CN107257254B - Constant modulus beam forming method in secure and secret communication system - Google Patents

Constant modulus beam forming method in secure and secret communication system Download PDF

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CN107257254B
CN107257254B CN201710599199.8A CN201710599199A CN107257254B CN 107257254 B CN107257254 B CN 107257254B CN 201710599199 A CN201710599199 A CN 201710599199A CN 107257254 B CN107257254 B CN 107257254B
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objective function
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function value
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CN107257254A (en
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利强
李超
林静然
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University of Electronic Science and Technology of China
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0426—Power distribution
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0456—Selection of precoding matrices or codebooks, e.g. using matrices antenna weighting
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/06—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0613—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission
    • H04B7/0615—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal
    • H04B7/0617—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal for beam forming
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

Abstract

The invention discloses a constant modulus wave beam shaping method in a security and secrecy system, which comprises the steps of S1, initializing parameters, S2, adopting a Dinkelbach algorithm to carry out objective function value η according to the initialized parametersiSolving the formula for iteration to obtain and store an objective function value ηi(ii) a S3, adopting ADMM algorithm to iteratively solve the next generation constant mode wave beam wi+1And corresponding objective function value ηi+1S4 η obtained according to step S2iAnd step S3 obtaining ηi+1Performing judgment and outer layer iteration to output constant modulus wave beam wi+1. The invention has higher frequency band utilization efficiency, and utilizes the multi-antenna technology to improve the utilization efficiency of frequency spectrum; the modular length of each element in the precoding matrix is constant and equal, so that information precoding can be realized only by using a single radio frequency link and a simple phase shifter, and the hardware cost of a large-scale antenna array is greatly reduced. The invention fixes the transmitting power of each antenna, can reduce the operation consumption of the whole system to the power and improves the utilization efficiency of the energy.

Description

Constant modulus beam forming method in secure and secret communication system
Technical Field
The invention relates to the field of wireless communication, in particular to a constant modulus beam forming method in a secure and secret communication system.
Background
MISOME (Multiple-Input Single-Output Multiple-Eresuppler) is one of the commonly studied scenarios in the field of physical layer security. In the past study [1], Khisti and Wornell derived a mathematical representation of MISOME privacy capacity, and demonstrated that beamforming techniques can optimize the implementation of this privacy capacity. The transmit beamforming technique is a simple and efficient technique for transmitting information. Recent research shows that the large-scale antenna array beamforming technology can effectively improve the utilization efficiency of frequency bands, but when the number of transmitting antennas is increased, the hardware implementation cost is greatly improved. In order to make more efficient use of large-scale antenna arrays, the cost of hardware equipment must be reduced.
Disclosure of Invention
Aiming at the defects in the prior art, the constant modulus beamforming method in the secure and confidential communication system provided by the invention maximizes the secure communication rate of the system and effectively reduces the cost of hardware equipment.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that:
a constant modulus beamforming method in a secure and secret communication system is provided, which comprises the following steps:
s1, initializing parameters;
s2, according to the initialized parameters, adopting Dinkelbach algorithm to carry out the objective function value η by the following formulaiIterate to obtain and store an objective function value ηi；
Figure GDA0001390697790000011
Wherein
Figure GDA0001390697790000021
Is the constant modulus beam obtained by the ith iteration;
Figure GDA0001390697790000022
p > 0 represents the transmission power of each antenna of the information source;
Figure GDA0001390697790000023
respectively are transmission channel matrixes from a signal source to a legal signal sink and from the signal source to an eavesdropper, N is the number of signal source antennas, and M is the number of eavesdropper receiving antennas; i is a unit array; h is conjugate transposition;
s3, adopting ADMM algorithm to iteratively solve the next generation constant mode wave beam wi+1And corresponding objective function value ηi+1；
S4, η according to step S2iAnd η obtained in step S3i+1Or a limited relation of iteration times i, and outputting a corresponding constant-mode beam wi+1。
Further, the method for initializing the parameters in step S1 includes:
setting i to be 0, the maximum iteration time T of the Dinkelbach algorithm and the minimum difference epsilon of target values before and after iteration, and initializing the beam w on the feasible regioniA channel matrix h and a channel matrix G;
further, in step S2, according to the initialized parameters, the target function value η is calculated by using the Dinkelbach algorithm according to the following formulaiIterate to obtain and store an objective function value ηiThe method comprises the following steps:
according to the initialized parameters of the system, iteration is carried out by adopting a Dinkelbach algorithm to obtain an ith objective function value ηiAnd recording the current objective function value to the storage variable fiMedium, objective function value ηiIteration is performed using the following formula:
Figure GDA0001390697790000024
wherein
Figure GDA0001390697790000025
Is the constant modulus beam obtained by the ith iteration;
Figure GDA0001390697790000026
p > 0 represents the transmission power of each antenna of the information source;
Figure GDA0001390697790000027
respectively are transmission channel matrixes from a signal source to a legal signal sink and from the signal source to an eavesdropper, N is the number of signal source antennas, and M is the number of eavesdropper receiving antennas; i is a unit array; h is conjugate transposition;
further, in step S3, the ADMM algorithm is used to iteratively solve the next generation constant modulus beam wi+1And corresponding objective function value ηi+1The method comprises the following steps:
s3-1, carrying out non-convex optimization on the objective function by adding a constant term:
Figure GDA0001390697790000031
conversion to an objective functionFor convex optimization problems:
Figure GDA0001390697790000032
wherein A ═ GGH-ηhhH-λmin(GGH-ηhhH)I≥0，λmin() represents the minimum eigenvalue of the matrix; i is a unit array;
s3-2, converting the objective function in the step S3-1 into a convex optimization problem, wherein the specific conversion process is as follows:
adopting an auxiliary variable x and an original variable α, and according to an ADMM algorithm framework, taking an objective function as a convex optimization problem:
Figure GDA0001390697790000033
the following optimization problem is converted into:
Figure GDA0001390697790000034
s3-3, optimizing the problem according to the step S3-2
Figure GDA0001390697790000035
Constraint of (8) | α m1, …, N and α yielding the augmented lagrangian function:
Figure GDA0001390697790000036
wherein
Figure GDA0001390697790000037
Is a lagrange multiplier;
s3-4, initialization parameters:
setting K to 0, maximum iteration number K of the ADMM algorithm and error ξ, and initializing on a feasible domain (xk,αk,vk)，
Figure GDA0001390697790000038
S3-5, updating the auxiliary variable x according to the initialization parameterk+1And is combined withFor the auxiliary variable x according to the following formulak+1Updating:
Figure GDA0001390697790000041
wherein m is 1, …, N;
s3-6, updating original variables α according to initialization parametersk+1And the original variable α is corrected according to the following formulak+1Updating:
αk+1＝(ρI+A)-1(ρxk+1+vk)
s3-7, updating Lagrange multiplier v according to initialization parametersk+1And according to the following formula for Lagrange multiplier vk +1Updating:
Figure GDA0001390697790000042
s3-8, carrying out inner layer iteration according to the variable x and the variable α, and carrying out inner layer iteration according to the following steps:
s3-8-1, repeating the steps S3-5 to S3-7 until the error precision reaches
Figure GDA0001390697790000043
Or the iteration number K +1 is more than or equal to K, and (x) is outputk+1,αk+1,vk+1) And obtaining the next solution wi+1＝αk+1Let update k equal to k + 1;
s3-8-2, the solution obtained in the step S3-8-1 is brought into the formula
Figure GDA0001390697790000044
η is obtainedi+1And record fi+1＝ηi+1；
Further, step S4 is performed according to η obtained in step S2iAnd η obtained in step S3i+1Or a limited relation of iteration times i, and outputting a corresponding constant-mode beam wi+1The method comprises the following steps:
the steps S2 to S3 are repeated,updating i to i +1 until the outer layer iteration frequency i +1 is more than or equal to T or the difference value | f of the optimized function value obtained by two iterationsi+1-fiIf is greater than epsilon, ending the algorithm and outputting a constant modulus beam wi+1To obtain the required constant modulus wave beam wi +1。
The invention has the beneficial effects that:
1. the invention has higher frequency band utilization efficiency, and utilizes the multi-antenna technology to improve the utilization efficiency of frequency spectrum;
2. the modular length of each element in the precoding matrix is constant and equal, so that information precoding can be realized only by using a single radio frequency link and a simple phase shifter, and the hardware cost of a large-scale antenna array is greatly reduced.
3. The invention fixes the transmitting power of each antenna, can reduce the operation consumption of the whole system to the power and improves the utilization efficiency of the energy.
Drawings
FIG. 1 is a flow chart of the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
As shown in fig. 1, the method for constant modulus beamforming in a secure and secure communication system includes the following steps:
s1, initializing parameters;
s2, according to the initialized parameters, adopting Dinkelbach algorithm to carry out the objective function value η by the following formulaiIterate to obtain and store an objective function value ηi；
Figure GDA0001390697790000051
Wherein
Figure GDA0001390697790000052
Is the constant modulus beam obtained by the ith iteration;
Figure GDA0001390697790000061
p > 0 represents the transmission power of each antenna of the information source;
Figure GDA0001390697790000062
respectively are transmission channel matrixes from a signal source to a legal signal sink and from the signal source to an eavesdropper, N is the number of signal source antennas, and M is the number of eavesdropper receiving antennas; i is a unit array; h is conjugate transposition;
s3, adopting ADMM algorithm to iteratively solve the next generation constant mode wave beam wi+1And corresponding objective function value ηi+1；
S4, η according to step S2iAnd η obtained in step S3i+1Or a limited relation of iteration times i, and outputting a corresponding constant-mode beam wi+1。
The method for initializing the parameters in step S1 includes:
setting i to be 0, the maximum iteration time T of the Dinkelbach algorithm and the minimum difference epsilon of target values before and after iteration, and initializing the beam w on the feasible regioniA channel matrix h and a channel matrix G;
in step S2, according to the initialized parameters, the target function value η is adjusted by the Dinkelbach algorithm according to the following formulaiIterate to obtain and store an objective function value ηiThe method comprises the following steps:
according to the initialized parameters of the system, iteration is carried out by adopting a Dinkelbach algorithm to obtain an ith objective function value ηiAnd recording the current objective function value to the storage variable fiMedium, objective function value ηiIteration is performed using the following formula:
Figure GDA0001390697790000063
wherein
Figure GDA0001390697790000064
Is the constant modulus beam obtained by the ith iteration;
Figure GDA0001390697790000065
p > 0 represents the transmission power of each antenna of the information source;
Figure GDA0001390697790000066
respectively are transmission channel matrixes from a signal source to a legal signal sink and from the signal source to an eavesdropper, N is the number of signal source antennas, and M is the number of eavesdropper receiving antennas; i is a unit array; h is conjugate transposition;
in step S3, the ADMM algorithm is adopted to iteratively solve the next generation of constant modulus wave beam wi+1And corresponding objective function value ηi+1The method comprises the following steps:
s3-1, carrying out non-convex optimization on the objective function by adding a constant term:
Figure GDA0001390697790000071
converting into an optimization problem with a convex objective function:
Figure GDA0001390697790000072
wherein A ═ GGH-ηhhH-λmin(GGH-ηhhH)I≥0，λmin() represents the minimum eigenvalue of the matrix; i is a unit array;
s3-2, converting the objective function in the step S3-1 into a convex optimization problem, wherein the specific conversion process is as follows:
adopting an auxiliary variable x and an original variable α, and according to an ADMM algorithm framework, taking an objective function as a convex optimization problem:
Figure GDA0001390697790000073
the following optimization problem is converted into:
Figure GDA0001390697790000074
s3-3, optimizing the problem according to the step S3-2
Figure GDA0001390697790000075
Constraint of (8) | α m1, …, N and α yielding the augmented lagrangian function:
Figure GDA0001390697790000076
wherein
Figure GDA0001390697790000077
Is a lagrange multiplier;
s3-4, initialization parameters:
setting K to 0, maximum iteration number K of the ADMM algorithm and error ξ, and initializing on a feasible domain (xk,αk,vk)，
Figure GDA0001390697790000078
S3-5, updating the auxiliary variable x according to the initialization parameterk+1And the auxiliary variable x is set according to the following formulak+1Updating:
Figure GDA0001390697790000081
wherein m is 1, …, N;
s3-6, updating original variables α according to initialization parametersk+1And the original variable α is corrected according to the following formulak+1Updating:
αk+1＝(ρI+A)-1(ρxk+1+vk)
s3-7, updating Lagrange multiplier v according to initialization parametersk+1And according to the following formula for Lagrange multiplier vk +1Updating:
Figure GDA0001390697790000082
s3-8, carrying out inner layer iteration according to the variable x and the variable α, and carrying out inner layer iteration according to the following steps:
s3-8-1, repeating the steps S3-5 to S3-7 until the error precision reaches
Figure GDA0001390697790000083
Or the iteration number K +1 is more than or equal to K, and (x) is outputk+1,αk+1,vk+1) And obtaining the next solution wi+1＝αk+1Let update k equal to k + 1;
s3-8-2, the solution obtained in the step S3-8-1 is brought into the formula
Figure GDA0001390697790000084
η is obtainedi+1And record fi+1＝ηi+1；
η obtained in step S2 in step S4iAnd η obtained in step S3i+1Or a limited relation of iteration times i, and outputting a corresponding constant-mode beam wi+1The method comprises the following steps:
repeating the steps S2 to S3, updating i to i +1 until the outer layer iteration number i +1 is more than or equal to T or the difference value | f of the optimization function values obtained by two iterationsi+1-fiIf is greater than epsilon, ending the algorithm and outputting a constant modulus beam wi+1To obtain the required constant modulus wave beam wi +1。
In one embodiment of the present invention, it is assumed that there is a multi-antenna source A, a single-antenna legitimate sink B and a multi-antenna eavesdropper E in a multi-antenna secure communication system, wherein the source A sends a signal to the legitimate sink B as
x(t)＝ws(t) (1)
Wherein
Figure GDA0001390697790000091
Is the secret information of the unit energy,
Figure GDA0001390697790000092
is a constant mode transmitting beam, each element of which needs to be satisfied
Figure GDA0001390697790000093
Where P > 0 represents the transmit power per antenna of source a.
The information source A and the eavesdropper E are respectively provided with N transmitting antennas and M receiving antennas (N > M), and the legal information sink B is a single antenna.
Figure GDA0001390697790000094
The transmission channel matrixes from the source A to the legal sink B and from the source A to the eavesdropper E are respectively. The received signals of the legitimate sink B and the eavesdropper E are respectively represented as
Figure GDA0001390697790000099
Where x is the information transmitted by source A, where
Figure GDA0001390697790000095
Suppose nb～CN(0,1),neCN (0, I) is the reception noise of the legitimate sink B and the eavesdropper E, respectively, where CN represents a complex gaussian distribution and the I covariance matrix is a unit matrix. According to the document [1]]The security rate of a secure communication system can be expressed as:
Rs(w)＝[log(1+|hHw|2)-log(1+||GHw||2)]+(4)
wherein
Figure GDA0001390697790000096
Therefore, the main task of the system is to realize the maximization of the security rate of the secure communication system by designing the transmission beam w of the information source a under the constraint condition that the transmission power of each antenna of the information source a is determined, and the final optimization problem can be described as:
Figure GDA0001390697790000097
it can be seen that the safe rate problem under constant modulus constraint (5) is non-convex and can prove to be an NP-hard problem.
Next, the ADMM-Dinkelbach method is used to solve the problem (5). First, we can transform the optimization problem (5) into the following form:
Figure GDA0001390697790000098
will wiNormalization, problem (6) can be further translated into:
Figure GDA0001390697790000101
from the ratio of the two quadratic form functions of the main function of the problem (7), the classical Dinkelbach method can be used to solve iteratively, first fixing the function value η transforms the fractional form problem into a quadratic programming problem (8), which can be solved using the ADMM algorithm:
Figure GDA0001390697790000102
wherein η is the target value of the problem (7) solved by the Dinkelbach algorithm in the previous iteration:
Figure GDA0001390697790000103
in solving the quadratic programming sub-problem (8), adding some constant terms can transform the problem into a convex problem:
Figure GDA0001390697790000104
wherein A ═ GGH-ηhhH-λmin(GGH-ηhhH)I≥0，λmin(. indicates the minimum eigenvalue of the matrix.
The ADMM algorithm framework solves a problem (10), the algorithm of which can be described as:
Figure GDA0001390697790000105
combining two equality constraints with an augmented Lagrangian function of
Figure GDA0001390697790000106
Where ρ > 0 and
Figure GDA0001390697790000107
is a lagrange multiplier. The ADMM algorithm adopts iteration to alternately update an original variable and a dual variable, the updating of each variable has a closed-form solution, wherein the formula for updating the variable x is as follows:
Figure GDA0001390697790000111
the formula for the update variable α is:
αk+1＝(ρI+A)-1(ρxk+1+vk) (14)
the formula for the final update v is:
Figure GDA0001390697790000112
in the specific implementation process, according to constant modulus constraint and power limitation, the Dinkelbach algorithm has the maximum iteration time T of 50 and the minimum difference epsilon of target values before and after iteration of 0.01, P of 10/N, i of 0, and adopts a standard complex Gaussian random initialization beam wiSetting the modular length of each element as 1, and generating channel matrixes h and G by using standard complex Gaussian;
solving by adopting an ADMM algorithm, firstly, obtaining a feature vector corresponding to the minimum feature value of A, setting K to be 0, and setting the maximum iteration number K of the ADMM algorithm to be 104Sum error ξ -0.01, new vector initialization α with each element projected to unit circlei，xiAt 0, an initial v is additionally randomly generatedi，
Figure GDA0001390697790000113
Wherein λmax(A) Is the maximum eigenvalue of a.
The invention has higher frequency band utilization efficiency, and utilizes the multi-antenna technology to improve the utilization efficiency of frequency spectrum; the modular length of each element in the precoding matrix is constant and equal, so that information precoding can be realized only by using a single radio frequency link and a simple phase shifter, and the hardware cost of a large-scale antenna array is greatly reduced. The invention fixes the transmitting power of each antenna, can reduce the operation consumption of the whole system to the power and improves the utilization efficiency of the energy.

Claims (2)

1. A constant modulus beam shaping method in a secure and secret communication system is characterized in that: the method comprises the following steps:
s1, initializing parameters;
s2, according to the initialized parameters, adopting Dinkelbach algorithm to carry out the objective function value η by the following formulaiIterate to obtain and store an objective function value ηi；
Figure FDA0002441859770000011
Wherein
Figure FDA0002441859770000012
Is the constant modulus beam obtained by the ith iteration;
Figure FDA0002441859770000013
P>0 represents the transmission power of each antenna of the source;
Figure FDA0002441859770000014
respectively are transmission channel matrixes from a signal source to a legal signal sink and from the signal source to an eavesdropper, N is the number of signal source antennas, and M is the number of eavesdropper receiving antennas; i is a unit array; h is conjugate transposition;
s3, using ADMM to calculateIterative solution of next generation constant mode wave beam wi+1And corresponding objective function value ηi+1；
S4, η according to step S2iAnd η obtained in step S3i+1Or a limited relation of iteration times i, and outputting a corresponding constant-mode beam wi+1；
The method for initializing the parameters in step S1 includes:
setting i to be 0, the maximum iteration time T of the Dinkelbach algorithm and the minimum difference epsilon of target values before and after iteration, and initializing the beam w on the feasible regioniA channel matrix h and a channel matrix G;
in step S2, according to the initialized parameters, the target function value η is adjusted by the Dinkelbach algorithm according to the following formulaiIterate to obtain and store an objective function value ηiThe method comprises the following steps:
according to the initialized parameters of the system, iteration is carried out by adopting a Dinkelbach algorithm to obtain an ith objective function value ηiAnd recording the current objective function value to the storage variable fiMedium, objective function value ηiIteration is performed using the following formula:
Figure FDA0002441859770000021
wherein
Figure FDA0002441859770000022
Is the constant modulus beam obtained by the ith iteration;
Figure FDA0002441859770000023
P>0 represents the transmission power of each antenna of the source;
Figure FDA0002441859770000024
respectively are transmission channel matrixes from a signal source to a legal signal sink and from the signal source to an eavesdropper, N is the number of signal source antennas, and M is the number of eavesdropper receiving antennas; i is a unit array; h is conjugate transposition;
ADM is adopted in step S3Iterative solution of next generation constant mode wave beam w by M algorithmi+1And corresponding objective function value ηi+1The method comprises the following steps:
s3-1, carrying out non-convex optimization on the objective function by adding a constant term:
Figure FDA0002441859770000025
converting into an optimization problem with a convex objective function:
Figure FDA0002441859770000026
wherein A ═ GGH-ηhhH-λmin(GGH-ηhhH)I≥0，λmin() represents the minimum eigenvalue of the matrix; i is a unit array;
s3-2, converting the objective function in the step S3-1 into a convex optimization problem, wherein the specific conversion process is as follows:
adopting an auxiliary variable x and an original variable α, and according to an ADMM algorithm framework, taking an objective function as a convex optimization problem:
Figure FDA0002441859770000027
the following optimization problem is converted into:
Figure FDA0002441859770000028
s3-3, optimizing the problem according to the step S3-2
Figure FDA0002441859770000029
Constraint of (8) | αm1, …, N and α yielding the augmented lagrangian function:
Figure FDA0002441859770000031
wherein
Figure FDA0002441859770000032
Is a lagrange multiplier;
s3-4, initialization parameters:
setting K to 0, maximum iteration number K of the ADMM algorithm and error ξ, and initializing on a feasible domain (xk,αk,vk)，
Figure FDA0002441859770000033
S3-5, updating the auxiliary variable x according to the initialization parameterk+1And the auxiliary variable x is set according to the following formulak+1Updating:
Figure FDA0002441859770000034
wherein m is 1, …, N;
s3-6, updating original variables α according to initialization parametersk+1And the original variable α is corrected according to the following formulak+1Updating:
αk+1＝(ρI+A)-1(ρxk+1+vk)
s3-7, updating Lagrange multiplier v according to initialization parametersk+1And according to the following formula for Lagrange multiplier vk+1Updating:
Figure FDA0002441859770000035
s3-8, carrying out inner layer iteration according to the variable x and the variable α, and carrying out inner layer iteration according to the following steps:
s3-8-1, repeating the steps S3-5 to S3-7 until the error precision reaches
Figure FDA0002441859770000036
Or the iteration number is more than or equal to K, and (x) is outputk+1,αk+1,vk+1) And obtaining the next solution wi+1＝αk+1；
S3-8-2, the solution obtained in the step S3-8-1 is brought into the formula
Figure FDA0002441859770000037
η is obtainedi+1And record fi+1＝ηi+1。
2. The method for constant modulus beamforming in a secure and secure communication system according to claim 1, wherein the step S4 is performed according to η obtained in step S2iAnd η obtained in step S3i+1Or a limited relation of iteration times i, and outputting a corresponding constant-mode beam wi+1The method comprises the following steps:
repeating the steps S2 to S3, and updating the iteration number i until the outer iteration number is more than or equal to T or the difference | f of the optimization function values obtained by two iterations before and afteri+1-fi｜>Epsilon, finish the algorithm and output a constant modulus beam wi+1To obtain the required constant modulus wave beam wi +1。
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