CN108712199B - Interruption probability constraint-based two-dimensional robust beamforming method under MISO eavesdropping channel - Google Patents

Interruption probability constraint-based two-dimensional robust beamforming method under MISO eavesdropping channel Download PDF

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CN108712199B
CN108712199B CN201810414582.6A CN201810414582A CN108712199B CN 108712199 B CN108712199 B CN 108712199B CN 201810414582 A CN201810414582 A CN 201810414582A CN 108712199 B CN108712199 B CN 108712199B
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CN108712199A (en
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徐静
刘智英
孙黎
任品毅
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Xian Jiaotong University
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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
    • H04B7/06—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
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L63/00—Network architectures or network communication protocols for network security
    • H04L63/16—Implementing security features at a particular protocol layer

Abstract

The invention discloses a two-dimensional robust beam forming method under a MISO interception channel based on interruption probability constraint, which provides a robust beam forming scheme of the MISO interception channel, converts a non-convex problem with interruption probability constraint into a series of semi-definite programming problems by utilizing a bisection method, semi-definite relaxation and Bernstein-type inequality under the condition of single-group multicast, obtains an optimal robust beam forming design by means of Gaussian variables, effectively improves the confidentiality rate of a communication system, reduces the interception capability of an intercepted user, and improves the safety of the communication system.

Description

Interruption probability constraint-based two-dimensional robust beamforming method under MISO eavesdropping channel
Technical Field
The invention belongs to the technical field of physical layer security in wireless communication, and particularly relates to a two-dimensional robust beam forming method under a MISO (multiple input single output) eavesdropping channel based on interruption probability constraint.
Background
In recent years, the continuous development of wireless communication technology has enabled people to access communication networks anytime and anywhere. However, the broadcast nature of wireless propagation means that all nodes within the transmission range can receive the signal transmitted by the source, and the information of a legitimate user is easily obtained by an eavesdropping user. Therefore, it becomes important to ensure secure transmission of wireless communication [1 ]. Compared with the traditional encryption method, the physical layer security aims to ensure the security of the wireless communication based on the theory in the information theory by utilizing the physical layer characteristics of the wireless communication, such as noise, fading and the like.
Wyner in 1975 first defined a noisy eavesdropping channel model and demonstrated that fully secure communication could be achieved when the eavesdropping channel had a lower signal-to-noise ratio than the main channel, and defined the rate at which reliable transmission of the main channel information could still be guaranteed without relying on a key as the security rate [2 ]. In the end of the 20 th century, the development of wireless communication technology is greatly promoted by the emergence of MIMO technology, the multi-antenna technology brings new opportunities and challenges for the security of the wireless communication physical layer, and in recent years, a great deal of literature research is available for improving the security rate of the physical layer security by using the multi-antenna technology [3] to [5 ]. For multi-antenna systems, the objective of secret rate is to design the spatial distribution of the transmitted signals. In such designs, beamforming techniques are an important design parameter therein. Beamforming techniques improve communication efficiency by directing information streams to legitimate users.
However, the physical layer safe transmission performance of beamforming depends heavily on the channel state information acquired by the transmitter. In the MISO eavesdropping channel model, the existing research is mostly based on the assumption [6] - [7] that the transmitter can acquire the ideal main channel and eavesdrop the channel state information. However, in practical applications, due to the dynamic change of the channel and the influence of factors such as channel estimation error, quantization error, feedback delay, etc., errors may occur in the channel state information acquired by the transmitting end [8 ]. Many researchers also consider the case that the channel state information of the legal user acquired by the transmitting end has errors. However, the existing physical layer security robust beamforming technology is mostly developed from the worst case point of view to improve the privacy rate [9] - [10 ]. Due to the fact that the occurrence probability of extreme conditions is low, the design mode is conservative, and the system performance cannot be improved well. It is therefore necessary to design a robust physical layer security transmission algorithm based on outage probability for non-ideal channel state information.
However, in the case of a large number of users, the performance of conventional beamforming methods is often severely degraded due to the limited freedom to design spatially selective beamforming [11 ]. Based on this, we propose a new method to increase the flexibility, and the idea of single group beamforming is combined with orthogonal space-time block coding. In semi-deterministic relaxation solutions with rank one and rank two, the traditional approach is optimal only for the rank one solution. Simulation results show that the two-dimensional beamforming method combined with the semi-deterministic relaxation technology is greatly improved compared with the traditional method.
Document [11] considers that two-dimensional beamforming single-group multicast networks use orthogonal space-time block codes. However, with respect to the channel state information acquired by the transmitting end, they do not consider such a case as follows: 1) the channel state information of the legal user has errors; 2) physical layer security, eavesdropping on the channel, and information security rates are not considered.
Reference documents:
[1]Chen X,Li F,Xue Z,et al.Research on the Security of MISO Wireless Channel with Artificial Noise[C]//International Conference on Computational&Information Sciences.2013:1533-1536.
[2]Wyner A D.The wire-tap channel[J].Bell System Technical Journal,1975,54(8):1355-1387.
[3]Zhang,Haiyang,Wang,et al.The achievable secrecy rate of MISO wiretap channels[C]//Wireless Communications and Signal Processing(WCSP),2011International Conference on.IEEE,2011:1-4.
[4]Xiong Q,Gong Y,Liang Y C.Achieving secrecy capacity of MISO fading wiretap channels with artificial noise[J].2013:2452-2456.
[5]Li,Q,Ma,et al.Optimal and Robust Transmit Designs for MISO Channel Secrecy by Semidefinite Programming[J].IEEE Transactions on Signal Processing,2011,59(8):3799-3812.
[6]Negi R,Goel S.Secret communication using artificial noise[J].2005,3:1906-1910.
[7]Xiong Q,Gong Y,Liang Y C.Achieving secrecy capacity of MISO fading wiretap channels with artificial noise[J].2013:2452-2456.
[8]Pascual-Iserte A,Palomar D P,Perez-Neira AI,et al.A robust maximin approach for MIMO communications with imperfect channel state information based on convex optimization[J].Signal Processing IEEE Transactions on,2006,54(1):346-360.
[9]Shi W,Ritcey J.Robust beamforming for MISO wiretap channel by optimizing the worst-case secrecy capacity[C]//2010:300-304.
[10]Li J,Petropulu AP.Explicit Solution of Worst-Case Secrecy Rate for MISO Wiretap Channels With Spherical Uncertainty[J].IEEE Transactions on Signal Processing,2011,60(60):3892-3895.
[11]Wen X,Law K L,Alabed S J,et al.Rank-two beamforming for single-group multicasting networks using OSTBC[C]//Sensor Array and Multichannel Signal Processing Workshop.IEEE,2012:69-72.
disclosure of Invention
The invention aims to provide a two-dimensional robust beam forming method under a MISO (multiple input single output) eavesdropping channel based on interruption probability constraint.
The invention is realized by adopting the following technical scheme:
the two-dimensional robust beamforming method under the MISO eavesdropping channel based on the interruption probability constraint comprises the following steps:
1) aiming at a MISO (MISO) eavesdropping channel model, an information sending end Alice combines Alamouti coding, namely two-dimensional beamforming, when beamforming is carried out on sent information, and Alice only has estimated channel state information of a legal user Bobs and an eavesdropping user Eve;
2) converting the non-convex problem of the maximum safe rate with the interruption probability constraint into a series of problems of minimizing the power of a transmitting end by utilizing a bisection method, a semi-definite relaxation and a Berstein-type inequality so as to obtain a covariance matrix of a beam forming vector of the transmitting end;
3) solving a beamforming vector according to the rank of the covariance matrix of the beamforming vector at the sending end, and solving a two-dimensional beamforming vector by means of a characteristic value decomposition method when the rank is 1 or 2; and when the rank is more than 2, solving the two-dimensional beamforming vector by means of a Gaussian random variable method.
The invention is further improved in that the step 1) specifically comprises the following steps: the method is characterized in that Alice carries out two-dimensional beamforming design on signals sent by a sending end of two continuous time slots, and due to the property of Alamouti coding, the power sent by the two continuous time slots of Alice is the same, so that the signal-to-noise ratios of information received by Bobs and Eve are the same, and the maximum safe rate of the introduced interruption probability is shown as the following formula:
Figure BDA0001648996380000041
wherein, R, W, hm、
Figure BDA0001648996380000042
g、
Figure BDA0001648996380000043
pm,outP is a secret rate, a covariance matrix of a beamforming vector of Alice, channel state information of an mth Bob, noise power of the mth Bob, channel state information of Eve, noise power of Eve, interruption probability and transmission power; wherein
Figure BDA0001648996380000044
w1And w2Is a two-dimensional beam-forming column,
Figure BDA0001648996380000045
Ntis the number of Alice's antennas.
The further improvement of the invention is that in step 2), when solving the problem of maximizing the safe rate, the problem is converted into a series of problems of minimizing the power of the transmitting end by means of a dichotomy, and the following steps are carried out:
Figure BDA0001648996380000051
Figure BDA0001648996380000059
(12b)
rank(W)≤2 (12c)
then removing the non-convex condition (12c) by using a semi-definite relaxation algorithm, rank (W) is less than or equal to 2;
the theory of the Bernstein-Type inequality: order to
Figure BDA0001648996380000052
Wherein the content of the first and second substances,
Figure BDA0001648996380000053
is a complex Hermitian matrix, for any sigma ≧ 0, there are:
Figure BDA0001648996380000054
wherein s is-(A)＝max(λmax(-A),0)，λmax(-A) represents the largest eigenvector of matrix A;
the non-convex problem with the probability of interruption is converted into a convex problem as shown below:
Figure BDA0001648996380000055
wherein, assuming that all the interruption probability values are consistent, let p be p1,out＝...＝pM,out，σ＝σ1＝...＝σM，σ＝ln(pout)，umAnd vmIs a relaxation variable;
Figure BDA0001648996380000056
Figure BDA0001648996380000057
Figure BDA0001648996380000058
wherein EbIs the covariance matrix of the channel estimated by Bob, EeIs the covariance matrix of the Eve estimated channel;
in order to tighten the result, aiming at the parameter of the interruption probability, carrying out Bayesian dichotomy; by continuously adjusting the probability of interruption and R*To obtain the maximum safe rate and covariance matrix W of the beamforming vector*The dichotomy is as follows:
TABLE 1 search for optimum W by dichotomy*Algorithm process of
Figure BDA0001648996380000061
The further improvement of the invention is that the specific implementation method of the step 3) is as follows:
when W is*When the rank of (1) is equal to w1The approximation of the optimal beamforming vector of (b) is W*The product of the evolution of the eigenvalue of (a) and the eigenvector; w is a2Is NtThe vector of dimension columns, the value of which is all 0; when W is*When the rank of (2) is equal to w1And w2The approximation of the optimal beamforming vector of (b) is W*The evolution of the two eigenvalues is respectively multiplied by the corresponding eigenvector;
when W is*When the rank is more than 2, solving the two-dimensional beamforming vector w by means of a Gaussian random method1And w2The following are:
TABLE 2 two-dimensional Gaussian random algorithm
Figure BDA0001648996380000062
Figure BDA0001648996380000071
The invention has the following beneficial technical effects:
under the condition of single-group multicast, the invention converts the non-convex problem with interruption probability constraint into a series of semi-definite programming problems by utilizing a bisection method, semi-definite relaxation and a Bernstein-type inequality, and obtains the optimal robust beam forming design by means of a Gaussian variable.
The invention provides a two-dimensional beam forming method under the model that a transmitter can acquire the non-ideal instantaneous channel state information of a legal user and an eavesdropping user, which effectively improves the secrecy rate of a communication system, reduces the eavesdropping capability of the eavesdropping user and improves the safety of the communication system.
Drawings
FIG. 1 is a diagram of a system model of the present invention;
fig. 2 is a schematic diagram of the power of a transmitting end at a safe rate vs;
FIG. 3 is a diagram of safe rate vs Bob channel error;
FIG. 4 is a diagram illustrating the number of users at a safe rate vs Bob;
fig. 5 is a diagram illustrating the number of distributed vs Bob users of W rank.
Detailed Description
The technical solutions of the present invention will be described clearly and completely with reference to the accompanying drawings and embodiments, and it is to be understood that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in fig. 1, the MISO eavesdropping channel model for the present invention includes a transmitter Alice, M legitimate users Bob 1. Transmitter configuration NtThe root antenna, the legal user and the eavesdropping user configure a single antenna.
Alamouti coding, two continuous symbols can be coded by synthesis, and the symbol vector is shown as the formula (1):
s＝[s1,s2]T (1)
wherein s is a symbol sent by the transmitter to a legal user, E { | s2}＝1；
The related coding matrix is shown as formula (2):
Figure BDA0001648996380000081
x1and x2Is a vector w of messages to be sent by the sender of two consecutive time slots1And w2Is an N-dimensional beamformed column vector
Figure BDA0001648996380000082
It is assumed that the channel fading of two consecutive time slots is irrelevant and that the channels of both the information receiver and the eavesdropper are non-ideal channels.
Figure BDA0001648996380000083
Is the channel estimation vector Δ h of the information receivermIs a channel error vector, obeying the mean value of 0EbThe variance is Gaussian distribution;
Figure BDA0001648996380000084
is the channel estimation vector of the eavesdropper, Δ g is the channel error vector of the eavesdropper, obeys 0 mean variance as EgGaussian distribution, as follows:
Figure BDA0001648996380000085
the information received by the mth information receiver is shown in formula (5) in which n is two consecutive time slots1,mAnd n2,mIs the noise in the information received by the mth information receiver in two consecutive time slots, assuming that the noise obeys a mean variance of 0 to
Figure BDA0001648996380000091
A gaussian distribution of (a).
Figure BDA0001648996380000092
Information received by an eavesdropper for two consecutive time slots. Wherein n is1And n2Is the noise in the information received in two consecutive time slots, assuming that the noise obeys a mean variance of 0 to
Figure BDA0001648996380000093
A gaussian distribution of (a).
Figure BDA0001648996380000094
Due to the orthogonality of the Almouti codes, the power consumed by the transmitting end of the first time slot is the same as the power consumed by the transmitting end of the second time slot.
Figure BDA0001648996380000095
The decoding of the Alamouti coding is single symbol detection, the detection of each symbol can be obtained through independent linear operation, and the detection scheme can enable the decoding performance to be optimal. In this case, the snr is the snr, and the snr at the receiving sites of the mth legal user and the eavesdropping user is respectively:
Figure BDA0001648996380000096
wherein
Figure BDA0001648996380000097
The instantaneous safe rate is shown in equation (9):
Figure BDA0001648996380000098
limited by the power of the transmitting end, the maximum safe rate is as shown in equation (10):
Figure BDA0001648996380000101
wherein, P is transmission power; r is the secret rate.
It is very difficult to know the channel state information in real time, and the time slot cannot guarantee that R is more than 0, and the interruption probability is introduced in the actual analysis. The maximum safe rate of introducing the probability of interruption is shown in equation (11):
Figure BDA0001648996380000102
pm,out∈(0,1]is the maximum interruption probability allowed;
the first step is as follows: transformation of optimization problem
Equation (11) is an optimization problem that maximizes the secret rate R with the transmission power P as a parameter, which is converted into a problem (12);
Figure BDA0001648996380000103
Figure BDA0001648996380000105
rank(W)≤2 (12c)
equation (12) is an optimization problem that minimizes the transmission power P with the secret rate R as a parameter;
the following constraints can be obtained by substituting (4), (8) into (12 a):
Figure BDA0001648996380000104
further decomposition gives the following formula:
Figure BDA0001648996380000111
the limitation (12a) is equivalent to the following formula:
Figure BDA0001648996380000112
second step semi-definite relaxation
Since the constraint (12c) is not convex. And removing the non-convex strip parts according to a semi-definite relaxation algorithm.
The probabilistic constraint is converted to a definite form using the Bernstein-Type inequality.
Order to
Figure BDA0001648996380000113
Wherein xh,m～CN(0,I)，xe～CN(0,I)，
Figure BDA0001648996380000114
The probabilistic constraint of the problem (12a) is expressed as:
Figure BDA0001648996380000115
wherein
Figure BDA0001648996380000116
Figure BDA0001648996380000117
Figure BDA0001648996380000118
The theory of the Bernstein-Type inequality: order to
Figure BDA0001648996380000119
Wherein the content of the first and second substances,
Figure BDA00016489963800001110
is a complex Hermitian matrix. For any sigma is greater than or equal to 0,
Figure BDA00016489963800001111
wherein s is-(A)＝max(λmax(-A),0)，λmax(-A) represents the largest eigenvector of matrix A;
the problem of equation (12) after semi-definite relaxation is finally transformed into the semi-definite programming problem by means of the Bernstein-Type inequality:
Figure BDA0001648996380000121
wherein sigmam＝ln(pm,out) Wherein u ismAnd vmIs a relaxation variable; the problem of the formula (13) is a convex problem. Let p bem,outIs given. Continuously adjusting R by Bayesian dichotomy*To obtain the optimum W.
Third step tightening scheme
We can obtain the maximum safe rate by constantly adjusting the value of the outage probability. Let p be p, assuming that all the interruption probabilities have the same value1,out＝...＝pM,outI.e. σ ═ σ1＝...＝σM. Let R beoptIs the solution of the maximum safe rate of equation (13).
Figure BDA0001648996380000122
The transmit power is minimized by successive solution (14) by bayesian bisection.
TABLE 3-1 Algorithm flow for finding the best W by dichotomy
Figure BDA0001648996380000123
Figure BDA0001648996380000131
Step four, solving the optimal beam forming vector
W*The optimal solution of equation (14) is shown.
When W is*When the rank of (1) is equal to w1The approximation of the optimal beamforming vector of (b) is W*The product of the evolution of the eigenvalue of (a) and the eigenvector; w is a2Is NtAnd a victory vector, all of which have a value of 0. When W is*When the rank of (2) is equal to w1And w2The approximation of the optimal beamforming vector of (b) is W*The evolution of the two eigenvalues of (a) is multiplied by the corresponding eigenvector.
However, when the number of users of Bobs increases, W*The probability of a rank of greater than 2 is increasing,this chapter uses a gaussian random approach to solve for two-dimensional beamforming vectors.
TABLE 3-2 two-dimensional Gaussian random algorithm
Figure BDA0001648996380000132
Figure BDA0001648996380000141
So far, the present invention has been described in detail with specific embodiments thereof.
The foregoing is a preferred embodiment of the present invention, and various modifications and substitutions can be made by those skilled in the art without departing from the technical principle of the present invention, and should be considered as the protection scope of the present invention.
Simulation experiment and effect analysis:
the simulation model parameters are as follows: antenna N of transmitting endtThe number of eavesdroppers Eve is 1, and the error covariance of the channel state information of the eavesdropping channel is 0.01. Noise power in information dissemination process
Figure BDA0001648996380000142
Fig. 2 shows the variation of the safe rate at different transmit end powers for four different schemes. It can be derived from the figure that the safe rate under robust two-dimensional beamforming design is optimal because two-dimensional beamforming has a higher degree of freedom than one-dimensional beamforming. The safe rate under the robust two-dimensional beamforming design is better than the safe rate under the non-robust (non-robust) beamforming design because the non-robust (non-robust) beamforming design does not take into account channel errors. The safe rate of the non-safe (Nonsecurity) beamforming design is worst, because this design does not take into account either the presence of an eavesdropper or the outage probability, i.e. the transmission rate cannot be larger than a given threshold at any moment.
Fig. 3 shows the variation of the safe rate for different channel errors by bob, where the transmit end power P is 10 dB. As can be seen from the figure, the safe rate is reduced under the four schemes as the error value of the channel state information is increased. Similarly, since two-dimensional beamforming has a higher degree of freedom than one-dimensional beamforming, the safe rate under the robust two-dimensional beamforming design is higher than the safe rate under the robust one-dimensional beamforming design. Channel errors are not considered in the non-robust beamforming design, so the safe rate under the design condition is less than that under the robust two-dimensional beamforming design. The non-secure beamforming design does not consider security and interruption, and the secure rate obtained under the design is the worst secure rate, i.e. all rates must be greater than a given threshold, so the secure rate under the design is the worst compared with the secure rates under the other three designs.
FIG. 4 shows W for different numbers of Bobs*And the distribution case of the rank, wherein the power P of the transmitting end is 10 dB. From the figure, W is shown as the number of Bobs increases*The probability of a rank greater than 2 is also increasing. As can be seen from the figure, W is 80% of the total number of Bobs (. gtoreq.20)*The rank is greater than 2.
Fig. 5 shows the variation of the safety rate for different numbers of Bobs, where P is 20 dB. Because the power value of the transmitting end is constant, the power received by the receiving end will decrease as the number of Bobs increases. Although the rate is reduced with the increase of the number of Bobs, the safe rate under the robust two-dimensional beamforming design is higher than that under the robust one-dimensional beamforming design due to the higher degree of freedom. Also the safe rate is lower in non-robust and non-safe designs that do not take into account channel errors and do not take into account safety and outage probabilities.

Claims (1)

1. The two-dimensional robust beamforming method under the MISO eavesdropping channel based on the interruption probability constraint is characterized by comprising the following steps of:
1) aiming at a MISO (MISO) eavesdropping channel model, an information sending end Alice combines Alamouti coding, namely two-dimensional beamforming, when beamforming is carried out on sent information, and the Alice only has estimated channel state information of a legal user Bobs and an eavesdropping user Eve; the method specifically comprises the following steps: the method is characterized in that Alice carries out two-dimensional beamforming design on signals sent by a sending end of two continuous time slots, and due to the property of Alamouti coding, the power sent by the two continuous time slots of Alice is the same, so that the signal-to-noise ratios of information received by Bobs and Eve are the same, and the maximum safe rate of the introduced interruption probability is shown as the following formula:
Figure FDA0002974050450000011
Figure FDA0002974050450000012
Tr(W)≤P，rank(W)≤2 (11)
its mountain R, W, hm、
Figure FDA0002974050450000013
g、
Figure FDA0002974050450000014
pm，outP is respectively a target secret rate variable to be optimized, a covariance matrix of a beamforming vector of Alice, channel state information of an mth Bob, noise power of the mth Bob, channel state information of Eve, noise power of Eve, an interruption probability threshold of the mth Bob and transmission power; wherein
Figure FDA0002974050450000015
w1And w2Is a two-dimensional beamforming column vector,
Figure FDA0002974050450000016
Ntis the number of antennas of Alice, M is the number of legitimate users, {1, 2, …, M } represents the index set of legitimate users Bobs；R*(P) represents the optimal target privacy rate solved by the problem (11) under the constraint of power P;
2) converting the non-convex problem of the maximum safe rate with the interruption probability constraint into a series of problems of minimizing the power of a transmitting end by utilizing a bisection method, a semi-definite relaxation and a Berstein-type inequality so as to obtain a covariance matrix of a beam forming vector of the transmitting end; when solving the problem of the maximum safe rate, the problem is converted into a series of problems of the minimum transmitting end power by means of a dichotomy, and the method comprises the following steps:
Figure FDA0002974050450000021
Figure FDA0002974050450000022
Figure FDA0002974050450000023
rank(W)≤2 (12c)
then removing the non-convex condition (12c) by a semi-definite relaxation algorithm;
the theory of the Bernstein-Type inequality: order to
Figure FDA0002974050450000024
Wherein the elements in the column vector x are independently identically distributed, each element obeying a standard complex Gaussian distribution,
Figure FDA0002974050450000025
a is a complex Hermitian matrix, a is a complex constant column vector, and for any sigma larger than or equal to 0, the following are provided:
Figure FDA0002974050450000026
where vec (a) denotes the column-wise accumulation of the elements of matrix a into a larger column vector,
s-(A)＝max(λmax(-A)，0)，λmax(-A) represents the largest eigenvector of matrix A;
the non-convex problem with the probability of interruption is converted into a convex problem as shown below:
Figure FDA0002974050450000027
Figure FDA0002974050450000028
Figure FDA0002974050450000029
Figure FDA00029740504500000210
Figure FDA00029740504500000211
wherein, let p be the same for all the interruption probability constraint valuesout＝pl，out＝...＝pM，out，σ＝σ1＝...＝σM，σ＝ln(pout)，umAnd vmFor the relaxation variable, here vec (A)m) Represents the matrix AmAre accumulated column by column into a larger column vector,
Figure FDA0002974050450000031
Figure FDA0002974050450000032
Figure FDA0002974050450000033
wherein Eb，mIs the covariance matrix of the mth Bob estimated channel, EeIs the covariance matrix of the Eve estimated channel;
in order to tighten the result, aiming at the parameter of the interruption probability, carrying out Bayesian dichotomy; by continuously adjusting the probability of interruption and R*To obtain the maximum safe rate and the covariance matrix W of the optimal beamforming vector★Bayesian dichotomy to find the best W★The algorithm flow of (1) is as follows:
1. the selection termination parameter xi > 0, lower bound sigmalAnd an upper bound σuLet σ ∈ [ ]l，σu](ii) a Selecting a termination parameter ε > 0, a lower bound RlAnd an upper bound RuLet R*∈[Rl，Ru]；
2. Let sigmamid＝(σl+σu)/2，σ＝σmid；
3. Let Rmid＝(Rl+Ru)/2，R*＝Rmid；
4. Will be sigma, R*Substituting (14) to solve the problem (14), if (14) is not feasible, let Ru＝RmidGo to step 6; otherwise, obtaining solution W and substituting probability constraint condition (12a) to check feasibility, if (12a) is not feasible, making Ru＝RmidGo to step 6; otherwise, go to step 5;
5. verifying whether the condition Tr (W) is not more than P, if so, making Rl＝RmidAnd W★W; otherwise let Ru＝Rmid；
6. If R isu-RlIf is greater than epsilon, turning to step 3; otherwise when R isu-RlWhen ≦ ε, if σ ≦ σmid，R*∈[Rl，Ru]The time problem (14) is solved to σu＝σmidElse, let σl＝σmid；
7. If σ isu-σlXi is not more than terminated, otherwise, the step 2 is switched to;
8. output W★；
3) Solving a beamforming vector according to the rank of the covariance matrix of the beamforming vector at the sending end, and solving a two-dimensional beamforming vector by means of a characteristic value decomposition method when the rank is 1 or 2; when the rank is more than 2, solving a two-dimensional beamforming vector by means of a Gaussian random variable method; the specific implementation method comprises the following steps:
when W is★When the rank of (1) is equal to w1The approximation of the optimal beamforming vector of (b) is W★The product of the evolution of the eigenvalue of (a) and the eigenvector; w is a2Is NtThe vector of dimension columns, the value of which is all 0; when W is★When the rank of (2) is equal to w1And w2The approximation of the optimal beamforming vector of (b) is W★The evolution of the two eigenvalues is respectively multiplied by the corresponding eigenvector;
when W is★When the rank is more than 2, solving the two-dimensional beamforming vector w by means of a Gaussian random method1And w2。
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