CN105228233B - A kind of Poewr control method based on monotonicity optimization and linear search in cognition wireless network - Google Patents
A kind of Poewr control method based on monotonicity optimization and linear search in cognition wireless network Download PDFInfo
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Abstract
A kind of Poewr control method based on monotonicity optimization and linear search in cognition wireless network, includes the following steps：(1) consider to include two parts interference between PU and SUs and between different SUs, optimization problem is described as a multivariable nonconvex property optimization problem；(2) it is bilevel optimization problem by problem (P1) orthogonal decomposition；(3) according to bottom problem, it is proposed that the Poewr control method of monotonicity optimization optimizes the transimission power of SUs in the case where the transimission power of PU is given；(4) it is based on bottom problem, the method for proposing linear search advanced optimizes the transimission power of PU；(5) it by the interactive iteration of bottom problem and top layer problem, finally solves the problems, such as (P1).The present invention provides a kind of effective and efficient optimization method in the net profit for ensureing the QoS of PU while maximizing PU, to improve system spectrum utilization rate, the configuration of optimization system resource.
Description
Technical field
The present invention relates in cognitive radio networks, it is a kind of optimized based on monotonicity carried out with linear search algorithm it is optimal
Poewr control method.
Background technology
With the rapid growth of mobile data service, the problem of finiteness of usable spectrum resource makes frequency spectrum congestion, is increasingly
It is prominent.Dynamic spectrum access (DSA) passes through intelligence as effective supplement of fixed frequency spectrum distribution method traditional in mobile network
Energyization ground recycling unauthorized system (Primary System-PS) or authorized user (Primary User-PU) are abundant
The mandate frequency spectrum resource utilized so that unauthorized user (Secondary User-SU) can in time access the mandate frequency of PU
Spectrum carries out data transmission, so that the availability of frequency spectrum is effectively promoted.DSA is with its superiority, it is considered to be Yi Zhongneng
It enough realizes flexible, and the model of the frequency spectrum supply mode of demand at present can be responded, have a extensive future.However, DSA nets
When carrying out frequency spectrum share in network, interference can be inevitably generated while PU serves SUs, including：1) PU and SUs
Between cochannel interference 2) interfering with each other between different SUs.In order to ensure PU QoS under the premise of serve SUs with
Additional income is obtained, during designing frequency spectrum share scheme, it is very reasonably to carry out resource allocation with interference management
It is necessary to.However above-described interference often makes problem have the problem of non-convex optimization and be difficult to solve, because
And it proposes a kind of meaningful in the effective and efficient optimization method for the net profit for ensureing the QoS of PU while maximizing PU.
Invention content
In order to ensure that frequency spectrum share can optimize the frequency spectrum resource configuration in DSA networks, present invention consideration includes PU and SUs
Between and difference SUs between two parts interference, it is proposed that it is a kind of while the QoS of PU is protected, pass through maximization
The net profit of PU is to realize the Poewr control method of optimization.The power control algorithm proposed has double-layer structure, is reducing
The validity and high efficiency of this method are improved while computation complexity.
The technical solution adopted by the present invention to solve the technical problems is：
A kind of Poewr control method based on monotonicity optimization and linear search in cognition wireless network, the control method
Include the following steps：
(1) it in cognitive radio networks, is controlled by the transmission power of authorized user PU and unauthorized user SUs,
While consideration includes two parts interference between PU and SUs and between different SUs, ensure maximum in the case of the QoS of PU
The optimization problem for changing the net profit of PU describes the nonconvex property optimization problem being as follows：
P1：max∑s∈ΩαsRs-β(p0-p0 min)
WhereinIndicate each unauthorized user
The handling capacity of SUs,Indicate the uplink throughput of authorized user PU,What Ω={ 1,2...S } indicated is the set of all unauthorized user SUs；
In problem P1, parameters are defined as follows：
αs:The marginal coefficient charged for each SU s unit handling capacity PU realized；
β：The marginal power consumption cost of PU, unit are $/Watt；
Rs:The handling capacity of each SU s；
p0:The transmission power of PU；
p0 min：The minimum transmission power of PU consumes；
n:Background Noise Power；
qs:The transmission power of SU s；
gsB:Channel power gain between SU-Tx and BS；
g0B:Channel power gain between PU-Tx and BS；
The throughput demands of each SU s；
The transmission power upper limit of PU；
The maximum transmission power upper limit of SU；
g0s:Channel power gain between PU-Tx and SU-Rx s；
gss:Channel power gain between SU-Tx s and SU-Rx s；
gjs:Channel power gain between SU-Tx j and SU-Rx s；
W:The bandwidth of PU channels；
Optimal value of subscript " * " expression parameter in optimization problem in pa-rameter symbols；
(2) formula is usedBy the R in constraintss
Expansion, the Section 2 of constraints are equivalent to Wherein
The decision variable of problem P1 translates into p0And { qs}s∈Ω, useWithThe optimal solution of problem of representation P1 respectively；
(3) feasibility of decision problem P1
By formulaIn p0With { qs}s∈ΩReplacement, to differ this
One group of linear restriction that formula is expressed as again：
And s ≠ j
M indicates that a S × s-matrix, S indicate the sum of SUs in Ω, and the item in M indicates as follows：
In addition, also defining the vectorial u of S × 1, each of which item is expressed as
Enable vectorIt indicates that SUs disclosure satisfy that the set of the transimission power of above-mentioned Linear Constraints, remembers condition C 1：
gssg0B-θsθ0gsBg0s> 0And condition C 2：Define the frequency spectrum radius of matrix M, ρ (M)=max {s |λ||λ is M
Characteristic value }, meet ρ (M) < 1；If condition C 1 and C2 disclosure satisfy that,Wherein I indicates S × S
Unit matrix；VectorThat is ((qs}s∈Ω) each element representation the minimum transmission power of each SU s, SUs's is each
Item (θs}s∈ΩIt is satisfied by requirement；Further from { qs}s∈ΩThe middle minimum transmission power for releasing PU
Then problem (P1) feasible adequate condition i.e. condition (C3) is obtained：
(C3)：And
(4) vertical demixing of problem P1, due to always having in the optimization of problem of implementation P1
That is, PU no longer needs to consumption more transimission power, problem (P1) orthogonal decomposition while meeting throughput demands
For double-layer structure, respectively problem (P1- bottoms) and problem (P1- top layers), the transmission work(of PU fixed first in bottom problem
Rate p0, correspondingly, bottom problem becomes the transmission power p in given PU0In the case of optimize SUs transmission power qs；
(P1- bottoms)：
By calculating F (p in bottom0) value, by F (p0) value be updated to top layer problem to optimize the transmission work(of PU
Rate；
(P1- top layers)：Wherein
(5) feasibility of decision problem (P1- bottoms)
Work as p0When determining, in order to meet { θs}s∈Ω, the power of SUs is required to meet formulaIt is equivalent to solution equationIndicate that a S × s-matrix, S indicate SUs in Ω with N
Sum, the item in N indicate as follows：
In addition, also defining the vector v and vector w of S × 1, each of which item is expressed as
Thus SUs meets its respective throughput demand { θs}s∈ΩTransmission power be expressed as
Work as p0When > 0,In item be non-negative, it will be seen thatWhen, problem (P1- bottoms)
It is feasible；With (x)sIndicate s of vector x, it will Substitute into inequalityThen can further clear problem (P1- bottoms) in p0Meet inequalityThe right of the inequality indicates p0Lower bound, be denoted asP；Meanwhile pass through by Compared with Qsmaxs ∈ Ω, the upper bound of p0 is solved,Wherein QmaxThe vector for indicating S × 1, is expressed as
By p0The upper bound be denoted as(P1- bottoms) the feasible adequate condition that must go wrong is
(6) solution of problem (P1- bottoms), for bottom problem, using the power control algorithm optimized based on monotonicity,
Process is as follows：
Step 6.1：Introduce the signal-to-noise ratio of auxiliary variable unauthorized user
Bottom problem is converted into one about unauthorized user signal-to-noise ratio ysMonotonicity optimization problem；
Wherein
Step 6.2：The initial optimal unauthorized user signal-to-noise ratio set of settingWhereinS=
1,2,3 ... current iterations k=1 is arranged in .S；
Step 6.3：For current optimal unauthorized user signal-to-noise ratio setThe target of all elements in set of computations
Functional valueIt is z to record the corresponding point of maximum target function valuek；
Step 6.4：According to bisection method datum point and zkLine withIntersection point
Step 6.5：IfThen algorithm terminates, and goes to step 6.9；Otherwise step 6.6 is gone to；
Step 6.6：According to formulaI=1,2,3....S, which calculate new non-of S, awards
Weigh the optional optimal solution of user's signal-to-noise ratio, wherein eiIt is S mutually orthogonal unit vectors；
Step 6.7：Z is replaced using calculated S in step 6.6 optional optimal solutionskTo update, currently optimal is non-to be awarded
User's signal-to-noise ratio set is weighed, remembers that the collection is combined into
Step 6.8：Iterations k=k+1, into recycling next time, return to step 6.2 are set；
Step 6.9：Algorithm terminates, and exits algorithm cycle, exports unauthorized user signal-to-noise ratio optimal solutionCurrently to collect
The maximum signal-to-noise ratio of target function value in conjunction；
Step 6.10：According to formulaS dimensional vector r are set, according to formula q*=(I-N)-1R is calculated
Best unauthorized user transmission power, wherein matrix
Step 6.11：According to formulaIt calculates in fixed p0In the case of
Bottom optimal objective function value for top layer use；
(7) threshold value PthSolution
According to the property of problem (P1- bottoms), findIt is upper that there are a special threshold value Pth, whenP≤p0≤Pth
When, inequalityIt is just set up, thus solves threshold value PthThe search of optimal solution can be reduced
Domain, solution procedure are as follows：
Step 7.1：Initialize installation, setting two close to 0 very little positive number as allow calculating error, distinguish
It is denoted as η and ε, enables plower=P,
Step 7.2：Calculate |plower-pupper|If the difference is smaller than permitted calculating error ε, indicate obtained
For value in the range of error allows, then algorithm termination, gos to step 7.6, otherwise, continues step 7.3；
Step 7.3：By the transmission power p of PU0It is set as plowerWith pupperIntermediate value, i.e.,
Step 7.4：Due to giving p in step 7.30, problem (P1- bottoms) is solved by step 6 and is obtained corresponding
Optimal solution
Step 7.5：It calculatesFor judging existing p0Problem can be met
The constraints of (P1- bottoms)Thus Ru Guo |J(p0)|< η, then by p0Upper limit pupperMore
It is newly existing p0, otherwise by plowerIt is updated to existing p0, return to step 7.2；
Step 7.6：By obtained p0As special threshold value Pth；
(8) solution of problem (P1- top layers), the optimal solution obtained according to problem (P1- bottoms)And it is optimal
Target function value F (p0), upper layer issue translates into one about authorized user's transmission power p0One Dimension Optimization Problems, use
Linear search algorithm solution problem (P1- top layers), process is as follows：
Step 8.1：Carry out Initialize installation：By the transmission power p of PU0It is initialized asWherein P is PU
Transmission power p0Lower bound；
Step 8.2：If the transmission power of PUIt is more preferably solved due to being not present in the region,
Then algorithm terminates, and gos to step 8.6, otherwise, continues step 8.3, whereinFor the transmission power p of PU0The upper bound, obtain
For details, reference can be made to step 5, PthFor by special threshold value calculated in step 7；
Step 8.3：Due to the transmission power p of PU0It has been provided that, F (p are calculated according to the underlying algorithm of step 6 provided0)
And it is calculated corresponding
Step 8.4：JudgeWhether it is more than what current search arrived
The maximum return F of PU*, if so, PU optimal transmission power is then setI.e. as existing optimal
PU transmission power, otherwise the optimization transimission power of SUs is denoted as
Step 8.5：Update the maximum return F of PU*ForUpdate p0=p0+ λ returns to step
Rapid 8.2；
Step 8.6：The transmission power of PU when optimization collocation is realized in outputThe transmission power of SUsAnd
The maximum net profit F that PU is obtained by servicing SUs*。
The present invention technical concept be：First, in considering cognitive radio networks, authorized user (PU) is by the frequency of oneself
Spectrum shares to unauthorized user (SUs) to obtain the scene of extra returns.Here, it is believed that authorized user (PU) needs protecting
Unauthorized user (SUs) is served while demonstrate,proving oneself handling capacity and meets its respective throughput demand respectively, it is also contemplated that
To authorized user (PU) in order to overcome corresponding interference to additional transmission power cost, while maximizing authorized user (PU)
Net profit.Then, it is analyzed by the characteristic to the problem, converting the problem to two layers of problem solves.Then,
According to the characteristic of two layers of problem, the Poewr control method based on monotonicity optimization and linear search is proposed, ensureing to realize
PU is maximized with PU net profit when SUs handling capacities.
Beneficial effects of the present invention are mainly manifested in:1, for total system, the implementation of frequency spectrum share can pass through
The secondary use for authorizing frequency spectrum, to improve the availability of frequency spectrum；2, for authorized user (PU), ensureing itself QoS's
Meanwhile, it is capable to obtain additional economic well-being of workers and staff；3, for unauthorized user (SUs), by the transaction of frequency spectrum secondary market,
The QoS demand that can realize itself obtains satisfied service.
Description of the drawings
Fig. 1 is showing comprising an authorized user (PU) and several unauthorized users (SUs) in cognitive radio networks
It is intended to.
Specific implementation mode
Present invention is further described in detail below in conjunction with the accompanying drawings.
Referring to Fig.1, the Poewr control method based on monotonicity optimization and linear search in a kind of cognition wireless network, is carried out
Under the premise of this method can meet PU and SUs at the same time so that PU net profit maximizes, while improving the frequency spectrum of whole system
Resource utilization.The present invention is applied in cognitive radio networks (as shown in Figure 1).Authorized user PU is by the frequency spectrum share of oneself
To unauthorized user SUs to obtain the scene of extra returns.Here, authorized user PU needs ensureing oneself handling capacity
Simultaneously serve unauthorized user SUs and meet its respective throughput demand respectively, it is also contemplated that authorized user PU in order to
Overcome corresponding interference to additional transmission power cost.For problem proposition according to control transmission power optimization system
Method mainly include steps are as follows：
(1) it in cognitive radio networks, is controlled by the transmission power of authorized user PU and unauthorized user SUs,
While consideration includes two parts interference between PU and SUs and between different SUs, ensure maximum in the case of the QoS of PU
The optimization problem for changing the net profit of PU describes the nonconvex property optimization problem being as follows：
P1：max∑s∈ΩαsRs-β(p0-p0 min)
WhereinIndicate handling up for each SU s
Amount,Indicate the uplink throughput of PU,Ω=1,
2...S what is } indicated is the set of all unauthorized user SUs.
In problem P1, parameters are defined as follows：
αs:The marginal coefficient charged for each SU s unit handling capacity PU realized；
β：The marginal power consumption cost of PU (unit is $/Watt)；
Rs:The handling capacity of each SU s；
p0:The transmission power of PU；
p0 min：The minimum transmission power of PU consumes；
n:Ambient noise (being assumed to be additive white Gaussian noise) power；
qs:The transmission power of SU s；
gsB:Channel power gain between SU-Tx and BS；
g0B:Channel power gain between PU-Tx and BS；
The throughput demands of each SU s；
The transmission power upper limit of PU；
The maximum transmission power upper limit of SU；
g0s:Channel power gain between PU-Tx and SU-Rx s；
gss:Channel power gain between SU-Tx s and SU-Rx s；
gjs:Channel power gain between SU-Tx j and SU-Rx s；
W:The bandwidth of PU channels；
Note：Optimal value of subscript " * " expression parameter occurred in pa-rameter symbols in optimization problem.
(2) formula is usedBy the R in constraintss
Expansion, the Section 2 of constraints are equivalent to Wherein
The decision variable of problem (P1) translates into p0And { qs}s∈Ω, useWithProblem of representation (P1) is optimal respectively
Solution.
(3) feasibility of decision problem (P1).
Indicate that a S × s-matrix (S indicates the sum of SUs in Ω), the item in M indicate as follows with M：
In addition, also defining the vectorial u of S × 1, each of which item is expressed as
Enable vectorIndicate that SUs disclosure satisfy that Linear Constraints And the set of the transimission power of s ≠ j, note
Condition C 1：gssg0B-θsθ0gsBg0s> 0,And condition C 2：Define the frequency spectrum radius of matrix M, ρ (M)=max {s |λ||
λ is the characteristic value of M }, meet ρ (M) < 1, if condition C 1 and C2 disclosure satisfy that,Wherein I tables
Show the unit matrix of S × S.VectorThat is ({ qs}s∈Ω) each element representation the minimum transmission power of each SU s,
Each single item { the θ of SUss}s∈ΩIt is satisfied by requirement.It may further be from { qs}s∈ΩIt is (i.e. vectorial) in release PU minimum transfer work(
RateThen it can be obtained problem (P1) feasible adequate condition i.e. condition (C3)：
(C3)：And
(4) vertical demixing of problem P1
Due to always having in the optimization of problem of implementation P1That is, PU is being met
Consumption more transimission power is no longer needed to while throughput demands, problem (P1) orthogonal decomposition is double-layer structure, is respectively asked
Inscribe (P1- bottoms) and problem (P1- top layers).The transmission power p of PU fixed first in bottom problem0, correspondingly, bottom problem
Become the transmission power p in given PU0In the case of optimize SUs transmission power qs。
(P1- bottoms)：
By calculating F (p in bottom0) value, by F (p0) value be updated to top layer problem to optimize the transmission work(of PU
Rate.
(P1- top layers)：
Wherein
It is final to solve the problems, such as former (P1) by the interactive iteration of bottom problem and top layer problem.It should be noted that logical
It crosses before two layers of algorithm iteration, needs to carry out feasibility judgement to problem (P1) using step 3, if condition is unsatisfactory for, entirely
Algorithm terminates, and is called without two layers of algorithm.
(5) feasibility of decision problem (P1- bottoms)
Problem (P1- bottoms) feasible adequate condition is
(6) solution of problem (P1- bottoms), using the power control algorithm optimized based on monotonicity, process is as follows：
Step 6.1：Introduce the signal-to-noise ratio of auxiliary variable unauthorized user
Bottom problem is converted into one about unauthorized user signal-to-noise ratio ysMonotonicity optimization problem.
Wherein
Step 6.2：The initial optimal unauthorized user signal-to-noise ratio set of settingWhereinS=
1,2,3 ... S.Current iterations k=1 is set；
Step 6.3：For current optimal unauthorized user signal-to-noise ratio setThe target of all elements in set of computations
Functional valueIt is z to record the corresponding point of maximum target function valuek；
Step 6.4：According to bisection method datum point and zkLine withIntersection point
Step 6.5：IfThen algorithm terminates, and goes to step 6.9；Otherwise step 6.6 is gone to；
Step 6.6：According to formulaI=1,2,3 ... S calculates new non-of S and awards
Weigh the optional optimal solution of user's signal-to-noise ratio.Wherein eiIt is S mutually orthogonal unit vectors；
Step 6.7：Z is replaced using calculated S in step 6.6 optional optimal solutionskTo update, currently optimal is non-to be awarded
User's signal-to-noise ratio set is weighed, remembers that the collection is combined into
Step 6.8：Iterations k=k+1, into recycling next time, return to step 6.2 are set；
Step 6.9：Algorithm terminates, and exits algorithm cycle, exports unauthorized user signal-to-noise ratio optimal solutionCurrently to collect
The maximum signal-to-noise ratio of target function value in conjunction；
Step 6.10：According to formulaS dimensional vectors r is set.According to formula q*=(I-N)-1R is calculated
Best unauthorized user transmission power, wherein matrix
Step 6.11：According to formulaIt calculates in fixed p0In the case of
Bottom optimal objective function value is used for top layer；
It should be noted that before using the power control algorithm based on monotonicity optimization, need to be carried out according to step 5
The feasibility of problem (P1- bottoms) judges, if condition is unsatisfactory for, entire algorithm terminates, and does not carry out power control by the method
System.
(7) threshold value PthSolution
According to the property of problem (P1- bottoms), it can be found thatIt is upper that there are a special threshold value Pth, whenP≤p0≤
PthWhen, inequalityIt is just set up, thus solves threshold value PthIt can largely reduce
The region of search of optimal solution, solution procedure are as follows：
Step 7.1：Initialize installation, setting two close to 0 very little positive number as allow calculating error, distinguish
It is denoted as η and ε.Enable plower=P,
Step 7.2：Calculate |plower-pupper|If the difference is smaller than permitted calculating error, indicate obtained
For value in the range of error allows, then algorithm termination, gos to step 7.6, otherwise, continues step 7.3；
Step 7.3：By the transmission power P of PU0It is set as plowerWith pupperIntermediate value, i.e.,
Step 7.4：Due to giving p in step 7.30, problem (P1- bottoms) is solved by step 6 and is obtained corresponding
Optimal solution
Step 7.5：It calculatesFor judging existing p0Problem can be met
The constraints of (P1- bottoms)Thus Ru Guo |J(p0)|< η, then by p0Upper limit pupperMore
It is newly existing p0, otherwise by plowerIt is updated to existing p0, return to step 7.2；
Step 7.6：By obtained p0As special threshold value Pth；
(8) solution of problem (P1- top layers)
The optimal solution obtained according to problem (P1- bottoms)And optimal target function value F (p0), upper layer is asked
Topic translates into one about authorized user's transmission power p0One Dimension Optimization Problems, using linear search algorithm solution problem (P1-
Top layer), process is as follows：
Step 8.1：Carry out Initialize installation：By the transmission power p of PU0It is initialized asWhereinPFor PU
Transmission power p0Lower bound, obtain that for details, reference can be made to steps 5.The income F of frequency spectrum share is carried out for PU*0 is initialized as,
And set step-size in search to a very small value；
Step 8.2：If the transmission power of PUIt is more preferably solved due to being not present in the region,
Then algorithm terminates, and gos to step 8.6, otherwise, continues step 8.3, whereinFor the transmission power p of PU0The upper bound,
For details, reference can be made to step 5, PthFor by special threshold value calculated in step 7；
Step 8.3：Due to the transmission power p of PU0It has been provided that, F (p are calculated according to the underlying algorithm of step 6 provided0)
And it is calculated corresponding
Step 8.4：JudgeWhether it is more than what current search arrived
The maximum return F of PU*, if so, PU optimal transmission power is then setI.e. as existing optimal
PU transmission power, otherwise the optimization transimission power of SUs is denoted as
Step 8.5：Update the maximum return F of PU*ForUpdate p0=p0+ λ returns to step
Rapid 8.2；
Step 8.6：The transmission power of PU when optimization collocation is realized in outputThe transmission power of SUsAnd
The maximum net profit F that PU is obtained by servicing SUs*。
In the present embodiment, Fig. 1 be in the cognitive radio networks that consider of the present invention comprising an authorized user (PU) and
The system of several unauthorized users (SUs).Within the system, the interference mainly considered includes two parts：1) between PU and SUs
Cochannel interference 2) interfering with each other between different SUs.In order to overcome due to the access SUs interference generated and meet itself
QoS demand, PU generally requires to promote the transimission power (compared with the case where not accessing any SU) of oneself, thus the ends SUs
It will produce the interference of bigger.Since " counter-measure " SUs of PU has to adjust the transimission power of oneself to meet the QoS of oneself
Demand, to increase the interference of PU.In order to preferably manage the positive feedback loop, reach the income of frequency spectrum share, it is proposed that
The present invention carries out the solution of problem.
The present embodiment is conceived to meet authorized user PU and unauthorized user SUs service quality at the same time under the premise of, most
The net profit of bigization PU, excitation PU service SUs, realize the raising of system spectrum utilization rate.Work can make interference management
It is able to efficiently and effectively be realized with the method for low computation complexity.So as to realize that the frequency spectrum resource of whole system is matched
Set more optimized, utilization rate higher.
Claims (1)
1. the Poewr control method based on monotonicity optimization and linear search in a kind of cognition wireless network, it is characterised in that：Institute
Control method is stated to include the following steps：
(1) it in cognitive radio networks, is controlled, is being considered by the transmission power of authorized user PU and unauthorized user SUs
While interference including two parts between PU and SUs and between different SUs, PU is maximized in the case of the QoS for ensureing PU
The optimization problem of net profit the nonconvex property optimization problem that is as follows is described：
P1：max∑s∈ΩαsRs-β(p0-p0 min)
WhereinIndicate each unauthorized user SUs's
Handling capacity,Indicate the uplink throughput of authorized user PU,What Ω={ 1,2...S } indicated is the set of all unauthorized user SUs；
In problem P1, parameters are defined as follows：
αs:The marginal coefficient charged for each SU s unit handling capacity PU realized；
β：The marginal power consumption cost of PU, unit are $/Watt；
Rs:The handling capacity of each SU s；
p0:The transmission power of PU；
p0 min：The minimum transmission power of PU consumes；
n:Background Noise Power；
qs:The transmission power of SU s；
gsB:Channel power gain between SU-Tx and BS；
g0B:Channel power gain between PU-Tx and BS；
The throughput demands of each SU s；
The transmission power upper limit of PU；
The maximum transmission power upper limit of SU；
g0s:Channel power gain between PU-Tx and SU-Rx s；
gss:Channel power gain between SU-Tx s and SU-Rx s；
gjs:Channel power gain between SU-Tx j and SU-Rx s；
W:The bandwidth of PU channels；
Optimal value of subscript " * " expression parameter in optimization problem in pa-rameter symbols；
(2) formula is usedBy the R in constraintssExhibition
It opens, the Section 2 of constraints is equivalent to WhereinIt asks
The decision variable of topic P1 translates into p0And { qs}s∈Ω, Wo MenyongWithThe optimal solution of problem of representation P1 respectively；
(3) feasibility of decision problem P1
By formulaIn p0With { qs}s∈ΩReplacement, thus again by the inequality
One group of linear restriction being expressed as：
And s ≠ j
M indicates that a S × s-matrix, S indicate the sum of SUs in Ω, and the item in M indicates as follows：
In addition, also defining the vectorial u of S × 1, each of which item is expressed as
Enable vectorIt indicates that SUs disclosure satisfy that the set of the transimission power of above-mentioned Linear Constraints, remembers condition C 1：And condition C 2：We define the frequency spectrum radius of matrix M, ρ (M)=max
{|λ||λ is the characteristic value of M }, meet ρ (M) < 1；If condition C 1 and C2 disclosure satisfy that,Its
Middle I indicates the unit matrix of S × S；VectorThat is ({ qs}s∈Ω) each element representation the minimum transfer work(of each SU s
Rate, each single item { θ of SUss}s∈ΩIt is satisfied by requirement；Further from { qs}s∈ΩThe middle minimum transmission power for releasing PUThen problem (P1) feasible adequate condition i.e. condition (C3) is obtained：
(C3)：And
(4) vertical demixing of problem P1, due to always having in the optimization of problem of implementation P1Namely
It says, PU no longer needs to consumption more transimission power while meeting throughput demands, and problem (P1) orthogonal decomposition is two layers
Structure, respectively problem (P1- bottoms) and problem (P1- top layers), the transmission power p of PU fixed first in bottom problem0, phase
It answers, bottom problem becomes the transmission power p in given PU0In the case of optimize SUs transmission power qs；
(P1- bottoms)：
By calculating F (p in bottom0) value, by F (p0) value be updated to top layer problem to optimize the transimission power of PU；
(P1- top layers)：
Wherein
(5) feasibility of decision problem (P1- bottoms)
Work as p0When determining, in order to meet { θs}s∈Ω, the power of SUs is required to meet formulaIt is equivalent to solution equationA S × s-matrix is indicated with N, and S indicates that SUs's in Ω is total
It counts, the item in N indicates as follows：
In addition, also defining the vector v and vector w of S × 1, each of which item is expressed as
Thus SUs meets its respective throughput demand { θs}s∈ΩTransmission power be expressed as
Work as p0When > 0,In item be non-negative, it will be seen thatWhen, problem (P1- bottoms) is can
Capable；With (x)sIndicate s of vector x, it willSubstitute into inequalityThen can further clear problem (P1- bottoms) in p0Meet inequalityThe right of the inequality indicates p0Lower bound, be denoted asP；Meanwhile pass through by WithIt compares, solves p0The upper bound, Wherein QmaxThe vector for indicating S × 1, is expressed as
By p0The upper bound be denoted as(P1- bottoms) the feasible adequate condition that must go wrong is
(6) solution of problem (P1- bottoms), for bottom problem, using the power control algorithm optimized based on monotonicity, process
It is as follows：
Step 6.1：Introduce the signal-to-noise ratio of auxiliary variable unauthorized user
Bottom problem is converted into one about unauthorized user signal-to-noise ratio ysMonotonicity optimization problem；
Wherein
Step 6.2：The initial optimal unauthorized user signal-to-noise ratio set of settingWhereinCurrent iterations k=1 is set；
Step 6.3：For current optimal unauthorized user signal-to-noise ratio setThe object function of all elements in set of computations
ValueIt is z to record the corresponding point of maximum target function valuek；
Step 6.4：According to bisection method datum point and zkLine withIntersection point
Step 6.5：IfThen algorithm terminates, and goes to step 6.9；Otherwise step 6.6 is gone to；
Step 6.6：According to formulaCalculate S new unauthorized use
The optional optimal solution of family signal-to-noise ratio, wherein eiIt is S mutually orthogonal unit vectors；
Step 6.7：Z is replaced using calculated S in step 6.6 optional optimal solutionskTo update current optimal unauthorized user
Signal-to-noise ratio set remembers that the collection is combined into
Step 6.8：Iterations k=k+1, into recycling next time, return to step 6.2 are set；
Step 6.9：Algorithm terminates, and exits algorithm cycle, exports unauthorized user signal-to-noise ratio optimal solutionFor in current collection
The maximum signal-to-noise ratio of target function value；
Step 6.10：According to formulaS dimensional vector r are set, according to formula q*=(I-N)-1R calculates best non-
Authorized user's transmission power, wherein matrix
Step 6.11：According to formulaIt calculates in fixed p0In the case of bottom
Optimal objective function value is used for top layer；
(7) threshold value PthSolution
According to the property of problem (P1- bottoms), findIt is upper that there are a special threshold value Pth, whenP≤p0≤PthWhen, no
EquationIt is just set up, thus solves threshold value PthThe region of search of optimal solution can be reduced, is solved
Process is as follows：
Step 7.1：Initialize installation, setting two close to 0 very little positive number as permission calculating error, be denoted as respectively
η and ε, enables plower=P,
Step 7.2：Calculate |plower-pupper|If the difference is smaller than permitted calculating error ε, indicate that obtained value exists
In the range of error allows, then algorithm terminates, and gos to step 7.6, otherwise, continues step 7.3；
Step 7.3：By the transmission power p of PU0It is set as plowerWith pupperIntermediate value, i.e.,
Step 7.4：Due to giving p in step 7.30, problem (P1- bottoms) is solved by step 6 and is obtained corresponding optimal
Solution
Step 7.5：It calculatesFor judging existing p0Problem (the bottoms P1- can be met
Layer) constraintsThus Ru Guo |J(p0)|< η, then by p0Upper limit pupperIt is updated to existing
P0, otherwise by plowerIt is updated to existing p0, return to step 7.2；
Step 7.6：By obtained p0As special threshold value Pth；
(8) solution of problem (P1- top layers), the optimal solution obtained according to problem (P1- bottoms)And optimal target
Functional value F (p0), upper layer issue translates into one about authorized user's transmission power ppOne Dimension Optimization Problems, using linear
Searching algorithm solution problem (P1- top layers), process is as follows：
Step 8.1：Carry out Initialize installation：By the transmission power p of PU0It is initialized asWhereinPFor the hair of PU
Send power p0Lower bound；
Step 8.2：If the transmission power of PUDue to there is no more preferably solving, then being calculated in the region
Method terminates, and gos to step 8.6, otherwise, continues step 8.3, whereinFor the transmission power p of PU0The upper bound, obtain specifically
It can be found in step 5, PthFor by special threshold value calculated in step 7；
Step 8.3：Due to the transmission power p of PU0It has been provided that, F (p are calculated according to the underlying algorithm of step 6 provided0) and
It is calculated corresponding
Step 8.4：JudgeWhether it is more than what current search arrived
The maximum return F of PU*, if so, PU optimal transmission power is then setI.e. as existing optimal PU
Transmission power, otherwise the optimization transimission power of SUs is denoted as
Step 8.5：Update the maximum return F of PU*ForUpdate p0=p0+ λ, return to step 8.2；
Step 8.6：The transmission power of PU when optimization collocation is realized in outputThe transmission power of SUsAnd PU passes through
The maximum net profit F that service SUs is obtained*。
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