CN113055064B - Steady beam forming design method for hidden communication of Internet of things - Google Patents

Abstract

The invention provides a robust beam forming design method for hidden communication of the Internet of things, which considers hidden beam forming design for an Intelligent Reflector (IRS) assisted Internet of things (IoT) network, researches the design of a joint beam forming device of Alice and can improve the hiding rate of Bob to the maximum extent when Alice does not completely know Willie Channel State Information (WCSI). For WCSI in a non-ideal state, an optimal decision threshold of Willie is deduced, and false alarm and missed detection probabilities in the case are analyzed. Furthermore, in this case, a robust beamformer based on relaxation, S-procedure and alternate iteration methods is built, taking advantage of the property of Kullback-Leibler dispersion.

Description

Steady beam forming design method for hidden communication of Internet of things

Technical Field

The invention relates to a robust beam forming design method for covert communication of the Internet of things.

Background

In the past few years, the internet of things (IoT) has been widely used in various fields of industry, agriculture, medicine, and the like. The number of intelligent communication devices has increased explosively, and data-hungry wireless applications have steadily increased, which requires internet-of-things networks to have higher spectrum and energy efficiency (cited documents: s.gong, x.lu, d.t.hoang, d.niyato, l.shu, d.i.kim, and y.c.liang, "heated small wireless communication vision in the illuminating surface: a coordinated overview," IEEE com.surv.turbine, vol.22, No.4, pp.2283-2314,2020.). Fortunately, Intelligent Reflective Surfaces (IRS) have recently been identified as a promising solution that can improve the spectral and energy efficiency of wireless networks by reconfiguring the wireless propagation environment.

IRS, also known as reconfigurable intelligent surface, is receiving wide attention in wireless communication applications. In particular, IRS is a planar surface consisting of a large number of low-cost passive reflective elements, each of which can independently reshape the phase, amplitude and reflection angle of an incident signal (cited in Q. Wu and R. Zhang, "transmitted small and reliable environment: Intelligent reflecting surface available wireless network," IEEE Commin. Magazine, vol.58, No.1, pp.106-112,2020 "), thereby intelligently adjusting the propagation channel to serve its own target. For example, signals reflected by IRS may be added or summed with signals reflected by non-IRS at the receiver to enhance the desired signal or suppress the undesired signal by adaptively adjusting the phase shift of the reflecting elements (cited references: X.Tan, Z.Sun, J.M.Jornet, and D.Pados, "incoming antenna reflected signal-array," in Proc. IEEE International Conference on Communications (ICC), 2016.).

The IRS auxiliary Internet of Things has the advantages of low hardware cost, low power consumption, simple structure and the like, and improves the quality of received signals by the unique electromagnetic characteristics (such as negative refraction, cited documents: G.Y u, X.Chen, C.ZHong, D.W.KWan Ng, and Z.Zhang, "Design, analysis, and optimization of a large intersecting reflecting surface-aid B5G cellular Internet of Things," IEEE Internet Things J., vol.7, No.9, pp.8902-8916,2020.).

However, due to the broadcast nature of wireless communication, IRS helps the internet of things to be easily eavesdropped, especially in some public areas such as classrooms, shopping malls and libraries. Recently, a large number of researchers have studied the safety of physical layers (cited document: M. Cui, G. Zhang, and R. Zhang, "Current Wireless communication via interacting communication interface," IEEE Wireless communication interface, "volume.8, No.5, pp.1410-1414,2019.) (cited document: L.Dong and H.Wang," Current MIMO transmission interface, "IEEE Wireless communication interface," volume.9, No.6, pp.787-790,2020.) (cited document: X.Y.u, D.xu, Y.Sun, D.W.K.Ng, and R.Scherer, "Rost and MIMO communication interface," MIMO.S.J.S.P.J.12, "Heart communication interface, C.S.J.1, C.S.12, C.S.N.N.N.N.P.N.N.P.C.S.S.J.S.C.S.C.S.C.C.C.C.S.S.C.C.C.S.C.S.S.C.S.S.S.S.S.S.C.S.S.S.S.S.S.S.S.S.S.C.S.S.S.S.S.S.S.S.C.S.S.S.S.S.S.C.S.S.S.S.S.S.S.S.S.S.No. Pat. No.6, C.S.S.S.S.S.S.No. Pat. No. 7.S.No. 7.S. No. 7.S.S.No. 7.S.S.S.S.S.S.S. No. 7.S. 7.S.S.S. 7.S. Pat. No. 7.S. No. 7. No. Pat. 7. No. 7. No.6, and S. 7. D.S.S.S.S. No. 7. D.S. No.7, S. K.S. K.S.S. K.S. K.S.S.S. K.S. K. 4, and R.S. K.S. 4, and R.S. S. K.S. K.S.S.S. K.S. S. K.S. S. K.S. K. K.S. S. of the cited document," S. K.S. S. K.S. S. S.S.S. S. K.S. K. K.S. S. K.S. S. S.S. S. K.S. S., "IEEE Trans. Inf. F organs Security, vol.16, pp.1655-1669,2021.) and covert communications (cited documents: lu, e.hossain, t.shafique, s.feng, h.jiang, and d.niyato, "Intelligent reflecting surface enabled coverage communications in wireless networks," IEEE net, vol.34, No.5, pp.148-155,2020 "). Physical Layer Security is primarily to prevent the transmitted wireless signal form From being decoded by an undesired user (cited documents: m.bloch and j.bars, Physical-Layer Security From Information Security to Security Engineering, u.k.: Cambridge univ, 2011), while covert communication is to hide the wireless signal From being discovered by an eavesdropper. In the literature: in m.cui, g.zhang, and r.zhang, "Secure Wireless communication view in reflecting surface," IEEE Wireless communication.lett., vol.8, No.5, pp.1410-1414,2019, the IRS may adaptively adjust the phase shift of the reflecting unit to enhance the desired signal and suppress the undesired signal, thereby maximally improving the security ratio. Researchers consider an IRS-assisted gaussian multiple-input multiple-output (MIMO) listening channel (cited document: l.dong and h.wang, "Secure MIMO transmission via adaptive redirection surface," IEEE Wireless communication.lett., vol.9, No.6, pp.787-790,2020.), whereas in documents: in x.y u, d.xu, y.sun, d.w.k.ng, and r.schober, "Robust and secure wireless communications via intersecting deflecting surfaces," IEEE j.sel.areas communications, vol.38, No.11, pp.2637-2652,2020 "), the joint design of a beamformer, an artificial noise covariance matrix, and a phase shifter in an infrared receiver was studied, and the influence of incomplete Channel State Information (CSI) of a intercepted channel was considered. By using IRS to improve safety performance, researchers are in the literature: one block coordinate descent optimization minimization (BCDMM) algorithm for MIMO secure communication systems is proposed in s.hong, c.pan, h.ren, k.wang, and a.nalalanathan, "architecture-noise-aided secure MIMO wireless communication view intersection deflecting surface," IEEE trans.communication ", vol.68, No.12, pp.7851-7866,2020. Furthermore, there are documents z.chu, w.hao, p.xiao, d.mi, z.liu, m.khalily, j.r.kelly, and a.p.fereidis, "correlation rate optimization for interpolating surface-applied MIMO systems," IEEE trans.inf.f. identities, vol.16, pp.1655-1669,2021 derive closed-form expressions of the Security and interference precoders by the Weighted Minimum Mean Square Error (WMMSE) algorithm and the Karush-Kuhn-tuner (kkt) condition, and derive phase shifts by the MM algorithm to obtain closed-form solutions thereof.

Furthermore, as evolving wireless systems face more and more security threats, even if the transmitted information is encrypted, the potential eavesdropping path is physically restricted, and the original data itself may expose confidential information. In the literature: lu, e.hossain, t.shafique, s.feng, h.jiang, and d.niyato, "Intelligent deflecting surface enabled covert communications in wireless networks," IEEE net, vol.34, No.5, pp.148-155,2020, the authors introduce covert communications techniques with IRS.

Disclosure of Invention

The purpose of the invention is as follows: in order to solve the technical problems in the background art, the invention provides a robust beam forming design method for covert communication of the Internet of things, which comprises the following steps:

step 1, establishing a covert communication environment;

and step 2, under the condition of imperfect WCSI (Willie channel state information), carrying out concealed beam forming design.

Has the advantages that: for the imperfect WCSI case, in the case of imperfect concealment constraints, the optimal threshold for Willie detection is derived and the corresponding detection error probability is proposed based on the robust beamforming vector. This result can be used as a theoretical basis for evaluating the concealment performance of the beamformer design. Furthermore, when Alice's WCSI is imperfect, the joint design of robust beamformer and IRS reflection beamformer targeting maximum achievable rates is studied and takes into account perfect blind transmission constraints, Alice's total transmit power constraints and IRS's QoS.

Drawings

The foregoing and/or other advantages of the invention will become further apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.

Fig. 1 is a schematic diagram of a covert communication scenario.

FIG. 2a shows a concealment threshold 2 ε according to the present invention²0.02, CSI error υ_w＝2×10^-4Then, D (p)₀||p₁) And the cumulative density function CDF.

FIG. 2b shows the concealment threshold 2 ε according to the present invention²0.02, CSI error υ_w＝2×10^-4Then, D (p)₀||p₁) And the cumulative density function CDF.

FIG. 3a shows the proposed epsilon value and concealment rate R_bAnd (5) a relational graph.

FIG. 3b shows the epsilon value and the probability of detection error according to the present invention

And (5) a relational graph.

FIG. 4a is a diagram of the proposed concealment rate R of the present invention_bAnd CSI error v_wAnd (5) a relational graph.

FIG. 4b is a schematic diagram of the false alarm probability of the present invention

And probability of missed detection

And CSI error v_wAnd (5) a relational graph.

FIG. 5 is a graph of the coverage rate R proposed by the present invention_bAnd CSI error v_w＝2×10^-4The relationship of the number of antennas N is shown schematically.

Detailed Description

In the invention, the following representation method is adopted: black bodyLower case letters and upper case letters represent vectors and matrices, respectively.

| | |, Tr (·), Re (·) and Im (·) represent the desired, Frobenius norm, trace, real and imaginary parts of the parameter, respectively. Operator

Indicating that a is semi-positive.

Expressed as mean μ and variance σ²A complex value circularly symmetric gaussian distribution.

The scenario considered by the present invention is shown in fig. 1, where Alice (base station) will transmit a private data stream x_bSent to Bob (hidden user). The invention provides a robust beam forming design method for covert communication of the Internet of things, which comprises the following steps:

step 1, establishing a covert communication environment;

and step 2, under the condition of imperfect WCSI (Willie channel state information), carrying out concealed beam forming design.

The step 1 comprises the following steps: alice represents the base station, Willie represents the eavesdropper, Bob represents the hidden user, and Alice represents the private data stream x_bIs sent to Bob, where

Representing a null hypothesis, that is, Alice does not send a private data stream to Bob;

represents another assumption that Alice sends a private data stream to Bob;

meanwhile, Willie is observing the communication environment as an eavesdropper and tries to identify whether Alice is transmitting to Bob; in order to protect the confidential signals from eavesdropping, an Intelligent Reflection Surface (IRS) with an intelligent controller is adopted to assist in covert transmission.

In step 1, it is set that Alice is provided withN antennas, wherein N is a natural number; bob and Willie each have an antenna; order to

Representing a signal x_b(x_bSignal sent by Alice to Bob); make it

Wherein the content of the first and second substances,

representing a set of complex matrices; h is_ABIs an Nx 1 complex matrix, which means the channel coefficient from Alice to Bob; h is_AWIs an Nx 1 complex matrix, which means the channel coefficients from Alice to Willie; h is_IBIs an Mx 1 complex matrix, which means IRS to Bob channel coefficients; h is_IWIs an M multiplied by 1 complex matrix, which refers to IRS to Willie channel coefficients;

by using

Representing the channel coefficients from Alice to IRS;

by using

Events representing the sending of information by Alice to Bob, using

An event indicating that Alice does not send information to Bob.

In step 1, from Willie's perspective, Alice's transmission signal x is as follows:

wherein w_bIs x_bSets Alice at

No signal is transmitted at the bottom and the beamforming vector w_bIn that

The following constraints are satisfied:

||w_b||²≤P_total (2)

in the formula P_totalIs the maximum transmit power of Alice;

the phase shift matrix Q is determined by

Given that Q represents a phase shift matrix whose diagonal elements are the corresponding elements Q of the vector; use of

Modeling the reflection of IRS units, wherein

q_mReflecting model of the M-th unit, j is an imaginary number, and when M is 1, … M, theta_mE [0,2 π) and β_m∈[0,1]Respectively representing the controllable phase shift and the amplitude reflection coefficient introduced by the mth unit;

is provided with

To achieve maximum reflected power gain, q should satisfy:

|q_m|＝1,m＝1,…M (3)

due to severe path loss, the signal reflected by IRS is ignored twice or more, the signal y received at Bob_bWriting into:

wherein z is_bFor the received noise at Bob to be the received noise,

signal noise z representing Bob_bObeying a mean of 0 and a variance of

Complex gaussian distribution. h is_IBIs the channel coefficient from IRS to Bob,

is h_IBThe conjugate transpose of (1); h is_ABThe channel coefficients for Alice to Bob are,

is h_ABThe conjugate transpose of (c).

In step 1, Willie receives signal y_wWriting into:

wherein z is_wIs the noise received at Willie and,

signal noise z representing Willie_wObeying a mean of 0 and a variance of

Complex gaussian distribution. h is_IWIs the channel coefficient from IRS to Willie,

is h_IWThe conjugate transpose of (1); h is_AWFor the Alice to Willie channel coefficients,

is h_AWThe conjugate transpose of (c).

In step 1, R is set_bIs a null hypothesis

The instantaneous rate of the next Bob, written as:

willie in

And

the likelihood functions of the lower received signals are respectively expressed as p₀(y_w) And p₁(y_w)；

According to formula (5), p₀(y_w) And p₁(y_w) Respectively as follows:

wherein

In

Signal noise z representing Willie_wOf the noise variance, λ₀And λ₁Representing the auxiliary variable.

In step 1, Willie wants to minimize the detection error probability ξ by applying the optimal detector, setting:

ξ＝1-V_T(p₀,p₁), (8)

wherein V_T(p₀,p₁) Is p₀(y_w) And p₁(y_w) Total change therebetween, usingThe Pinsker inequality (Flat-Stark inequality) (ref. T.M.Cover and J.A.Thomas, Elements of Information Theory, New York: Wiley,2006.) yields:

wherein D (p)₀||p₁) Represents from p₀(y_w) To p₁(y_w) KL (Kullback-Leibler) divergence (Kullback-Leibler divergence relative entropy), D (p)₁||p₀) Is from p₁(y_w) To p₀(y_w) KL divergence of (1);

D(p₀||p₁) And D (p)₁||p₀) Respectively as follows:

to achieve implicit communication with a given ξ, i.e., ξ ≧ 1- ε, the KL divergence of the likelihood function satisfies one of the following constraints:

D(p₀||p₁)≤2ε², (11a)

D(p₁||p₀)≤2ε² (11b)。

the step 2 comprises the following steps: the imperfect WCSI case refers to: willie is an ordinary user (i.e., Willie is not a legitimate user), Alice does not know channel state information about Willie, and wants to acquire personal information of Bob, in which case Alice is a passive gatekeeper (references D. goeckel, b. base, s. guha, and D. towsley, "cover communications places side not to point the background noise power," IEEE communications. let, vol.20, No.2, pp.236-239, feb.2016) and channel estimation errors (references m.forouzesh, p.azmi, n.mokari, and D. goeckel, "cover ni-use node and 3D. waiting information," CSI, 8568, IEEE 8525. 858, and No. 25. 12. c.25. for perfect;

imperfect WCSI is modeled as:

and:

wherein h is_AWIs the channel coefficient h from Alice to Willie_IWRefer to the IRS to Willie channel coefficients;

and

respectively representing a CSI (channel state information) estimation vector between Alice and Willie and a CSI estimation vector between Willie and IRS;

Δh_AWrepresenting the CSI error vectors of Alice to Willie;

Δh_IWa CSI error vector representing IRS to Willie;

CSI error vector Δ h_AWAnd Δ h_IWIs characterized by an elliptical area, namely:

and

wherein epsilon_AWA range expression referring to the CSI error vector from Alice to Willie;

represents Δ h_AWThe conjugate transpose of (1); epsilon_IWRefers to the range representation of the IRS to Willie CSI error vector;

represents Δ h_IWThe conjugate transpose of (1);

axis of control ellipsoid, v_AW>0,υ_IW>0 determines the volume of an ellipsoid (ref. x.y u, d.xu, y.sun, d.w.k.ng, and r.schober, "Robust and secure wireless communication via intersecting reflecting surfaces," IEEE j.sel.areas communication, vol.38, No.11, pp.2637-2652,2020.);

willies assay performance:

for the imperfect WCSI condition, Willie's optimal decision threshold is studied, and corresponding false alarm and missed detection probabilities are given. In which the worst case of blind transmission is considered, in which case Willie knows the beamformer w_b。

According to the Neyman-Pearson criterion (ref. E.L.Lehmann and J.P.Romano, Testing Statistical Hypotheses, spring New Y ork,2005.), the optimal rule for Willie to minimize detection errors is the likelihood ratio test (ref. E.L.Lehmann and J.P.Romano, Testing Statistical Hypotheses, spring New Y ork, 2005.):

according to

And

are respectively corresponding hypothesis

And

the binary decision of equation (16) is rewritten as:

wherein phi^*Is y_w|²The optimal detection threshold of (a) is as follows:

λ₀and λ₁Dependent on the beamforming vector w_bAnd an IRS reflected beam forming vector q;

and

lower y_w|²Respectively, as the cumulative density function CDFs of

Based on optimal detection threshold phi^*GiveFalse alarm

And probability of missed detection

Comprises the following steps:

by using

And

the expression of (a) to characterize the ideal detection performance of Willie;

D(p₀||p₁)≤2ε²the case (2) is as follows:

in practical applications, the obtained CSI is often corrupted by certain estimation errors (references l.wang, w.wornell, and l.zheng, "Fundamental limits of communication with low mobility of detection," IEEE trans. inf.theory, vol.62, No.6, pp.3493-3503, jun.2016.). Therefore, a robust beamforming design is proposed to maximize Bob's concealment rate R_b. In this case, it is difficult to achieve perfect blind transmission, i.e., D (p)₀||p₁) 0. On this basis, D (p) given by (11) is employed₀||p₁)≤2ε²And D (p)₁||p₀)≤2ε²As covert constraints (references s.yan, y.cong, s.v. handle, and x.zhou, "Gaussian signalling for coverage communications," IEEE trans. wireless communications, vol.18, No.7, pp.3542-3553,2019.);

the robust hiding rate maximization problem is expressed as the following problem (21):

s.t D(p₀||p₁)≤2ε² (21b)

||w_b||²≤P_total, (21c)

using functions

For x>0 to restate the hidden constraint (21b), the hidden constraint

Equivalent transformation into:

wherein

And

is the equation

Two roots of (c); the constraint (21b) is equivalently restated as:

due to delta h_AW∈ε_AWAnd Δ h_IW∈ε_IWIn constraints (21e) and (21f), Δ h_AWOr Δ h_IWThere are infinite options. This makes the problem (21) non-convex and difficult to solve. To overcome this challenge, a method of relaxing and constraining was proposed.

W in the alternative optimization problem (21)_bAnd q, decomposed into two subproblems 1 and 2 as follows:

sub-problem 1:

optimizing w_bGiven q: to solve the problem (21), the beamformer w is optimized by fixing q under the constraints (21b), (21c), (21e) and (21f)_bDefining the auxiliary variable g_B、g_WAnd

problem (24) was obtained:

wherein Q represents a phase shift matrix;

denotes g_BThe conjugate transpose of (1);

signal noise z representing Willie_wThe noise variance of (2);

and

is the equation

Two roots of (c); p_totalIs the maximum transmit power of Alice;

denotes g_WThe CSI estimation vector of (1); Δ g_WDenotes g_WThe CSI vector error of (1);

ε_Wfinger g_WA range expression of the CSI error vector of (a); relaxing the constraint (24b) to a convex form by applying SDR, by applying auxiliary variables

Is relaxed to

The constraint is equivalently re-expressed as:

wherein

Represents Δ g_WConjugate transpose;

to represent

The conjugate transpose of (1);

representing a CSI estimate vector between IRS and Willie;

representing a CSI estimation vector between Alice and Willie;

application of SDR to

Then, the problem (24) is relaxed to obtain the following problem (26):

Δg_W∈ε_W， (26d)

(25a)，(25b)；

note that the objective function and constraints are

Is linear and thus the SDR problem (26) is quasi-convex. However, the problem (26) remains computationally difficult to solve because of Δ g_W∈ε_WIt contains an infinite number of constraints. An infinite number of constraints are recast into a set of Linear Matrix Inequalities (LMIs) using the S-Procedure, which is an easy-to-handle approximation.

Quotation 1(S-Procedure, ref. D.W.K.Ng, E.S.Lo, and R.Schober, "Robust beamforming for secure communication in systems with wireless information and power transfer," IEEE transactions. Wireless communication, vol.13, No.8, pp.4599-4615,2014.): setting function

Is defined as:

wherein

Is a complex-early-late-matrix,

representing an N × 1-dimensional complex vector;

represents a one-dimensional real number;

if and only if there is a variable η ≧ 0,

such that:

constraints (25a) and (25b) are converted into a finite number of LMIs (Linear matrix inequality) respectively by S-Procedure (S-lemma):

a conservative approximation of the problem (26) is obtained, namely the following problem (30):

s.t(26b)，(26c)，(29a)，(29b)；

the problem (30) is a convex SDP problem and can therefore be solved optimally using the interior point method. In the same way, let

Represents an optimal solution to the problem (30). If it is not

Then

Is an optimal solution to the problem (30) and the optimal beamformer is derived by SVD

That is to say that the first and second electrodes,

otherwise, if

Then a gaussian randomization process is employed to produce a high quality rank 1 solution to the problem (30) (referenceThe documents Z.Luo, W.Ma, A.M.so, Y.Ye, and S.Zhang, "Semidefinite repetition of quadrature optimization schemes," IEEE Signal Process.Mag., vol.27, No.3, pp.20-34,2010 ].

Sub-problem 2:

given w_bOptimizing q: at a fixed w_bConsidering the design of q on the basis of (1), in this case the problem (21) translates into a problem (31) of the form:

applying SDR techniques to

And

to overcome non-convexity by removing

Re-expressing the problem (31) in relaxed form, i.e. the following problem (32):

wherein

E_mIs an M +1 dimensional matrix, and the (i, j) th element is marked as [ E_m]_i,jSatisfy the following requirements

The problem (32) is a convex SDP problem that can be solved optimally using the interior point method. It should be noted that due to the relaxation of SDR, the optimal solution

May not be the optimal solution to the problem (32).

D(p₁||p₀)≤2ε²The case (2) is as follows:

considering the constraint D (p)₁||p₀)≤2ε²The case (1). The corresponding robust hiding rate maximization problem is expressed as the following problem (33):

s.t D(p₁||p₀)≤2ε², (33b)

||w_b||²≤P_total, (33c)

wherein

Note that problem (33) is similar to problem (21), except for the hidden constraints. Covert restraint

Equivalent transformation is as follows:

wherein

Is an equation

Two roots of (2).

The problem is solved (33) using an alternating iterative, relaxation and constraint approach.

In step 2, the following procedure was used to address problem (21):

step a1, initialize setting

1_NRefers to an N × 1 vector whose elements are all 1;

indicating that the problem (21) has in iteration kWith variable quantity

And

a target value of (d);

step a2, when

Repeating steps a3 through a 6;

step a3, setting k to k + 1;

step a4, given q^(k-1)Solving the problem (30);

step a5, given

Solving a problem (32);

step a6, providing

Step a7, until

E denotes a threshold value, typically set at 10^-3The effect is the accuracy of the control parameter, E>0；

Step a8, solving

And q is^(k)Performing Gaussian randomization to obtain an approximate solution

And q is^(k)Then updated

3.1 evaluation of scene 2

First, the robust beamformer design proposed in the scenario of Alice and imperfect WCSI (Willie's channel state information) was evaluated.

FIGS. 2a and 2b show the concealment threshold 2 ε according to the invention²0.02, CSI error υ_w＝2×10^-4Then, D (p)₀||p₁) And D (p)₀||p₁) And the cumulative density function CDF. FIGS. 2a and 2b show the error parameter for CSI as upsilon, respectively_w＝2×10^-4D (p) of₀||p₁) And D (p)₁||p₀) The empirical Cumulative Density Function (CDF). The concealment threshold values of both robust and non-robust designs are 2 epsilon²0.02, i.e. D (p)₀||p₁) Less than or equal to 0.02 and D (p)₁||p₀) Less than or equal to 0.02. As can be seen from fig. 2a and 2b, the CDF in the non-robust design KL divergence cannot satisfy the constraint. On the other hand, the robust beamforming design ensures the requirement of KL divergence, namely, the requirement of Willie on the probability of false detection is met. Non-robust design here means that under the same conditions, use is made of

And

the proposed implicit design. Overall, fig. 2a and 2b verify the necessity and effectiveness of the proposed robust design.

Fig. 3a and 3b are graphs of the value of epsilon versus the concealment rate and the detection error probability. FIG. 3a plots the covert constraint D (p)₀||p₁)≤2ε²And D (p)₁||p₀)≤2ε²At the value of epsilon of_bPlot against ε value, where CSI error υ_w＝2×10^-4，P _total5 dBm. Wherein

Represents D (p)₀||p₁)≤2ε²Probability of false alarm of time

The other notation is defined as such. Simulation results and theoryThe theoretical analysis is consistent, i.e. as ε becomes larger, the blind constraint becomes loose, resulting in R_bBecomes larger. FIG. 3b plots the false alarm probability

And probability of missed detection

And CSI error v_w＝2×10^-4The value of epsilon. It can be observed that under two different covert constraints, the false alarm probability

And probability of missed detection

Decreases with increasing epsilon, wherein

Is always less than

This indicates that the more relaxed the transformation constraints, the better Willie detection performance. In addition, fig. 3b also verifies the effectiveness of the proposed robust beamformer design in covert communications, i.e.

Thus, from fig. 3a and 3b, the trade-off between Willie's detection performance and Bob's coverage is revealed, and the desired trade-off can be achieved by a suitably robust beamforming design.

FIGS. 4a and 4b show the probability of detection error versus CSI error upsilon for a concealment rate and epsilon of 0.1_wAnd (4) a relation graph of the ratio. FIG. 4a depicts D (p) at two KL divergence cases₀||p₁)≤2ε²And D (p)₁||p₀)≤2ε²Lower hiding ratio R_bAnd CSI error v_wThe relationship (2) of (c). As can be seen from FIG. 4a, the CSI error v_wThe higher the implemented hiding rate R_bThe lower. FIG. 4b is a graph showingHidden under-constraint false alarm probability

And probability of missed detection

And CSI error v_wThe relationship (2) of (c). We have observed that in both cases of covert constraints, the false alarm probability

And probability of missed detection

Are all dependent on upsilon_wIs reduced and reduced, wherein

Is always less than

Furthermore, as can be seen from fig. 4a and 4b, the larger error v_wMay result in the beamformer at coverage rate R_bThe design of the aspect is poor. However, such a beamformer may interfere with Willie detection, which is also advantageous for Bob. Therefore, this trade-off should also be noted in the design of the robust beamformer.

FIG. 5 is a coverage ratio R_bAnd CSI error v_w＝2×10^-4The relationship of the number of antennas N is shown schematically. Fig. 5 observes that the concealment rate R of two concealment constraints increases with the number of antennas N_bThis is increased, similarly to the case of fig. 5. As can be seen from FIGS. 3a, 3b, 4a, 4b and 5, the blind constraint D (p)₀||p₁)≤2ε²Is higher than the two KL divergence cases D (p)₁||p₀)≤2ε²The rate of (c).

The invention provides a robust beamforming design method for covert communication of the internet of things, and a plurality of methods and approaches for implementing the technical scheme, where the foregoing is only a preferred embodiment of the invention, it should be noted that, for those skilled in the art, a plurality of improvements and modifications may be made without departing from the principle of the invention, and these improvements and modifications should also be considered as the protection scope of the invention. All the components not specified in the present embodiment can be realized by the prior art.

Claims (1)

1. A robust beam forming design method for hidden communication of the Internet of things is characterized by comprising the following steps:

step 1, establishing a covert communication environment;

step 2, under the condition of imperfect WCSI, carrying out hidden beam forming design;

the step 1 comprises the following steps: alice represents the base station, Willie represents the eavesdropper, Bob represents the hidden user, and Alice represents the private data stream x_bIs sent to Bob, where

Representing a null hypothesis, that is, Alice does not send a private data stream to Bob;

represents another assumption that Alice sends a private data stream to Bob;

meanwhile, Willie is observing the communication environment as an eavesdropper and tries to identify whether Alice is transmitting to Bob; in order to protect the confidential signals from eavesdropping, an IRS intelligent reflecting surface with an intelligent controller is adopted to assist in hidden transmission;

in the step 1, Alice is set to be provided with N antennae, and N is a natural number; bob and Willie each have an antenna; order to

Representing a signal x_bPower of x_bA signal sent to Bob for Alice; make it

Wherein the content of the first and second substances,

representing a set of complex matrices; h is_ABIs an Nx 1 complex matrix, which means the channel coefficient from Alice to Bob; h is_AWIs an Nx 1 complex matrix, which means the channel coefficients from Alice to Willie; h is_IBIs an Mx 1 complex matrix, which means IRS to Bob channel coefficients; h is_IWIs an M multiplied by 1 complex matrix, which refers to IRS to Willie channel coefficients;

by using

Representing the channel coefficients from Alice to IRS;

by using

Events representing the sending of information by Alice to Bob, using

An event indicating that Alice does not send information to Bob;

in step 1, from Willie's perspective, Alice's transmission signal x is as follows:

wherein w_bIs x_bSets Alice at

No signal is transmitted at the bottom and the beamforming vector w_bIn that

The following constraints are satisfied:

||w_b||²≤P_total (2)

in the formula P_totalIs the maximum transmit power of Alice;

the phase shift matrix Q is determined by

Given that Q represents a phase shift matrix whose diagonal elements are the corresponding elements Q of the vector; use of

Modeling the reflection of IRS units, wherein

q_mReflecting model of the M-th unit, j is an imaginary number, and when M is 1, … M, theta_mE [0,2 π) and β_m∈[0,1]Respectively representing the controllable phase shift and the amplitude reflection coefficient introduced by the mth unit;

is provided with

To achieve maximum reflected power gain, q should satisfy:

|q_m|＝1,m＝1,…M (3)

signal y received at Bob_bWriting into:

wherein z is_bFor the received noise at Bob to be the received noise,

signal noise z representing Bob_bObeying a mean of 0 and a variance of

Complex gaussian distribution of (a); h is_IBIs the channel coefficient from IRS to Bob,

is h_IBThe conjugate transpose of (1); h is_ABThe channel coefficients for Alice to Bob are,

is h_ABThe conjugate transpose of (1);

in step 1, Willie receives signal y_wWriting into:

wherein z is_wIs the noise received at Willie and,

signal noise z representing Willie_wObeying a mean of 0 and a variance of

Complex gaussian distribution of (a); h is_IWIs the channel coefficient from IRS to Willie,

is h_IWThe conjugate transpose of (1); h is_AWFor the Alice to Willie channel coefficients,

is h_AWThe conjugate transpose of (1);

in step 1, R is set_bIs a null hypothesis

The instantaneous rate of the next Bob, written as:

willie in

And

the likelihood functions of the lower received signals are respectively expressed as p₀(y_w) And p₁(y_w)；

According to formula (5), p₀(y_w) And p₁(y_w) Respectively as follows:

wherein

In

Signal noise z representing Willie_wOf the noise variance, λ₀And λ₁Representing an auxiliary variable;

in step 1, Willie wants to minimize the detection error probability ξ by applying the optimal detector, setting:

ξ＝1-V_T(p₀,p₁), (8)

wherein V_T(p₀,p₁) Is p₀(y_w) And p₁(y_w) The total change between them is obtained by using the Pincker inequality:

wherein D (p)₀||p₁) Represents from p₀(y_w) To p₁(y_w) KL of (a) D (p)₁||p₀) Is from p₁(y_w) To p₀(y_w) KL divergence of (1);

D(p₀||p₁) And D (p)₁||p₀) Respectively as follows:

to achieve implicit communication with a given ξ, i.e., ξ ≧ 1- ε, the KL divergence of the likelihood function satisfies one of the following constraints:

D(p₀||p₁)≤2ε², (11a)

D(p₁||p₀)≤2ε² (11b)；

the step 2 comprises the following steps: the imperfect WCSI case refers to: willie is a common user, Alice does not know the channel state information about Willie and wants to acquire Bob's personal information, in which case Alice does not have perfect knowledge of CSI because Alice is a passive guard and channel estimation error;

imperfect WCSI is modeled as:

and:

wherein h is_AWIs the channel coefficient h from Alice to Willie_IWRefer to the IRS to Willie channel coefficients;

and

respectively representing a CSI channel state information estimation vector between Alice and Willie and a CSI estimation vector between Willie and IRS;

Δh_AWrepresenting the CSI error vectors of Alice to Willie;

Δh_IWa CSI error vector representing IRS to Willie;

CSI error vector Δ h_AWAnd Δ h_IWIs characterized by an elliptical area, namely:

and

wherein epsilon_AWA range expression referring to the CSI error vector from Alice to Willie;

represents Δ h_AWThe conjugate transpose of (1); epsilon_IWRefers to the range representation of the IRS to Willie CSI error vector;

represents Δ h_IWThe conjugate transpose of (1);

axis of control ellipsoid, v_AW>0,υ_IW>0 determining ellipseA sphere volume;

the optimal rule for Willie to minimize detection error is the likelihood ratio test according to Neyman-Pearson's criterion:

according to

And

are respectively corresponding hypothesis

And

the binary decision of equation (16) is rewritten as:

wherein phi^*Is y_w|²The optimal detection threshold of (a) is as follows:

λ₀and λ₁Dependent on the beamforming vector w_bAnd an IRS reflected beam forming vector q;

and

lower latticey_w|²Respectively, as the cumulative density function CDFs of

And

based on optimal detection threshold phi^*Giving false alarm

And probability of missed detection

Comprises the following steps:

by using

And

the expression of (a) to characterize the ideal detection performance of Willie;

D(p₀||p₁)≤2ε²the case (2) is as follows:

using D (p) given by (11)₀||p₁)≤2ε²And D (p)₁||p₀)≤2ε²As a covert constraint;

the robust hiding rate maximization problem is expressed as the following problem (21):

s.t D(p₀||p₁)≤2ε² (21b)

||w_b||²≤P_total, (21c)

using functions

For x>0 to restate the hidden constraint (21b), the hidden constraint

Equivalent transformation into:

wherein

And

is the equation

Two roots of (c); the constraint (21b) is equivalently restated as:

w in the alternative optimization problem (21)_bAnd q, decomposed into two subproblems 1 and 2 as follows:

sub-problem 1:

optimizing w_bGiven q: to solve the problem (21), the beamformer w is optimized by fixing q under the constraints (21b), (21c), (21e) and (21f)_bDefining the auxiliary variable g_B、g_WAnd

problem (24) was obtained:

wherein Q represents a phase shift matrix;

denotes g_BThe conjugate transpose of (1);

signal noise z representing Willie_wThe noise variance of (2);

and

is the equation

Two roots of (c); p_totalIs the maximum transmit power of Alice;

denotes g_WThe CSI estimation vector of (1); Δ g_WDenotes g_WThe CSI vector error of (1);

υ_W＝υ_AW+υ_IW；ε_Wfinger g_WA range expression of the CSI error vector of (a);

relaxing the constraint (24b) to a convex form by applying SDR, by applying auxiliary variables

Is relaxed to

The constraint is equivalently re-expressed as:

wherein

Represents Δ g_WConjugate transpose;

to represent

The conjugate transpose of (1);

representing a CSI estimate vector between IRS and Willie;

representing a CSI estimation vector between Alice and Willie;

application of SDR to

Then, the problem (24) is relaxed to obtain the following problem (26):

Δg_W∈ε_W， (26d)

(25a)，(25b)；

let function f_m(x),m∈{1,2},

Is defined as:

wherein

Is a complex-early-late-matrix,

representing an N × 1-dimensional complex vector;

represents a one-dimensional real number;

if and only if there is a variable η ≧ 0,

such that:

constraints (25a) and (25b) are respectively converted into a finite number of LMIs linear matrix inequalities by S-Procedure:

therein, a conservative approximation of the problem (26) is obtained, namely the following problem (30):

s.t(26b)，(26c)，(29a)，(29b)；

let

Represents an optimal solution to the problem (30) if

Then

Is an optimal solution to the problem (30) and the optimal beamformer is derived by SVD

That is to say that the first and second electrodes,

otherwise, if

A gaussian randomization process is employed to produce a high quality rank 1 solution to the problem (30);

sub-problem 2:

given w_bOptimizing q: at a fixed w_bConsidering the design of q on the basis of (1), in this case the problem (21) translates into a problem (31) of the form:

applying SDR techniques to

And

to overcome non-convexity by removing

Re-expressing the problem (31) in relaxed form, i.e. the following problem (32):

wherein

E_mIs an M +1 dimensional matrix, and the (i, j) th element is marked as [ E_m]_i,jSatisfy the following requirements

D(p₁||p₀)≤2ε²The case (2) is as follows:

the corresponding robust hiding rate maximization problem is expressed as the following problem (33):

s.t D(p₁||p₀)≤2ε², (33b)

||w_b||²≤P_total, (33c)

wherein

Covert restraint

Equivalent transformation is as follows:

wherein

Is an equation

Two roots of (2).

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