Disclosed is a method for interference pricing and power control based on hierarchical game in user-assisted multi-band covert communication

Abstract

The application discloses a method for interference pricing and power control based on hierarchical game in user-assisted multi-band covert communication. The method discloses a scenario that user-assisted multi-covert users transmit on multiple frequency bands to cope with threats from multiple detectors. When the covert users transmit, the transmission of the covert users will cause co-frequency interference to the user, and the transmission rate of the user is reduced. The covert users need to pay the user when transmitting, and the interference price is determined by the user. The covert users decide their transmission power according to the interference pricing under the constraint of transmission concealment. The process that the user prices the power of the covert users is modeled as a hierarchical game, wherein the user is the leader, and the multiple pairs of covert users are the followers. The optimal pricing decision process of the user is obtained by using a backward recursion method, and the optimal pricing of the user and the optimal transmission power of the covert users are obtained by using an iterative algorithm. The model is complete, reasonable and effective, and can obtain the game equilibrium solution of the user and the covert users.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of wireless communication, in particular to a method for interference pricing and power control based on hierarchical game in user-assisted multi-band covert communication. BACKGROUND

[0002] In the field of wireless communication, information is transmitted through electromagnetic waves in space, and the receiving party can demodulate the message of the sending party by changing the amplitude, frequency, phase and other parameters of the electromagnetic wave. However, the open transmission characteristics of electromagnetic waves in space make wireless information transmission easy to be detected, eavesdropped and attacked by third parties. In the field of medical health, if the electromagnetic signals of medical devices are detected by third parties, the privacy of patients will be exposed; in the field of communication confrontation, if the electromagnetic signals of key communication nodes are detected by the enemy, their positions will be exposed, and they may be interfered or attacked. Therefore, once the wireless communication signal is found by the outside detector, there is a security risk. The wireless covert communication technology aims to hide the communication signal in the electromagnetic environment, so that the detector cannot accurately find the existence of the communication signal, and can improve the security of wireless communication.

[0003] When the public communication user assists in covert transmission, the transmission of the covert user will interfere with the public communication user at the same frequency, thereby reducing the transmission rate of the public communication user. In the existing research on public signal-assisted covert transmission, there is little negative impact of covert transmission on public users. In order to encourage public users to participate in covert transmission, a reasonable compensation mechanism for public users is needed. At the same time, the transmission concealment requirement and the interference pricing of the public user will affect the transmission power of the covert user. In order to realize the reasonable use of system resources, the pricing of the public user and the transmission power of the covert user need to be optimized. SUMMARY

[0004] The present application provides a method for interference pricing and power control based on hierarchical game in user-assisted multi-band covert communication, which can be used to solve the technical problem of unreasonable arrangement of system resources.

[0005] The present application provides a method for interference pricing and power control based on hierarchical game in user-assisted multi-band covert communication, which comprises the following steps:

[0006] Step 1, setting up a multi-band covert transmission system model assisted by public communication users;

[0007] Step 2, determining the transmission concealment index of each pair of covert users;

[0008] Step 3, determining the transmission effectiveness index of the covert user;

[0009] Step 4, formulating an interference pricing mechanism based on hierarchical game and performing power control;

[0010] Step 5, the equilibrium solution of the Stackelberg game composed of the first optimization problem P1 and the second optimization problem P2 is solved by using a backward recursion method.

[0011] Further, step 1, a model of a multi-band covert transmission system assisted by open communication users is set, comprising:

[0012] The multi-band covert transmission model comprises one pair of open users and M pairs of covert users and a detector; the open communication users and the M pairs of covert users share a frequency band, and the corresponding frequency band is divided into M mutually orthogonal sub-bands, and each sub-band is allocated to a pair of covert users; the open communication users transmit on the entire frequency band, and the transmission power is uniformly distributed on the frequency band, and the transmission power of the open communication users on each sub-band is P l ; the detector is a manager of the corresponding frequency band and has granted the frequency band usage right to the open communication users, and the covert users are not authorized by the detector, and the goal of the detector is to detect whether there is covert transmission on each sub-band; the detector does not know the instantaneous gain of the open user to itself channel on each sub-band, so the covert users can transmit information under the cover of the open user signal without being accurately detected by the detector.

[0013] Consider that the open user continuously transmits within a period of time, at this time the covert user selects a time slot to secretly transmit covert information within the period of time; however, the transmission of the covert user will bring co-frequency interference to the open communication user, in order to compensate for the covert transmission, the covert user pays a fee to the open communication user when transmitting; the pricing of the interference is decided by the open communication user, and the covert user transmits power according to the interference pricing decision of the open communication user; all wireless channels in the system are Rayleigh channels, so the channel gain is independently and identically distributed in different time slots, and only one time slot needs to be considered when analyzing system performance and optimizing parameters.

[0014] Further, step 2, a transmission concealment index of each pair of covert users is determined, comprising:

[0015] It is considered that the transmission power of the open communication user on different sub-bands is the same, and is P l ; let and represent the assumption that the i-th pair of covert users transmits information and does not transmit information respectively, and the signal received by the detector on the frequency band i is represented as:

[0016]

[0017] wherein p i represents the transmission power of the i-th pair of covert users, and represent the channel response from the open communication user and the covert user to the detector, x p and x crespectively, and satisfy E{|x p (j)| 2} = E{|x c (j)| 2} = 1, denotes the additive white Gaussian noise signal of the detector in frequency band i and obeys a complex Gaussian distribution with mean 0 and variance , j is the serial number of channel use;

[0018] The detector adopts a threshold-based average power detection method, and the detection rule is expressed as follows

[0019]

[0020] wherein is the average received power of the detector in frequency band i in the corresponding time slot; τ i is the detection threshold of the detector in sub-band i; and are the decision results of the detector accepting the original hypothesis and rejecting the alternative hypothesis, and accepting the alternative hypothesis and rejecting the original hypothesis, respectively. When the number of channel uses is large enough, the average received power of the detector is:

[0021]

[0022] Since the detector is unknown from the instantaneous channel gain of the public user to itself, the detector has uncertainty about the background noise power, and there is a probability of false decision. False decisions are divided into two cases: one is that the detector thinks that there is covert information transmission when there is no transmission condition of the covert user, which is called false alarm, and the corresponding conditional probability is denoted as The second is that the detector thinks that there is no covert user transmission when there is transmission condition of the covert user, which is called missed detection, and the corresponding conditional probability is denoted as The definition of false alarm probability and missed detection probability is as follows:

[0023]

[0024] The detector adopts the detection error probability as the detection performance evaluation index, and considers that the detector can adjust its own detection threshold to minimize the detection error probability, and the optimal detection threshold and the corresponding minimum error decision probability are as follows:

[0025]

[0026] Since the instantaneous channel gain from the covert user to the detector is unknown , the is adopted about The expected value of the detection probability is taken as the evaluation index of the transmission concealment, and the evaluation method is as follows:

[0027]

[0028] wherein, is the probability density function of the detection probability; it can be seen from the above formula that the greater the power of the open user, the greater the expected minimum error detection probability of the detector, which can provide better cover for the transmission of the concealed user; when the following conditions are met, it is said that the information transmission meets the concealment requirement:

[0029]

[0030] wherein ∈ is the maximum exposure probability that the detector can tolerate, i.e. the probability of accurate detection by the detector, and therefore, in order to meet the transmission concealment requirement, the maximum transmission power of the concealed user is:

[0031]

[0032] Further, step 3, determining the effectiveness index of the transmission of the concealed user, includes:

[0033] The signal-to-interference-and-noise ratio of the ith concealed receiving end is expressed as:

[0034]

[0035] wherein is the additive white noise power of the concealed receiving end, is the channel gain from the concealed sender to the receiving end in the frequency band i, is the channel gain from the open user to the concealed receiving end in the frequency band i; the transmission channel capacity of the concealed transmission is expressed as:

[0036] R i = ln (1 + γ i (p i )) (10)

[0037] The unit is nats / second / hertz.

[0038] Further, step 4, formulating an interference pricing mechanism based on hierarchical game and performing power control, includes:

[0039] Since the transmission of the concealed user will bring co-frequency interference to the open user, in order to compensate the open communication user, the concealed user needs to pay a fee to the open communication user during transmission, wherein the price is formulated by the open communication user, and the concealed user decides its transmission power according to the price. Specifically, at the beginning of each time slot, the open communication user formulates the interference price for each concealed user and broadcasts it to all the concealed users, and the price vector is c=(c1, c2,..., cn). The concealed user decides its transmission power according to the price, and the transmission power vector is p=(p1, p2,..., pn).M ) T , and then the hidden user decides its transmit power according to the price;

[0040] The order of the user actions exists in the process that the open user first prices and the hidden user accesses according to the pricing, and the decision of the open user influences the decision of the hidden user, so the hierarchical game model is suitable for modeling. The Stackelberg game is a non-cooperative hierarchical game, in which a part of game participants act before other participants, have the dominance and are called leaders, and other participants who act later observe the actions of the leaders and their actions are influenced by the leaders, and these participants are called followers. In the present application, the open user who first prices is the leader, and the hidden user who prices later is the follower; the utility function and the decision problem of the user are as follows:

[0041] For the ith pair of hidden users, the utility function is influenced by the hidden transmission rate and the cost paid to the open user, and the utility is expressed as:

[0042] U i (p i ,c i )=ln(1+γ i (p i ))-c i I i (p i ) (11)

[0043] Wherein is the interference power caused by the hidden user to the open user during transmission; it should be pointed out that although the transmit power of other hidden users does not appear in the utility function of the ith hidden user, the transmit power of other hidden users will affect the total interference power received by the open user, so the open user changes its pricing vector with probability; therefore, the transmit power of different hidden users influences each other; for the hidden user i, the first optimization problem to be solved is to decide the transmit power to maximize its utility, and is as follows:

[0044]

[0045] s.t.p i ≤P max

[0046] For the open user, the revenue comes from the amount paid by the hidden user, and the utility function is expressed as:

[0047]

[0048] In order not to affect its own transmission, the primary user selects a suitable pricing vector to make the total received interference power less than the maximum tolerable interference power value Q; the second optimization problem faced by the primary user is in the form of:

[0049]

[0050] The first optimization problem P1 and the second optimization problem P2 together constitute a Stakelberg game, in which it is crucial to obtain the Stakelberg equilibrium. The definition of the Stakelberg equilibrium is as follows:

[0051] Let and c * be the solutions of the first optimization problem P1 and the second optimization problem P2 respectively, then (p * ,c * ) is called a Stakelberg equilibrium of the game when the following conditions are met:

[0052]

[0053] Where represents the vector composed of the transmission power of other covert users except the covert user i; from the definition, when the game reaches the Stakelberg equilibrium, any participant cannot obtain greater utility by unilaterally changing its own strategy, that is, the leader and each follower will not leave the current equilibrium state, at which time the system reaches a stable state.

[0054] Further, step 5, the equilibrium solution of the Stakelberg game composed of the first optimization problem P1 and the second optimization problem P2 is solved by using a backward recursion method, including:

[0055] When the open user pricing is given, according to the characteristics of the objective function, the optimal transmission power of the covert user is:

[0056]

[0057] Where (·) + =max(·,0) is a positive function; then the optimal transmission power of the covert user is substituted back into the decision problem of the open user, and the optimal pricing problem of the open user is obtained as:

[0058]

[0059] Observing the optimal pricing problem, min(.) and (·) +The existence of the causes the objective function to be a non-convex function, so the optimal pricing problem is a non-convex optimization problem; therefore it is very difficult to solve by using existing optimization methods. In order to solve the optimal pricing problem, first consider the case that the hidden users have no maximum transmit power limit, and then consider the maximum transmit power limit of the hidden users after obtaining the solution of the optimization problem without the maximum transmit power limit condition;

[0060] When the hidden users have no maximum transmit power limit, the optimal pricing problem P3 is transformed into:

[0061]

[0062] At this time, the fourth optimization problem P4 is a convex problem, and the closed-form optimal solution is obtained by using the KKT condition; arrange the hidden users in the order of μ1≥μ2≥…≥μ M , and the optimal solution of the fourth optimization problem P4 is:

[0063]

[0064] Wherein The optimal transmit power of the corresponding hidden user is:

[0065]

[0066] Next, consider the transmit power limit of the hidden users;

[0067] First, when , the interference allowance is excessive, and the optimal transmit power of all hidden users is P max At this time, the optimal pricing of the public user to the hidden user i is:

[0068]

[0069] When the maximum transmit power in formula (20) is less than P max , the concealment constraint is naturally satisfied, and the result of formula (19) is the optimal pricing of the public user, and at this time the total interference allowance satisfies:

[0070]

[0071] When the interference allowance does not satisfy the above two conditions, the public user uses the optimal pricing algorithm to obtain the optimal pricing vector, and the algorithm includes the following steps:

[0072] Initialization: arrange the hidden users in the order of μ1≥μ2≥…≥μ M ;

[0073] Step 51, compare the public user interference allowance Q and If If yes, the algorithm ends and outputs the optimal pricing in formula (21), otherwise go to step 52;

[0074] Step 52, calculate the optimal solution of the fourth optimization problem (i.e. in formula 19) to obtain the optimal pricing without the concealment constraint and the corresponding optimal transmit power in formula (20);

[0075] Step 53, if the optimal transmit power without the concealment constraint is less than P max , the result of the optimal solution of the fourth optimization problem in formula (19) is the optimal pricing vector, the algorithm ends and outputs the optimal pricing vector; otherwise go to step 54;

[0076] Step 54, find the pair of concealed users with the maximum transmit power in formula (20), mark the serial number as U, and set the pricing of the pair of concealed users to the public user as Therefore, the optimal transmit power of the concealed user U is P max ;

[0077] Step 55, update the total interference quota Go to step 51 to allocate the remaining interference quota to other concealed users;

[0078] The above iterative process ensures that the public user reaches the Stearnberg equilibrium.

[0079] The present application has the following beneficial effects:

[0080] (1) The interference compensation mechanism in the public user assisted multi-band concealed communication is described, and a suitable utility function is selected to describe the process of the public user decision pricing and the concealed user decision transmit power;

[0081] (2) The model is complete and has clear physical meaning, the interference pricing and power control method based on the hierarchical game is proposed, the public user is modeled as a leader and the concealed user is modeled as a follower, and the optimal decision optimization problem of the public user is obtained by the backward recursion method;

[0082] (3) The means + the proposed iterative algorithm can obtain the optimal pricing of the public user and the optimal transmit power of the concealed user, so that the public user and the concealed user both reach a stable state. BRIEF DESCRIPTION OF DRAWINGS

[0083] Figure 1 is a model diagram of the public user assisted multi-band concealed transmission system of the present application;

[0084] Figure 2 is a frequency band usage diagram and a received power diagram of the detector on different sub-bands in the system model of the present application.

[0085] Figure 3is a curve of the maximum total revenue of the open user versus the total interference quota in the embodiment of the present application.

[0086] Figure 4 is a curve of the optimal transmission power of different covert users versus the total interference quota in the embodiment of the present application. DETAILED DESCRIPTION

[0087] In order to make the purpose, technical scheme and advantages of the present application more clear, the embodiments of the present application will be further described in detail below with reference to the drawings.

[0088] Firstly, the embodiments of the present application will be introduced below with reference to the drawings.

[0089] The embodiments of the present application are described as follows, the system simulation adopts MATLAB software, and the setting of parameters does not affect the generality. Figure 3 The effectiveness of the anti-interference under the existence of wideband and multiple interference is verified. The parameter settings are as follows: the number of covert users is M=6, the covert users have been arranged in the order from large to small according to μ, the mean of all wireless channels is 1, the channel gain adopts random number generation, the additive white Gaussian noise power at the user receiver is The transmission power of the open user is P l =30dBm.

[0090] Figure 3 is a curve of the maximum total revenue of the open user versus the total interference quota in the embodiment of the present application. When the maximum transmission power of the covert user is not limited, the maximum total revenue of the open user always increases with the increase of the interference quota. When the maximum transmission power of the covert user exists, the maximum total revenue of the open user first increases and then remains unchanged with the increase of the interference quota. This is because the existence of the maximum transmission power limitation leads to that the interference quota that can be sold by the open user is limited, when the total interference quota has made all the covert users reach the maximum transmission power, the increase of the total interference quota cannot make the open user's revenue increase. When the revenue of the open user does not change with the total interference quota, the greater the maximum transmission power of the covert user is, the greater the interference quota that can be sold by the open user is, and the greater the total revenue of the open user is.

[0091] Figure 4is the curve of the optimal transmit power of the hidden user in the embodiment of the present application varying with the total interference quota. When the total interference quota is small, the optimal transmit power of the hidden user to 1 and 2 is not zero, when the total interference quota continues to increase, the optimal transmit power of all the hidden users increases and finally converges to a constant value. This is because when the total interference quota is small, the open user will prefer to allocate the interference quota to the hidden user with smaller μ value to maximize the total revenue, when the total interference quota continues to increase, the open user adjusts the interference pricing so that other hidden users can also allocate the interference quota, and due to the existence of the hidden constraint, the maximum transmit power of the hidden user cannot exceed the maximum value.

[0092] In summary, the interference pricing and power control method based on hierarchical game in the open user assisted multi-band hidden transmission proposed by the present application fully considers the mutual interference brought by the hidden user transmission to the open user, designs reasonable utility functions for the open user and the hidden user, and can make the open user and the hidden user reach a stable state; the optimal interference pricing iterative solution algorithm based on backward recursion proposed by the present application can realize effective solution of the proposed model, obtain the optimal interference pricing of the open user and the optimal transmit power of the hidden user, and effectively cope with the problem of complex optimization solution caused by the transmission hidden constraint and the total interference quota constraint.

[0093] The present application has been described above by way of example, and those skilled in the art should understand that the present disclosure is not limited to the above-described embodiments, and various changes, modifications and replacements can be made without departing from the scope of the present application.

[0094] The above-described embodiments of the present application do not constitute a limitation on the protection scope of the present application.

Claims

1. A method for interference pricing and power control based on hierarchical game in user-assisted multi-band covert communication, characterized in that, The method comprises: Step 1, setting a multi-band covert transmission system model assisted by open communication users; Step 2, determining a transmission concealment index of each pair of covert users; Step 3, determining a transmission effectiveness index of the covert users; Step 4, formulating an interference pricing mechanism based on a hierarchical game and performing power control; Step 5, solving an equilibrium solution of a Stackelberg game formed by the first optimization problem P1 and the second optimization problem P2 by using a backward recursion method; The transmission concealment index is an expected probability value that information transmission of the pair of covert users is not accurately detected, and is expressed as: wherein, wherein P l is the transmit power of the public user, p i is the transmit power of the i-th pair of hidden users; The transmission effectiveness index of the covert users is a covert transmission channel capacity on a unit frequency band, and is expressed as: wherein is the instantaneous gain of the sender-to-receiver wireless channel in the i-th pair of covert users, is the instantaneous gain of the public user-to-receiver wireless channel in the i-th pair of covert users, is the additive white noise power at the covert receiver; the greater the covert transmission channel capacity per unit frequency band indicates the better effectiveness of the covert user transmission; Step 4, formulating an interference pricing mechanism based on a hierarchical game and performing power control, comprising: At the beginning of each time slot, the open users determine the interference price for each hidden user and broadcast to all hidden users, the price vector is c = (c1, c2,..., cN)T. M ) T Then the hidden users decide their transmit power according to the price. The open communication users are priced as leaders first, and the M pairs of covert users are priced as followers subsequently; a utility function and a decision problem of the users are determined as follows: For the ith pair of covert users, the utility function is affected by a covert transmission rate and a fee paid to the open users, and the utility is expressed as: U i (p i ,c i )=ln(1+γ i (p i ))-c i I i (p i ) (11) where is the interference power caused to the open user when the hidden user transmits; the transmit power of other hidden users also affects the total interference power received by the open user, so the open user changes its pricing vector according to the existence probability; therefore, the transmit power of different hidden users affects each other; for the hidden user i, to maximize its own utility by deciding its own transmit power, the first optimization problem to be solved is as follows: For the open communication users, a revenue comes from an amount paid by the covert users, and a utility function is expressed as: In order not to affect the transmission of the open users, the primary users select a suitable pricing vector to make the total received interference power less than a maximum tolerable interference power value Q; a form of the second optimization problem faced by the primary users is: The first optimization problem P1 and the second optimization problem P2 jointly form a Stackelberg game, and a definition of a Stackelberg equilibrium is as follows: Let p i * and c * be the solutions of the first and second optimization problems P1 and P2, respectively, then (p * ,c * ) is called a Nash equilibrium of the game when the following conditions are satisfied: wherein denotes the vector of transmit powers of the other concealed users than the concealed user i; it is clear from the definition that when the game reaches the Steinerberg equilibrium, no participant can obtain a greater utility by unilaterally changing his strategy, i.e. the leader and each follower will not leave the current equilibrium state, at which the system reaches a stable state.

2. The method of claim 1, wherein, Step 1, setting a multi-band covert transmission system model assisted by open communication users, comprising: The multi-band covert transmission model includes a pair of open users and M pairs of covert users and a detector; the open communication users and the M pairs of covert users share a frequency band, the corresponding frequency band is divided into M mutually orthogonal sub-bands, and each sub-band is allocated to a pair of covert users; the open communication users transmit on the entire frequency band, and the transmission power is uniformly distributed on the frequency band, and the transmission power of the open communication users on each sub-band is P l ; the detector is the manager of the corresponding frequency band and has granted the frequency band usage right to the open communication users, and the covert users are not authorized by the detector, and the goal of the detector is to detect whether there is a covert transmission on each sub-band; the detector does not know the instantaneous gain of the open user to itself channel on each sub-band, so the covert user can transmit information under the cover of the open user signal without being accurately detected by the detector; The open users continuously transmit in a time period, and the covert users select time slots to transmit covert information in the time period; the covert users pay fees to the open communication users in the transmission; pricing of the interference is determined by the open communication users, and the covert users determine transmission power according to the interference pricing decision of the open communication users; all wireless channels in the system are Rayleigh channels, and therefore channel gains are independently and identically distributed in different time slots.

3. The method of claim 1, wherein, Step 2, determining a transmission concealment index of each pair of covert users, comprising: Consider that the transmitting power of the open communication users on different sub-bands is the same, P l ; record and respectively represent the assumption of the i-th pair of hidden users transmitting information and not transmitting information, and the signal received by the detector on the frequency band i is represented as: where p i represents the transmission power of the i-th pair of hidden users, and represents the channel response from the open and hidden users to the detector, x p and x c respectively represent the signals generated by the open and hidden users and satisfy E{|x p (j)| 2} = E{|x c (j)| 2} = 1, represents the additive white Gaussian noise signal of the detector on the frequency band i and obeys a complex Gaussian distribution with mean 0 and variance , and j is the channel use number. A detector adopts a threshold-based average power detection method, and a detection rule is expressed as follows wherein is the average received power of the detector in the corresponding time slot on the frequency band i; τ i is the detection threshold of the detector on the sub-band i; and are the decision results of the detector accepting the original hypothesis and rejecting the alternative hypothesis, and accepting the alternative hypothesis and rejecting the original hypothesis, respectively, and when the number of channel uses is large enough, the average received power of the detector is: There is a probability of false decision results because of the uncertainty of the background noise power of the detector; false decision results are divided into two cases: one is that the detector considers that there is covert information transmission when there is no transmission condition of the covert user, which is called false alarm, and the corresponding conditional probability is denoted as The other is that the detector considers that there is no covert user transmission when there is transmission condition of the covert user, which is called missed detection, and the corresponding conditional probability is denoted as The definition of the false alarm probability and the missed detection probability is as follows: The detector adopts a detection error probability As a detection performance evaluation index, the detector can adjust its own detection threshold to minimize the detection error probability, and the optimal detection threshold And the corresponding minimum error decision probability As follows: Since the cover user does not know the instantaneous channel gain from itself to the detector Adopting As to the evaluation index of the self-transmission concealability of the expectation value of The evaluation mode is as follows: wherein, is the probability density function; the greater the public user power, the greater the expected minimum probability of error decision of the detector, which can provide better cover for covert user transmission; when the following conditions are met, it is said that the information transmission meets the concealment requirement: Wherein, ∈ is a maximum exposure probability that can be tolerated by the detector, that is, a probability of being accurately detected by the detector, and therefore the maximum transmission power of the covert users is:

4. The method of claim 1, wherein, Step 3, determining a transmission effectiveness index of the covert users, comprising: A signal-to-interference-and-noise ratio of the ith covert receiving end is expressed as: wherein is the additive white noise power at the covert receiver, is the channel gain from the covert transmitter to the receiver in frequency band i, is the channel gain from the open user to the covert receiver in frequency band i; the covert transmission channel capacity is represented as: R i = ln(l + γ i (p i )) (10) The unit is nats / second / hertz.

5. The method of claim 1, wherein, Step 5, solving an equilibrium solution of a Stackelberg game formed by the first optimization problem P1 and the second optimization problem P2 by using a backward recursion method, comprising: When the pricing of the open users is given, the optimal transmission power of the covert users is obtained according to characteristics of the objective function as: where (·) + = max(·, 0) is the take positive function; then the optimal transmit power of the hidden user is substituted back into the decision problem of the public user, and the optimal pricing problem of the public user is obtained as Observing the optimal pricing problem, min(·) and (·) + The existence of causes the objective function to be non-convex function, so the optimal pricing problem is a non-convex optimization problem; first consider the case of hidden users without maximum transmit power limit, and then consider the maximum transmit power limit of hidden users after getting the solution of the optimization problem without maximum transmit power limit When the transmission power of the covert users is not limited, the optimal pricing problem P3 is transformed into At this time, the fourth optimization problem P4 is a convex problem, and the closed optimal solution is obtained by using the KKT condition; the hidden users are arranged in the order of μ1≥μ2≥…≥μ M , and the optimal solution of the fourth optimization problem P4 is: wherein The optimal transmit power of the corresponding hidden user is: Next, the transmission power limitation of the covert users is considered; First, when the interference margin is excessive, the optimal transmit power of all concealed users is P max At this time, the optimal pricing of the public user to the concealed user i is: When the maximum transmit power in formula (20) is less than P max When the maximum transmit power in formula (20) is less than P max When the maximum transmit power in formula (20) is less than P max When the maximum transmit power in formula (20) is less than P max When the maximum transmit power in formula (20) is less than P max When the maximum transmit power in formula (20) is less than P <000003 When the interference quota does not satisfy the above two conditions, the open users adopt an optimal pricing algorithm to obtain an optimal pricing vector, and the algorithm comprises the following steps: Initialization: The concealed users are arranged in the order of μ1≥ μ2≥... ≥ μ M n. Step 51, compare the public user interference quota Q and If End of algorithm, output the optimal pricing in formula (21), otherwise go to step 52; Step 52, calculate the optimal pricing and optimal transmit power without the constraint of concealment; Step 53, if none of the optimal transmit powers under the concealment constraint condition exceeds P max , the optimal solution of the fourth optimization problem is the optimal pricing vector, the algorithm ends and outputs the optimal pricing vector; otherwise, go to step 54. Step 54, find the pair of the hidden user with the maximum transmit power in formula (20), mark the serial number as U, and set the pricing of the pair of the public user and the hidden user as formula (21) Therefore, the optimal transmit power of the hidden user U is P max ; Step 55, update total interference quota Go to step 51 to allocate the remaining interference quota to other hidden users. The above iterative process is performed to ensure that the public user reaches the Stiglitz equilibrium.

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