CN108173800A - OFDM peak-to-average ratio suppressing method based on alternating direction multipliers method - Google Patents
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Abstract
The invention discloses a kind of OFDM peak-to-average ratio suppressing methods based on alternating direction multipliers method, solve the problems, such as that existing reduction OFDM peak-to-average ratio optimization method computation complexity is high.Implementation includes：It is interfered with frequency domain ofdm signal data carrier as object function, peak-to-average force ratio and free carrier power are constraints, establish non-convex Optimized model；Optimization process is participated in alternating direction multipliers method ADMM, seeks optimal solution, optimization process includes two schemes, and scheme 1 is directly solved using ADMM methods, can ensure to converge to the KKT points of Optimized model if convergence.Scheme 2 has carried out Optimized model relaxation and has obtained relaxation model, ensures convergence, and in the case where parameter selection is appropriate unlimited optimization model KKT points.With existing method for suppressing peak to average ratio ratio, the present invention can obtain smaller and constant peak-to-average ratio, improve system error performance, reduce computation complexity, for field of communication technology, improve the transmission quality of communication system signal.
Description
Technical Field
The invention belongs to the field of communication, in particular relates to a sending end technology suitable for Orthogonal Frequency Division Multiplexing (OFDM), and particularly relates to an OFDM peak-to-average power ratio restraining method based on an alternating direction multiplier method, which can be used in the field of wireless communication.
Background
Orthogonal Frequency Division Multiplexing (OFDM) is an important multi-carrier modulation technology, and due to the high spectrum utilization rate and the capability of resisting multipath fading, OFDM is widely applied to modern wireless communication systems. However, since OFDM is a multi-carrier modulation scheme and data of different carriers are independent, the same or similar phase of frequency domain carriers may result in a larger time domain signal peak and a higher peak-to-average ratio. However, the power amplifier of the transmitter is power-limited, and a large peak-to-average ratio can cause the power amplifier to enter a nonlinear saturation region, so that signals are subjected to in-band interference and out-of-band radiation.
To solve this problem, the scholars propose a series of methods for reducing the peak-to-average ratio, among which the repetitive cut filtering technique RCF is the simplest method with the lowest computational complexity. The method leads the OFDM signal to approach the determined PAPR threshold value by introducing the interference signal. The classical RCF method and some variants thereof, however, do not meet more complex system requirements, such as having the free carrier power below a certain threshold or minimizing signal interference, which are not met by RCFs.
On the basis, the optimization method is widely applied in recent years, and the peak-to-average ratio of the OFDM signal is reduced by the optimization method, so that the system parameters can be optimized. Typical optimization methods, such as second order cone planning and semi-definite planning, are widely studied and used. The first applied second-order cone programming method is Aggarwal, in the model, the objective function is to minimize the peak value of a time domain signal, the constraint conditions are error vector amplitude and free carrier power, and Aggarwal proves that the model has a global optimal solution, but the computation complexity is higher and is about O (N)3). After that, different second order cone planning methods are proposed to reduce the peak-to-average ratio of OFDM. Semi-definite programming is another important optimization method, Yongchao Wang and the like utilize the method to relax a non-convex quadratic optimization model, an optimized OFDM signal has a constant peak-to-average ratio, and a better system error rate is obtained at the same time, but the calculation complexity is higher.
The methods can effectively reduce the peak-to-average ratio of the OFDM signals and obtain a better system error rate, but the calculation complexity of the methods is higher.
Disclosure of Invention
The present invention is directed to provide a method for suppressing the peak-to-average power ratio of OFDM based on the alternating direction multiplier method ADMM, which has a low complexity and a smaller peak-to-average power ratio, in view of the above-mentioned disadvantages of the prior art.
The invention relates to an OFDM peak-to-average power ratio restraining method based on an alternating direction multiplier method, which is characterized by comprising the following steps:
(1) modulating an input OFDM signal to obtain a frequency domain signal c;
(2) carrying out series-parallel transformation on the frequency domain signal c, and then carrying out lN point inverse Fourier transform to obtain a time domain signal x;
(3) and optimizing the frequency domain signal c and the time domain signal x by using one of two optimization schemes, wherein the scheme 1 is optimized by using a direct model, and the scheme 2 is optimized by using a relaxation model.
(3.1) scheme 1: the method comprises the steps of establishing a non-convex direct model by taking data carrier interference as an objective function and peak-to-average power ratio and free carrier power as constraint conditions, directly applying an ADMM method to the direct model, respectively solving a frequency domain signal c and a time domain signal x, and finally obtaining the optimal solution of the model.
(3.2) scheme 2: the method comprises the steps of introducing two auxiliary variables, namely a first auxiliary variable u and a second auxiliary variable w, into an objective function on the basis of taking data carrier interference as an objective function and taking a peak-to-average ratio and free carrier power as constraint conditions, adding u-w as a penalty term, establishing a non-convex relaxation model, applying an ADMM method to the relaxation model, solving a frequency domain signal c, a time domain signal x, the auxiliary variables u and w respectively, and finally obtaining the optimal solution of the model.
(4) Optimizing the time domain signal x in the optimal solutionoptAdding a cyclic prefix to obtain x1And after the suppression of the peak-to-average power ratio of the OFDM signal is finished, the OFDM signal is sent to subsequent equipment such as a digital-to-analog converter and the like for transmission.
The invention optimizes the peak-to-average power ratio of the OFDM to a smaller constant value with lower calculation complexity, and meets the requirements of the modern wireless communication system.
Compared with the prior art, the invention has the following advantages:
1. the peak-to-average ratio is reduced.
Compared with the traditional OFDM peak-to-average ratio inhibition method, the method takes the minimized data carrier interference as a target function, and takes the peak-to-average ratio and the free carrier power as constraint conditions to establish a non-convex optimization model; and (4) participating in an optimization process by using an ADMM method to solve an optimal solution. Because the peak-to-average ratio is taken as a constraint condition, the peak-to-average ratio of the OFDM is reduced to a smaller constant value which is 4 dB.
2. The error rate is reduced.
The proposal provided by the invention takes the minimized data carrier interference as an objective function, so that the signal distortion reaches the minimum, and the proposal can ensure that the error code performance is better than that of other methods.
3. Reduces the complexity of calculation
The two optimization schemes adopted by the method are both optimized by using an ADMM method, the optimal solution of the optimization model is solved, and the ADMM method is low in calculation complexity, so that the complexity is greatly reduced compared with the traditional optimization method, and the calculation complexity of each iteration of the ADMM method is about O (2 lNlog)2lN)。
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a graph showing the convergence of scheme 1 in the present invention;
FIG. 3 is a graph showing the convergence of scheme 2 in the present invention;
FIG. 4 is a graph comparing the peak-to-average ratio inhibition performance of the present invention with that of the prior art peak-to-average ratio inhibition method;
fig. 5 is a graph comparing the error rate performance of the peak-to-average ratio suppression method of the present invention with that of the prior art.
Detailed Description
The following detailed description of the embodiments and effects of the present invention will be made with reference to the accompanying drawings.
Example 1
Orthogonal frequency division multiplexing, OFDM, is an important multi-carrier modulation technique that has found wide application in modern wireless communication systems. One of the biggest disadvantages of OFDM is that it results in a large peak-to-average ratio of the signal. However, the power amplifier of the transmitter is power-limited, and a large peak-to-average ratio can cause the power amplifier to enter a nonlinear saturation region, so that signals are subjected to in-band interference and out-of-band radiation. Some methods in the prior art can effectively reduce the peak-to-average ratio of the OFDM signal and obtain a better system error rate, but the calculation complexity of the methods is higher, so that the practical engineering application is hindered. The invention develops research on the method, and establishes a non-convex optimization model by taking minimized data carrier interference as an objective function and peak-to-average ratio and free carrier power as constraint conditions; and (5) participating in an optimization process by using an ADMM method, and solving an optimal solution. The specific implementation method, as shown in fig. 1, comprises the following steps:
(1) and modulating the information bit stream of the input OFDM signal in a modulation mode including 16QAM, QPSK and the like to obtain an orthogonal frequency division multiplexing OFDM frequency domain signal c.
(2) And performing series-parallel transformation on the frequency domain signal c, and then performing lN point inverse Fourier transform to obtain a time domain signal x.
(3) Then, optimizing the frequency domain signal c and the time domain signal x by using one of two optimization schemes, wherein the scheme 1 is that a direct model is used for optimization, and has a better error rate, namely the error rate is lower; in the scheme 2, the optimization is performed by using a relaxation model, and the convergence characteristic is better, namely the convergence speed is higher.
(3.1) scheme 1: the method comprises the steps of establishing a non-convex direct model by taking data carrier interference as an objective function and peak-to-average power ratio and free carrier power as constraint conditions, directly applying an ADMM method to the direct model, respectively solving a frequency domain signal c and a time domain signal x, and finally obtaining the optimal solution of the model.
(3.2) scheme 2: on the basis of a direct model of a scheme 1, namely, two auxiliary variables are introduced under the conditions that data carrier interference is taken as an objective function, a peak-to-average ratio and free carrier power are taken as constraints, a first auxiliary variable u and a second auxiliary variable w are added into the objective function as penalty terms, a non-convex relaxation model is established, an ADMM method is applied to the relaxation model, a frequency domain signal c, a time domain signal x, the auxiliary variables u and w are solved respectively, and finally the optimal solution of the model is obtained.
The two optimization schemes adopted by the method are both the optimal solutions of the non-convex model solved by the ADMM method, and the ADMM method is low in computational complexity, so that the complexity is greatly reduced compared with the traditional optimization method, and the computational complexity of each iteration of the ADMM method is about O (2 lNlog)2lN)。
(4) Optimizing the time domain signal x in the optimal solutionoptAdding cyclic prefix x1And after the suppression of the peak-to-average power ratio of the OFDM signal is finished, the OFDM signal is sent to subsequent equipment such as a digital-to-analog converter and the like for transmission.
The basic idea of the invention is as follows: establishing a non-convex optimization model by taking the minimized data carrier interference as a target function and taking the peak-to-average ratio and the free carrier power as constraint conditions; and (5) participating in an optimization process by using an ADMM method, and solving an optimal solution. Compared with the traditional OFDM peak-to-average ratio restraining method, the two schemes provided by the invention both take the peak-to-average ratio as a constraint condition, further reduce the peak-to-average ratio of the OFDM and enable the peak-to-average ratio of the OFDM to reach a constant value.
Example 2
The method for suppressing the peak-to-average power ratio of the OFDM based on the alternating direction multiplier method is the same as that in the embodiment 1, and the optimization by using one of the two schemes in the step (3) is modeling and solving to obtain an optimal solution, wherein the modeling solving process of the scheme 1 comprises the following steps:
(3.1.a) establishing a direct model: the objective function is set to minimize data carrier interference, and the following direct model is established by taking the peak-to-average ratio and the free carrier power as constraint conditions, and is a model <1 >:
an objective function:
constraint conditions are as follows:
Ac＝x ＜1d＞
wherein N is the number of subcarriers, l is an oversampling factor, C belongs to CNRepresenting an OFDM frequency domain signal, x ∈ ClNRepresenting the corresponding time domain signal. In the objective function, c denotes an optimized OFDM signal, coIs the original OFDM signal; sDIs a binary diagonal matrix, and the corresponding set D ═ im|m＝1,…,M}，imRepresents the mth data carrier, and when i ∈ D, SDii1, otherwise, SDii0; matrix SFAnd the corresponding set F have similar definitions except that it corresponds to a free carrier; according to SDAnd SFCan be defined as SD+SF＝I，SDSFthe first constraint and the second constraint respectively represent peak-to-average ratio and free carrier power constraint, α and β are preset threshold values and satisfy α > 1 and β > 0, and A belongs to ClN ×NRepresenting the first N columns of the inverse fourier transform twiddle factor matrix.
(3.1.b) applying the ADMM method to the direct model to obtain the direct ADMM model: applying the ADMM method to the direct model obtained in (3.1.a) to obtain the following direct ADMM model, model <2 >:
yk+1＝yk+ρ(Ack+1-xk+1) ＜2c＞
in the above model, Lρ(. cndot.) represents a Lagrange function, x belongs to x, C belongs to C and respectively represents a peak-to-average ratio constraint and a free carrier constraint, and y belongs to ClNIs a lagrange multiplier and k represents the number of iterations.
The augmented Lagrangian function <3> for establishing the direct ADMM model is as follows:
wherein, rho > 0 represents a penalty parameter, Re represents a real part, and H represents the conjugate transpose of the vector.
(3.1.c) establishing a solving model of the frequency domain signal c: according to the augmented Lagrange function of the direct ADMM model, obtaining a specific form of solving the frequency domain signal c in the direct ADMM model, namely a model <4 >:
an objective function:
constraint conditions are as follows:
(3.1.d) establishing a solving model of the time domain signal x: obtaining a specific form of solving a time domain signal x in the direct ADMM model according to an augmented Lagrange function of the direct ADMM model, wherein the model <5 >:
an objective function:
constraint conditions：
And the constraint condition of the solving model of the time domain signal x in the direct ADMM model is peak-to-average ratio constraint.
(3.1.e) updating Lagrangian multiplier: and participating in the solution of the frequency domain signal c and the time domain signal x by using Lagrange multipliers of the updated direct ADMM model.
(3.1.f) iterative solution: and (5) repeatedly executing the steps (3.1.c) - (3.1.e) to obtain the optimal solution of the direct model.
The modeling solving process has lower complexity, and if the method converges, the converged point must be the Karush-Kuhn-Tucker (KKT) point of the direct model <1 >.
Example 3
The OFDM peak-to-average ratio restraining method based on the alternative direction multiplier method is the same as the embodiment 1-2, and the modeling of the scheme 2 of the invention comprises the following steps:
(3.2.a) establishing a relaxation model: two auxiliary variables u and w are introduced, and the direct model in (3.1.a) is relaxed with the first and second auxiliary variables, resulting in the following relaxed model, model <6 >:
an objective function:
constraint conditions are as follows:
Ac＝u ＜6d＞
x＝w ＜6e＞
wherein,is a penalty parameter.
The augmented Lagrangian function <7> of the relaxation model is established as follows:
wherein, y1∈ClN,y2∈ClNLagrange multipliers corresponding to the constraints Ac ═ u and x ═ w, respectively.
(3.2.b) applying the ADMM method to the relaxation model to obtain a relaxation ADMM model: applying the ADMM method to the relaxation model obtained in (3.2.a) to obtain the relaxation ADMM model of scheme 2 as model <8 >:
(3.2.c) establishing a solving model of the frequency domain signal c: obtaining a specific form of solving the frequency domain signal c in the relaxed ADMM model according to the augmented Lagrange function of the relaxed ADMM model, namely a model <9 >:
an objective function:
constraint conditions are as follows:
(3.2.d) establishing a solving model of the time domain signal x: obtaining a specific form of solving a time domain signal x in a relaxed ADMM model according to an augmented Lagrange function of the relaxed ADMM model, wherein the model <10 >:
an objective function:
constraint conditions are as follows:
the constraint condition of the solution model of the time domain signal x in the relaxed ADMM model is peak-to-average ratio constraint.
(3.2.e) solving the subproblem <8c > of the relaxed ADMM model according to the formula <7>, and solving the auxiliary variables u and w; and updating the Lagrange multiplier, and using the Lagrange multiplier of the updated relaxed ADMM model to participate in the solution of the frequency domain signal c and the time domain signal x.
(3.2.f) iterative solution: and (5) repeatedly executing the steps (3.2.c) - (3.2.e) to obtain the optimal solution of the relaxation model.
The modeling solving process has low computational complexity, the relaxation model can theoretically prove convergence, the obtained point is the KKT point of the relaxation model, and in addition, if the initial value is positioned in the feasible domain of the model <1>, the optimal point (c, x) approaches the KKT point of the model <1> along with the increase of the penalty factor.
Example 4
Similar to the embodiments 1 to 3, in the method for suppressing the peak-to-average power ratio of the OFDM based on the alternating direction multiplier method, the solution of the specific solution model <4> of the frequency domain signal c in the step (3.1.c) can also obtain an optimal solution by the lagrangian multiplier method:
(3.1.c) solving the frequency domain signal c, using its optimal solution ck+1And lagrange multiplier muk*Should satisfyNamely:
from this, can findWherein,
then solve muk*. If the free carrier constraint is not an aggressive constraint, then μk*0, if the free carrier constraint is an aggressive constraint, then μk*Satisfy the requirement ofAnd is non-negative, addingk+1Carry in | | SFck+1||2And SDck+1||2To obtain
The specific solving model <5> of the time domain signal x in the step (3.1.d) can be solved by introducing an auxiliary variable t, z of the peak-to-average ratio, and the specific process comprises the following steps:
(3.1.d.1) to simplify the peak-to-average ratio constraint, an auxiliary peak-to-average variable t, z is introduced, x ═ tz andbring them into the model of (3.1.d)<5>In (1), obtaining a model<11>：
An objective function:
constraint conditions are as follows:
wherein,
(3.1.d.2) subsequently, using zk+1,tk+1Representation model<11>Because t is a real number and is unconstrained, its value does not affect the optimal solution zk+1Is taken from the model<11>Extracting t to obtain the following model<12>：
An objective function:
constraint conditions are as follows:
(3.1.d.3) because α > 1, the model <13> is further obtained at the lower side, and the optimal solutions of the two are the same because the optimal solution of the model <13> is always obtained when the last constraint satisfies the equality sign.
An objective function:
constraint conditions are as follows:
in actual practice, the solution complexity of model <13> is still high,
(3.1.d.4) A non-exact solution algorithm is proposed by introducing the Lagrange multiplier γkIs greater than 0 and is decomposed into lN parallel subproblems, the following model is obtained<14>：
An objective function:
constraint conditions are as follows:
the objective function of the model is a convex function, and only one constraint condition is provided, so that the gradient of the objective function is 0, and the static point of the objective function is projected into a feasible domain, so that the optimal solution of the objective function can be obtained:
among them, the optimum Lagrange multiplier γk*The result was obtained by the dichotomy method.
(3.1.d.5) subsequently, t is obtained by solvingk+1＝max{0,Re(zk+1Hbk)}. Will zk+1And tk+1Bringing into x-tz to give xk+1。
The modeling solving process has lower complexity, and can obtain an analytic solution, and meanwhile, if the method converges, the converged point is necessarily the KKT point of the direct model <1 >.
Example 5
Similar to embodiments 1 to 4, in the method for suppressing the peak-to-average power ratio of the OFDM based on the alternating direction multiplier method, the solution of the specific solution model <9> of the frequency domain signal c in step (3.2.c) can also obtain an optimal solution by the lagrangian multiplier method:
in step (3.2.c) the frequency domain signal c is solved, and its optimum solution c is usedk+1And lagrange multiplier muk*should satisfy ▽ L (c)k+1,μk*) 0, namely:
from this, can findWherein,
then solve muk*. If free loadedThe wave constraint is non-positive, then μk*0, if the free carrier constraint is an aggressive constraint, then μk*Satisfy the requirement ofAnd is non-negative, addingk+1Carry in | | SFck+1||2And SDck+1||2To obtain
The solving of the model <10> of the time domain signal x in the step (3.2.d) can also be realized by introducing an auxiliary variable t, z of the peak-to-average ratio, and the specific process comprises the following steps:
(3.2.d.1) to simplify the peak-to-average ratio constraint, an auxiliary peak-to-average variable t, z is introduced, x ═ tz andbring them into the model of (3.1.d)<10>In (1), obtaining a model<15>：
An objective function:
constraint conditions are as follows:
wherein,
(3.2.d.2) subsequently, using zk+1,tk+1To representModel (model)<15>Because t is a real number and is unconstrained, its value does not affect the optimal solution zk+1Is taken from the model<15>Extracting t to obtain the following model<16>：
An objective function:
constraint conditions are as follows:
(3.2.d.3) because α > 1, the lower model <17> is further obtained, and the optimal solutions of the two are the same because the optimal solution of the model <17> is always obtained when the last constraint satisfies the equality.
An objective function:
constraint conditions are as follows:
in actual practice, the solution complexity of model <17> is still high,
(3.2.d.4) A non-exact solution algorithm is proposed by introducing the Lagrange multiplier γkIs greater than 0 and is decomposed into lN parallel subproblems, the following model is obtained<18>：
An objective function:
constraint conditions are as follows:
the objective function of the model is a convex function, and only one constraint condition is provided, so that the gradient of the objective function is 0, and the static point of the objective function is projected into a feasible domain to obtain the optimal solution:
among them, the optimum Lagrange multiplier γk*The result was obtained by the dichotomy method.
(3.2.d.5) subsequently, t is obtained by solvingk+1＝max{0,Re(zk+1Hbk)}. Will zk+1And tk+1Bringing into x-tz to give xk+1。
Solving the values of the auxiliary variables u and w in step (3.2.e), and solving u and w in the relaxed ADMM model is an unconstrained problem, so that the optimal solution u of the method isk+1,wk+1Satisfy the requirement of
Solving the system of equations to obtain
And obtaining the optimal solution of the auxiliary variable.
The solving process has low calculation complexity, the analytic solution of the model can be obtained, meanwhile, the relaxation model can theoretically prove convergence, the obtained point is the KKT point of the relaxation model, and in addition, if the initial value is located in the feasible domain of the model <1>, the optimal point (c, x) approaches to the KKT point of the model <1> along with the increase of the penalty factor.
A more detailed example is given below to further illustrate the invention from another perspective
Example 6
The OFDM peak-to-average power ratio suppressing method based on the alternative direction multiplier method is the same as that of the embodiment 1-5,
the optimization process of the frequency domain signal c and the time domain signal x is divided into two schemes, and can be optimized by a scheme 1 or a scheme 2, wherein the scheme 1 is as follows: the method comprises the steps of establishing a non-convex direct model by taking data carrier interference as an objective function and peak-to-average power ratio and free carrier power as constraint conditions, directly applying an ADMM method to the model, respectively solving a frequency domain signal c and a time domain signal x, and finally obtaining the optimal solution of the model.
Step 1: the objective function is set to minimize the data carrier interference, and a model <1> is established by taking the peak-to-average ratio and the free carrier power as constraint conditions, wherein the objective function is represented by the formula <1a >, and the constraint conditions are represented by the formula <1b >, <1c >, <1d >.
Initialization variable (c)1,x1,y1) Randomly generating OFDM signals and carrying out 16QAM modulation to obtain c1To c for1Performing lN point inverse Fourier transform to obtain x1Set up y1＝0selecting parameters α -4 dB, β 0.15, rho 100, setting matrix SD,SFThe value of (c).
Step 2: the ADMM method is directly applied to a direct model <1> to obtain a model <2>, the formula <2a > is a subproblem for solving c, the formula <2b > is a subproblem for solving x, and the formula <2c > is an updated Lagrangian multiplier.
And step 3: and establishing an augmented Lagrangian function <3> of the model <2 >.
And 4, step 4: according to the formula <3>, a concrete form of the sub-problem <2a > for solving the frequency domain signal c in the direct ADMM model, namely the model <4> is written.
(4a) In the model <4> for solving the specific form of the frequency domain signal c, the formula <4a > is an objective function, the formula <4b > is a constraint condition of the free carrier power, and the optimal solution for solving the model <4> for solving the specific form of the frequency domain signal c can be solved by a lagrange multiplier method because only one constraint condition exists. Solving the lagrangian function corresponding to <4> of the specific form of the frequency domain signal c as follows:
wherein, mukA value of ≧ 0 indicates the Lagrangian multiplier.
Model for solving specific form of frequency domain signal c<4>Of (c) an optimal solutionk+1And lagrange multiplier muk*Should satisfyNamely:
from this equation, can obtainWherein,
(4b) then solve muk*。
If free carrier power constraints<4b>Is not positively constrained, then μk*0 if free carrier power constraint<4b>Is a positive constraint, then μk*Satisfy the requirement ofAnd is non-negative, addingk+1Carry in | | SFck+1||2And SDck+1||2Can obtain
Thus, can obtain
And 5: according to the formula <3>, a concrete form of the sub-problem <2b > for solving the time-domain signal x in the direct ADMM model, namely the model <5> is written.
Concrete form model for solving time domain signal x<5>In the formula<5a>Is an objective function of the formula<5b>For peak-to-average ratio constraints, for simplifying peak-to-average ratio constraints<5b>Introducing an auxiliary variable t, z, satisfying x ═ tz andby substituting them into a model for solving for a particular form of the time-domain signal x<5>To obtain a model<5>Equivalent model of<11>：
An objective function:
constraint conditions are as follows:
wherein,zie.g. z. Then, with zk+1,tk+1Representation model<11>Because t is a real number and is unconstrained, its value does not affect the optimal solution zk+1Is taken from the model<11>Extracting t to obtain the equivalent model<12>：
An objective function:
constraint conditions are as follows:
further, because α > 1, the equivalent model <13> is obtained, the optimal solutions for model <13> and model <12> are the same, because the optimal solution for model <13> is always obtained when <13c > satisfies the equality sign.
An objective function:
constraint conditions are as follows:
in actual operation, the model<13>The solution complexity is still high, and therefore, a non-precise solution algorithm is provided, and the lagrange multiplier gamma is introducedkIs greater than 0 and is decomposed into lN parallel subproblems, the following model is obtained<14>：
An objective function:
constraint conditions are as follows:
in the model <14>, the objective function is a convex function, and there is only one constraint condition, making the gradient of the objective function 0, and projecting the static point into the feasible domain to obtain the optimal solution:
among them, the optimum Lagrange multiplier γk*The result was obtained by the dichotomy method.
Then, z obtained isk+1Brought into<11>In (1) obtaining tk+1＝max{0,Re(zk+1Hbk)}. Will zk+1And tk+1Bringing into x-tz to give xk+1。
Step 6: updating Lagrange multiplier: y isk+1＝yk+ρ(IFFTl(ck+1)-xk+1). Wherein the IFFTl(ck+1) Denotes ck +1lN point inverse fourier transform.
And repeating the steps 4-6 until the maximum iteration number is reached, wherein the maximum iteration number is 5 in the example, and the obtained peak-to-average ratio value is not changed along with the increase of the iteration number, so that the signal error rate is reduced, but the processing complexity is increased, therefore, the selection of the maximum iteration number of 5 is a good compromise between the signal error rate and the processing complexity.
The optimization scheme 2 of the invention is as follows: the method comprises the steps of introducing two auxiliary variables, namely a first auxiliary variable u and a second auxiliary variable w, into an objective function on the basis of taking data carrier interference as an objective function and taking a peak-to-average ratio and free carrier power as constraint conditions, adding u-w as a penalty term, establishing a non-convex relaxation model, applying an ADMM method to the relaxation model, solving a frequency domain signal c, a time domain signal x, the auxiliary variables u and w respectively, and finally obtaining the optimal solution of the model.
Step 1: introducing auxiliary variables u and w to direct model<1>Performing relaxation to obtain a relaxation model<6>Wherein, formula<6a>Is an objective function of the formula<6b>、<6c>、<6d>、<6e>Are constraints. Initializing variablesRandomly generating OFDM signals, and carrying out 16QAM modulation to obtain c1To c for1Performing lN point inverse Fourier transform to obtain x1Set up u1＝IFFTl(c1)，w1＝x1，the parameters α is 4dB, β is 0.15,ρ is 100; setting matrix SD,SFThe value of (c).
Step 2: and establishing an augmented Lagrangian function <7> of the model <6 >.
And step 3: applying the ADMM method to the model <6> to obtain a relaxed ADMM model <8> of the scheme 2, wherein the formula <8a > is a subproblem for solving c, the formula <8b > is a subproblem for solving x, the formula <8c > is a subproblem for solving u and w, and the formula <8d > <8e > is an updated Lagrangian multiplier.
And 4, step 4: according to the augmented Lagrange function formula <7>, a concrete formal model <9> of the subproblem <8a > for solving the relaxed ADMM model frequency domain signal c is written out.
(4a) In the model <9> for solving the specific form of the sub-problem <8a > of the frequency domain signal c, the formula <9a > is an objective function, the formula <9b > is a constraint condition of free carrier power, and the optimal solution of the model <9> is solved by a Lagrange multiplier method because only one constraint condition exists. The lagrangian function for model <9> is:
wherein, mukA value of ≧ 0 indicates the Lagrangian multiplier.
Model (model)<9>Of (c) an optimal solutionk+1And lagrange multiplier muk*Should satisfyNamely:
from this equation, it can be found:wherein,
(4b) solving for muk*The method is the same as the solution method in step (4b) in scheme 1.
And 5: writing a specific formal model <10> of the sub-problem <8b > for solving the time-domain signal x according to the formula <7>
Formula (II)<8b>Is the same as the solving method of step 5 in scheme 1, and the only difference is bkIs different in the value of (1), in scheme 2
Step 6: and solving a subproblem formula <8c > according to the formula <7 >.
Formula (II)<8c>Is an unconstrained problem, so its optimal solution uk+1,wk+1Satisfy the requirement of
I.e. uk+1,wk+1Is a solution of the following equation:
solving the system of linear equations to obtain
And 7: the lagrange multiplier is updated.
In scheme 2, steps 4-7 are also repeated until the maximum number of iterations is reached, in this example, the maximum number of iterations is 5. With the increase of the iteration times, the obtained peak-to-average ratio value is not changed, the signal error rate is reduced, but the processing complexity is increased, so that the maximum iteration time is selected to be 5, and the method is a good compromise between the signal error rate and the processing complexity.
The effect of the invention is further illustrated by the following simulation results:
example 7
The method for suppressing the peak-to-average power ratio of OFDM based on the alternative direction multiplier method is the same as that of the embodiments 1-6,
the simulation method comprises the following steps: the invention is used for second-order cone planning, repeated cutting filtering, low-complexity pre-coding, simplified repeated cutting filtering optimization method and improved selective mapping method in the existing method. The invention has two optimization schemes, and the two schemes are respectively used for participating in simulation.
Simulation 1: the convergence characteristic of the scheme 1 in the invention is shown in fig. 2, the convergence characteristic of the scheme 2 is shown in fig. 3, and as can be seen from fig. 2 and fig. 3, the invention can converge after several iterations in two optimization schemes, especially, under the condition that the invention uses the optimization scheme 2, the convergence starts rapidly after the first several iterations, and the convergence rate is very fast, and after 5 iterations, the convergence rate becomes flat.
Example 8
The OFDM peak-to-average power ratio suppressing method based on the alternative direction multiplier method is the same as the embodiment 1-6, the simulation condition is the same as the embodiment 7,
simulation 2: the comparison between the method of the present invention and several methods for reducing the peak-to-average power ratio of the OFDM is performed to compare the optimal peak-to-average power ratio values, and the result is shown in fig. 4.
Fig. 4 shows a complementary cumulative distribution function CCDF curve of the peak-to-average ratio of the OFDM signal, and as can be seen from fig. 4, the peak-to-average ratio of the OFDM signal is reduced by all the methods in the simulation, and the peak-to-average ratios obtained by using the two optimization schemes of the present invention are the same, which are the most optimal and constant among all the simulation methods, and the constant value is 4dB, which is not achieved by all the participating simulation methods. The CCDF curve of the second-order cone planning, the low-complexity precoding and the improved selective mapping method is gradually reduced, namely the change range of the peak-to-average ratio after the suppression is larger, and the peak-to-average ratio of the OFDM obtained by the two schemes, the repeated cutting filtering and the simplified repeated cutting filtering optimization method through the suppression is more constant; the peak-to-average ratio value obtained by adopting the two optimization schemes of the invention is 1dB less than that of the repeated cutting filtering optimization method, and only 5 iterations are needed, while 10 iterations are needed when the repeated cutting filtering reaches approximate performance.
Example 9
OFDM peak-to-average ratio suppression method based on alternating direction multiplier method is the same as embodiments 1-6, and simulation conditions are the same as embodiment 7
Simulation 3: the error rate characteristics of the OFDM signal are compared by comparing the method of the invention with the prior methods for reducing the peak-to-average ratio of the OFDM signal, and the result is shown in figure 5.
Fig. 5 shows an error rate curve of an OFDM time domain signal x after passing through a gaussian white noise channel, and as can be seen from fig. 5, the low complexity precoding method and the improved selective mapping method can obtain the best error rate performance, but their peak-to-average ratio performance is worse compared with the two optimization schemes of the present invention. Compared with other methods, such as a repeated cutting filtering method, a second-order cone planning method and a simplified repeated cutting filtering optimization method, the error code performance of the method is optimal no matter which of two optimization schemes is adopted, namely the error code rate is lowest, and the error code performance of the optimization scheme 1 is better than that of the optimization scheme 2 under the simulation condition.
In short, the invention discloses a method for reducing the PAPR of OFDM (orthogonal frequency division multiplexing) based on an alternative direction multiplier method ADMM (amplitude modulation method), which mainly solves the problem of high computational complexity of the conventional optimization method for reducing the PAPR of OFDM. The implementation scheme is as follows: firstly, frequency domain OFDM signal data carrier interference is taken as an objective function, peak-to-average ratio and free carrier power are taken as constraint conditions, a non-convex optimization model is established, then an ADMM method is used for participating in an optimization process, and an optimal solution of the model is solved. The optimization process comprises two ADMM solution schemes, both the two schemes can obtain the analytic solution of the model, the scheme 1 directly utilizes the ADMM method to solve, and if the scheme 1 converges, the convergence to the KKT point of the model can be ensured. The scheme 2 relaxes the direct model, ensures convergence, and can infinitely approach the KKT point of the model under the condition of proper parameter selection; optimizing the time domain signal x in the optimal solutionoptAdding a cyclic prefix to obtain x1And after the suppression of the peak-to-average power ratio of the OFDM signal is finished, the OFDM signal is sent to subsequent equipment such as a digital-to-analog converter and the like for transmission. Compared with the existing peak-to-average ratio restraining method based on the optimization method, the method can obtain smaller and constant peak-to-average ratio, improve the error code performance of the system, reduce the calculation complexity, and can be used in the technical field of communication to improve the transmission quality of the signals of the communication system.
Claims (5)
1. An OFDM peak-to-average ratio restraining method based on an alternating direction multiplier method is characterized by comprising the following steps:
(1) modulating an input OFDM signal to obtain a frequency domain signal c;
(2) carrying out series-parallel transformation on the frequency domain signal c, and then carrying out lN point inverse Fourier transform to obtain a time domain signal x;
(3) optimizing the frequency domain signal c and the time domain signal x by using one of two optimization schemes, wherein the scheme 1 is to optimize by using a direct model, and the scheme 2 is to optimize by using a relaxation model;
(3.1) scheme 1: establishing a non-convex direct model by taking data carrier interference as a target function and taking a peak-to-average ratio and free carrier power as constraint conditions, directly applying an ADMM method to the direct model, respectively solving a frequency domain signal c and a time domain signal x, and finally obtaining an optimal solution of the model;
(3.2) scheme 2: the method comprises the steps of introducing two auxiliary variables, namely a first auxiliary variable u and a second auxiliary variable w, into an objective function on the basis of taking data carrier interference as an objective function and taking a peak-to-average ratio and free carrier power as constraint conditions, adding u-w as a penalty term, establishing a non-convex relaxation model, applying an ADMM method to the relaxation model, solving a frequency domain signal c, a time domain signal x, the auxiliary variables u and w respectively, and finally obtaining the optimal solution of the model;
(4) optimizing the time domain signal x in the optimal solutionoptAdding a cyclic prefix to obtain x1And after the suppression of the peak-to-average power ratio of the OFDM signal is finished, the OFDM signal is sent to subsequent equipment such as a digital-to-analog converter and the like for transmission.
2. The method according to claim 1, wherein the optimization by one of the two schemes in step (3) is modeling and solving to obtain an optimal solution, and the modeling solution process of scheme 1 includes the following steps:
(3.1.a) establishing a direct model: the objective function is set to minimize data carrier interference, and the following direct model is established by taking the peak-to-average ratio and the free carrier power as constraint conditions:
an objective function:
constraint conditions are as follows:
wherein N is the number of subcarriers, l is an oversampling factor, C belongs to CNRepresenting an OFDM frequency domain signal, x ∈ ClNTo representA corresponding time domain signal. In the objective function, c denotes an optimized OFDM signal, coIs the original OFDM signal; sDIs a binary diagonal matrix, and the corresponding set D ═ im|m＝1,…,M}，imRepresents the mth data carrier, and when i ∈ D, SDii1, otherwise, SDii0; matrix SFAnd the corresponding set F have similar definitions except that it corresponds to a free carrier; according to SDAnd SFCan be defined as SD+SF＝I，SDSFthe first constraint and the second constraint respectively represent peak-to-average ratio and free carrier power constraint, α and β are preset threshold values and satisfy α > 1 and β > 0, and A belongs to ClN×NRepresenting the first N columns of the inverse Fourier transform twiddle factor matrix;
(3.1.b) applying the ADMM method to the direct model to obtain the direct ADMM model: applying the ADMM method directly to the direct model obtained in (3.1.a) to obtain the following direct ADMM model:
yk+1＝yk+ρ(Ack+1-xk+1)
in the above model, Lρ(. cndot.) represents a Lagrange function, x belongs to x, C belongs to C and respectively represents a peak-to-average ratio constraint and a free carrier constraint, and y belongs to ClNIs a lagrange multiplier and k represents the number of iterations.
The augmented lagrangian function for establishing the direct ADMM model is as follows:
wherein rho > 0 represents a penalty parameter, Re represents a real part, and H represents the conjugate transpose of the solved vector;
(3.1.c) establishing a solving model of the frequency domain signal c: obtaining a specific model c1 model for solving the frequency domain signal c in the direct ADMM model according to the augmented Lagrange function of the direct ADMM model:
an objective function:
constraint conditions are as follows:
(3.1.d) establishing a solving model of the time domain signal x: obtaining a specific model x1 model for solving the time domain signal x in the direct ADMM model according to the augmented Lagrange function of the direct ADMM model:
an objective function:
constraint conditions are as follows:
(3.1.e) updating Lagrangian multiplier: using Lagrange multipliers of the updated direct ADMM model to participate in the solution of the frequency domain signal c and the time domain signal x;
(3.1.f) iterative solution: and (5) repeatedly executing the steps (3.1.c) - (3.1.e) to obtain the optimal solution of the direct model.
3. The method for suppressing the peak-to-average power ratio of the OFDM based on the alternative direction multiplier method as claimed in claim 1, wherein the modeling of the scheme 2 comprises the steps of:
(3.2.a) establishing a relaxation model: two auxiliary variables u and w are introduced, and the direct model in (3.1.a) is relaxed by the two auxiliary variables u and w, resulting in the following relaxation model:
an objective function:
constraint conditions are as follows:
Ac＝u
x＝w
wherein,is a penalty parameter.
The augmented Lagrangian function for establishing the relaxation model is as follows:
wherein, y1∈ClN,y2∈ClNLagrange multipliers corresponding to constraints Ac ═ u and x ═ w, respectively;
(3.2.b) applying the ADMM method to the relaxation model to obtain a relaxation ADMM model: applying the ADMM method to the relaxation model obtained in (3.2.a) to obtain the relaxation ADMM model of scheme 2 as:
the solving process is similar to the solving steps of (3.1.c) - (3.1.f) in scheme 1, and the solving steps are (3.2.c) - (3.2.f) correspondingly in scheme 2. And establishing a model of the frequency domain signal c and the time domain signal x and updating a Lagrange multiplier to obtain an optimal solution.
4. The OFDM peak-to-average ratio suppressing method according to claim 2, wherein the solution of the c1 model of the frequency domain signal c in step (3.1.c) can further obtain an optimal solution by a lagrange multiplier method:
(3.1.c) solving the frequency domain signal c, using its optimal solution ck+1And lagrange multiplier muk*Should satisfyNamely:
from this, can findWherein,
then solve muk*If the free carrier constraint is not an aggressive constraint, then μk*0, if the free carrier constraint is an aggressive constraint, then μk*Satisfy the requirement ofAnd is non-negative, addingk+1Carry in | | SFck+1||2And SDck+1||2To obtain
The solving of the x1 model of the time domain signal x in the step (3.1.d) can also be realized by introducing an auxiliary variable t, z of the peak-to-average ratio, and the specific process comprises the following steps:
(3.1.d.1) to simplify the peak-to-average ratio constraint, an auxiliary peak-to-average variable t, z is introduced, x ═ tz andthese were fit into the x1 model of (3.1.d), yielding the x2 model:
an objective function:
constraint conditions are as follows:
wherein,
(3.1.d.2) subsequently, usingRepresents the optimal solution of the x2 model, and because t is real and unconstrained, its value does not affect the optimal solution zk+1Extracting t from the x2 model to obtain the following x3 model:
an objective function:
constraint conditions are as follows:
(3.1.d.3) because α > 1, further get the lower x4 model, the optimal solution of the two is the same, because the optimal solution of the x4 model is always obtained when the last constraint satisfies the equal sign,
an objective function:
constraint conditions are as follows:
in actual operation, the solving complexity of the x4 model is still high;
(3.1.d.4) A non-exact solution algorithm is proposed by introducing the Lagrange multiplier γk> 0 and decompose it into lN parallel subproblems, yielding the following x5 model:
an objective function:
constraint conditions are as follows:
the objective function of the model is a convex function, and only one constraint condition is provided, so that the gradient of the objective function is 0, and the static point of the objective function is projected into a feasible domain to obtain the optimal solution:
among them, the optimum Lagrange multiplier γk*Obtaining by a dichotomy;
(3.1.d.5) subsequently, t is obtained by solvingk+1＝max{0,Re(zk+1Hbk)}. Will zk+1And tk+1Bringing into x-tz to give xk +1。
5. The method for suppressing OFDM peak-to-average power ratio based on the alternative direction multiplier method as claimed in claim 3, wherein the solution of the c1 model of the frequency domain signal c in step (3.2.c) can further obtain an optimal solution by Lagrange multiplier method:
the solution of the frequency domain signal c in step (3.2.c) is similar to the solution of the frequency domain signal c in step (3.1.c), with the difference that v iskIn (3.2.c)
The step of solving for the time domain signal x in step (3.2.d) is similar to the step of solving for the time domain signal x in step (3.1.d), with the difference being that b iskIn (3.2.d) are different
Next, the values of the auxiliary variables u and w need to be solved, and the solution of u and w in the relaxed ADMM model is an unconstrained problem, so that the optimal solution u of the ADMM model isk+1,wk+1Satisfy the requirement of
Solving the system of equations to obtain
And obtaining the optimal solution of the auxiliary variable.
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