CN105228233A - Based on the Poewr control method of monotonicity optimization and linear search in a kind of cognition wireless network - Google Patents

Abstract

A power control method based on monotonicity optimization and linear search in a cognitive wireless network, comprising the following steps: (1) Considering two-part interference including between PUs and SUs and between different SUs, the optimization problem is described as a multivariate Non-convex optimization problem; (2) The problem (P1) is vertically decomposed into two-layer optimization problems; (3) According to the underlying problem, a power control method for monotonic optimization is proposed, and the optimized power is optimized when the transmission power of the PU is given. The transmission power of SUs; (4) Based on the underlying problem, a linear search method is proposed to further optimize the transmission power of the PU; (5) Through the interactive iteration of the bottom problem and the top problem, the problem is finally solved (P1). The present invention provides an effective and efficient optimization method that maximizes the net income of the PU while guaranteeing the QoS of the PU, so as to improve the utilization rate of the system frequency spectrum and optimize the configuration of system resources.

Description

A Power Control Method Based on Monotonicity Optimization and Linear Search in Cognitive Wireless Networks

technical field

The invention relates to an optimal power control method based on monotonic optimization and linear search algorithm in a cognitive radio network.

Background technique

With the rapid growth of mobile data services, the limited available spectrum resources make the problem of spectrum congestion increasingly prominent. Dynamic Spectrum Access (DSA), as an effective supplement to the traditional fixed spectrum allocation method in the mobile network, intelligently reuses the licensed spectrum fully utilized by the unlicensed system (PrimarySystem-PS) or licensed user (PrimaryUser-PU) Resources, so that unlicensed users (SecondaryUser-SU) can access the licensed spectrum of the PU for data transmission in a timely manner, so that the spectrum utilization rate is effectively improved. Due to its superiority, DSA is considered to be a model of a spectrum supply method that can be flexible and respond to current needs, and has a broad prospect. However, when spectrum sharing is performed in a DSA network, interference will inevitably occur when PUs serve SUs, including: 1) co-channel interference between PUs and SUs 2) mutual interference between different SUs. In order to obtain additional benefits from serving SUs under the premise of guaranteeing the QoS of PUs, it is very necessary to rationally perform resource allocation and interference management in the process of designing a spectrum sharing scheme. However, the interference mentioned above often makes the problem difficult to solve due to non-convex optimization. Therefore, it is meaningful to propose an effective and efficient optimization method that guarantees the QoS of the PU and maximizes the net benefit of the PU. .

Contents of the invention

In order to ensure that spectrum sharing can optimize the allocation of spectrum resources in the DSA network, the present invention considers two parts of interference including between PUs and SUs and between different SUs, and proposes a way to ensure the QoS of the PU while maximizing the PU net gain to achieve an optimized power control method. The proposed power control algorithm has a two-layer structure, which improves the effectiveness and efficiency of the method while reducing the computational complexity.

The technical solution adopted by the present invention to solve its technical problems is:

A power control method based on monotonicity optimization and linear search in a cognitive wireless network, the control method comprising the following steps:

(1) In a cognitive radio network, through the transmission power control of authorized users PU and unlicensed users SUs, the QoS of PU is guaranteed while considering two parts of interference including between PU and SUs and between different SUs The optimization problem of maximizing the net benefit of the PU is described as a non-convex optimization problem as shown below:

P1: maxΣ _s∈Ω α _s R _s -β(p ₀ -p ₀ ^min

in represents the throughput of each unauthenticated user SUs, Indicates the uplink throughput of the authorized user PU, Ω={1,2...S} represents the set of all unauthorized users SUs;

In problem P1, each parameter is defined as follows:

α _s : the marginal coefficient of charging for the unit throughput PU realized by each SUs;

β: The marginal power consumption cost of the PU, in $/Watt;

R _s : throughput of each SUs;

p ₀ : transmit power of PU;

p ₀ ^min : the minimum transmit power consumption of the PU;

n: background noise power;

q _s : transmit power of SUs;

g _sB : channel power gain between SU-Tx and BS;

g _0B : channel power gain between PU-Tx and BS;

Throughput requirements for each SUs;

PU transmit power upper limit;

The upper limit of the maximum transmission power of the SU;

g _0s : channel power gain between PU-Tx and SU-Rxs;

g _ss : channel power gain between SU-Txs and SU-Rxs;

g _js : channel power gain between SU-Txj and SU-Rxs;

W: bandwidth of PU channel;

The superscript "*" in the parameter symbol indicates the optimal value of the parameter in the optimization problem;

(2) use the formula Expand the R _s in the constraints, the second term of the constraints is equivalent to in The decision variable of problem P1 is transformed into p ₀ and {q _s } _s∈Ω , using and respectively represent the optimal solution of problem P1;

(3) Judging the feasibility of problem P1

the formula p ₀ in is replaced by {q _s } _s∈Ω , so that the inequality can be re-expressed as a set of linear constraints as follows:

And s≠j

M represents an S×S matrix, S represents the total number of SUs in Ω, and the entries in M are expressed as follows:

In addition, a vector u of S×1 is defined, each item of which is expressed as

order vector Indicates the set of transmission power of SUs that can satisfy the above linear constraints, record condition C1: g _ss g _0B -θ _s θ ₀ g _sB g _0s >0, And condition C2: define the spectral radius of matrix M, ρ(M)=max{|λ||λ is the eigenvalue of M}, and satisfy ρ(M)<1; if conditions C1 and C2 can be satisfied, then where I represents the identity matrix of S×S; the vector That is, each element of ({q _s } _s∈Ω ) represents the minimum transmission power of each SUs, and each item {θ _s } _s∈Ω of SUs meets the requirements; further from {q _s } _s∈Ω Introduce the minimum transmit power of the PU Then get the sufficient condition (C3) that the problem (P1) is feasible:

(C3): and

(4) The vertical layering of problem P1, since there is always That is to say, the PU does not need to consume more transmission power while meeting the throughput requirements. The problem (P1) is vertically decomposed into two layers, namely the problem (P1-bottom layer) and the problem (P1-top layer). In the underlying problem, the transmission power p ₀ of the PU is first fixed, and accordingly, the underlying problem becomes to optimize the transmission power q _s of the SUs given the transmission power p ₀ of the PU;

(P1-bottom):

By calculating the value of F(p ₀ ) in the bottom layer and substituting the value of F(p ₀ ) into the top-level problem to optimize the transmission power of the PU;

(P1-top level):

in

(5) Judging the feasibility of the problem (P1-bottom layer)

When p ₀ is determined, in order to satisfy {θ _s } _s∈Ω , the power of SUs needs to be able to satisfy the formula Equivalent to solving the equation

Let N denote an S×S matrix, S denote the total number of SUs in Ω, and the entries in N are expressed as follows:

In addition, S×1 vector v and vector w are also defined, each of which is expressed as

Thus the transmit power of SUs to meet their respective throughput requirements {θ _s } _s∈Ω is expressed as

When p ₀ >0, The terms in are non-negative, and it can be seen that , the problem (P1-bottom) is feasible; denoting the sth item of the vector x by (x) _s , the (IN) ^-1 (v+wp ₀ ) into the inequality Then it can be further clarified that the problem (P1-bottom) satisfies the inequality at p ₀ The right side of the inequality represents the lower bound of p ₀ , denoted as P ; at the same time, by Solve for an upper bound on p0 compared to Qsmaxs ∈ Ω, where Q ^max represents a vector of S×1 expressed as Denote the upper bound of p ₀ as Deriving the problem (P1-bottom) is feasible enough

(6) To solve the problem (P1-bottom layer), a power control algorithm based on monotonic optimization is adopted for the bottom layer problem. The process is as follows:

Step 6.1: Introduce the auxiliary variable SNR of unauthorized users

Transform the underlying problem into a monotonic optimization problem about the signal-to-noise ratio y _s of unauthorized users;

in

Step 6.2: Set the initial optimal unlicensed user SNR set in

Set the current number of iterations k=1;

Step 6.3: For the current optimal unlicensed user SNR set Computes the value of the objective function for all elements in the set Record the point corresponding to the maximum objective function value as z ^k ;

Step 6.4: According to the bisection method, calculate the connection line between the origin and z ^k and intersection of

Step 6.5: If Then the algorithm is terminated, go to step 6.9; otherwise go to step 6.6;

Step 6.6: According to the formula Calculate the optional optimal solution of S new SNRs of unauthorized users, where e _i are S mutually orthogonal unit vectors;

Step 6.7: Use the S optional optimal solutions calculated in step 6.6 to replace z ^k to update the current optimal unlicensed user signal-to-noise ratio set, which is recorded as

Step 6.8: Set the number of iterations k=k+1, enter the next cycle, and return to step 6.2;

Step 6.9: The algorithm terminates, exits the algorithm loop, and outputs the optimal solution for the signal-to-noise ratio of unauthorized users is the signal-to-noise ratio with the largest objective function value in the current set;

Step 6.10: According to the formula Set the S-dimensional vector r, and calculate the optimal unlicensed user transmit power according to the formula q ^* = (IN) ^-1 r, where the matrix

Step 6.11: According to the formula Calculate the optimal objective function value of the bottom layer under the condition of fixing p ₀ for use by the top layer;

(7) Solution of threshold P _th

Depending on the nature of the problem (P1-bottom), it is found that There is a special threshold P _th on , when P ≤ p ₀ ≤ P _th , the inequality Only then can it be established, so solving the threshold P _th can narrow the search domain of the optimal solution, the solution process is as follows:

Step 7.1: Initialize the settings, set two small positive numbers close to 0 as the allowable calculation error, denoted as η and ε respectively, let

Step 7.2: Calculate |p _lower -p _upper |, if the difference is smaller than the allowable calculation error ε, it means that the obtained value is within the allowable range of error, then the algorithm terminates, and jumps to step 7.6, otherwise, continue Go to step 7.3;

Step 7.3: Set the transmit power p ₀ of the PU to the median value of p _lower and p _upper , namely

Step 7.4: Since p ₀ is given in Step 7.3, solve the problem (P1-bottom) by Step 6 and get the corresponding optimal solution

Step 7.5: Calculation It is used to judge whether the current p ₀ can satisfy the constraints of the problem (P1-bottom layer) Therefore, if |J(p ₀ )|<η, update the _upper limit p ₀ of p 0 to the current p ₀ , otherwise update p _lower to the current p ₀ , and return to step 7.2;

Step 7.6: use the obtained p ₀ as the special threshold P _th ;

(8) The solution of the problem (P1-top layer), the optimal solution obtained according to the problem (P1-bottom layer) And the optimal objective function value F(p ₀ ), the upper layer problem is transformed into a one-dimensional optimization problem about the authorized user’s transmit power p ₀ , and the linear search algorithm is used to solve the problem (P1-top layer), the process is as follows:

Step 8.1: Perform initialization settings: initialize the transmit power p ₀ of the PU to Where P is the lower bound of the transmit power p ₀ of the PU;

Step 8.2: If the transmit power of the PU Since there is no better solution in this area, the algorithm terminates and jumps to step 8.6, otherwise, proceeds to step 8.3, where For the upper bound of the transmit power _p0 of the PU, refer to step 5 for details, and _Pth is the special threshold obtained in step 7;

Step 8.3: Since the transmit power p ₀ of the PU has been given, calculate F(p ₀ ) according to the underlying algorithm given in step 6 and calculate the corresponding

Step 8.4: Judgment Is it greater than the currently searched

The maximum income F ^* of the PU, if established, set the optimal transmission power of the PU That is to say, it will be used as the current optimal transmission power of PU, otherwise, the optimal transmission power of SUs will be denoted as

Step 8.5: Update the maximum benefit F ^* of the PU as Update p ₀ =p ₀ +λ, return to step 8.2;

Step 8.6: Output the transmit power of the PU when the optimal configuration is achieved Transmit power of SUs and the maximum net benefit F ^* that PU obtains by serving SUs.

The technical idea of the present invention is as follows: firstly, in a cognitive radio network, licensed users (PUs) share their own spectrum with unlicensed users (SUs) to obtain additional benefits. Here, it is considered that authorized users (PUs) need to serve unlicensed users (SUs) while ensuring their own throughput and meet their respective throughput requirements, and it is also considered that authorized users (PUs) need to overcome the corresponding interference and thus Additional transmit power expense while maximizing the net benefit to authorized users (PUs). Then, by analyzing the characteristics of the problem, the problem is transformed into a two-layer problem for solution. Then, according to the characteristics of the two-layer problem, a power control method based on monotonic optimization and linear search is proposed, so as to maximize the net benefit of PU while ensuring the throughput of PU and SUs.

The beneficial effects of the present invention are mainly manifested in: 1. For the overall system, the implementation of spectrum sharing can improve the spectrum utilization rate through the secondary utilization of authorized spectrum; At the same time of QoS, additional economic benefits can be obtained; 3. For unlicensed users (SUs), they can realize their own QoS requirements and obtain satisfactory services through transactions in the spectrum secondary market.

Description of drawings

Fig. 1 is a schematic diagram of a cognitive radio network including a licensed user (PU) and several unlicensed users (SUs).

detailed description

The present invention will be described in further detail below in conjunction with the accompanying drawings.

Referring to Figure 1, a power control method based on monotonicity optimization and linear search in a cognitive wireless network, the implementation of this method can maximize the net benefit of the PU and improve the spectrum of the entire system while satisfying the premise of both PUs and SUs resource utilization. The present invention is applied in a cognitive radio network (as shown in FIG. 1 ). A scenario in which licensed users PU share their own spectrum with unlicensed users SUs to obtain additional revenue. Here, the authorized user PU needs to serve the unlicensed users SUs and meet their respective throughput requirements while guaranteeing their own throughput, and also consider the additional transmission power consumption of the authorized user PU to overcome the corresponding interference. Aiming at this problem, the method of optimizing the system according to the control transmission power mainly includes the following steps:

(1) In a cognitive radio network, through the transmission power control of authorized users PU and unlicensed users SUs, the QoS of PU is guaranteed while considering two parts of interference including between PU and SUs and between different SUs The optimization problem of maximizing the net benefit of the PU is described as a non-convex optimization problem as shown below:

P1: maxΣ _s∈Ω α _s R _s -β(p ₀ -p ₀ ^min )

in represents the throughput of each SUs, Indicates the uplink throughput of the PU, Represents the collection of all unauthorized user SUs.

In problem P1, each parameter is defined as follows:

α _s : the marginal coefficient of charging for the unit throughput PU realized by each SUs;

β: marginal power consumption cost of PU (in $/Watt);

R _s : throughput of each SUs;

p ₀ : transmit power of PU;

p ₀ ^min : the minimum transmit power consumption of the PU;

n: background noise (assumed to be additive Gaussian white noise) power;

q _s : transmit power of SUs;

g _sB : channel power gain between SU-Tx and BS;

g _0B : channel power gain between PU-Tx and BS;

Throughput requirements for each SUs;

PU transmit power upper limit;

The upper limit of the maximum transmission power of the SU;

g _0s : channel power gain between PU-Tx and SU-Rxs;

g _ss : channel power gain between SU-Txs and SU-Rxs;

g _js : channel power gain between SU-Txj and SU-Rxs;

W: bandwidth of PU channel;

Note: The superscript "*" appearing in the parameter symbol indicates the optimal value of the parameter in the optimization problem.

(2) use the formula Expand the R _s in the constraints, the second term of the constraints is equivalent to in The decision variables of the problem (P1) are transformed into p ₀ and {q _s } _s∈Ω , using and Respectively represent the optimal solution of problem (P1).

(3) Judge the feasibility of the problem (P1).

Let M represent an S×S matrix (S represents the total number of SUs in Ω), and the entries in M are expressed as follows:

In addition, a vector u of S×1 is defined, each item of which is expressed as

order vector Indicates that SUs can satisfy the linear constraints And s≠j set of transmission power, record condition C1: g _ss g _0B -θ _s θ ₀ g _sB g _0s >0, And condition C2: define the spectral radius of matrix M, ρ(M)=max{|λ||λ is the eigenvalue of M}, satisfy ρ(M)<1, if conditions C1 and C2 can be satisfied, then where I represents the identity matrix of S×S. vector That is, each element of ({q _s } _s∈Ω ) represents the minimum transmission power of each SUs, and each item {θ _s } _s∈Ω of SUs meets the requirements. Further, from {q _s } _s∈Ω (that is, the vector ) to introduce the minimum transmission power of the PU Then we can get the feasible sufficient condition of the problem (P1), that is, the condition (C3):

(C3): and

(4) Vertical layering of problem P1

Since when implementing the optimization of problem P1, there is always That is to say, the PU does not need to consume more transmission power while meeting the throughput requirements. The problem (P1) is vertically decomposed into two layers, namely the problem (P1-bottom layer) and the problem (P1-top layer). In the underlying problem, the transmission power p ₀ of the PU is first fixed, and accordingly, the underlying problem becomes to optimize the transmission power q _s of the SUs given the transmission power p ₀ of the PU.

(P1-bottom):

By calculating the value of F(p ₀ ) in the bottom layer and substituting the value of F(p ₀ ) into the top-level problem to optimize the transmission power of the PU.

(P1-top level):

in

Through the interactive iteration of the bottom problem and the top problem, the original problem (P1) is finally solved. It should be noted that before iterating through the two-layer algorithm, it is necessary to use step 3 to judge the feasibility of the problem (P1). If the condition is not satisfied, the entire algorithm is terminated and the two-layer algorithm is not called.

(5) Judging the feasibility of the problem (P1-bottom layer)

The sufficient condition for the problem (P1-bottom) to be feasible is

(6) To solve the problem (P1-bottom layer), a power control algorithm based on monotonic optimization is used, and the process is as follows:

Step 6.1: Introduce the auxiliary variable SNR of unauthorized users Transform the underlying problem into a monotonic optimization problem about the signal-to-noise ratio y _s of unauthorized users.

in

Step 6.2: Set the initial optimal unlicensed user SNR set in Set the current number of iterations k=1;

Step 6.3: For the current optimal unlicensed user SNR set Computes the value of the objective function for all elements in the set Record the point corresponding to the maximum objective function value as z ^k ;

Step 6.4: According to the bisection method, calculate the connection line between the origin and z ^k and intersection of

Step 6.5: If Then the algorithm is terminated, go to step 6.9; otherwise go to step 6.6;

Step 6.6: According to the formula An optional optimal solution for S new SNRs of unauthorized users is calculated. where e _i are S mutually orthogonal unit vectors;

Step 6.7: Use the S optional optimal solutions calculated in step 6.6 to replace z ^k to update the current optimal unlicensed user signal-to-noise ratio set, which is recorded as

Step 6.8: Set the number of iterations k=k+1, enter the next cycle, and return to step 6.2;

Step 6.9: The algorithm terminates, exits the algorithm loop, and outputs the optimal solution for the signal-to-noise ratio of unauthorized users is the signal-to-noise ratio with the largest objective function value in the current set;

Step 6.10: According to the formula Set the S-dimensional vector r. Calculate the optimal unlicensed user transmit power according to the formula q ^* = (IN) ^-1 r, where the matrix

Step 6.11: According to the formula Calculate the optimal objective function value of the bottom layer under the condition of fixing p ₀ for use by the top layer;

It should be noted that before adopting the power control algorithm based on monotonic optimization, it is necessary to judge the feasibility of the problem (P1-bottom layer) according to step 5. If the condition is not satisfied, the entire algorithm is terminated, and power control is not performed through this method .

(7) Solution of threshold P _th

Depending on the nature of the problem (P1-bottom), one can find There is a special threshold P _th on , when P ≤ p ₀ ≤ P _th , the inequality Therefore, solving the threshold P _th can greatly reduce the search domain of the optimal solution. The solution process is as follows:

Step 7.1: Initialize the settings, set two small positive numbers close to 0 as the allowable calculation error, which are respectively recorded as η and ε. make

Step 7.2: Calculate |p _lower -p _upper |, if the difference is smaller than the allowable calculation error ε, it means that the obtained value is within the allowable range of error, then the algorithm terminates, and jumps to step 7.6, otherwise, continue Go to step 7.3;

Step 7.3: Set the transmit power p ₀ of the PU to the median value of p _lower and p _upper , namely

Step 7.4: Since p ₀ is given in Step 7.3, solve the problem (P1-bottom) by Step 6 and get the corresponding optimal solution

Step 7.5: Calculation It is used to judge whether the current p ₀ can satisfy the constraints of the problem (P1-bottom layer) Therefore, if |J(p ₀ )|<η, update the _upper limit p ₀ of p 0 to the current p ₀ , otherwise update p _lower to the current p ₀ , and return to step 7.2;

Step 7.6: use the obtained p ₀ as the special threshold P _th ;

(8) Solution of problem (P1-top level)

The optimal solution obtained according to the problem (P1-bottom) And the optimal objective function value F(p ₀ ), the upper layer problem is transformed into a one-dimensional optimization problem about the authorized user’s transmit power p ₀ , and the linear search algorithm is used to solve the problem (P1-top layer), the process is as follows:

Step 8.1: Perform initialization settings: initialize the transmit power p ₀ of the PU to Where P is the lower bound of the transmit power p ₀ of the PU, for details, refer to step 5. For the PU, the benefit F ^* of spectrum sharing is initialized to 0, and the search step size λ is set to a very small value;

Step 8.2: If the transmit power of the PU Since there is no better solution in this area, the algorithm terminates and jumps to step 8.6, otherwise, proceeds to step 8.3, where For the upper bound of the transmit power _p0 of the PU, refer to step 5 for details, and _Pth is the special threshold obtained in step 7;

Step 8.3: Since the transmit power p ₀ of the PU has been given, calculate F(p ₀ ) according to the underlying algorithm given in step 6 and calculate the corresponding

Step 8.4: Judgment Is it greater than the currently searched

The maximum income F ^* of the PU, if established, set the optimal transmission power of the PU That is to say, it will be used as the current optimal transmission power of PU, otherwise, the optimal transmission power of SUs will be denoted as

Step 8.5: Update the maximum benefit F ^* of the PU as Update p ₀ =p ₀ +λ, return to step 8.2;

Step 8.6: Output the transmit power of the PU when the optimal configuration is achieved Transmit power of SUs and the maximum net benefit F ^* that PU obtains by serving SUs.

In this embodiment, Fig. 1 is a system including one authorized user (PU) and several unauthorized users (SUs) in the cognitive radio network considered in the present invention. In this system, the interference mainly considered includes two parts: 1) co-channel interference between PU and SUs 2) mutual interference between different SUs. In order to overcome the interference caused by accessing SUs and meet its own QoS requirements, PUs often need to increase their transmission power (compared with the case of not accessing any SUs), so more interference will be generated on the SUs side. Since the "anti-interference" SUs of the PU have to adjust their transmission power to meet their own QoS requirements, the interference to the PU increases. In order to better manage the positive feedback loop and achieve the benefit of spectrum sharing, the present invention is proposed to solve the problem.

This embodiment focuses on maximizing the net income of PUs and encouraging PUs to serve SUs under the premise of satisfying the service quality of authorized users PU and unlicensed users SUs at the same time, so as to improve the utilization rate of system spectrum. The work of can enable interference management to be implemented effectively and efficiently with low computational complexity. Therefore, it is possible to achieve more optimized allocation of spectrum resources in the entire system and a higher utilization rate.

Claims (1)

1. A power control method based on monotonicity optimization and linear search in a cognitive wireless network, characterized in that: the control method comprises the following steps: (1) In a cognitive radio network, through the transmission power control of authorized users PU and unlicensed users SUs, the QoS of PU is guaranteed while considering two parts of interference including between PU and SUs and between different SUs The optimization problem of maximizing the net benefit of the PU is described as a non-convex optimization problem as shown below: P1: maxΣ _s∈Ω α _s R _s -β(p ₀ -p ₀ ^min ) $\begin{array}{cc}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&Greater\; Equal;</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\frac{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; B</span>}}}{{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; B</span>}}}\end{array}$ ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; \&Greater\; Equal;</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&ForAll;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}$ $\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}}$ $\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&ForAll;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}$ in represents the throughput of each unauthenticated user SUs, Indicates the uplink throughput of the authorized user PU, Ω={1,2...S} represents the set of all unauthorized users SUs; In problem P1, each parameter is defined as follows: α _s : the marginal coefficient of charging for the unit throughput PU realized by each SUs; β: The marginal power consumption cost of the PU, in $/Watt; R _s : throughput of each SUs; p ₀ : transmit power of PU; p ₀ ^min : the minimum transmit power consumption of the PU; n: background noise power; q _s : transmit power of SUs; g _sB : channel power gain between SU-Tx and BS; Channel power gain between PU-Tx and BS; Throughput requirements for each SUs; PU transmit power upper limit; The upper limit of the maximum transmission power of the SU; g _0s : channel power gain between PU-Tx and SU-Rxs; g _ss : channel power gain between SU-Txs and SU-Rxs; g _js : channel power gain between SU-Txj and SU-Rxs; W: bandwidth of PU channel; The superscript "*" in the parameter symbol indicates the optimal value of the parameter in the optimization problem; (2) use the formula $R_{\mathrm{the\; s}}={\mathrm{wlog}}_2\left(1+\frac{q_{\mathrm{the\; s}}{g}_{\mathrm{the\; s}\mathrm{the\; s}}}{\mathrm{no}+p_0{g}_{0\mathrm{the\; s}}+{\&Sigma;}_{j\&NotEqual;\mathrm{the\; s},j\&Element;\mathrm{\&Omega;}}{q}_j{g}_{j\mathrm{the\; s}}}\right),\&ForAll;\mathrm{the\; s}\&Element;\mathrm{\&Omega;}$ Expand the R _s in the constraints, the second term of the constraints is equivalent to in The decision variable of problem P1 is transformed into p ₀ and {q _s } _s∈Ω , we use and respectively represent the optimal solution of problem P1; (3) Judging the feasibility of problem P1 the formula p ₀ in is replaced by {q _s } _s∈Ω , so that the inequality can be re-expressed as a set of linear constraints as follows: And s≠j M represents an S×S matrix, S represents the total number of SUs in Ω, and the entries in M are expressed as follows: In addition, a vector u of S×1 is defined, each item of which is expressed as order vector Indicates the set of transmission power of SUs that can satisfy the above linear constraints, record condition C1: And condition C2: We define the spectral radius of matrix M, ρ(M)=max{|λ||λ is the eigenvalue of M}, satisfying ρ(M)<1; if conditions C1 and C2 can be satisfied, then where I represents the identity matrix of S×S; the vector That is, each element of ({q _s } _s∈Ω ) represents the minimum transmission power of each SUs, and each item {θ _s } _s∈Ω of SUs meets the requirements; further from {q _s } _s∈Ω Introduce the minimum transmit power of the PU Then get the sufficient condition (C3) that the problem (P1) is feasible: $\left(C3\right):{\hat{q}}_{\mathrm{the\; s}}\&le;Q_{\mathrm{the\; s}}^{\max },\&ForAll;\mathrm{the\; s}\&Element;\mathrm{\&Omega;},$ and ${\hat{p}}_0\&le;P_0^{\max }$ (4) The vertical layering of problem P1, since there is always That is to say, the PU does not need to consume more transmission power while meeting the throughput requirements. The problem (P1) is vertically decomposed into two layers, namely the problem (P1-bottom layer) and the problem (P1-top layer). In the underlying problem, the transmission power p ₀ of the PU is first fixed, and accordingly, the underlying problem becomes to optimize the transmission power q _s of the SUs given the transmission power p ₀ of the PU; (P1-bottom): $f\left(p_0\right)=\underset{{\left\{q_{\mathrm{the\; s}}\right\}}_{\mathrm{the\; s}\&Element;\mathrm{\&Omega;}}}{\max }\underset{\mathrm{the\; s}\&Element;\mathrm{\&Omega;}}{\&Sigma;}{\mathrm{\&alpha;}}_{\mathrm{the\; s}}{R}_{\mathrm{the\; s}}$ $\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&ForAll;</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}$ By calculating the value of F(p ₀ ) in the bottom layer and substituting the value of F(p ₀ ) into the top-level problem to optimize the transmission power of the PU; (P1-top level): $\underset{p_0^{\min }\&le;p_0\&le;p_0^{\max }}{\max }f\left(p_0\right)-\mathrm{\&beta;}\left(p_0-p_0^{\min }\right)$ in (5) Judging the feasibility of the problem (P1-bottom layer) When p ₀ is determined, in order to satisfy {θ _s } _s∈Ω , the power of SUs needs to be able to satisfy the formula Equivalent to solving the equation Let N denote an S×S matrix, S denote the total number of SUs in Ω, and the entries in N are expressed as follows: In addition, S×1 vector v and vector w are also defined, each of which is expressed as Thus the transmit power of SUs to meet their respective throughput requirements {θ _s } _s∈Ω is expressed as $\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; wp</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>$ When p ₀ >0, The terms in are non-negative, and it can be seen that , the problem (P1-bottom) is feasible; denoting the sth item of the vector x by (x) _s , the $\hat{q}\left(p_0\right)={(I-N)}^{-1}(v+{\mathrm{wp}}_0)$ Substitute into the inequality Then it can be further clarified that the problem (P1-bottom) satisfies the inequality at p ₀ The right side of the inequality represents the lower bound of p ₀ , denoted as P ; at the same time, by and In contrast, solving for an upper bound on p ₀ , where Q ^max represents a vector of S×1 expressed as Denote the upper bound of p ₀ as The sufficient condition for the problem (P1-bottom) to be feasible is $\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; ,</span>\underset{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; \{</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; \}</span>$ (6) To solve the problem (P1-bottom layer), a power control algorithm based on monotonic optimization is adopted for the bottom layer problem. The process is as follows: Step 6.1: Introduce the auxiliary variable SNR of unauthorized users Transform the underlying problem into a monotonic optimization problem about the signal-to-noise ratio y _s of unauthorized users; in Step 6.2: Set the initial optimal unlicensed user SNR set in s=1,2,3,...S, set the current number of iterations k=1; Step 6.3: For the current optimal unlicensed user SNR set Computes the value of the objective function for all elements in the set Record the point corresponding to the maximum objective function value as z ^k ; Step 6.4: According to the bisection method, calculate the connection line between the origin and z ^k and intersection of Step 6.5: If Then the algorithm is terminated, go to step 6.9; otherwise go to step 6.6; Step 6.6: According to the formula i=1, 2, 3, ... S calculates the optional optimal solution of S new unlicensed user signal-to-noise ratios, where e _i are S mutually orthogonal unit vectors; Step 6.7: Use the S optional optimal solutions calculated in step 6.6 to replace z ^k to update the current optimal unlicensed user signal-to-noise ratio set, which is recorded as Step 6.8: Set the number of iterations k=k+1, enter the next cycle, and return to step 6.2; Step 6.9: The algorithm terminates, exits the algorithm loop, and outputs the optimal solution for the signal-to-noise ratio of unauthorized users is the signal-to-noise ratio with the largest objective function value in the current set; Step 6.10: According to the formula Set the S-dimensional vector r, and calculate the optimal unlicensed user transmit power according to the formula q ^* = (IN) ^-1 r, where the matrix $N_{\mathrm{the\; s}j}=\left\{\begin{array}{c}0,\mathrm{the\; s}=j;\\ \frac{g_{j\mathrm{the\; s}}{\mathrm{\&theta;}}_{\mathrm{the\; s}}}{g_{\mathrm{the\; s}\mathrm{the\; s}}},\mathrm{the\; s}\&NotEqual;j.\end{array}\right.;$ Step 6.11: According to the formula Calculate the optimal objective function value of the bottom layer under the condition of fixing p ₀ for use by the top layer; (7) Solution of threshold P _th Depending on the nature of the problem (P1-bottom), it is found that There is a special threshold P _th on , when P ≤ p ₀ ≤ P _th , the inequality Only then can it be established, so solving the threshold P _th can narrow the search domain of the optimal solution, the solution process is as follows: Step 7.1: Initialize the settings, set two very small positive numbers close to 0 as the allowable calculation error, denoted as η and ε respectively, let p _lower = P , Step 7.2: Calculate |p _lower -p _upper |, if the difference is smaller than the allowable calculation error ε, it means that the obtained value is within the allowable range of error, then the algorithm terminates, and jumps to step 7.6, otherwise, continue Go to step 7.3; Step 7.3: Set the transmit power p ₀ of the PU to the median value of p _lower and p _upper , namely Step 7.4: Since p ₀ is given in Step 7.3, solve the problem (P1-bottom) by Step 6 and get the corresponding optimal solution Step 7.5: Calculation It is used to judge whether the current p ₀ can satisfy the constraints of the problem (P1-bottom layer) Therefore, if |J(p ₀ )|<η, update the _upper limit p ₀ of p 0 to the current p ₀ , otherwise update p _lower to the current p ₀ , and return to step 7.2; Step 7.6: use the obtained p ₀ as the special threshold P _th ; (8) The solution of the problem (P1-top layer), the optimal solution obtained according to the problem (P1-bottom layer) And the optimal objective function value F(p ₀ ), the upper layer problem is transformed into a one-dimensional optimization problem about the authorized user’s transmit power p ₀ , and the linear search algorithm is used to solve the problem (P1-top layer), the process is as follows: Step 8.1: Perform initialization settings: initialize the transmit power p ₀ of the PU to Where P is the lower bound of the transmit power p ₀ of the PU; Step 8.2: If the transmit power of the PU Since there is no better solution in this area, the algorithm terminates and jumps to step 8.6, otherwise, proceeds to step 8.3, where For the upper bound of the transmit power _p0 of the PU, refer to step 5 for details, and _Pth is the special threshold obtained in step 7; Step 8.3: Since the transmit power p ₀ of the PU has been given, calculate F(p ₀ ) according to the underlying algorithm given in step 6 and calculate the corresponding Step 8.4: Judgment Is it greater than the currently searched The maximum income F ^* of the PU, if established, set the optimal transmission power of the PU That is to say, it will be used as the current optimal transmission power of PU, otherwise, the optimal transmission power of SUs will be denoted as Step 8.5: Update the maximum benefit F ^* of the PU as Update p ₀ =p ₀ +λ, return to step 8.2; Step 8.6: Output the transmit power of the PU when the optimal configuration is achieved Transmit power of SUs and the maximum net benefit F ^* that PU obtains by serving SUs.