CN106130611B - The codebook design method based on weighted Voronoi diagrams figure of passive wiretapping is fought in MISO system - Google Patents
The codebook design method based on weighted Voronoi diagrams figure of passive wiretapping is fought in MISO system Download PDFInfo
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Abstract
The codebook design method based on weighted Voronoi diagrams figure of passive wiretapping is fought in MISO system of the present invention, including, one with optimum beam figuration on theory significance and its corresponding traversal safe rate, as the benchmark for measuring security performance loss；Two calculate using the traversal safe rate after quantization wave beam forming, in conjunction with traversal safe rate, obtain traversal safe rate loss；Three according to traversal safe rate loss, defines and calculates traversal non-zero safe rate loss, and optimizes and obtain its upper bound and corresponding codebook design standard；Four, according to codebook design standard, construct the weighted Voronoi diagrams figure of complex field grassmann manifold spatially；Five, to minimize traversal non-zero safe rate loss as target, construct iterative algorithm VQ-WVD by training method under the line of iteration, realize the weighted Voronoi diagrams figure of complex field grassmann manifold spatially, complete the generation of code word, obtain the code book based on VQ-WVD.
Description
Technical Field
The invention belongs to the field of wireless communication, and particularly relates to codebook design mechanism research in a limited feedback MISO system oriented to physical layer security, in particular to a codebook design method based on a weighted Voronoi diagram for resisting passive eavesdropping in the MISO system.
Background
With the rapid development of wireless communication technology, security risks are gradually threatening and restricting the application and development of wireless communication technology. The broadcast nature of the wireless channel may cause a harmful signal source to eavesdrop on other nodes' confidential and private information or to jeopardize the communication links that interfere with other normal nodes. At present, the security of a wireless communication environment can be ensured to a certain extent by an upper-layer password encryption and decryption system, but with the development of computer technology, a harmful signal source often has strong calculation, eavesdropping and attacking capabilities, and even can completely decode the upper-layer encryption system, so that information of a legal receiver is completely exposed to the harmful signal source. For this reason, the physical layer security mechanism attracts a lot of attention and research, and the conventional physical layer security research is often based on perfect channel state information and performs corresponding mechanism design and security performance analysis. However, it is often difficult for an actual communication system to obtain perfect channel state information, and therefore, in recent years, physical layer security research oriented to physical layer security mechanism design of the actual communication system, especially a limited feedback system, has become a hot spot.
The limited feedback system usually requires that a legal transceiving end maintains normal information transmission operation, and the receiving end provides certain feedback information to the transmitting end, where the information carries a part of channel state information of the receiving end. Generally, a receiving end quantizes a self channel through a codebook known by a transmitting end and a receiving end, and transmits the part of information to a transmitting end through a low-rate feedback channel, and a transmitting end acquires feedback information, recovers a part of required channel state information, and is used for beamforming design of the transmitting end. However, such information poses challenges to the design of the transmitting end, especially causes inaccuracy of beamforming at the transmitting end, and further causes more user information to be leaked. When an eavesdropper appears around, the information can be acquired by the eavesdropper, and the security rate of the system is seriously influenced. Generally speaking, whether quantized information depends on a design method of a codebook accurately, a traditional codebook only considers a single-user channel model, and inevitable user information leakage is caused.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a codebook design method for resisting passive eavesdropping in a MISO system based on a weighted Voronoi diagram, which can improve the traversal security rate of the limited feedback MISO system without losing beam forming gain and increasing additional feedback overhead.
The invention is realized by the following technical scheme:
a method for weighted Voronoi diagram-based codebook design for protection against passive eavesdropping in MISO systems, comprising the steps of,
step1, obtaining theoretically optimal beam forming and corresponding traversal safety rate according to a received signal model in an MISO system, and taking the optimal beam forming and the corresponding traversal safety rate as a reference for measuring safety performance loss;
step2, calculating the traversal safety rate after the adoption of the quantized beam forming, and combining the traversal safety rate to obtain the traversal safety rate loss;
step3, defining and calculating traversal non-zero safe rate loss according to traversal safe rate loss, and optimizing to obtain the upper bound and corresponding codebook design standard;
step4, constructing a weighted Voronoi diagram on a complex field Grassmann flow pattern space according to a codebook design standard;
and 5, aiming at minimizing the traversing non-zero safe rate loss, constructing an iterative algorithm VQ-WVD in an iterative offline training mode, realizing a weighted Voronoi diagram on a complex field Grassmann flow pattern space, completing the generation of code words, and obtaining a codebook based on VQ-WVD.
Preferably, said MISO system comprises a legitimate transceiver pair and an eavesdropper, the transmitter having NtThe root antenna, the legal receiver and the eavesdropper all have single antennas, and the received signal models of the legal receiver and the eavesdropper are respectively as follows:
wherein, andthe channels from the transmitting end to a legal receiver and an eavesdropper are respectively, and the elements of the channels are subjected to circularly symmetric complex Gaussian distribution with the mean value of 0 and the variance of 1; sigmaL＝[1 0],ΣE＝[0 1](ii) a s is legal receiver information, and is subjected to circularly symmetric complex Gaussian distribution with a mean value of 0 and a variance of 1; n isL,nEAdditive white Gaussian noise is respectively applied to a legal receiver and a receiving end of an eavesdropper, and the cyclic symmetric complex Gaussian distribution with the mean value of 0 and the variance of 1 is obeyed; p is information transmission power and beam forming vectorThe feedback information is generated according to the feedback information of the legal receiver, the feedback information is obtained by the legal receiver quantizing the legal channel thereof through a codebook, and the signal-to-noise ratio of the legal receiver and the signal-to-noise ratio of an eavesdropper are respectivelyWhereinAndrespectively, the legitimate receiver and the receiver noise of the eavesdropper.
Further, in step1, according to a received signal model in the MISO system, an optimal beamforming w in a theoretical sense is obtainedoptAs follows below, the following description will be given,
its corresponding traversal security rateAs follows below, the following description will be given,
wherein, γLAnd gammaESignal-to-noise ratio, lambda, of legitimate receivers and eavesdroppers, respectivelymax(A(G,γL),B(G,γE) Is a matrix B-1The maximum characteristic value of a is set as,is a matrix B-1A maximum eigenvalue λmax(A, B) corresponding feature vectors; ΣL＝[1 0],ΣE＝[0 1]， andchannels from the transmitting end to a legitimate receiver and an eavesdropper, N, respectivelytThe number of the antennas at the transmitting end.
Still further, in step2, the quantized beamforming is represented as vL＝Q(wopt) (ii) a Wherein v isLIs the code word vector in the codebook, and Q is the quantization function.
Further, in step3, the expression of traversing non-zero safe rate loss is obtained as follows:
its upper boundComprises the following steps:
the codebook design criteria are:
wherein, C ═ λmax(A(G,γL))B(G,γE)，λmax(A(G,γL) Represents the matrix A (G, γ)L) The maximum eigenvalue of (c).
Further, in step4, according to the codebook design criteria, the specific steps of constructing a weighted Voronoi diagram in the complex field grassmann flow pattern space are as follows:
defining a one-dimensional complex-field Grassmann flow pattern according to codebook design criteriaThe space above is divided into:
wherein,||hLi represents hLMolding.Is thatAny vector of (a);
according to one-dimensional complex-domain Grassmann flow patternThe weighted Voronoi diagram on the complex field Grassmann flow pattern is obtained by the following method:
wherein,
is an additive weight, f (x) is a multiplicative weight, and is a monotonic function satisfying f (x) x γL||hL||2，
Furthermore, in the step5, with the aim of minimizing the traversing non-zero safe rate loss, the specific steps of constructing the iterative algorithm VQ-WVD in an iterative offline training mode are as follows,
step 5.1, initializing a codebook: v(0)＝{v(0),L,1,v(0),L,2...,v(0),L,N},N＝2B，
Wherein,i is 1, …, N represents the code word vector in the initial codebook, B represents the size of the codebook or the quantization bit, and the initial iteration number is set to k is 1;
step 5.2, solving each space division R in the Voronoi diagram weighted on the complex field Grassmann flow pattern(k),i(W(k-1)) The corresponding codeword vector is a vector of words of,
wherein,
step 5.3, iteratively updating the code words and the weights;
updating the code word; the code word v obtained by the k iteration is obtained according to the step 5.2(k),L,iN, obtaining the codebook V formed in the k iteration process(k)＝{v(k),L,1,v(k),L,2...,v(k),L,NAfter V(k)Substituting codebook V used in the k-1 iteration(k-1)；
Updating the weight; weight W generated with the kth iteration(k)In place of the firstWeight W used k-1 times(k-1)Whereineach weight in (1) is represented by v(k),L,iN is calculated as 1.;
step 5.4, when the iteration in the step 5.3 meets the iteration termination condition, the iteration is terminated, and the final codebook V of the current iteration is output(k)Otherwise, the iteration times are increased by 1, and the system carries out the next iteration until the iteration termination condition is met;
the iteration termination condition is as follows,
{S(v(k-1),L)-S(v(k),L)}/S(v(k),L)≤0.0005；
wherein: for a randomly generated training set, each GiAre all independent co-distributed channel matrices randomly generated in the training setT represents the number of channel matrices used for training.
Still further, in step5, the final codebook V generated according to the iterative algorithm VQ-WVD is V ═ V(k)The code word selection criteria for a legitimate receiver are designed as follows:
still further, also include the choice step of the legal receiver to the code word; the receiving end obtains the optimal code word vector through the designed code word selection standard according to the newly generated VQ-WVD codebookAnd feeds back the information to the transmitting terminal for improving the legal informationThe traversal safety rate of the information at the transmitting end.
Compared with the prior art, the invention has the following beneficial technical effects:
the invention solves the problem of low safe transmission rate of legal receiver information in the limited feedback MISO system caused by the stealing of information by an eavesdropper by redesigning the quantization codebook in the limited feedback MISO system. The method is based on a weighted Voronoi diagram theory, the statistical characteristics of a legal channel and an eavesdropping channel are integrated into codebook design, and a novel codebook is generated in a codebook training mode under a line according to the minimum traversal non-zero safe rate loss principle. Firstly, defining a QLES index, and obtaining a corresponding upper bound value; then, an effective codebook design standard is provided by minimizing the QLES upper bound value; by optimizing the standard, a weighted Voronoi diagram is further constructed on a one-dimensional complex field Grassmann flow pattern, and the rationality of the existence of the codebook is proved; finally, we propose a VQ-WVD algorithm to train and implement the codebook.
The invention aims at minimizing the traversing non-zero safe rate loss, and constructs a novel codebook based on VQ-WVD in an iterative offline training mode. Compared with the traditional codebook, the codebook can improve the traversal safety rate of a limited feedback MISO system under the conditions of not losing beamforming gain and not increasing extra feedback overhead.
Drawings
FIG. 1 is a schematic diagram of a system configuration in use of the method of the present invention.
Fig. 2 is a graph of the safe rate versus the signal-to-noise ratio of the legitimate receiver corresponding to different codebooks in the example of the present invention.
Figure 3 is a graph of the safe rate versus the legitimate receiver snr for the RVQ codebook and its corresponding modified codebook as described in the examples of the present invention.
Fig. 4 is a graph of system transmission rate versus legitimate receiver signal-to-noise ratio for the three codebooks and their corresponding modified codebooks in accordance with an embodiment of the present invention.
Detailed Description
The present invention will now be described in further detail with reference to specific examples, which are intended to be illustrative, but not limiting, of the invention.
The system model of the present invention is schematically illustrated in FIG. 1. the present invention is based on a legitimate transceiver pair and an eavesdropper, the transmitter having NtThe root antenna, the legal receiver and the eavesdropper all have single antennas, and the received signal models of the legal receiver and the eavesdropper are respectively as follows:
wherein, andthe channels from the transmitting end to a legal receiver and an eavesdropper are respectively, and the elements of the channels are subjected to circularly symmetric complex Gaussian distribution with the mean value of 0 and the variance of 1; sigmaL＝[1 0],ΣE＝[0 1](ii) a s is legal receiver information, and is subjected to circularly symmetric complex Gaussian distribution with a mean value of 0 and a variance of 1; n isL,nEAdditive white Gaussian noise is respectively applied to a legal receiver and a receiving end of an eavesdropper, and the cyclic symmetric complex Gaussian distribution with the mean value of 0 and the variance of 1 is obeyed; p is information transmission power and beam forming vectorGenerating according to the feedback information of a legal receiver; the feedback information is obtained by a legal receiver quantizing a legal channel per se through a codebook, and for convenient calculation, the signal-to-noise ratio of the legal receiver and the signal-to-noise ratio of an eavesdropper are respectively defined asWhereinAndrespectively, the legitimate receiver and the receiver noise of the eavesdropper.
The codebook design method of the present invention includes the steps of,
(1) according to the received signal model, the optimal wave beam shaping w in theory is solvedoptAnd its corresponding traversal safety rateThis is used as a benchmark to measure the loss of security performance.
(2) Computing beamforming with quantization vL＝Q(wopt) Post traversal security rateIncorporating traversal security ratesObtaining traversal safety rate lossThe theoretical expression of (1);
(3) according toDefining and calculating a traversal non-zero safe rate loss RQLESFurther optimizing to obtain the upper boundAnd obtaining a corresponding codebook design standard;
(4) constructing a weighted Voronoi graph on a complex field Grassmann flow pattern space according to a codebook design standard;
(5) and constructing an iterative algorithm VQ-WVD, and further realizing the weighted Voronoi graph, thereby completing the generation of code words and obtaining a codebook based on the VQ-WVD.
Specifically, the invention solves the theoretically optimal beamforming w according to the model of the received signal of the legal receiveroptAs follows:
its corresponding traversal security rateThe theoretical expression is as follows:
this is used as a benchmark to measure the loss of security performance. Wherein, γLAnd gammaESignal-to-noise ratio, lambda, of legitimate receivers and eavesdroppers, respectivelymax(A(G,γL),B(G,γE) Is a matrix B-1The maximum characteristic value of a is set as,is a matrix B-1A maximum eigenvalue λmaxAnd (A, B) corresponding feature vectors.
In order to measure the loss of the safe rate and construct a novel codebook, the codebook design method is carried out according to the following three processes:
1. defining traversal non-zero safe rate loss QLES;
for code word vector v in codebookLThe following design methods are defined:
vL＝Q(wopt)；
wherein Q is a quantization function,
ΣL＝[1 0],ΣE＝[0 1]，
andthe channels from the transmitting end to the legitimate receiver and the eavesdropper respectively,
Ntboth the legitimate receiver and the eavesdropper are single antennas for the number of antennas at the transmitting end.
Our goal is to follow woptBased on the principle of minimizing QLES, an algorithm is designed to calculate and realize a quantization function Q, and further the calculated code word and the corresponding codebook are obtained.
First, define traversal security rate loss:
secondly, by optimizing the traversal safety rate loss definition, the expression of the traversal non-zero safety rate loss is obtained as follows:
2. a codebook design criterion;
by optimizing RQLESTo obtain its upper boundComprises the following steps:
wherein, C ═ λmax(A(G,γL))B(G,γE)，λmax(A(G,γL) Represents the matrix A (G, γ)L) The maximum eigenvalue of (d);
further approximating this upper bound is:
finally, the codebook design criteria are:
3. a codebook design algorithm;
3.1, defining safety quality measure: to measure security performance, according to codebook design criteria, the following measure is defined to measure security performance:
3.2, codebook training algorithm: according to the codebook design standard and the corresponding safety quality measure, the codebook can be realized through iterative offline training, and the specific realization mode is as follows:
step 1: initializing a codebook: v(0)＝{v(0),L,1,v(0),L,2…,v(0),L,N},N＝2B，
Wherein,i is 1, …, N represents the codeword vector in the initial codebook, B represents the size of the codebook or the quantization bits, and the initial iteration number is set to k 1.
Step2, defining one-dimensional complex field Grassmann flow pattern according to codebook design standardThe space above is divided into:
wherein,||hLII denotes hLMolding.Is thatAny vector of (1).
Based on the above equation, a complex-field grassmann flow-type up-weighted Voronoi diagram can be constructed as follows:
wherein,
is an additive weight, f (x) is a multiplicative weight, and is a monotonic function satisfying f (x) x γL‖hL‖2，
Step 3. Each partitioned space R in a complex field Grassmann flow-type up-weighted Voronoi diagram(k),i(W(k-1)) The corresponding codeword vector:
by means of the simplification, the method has the advantages that,
wherein,
and Step4, updating the code words and the weights:
according to Step3, the code word obtained by the k iteration is v(k),L,iN, so we can get the k-th iteration formedCodebook V(k)＝{v(k),L,1,v(k),L,2…,v(k),L,NAfter V(k)Substituting codebook V used in the k-1 iteration(k-1)(ii) a Updating the weight: i.e. the weight W generated by the kth iteration(k)Weight W used instead of k-1(k-1)Wherein: w(k)＝{w(k),1…w(k),NTEach weight in (v) may be represented by v(k),L,iN is calculated as 1.
And Step5, calculating an iteration termination condition: in the actual iterative simulation of codebook generation, a training set needs to be randomly generated firstEach GiAre all independent co-distributed channel matrices randomly generated in the training setT represents the number of channel matrices used for training, all the legal receiver channels in the training set PThe set of compositions is divided into different R in Step2(k),i. Based on all legal receiver channels in training set PSet of components, we define the following iteration termination conditions:
{S(v(k-1),L)-S(v(k),L)}/S(v(k),L)≤0.0005；
wherein:
if the above conditions are met, the iteration is terminated, and the final codebook V of the current iteration is output(k)Otherwise, the iteration number is increased by 1, and the system carries out the next iteration until the termination condition is met.
3.3, code word selection algorithm: generating from an iterative algorithmIs the final codebook V ═ V(k)The code word selection criteria for a legitimate receiver are designed as follows:
finally, the choice of the codeword by the legitimate receiver: the receiving end adopts the designed code word selection standard according to the newly generated code book to obtain the optimal code word vectorAnd the information is fed back to the transmitting end, so that the traversal safety rate of the legal information at the transmitting end is further improved.
The simulation verification of the present invention is shown in fig. 2, 3 and 4, respectively, with the simulation parameters set as follows. The invention is based on T10000 signal channel training sets, and keeps the same signal-to-noise ratio between the eavesdropper and the legal receiver in the codebook generating process, namely gammaE＝γLWhile initiating codebook V(0)Respectively setting the code books as DFT, RVQ and Grassmann code books, correspondingly generating novel code books as DFT code book based on VQ-WVD, RVQ code book based on VQ-WVD and Grassmann code book based on VQ-WVD, setting the size of the code book as N2BAnd B is 4. The number of the transmitter antennas is set to be Nt＝4。
FIGS. 2 and 3 show that three VQ-WVD-based codebooks generated by the invention can effectively bring about the improvement of the traversal security rate of the limited-feedback MISO system, and the security rates under the traditional DFT, RVQ and Grassmann codebooks are drawn as reference. Fig. 4 shows that, compared with the conventional codebook, the three codebooks generated by the proposed codebook design method can slightly increase the average transmission rate of the system even in the absence of an eavesdropper, thereby increasing the beamforming gain in the traversal sense. The simulation results of fig. 2, 3 and 4 further show that compared with the conventional codebook, the novel codebooks can improve the traversal safety rate of the limited feedback MISO system without losing beamforming gain and adding additional feedback overhead.
Claims (1)
- Method for weighted Voronoi diagram-based codebook design against passive eavesdropping in a MISO system, characterized in that it comprises the steps of,step1, obtaining theoretically optimal beam forming and corresponding traversal safety rate according to a received signal model in an MISO system, and taking the optimal beam forming and the corresponding traversal safety rate as a reference for measuring safety performance loss;step2, calculating the traversal safety rate after the adoption of the quantized beam forming, and combining the traversal safety rate to obtain the traversal safety rate loss;step3, defining and calculating traversal non-zero safe rate loss according to traversal safe rate loss, and optimizing to obtain the upper bound and corresponding codebook design standard;step4, constructing a weighted Voronoi diagram on a complex field Grassmann flow pattern space according to a codebook design standard;step5, aiming at minimizing the traversing non-zero safe rate loss, constructing an iterative algorithm VQ-WVD in an iterative offline training mode, realizing a weighted Voronoi diagram on a complex field Grassmann flow pattern space, completing the generation of code words, and obtaining a codebook based on VQ-WVD; the method comprises the following specific steps;3.1, defining safety quality measure: to measure security performance, according to codebook design criteria, the following measure is defined to measure security performance:3.2, codebook training algorithm: according to the codebook design standard and the corresponding safety quality measure, the codebook can be realized through iterative offline training, and the specific realization mode is as follows:step 1: initializing a codebook: v(0)＝{v(0),L,1,v(0),L,2...,v(0),L,N},N＝2b，Wherein,i is 1, …, N represents the code word vector in the initial code book, b is tableShown as the size of the codebook or quantization bits, the initial number of iterations is set to k-1;step2 defining one-dimensional complex-field Grassmann flow pattern (N) according to codebook design standardtAnd 1) space division:wherein,||hLi represents hLMolding of (2);is (N)tAny vector in 1);based on the above equation, a complex-field grassmann flow-type up-weighted Voronoi diagram can be constructed as follows:wherein,is an additive weight, f (x) is a multiplicative weight, and is a monotonic function satisfying f (x) x γL||hL||2，Step3 Complex field Grassmann flow-type upward weighted Voronoi diagram with each partitioned space R(k),i(W(k-1)) The corresponding codeword vector:by means of the simplification, the method has the advantages that,wherein,update codewords and weights, Step4:according to Step3, the code word obtained in the k iteration is v(k),L,iN, so we can get the codebook V formed during the kth iteration(k)＝{v(k),L,1,v(k),L,2…,v(k),L,NAfter V(k)Substituting codebook V used in the k-1 iteration(k-1)(ii) a Updating the weight: i.e. the weight W generated by the kth iteration(k)Weight W used instead of k-1(k-1)Wherein:each weight in (1) may be represented by v(k),L,iN is calculated as 1.;step5 calculating iteration termination conditions: in the actual iterative simulation of codebook generation, a training set needs to be randomly generated firstEach GiAre all independent co-distributed channel matrices randomly generated in the training setT represents the number of channel matrices used for training, all legal receiver channels in the training setThe set of compositions is divided into different R in Step2(k),i(ii) a Based on all legal receiver channels in the training setSet of components, we define the following iteration termination conditions:{S(v(k-1),L)-S(v(k),L)}/S(v(k),L)≤0.0005；wherein:if the above conditions are met, the iteration is terminated, and the final codebook V of the current iteration is output(k)Otherwise, the iteration times are increased by 1, and the system carries out the next iteration until the termination condition is met;3.3, code word selection algorithm: generating a final codebook V ═ V according to an iterative algorithm(k)The code word selection criteria for a legitimate receiver are designed as follows:finally, the choice of the codeword by the legitimate receiver: the receiving end adopts the designed code word selection standard according to the newly generated code book to obtain the optimal code word vectorThe information is fed back to the transmitting end to further improve the traversal safety rate of legal information at the transmitting end;said MISO system comprising a legitimate transceiver pair and an eavesdropper, the transmitter having NtThe root antenna, the legal receiver and the eavesdropper all have single antennas, and the received signal models of the legal receiver and the eavesdropper are respectively as follows:wherein, andthe channels from the transmitting end to a legal receiver and an eavesdropper are respectively, and the elements of the channels are subjected to circularly symmetric complex Gaussian distribution with the mean value of 0 and the variance of 1; sigmaL＝[1 0],ΣE＝[0 1](ii) a s is legal receiver information, and is subjected to circularly symmetric complex Gaussian distribution with a mean value of 0 and a variance of 1; n isL,nEAdditive white Gaussian noise is respectively applied to a legal receiver and a receiving end of an eavesdropper, and the cyclic symmetric complex Gaussian distribution with the mean value of 0 and the variance of 1 is obeyed; p is information transmission power and beam forming vectorThe feedback information is generated according to the feedback information of the legal receiver, the feedback information is obtained by the legal receiver quantizing the legal channel thereof through a codebook, and the signal-to-noise ratio of the legal receiver and the signal-to-noise ratio of an eavesdropper are respectivelyWhereinAndthe noise of a legal receiver and the noise of a receiving end of an eavesdropper are respectively received;in step1, according to a received signal model in a MISO system, the theoretically optimal beamforming w is obtainedoptAs follows below, the following description will be given,its corresponding traversal security rateAs follows below, the following description will be given,wherein, γLAnd gammaESignal-to-noise ratio, lambda, of legitimate receivers and eavesdroppers, respectivelymax(A(G,γL),B(G,γE) Is a matrix B (G, γ)E)-1A(G,γL) Is determined by the maximum characteristic value of the image,is matrix B (G, γ)E)-1A(G,γL) Maximum eigenvalue λmax(A(G,γL),B(G,γE) Corresponding feature vectors; ΣL＝[1 0],ΣE＝[0 1]， andchannels from the transmitting end to a legitimate receiver and an eavesdropper, N, respectivelytThe number of antennas at the transmitting end;in step2, quantized beamformingIs represented by vL＝Q(wopt) (ii) a Wherein v isLA code word vector in a codebook is taken, and Q is a quantization function;in step3, the expression for traversing the non-zero safe rate loss is obtained as follows:its upper boundComprises the following steps:the codebook design criteria are:wherein, C ═ λmax(A(G,γL))B(G,γE)，λmax(A(G,γL) Represents the matrix A (G, γ)L) The maximum eigenvalue of (c).
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