follower optimization problem can be approximated by continuous convex to convex form, so it has an asymptotic optimal solution, which is recorded as
Figure BDA0003253573840000168
Thus, given any S-strategy psThe following is true:
Figure BDA0003253573840000169
by using
Figure BDA00032535738400001610
Substituting to obtain
Figure BDA00032535738400001611
Similarly, the leader optimization problem can be approximated to be convex by continuous convex, and the progressive optimal solution of the leader optimization problem is recorded as
Figure BDA00032535738400001612
Thus, for any given R-policy prExistence of
Figure BDA00032535738400001613
By using
Figure BDA00032535738400001614
Is substituted to obtain
Figure BDA00032535738400001615
Step 3-3: and transforming the non-convex problem of the optimization problem model by using continuous convex approximation, decomposing the non-convex problem into a series of sub-convex functions, and solving the sub-convex functions through the KKT condition.
The objective function of P1 is non-convex, so solving for P1 is difficult. To effectively solve this problem, the present invention decomposes non-convex P1 into a series of sub-convex functions using a continuous convex approximation.
The basic idea is to approximate the original optimization problem P1 with a sub-convex function. To solve for P1, the following two steps are required:
1) the problem is resolved. By first order Taylor expansion of the utility function of P1, the present invention iteratively approximates the sub-convex function of the utility function. In each iteration, Ur(pr，ps) Can be expressed as
Figure BDA00032535738400001616
Wherein, thetarIs that
Figure BDA00032535738400001617
In that
Figure BDA00032535738400001618
In the form of a first-order Taylor expansion, in particular
Figure BDA0003253573840000171
Wherein phi1Is that
Figure BDA0003253573840000172
In that
Figure BDA0003253573840000173
Function value of (phi)2Is that
Figure BDA0003253573840000174
In that
Figure BDA0003253573840000175
The first derivative value of (a). Therefore, one approximation problem to further derive P1 is
SCP1:
Figure BDA0003253573840000176
s.t.pr≤pr,max
By solving for SCP1, the newly obtained
Figure BDA0003253573840000177
As the next taylor expansion point and start a new iteration until the maximum number of iterations is reached or remains unchanged. When iteration stops, p of the last iteration roundrIs assigned to
Figure BDA0003253573840000178
2) Solving the sub convex function. An expansion of any concave function at any point is a global upper bound. Therefore, U'r(pr，ps) As an original objective function Ur(pr，ps) The upper bound of (c). By iteratively solving SCP1, the original objective function in P1 can be approximated, thereby approximating P1.
For the convex optimization problem SCP1, the lagrange function is introduced as follows:
Lr(pr，ps)＝U′r(pr，ps)+λr(pr，max-pr)-μrpr
wherein λrAnd murIs the lagrange multiplier.
Due to Lr(pr，ps) Is a concave function, so the dual gap between the lagrange dual problem and the original problem is zero; then, under KKT conditions, a solution of SCP1 is obtained
Figure BDA0003253573840000179
Likewise, the same steps are taken to solve for P2.
Step 3-4: and solving the Stackelberg balance by adopting an inverse induction method based on the solution of the sub-convex function.
Specifically, the solution is subjected to an iterative process according to the following steps:
a) setting S initial iteration power, an initial Taylor expansion point of S and an initial Taylor expansion point of R;
b) solving a follower sub-convex function according to the S initial iteration power and the R initial Taylor expansion point to obtain a solution of the follower sub-convex function of the current round;
c) taking the solution of the previous follower sub-convex function as a new Taylor expansion point, and iteratively solving the follower sub-convex function until convergence;
d) taking the solution of the follower sub-convex function of the last round as input, and solving the leader sub-convex function through combining the initial Taylor expansion point of the sum S to obtain the solution of the leader sub-convex function of the round;
e) taking the solution of the previous leader sub-convex function as a new round of Taylor expansion point, iteratively solving the leader sub-convex function until convergence, and replacing the S initial iteration power with the solution of the leader sub-convex function of the last round;
f) repeating the steps b) and e) until convergence, and obtaining a Stackelberg equilibrium solution
Figure BDA0003253573840000181
Simulation analysis was performed based on the values of table 1 and table 2:
TABLE 1
Figure BDA0003253573840000182
Figure BDA0003253573840000191
Figure BDA0003253573840000201
TABLE 2
Figure BDA0003253573840000202
Figure BDA0003253573840000211
FIG. 3 illustrates the convergence process of the successive convex approximations employed by the present invention. It can be seen that when the convergence reaches about 10 th time, the interference power of R and the transmitting power of S both reach convergence, which shows that the method adopted by the invention has good convergence and can accelerate the decision speed of the two parties.
FIG. 4 shows the iterative process of the present invention in solving Stackelberg equalization. It can be seen that in the process of solving in the Stackelberg equalization, when iteration is performed for the 7 th time, convergence is already finished by both sides, and at this time, both sides cannot easily change their own decisions, so that the utility reaches the maximum. Meanwhile, the convergence results of the two parties meet the power constraint set by the invention, which shows the effectiveness of the method adopted by the invention.
Fig. 5 shows the utility values obtained by the method of the present invention compared to utility values that are not approximated in practice, and nash equalization. It can be seen that no matter how the power throttling factor and the power cost are taken, the method adopted by the invention is very close to the actual utility value, which shows the approximate reliability and accuracy of the method adopted by the invention. Meanwhile, the utility values of both sides are superior to the utility value of Nash balance, which illustrates the superiority of Stackelberg balance.
Figure 6 shows a comparison of the simultaneous jamming and eavesdropping method employed by the invention with several other reference schemes. It can be seen that as the power factor is increased, the effectiveness of R decreases in either case. This is because as the power throttle factor increases, R will focus more on hostile interference throttling, increasing the own interfering power. Since the interference power increases, the self-interference and the power cost increase, which results in increased utility. In addition, no matter how the value of the power suppression factor is taken, the simultaneous interference and interception method adopted by the invention is superior to other reference schemes, and the necessity of the simultaneous interference and interception method is illustrated.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (10)

1. A simultaneous interference and eavesdropping method based on a Bayesian Stackelberg game is characterized by comprising the following steps:
step 1: scene modeling: establishing an confrontation scene model based on the intelligent jammer of the party and the communication user pair of the enemy;
step 2: game modeling: based on the confrontation scene model in the step 1, by utilizing a full-duplex technology, the communication confrontation of the users of the two enemies and the two parties under the incomplete information condition is modeled into a Bayesian Stackelberg game model, a leader and a follower are determined, the problem of simultaneously implementing interference and eavesdropping is converted into a game optimization problem, and the game optimization problem is a non-convex optimization problem of the leader and the follower;
and step 3: and (3) optimizing and solving: and transforming a non-convex optimization problem of the leader and the follower by adopting a continuous convex approximation SCA, and solving a Bayesian Stackelberg game equilibrium solution by using a KKT condition.
2. The simultaneous interference and eavesdropping method based on the Bayesian Stackelberg game as claimed in claim 1, wherein in the confrontation scene model of step 1, the game parties are pairs of the intelligent jammer of my party and the communication user of the enemy party, wherein the working mode of the intelligent jammer of my party is full duplex, and the eavesdropping is performed on the communication user of the enemy party while the interference is released, and the communication user of the enemy party has a pair of communication transceiving pairs to transmit information in real time, specifically:
r represents the number of the intelligent jammer of our party, S-D represents the number of the enemy communication transceiving pair, in the countermeasure scene, the signal transmission between any two nodes has path loss and small-scale fading at the same time, the two users hardly know the exact channel state information of the opposite party, and the two users have incomplete cognition on the small-channel fading;
for the same channel, two different probabilities are needed to represent the uncertainty of the channel power gain, b and e respectively represent the R and the S-D communication pair of the enemy, and a belongs to the group of b and e;
a is the cognitive set of small-scale fading gains from node x to node y:
Figure FDA0003253573830000011
Figure FDA0003253573830000012
the small-scale fading gain value of the Z th under a certain probability is represented, Z belongs to {1,2, …, Z } and represents a set index number, and x belongs to { s, d, r };
Figure FDA0003253573830000013
the nodes S, d and R respectively represent a transmitter S, a receiver R and a full-duplex node R; "-" is an exclude operator;
therefore, based on the knowledge of a, the channel gains from node x to node y are:
Figure FDA0003253573830000014
wherein the content of the first and second substances,
Figure FDA0003253573830000015
for free space path loss, α is the path loss coefficient, dx，yIs the distance from node x to node y;
because intelligent jammer R adopts full-duplex communication mode, can have certain self-interference, and the cognitive set of a to R's self-interference channel gain is:
Figure FDA0003253573830000021
where N ∈ {1, 2., N } represents a set index number,
Figure FDA0003253573830000022
taking the value of the nth self-interference fading gain;
thus, based on the knowledge of a, the self-interference channel gain is defined as:
Figure FDA0003253573830000023
wherein k is0Is a self-interference cancellation factor.
3. The Bayesian Stackelberg game-based simultaneous interference and interception method according to claim 2, wherein the confrontation scenario model in step 1 further defines a cognition-based SINR in the confrontation scenario where interference and interception exist, specifically:
in the S-D tactical communication process, the node D can be interfered and intercepted by a full-duplex jammer R, and based on the cognition of a, the signals received by the node D are as follows:
Figure FDA0003253573830000024
wherein
Figure FDA0003253573830000025
Is the signal received from S;
Figure FDA0003253573830000026
is an interference signal of node R; i belongs to {1,2, …, I }, I is a discrete sample of the small fading gain between S and D, J belongs to {1,2, …, J }, and J is a discrete sample of the small fading gain between R and D; p is a radical ofsAnd prRespectively, the transmission power of the node S and the interference power of the node R; x is the number ofsAnd xrTransmit signals of S and R, respectively; n is1Is additive white gaussian noise;
thus, a 'S knowledge of the received SINR at node D is a two-dimensional random variable, depending on a' S knowledge of the S-D and R-D channel gains, respectively denoted as
Figure FDA0003253573830000027
And
Figure FDA0003253573830000028
therefore, a considers the received signal-to-interference-and-noise ratio at node D to be:
Figure FDA0003253573830000029
wherein the content of the first and second substances,
Figure FDA00032535738300000210
N1is the unilateral power spectral density of the gaussian noise at node D; b issIs the S-D channel bandwidth;
to monitor what tactical data is sent by S to its target recipient D, the full-duplex node R eavesdrops on the S-D transmission signal, so the signal received at R consists of an eavesdropping signal and a self-interference signal, and the knowledge of a on the signal received at R is expressed as:
Figure FDA0003253573830000031
wherein
Figure FDA0003253573830000032
Is an eavesdropping signal;
Figure FDA0003253573830000033
is a self-interference signal; m belongs to {1,2, …, M }, and M is a discrete sample of the S-R channel gain; n is2Is additive white gaussian noise;
since the knowledge of a on S-R and self-interference channel gains are respectively
Figure FDA0003253573830000034
And
Figure FDA0003253573830000035
thus, the eavesdropping signal-to-interference-and-noise ratio of node R is:
Figure FDA0003253573830000036
in the formula
Figure FDA0003253573830000037
N2A single-sided power spectral density that is gaussian noise;
assuming that the R-D channel bandwidth is the same as the S-D channel, denoted BsIn addition, with
Figure FDA0003253573830000038
In a similar manner to that described above,
Figure FDA0003253573830000039
is a two-dimensional random variable that depends on the knowledge of a on the S-R and self-interference channel gains.
4. The Bayesian Stackelberg game-based simultaneous jamming and eavesdropping method according to claim 3, wherein the step 2 of communication countermeasure of the friend or foe user under incomplete information condition is a hierarchical countermeasure process of the following power domain:
the behavior of S-D is taken as a leader, action is taken firstly, and the intelligent jammer R of the client is a follower, and action is taken after S-D, specifically:
the leader S executes tactical communications and adjusts its power policy, with the goal of ensuring secure communications by reducing the data rate overheard by jammers, and facing interference threats by increasing tactical communications capacity;
after observing the power policy of the leader S, the follower R releases the interfering signal to force S to increase its transmit power, allowing R to eavesdrop on the S-D transmission and also force D to receive its intended signal at a lower rate;
the intelligent jammer R learns the transmitting power of the S and adaptively adjusts the interference power of the intelligent jammer R so as to maximize the utility;
in addition, both enemy and my parties have only incomplete channel condition information, including interference and eavesdropping on the link.
5. The Bayesian Stackelberg game-based simultaneous interference and eavesdropping method according to claim 4, wherein in the step 2, based on channel cognition of the user and the opponent, a communication countermeasure process of the user of the two sides of the enemy and the me under the incomplete information condition is described through the Bayesian Stackelberg game, and a Bayesian Stackelberg game model is constructed, specifically:
the goal of node S is to increase the S-D tactical communication rate while reducing the data rate eavesdropped by R at a lower data transmission power cost, so the utility of transmitting node S is related to the S-D tactical communication rate, the data transmission power and the data rate eavesdropped by R;
the utility of S is defined as:
Figure FDA0003253573830000041
wherein D issIs a normal number, guaranteed UsIf the S value is positive, the S value can be independently and autonomously determined;
Figure FDA0003253573830000042
expected S-D channel capacity for an adversary;
Figure FDA0003253573830000043
is the expected S-R eavesdropping channel rate of enemy e; thetasIs an interception factor which represents the attention degree of an enemy to the data rate intercepted by the R; etaspsIs the power cost at S; etasIs the cost per unit power at S;
compared with S, the full-duplex node R aims to realize higher S-R eavesdropping rate and reduce S-D transmission with lower power cost;
the utility of the interfering node R is defined as:
Figure FDA0003253573830000044
wherein D isrIs a normal number to ensure UrIs positive;
Figure FDA0003253573830000045
is the expected eavesdropping rate of R;
Figure FDA0003253573830000046
is the expected channel capacity of R versus S-D; thetarIs a rate suppression factor for R concerned about the extent to which the communication quality of its opponent is reduced, larger thetarIt is stated that R focuses more on suppressing S-D communication; etarprIs the power cost of R; etarA cost per unit power of R;
based on Bayes formula, gives
Figure FDA0003253573830000047
And
Figure FDA0003253573830000048
the specific definition of (1):
is provided with
Figure FDA0003253573830000049
Denotes the cognitive acquisition of a for the received signal to interference and noise ratio at R
Figure FDA00032535738300000410
Of value of (a), wherein
Figure FDA00032535738300000411
Thus, definition a for
Figure FDA00032535738300000412
The expected values of (c) are:
Figure FDA00032535738300000413
wherein the content of the first and second substances,
Figure FDA0003253573830000051
and
Figure FDA0003253573830000052
fractional S-D and R-D small channel fading gain acquisition
Figure FDA0003253573830000053
And
Figure FDA0003253573830000054
the probability of (d);
suppose that
Figure FDA0003253573830000055
And
Figure FDA0003253573830000056
is independent, therefore
Figure FDA0003253573830000057
According to
Figure FDA0003253573830000058
Defining eavesdropping rate
Figure FDA0003253573830000059
Comprises the following steps:
Figure FDA00032535738300000510
wherein the content of the first and second substances,
Figure FDA00032535738300000511
for eavesdropping signal-to-interference-and-noise ratio acquisition
Figure FDA00032535738300000512
Probability of value, and
Figure FDA00032535738300000513
suppose that
Figure FDA00032535738300000514
And
Figure FDA00032535738300000515
is independent, therefore
Figure FDA00032535738300000516
Figure FDA00032535738300000517
And
Figure FDA00032535738300000518
respectively S-R and self-interference channel gain taking
Figure FDA00032535738300000519
And
Figure FDA00032535738300000520
the probability of (c).
6. The Bayesian Stackelberg-based simultaneous jamming and eavesdropping method according to claim 1,
the step 3 specifically comprises the following steps:
step 3-1: constructing an optimization problem model: respectively establishing an optimization problem model of a follower intelligent jammer R and an optimization problem model of a leader S based on the interference power of the R and the data transmission power of the S;
step 3-2: establishing a Stackelberg balance based on an optimization problem model, and proving the existence of the Stackelberg balance;
step 3-3: transforming a non-convex problem of the optimization problem model by using continuous convex approximation, decomposing the non-convex problem into a series of sub-convex functions, and solving the sub-convex functions through a KKT condition;
step 3-4: and solving the Stackelberg balance by adopting an inverse induction method based on the solution of the sub-convex function.
7. The Bayesian Stackelberg game-based simultaneous jamming and eavesdropping method according to claim 6, wherein the step 3-1 is to construct an optimization problem model: respectively establishing an optimization problem model of a follower intelligent jammer R and an optimization problem model of a leader S based on the interference power of the follower intelligent jammer R and the data transmission power of the leader S, and specifically comprising the following steps of:
for the follower my intelligent jammer R, the optimization problem is defined as follows:
P1：
Figure FDA00032535738300000521
s.t.0＜pr≤pr，max
wherein p isr，maxIs the maximum interference power;
optimizing problem for leader S due to channel expected capacity of enemy communication user
Figure FDA0003253573830000061
Requiring more than a threshold value gamma0Namely:
Figure FDA0003253573830000062
due to the fact that
Figure FDA0003253573830000063
Relative to psIs monotonically increasing, so that p is presents，minWhen p iss≥ps，minWhen the temperature of the water is higher than the set temperature,
Figure FDA0003253573830000064
furthermore, psLess than maximum transmission power ps，max；
Thus, the optimization problem defining the leader S is:
P2：
Figure FDA0003253573830000065
s.t.ps，min＜ps≤ps，max。
8. the Bayesian Stackelberg game-based simultaneous jamming and eavesdropping method according to claim 6, wherein the step 3-2 of constructing the Stackelberg balance based on the optimization problem model and proving the existence of the Stackelberg balance specifically comprises:
by using
Figure FDA0003253573830000066
And
Figure FDA0003253573830000067
represent the solutions of the optimization problems P1 and P2, respectively
Figure FDA0003253573830000068
And
Figure FDA0003253573830000069
satisfies the following conditions:
Figure FDA00032535738300000610
Figure FDA00032535738300000611
Figure FDA00032535738300000612
forming a Stackelberg balance;
thus, the existence of the above Stackelberg game balance is demonstrated as follows:
the follower optimization problem can be approximated by continuous convex to convex form, so it has an asymptotic optimal solution, which is recorded as
Figure FDA00032535738300000613
Thus, given any S-strategy psThe following is true:
Figure FDA00032535738300000614
by using
Figure FDA00032535738300000615
Substituting to obtain
Figure FDA00032535738300000616
Similarly, the leader optimization problem can be approximated to be convex by continuous convex, and the progressive optimal solution of the leader optimization problem is recorded as
Figure FDA00032535738300000617
Thus, for any given R-policy prExistence of
Figure FDA00032535738300000618
By using
Figure FDA00032535738300000619
Is substituted to obtain
Figure FDA00032535738300000620
9. The Bayesian Stackelberg game-based simultaneous jamming and eavesdropping method according to claim 6, wherein the non-convex problem of the optimization problem model is transformed by using continuous convex approximation in step 3-3, and is decomposed into a series of sub-convex functions, and the sub-convex functions are solved by KKT conditions, specifically:
1) and (3) problem decomposition: and (3) performing first-order Taylor expansion on the utility function of P1 to iteratively approximate a convex function of the utility function:
in each iteration, Ur(pr，ps) The approximate value of (d) is expressed as:
Figure FDA0003253573830000071
wherein, thetarIs that
Figure FDA0003253573830000072
In that
Figure FDA0003253573830000073
The first-order Taylor expansion form is as follows:
Figure FDA0003253573830000074
wherein phi1Is that
Figure FDA0003253573830000075
In that
Figure FDA0003253573830000076
Function value of (phi)2Is that
Figure FDA0003253573830000077
In that
Figure FDA0003253573830000078
A first derivative value of (d);
therefore, one further approximation problem to get P1 is:
SCP1:
Figure FDA0003253573830000079
s.t.pr≤pr,max
by solving for SCP1, the newly obtained
Figure FDA00032535738300000710
Taking the point as the next Taylor expansion point, and starting new iteration until the maximum iteration number is reached, or keeping the point unchanged;
when iteration stops, p of the last iteration roundrIs assigned to
Figure FDA00032535738300000711
2) Solving the sub-convex problem: the spread of any concave function at any point is the global upper bound, hence, U'r(pr，ps) As an original objective function Ur(pr，ps) By iteratively solving SCP1, the original objective function in P1 is approximated, thereby approximating P1:
for the convex optimization problem SCP1, the lagrange function is introduced as follows:
Lr(pr，ps)＝U′r(pr，ps)+λr(pr，max-pr)-μrpr
wherein λrAnd murIs the Lagrangian multiplier;
due to Lr(pr，ps) Is a concave function, so the dual gap between the lagrange dual problem and the original problem is zero;
then, under KKT conditions, a solution of SCP1 is obtained
Figure FDA0003253573830000081
Likewise, the same steps are taken to solve for P2.
10. The Bayesian Stackelberg game-based simultaneous interference and eavesdropping method according to claim 6, wherein the solution based on the sub-convex function in steps 3-4 is solved for Stackelberg equalization by an inverse induction method, and specifically the solution is subjected to an iterative process according to the following steps:
a) setting S initial iteration power, an initial Taylor expansion point of S and an initial Taylor expansion point of R;
b) solving a follower sub-convex function according to the S initial iteration power and the R initial Taylor expansion point to obtain a solution of the follower sub-convex function of the current round;
c) taking the solution of the previous follower sub-convex function as a new Taylor expansion point, and iteratively solving the follower sub-convex function until convergence;
d) taking the solution of the follower sub-convex function of the last round as input, and solving the leader sub-convex function through combining the initial Taylor expansion point of the sum S to obtain the solution of the leader sub-convex function of the round;
e) taking the solution of the previous leader sub-convex function as a new round of Taylor expansion point, iteratively solving the leader sub-convex function until convergence, and replacing the S initial iteration power with the solution of the leader sub-convex function of the last round;
f) repeating the steps b) and e) until convergence, and obtaining a Stackelberg equilibrium solution
Figure FDA0003253573830000082
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