CN113726472B - Simultaneous interference and monitoring method based on Bayesian Stackelberg game - Google Patents

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

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CN113726472B
CN113726472B CN202111053174.0A CN202111053174A CN113726472B CN 113726472 B CN113726472 B CN 113726472B CN 202111053174 A CN202111053174 A CN 202111053174A CN 113726472 B CN113726472 B CN 113726472B
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CN113726472A (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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    • 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
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Abstract

The invention discloses a simultaneous interference and monitoring method based on Bayesian Stackelberg game, which comprises the following steps: scene modeling: establishing an countermeasure scene model based on communication transceiver pairs of the intelligent jammer and the enemy; game modeling: modeling communication countermeasures of the users of the two parties of the friend and foe under the condition of incomplete information as a Bayesian Stackelberg game model by using a full duplex technology, and converting the problem of simultaneously implementing interference and monitoring into a game optimization problem; and (3) optimizing and solving: and adopting continuous convex approximation SCA to convert non-convex optimization problems of a leader and a follower, and solving a Bayesian Stackelberg game equilibrium solution through KKT conditions. Compared with half duplex, independent interference and independent monitoring schemes, the simultaneous interference and monitoring simulation method provided by the invention shows that the method has good accuracy and convergence and is superior to other schemes.

Description

Simultaneous interference and monitoring 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 Bayesian Stackelberg game-based simultaneous interference and monitoring method.
Background
In recent years, in the field of military wireless communications, there has been an urgent need to monitor in time tactical information transmitted from an adversary transmitter to a target receiver and to interrupt transmission immediately when needed. The information advantage is an important factor for determining the battlefield advantage, and the pure electromagnetic interference is insufficient to exert effective killing power in the face of opponents with intelligent anti-interference capability, and meanwhile, the effective information of the opponent users is difficult to obtain by pure monitoring.
Full duplex technology has great advantages in meeting the above-mentioned needs, as it facilitates simultaneous interference and listening. Therefore, the formulation of power strategies to study simultaneous interference and listening is also a current research hotspot. Based on the full duplex simultaneous interference and listening technology, the method is still an underexplored research direction as an emerging hot spot problem. Some existing studies can be divided into two directions of study, namely theoretical and experimental.
In theory, t.riihonen introduced an aggressive application of simultaneous transmit and receive capability in 2017, and this capability enabled joint interference and perception in hostile situations. Mietzner studied responsive attack applications in 2012 to protect vehicles from radio controlled explosives. Kong studied in 2016 the physical security problem of the existence of an active listener listening to user data transmissions while releasing interfering signals, and deduced therein the probability of privacy interruption of the victim node.
Experimentally, several laboratory experiments were performed on the radio defined by the generic software radio, which verifies the feasibility of full duplex simultaneous interference and listening technology. However, the work currently published focuses on listening to 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 in 2013 as a jackberg game. The optimal response strategy of the jammer (as follower) is first estimated, on the basis of which the optimal strategy of the leader is determined. Tang studied in 2017 that energy-saving transmission with security has the problem of full duplex active listeners under a Stackelberg game framework, which improves the defense against simultaneous listening and interference, but these works do not stand on the standpoint of an attacker, and study the formulation of simultaneous interference and listening strategies.
In electronic countermeasure, electromagnetic interference alone is not sufficient to exert an effective killing power. Meanwhile, the effective information of the opposite user is difficult to obtain by simple monitoring. Conventional electromagnetic countermeasure techniques are difficult to meet the requirements of a communication battlefield. In general, the existing wireless communication countermeasure model and angle mainly have the following problems: 1) The incompleteness of the channel information in the countermeasure environment is not considered. 2) Without considering the intelligence of the user, the intelligent user is often able to dynamically change own policies according to the policies of the other party. 3) From the perspective of the non-standing attacker, consider how to efficiently formulate the simultaneous interference and listening policy.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a simultaneous interference and monitoring method based on a Bayesian Stackelberg game, which utilizes a full duplex technology to model communication antagonism of two users of the friend and foe under the condition of incomplete information as the Bayesian Stackelberg game, realizes simultaneous interference and monitoring, converts the non-convex optimization problem of a leader and a follower through continuous convex approximation, and solves the balance of the Stackelberg game through a KKT condition.
In order to achieve the technical purpose, the invention adopts the following technical scheme:
a Bayesian Stackelberg game-based simultaneous interference and monitoring method comprises the following steps:
step 1: scene modeling: establishing an countermeasure scene model based on communication transceiver pairs of the intelligent jammer and the enemy;
step 2: game modeling: based on the countermeasure scene model in the step 1, communication countermeasure between the two users of the friend and foe is modeled as a Bayesian Stackelberg game model under the condition of incomplete information by utilizing a full duplex technology, a leader and a follower are determined, the problem of simultaneously implementing interference and monitoring is converted into a game optimization problem, and the game optimization problem is a non-convex optimization problem of the leader and the follower;
step 3: and (3) optimizing and solving: and adopting continuous convex approximation SCA to convert non-convex optimization problems of a leader and a follower, and solving a Bayesian Stackelberg game equilibrium solution through KKT conditions.
In order to optimize the technical scheme, the specific measures adopted further comprise:
in the countermeasure scene model in the above step 1, the two parties in the game are communication receiving and transmitting pairs of the my intelligent jammer and the enemy, wherein the working mode of the my intelligent jammer is full duplex, and the enemy communication user is monitored while the interference is released, and the enemy communication user has a pair of communication receiving and transmitting pairs, and the information is specifically transmitted in real time:
r is used for representing the number of the intelligent jammer on the my side, S-D is used for representing the number of the receiving and transmitting pair of the communication of the enemy side, in the countermeasure scene, the signal transmission between any two nodes has path loss and small-scale fading at the same time, the users of both sides hardly know the exact channel state information of the other side, and the users of both sides have incomplete knowledge on the small-channel fading;
for the same channel, two different probabilities are needed to represent the uncertainty of the channel power gain, and b and e are used to represent the intelligent interference machine R and the enemy communication receiving and transmitting pair S-D respectively, so that a is E (b and e);
the cognitive set sum of a to node x to node y small scale fading gains is:
Figure GDA0004177737010000021
Figure GDA0004177737010000031
represents the Z-th small-scale fading gain value under a certain probability, and Z epsilon {1,2, …, Z } represents the set index number,/->
Figure GDA0004177737010000032
Nodes S, D and R respectively represent a transmitter S, a receiver D and an intelligent jammer R; "-" is an exclude operator;
thus, based on knowledge of a, the node x to node y channel gain is:
Figure GDA0004177737010000033
wherein,,
Figure GDA0004177737010000034
for free space path loss, α is the path loss coefficient, d x，y Is the distance from node x to node y;
because the intelligent jammer R adopts a full duplex communication mode, certain self interference exists, and the cognitive set of the self interference channel gain of the intelligent jammer R is assumed to be:
Figure GDA0004177737010000035
where n.epsilon.1, 2.,. N } represents the set index number,
Figure GDA0004177737010000036
the nth self-interference fading gain is valued;
thus, based on the knowledge of a, the self-interference channel gain is defined as:
Figure GDA0004177737010000037
wherein k is 0 Is a self-interference cancellation factor.
In the countermeasure scene model in the above step 1, a cognition-based signal-to-interference-and-noise ratio under the countermeasure scene with interference and monitoring is also defined, specifically:
in the S-D tactical communication process, the intelligent jammer R can interfere and monitor, and based on the cognition of a, the signal received by the node D is as follows:
Figure GDA0004177737010000038
wherein the method comprises the steps of
Figure GDA0004177737010000039
Is a signal received from S; />
Figure GDA00041777370100000310
Is the interference signal of node R; i e {1,2, …, I }, I is a discrete sample of the small fading gain between S and D, J e {1,2, …, J }, J is a discrete sample of the small fading gain between R and D; p is p s And p r The transmitting power of the node S and the interference power of the node R are respectively; x is x s And x r Respectively SAnd R's transmit signal; n is n 1 Is additive white gaussian noise;
thus, a is a two-dimensional random variable for the knowledge of the received signal-to-interference-and-noise ratio at node D, expressed as
Figure GDA0004177737010000041
And->
Figure GDA0004177737010000042
Thus, a considers the received signal-to-interference-and-noise ratio at node D as:
Figure GDA0004177737010000043
wherein,,
Figure GDA0004177737010000044
N 1 a single-sided power spectral density that is gaussian noise at node D; b (B) s Is the S-D channel bandwidth;
in order to monitor what tactical data is sent by S to its target receiver D, the intelligent jammer R listens to the S-D transmission signal, so that the signal received at R consists of a listening signal and a self-interfering signal, a' S knowledge of the signal received at R is expressed as:
Figure GDA0004177737010000045
wherein the method comprises the steps of
Figure GDA0004177737010000046
Is a monitor signal; />
Figure GDA0004177737010000047
Is a self-interference signal; m is {1,2, …, M }, M is
The S-R channel gains one discrete sample; n is n 2 Is additive white gaussian noise;
due to a vs S-RAnd the knowledge of the self-interference channel gain are respectively
Figure GDA0004177737010000048
And->
Figure GDA0004177737010000049
Therefore, the listening signal-to-interference-and-noise ratio of the node R is:
Figure GDA00041777370100000410
in the middle of
Figure GDA00041777370100000411
N 2 Single-sided power spectral density, which is gaussian noise;
let R-D channel be the same bandwidth as S-D channel, denoted B s In addition, with
Figure GDA00041777370100000412
Similarly, is->
Figure GDA00041777370100000413
Is a two-dimensional random variable that depends on a' S knowledge of S-R and self-interference channel gain.
The communication countermeasure of the two friend-foe users in the step 2 is a layered countermeasure process of the following power domains under the condition of incomplete information:
S-D acts as a leader, first taking action, and My Smart jammer R is a follower, taking action after S-D, specifically:
the leader S performs tactical communications and adjusts its power policy with the aim of ensuring secure communications by reducing the data rate listened to by the jammer and facing interference threats by increasing tactical communications capacity;
after observing the power policy of the leader S, the intelligent jammer R releases the interfering signal to force S to increase its transmit power, causing R to listen to S-D transmissions, and also forcing D to receive its intended signal at a lower rate;
the intelligent jammer R learns the transmitting power of S, and adaptively adjusts the interference power of the intelligent jammer R so as to maximize the utility;
in addition, the two parties of the friend and the foe have incomplete channel condition information, wherein the incomplete channel condition information is the incomplete information condition, and the incomplete information comprises interference and a monitoring link.
In the step 2, based on the channel cognition of the self and opponents, describing the communication countermeasure process of the users of the two sides of the friend and foe under the condition of incomplete information by using the Bayesian Stackelberg game, and constructing a Bayesian Stackelberg game model, in particular:
the goal of node S is to increase the S-D tactical communication rate while reducing the data rate being monitored 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 being monitored by R;
the utility of S is defined as:
Figure GDA0004177737010000051
wherein D is s Is a normal number, ensure U s For positive, the value of the value can be independently and autonomously determined by S;
Figure GDA0004177737010000052
expected S-D channel capacity for adversaries; />
Figure GDA0004177737010000053
The intended S-R listening channel rate for enemy e; θ s Is a monitor factor, and represents the attention degree of enemy to the data rate monitored by R; η (eta) s p s Is the power cost at S; η (eta) s Is the unit power cost at S;
compared with S, the intelligent jammer R aims to realize larger S-R monitoring rate and reduce S-D transmission with lower power cost;
the utility of the intelligent jammer R is defined as:
Figure GDA0004177737010000054
wherein D is r Is a positive constant to ensure U r Is positive;
Figure GDA0004177737010000055
is the expected listening rate of R; />
Figure GDA0004177737010000056
Is the expected channel capacity of R versus S-D; θ r Is a rate-suppressing factor for representing the degree of interest of R in reducing the communication quality of an opponent thereof, a larger theta r The description R focuses more on suppressing S-D communications; η (eta) r p r Is the power cost of R; η (eta) r Unit power cost for R;
based on Bayes formula, give
Figure GDA0004177737010000061
And->
Figure GDA0004177737010000062
Is defined in detail by (a):
is provided with
Figure GDA0004177737010000063
Representing a awareness of the signal to interference plus noise ratio received at R>
Figure GDA0004177737010000064
Of the value of (1), wherein
Figure GDA0004177737010000065
Thus, definition a is for
Figure GDA0004177737010000066
The expected values of (2) are:
Figure GDA0004177737010000067
wherein,,
Figure GDA0004177737010000068
taking +.about.f for S-D small channel fading gain>
Figure GDA0004177737010000069
Probability of (2);
in a corresponding manner,
Figure GDA00041777370100000610
taking +.about.f for R-D small channel fading gain>
Figure GDA00041777370100000611
Probability of (2);
Figure GDA00041777370100000612
and->
Figure GDA00041777370100000613
Is independent, so->
Figure GDA00041777370100000614
According to
Figure GDA00041777370100000615
Defining a listening rate->
Figure GDA00041777370100000616
The method comprises the following steps:
Figure GDA00041777370100000617
wherein,,
Figure GDA00041777370100000618
get +.>
Figure GDA00041777370100000619
Probability of value, and->
Figure GDA00041777370100000620
Assume that
Figure GDA00041777370100000621
And->
Figure GDA00041777370100000622
Is independent, so->
Figure GDA00041777370100000623
Figure GDA00041777370100000624
And->
Figure GDA00041777370100000625
S-R and self-interference channel gain are respectively +.>
Figure GDA00041777370100000626
And->
Figure GDA00041777370100000627
Is a probability of (2).
The step 3 specifically includes the following steps:
step 3-1: constructing an optimization problem model: based on the interference power of R and the data transmission power of S, respectively establishing an optimization problem model of the intelligent jammer R on the part of the follower and an optimization problem model of the leader S;
step 3-2: establishing the balance of the Stackelberg based on the optimization problem model, and proving the existence of the balance of the Stackelberg;
step 3-3: decomposing the non-convex problem of the continuous convex approximation transformation optimization problem model into a series of sub-convex functions, and solving the sub-convex functions through KKT conditions;
step 3-4: based on the solution of the sub convex function, solving the balance of the Stackelberg by adopting an inverse induction method.
The optimization problem model is constructed as described in the step 3-1: based on the interference power of R and the data transmission power of S, respectively establishing an optimization problem model of the intelligent jammer R on the part of the follower and an optimization problem model of the leader S, wherein the optimization problem model specifically comprises the following steps:
for the following intelligent jammer R, defining the optimization problem as follows:
Figure GDA0004177737010000071
s.t.0＜p r ≤p r，max
wherein p is r，max Is the maximum interference power;
for the leader S optimization problem, channel expected capacity due to adversary communication users
Figure GDA0004177737010000072
Requiring greater than or equal to a threshold gamma 0 The method comprises the following steps:
Figure GDA0004177737010000073
due to
Figure GDA0004177737010000074
Relative to p s Is monotonically increasing, so p is present s，min When p is s ≥p s，min When (I)>
Figure GDA0004177737010000075
Furthermore, p s Less than maximum transmission power p s，max ；
Thus, the optimization problem defining the leader S is:
Figure GDA0004177737010000076
the step 3-2 builds the balance of the Stackelberg based on the optimization problem model, and proves the existence of the balance of the Stackelberg, and the method specifically comprises the following steps:
by using
Figure GDA0004177737010000077
And->
Figure GDA0004177737010000078
Solutions of the optimization problems P1 and P2 are shown, respectively, when +.>
Figure GDA0004177737010000079
And->
Figure GDA00041777370100000710
The method meets the following conditions:
Figure GDA00041777370100000711
Figure GDA00041777370100000712
Figure GDA00041777370100000713
constructing a Stackelberg equilibrium;
thus, the following demonstrates the equilibrium existence of the above described Stackelberg game:
the follower optimization problem can be approximated by a continuous convex to a convex form, so that it has an asymptotically optimal solution, denoted as
Figure GDA00041777370100000714
Thus, given any S policy p s The following holds true:
Figure GDA00041777370100000715
by using
Figure GDA0004177737010000081
Substituted to obtain +.>
Figure GDA0004177737010000082
Similarly, the leader optimization problem may be approximated by a continuous convex form, with its progressive optimal solution noted as
Figure GDA0004177737010000083
Thus, for a given R any policy p r There is->
Figure GDA0004177737010000084
Use->
Figure GDA0004177737010000085
Substitution to obtain
Figure GDA0004177737010000086
The non-convex problem using the continuous convex approximation transformation optimization problem model in the step 3-3 is decomposed into a series of sub-convex functions, and the sub-convex functions are solved by the KKT condition, specifically:
1) Problem resolution: expanding the first-order Taylor of the utility function of P1, and iteratively approximating the sub-convex function of the utility function:
in each iteration, U r (p r ，p s ) The approximation of (2) is expressed as:
Figure GDA0004177737010000087
wherein Θ is r Is that
Figure GDA0004177737010000088
At->
Figure GDA0004177737010000089
The first-order Taylor expansion form is as follows:
Figure GDA00041777370100000810
wherein phi is 1 Is that
Figure GDA00041777370100000811
At->
Figure GDA00041777370100000812
Function value of the point phi 2 Is->
Figure GDA00041777370100000813
At->
Figure GDA00041777370100000814
A first derivative value at;
thus, one approximate problem further resulting in P1 is:
Figure GDA00041777370100000815
s.t.p r ≤p r,max
by solving SCP1, the new result is obtained
Figure GDA00041777370100000816
As the next taylor expansion point, and starting a new iteration, stopping the iteration until the maximum iteration number is reached or the utility remains unchanged;
when iteration stops, p of the last iteration round r Is assigned to
Figure GDA00041777370100000817
2) Solving the sub convex problem: an expansion of any concave function at any point is a global upper bound, therefore U' r (p r ，p s ) As an original objective function U r (p r ，p s ) By iteratively solving SCP1, approximating the original objective function in P1 by successive approximationSolving P1:
for the sub-convex optimization problem SCP1, a Lagrangian function is introduced as follows:
L r (p r ，p s )＝U′ r (p r ，p s )+λ r (p r，max -p r )-μ r p r
wherein lambda is r Sum mu r Is a Lagrangian multiplier;
due to L r (p r ，p s ) As a concave function, the dual gap between the Lagrangian dual problem and the original problem is zero;
then, under KKT conditions, a solution of SCP1 is obtained
Figure GDA0004177737010000091
Similarly, the same procedure is taken to solve for P2.
The solution based on the sub convex function in the step 3-4 adopts an inverse induction method to solve the balance of the Stackelberg, and the specific solution is an iterative process according to the following steps:
a) Setting an initial S 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 initial Taylor expansion point of R to obtain a solution of the follower sub-convex function of the current round;
c) Taking the solution of the last 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 last round of follower sub-convex function as input and solving the leader sub-convex function through combining with the initial Taylor expansion point of S to obtain the solution of the leader sub-convex function of the round;
e) Taking the solution of the last leader sub-convex function as a new Taylor expansion point, iteratively solving the leader sub-convex function until convergence, and replacing the S initial iteration power with the solution of the last leader sub-convex function;
f) Repeating the steps b) -e) until convergence to obtain a Stackelberg equilibrium solution
Figure GDA0004177737010000092
The invention has the following beneficial effects:
the invention researches an countermeasure game between an intelligent jammer with full duplex technology and a counterpart user, and the specific form of the simultaneous interference and monitoring strategy is as follows: the intelligent jammer releases interference to reduce data transmission of users of the adversary and monitor data transmission of the other party. In order to describe the countermeasure relation of the two parties under the condition of incomplete information, the invention provides a Bayesian Stackelberg game framework model of a power domain, which is characterized in that a continuous convex approximation SCA (successive convex approximation) is adopted to convert a non-convex problem and a KKT (Karush-Kuhn-Tucker) condition is adopted to solve a Stackelberg game equilibrium solution, namely, how to effectively and simultaneously implement interference and monitoring by using a full duplex technology in communication countermeasure is constructed as the Bayesian Stackelberg game model, so that the problem is converted into a game optimization problem, the non-convex optimization problem of a leader and a follower is converted through the continuous convex approximation, and the Starelberg game equilibrium is solved through the KKT condition.
Meanwhile, the invention proves that the proposed game equilibrium solution exists and is superior to Nash equilibrium, and the invention researches the influence of the rate suppression weight and the power cost coefficient of the intelligent jammer on the power strategy and the utility. Compared with half duplex, independent interference and independent monitoring schemes, the simultaneous interference and monitoring simulation method provided by the invention shows that the method has good accuracy and convergence and is superior to other schemes.
Drawings
FIG. 1 is a schematic diagram of the present invention;
FIG. 2 is a communication scene diagram of both parties of a friend and foe;
FIG. 3 is a convergence diagram of a continuous convex approximation method;
FIG. 4 is a chart of a Stackelberg equalization iteration convergence;
FIG. 5 is a graph of performance analysis;
FIG. 6 is a graph of utility versus several baseline schemes.
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 communication countermeasure distribution of the two sides of the friend and foe shown in figure 2.
Fig. 1 is a flowchart of the present invention, which is a bayesian-based method for simultaneous interference and listening based on bayesian-based gaming, comprising the following steps:
step 1: scene modeling: establishing an countermeasure scene model based on communication transceiver pairs of the intelligent jammer and the enemy;
step 2: game modeling: based on the countermeasure scene model in the step 1, communication countermeasure between the two users of the friend and foe is modeled as a Bayesian Stackelberg game model under the condition of incomplete information by utilizing a full duplex technology, a leader and a follower are determined, the problem of simultaneously implementing interference and monitoring is converted into a game optimization problem, and the game optimization problem is a non-convex optimization problem of the leader and the follower;
step 3: and (3) optimizing and solving: and adopting continuous convex approximation SCA to convert non-convex optimization problems of a leader and a follower, and solving a Bayesian Stackelberg game equilibrium solution through KKT conditions.
In the countermeasure scene model described in step 1, the two parties in the game are communication transceiver pairs of the intelligent jammer and the enemy, wherein the working mode of the intelligent jammer is full duplex, so that the enemy communication user can be monitored while interference can be released, the enemy communication user has a pair of communication transceiver pairs, information is transmitted in real time, and the embodiment is as follows:
r is used for representing the number of the intelligent jammer on the my side, S-D is used for representing the number of the receiving and transmitting pair of the communication of the enemy side, in the countermeasure scene, the signal transmission between any two nodes has path loss and small-scale fading at the same time, the users of both sides hardly know the exact channel state information of the other side, and the users of both sides have incomplete knowledge on the small-channel fading;
for the same channel, two different probabilities are required to represent the uncertainty of the channel power gain, b, e represent the my intelligent jammer R and the adversary communication transceiver pair S-D, respectively, let a e (b, e),
wherein the sum of a cognitive set of small-scale fading gains from node x to node y is:
Figure GDA0004177737010000111
Figure GDA0004177737010000112
represents the Z-th small-scale fading gain value under a certain probability, and Z epsilon {1,2, …, Z } represents the set index number,/->
Figure GDA0004177737010000113
Nodes S, D and R respectively represent a transmitter S, a receiver D and an intelligent jammer R; "-" is an exclude operator;
thus, based on knowledge of a, the node x to node y channel gain is:
Figure GDA0004177737010000114
wherein,,
Figure GDA0004177737010000115
for free space path loss, alpha is the path loss coefficient, and without loss of generality, the invention is set as 2, d x，y Is the distance from node x to node y;
because the intelligent jammer R adopts a full duplex communication mode, certain self interference exists, and the cognitive set of the self interference channel gain of the intelligent jammer R is assumed to be:
Figure GDA0004177737010000116
where n.epsilon.1, 2.,. N } represents the set index number,
Figure GDA0004177737010000117
the nth self-interference fading gain is valued;
thus, based on the knowledge of a, the self-interference channel gain is defined as:
Figure GDA0004177737010000118
wherein k is 0 Is a self-interference cancellation factor.
In an embodiment, in the countermeasure scene model described in step 1, a cognition-based signal-to-interference-and-noise ratio under a countermeasure scene where interference and listening exist is further defined, and the cognition-based signal-to-interference-and-noise ratio is specifically:
in the S-D tactical communication process, the intelligent jammer R can interfere and monitor, and based on the cognition of a, the signal received by the node D is as follows:
Figure GDA0004177737010000121
wherein the method comprises the steps of
Figure GDA0004177737010000122
Is a signal received from S; />
Figure GDA0004177737010000123
Is the interference signal of node R; i e {1,2, …, I }, I is a discrete sample of the small fading gain between S and D, J e {1,2, …, J }, J is a discrete sample of the small fading gain between R and D; p is p s And p r The transmitting power of the node S and the interference power of the node R are respectively; x is x s And x r The transmitted signals are S and R, respectively; n is n 1 Is additive white gaussian noise;
thus, a is a two-dimensional random variable for the knowledge of the received signal-to-interference-and-noise ratio at node D, expressed as
Figure GDA0004177737010000124
And->
Figure GDA0004177737010000125
Thus, a considers the received signal-to-interference-and-noise ratio at node D as:
Figure GDA0004177737010000126
wherein,,
Figure GDA0004177737010000127
N 1 a single-sided power spectral density that is gaussian noise at node D; b (B) s Is the S-D channel bandwidth;
in order to monitor what tactical data is sent by S to its intended recipient D, the intelligent jammer R listens to the S-D transmission signal, so that the signal received at R consists of a listening signal and a self-interfering signal;
thus, a's knowledge of the signal received at R is expressed as:
Figure GDA0004177737010000128
wherein the method comprises the steps of
Figure GDA0004177737010000129
Is a monitor signal; />
Figure GDA00041777370100001210
Is a self-interference signal; m is {1,2, …, M }, M is
The S-R channel gains one discrete sample; n is n 2 Is additive white gaussian noise;
since a is aware of the S-R and self-interference channel gains, respectively
Figure GDA00041777370100001211
And->
Figure GDA00041777370100001212
Therefore, the listening signal-to-interference-and-noise ratio of the node R is:
Figure GDA00041777370100001213
in the middle of
Figure GDA0004177737010000131
N 2 Single-sided power spectral density, which is gaussian noise;
without loss of generality, assume that the R-D channel bandwidth is the same as the S-D channel, denoted B s In addition, with
Figure GDA0004177737010000132
Similarly to this, the process is carried out,
Figure GDA0004177737010000133
and is also a two-dimensional random variable depending on a' S knowledge of S-R and self-interference channel gain.
In an embodiment, the communication countermeasure of the two friend or foe users in step 2 under the condition of incomplete information is a layered countermeasure process of the following power domains:
S-D acts as a leader, first taking action, and My Smart jammer R is a follower, taking action after S-D, specifically:
the leader S performs tactical communications and adjusts its power policy with the aim of ensuring secure communications by reducing the data rate listened to by the jammer and facing interference threats by increasing tactical communications capacity;
after observing the power policy of the leader S, the intelligent jammer R releases the interfering signal to force S to increase its transmit power, which helps R to listen for S-D transmissions, as well as forcing D to receive its intended signal at a lower rate;
the intelligent jammer R can quickly learn the transmitting power of S, adaptively adjust the interference power of the intelligent jammer R, and maximize the utility;
the above procedure may be expressed as a layered countermeasure procedure for one power domain.
In addition, the two parties of the friend and the foe have incomplete channel condition information, wherein the incomplete channel condition information is the incomplete information condition, and the incomplete information comprises interference and a monitoring link.
In the embodiment, in the step 2, based on the channel cognition of the self and opponents, the present invention describes the communication countermeasure process of the two users of the friend and foe under the condition of incomplete information by using bayesian stack game, and builds a bayesian stack game model, and the specific is that:
the goal of node S is to increase the S-D tactical communication rate while reducing the data rate being monitored 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 being monitored by R;
the utility of S is defined as:
Figure GDA0004177737010000134
wherein D is s Is a normal number, ensure U s For positive, the value of the value can be independently and autonomously determined by S;
Figure GDA0004177737010000135
expected S-D channel capacity for adversaries; />
Figure GDA0004177737010000141
The intended S-R listening channel rate for enemy e; θ s Is a monitor factor, and represents the attention degree of enemy to the data rate monitored by R; η (eta) s p s Is the power cost at S; η (eta) s Is the unit power cost at S;
compared with S, the intelligent jammer R aims to realize larger S-R monitoring rate and reduce S-D transmission with lower power cost;
the utility of the intelligent jammer R is defined as:
Figure GDA0004177737010000142
wherein D is r Is a positive constant to ensure U r Is positive;
Figure GDA0004177737010000143
is the expected listening rate of R; />
Figure GDA0004177737010000144
Is the expected channel capacity of R versus S-D; θ r Is a rate-suppressing factor for representing the degree of interest of R in reducing the communication quality of an opponent thereof, a larger theta r The description R focuses more on suppressing S-D communications; η (eta) r p r Is the power cost of R; η (eta) r The unit power cost for R.
In an embodiment, based on a Bayesian formula, we give
Figure GDA0004177737010000145
And->
Figure GDA0004177737010000146
Is defined in detail by (a):
is provided with
Figure GDA0004177737010000147
Representing a awareness of the signal to interference plus noise ratio received at R>
Figure GDA0004177737010000148
Is a probability of a value of (c). Note that therein
Figure GDA0004177737010000149
Thus, definition a is for
Figure GDA00041777370100001410
The expected values of (2) are:
Figure GDA00041777370100001411
wherein,,
Figure GDA00041777370100001412
taking +.about.f for S-D small channel fading gain>
Figure GDA00041777370100001413
Probability of (2);
in a corresponding manner,
Figure GDA00041777370100001414
taking +.about.f for R-D small channel fading gain>
Figure GDA00041777370100001415
Probability of (2);
Figure GDA00041777370100001416
and->
Figure GDA00041777370100001417
Is independent, so->
Figure GDA00041777370100001418
According to
Figure GDA00041777370100001419
Defining a listening rate->
Figure GDA00041777370100001420
The method comprises the following steps:
Figure GDA00041777370100001421
wherein,,
Figure GDA0004177737010000151
get +.>
Figure GDA0004177737010000152
Probability of value, and->
Figure GDA0004177737010000153
Assume that
Figure GDA0004177737010000154
And->
Figure GDA0004177737010000155
Is independent, so->
Figure GDA0004177737010000156
Figure GDA0004177737010000157
And->
Figure GDA0004177737010000158
S-R and self-interference channel gain are respectively +.>
Figure GDA0004177737010000159
And->
Figure GDA00041777370100001510
Is a probability of (2).
The step 3 specifically comprises the following steps:
step 3-1: constructing an optimization problem model: based on the interference power of R and the data transmission power of S, respectively establishing an optimization problem model of the intelligent jammer R on the part of the follower and an optimization problem model of the leader S;
specific:
and constructing an optimization problem model. In gaming, both the interference power of R and the data transmission power of S need to be carefully designed. In particular the number of the elements,
for R, a blind increase in interference power can lead to severe self-interference, resulting in a decrease in listening rate. Thus, R needs to adjust its power to maximize utility.
In addition, for S, blind increases in transmit power to combat interference from R increase the risk that more data will be listened to, increasing power costs.
Thus, for the follower my intelligent jammer R, its optimization problem is defined as:
Figure GDA00041777370100001511
wherein p is r，max Is the maximum interference power;
for the leader S optimization problem, channel expected capacity due to adversary communication users
Figure GDA00041777370100001512
Requiring greater than or equal to a threshold gamma 0 The method comprises the following steps:
Figure GDA00041777370100001513
due to
Figure GDA00041777370100001514
Relative to p s Is monotonically increasing, so p is present s，min When p is s ≥p s，min When (I)>
Figure GDA00041777370100001515
Furthermore, p s Less than maximum transmission power p s，max ；
Thus, the optimization problem defining the leader S is:
Figure GDA0004177737010000161
s.t.p s，min ＜p s ≤p s，max
step 3-2: establishing the Stackelberg equilibrium based on the optimization problem model, and proving the existence of the Stackelberg equilibrium:
in gaming, both countering parties are intelligent, and therefore, both power strategies are mutually influencing.
R acts as a follower and can quickly observe the adversary's strategy and adjust its power using intelligent sensors and positioning devices to maximize its utility.
S acts as a leader and is able to predict the power policy of follower R and make decisions based on the predictions.
By using
Figure GDA0004177737010000162
And->
Figure GDA0004177737010000163
Solutions of the optimization problems P1 and P2 are shown, respectively, when +.>
Figure GDA0004177737010000164
And->
Figure GDA0004177737010000165
The method meets the following conditions:
Figure GDA0004177737010000166
Figure GDA0004177737010000167
this means that R and S cannot unilaterally change their power, otherwise their utility will drop, at which point,
Figure GDA0004177737010000168
constructing a Stackelberg equilibrium;
thus, the following demonstrates the equilibrium existence of the above described Stackelberg game:
the follower optimization problem can be approximated by a continuous convex to a convex form, so that it has an asymptotically optimal solution, denoted as
Figure GDA0004177737010000169
Thus, given any S policy p s The following holds true:
Figure GDA00041777370100001610
by using
Figure GDA00041777370100001611
Substitution, can obtain->
Figure GDA00041777370100001612
Similarly, the leader optimization problem may be approximated by a continuous convex form, with its progressive optimal solution noted as
Figure GDA00041777370100001613
Thus, for a given R any policy p r There is->
Figure GDA00041777370100001614
Use->
Figure GDA00041777370100001615
Substitution to obtain
Figure GDA00041777370100001616
Step 3-3: the non-convex problem of the optimization problem model is converted by utilizing continuous convex approximation, the non-convex problem is decomposed into a series of sub-convex functions, and the sub-convex functions are solved through KKT conditions.
The objective function of P1 is not convex, so solving P1 is difficult. To effectively solve this problem, the present invention utilizes a continuous convex approximation to decompose the non-convex P1 into a series of sub-convex functions.
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. The utility function of P1 is developed by Taylor in first order, and the invention iteratively approximates to the sub convex function of the utility function. In each iteration, U r (p r ，p s ) Can be expressed as an approximation of
Figure GDA0004177737010000171
Wherein Θ is r Is that
Figure GDA0004177737010000172
At->
Figure GDA0004177737010000173
The first-order Taylor expansion form of the position is specifically that
Figure GDA0004177737010000174
Wherein phi is 1 Is that
Figure GDA0004177737010000175
At->
Figure GDA0004177737010000176
Function value of the point phi 2 Is->
Figure GDA0004177737010000177
At->
Figure GDA0004177737010000178
A first derivative value at. Thus, an approximate problem of further obtaining P1 is
Figure GDA0004177737010000179
s.t.p r ≤p r,max
By solving SCP1, the new result is obtained
Figure GDA00041777370100001710
As the next taylor expansion point, and start a new iteration until the maximum number of iterations is reached or the utility remains unchanged. When iteration stops, p of the last iteration round r Is assigned to
Figure GDA00041777370100001711
2) And solving a sub convex function. An expansion of any concave function at any point is a global upper bound. Thus, U' r (p r ，p s ) As an original objective function U r (p r ，p s ) Is a lower bound of (c). By iteratively solving SCP1, the original objective function in P1 can be approximated, thereby approximating solution P1.
For the sub-convex optimization problem SCP1, a Lagrangian function is introduced as follows:
L r (p r ，p s )＝U′ r (p r ，p s )+λ r (p r，max -p r )-μ r p r
wherein lambda is r Sum mu r Is a lagrangian multiplier.
Due to L r (p r ，p s ) As a concave function, the dual gap between the Lagrangian dual problem and the original problem is zero; then, under KKT conditions, a solution of SCP1 is obtained
Figure GDA0004177737010000181
Similarly, the same procedure is taken to solve for P2.
Step 3-4: based on the solution of the sub convex function, solving the balance of the Stackelberg by adopting an inverse induction method.
The specific solution is to carry out an iterative process according to the following steps:
a) Setting an initial S 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 initial Taylor expansion point of R to obtain a solution of the follower sub-convex function of the current round;
c) Taking the solution of the last 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 last round of follower sub-convex function as input and solving the leader sub-convex function through combining with the initial Taylor expansion point of S to obtain the solution of the leader sub-convex function of the round;
e) Taking the solution of the last leader sub-convex function as a new Taylor expansion point, iteratively solving the leader sub-convex function until convergence, and replacing the S initial iteration power with the solution of the last leader sub-convex function;
f) Repeating the steps b) -e) until convergence to obtain a Stackelberg equilibrium solution
Figure GDA0004177737010000182
Simulation analysis was performed based on the values of tables 1 and 2:
TABLE 1
Figure GDA0004177737010000183
/>
Figure GDA0004177737010000191
/>
Figure GDA0004177737010000201
TABLE 2
Figure GDA0004177737010000202
/>
Figure GDA0004177737010000211
Fig. 3 illustrates the convergence process of the successive convex approximations employed by the present invention. It can be seen that when the convergence is about 10 th time, both the interference power of R and the transmitting power of S reach the 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 for the tuckelberg equalization. It can be seen that during the solving process of the Stackelberg equilibrium, both sides have completed convergence at the 7 th iteration, and at this time, both sides cannot easily change their decisions, so that the utility is maximized. Meanwhile, the convergence results of the two parties meet the power constraint set by the invention, and the effectiveness of the adoption method of the invention is illustrated.
Fig. 5 shows a comparison of utility values obtained by the method employed by the present invention with utility values that are not approximated in practice, and nash equalization. It can be seen that the method adopted by the invention is very close to the actual utility value no matter how the power suppression factor and the power cost take values, and the reliability and accuracy of the approximation of the method adopted by the invention are illustrated. Meanwhile, the utility of both parties is better than the utility value of Nash equilibrium, which demonstrates the superiority of Stackelberg equilibrium.
Fig. 6 shows a comparison of the simultaneous interference and listening method employed by the present invention with several other reference schemes. It can be seen that as the power suppression factor increases, the effectiveness of R decreases in either scenario. This is because as the power suppression factor increases, R will pay more attention to the hostile interference suppression and will increase the own interference power. As the interference power increases, both self-interference and power costs increase, resulting in an increase in utility. In addition, no matter how the power suppression factor is valued, the simultaneous interference and monitoring method adopted by the invention is superior to other reference schemes, and the necessity of the simultaneous interference and monitoring 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 examples, and all technical solutions belonging to the concept of the present invention belong to the protection scope of the present invention. It should be noted that modifications and adaptations to the invention without departing from the principles thereof are intended to be within the scope of the invention as set forth in the following claims.

Claims (1)

1. The simultaneous interference and monitoring method based on the Bayesian Stackelberg game is used for simultaneously interfering and monitoring enemy communication users, and is characterized by comprising the following steps:
step 1: scene modeling: establishing an countermeasure scene model based on communication transceiver pairs of the intelligent jammer and the enemy;
in the countermeasure scene model described in step 1, the two parties in the game are communication receiving and transmitting pairs of the intelligent interference machine and the enemy, wherein the working mode of the intelligent interference machine is full duplex, the enemy communication user is monitored while the interference is released, and the enemy communication user has a pair of communication receiving and transmitting pairs, and the information is specifically transmitted in real time:
r is used for representing the number of the intelligent jammer on the my side, S-D is used for representing the number of the receiving and transmitting pair of the communication of the enemy side, in the countermeasure scene, the signal transmission between any two nodes has path loss and small-scale fading at the same time, the users of both sides hardly know the exact channel state information of the other side, and the users of both sides have incomplete knowledge on the small-channel fading;
for the same channel, two different probabilities are used for representing the uncertainty of the power gain of the channel, b and e are used for respectively representing the intelligent interference machine R and the communication receiving and transmitting pair S-D of the enemy, so that a is E (b and e);
the cognitive set sum of a to node x to node y small scale fading gains is:
Figure FDA0004177737000000011
Figure FDA0004177737000000012
represents the Z-th small-scale fading gain value under a certain probability, Z epsilon {1,2, …, Z } represents the set index number,
Figure FDA0004177737000000013
nodes S, D and R respectively represent a transmitter S, a receiver D and an intelligent jammer R; />
Figure FDA0004177737000000018
Is an exclusion operator;
thus, based on knowledge of a, the node x to node y channel gain is:
Figure FDA0004177737000000014
wherein,,
Figure FDA0004177737000000015
for free space path loss, α is the path loss coefficient, d x，y Is the distance from node x to node y;
because the intelligent jammer R adopts a full duplex communication mode, certain self interference exists, and the cognitive set of the self interference channel gain of a to R is as follows:
Figure FDA0004177737000000016
where n.epsilon.1, 2.,. N } represents the set index number,
Figure FDA0004177737000000017
the nth self-interference fading gain is valued;
thus, based on the knowledge of a, the self-interference channel gain is defined as:
Figure FDA0004177737000000021
wherein k is 0 Is a self-interference cancellation factor;
in the countermeasure scene model described in step 1, a cognition-based signal-to-interference-and-noise ratio under the countermeasure scene with interference and monitoring is also defined, and the cognition-based signal-to-interference-and-noise ratio is specifically:
in the S-D tactical communication process, the intelligent jammer R can interfere and monitor, and based on the cognition of a, the signal received by the node D is as follows:
Figure FDA0004177737000000022
wherein the method comprises the steps of
Figure FDA0004177737000000023
Is a signal received from S; />
Figure FDA0004177737000000024
Is the interference signal of node R; i e {1,2, …, I }, I is a discrete sample of the small fading gain between S and D, J e {1,2, …, J }, J is a discrete sample of the small fading gain between R and D; p is p s And p r The transmitting power of the node S and the interference power of the node R are respectively; x is x s And x r The transmitted signals are S and R, respectively; n is n 1 Is additive white gaussian noise;
thus, a is a two-dimensional random variable for the knowledge of the received signal-to-interference-and-noise ratio at node D, expressed as
Figure FDA0004177737000000025
And->
Figure FDA0004177737000000026
Thus, a considers the received signal-to-interference-and-noise ratio at node D as:
Figure FDA0004177737000000027
wherein,,
Figure FDA0004177737000000028
N 1 a single-sided power spectral density that is gaussian noise at node D; b (B) s Is the S-D channel bandwidth;
in order to monitor what tactical data is sent by S to its target receiver D, the intelligent jammer R listens to the S-D transmission signal, so that the signal received at R consists of a listening signal and a self-interfering signal, a' S knowledge of the signal received at R is expressed as:
Figure FDA0004177737000000029
wherein the method comprises the steps of
Figure FDA00041777370000000210
Is a monitor signal; />
Figure FDA00041777370000000211
Is a self-interference signal; m is {1,2, …, M }, M is one discrete sample of the S-R channel gain; n is n 2 Is additive white gaussian noise;
since a is aware of the S-R and self-interference channel gains, respectively
Figure FDA0004177737000000031
And->
Figure FDA0004177737000000032
Therefore, the listening signal-to-interference-and-noise ratio of the node R is:
Figure FDA0004177737000000033
in the middle of
Figure FDA0004177737000000034
N 2 Single-sided power spectral density, which is gaussian noise;
the R-D channel has the same bandwidth as the S-D channel and is denoted as B s In addition,
Figure FDA0004177737000000035
is also a two-dimensional random variable, depending on a' S knowledge of S-R and self-interference channel gain;
step 2: game modeling: based on the countermeasure scene model in the step 1, communication countermeasure between the two users of the friend and foe is modeled as a Bayesian Stackelberg game model under the condition of incomplete information by utilizing a full duplex technology, a leader and a follower are determined, the problem of simultaneously implementing interference and monitoring 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 2, the communication countermeasure of the two friend-foe users under the condition of incomplete information is a layered countermeasure process of the following power domains:
S-D acts as a leader, first taking action, and My Smart jammer R is a follower, taking action after S-D, specifically:
the leader S performs tactical communications and adjusts its power policy with the aim of ensuring secure communications by reducing the data rate listened to by the jammer and facing interference threats by increasing tactical communications capacity;
after observing the power policy of the leader S, the intelligent jammer R releases the interfering signal to force S to increase its transmit power, causing R to listen to S-D transmissions, and also forcing D to receive its intended signal at a lower rate;
the intelligent jammer R learns the transmitting power of S, and adaptively adjusts the interference power of the intelligent jammer R so as to maximize the utility;
in addition, the two parties of the friend and foe have incomplete channel condition information, wherein the incomplete channel condition information is the incomplete information condition and comprises interference and monitoring links;
in the step 2, based on channel cognition of the self and opponents, describing a communication countermeasure process of the users of the two sides of the friend and foe under the condition of incomplete information by using a bayesian Stackelberg game, and constructing a bayesian Stackelberg game model, in particular:
the goal of node S is to increase the S-D tactical communication rate while reducing the data rate being monitored 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 being monitored by R;
the utility of S is defined as:
Figure FDA0004177737000000041
wherein D is s Is a normal number, ensure U s If yes, independently and autonomously determining the value by S;
Figure FDA0004177737000000042
channel expected capacity for adversary e communication users; />
Figure FDA0004177737000000043
The intended S-R listening channel rate for enemy e; θ s Is a monitor factor, which represents the attention degree of enemy e to the data rate monitored by R; η (eta) s p s Is the power cost at S; η (eta) s Is the unit power cost at S;
compared with S, the intelligent jammer R aims to realize larger S-R monitoring rate and reduce S-D transmission with lower power cost;
the utility of the intelligent jammer R is defined as:
Figure FDA0004177737000000044
wherein D is r Is a positive constant to ensure U r Is positive;
Figure FDA0004177737000000045
is the expected listening rate of R; />
Figure FDA0004177737000000046
Is the expected channel capacity of R versus S-D; θ r Is a rate-suppressing factor for representing the degree of interest of R in reducing the communication quality of an opponent thereof, a larger theta r The description R focuses more on suppressing S-D communications; η (eta) r p r Is the power cost of R; η (eta) r Unit power cost for R;
based on Bayes formula, give
Figure FDA0004177737000000047
And->
Figure FDA0004177737000000048
Is defined in detail by (a):
is provided with
Figure FDA0004177737000000049
Representing a awareness of the signal to interference plus noise ratio received at R>
Figure FDA00041777370000000410
Of the value of (1), wherein
Figure FDA00041777370000000411
Thus, definition a is for
Figure FDA00041777370000000412
The expected values of (2) are:
Figure FDA00041777370000000413
wherein,,
Figure FDA00041777370000000414
taking +.about.f for S-D small channel fading gain>
Figure FDA00041777370000000415
Probability of (2);
in a corresponding manner,
Figure FDA0004177737000000051
taking +.about.f for R-D small channel fading gain>
Figure FDA0004177737000000052
Probability of (2);
Figure FDA0004177737000000053
and->
Figure FDA0004177737000000054
Is independent, so->
Figure FDA0004177737000000055
According to
Figure FDA0004177737000000056
Defining a listening rate->
Figure FDA0004177737000000057
The method comprises the following steps:
Figure FDA0004177737000000058
wherein,,
Figure FDA0004177737000000059
get +.>
Figure FDA00041777370000000510
Probability of value, and->
Figure FDA00041777370000000511
Figure FDA00041777370000000512
And->
Figure FDA00041777370000000513
Is independent, so->
Figure FDA00041777370000000514
Figure FDA00041777370000000515
And->
Figure FDA00041777370000000516
S-R and self-interference channel gain are respectively +.>
Figure FDA00041777370000000517
And->
Figure FDA00041777370000000518
Probability of (2);
step 3: and (3) optimizing and solving: adopting continuous convex approximation SCA to convert non-convex optimization problems of a leader and a follower, and solving a Bayesian Stackelberg game equilibrium solution through KKT conditions;
the step 3 specifically comprises the following steps:
step 3-1: constructing an optimization problem model: based on the interference power of R and the data transmission power of S, respectively establishing an optimization problem model of the intelligent jammer R on the part of the follower and an optimization problem model of the leader S;
step 3-2: establishing the balance of the Stackelberg based on the optimization problem model, and proving the existence of the balance of the Stackelberg;
step 3-3: decomposing the non-convex problem of the continuous convex approximation transformation optimization problem model into a series of sub-convex functions, and solving the sub-convex functions through KKT conditions;
step 3-4: based on the solution of the sub convex function, solving the balance of the Stackelberg by adopting a reverse induction method;
and (3) constructing an optimization problem model in the step 3-1: based on the interference power of R and the data transmission power of S, respectively establishing an optimization problem model of the intelligent jammer R on the part of the follower and an optimization problem model of the leader S, wherein the optimization problem model specifically comprises the following steps:
for the following intelligent jammer R, defining the optimization problem as follows:
P1：
Figure FDA00041777370000000519
s.t.0＜p r ≤p r，max
wherein p is r，max Is the maximum interference power;
for the leader S optimization problem, channel expected capacity due to adversary communication users
Figure FDA00041777370000000520
Requiring greater than or equal to a threshold gamma 0 The method comprises the following steps:
Figure FDA0004177737000000061
due to
Figure FDA0004177737000000062
Relative to p s Is monotonically increasing, so p is present s，min When p is s ≥p s，min When (I)>
Figure FDA0004177737000000063
Furthermore, p s Less than maximum transmission power p s，max ；
Thus, the optimization problem defining the leader S is:
P2：
Figure FDA0004177737000000064
s.t.p s，min ＜p s ≤p s,max ；
step 3-2, constructing a Stackelberg equilibrium based on the optimization problem model, and proving the existence of the Stackelberg equilibrium, wherein the method specifically comprises the following steps:
by using
Figure FDA0004177737000000065
And->
Figure FDA0004177737000000066
Solutions of the optimization problems P1 and P2 are shown, respectively, when +.>
Figure FDA0004177737000000067
And->
Figure FDA0004177737000000068
The method meets the following conditions:
Figure FDA0004177737000000069
Figure FDA00041777370000000610
Figure FDA00041777370000000611
constructing a Stackelberg equilibrium;
thus, the following demonstrates the equilibrium existence of the above described Stackelberg game:
the follower optimization problem is approximated by a continuous convex form, so that it has an asymptotically optimal solution, denoted as
Figure FDA00041777370000000612
Thus, given any S policy p s The following holds true:
Figure FDA00041777370000000613
by using
Figure FDA00041777370000000614
Substituted to obtain +.>
Figure FDA00041777370000000615
Similarly, the leader optimization problem is approximated by a continuous convex form, the progressive optimal solution of which is noted as
Figure FDA00041777370000000616
Thus, for a given R any policy p r There is->
Figure FDA00041777370000000617
Use->
Figure FDA00041777370000000618
Substitution to get->
Figure FDA00041777370000000619
And 3-3, decomposing the non-convex problem of the continuous convex approximation transformation optimization problem model into a series of sub-convex functions, and solving the sub-convex functions through KKT conditions, wherein the method specifically comprises the following steps:
1) Problem resolution: expanding the first-order Taylor of the utility function of P1, and iteratively approximating the sub-convex function of the utility function:
in each iteration, U r (p r ，p s ) The approximation of (2) is expressed as:
Figure FDA0004177737000000071
wherein Θ is r Is that
Figure FDA0004177737000000072
At->
Figure FDA0004177737000000073
The first-order Taylor expansion form is as follows:
Figure FDA0004177737000000074
wherein phi is 1 Is that
Figure FDA0004177737000000075
At->
Figure FDA0004177737000000076
Function value of the point phi 2 Is->
Figure FDA0004177737000000077
At->
Figure FDA0004177737000000078
A first derivative value at;
thus, one approximate problem further resulting in P1 is:
SCP1:
Figure FDA0004177737000000079
s.t.p r ≤p r,max
by solving SCP1, the new result is obtained
Figure FDA00041777370000000710
As the next taylor expansion point, and starting a new iteration, stopping the iteration until the maximum iteration number is reached or the utility remains unchanged;
when iteration stops, p of the last iteration round r Is assigned to
Figure FDA00041777370000000711
2) Solving the sub convex problem: an expansion of any concave function at any point is a global upper bound, therefore U' r (p r ，p s ) As an original objective function U r (p r ，p s ) By iteratively solving SCP1, successive approximation of the original objective function in P1, fromAnd approximately solving for P1:
for the sub-convex optimization problem SCP1, a Lagrangian function is introduced as follows:
L r (p r ，p s )＝U′ r (p r ，p s )+λ r (p r，max -p r )-μ r p r
wherein lambda is r Sum mu r Is a Lagrangian multiplier;
due to L r (p r ，p s ) As a concave function, the dual gap between the Lagrangian dual problem and the original problem is zero;
then, under KKT conditions, a solution of SCP1 is obtained
Figure FDA0004177737000000081
Similarly, adopting the same steps to solve P2;
and 3-4, solving the balance of the Stackelberg by adopting an inverse induction method based on the solution of the sub convex function, and carrying out an iterative process by specifically solving the solution according to the following steps:
a) Setting an initial S 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 initial Taylor expansion point of R to obtain a solution of the follower sub-convex function of the current round;
c) Taking the solution of the last 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 last round of follower sub-convex function as input and solving the leader sub-convex function by combining the initial Taylor expansion point of S to obtain the solution of the leader sub-convex function of the round;
e) Taking the solution of the last leader sub-convex function as a new Taylor expansion point, iteratively solving the leader sub-convex function until convergence, and replacing the S initial iteration power with the solution of the last leader sub-convex function;
f) Repeating the steps b) -e) until convergence to obtain a Stackelberg equilibrium solution
Figure FDA0004177737000000082
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