Interference pricing and power control method based on hierarchical game in public user assisted multi-band covert communication

Abstract

The invention discloses an interference pricing and power control method based on a hierarchical game in public user-assisted multi-band covert communication, which is characterized in that a public user assists multiple pairs of covert users to transmit on multiple bands so as to cope with a scene threatened by multiple detectors, co-frequency mutual interference is brought to the public user during transmission of the covert users, and the transmission rate of the public user is reduced. The hidden user needs to pay the public user during transmission, the interference price is determined by the public user, and the hidden user decides the own transmitting power according to the interference pricing under the constraint of transmission concealment. The process of pricing hidden user decision power by a public user is modeled as a hierarchical game, the public user is a leader, and a plurality of pairs of hidden users are followers. And obtaining an optimal pricing decision process of the public user by using a reverse recursion method, and obtaining the optimal pricing of the public user and the optimal transmitting power of the hidden user through an iterative algorithm. The model is complete, reasonable and effective, and game equilibrium solutions of public users and hidden users can be obtained.

Description

Technical Field

[0001] The present application relates to the field of wireless communication technology, and in particular to an interference pricing and power control method based on layered game in public user-assisted multi-band covert communication. Background Art

[0002] In the field of wireless communications, information is transmitted in space through electromagnetic waves, and the receiver can demodulate the sender's message through changes in the amplitude, frequency, phase and other parameters of the electromagnetic waves. However, the open propagation 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 and health care, the electromagnetic signals of medical equipment can easily expose patient privacy if detected by a third party; in the field of communication confrontation, the electromagnetic signals of key communication nodes will expose their locations once detected by the enemy, and may be interfered with or attacked. Therefore, once wireless communication signals are discovered by external detectors, there is a security risk. Wireless covert communication technology aims to hide communication signals in an electromagnetic environment, making it impossible for detectors to accurately detect the existence of communication signals, which can improve the security of wireless communications.

[0003] While public communication users assist covert transmission, the transmission of covert users will cause co-frequency interference to public communication users, thereby reducing the transmission rate of public communication users. In the existing research work on public signal-assisted covert transmission, there is little research on the negative impact of covert transmission on public users. In order to encourage public users to participate in the covert transmission, a reasonable public user compensation mechanism is needed. At the same time, the transmission concealment requirements and the interference pricing of public users will affect the transmission power of covert users. In order to achieve the rational use of system resources, it is necessary to optimize the pricing of public users and the transmission power of covert users. Summary of the invention

[0004] The present application provides an interference pricing and power control method based on layered game in public 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, the method comprising:

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

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

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

[0009] Step 4: Develop an interference pricing mechanism based on layered game and perform power control;

[0010] Step 5, using the reverse recursive method to solve the equilibrium solution of the Steinberg game consisting of the first optimization problem P1 and the second optimization problem P2.

[0011] Furthermore, step 1 sets a multi-band covert transmission system model assisted by a public communication user, including:

[0012] The multi-band covert transmission model includes a pair of public users, M pairs of covert users and a detector; the public communication users and M pairs of covert users share a spectrum, and the corresponding frequency band is divided into M mutually orthogonal sub-bands, each of which is assigned to a pair of covert users; the public communication users transmit on the entire frequency band, and the transmission power is evenly distributed on the frequency band. The transmission power of the public communication users on each sub-band is P l ; The detector is the manager of the corresponding frequency band and has granted the right to use the frequency band to public communication users, while the hidden users are not authorized by the detector. The detector's goal is to detect whether there is any hidden transmission on each sub-band; the detector is unaware of the instantaneous gain of the public user to its own channel on each sub-band, so the hidden user can transmit information under the cover of the public user signal without being accurately detected by the detector.

[0013] Consider a period of continuous transmission by public users, and the hidden users select time slots within this period to secretly transmit covert information. However, the transmission of the hidden users will cause co-frequency interference to the public communication users. In order to compensate for the covert transmission, the hidden users pay fees to the public communication users when transmitting. The pricing of interference is determined by the public communication users, and the hidden users decide the transmission power based on the interference pricing of the public communication users. All wireless channels in the system are Rayleigh channels, so the channel gains are independent and identically distributed in different time slots. Only one time slot needs to be considered when analyzing the system performance and optimizing the parameters.

[0014] Furthermore, step 2 determines the transmission concealment index of each pair of concealed users, including:

[0015] Consider that the transmission power of public communication users in different sub-bands is the same, which is P l ;remember and denote the assumptions that the i-th pair of hidden users transmit information and do not transmit information, respectively. Then the signal received by the detector on frequency band i is expressed as:

[0016]

[0017] Among them, p i represents the transmission power of the i-th hidden user pair, and represents the channel response from the public communication user and the hidden user to the detector, x p and x cdenote the signals generated by the public user and the hidden user respectively and satisfy E{|x p (j)| 2}=E{|x c (j)| 2}=1, It represents the additive Gaussian white noise signal of the detector in frequency band i and has a mean of 0 and a variance of The complex Gaussian distribution of , j is the channel usage number;

[0018] The detector uses a threshold-based average power detection method. The detection rules are as follows:

[0019]

[0020] in is the average received power of the detector in the frequency band i in the corresponding time slot; τ i is the detection threshold of the detector in sub-band i; and The detector accepts the null hypothesis and rejects the alternative hypothesis, and accepts the alternative hypothesis and rejects the null hypothesis. When the channel usage times are large enough, the average receiving power of the detector is:

[0021]

[0022] Since the detector does not know the instantaneous channel gain from the public user to itself, the detector is uncertain about the background noise power and there is a probability of making an erroneous decision. There are two types of erroneous decisions: one is that the detector believes that there is a hidden message being sent when the hidden user does not transmit, which is called a false alarm. The corresponding conditional probability is recorded as Second, under the condition of hidden user transmission, the detector believes that there is no hidden user and no transmission, which is called missed detection. The corresponding conditional probability is recorded as The definitions of false alarm probability and missed detection probability are as follows:

[0023]

[0024] Detection Error Probability As a detection performance evaluation indicator, considering that the detector can adjust its own detection threshold to minimize the probability of detection error, the optimal detection threshold And the corresponding minimum probability of wrong decision as follows:

[0025]

[0026] Since the hidden user does not know the instantaneous channel gain from itself to the detector use about The expected value of is used as the evaluation index of its own transmission concealment, and the evaluation method is as follows:

[0027]

[0028] in, yes The probability density function of ; From the above formula, it can be seen that the greater the power of the public user, the greater the expected minimum error judgment probability of the detector, which can provide better cover for the hidden user transmission; when the following conditions are met, the information transmission is said to meet the concealment requirements:

[0029]

[0030] Where ∈ is the maximum exposure probability that the detector can tolerate, that is, the probability of accurate detection by the detected person. Therefore, in order to meet the transmission concealment requirements, the maximum transmission power of the concealed user is:

[0031]

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

[0033] The signal to interference noise ratio of the i-th hidden receiver is expressed as:

[0034]

[0035] in is the additive white noise power of the concealed receiver, is the channel gain from the hidden sender to the receiver on frequency band i, is the channel gain from the public user to the hidden receiver on frequency band i; the hidden transmission channel capacity is expressed as:

[0036] R i =ln(1+γ i (p i )), (10)

[0037] The unit is nit / second / Hertz.

[0038] Furthermore, in step 4, an interference pricing mechanism based on layered game is formulated and power control is performed, including:

[0039] Since hidden users will cause co-frequency interference to public users when transmitting, in order to compensate public communication users, hidden users need to pay fees to public communication users when transmitting. The price is set by the public communication users, and the hidden users decide their own transmission power based on the price. Specifically, at the beginning of each time slot, the public communication users set the interference price for each hidden user and broadcast it to all hidden users. The price vector is c = (c1, c2, ..., cM ) T ,Then the hidden user decides its own transmission power according to the price;

[0040] The public communication user is priced first, and the hidden user is accessed according to the pricing. In the process, the user actions have a sequence, and the decision of the public communication user affects the decision of the hidden user, which is suitable for modeling with a hierarchical game model. The Steinberg game is a non-cooperative hierarchical game, in which some game participants act before other participants, have a dominant position and are called leaders, and other participants who act later observe the actions of the leader and their actions are affected by the leader. These participants are called followers. In the present invention, the public communication user is priced as the leader first, and the hidden user is subsequently priced as a follower; the user's utility function and decision problem are clearly defined as follows:

[0041] For the i-th pair of hidden users, the utility function is affected by the hidden transmission rate and the fee paid to the public 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] in is the interference power caused by the hidden user to the public communication user when transmitting; it should be pointed out that although the transmission power of other hidden users does not appear in the utility function of hidden user pair i, the transmission power of other hidden users will also affect the total interference power received by the public user, so that the public user has the probability to change its own pricing vector; therefore, the transmission power between different hidden users affects each other; for hidden user pair i, the first optimization problem to be solved in deciding its own transmission power to maximize its own utility is as follows:

[0044]

[0045] For public communication users, the benefits come from the amount paid by hidden users, and the utility function is expressed as:

[0046]

[0047] In order not to affect its own transmission, the primary user selects a suitable pricing vector so that the total interference power received is less than the maximum tolerable interference power value Q; the second optimization problem faced by the primary user is in the form of:

[0048]

[0049] The first optimization problem P1 and the second optimization problem P2 together constitute a Steinberg game. In this type of game, it is very important to find the Steinberg equilibrium. The definition of Steinberg equilibrium is as follows:

[0050] remember and c * are the solutions to the first optimization problem P1 and the second optimization problem P2 respectively. Then, when the following conditions are met (p * ,c * ) is called a Steinberg equilibrium of the game:

[0051]

[0052] in Represents the vector composed of the transmission power of other hidden users except hidden user i; From the definition, it can be seen that when the game reaches the Steinberg equilibrium, no participant can obtain greater utility by unilaterally changing its own strategy, that is, the leader and each follower will not leave the current equilibrium state, and the system reaches a stable state.

[0053] Furthermore, in step 5, a reverse recursive method is used to solve the equilibrium solution of the Steinberg game consisting of the first optimization problem P1 and the second optimization problem P2, including:

[0054] When the public user pricing is given, according to the specific objective function, the optimal transmission power of the hidden user is obtained as:

[0055]

[0056] in(·) + =max(·,0) is a positive function; then the optimal transmission power of the hidden user is substituted into the decision problem of the public user, and the optimal pricing problem of the public user is obtained as follows:

[0057]

[0058] Observe the optimal pricing problem, min(·) and (·) + The existence of causes the objective function to be a non-convex function, and thus the optimal pricing problem is a non-convex optimization problem; therefore, it is very difficult to solve it using existing optimization methods. In order to solve the optimal pricing problem, we first consider the case where the hidden user has no maximum transmission power limit, and then consider the maximum transmission power limit of the hidden user after obtaining the solution of the optimization problem without the maximum transmission power limit;

[0059] When the hidden user has no maximum transmission power limit, the optimal pricing problem P3 is transformed into:

[0060]

[0061] At this time, the fourth optimization problem P4 is a convex problem, and the closed-form optimal solution is obtained using the KKT condition; the hidden users are divided according to μ1≥μ2≥…≥μ M The optimal solution of the fourth optimization problem P4 is:

[0062]

[0063] in The corresponding optimal transmission power of the hidden user is:

[0064]

[0065] Next, consider the transmit power limit of the hidden user;

[0066] First when When the interference quota is excessive, the optimal transmission power of all hidden users is P max , at this time, the optimal price of the public user to the hidden user i is:

[0067]

[0068] When the maximum transmit power in formula (20) is less than P max When , the concealment constraint is naturally satisfied, and the result of formula (19) is the optimal pricing for public users. At this time, the total interference quota satisfies:

[0069]

[0070] When the interference quota does not meet the above two conditions, the public user adopts the optimal pricing algorithm to obtain the optimal pricing vector. The algorithm includes the following steps:

[0071] Initialization: Set the hidden user pairs according to μ1≥μ2≥…≥μ M Arrange in order;

[0072] Step 51: compare the public user interference quota Q and if The algorithm ends and outputs the optimal pricing in formula (21), otherwise it goes to step 52;

[0073] Step 52, calculating the optimal pricing without hidden constraints in the optimal solution of the fourth optimization problem (i.e., in formula 19) and the optimal transmission power in the corresponding formula (20);

[0074] Step 53: If the optimal transmission power under the non-concealment constraint condition does not exceed P max , 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, it goes to step 54;

[0075] Step 54, find the hidden user pair with the largest transmission power in formula (20), record the serial number as U, and set the price of the public user to the hidden user to Therefore, the optimal transmission power of hidden user U is P max ;

[0076] Step 55: Update the total interference quota Go to step 51 to distribute the remaining interference quota to other hidden users;

[0077] By executing the above iterative process, it is ensured that the public user reaches the Steinberg equilibrium.

[0078] The beneficial effects of the present invention are:

[0079] (1) The interference compensation mechanism in public user-assisted multi-band covert communication is described, and the appropriate utility function is selected to describe the process of public user decision-making pricing and covert user decision-making transmission power;

[0080] (2) The model is complete and the physical meaning is clear. The proposed interference pricing and power control method based on hierarchical game models the public user as the leader and the hidden user as the follower. The optimal decision optimization problem of the public user is obtained through the reverse recursive method.

[0081] (3) Means + The proposed iterative algorithm can obtain the optimal pricing for public users and the optimal transmission power for hidden users, so that both public users and hidden users can reach a stable state. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] Figure 1 It is a model diagram of a user-assisted multi-band end concealed transmission system disclosed in the present invention;

[0083] Figure 2 It is a diagram of frequency band usage in the system model of the present invention and a schematic diagram of the receiving power of the detector in different sub-bands.

[0084] Figure 3 It is a diagram showing the change of the maximum total benefit of a public user along with the total interference amount in an embodiment of the present invention.

[0085] Figure 4 This is a graph showing how the optimal transmission power of different concealed users varies with the total interference quota in an embodiment of the present invention. DETAILED DESCRIPTION

[0086] In order to make the objectives, technical solutions and advantages of the present application clearer, the implementation methods of the present application will be further described in detail below with reference to the accompanying drawings.

[0087] The following first introduces the embodiments of the present application in conjunction with the accompanying drawings.

[0088] The embodiments of the present invention are described in detail as follows. The system simulation adopts MATLAB software, and the setting of parameters does not affect the generality. Figure 3 Verify the effectiveness of anti-interference in the presence of broadband and multiple interferences. The parameters are set as follows: the number of hidden user pairs is M = 6, the hidden users have been arranged in descending order according to μ, the mean of all wireless channels is 1, the channel gain is generated by random numbers, and the additive white Gaussian noise power at the user receiver is The transmission power of the public user is P l =30dBm.

[0089] Figure 3 It is a curve showing the change of the maximum total revenue of the public user with its interference quota in the embodiment of the present invention. When there is no maximum transmission power limit for the hidden user, the maximum total revenue of the public user increases with the increase of the interference quota. When there is a maximum transmission power limit for the hidden user, the maximum total revenue of the public user increases first and then remains unchanged as the interference quota increases. This is because the existence of the maximum transmission power limit leads to the limitation of the interference quota that the public user can sell. When the total interference quota has allowed all hidden users to reach the maximum transmission power, continuing to increase the total interference quota will not increase the revenue of the public users. When the revenue of the public users does not change with the total interference quota, the greater the maximum transmission power of the hidden user, the greater the interference quota that the public user can sell, and the greater its total revenue.

[0090] Figure 4 It is the curve of the optimal transmission power of the hidden user in the embodiment of the present invention changing with the total interference quota. When the total interference quota is small, only the optimal transmission power of the hidden user for 1 and 2 is not zero. When the total interference quota continues to increase, the optimal transmission power of all hidden users increases and eventually converges to a constant value. This is because when the total interference quota is small, the public user will give priority to allocating the interference quota to the hidden user with a smaller μ value to maximize the total benefit. When the total interference quota continues to increase, the public user adjusts the interference pricing so that other hidden users can also be allocated interference quotas. And due to the existence of the concealment constraint, the maximum transmission power of the hidden user cannot exceed the maximum value.

[0091] In summary, the interference pricing and power control method based on layered game in public user-assisted multi-band covert transmission proposed in the present invention fully considers the mutual interference caused to the public user by the hidden user during transmission, and designs reasonable utility functions for the public user and the hidden user, so that both the public user and the hidden user can reach a stable state; the proposed optimal interference pricing iterative solution algorithm based on reverse recursion can effectively solve the proposed model, calculate the optimal interference pricing for the public user and the optimal transmission power for the hidden user, and effectively deal with the complex optimization solution problems caused by the transmission concealment constraints and the total interference quota constraints.

[0092] The present invention has been described above in an illustrative manner using embodiments. Those skilled in the art should understand that the present disclosure is not limited to the embodiments described above, and that various changes, modifications and substitutions may be made without departing from the scope of the present invention.

[0093] 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 is disclosed, characterized in that: The method comprises: Step 1, setting a multi-band covert transmission system model assisted by public communication users; Step 2, determining the transmission concealment index of each pair of concealed users; Step 3, determining the concealed user transmission effectiveness index; Step 4: Develop an interference pricing mechanism based on layered game and perform power control; Step 5, using the reverse recursive method to solve the equilibrium solution of the Steinberg game consisting of the first optimization problem P1 and the second optimization problem P2.

2. The method according to claim 1, characterized in that Step 1: Setting a multi-band covert transmission system model assisted by public communication users, including: The multi-band covert transmission model includes a pair of public users, M pairs of covert users and a detector; the public communication users and M pairs of covert users share a spectrum, and the corresponding frequency band is divided into M mutually orthogonal sub-bands, each of which is assigned to a pair of covert users; the public communication users transmit on the entire frequency band, and the transmission power is evenly distributed on the frequency band. The transmission power of the public communication users on each sub-band is P l ; The detector is the manager of the corresponding frequency band and has granted the right to use the frequency band to public communication users, while the hidden users are not authorized by the detector. The detector's goal is to detect whether there is any hidden transmission on each sub-band; the detector is unaware of the instantaneous gain of the public user to its own channel on each sub-band, so the hidden user can transmit information under the cover of the public user signal without being accurately detected by the detector. Consider a period of continuous transmission by public users, at which time the hidden users select time slots within this period to transmit hidden information; the hidden users pay fees to the public communication users when transmitting; the pricing of interference is determined by the public communication users, and the hidden users decide the transmission power based on the interference pricing of the public communication users; all wireless channels in the system are Rayleigh channels, so the channel gains are independent and identically distributed in different time slots.

3. The method according to claim 1, characterized in that Step 2, determining the transmission concealment index of each pair of concealed users, including: Consider that the transmission power of public communication users in different sub-bands is the same, which is P l ;remember and denote the assumptions that the i-th pair of hidden users transmit information and do not transmit information, respectively. Then the signal received by the detector on frequency band i is expressed as: Among them, p i represents the transmission power of the i-th hidden user pair, and represents the channel response from the public communication user and the hidden user to the detector, x p and x c denote the signals generated by the public user and the hidden user respectively and satisfy E{|x p (j)| 2 }=E{|x c (j)| 2 }=1, It represents the additive Gaussian white noise signal of the detector in frequency band i and has a mean of 0 and a variance of The complex Gaussian distribution of , j is the channel usage number; The detector uses a threshold-based average power detection method. The detection rules are as follows: in is the average received power of the detector in the frequency band i in the corresponding time slot; τ i is the detection threshold of the detector in sub-band i; and The detector accepts the null hypothesis and rejects the alternative hypothesis, and accepts the alternative hypothesis and rejects the null hypothesis. When the channel usage times are large enough, the average receiving power of the detector is: The detector is uncertain about the background noise power, and there is a probability of making an erroneous judgment result. There are two kinds of erroneous judgments: one is that the detector believes that there is a hidden message when the hidden user does not transmit, which is called a false alarm. The corresponding conditional probability is recorded as Second, under the condition of hidden user transmission, the detector believes that there is no hidden user and no transmission, which is called missed detection. The corresponding conditional probability is recorded as The definitions of false alarm probability and missed detection probability are as follows: Detection Error Probability As a detection performance evaluation indicator, considering that the detector can adjust its own detection threshold to minimize the probability of detection error, the optimal detection threshold And the corresponding minimum probability of wrong decision as follows: Since the hidden user does not know the instantaneous channel gain from itself to the detector use about The expected value of is used as the evaluation index of its own transmission concealment, and the evaluation method is as follows: in, yes The probability density function of the public user is: the greater the power of the public user, the greater the expected minimum error judgment probability of the detector, which can provide better cover for the hidden user transmission; when the following conditions are met, the information transmission is said to meet the concealment requirements: Where ∈ is the maximum exposure probability that the detector can tolerate, that is, the probability of accurate detection by the detected person. Therefore, in order to meet the transmission concealment requirements, the maximum transmission power of the concealed user is:

4. The method according to claim 1, characterized in that: Step 3, determining the concealed user transmission effectiveness index, including: The signal to interference noise ratio of the i-th hidden receiver is expressed as: in is the additive white noise power of the hidden receiver, is the channel gain from the hidden sender to the receiver on frequency band i, is the channel gain from the public user to the hidden receiver on frequency band i; the hidden transmission channel capacity is expressed as: R i =ln(1+γ i (p i )), (10) The unit is nit / second / Hertz.

5. The method according to claim 1, characterized in that Step 4: Develop an interference pricing mechanism based on layered game and perform power control, including: At the beginning of each time slot, the public communication user sets the interference price for each hidden user and broadcasts it to all hidden users. The price vector is c = (c1, c2, ..., c M ) T ,Then the hidden user decides its own transmission power according to the price; The public communication user is first priced as a leader, and M subsequently prices the hidden user as a follower; the user's utility function and decision problem are defined as follows: For the i-th pair of hidden users, the utility function is affected by the hidden transmission rate and the fee paid to the public user, 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) in is the interference power caused by the hidden user to the public communication user when transmitting; the transmission power of other hidden users will also affect the total interference power received by the public user, so the public user has the probability to change its own pricing vector; therefore, the transmission power between different hidden users affects each other; for hidden user pair i, to decide its own transmission power to maximize its own utility, the first optimization problem that needs to be solved is as follows: For public communication users, the benefits come from the amount paid by hidden users, and the utility function is expressed as: In order not to affect its own transmission, the primary user selects a suitable pricing vector so that the total interference power received is less than the maximum tolerable interference power value Q; the second optimization problem faced by the primary user is in the form of: The first optimization problem P1 and the second optimization problem P2 together constitute a Steinberg game. The Steinberg equilibrium is defined as follows: remember and c * are the solutions to the first optimization problem P1 and the second optimization problem P2 respectively. Then, when the following conditions are met (p * ,c * ) is called a Steinberg equilibrium of the game: in Represents the vector composed of the transmission power of other hidden users except hidden user i; From the definition, it can be seen that when the game reaches the Steinberg equilibrium, no participant can obtain greater utility by unilaterally changing its own strategy, that is, the leader and each follower will not leave the current equilibrium state, and the system reaches a stable state.

6. The method according to claim 1, characterized in that Step 5, using a reverse recursive method to solve the equilibrium solution of the Steinberg game consisting of the first optimization problem P1 and the second optimization problem P2, including: When the public user pricing is given, according to the specific objective function, the optimal transmission power of the hidden user is obtained as: in(·) + =max(·,0) is a positive function; then the optimal transmission power of the hidden user is substituted into the decision problem of the public user, and the optimal pricing problem of the public user is obtained as follows: Observe the optimal pricing problem, min(·) and (·) + The existence of causes the objective function to be a non-convex function, so the optimal pricing problem is a non-convex optimization problem; first consider the case where the hidden user has no maximum transmission power limit, and then consider the maximum transmission power limit of the hidden user after obtaining the solution of the optimization problem under the condition of no maximum transmission power limit; When the hidden user has no maximum transmission power limit, the optimal pricing problem P3 is transformed into: At this time, the fourth optimization problem P4 is a convex problem, and the closed-form optimal solution is obtained using the KKT condition; the hidden users are divided according to μ1≥μ2≥…≥μ M The optimal solution of the fourth optimization problem P4 is: in The corresponding optimal transmission power of the hidden user is: Next, consider the transmit power limit of the hidden user; First when When the interference quota is excessive, the optimal transmission power of all hidden users is P max , at this time, the optimal price of the public user to the hidden user i is: When the maximum transmit power in formula (20) is less than P max When , the concealment constraint is naturally satisfied, and the result of formula (19) is the optimal pricing for public users. At this time, the total interference quota satisfies: When the interference quota does not meet the above two conditions, the public user adopts the optimal pricing algorithm to obtain the optimal pricing vector. The algorithm includes the following steps: Initialization: Set the hidden user pairs according to μ1≥μ2≥…≥μ M Arrange in order; Step 51: compare the public user interference quota Q and if The algorithm ends and outputs the optimal pricing in formula (21), otherwise it goes to step 52; Step 52, calculating the optimal pricing and optimal transmission power without concealment constraints; Step 53: If the optimal transmission power under the non-concealment constraint condition does not exceed P max , the result of the optimal solution to the fourth optimization problem is the optimal pricing vector, the algorithm ends, and the optimal pricing vector is output; otherwise, the process goes to step 54; Step 54, find the hidden user pair with the largest transmission power in formula (20), record the serial number as U, and set the price of the public user to the hidden user to Therefore, the optimal transmission power of hidden user U is P max ; Step 55: Update the total interference quota Go to step 51 to distribute the remaining interference quota to other hidden users; The above iterative process is performed to ensure that the public user reaches the Steinberg equilibrium.

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