CN111224702A - Minimum regularization transmitting power beam forming method based on Lagrange multiplier - Google Patents

Minimum regularization transmitting power beam forming method based on Lagrange multiplier Download PDF

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CN111224702A
CN111224702A CN201911025873.7A CN201911025873A CN111224702A CN 111224702 A CN111224702 A CN 111224702A CN 201911025873 A CN201911025873 A CN 201911025873A CN 111224702 A CN111224702 A CN 111224702A
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beam forming
privacy information
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CN111224702B (en
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束锋
朱玲玲
邹骏
沈桐
李嘉钰
王云天
刘林
桂林卿
陆锦辉
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Nanjing University of Science and Technology
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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
    • H04L27/00—Modulated-carrier systems
    • H04L27/26—Systems using multi-frequency codes
    • H04L27/2601—Multicarrier modulation systems
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    • 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
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    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

The invention discloses a minimum regularization transmitting power beam forming method based on a Lagrange multiplier. The method comprises the following steps: constructing a linear OFDM subcarrier set by utilizing a random subcarrier selection method, randomly selecting N subcarriers from the linear OFDM subcarrier set, distributing the N subcarriers to N transmitting antennas, and decoupling the correlation between the beam forming direction and the distance; setting orthogonal constraint conditions and phase alignment constraint conditions which need to be met by beam forming; removing orthogonal constraint conditions by using a null space projection method; adding a regularization penalty term into the target function after the orthogonal constraint condition is removed; constructing a privacy information beam forming vector and an artificial noise beam forming vector by utilizing a Lagrange multiplier method; and utilizing two-dimensional search to obtain the optimal regularization factor in the privacy information beam forming vector and the artificial noise beam forming vector. The invention weakens the ability of a sensitive eavesdropper to intercept the private information, improves the security of wireless communication and network, and enhances the stability of a communication system.

Description

Minimum regularization transmitting power beam forming method based on Lagrange multiplier
Technical Field
The invention relates to the technical field of wireless communication, in particular to a minimum regularization transmitting power beam forming method based on a Lagrange multiplier.
Background
In recent years, physical layer security has become a hot spot of wireless communication and network research, and some techniques for physical layer security, such as relay cooperation, full duplex and directional modulation, are being developed. To improve the secure transmission in fading channels, spatial modulation schemes combining transmit antenna selection, privacy information beamforming design, and artificial noise projection design are proposed. Furthermore, directional modulation has also attracted much research in recent years. Daly et al propose a directional modulation technique using a phased array, the basic idea of which is that by phase shifting each antenna, the transmitted signal is transmitted in a given direction. Ding et al propose a new idea of synthesizing directional modulation signals at the baseband end, introducing artificial noise and using an orthogonal vector method of null-space projection to interfere interception of private information by an eavesdropper. Hu et al designs a low-complexity artificial noise projection matrix algorithm and generalizes it to imperfect channel state information scenarios. Subsequently, a robust directional modulation algorithm applied to security in broadcast, multicast, etc. scenarios was also proposed in succession. The directional modulation can be directly applied to millimeter wave and drone channels, providing alternative security solutions for future wireless networks, especially in 5G and drone networks.
As a version of directional modulation for enhancing the security, a secure and accurate transmission technology is combined with a random frequency diversity array and the directional modulation, so that the problem that the beam forming only depends on a direction angle and not on a distance in the directional modulation is solved. The problem is particularly that private information in a desired direction can be easily intercepted when an eavesdropper moves in the main lobe direction of the desired direction. The safe and accurate wireless transmission technology establishes the characteristic related to the range angle, and forms the main peak of the private information around the expected user, so that the energy leakage of the private information outside the main peak can be greatly reduced due to artificial noise, the energy leakage can be ignored, and the safe transmission of the private information is guaranteed. In order to reduce the complexity of a safe and accurate wireless transmission receiver, Shu et al propose a safe and accurate wireless transmission structure based on random subcarrier selection of OFDM, and Lin et al optimize frequency offset through a blocking continuous upper bound minimization algorithm to realize the maximum safe rate in the scene that an eavesdropper approaches a user. Furthermore, Qiu et al extend the application scenario of a single eavesdropper studied by Lin to multiple eavesdropping user scenarios, and introduce artificial noise to achieve secure transmission of private information.
However, the above security studies on accurate communication are performed from the perspective that the introduced artificial noise does not interfere with Bob, and the statistical interference of the artificial noise to the illegal eavesdropper is dispersive, and cannot achieve efficient concentration or suppression interference, so the interference effect is not strong enough. At this time, if the sensitivity of the receiver of the eavesdropper is high and the detection capability is strong, the private information can still be recovered, so that the purpose of eavesdropping is achieved.
Disclosure of Invention
The invention aims to provide a minimum regularization transmitting power beam forming method based on a Lagrange multiplier, which can improve the safety of wireless communication and a network and enhance the communication stability.
The technical solution for realizing the purpose of the invention is as follows: a minimum regularization transmitting power beam forming method based on a Lagrange multiplier comprises the following steps:
step 1, constructing a linear OFDM subcarrier set by using a random subcarrier selection method, randomly selecting N subcarriers in the constructed linear OFDM subcarrier set and distributing the N subcarriers to N transmitting antennas, and decoupling the correlation between the beam forming direction and the distance;
step 2, setting orthogonal constraint conditions and phase alignment constraint conditions which need to be met by beam forming;
step 3, removing orthogonal constraint conditions by using a null space projection method;
step 4, adding a regularization penalty term into the target function after the orthogonal constraint condition is removed;
step 5, constructing a privacy information beam forming vector and an artificial noise beam forming vector by utilizing a Lagrange multiplier method;
and 6, solving the optimal regularization factor in the privacy information beam forming vector and the artificial noise beam forming vector by utilizing two-dimensional search.
Compared with the prior art, the invention has the following remarkable advantages: (1) by combining the technologies of random subcarrier selection, directional modulation, beam forming and the like, a privacy information beam forming vector and an artificial noise beam forming vector which can ensure accurate interference and accurate communication are designed, a main privacy information energy peak is formed in an expected direction, and a main artificial noise energy peak is formed in an eavesdropping direction, so that the capability of a sensitive eavesdropper for intercepting privacy information is weakened, and the security of wireless communication and a network is improved; (2) the privacy information beam forming vector and the artificial noise beam forming vector meet orthogonal constraint, namely, privacy is hardly or rarely leaked to an eavesdropping area, and artificial noise hardly causes interference to an expected user, so that the expected user can be ensured to correctly receive and judge the privacy information; (3) orthogonal constraint conditions are removed by using a null space projection method, and then a privacy information beam forming vector and an artificial noise beam forming vector are constructed by using a Lagrange multiplier method to obtain a closed expression of the privacy information beam forming vector and the artificial noise beam forming vector, so that the system performance is improved, and the performance in the aspects of error rate and safety rate is improved.
Drawings
Fig. 1 is a schematic flow chart of the minimum regularization transmit power beamforming method based on the lagrangian multiplier according to the present invention.
Fig. 2 is a safety rate and snr change curve diagram of a minimum normalized transmit power beamforming algorithm, an unregulated minimum transmit power beamforming algorithm, and a conventional constant amplitude beamforming algorithm according to an embodiment of the present invention.
Fig. 3 is a bit error rate and signal-to-noise ratio variation curve diagram of the minimum regularized transmit power beamforming algorithm, the non-regularized minimum transmit power beamforming algorithm, and the conventional constant amplitude beamforming algorithm in the embodiment of the present invention.
Detailed Description
The invention discloses a minimum regularization transmitting power beam forming method based on a Lagrange multiplier, which comprises the following steps:
step 1, constructing a linear OFDM subcarrier set by using a random subcarrier selection method, randomly selecting N subcarriers in the constructed linear OFDM subcarrier set and distributing the N subcarriers to N transmitting antennas, and decoupling the correlation between the beam forming direction and the distance;
step 2, setting orthogonal constraint conditions and phase alignment constraint conditions which need to be met by beam forming;
step 3, removing orthogonal constraint conditions by using a null space projection method;
step 4, adding a regularization penalty term into the target function after the orthogonal constraint condition is removed;
step 5, constructing a privacy information beam forming vector and an artificial noise beam forming vector by utilizing a Lagrange multiplier method;
and 6, solving the optimal regularization factor in the privacy information beam forming vector and the artificial noise beam forming vector by utilizing two-dimensional search.
Further, the method of selecting by using random subcarriers described in step 1 constructs a linear OFDM subcarrier set, randomly selects N subcarriers in the constructed linear OFDM subcarrier set and allocates the subcarriers to N transmitting antennas, and decouples the correlation between the direction and the distance of beamforming, specifically as follows:
step 1.1, establishing a system model, wherein a base station Alice adopts an N-array element uniform linear array, an expected user Bob and an eavesdropping user Eve are both single-antenna receivers, a channel between a transmitter and a receiver is set to be a line-of-sight channel, and a normalized steering vector h (theta, R) of a transmitting antenna is as follows:
Figure BDA0002248602550000031
wherein N is 0,1,2, …, N-1 [ · or]TRepresenting the transpose operation, R, theta representing the angle, distance from the receiver to the transmitter, respectively, phin(theta, R) denotes the phase psi of the nth antenna relative to the reference antenna0A phase shift of (θ, R); setting Bob and Eve corresponding steering vectors as h (theta)B,RB) And h (θ)E,RE) Wherein thetaBAnd thetaEThe direction angles, R, of Bob and Eve, respectivelyB、RERespectively the distance from Alice to Bob and the distance from Alice to Eve;
the baseband signal is represented as:
s＝vCMx+vANz (7)
wherein x is privacy information, z is artificial noise, and respectively satisfies average power constraint, namely E [ | x shading2]＝1，E[|z|2]＝1；vCMAnd vANRespectively forming a privacy information beam forming vector and an artificial noise beam forming vector;
transmission through the channel, received signals y (θ) at Bob and EveB,RB)、y(θE,RE) Respectively as follows:
Figure BDA0002248602550000041
Figure BDA0002248602550000042
wherein [ ·]HWhich is representative of a conjugate transpose operation,
Figure BDA0002248602550000043
the path loss coefficients g from Alice to Bob and Alice to Eve0Is a reference distance; n isBAnd nEIs additive white Gaussian noise, obeysMean 0 and variance σ2Is a Gaussian distribution of
Figure BDA0002248602550000044
Step 1.2, constructing linear OFDM subcarrier set SsubComprises the following steps:
Ssub＝{fm|fm＝fc+mΔf,m＝0,1,…,NS-1} (10)
wherein N isSFor all sub-carrier numbers, f, in a linear OFDM sub-carrier setcFor the carrier frequency, Δ f is the subchannel bandwidth, and the system bandwidth is defined as B ═ NSΔ f, frequency increment and center carrier frequency satisfy NSΔf≤fc；fmIs a sub-carrier in a linear OFDM sub-carrier set;
step 1.3, randomly selecting N subcarriers from the constructed linear OFDM subcarrier set and distributing the subcarriers to N antenna array elements for sending privacy information and artificial noise to Bob and Eve;
setting fnFor the sub-carriers allocated to the n-th antenna, fn∈SsubOn the basis of the original carrier frequency, a random frequency increment is added to the carrier frequency of each antenna array element, so that a transmitted beam has direction angle-distance dependence, and privacy information can be transmitted to a given position, therefore, in the formula (1):
Figure BDA0002248602550000045
where d is the spacing of each element of the uniform linear array and c is the speed of light.
Further, the setting of the orthogonal constraint condition and the phase alignment constraint condition that the beamforming needs to satisfy in step 2 is specifically as follows:
step 2.1, expressing the optimization problem of the privacy information beam forming vector as follows:
Figure BDA0002248602550000046
step 2.2, expressing the optimization problem of the artificial noise beam forming vector as follows:
Figure BDA0002248602550000051
wherein v isCM、vANRespectively, a privacy information beamforming vector and an artificial noise beamforming vector, OC represents an orthogonal constraint, and PAC represents a phase alignment constraint.
Further, the method for removing the orthogonal constraint condition by using the null space projection in step 3 specifically includes the following steps:
step 3.1, orthogonal constraint h is carried out when privacy information beam forming vectors are designedH(θE,RE)vCMThe privacy information is transmitted along the null space of Eve as 0, so the orthogonal constraint, OC condition, is simplified as:
vCM＝(IN-h(θE,RE)hH(θE,RE))uCM(9)
wherein, INIs an identity matrix of dimension N, will uCMIn place of vCMAs a new optimization variable, the original problem of the privacy information beam forming vector optimization is converted into a constraint by two constraints; let A be IN-h(θE,RE)hH(θE,RE) Then v isCM＝AuCMV of formula (7)CMBy vCM＝AuCMAnd (3) all replacing to obtain a simplified privacy information beam forming vector optimization problem:
Figure BDA0002248602550000052
step 3.2, utilizing OC condition h when designing artificial noise beam forming vectorH(θB,RB)vANWhen v is equal to 0ANExpressed as:
vAN＝(IN-h(θB,RB)hH(θB,RB))uAN(11)
will uANAs a new optimization variable; let B be IN-h(θB,RB)hH(θB,RB) Then v isAN＝BuANSimplifying the artificial noise beamforming vector optimization problem of equation (8) as follows:
Figure BDA0002248602550000053
further, a regularization penalty term is added to the objective function after the orthogonal constraint condition is removed in step 4, which specifically includes the following steps:
step 4.1, adding a regularization penalty term into the objective function of the formula (10), and converting the optimization problem of the privacy information beam forming vector into the following steps:
Figure BDA0002248602550000061
wherein
Figure BDA0002248602550000062
As a regularizing term, γCMIs the corresponding regularization factor;
step 4.2, adding a regularization penalty term into the objective function of the formula (12), and replacing the optimization problem of the artificial noise beam forming vector by:
Figure BDA0002248602550000063
wherein gamma isANAnd forming a corresponding regularization factor for the artificial noise beam forming vector.
Further, the step 5 of constructing the privacy information beamforming vector and the artificial noise beamforming vector by using a lagrange multiplier method specifically includes the following steps:
step 5.1, solving the optimization problem of the formula (13) by using a Lagrange multiplier method to obtain the following Lagrange function:
Figure BDA0002248602550000064
let the first derivative of Lagrangian be zero, we get:
Figure BDA0002248602550000065
wherein [ ·]*Representing a conjugation operation, further yielding:
uCM＝-λ((AHA+γCMIN)-1)*AHh(θB,RB) (17)
wherein [ ·]-1Represents the inversion, and the lagrange multiplier λ is represented by substituting equation (17) back into equation (12):
Figure BDA0002248602550000066
according to formula (17) and formula (18):
Figure BDA0002248602550000067
thus, the expression of the privacy information beamforming vector is:
Figure BDA0002248602550000068
step 5.2, the expression of the artificial noise beam forming vector is as follows:
Figure BDA0002248602550000071
the invention is described in further detail below with reference to the figures and the specific examples.
Examples
With reference to fig. 1, the minimum regularization transmit power beamforming method based on the lagrangian multiplier of the present invention includes the following steps:
step 1, constructing a linear Orthogonal Frequency Division Multiplexing (OFDM) subcarrier set by using the idea of random subcarrier selection, randomly selecting N subcarriers in the constructed linear OFDM subcarrier set and distributing the N subcarriers to N transmitting antennas, and decoupling the correlation between the beam forming direction and the distance, specifically as follows:
step 1.1, establishing a system model of accurate interference and accurate communication, wherein a base station adopts an N-array element uniform linear array, an expected user (Bob) and an eavesdropping user (Eve) are both single-antenna receivers, a channel between a transmitter (Alice) and the receivers is set to be a line-of-sight channel, and the normalized guide vector of a transmitting antenna is as follows:
Figure BDA0002248602550000072
wherein N is 0,1,2, …, N-1 [ · or]TRepresenting the transpose operation, R and theta denote the angle and distance, ψ, from the receiver to the transmitter, respectivelyn(theta, R) denotes the phase ψ of the nth antenna with respect to the reference antenna0A phase shift of (θ, R); setting Bob and Eve corresponding steering vectors as h (theta)B,RB) And h (θ)E,RE) Wherein thetaBAnd thetaEThe direction angles, R, of Bob and Eve, respectivelyBAnd RERespectively the distance from Alice to Bob and the distance from Alice to Eve;
the baseband signal is represented as:
s＝vCMx+vANz (12)
wherein x is privacy information, z is artificial noise, and respectively satisfy average power constraint (E [ | x tint [ ]2]＝1，E[|z|2]＝1)；
vCMAnd vANRespectively forming a privacy information beam forming vector and an artificial noise beam forming vector;
the received signals at the desired user Bob and the eavesdropping user Eve, transmitted over the channel, are:
Figure BDA0002248602550000073
Figure BDA0002248602550000074
wherein [ ·]HWhich is representative of a conjugate transpose operation,
Figure BDA0002248602550000081
path loss coefficients, g, from Alice to Bob and Alice to Eve, respectively0For reference distance, here set to 1 meter; n isBAnd nEIs Additive White Gaussian Noise (AWGN), obeys a mean value of 0 and a variance of σ2Is a Gaussian distribution of
Figure BDA0002248602550000082
Because the minimum regularization transmitting power beam forming method based on the Lagrange multiplier method is used for achieving the purpose that privacy information is accurately transmitted to Bob and artificial noise accurately interferes with Eve, the beam forming design needs to start from the aspects of random subcarrier selection, constraint conditions, constraint condition simplification and introduction of regularization penalty terms, then privacy information beam forming vectors and artificial noise beam forming vectors are designed through the Lagrange multiplier method, and finally the optimal regularization factor is obtained through two-dimensional search;
step 1.2, constructing a linear OFDM subcarrier set as follows:
Ssub＝{fm|fm＝fc+mΔf,m＝0,1,…,NS-1} (15)
wherein N isSFor all sub-carrier numbers, f, in a linear OFDM sub-carrier setcIs the carrier frequency, Δ f is the subchannel bandwidth, and the system bandwidth is defined as B-NSΔ f, frequency increment and center carrier frequency satisfy NSΔf≤fc；
Step 1.3, randomly selecting N subcarriers from the constructed linear OFDM subcarrier set and distributing the subcarriers to N antenna array elements for sending privacy information and artificial noise to Bob and Eve;
setting fnFor the sub-carriers allocated to the n-th antenna, fn∈SsubOn the basis of the original carrier frequency, a random frequency increment is added to the carrier frequency of each antenna array element, so that a transmitted beam has direction angle-distance dependence, privacy information can be accurately transmitted to a given position, and therefore, the formula (1) is
Figure BDA0002248602550000083
d is the pitch of each element of the uniform linear array and c is the speed of light.
Step 2, setting two constraint conditions of Orthogonal Constraint (OC) and Phase Alignment (PAC) which need to be met by beam forming design, specifically as follows:
safe transmission and correct reception of information require that the privacy information beamforming vector and the artificial noise beamforming vector meet an orthogonal constraint condition; the integration of accurate interference and accurate communication requires that a privacy information beam forming vector and an artificial noise beam forming vector meet a phase alignment constraint condition;
step 2.1, expressing the optimization problem of the privacy information beam forming vector as follows:
Figure BDA0002248602550000091
step 2.2, expressing the optimization problem of the artificial noise beam forming vector as follows:
Figure BDA0002248602550000092
and 3, removing the orthogonal constraint condition by using a null space projection method, specifically as follows:
step 3.1, orthogonal constraint h is carried out when privacy information beam forming vectors are designedH(θE,RE)v CM0, transmitting the privacy information along the zero space of the Eve, so that the privacy information is hardly leaked to the Eve area; to simplify the optimization problem, the orthogonal constraint can be expressed as:
vCM＝(IN-h(θE,RE)hH(θE,RE))uCM(18)
wherein, INIs an identity matrix of dimension N, will uCMIn place of vCMAs a new optimization variable, the original problem of the privacy information beam forming vector optimization is converted into a constraint by two constraints; let A be IN-h(θE,RE)hH(θE,RE) Then v isCM＝AuCMV of formula (6)CMBy vCM＝AuCMAll substitutions, a simplified optimization problem is obtained:
Figure BDA0002248602550000093
step 3.2, similarly, when designing the artificial noise beam forming vector, orthogonal constraint h is utilizedH(θB,RB)vANWhen v is equal to 0ANExpressed as:
vAN＝(IN-h(θB,RB)hH(θB,RB))uAN(20)
will uANAs a new optimization variable; let B be IN-h(θB,RB)hH(θB,RB) Then v isAN＝BuANThe optimization problem of equation (7) is simplified to:
Figure BDA0002248602550000094
since A and B are both singular matrices, solving for uCMAnd uANThe process requires pseudo-inverse operations and therefore leads to some stability problems or performance loss.
Step 4, setting a regularization penalty term, and eliminating singular value and stability problems in optimization, wherein the regularization penalty term specifically comprises the following steps:
step 4.1, adding a regularization penalty term into the objective function of the formula (9), and converting the optimization problem of the privacy information beam forming vector into the following steps:
Figure BDA0002248602550000101
wherein
Figure BDA0002248602550000102
As a regularizing term, γCMIs the corresponding regularization factor;
step 4.2, similarly, adding a regularization penalty term into the objective function of the formula (11), and converting the optimization problem of the artificial noise beam forming vector into the following steps:
Figure BDA0002248602550000103
wherein gamma isANAnd forming a corresponding regularization factor for the artificial noise beam forming vector.
Step 5, designing corresponding privacy information beam forming vectors and artificial noise beam forming vectors by using a Lagrange multiplier method, wherein the method specifically comprises the following steps:
step 5.1, solving the optimization problem of the formula (12) by using a Lagrange multiplier method to obtain the following Lagrange function:
Figure BDA0002248602550000104
let the first derivative of Lagrangian be zero, we get:
Figure BDA0002248602550000105
wherein [ ·]*Representing a conjugation operation, one can obtain:
uCM＝-λ((AHA+γCMIN)-1)*AHh(θB,RB) (26)
wherein [ ·]-1Representing the inversion operation, substituting (16) back into (12), the lagrange multiplier λ can be expressed as:
Figure BDA0002248602550000106
from (16) and (17):
Figure BDA0002248602550000107
thus, the expression of the privacy information beamforming vector is:
Figure BDA0002248602550000111
step 5.2, obtaining an expression of the artificial noise beam forming vector by using the same method, wherein the expression comprises the following steps:
Figure BDA0002248602550000112
step 6, utilizing two-dimensional search to obtain the optimal regularization factor gamma in the private information beam forming vector and the artificial noise beam forming vectorCMAnd gammaAN。
Fig. 2 is a safety rate versus snr curve of the minimum normalized transmit power beamforming algorithm, the non-normalized minimum transmit power beamforming algorithm, and the conventional constant amplitude beamforming algorithm according to the present invention, wherein there are 8 total antennas of the transmitter. It can be seen from the figure that in the middle and high signal-to-noise ratio regions, the proposed minimum regularized transmit power algorithm is superior to the conventional constant-amplitude beamforming algorithm. As the signal-to-noise ratio increases, the gain in safe rate becomes more significant relative to conventional equal amplitude beamforming algorithms. When the signal-to-noise ratio is 20dB, compared with a constant-amplitude beamforming algorithm, the beamforming algorithm with the minimum regularized transmitting power improves the safety rate performance by nearly 50%.
Fig. 3 shows the bit error rate and signal-to-noise ratio variation curves of the minimum normalized transmit power beamforming algorithm, the non-normalized minimum transmit power beamforming algorithm, and the conventional constant amplitude beamforming algorithm according to the present invention. As can be seen from the figure, the bit error rate performance of the proposed beamforming algorithm with the minimum regularized transmission power is obviously superior to that of the traditional constant-amplitude beamforming algorithm when the signal-to-noise ratio is greater than 6 dB. In addition, as the signal-to-noise ratio is larger and larger, the bit error rate performance of the proposed beamforming algorithm with the minimum normalized transmission power is better and better than that of the beamforming algorithm with the minimum transmission power which is not normalized. The regularization factor is introduced into the minimum regularized transmit power beam forming algorithm, so that the loss caused by the singular problem of the non-regularized minimum transmit power beam forming algorithm is reduced, and the error rate performance is improved. The error rate performance of the three algorithms is sequentially ordered from good to bad as follows: the invention provides beam forming of minimum regularized transmit power, minimum transmit power beam forming without regularization and traditional constant amplitude beam forming.

Claims (6)

1. A minimum regularization transmit power beamforming method based on a Lagrange multiplier is characterized by comprising the following steps:
step 1, constructing a linear OFDM subcarrier set by using a random subcarrier selection method, randomly selecting N subcarriers in the constructed linear OFDM subcarrier set and distributing the N subcarriers to N transmitting antennas, and decoupling the correlation between the beam forming direction and the distance;
step 2, setting orthogonal constraint conditions and phase alignment constraint conditions which need to be met by beam forming;
step 3, removing orthogonal constraint conditions by using a null space projection method;
step 4, adding a regularization penalty term into the target function after the orthogonal constraint condition is removed;
step 5, constructing a privacy information beam forming vector and an artificial noise beam forming vector by utilizing a Lagrange multiplier method;
and 6, solving the optimal regularization factor in the privacy information beam forming vector and the artificial noise beam forming vector by utilizing two-dimensional search.
2. The minimum regularization transmit power beamforming method according to claim 1, wherein the method of using random subcarrier selection in step 1 constructs a linear OFDM subcarrier set, randomly selects N subcarriers from the constructed linear OFDM subcarrier set and allocates the N subcarriers to N transmit antennas, and decouples the correlation between the beamforming direction and the distance as follows:
step 1.1, establishing a system model, wherein a base station Alice adopts an N-array element uniform linear array, an expected user Bob and an eavesdropping user Eve are both single-antenna receivers, a channel between a transmitter and a receiver is set to be a line-of-sight channel, and a normalized steering vector h (theta, R) of a transmitting antenna is as follows:
Figure FDA0002248602540000011
wherein N is 0,1, …, N-1 [. cndot. ]]TRepresenting the transpose operation, R, theta representing the angle, distance from the receiver to the transmitter, respectively, phin(theta, R) denotes the phase psi of the nth antenna relative to the reference antenna0A phase shift of (θ, R); setting Bob and Eve corresponding steering vectors as h (theta)B,RB) And h (θ)E,RE) Wherein thetaBAnd thetaEThe direction angles, R, of Bob and Eve, respectivelyB、RERespectively the distance from Alice to Bob and the distance from Alice to Eve;
the baseband signal is represented as:
s＝vCMx+vANz (2)
wherein x is privacy information, z is artificial noise, and respectively satisfies average power constraint, namely E [ | x shading2]＝1，E[|z|2]＝1；vCMAnd vANRespectively forming a privacy information beam forming vector and an artificial noise beam forming vector;
transmission through the channel, received signals y (θ) at Bob and EveB,RB)、y(θE,RE) Respectively as follows:
Figure FDA0002248602540000021
Figure FDA0002248602540000022
wherein [ ·]HWhich is representative of a conjugate transpose operation,
Figure FDA0002248602540000023
the path loss coefficients g from Alice to Bob and Alice to Eve0Is a reference distance; n isBAnd nEIs additive white Gaussian noise, obeys mean value of 0 and variance of sigma2Is a Gaussian distribution of
Figure FDA0002248602540000024
Step 1.2, constructing linear OFDM subcarrier set SsubComprises the following steps:
Ssub＝{fm|fm＝fc+mΔf,m＝0,1,…,NS-1} (5)
wherein N isSFor all sub-carrier numbers, f, in a linear OFDM sub-carrier setcFor the carrier frequency, Δ f is the subchannel bandwidth, and the system bandwidth is defined as B ═ NSΔ f, frequency increment and center carrier frequency satisfy NSΔf≤fc；fmIs a sub-carrier in a linear OFDM sub-carrier set;
step 1.3, randomly selecting N subcarriers from the constructed linear OFDM subcarrier set and distributing the subcarriers to N antenna array elements for sending privacy information and artificial noise to Bob and Eve;
setting fnFor the sub-carriers allocated to the n-th antenna, fn∈SsubOn the basis of the original carrier frequency, a random frequency increment is added to the carrier frequency of each antenna array element, so that a transmitted beam has direction angle-distance dependence, and privacy information can be transmitted to a given position, therefore, in the formula (1):
Figure FDA0002248602540000025
where d is the spacing of each element of the uniform linear array and c is the speed of light.
3. The method according to claim 1, wherein the setting of the orthogonal constraint and the phase alignment constraint that the beamforming needs to satisfy in step 2 is as follows:
step 2.1, expressing the optimization problem of the privacy information beam forming vector as follows:
Figure FDA0002248602540000031
step 2.2, expressing the optimization problem of the artificial noise beam forming vector as follows:
Figure FDA0002248602540000032
wherein v isCM、vANRespectively, a privacy information beamforming vector and an artificial noise beamforming vector, OC represents an orthogonal constraint, and PAC represents a phase alignment constraint.
4. The method of claim 1, wherein the method of utilizing the null-space projection to remove the orthogonal constraint is as follows:
step 3.1, orthogonal constraint h is carried out when privacy information beam forming vectors are designedH(θE,RE)vCMThe privacy information is transmitted along the null space of Eve as 0, so the orthogonal constraint, OC condition, is simplified as:
vCM＝(IN-h(θE,RE)hH(θE,RE))uCM(9)
wherein, INIs an identity matrix of dimension N, will uCMIn place of vCMAs a new optimization variable, the original problem of the privacy information beam forming vector optimization is converted into a constraint by two constraints;
let A be IN-h(θE,RE)hH(θE,RE) Then v isCM＝AuCMV of formula (7)CMBy vCM＝AuCMAnd (3) all replacing to obtain a simplified privacy information beam forming vector optimization problem:
Figure FDA0002248602540000033
step 3.2, utilizing OC condition h when designing artificial noise beam forming vectorH(θB,RB)vANWhen v is equal to 0ANExpressed as:
vAN＝(IN-h(θB,RB)hH(θB,RB))uAN(11)
will uANAs a new optimization variable; let B be IN-h(θB,RB)hH(θB,RB) Then v isAN＝BuANSimplifying the artificial noise beamforming vector optimization problem of equation (8) as follows:
Figure FDA0002248602540000041
5. the minimum regularization transmit power beamforming method according to claim 4 based on the Lagrangian multiplier, wherein the regularization penalty term is added to the objective function after the orthogonal constraint condition is removed in step 4, specifically as follows:
step 4.1, adding a regularization penalty term into the objective function of the formula (10), and converting the optimization problem of the privacy information beam forming vector into the following steps:
Figure FDA0002248602540000042
wherein
Figure FDA0002248602540000043
As a regularizing term, γCMIs the corresponding regularization factor;
step 4.2, adding a regularization penalty term into the objective function of the formula (12), and replacing the optimization problem of the artificial noise beam forming vector by:
Figure FDA0002248602540000044
wherein gamma isANAnd forming a corresponding regularization factor for the artificial noise beam forming vector.
6. The Lagrangian multiplier based minimum regularization transmit power beamforming method according to claim 5, wherein the Lagrangian multiplier method used in step 5 is used to construct the privacy information beamforming vector and the artificial noise beamforming vector, specifically as follows:
step 5.1, solving the optimization problem of the formula (13) by using a Lagrange multiplier method to obtain the following Lagrange function:
Figure FDA0002248602540000045
let the first derivative of Lagrangian be zero, we get:
Figure FDA0002248602540000046
wherein [ ·]*Representing a conjugation operation, further yielding:
uCM＝-λ((AHA+γCMIN)-1)*AHh(θB,RB) (17)
wherein [ ·]-1Represents inversion, and the Lagrange product is obtained by substituting equation (17) back into equation (12)The sub λ is expressed as:
Figure FDA0002248602540000051
according to formula (17) and formula (18):
Figure FDA0002248602540000052
thus, the expression of the privacy information beamforming vector is:
Figure FDA0002248602540000053
step 5.2, the expression of the artificial noise beam forming vector is as follows:
Figure FDA0002248602540000054
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