Terahertz near-field covert communication method based on TTD

Abstract

The invention discloses a TTD-based terahertz near-field covert communication method, and belongs to the technical field of wireless communication. The implementation method comprises the following steps of: enhancing the concealment of a communication system by utilizing near-field channel characteristics and unbounded noise uncertainty; analyzing system performance indexes under the unbounded noise uncertainty model, wherein the system performance indexes comprise an average hiding probability, a hiding interruption probability and a hiding rate; constructing a hybrid beam forming optimization problem under the conditions of concealment constraint, power constraint and hardware constraint by adopting a TTD-based hybrid beam forming architecture and taking concealment rate maximization as a target; in the simulation domain, the near-field array response is approximated by using the segmented far-field array response to obtain the optimal solution of the simulation beam former, so that the adverse effect of the beam splitting effect is reduced, and the concealment is improved to the maximum extent; in a digital domain, a non-convex optimization problem is converted into a convex form by adopting an SCA algorithm, and a CVX solver is used for solving an optimal solution of a digital beam former so as to maximize a system hiding rate.

Description

Technical Field

[0001] The invention relates to a terahertz near-field covert communication method based on TTD, belonging to the technical field of wireless communications. Background Art

[0002] With the development of wireless communication technology, more and more private data and sensitive information are transmitted through wireless channels, which poses a great challenge to the security and reliability of wireless communication. Traditional encryption-based methods mainly ensure the security of signals through application layer key sharing, and their security mainly depends on the computational complexity of cracking the generated key. However, with the rapid development of computer technology, many traditional encryption technologies are becoming increasingly unreliable. In addition, physical layer security technology uses the dynamic characteristics of wireless media to minimize the information that eavesdroppers can obtain. However, the above technologies are all aimed at ensuring the security of communication content, while ignoring that the exposure of user communication behavior will also lead to information leakage, such as node location, transmission mode, etc. In order to solve this problem, some scholars have proposed covert communication technology. Covert communication technology can not only protect the communication content from being eavesdropped, but also ensure that the communication behavior between the two cannot be detected by eavesdroppers.

[0003] With the rapid development of wireless communication technology from traditional microwave communication to millimeter wave (mmWave) and terahertz (THz) communication, transmission is becoming more and more directional, which is conducive to the concealment of wireless communication. THz communication, which has a frequency range of 0.1THz to 10THz and can achieve ultra-high transmission rates at the Tbps level, has attracted widespread attention. In addition to providing higher transmission rates, THz communication also has a natural security advantage, namely the high directivity of the THz frequency band. The highly directional transmission of THz communication can not only combat the high path loss caused by propagation, atmospheric attenuation and scattering effects, but also limit the transmission power to a narrow beam area, which naturally prevents the communication behavior from being detected by eavesdroppers outside the beam area. Therefore, combining THz communication technology with covert communication technology can effectively improve the concealment performance of communication.

[0004] Although the THz band has the advantage of ensuring secret communications, we still face some difficulties. The high directivity of the THz band can effectively limit the transmission power within the beam area. If the eavesdroppers are distributed outside the beam area, covert communication can be effectively carried out. But if the eavesdroppers are distributed in the fan-shaped area, the higher transmission power will seriously reduce the concealment of the communication behavior. Although some covert communication schemes have been proposed to combat eavesdroppers at a long distance in the beam area, they still cannot solve the threat of eavesdroppers at a close distance in the beam area. Therefore, it is urgent to study covert communication methods that can combat eavesdroppers at a close distance. Summary of the invention

[0005] In view of the problem that it is difficult to resist eavesdroppers at a close distance in THz covert communication scenarios, the main purpose of the present invention is to provide a TTD-based terahertz near-field covert communication method, which utilizes near-field channel characteristics and unbounded noise uncertainty to enhance the concealment of the communication system; analyzes system performance indicators under the unbounded noise uncertainty model, including average concealment probability, concealment interruption probability and concealment rate; adopts a TTD-based hybrid beamforming architecture, takes maximizing the concealment rate as the goal, and constructs a hybrid beamforming optimization problem under the conditions of concealment constraints, power constraints and hardware constraints; in the analog domain, the optimal solution of the analog beamformer is obtained by approximating the near-field array response by utilizing the segmented far-field array response, so as to reduce the adverse effects of the beam splitting effect and maximize the concealment; in the digital domain, the non-convex optimization problem is converted into a convex form by adopting the SCA algorithm, and the optimal solution of the digital beamformer of the CVX solver is used to maximize the system hiding rate.

[0006] The purpose of the present invention is achieved through the following technical solutions.

[0007] The TTD-based near-field covert communication method disclosed in the present invention comprises the following steps:

[0008] Step 1: Construct a covert communication system consisting of transmitter Alice, legitimate user Bob, and eavesdropper Willie. The analog beamformer of the covert communication system and digital beamformer The constraints include power constraint, constant modulus constraint and maximum delay constraint; on this basis, the near-field channel characteristics are applied to the covert communication system to determine Willie's statistical test quantity and judgment criteria; the expressions of the average concealment probability and concealment interruption probability of the covert communication system are obtained;

[0009] Step 1.A: Construct a covert communication system consisting of transmitter Alice, legitimate user Bob, and eavesdropper Willie;

[0010] In the covert communication system, Alice uses N t The uniform linear array ULA and N RF A radio frequency chain, RF chain, to transmit N s data streams, and Bob and Willie are equipped with a single antenna; the covert communication system uses an M-subcarrier orthogonal frequency division multiplexing system, and the frequency of the m-th subcarrier is f m =f c +B(2m-1-M) / (2M), m∈{1,2,...,M}, where f c is the center carrier frequency, B is the bandwidth; Ailce uses an analog beamformer and digital beamformer The TTD-based hybrid beamforming architecture is composed.

[0011] In the TTD-based hybrid beamforming architecture, the analog beamformer F RF,m =F PS T m It is composed of TTDs and PSs. is a frequency-independent partial analog beamformer implemented by PSs, is the frequency-dependent partial analog beamformer implemented by TTDs, N T The number of TTDs connected to each RF chain; F PS Specifically expressed as

[0012]

[0013] in, is a PSs-based partial analog beamformer connected to the kth RF chain, specifically expressed as

[0014]

[0015] in, is the PS-based partial analog beamformer of the s-th TTD connected to the k-th RF chain, The phase shift achieved for the pth antenna connected to the sth TTD pair connected to the kth RF chain, is the number of antennas connected to each TTD; T m Specifically expressed as

[0016]

[0017] in, The delay vector, t, is realized for the TTDs connected to the kth RF chain. k,s The delay achieved for the sth TTD connected to the kth RF chain. In addition, the simulated beamformer and digital beamformer The power constraint and hardware constraint need to be met. The hardware constraint includes constant modulus constraint and maximum delay constraint. The power constraint is

[0018]

[0019] P max is the maximum transmit power; the constant modulus constraint is

[0020]

[0021] The maximum delay constraint is

[0022]

[0023] t max is the maximum delay that TTD can achieve;

[0024] Step 1.B: Apply the near-field channel characteristics to the covert communication system of 1.A;

[0025] Alice is the origin of the reference coordinate system, r b ,θ b ∈[-1,1] are the distance and physical direction from Bob to Alice, r w ,θ w ∈[-1,1] are the distance and physical direction from Willie to Alice respectively. Substituting the near-field channel into the covert communication system constructed in step 1.A, the received signals of Bob and Willie at the mth subcarrier at time z∈[Z] are expressed as:

[0026]

[0027] in, is the signal sent by Alice at the mth subcarrier, n b,m (z) and n w,m (z) are the noises at Bob and Willie respectively, is a frequency domain near-field legal channel, It is a frequency domain near-field eavesdropping channel;

[0028] For simplicity, we use R = {b, w} to indicate that the receiver is Bob or Willie, and the near-field channel h R,m The frequency domain near-field legal channel and the frequency domain near-field eavesdropping channel are uniformly represented. The near-field channel is specifically represented as:

[0029]

[0030] L is the number of paths, α l is the complex gain of the lth (l=1,2,...,L)th path, is the near-field array response vector. Derived from spherical waves, it is expressed as:

[0031]

[0032] Where c is the speed of light, r R,l is the distance between the transmitter and the receiver or scattering point; is the distance between the qth transmitting antenna and the receiver in the lth path component, which can be specifically expressed as:

[0033]

[0034] in, is the spatial direction of the lth path component at the mth subcarrier, θ R,l ∈[-1,1] is the physical direction of the lth path component, d=c / (2f c ) is the antenna spacing, δ q =(2q-N t -1) / 2, q=1,2,...,N t is an intermediate variable;

[0035] Step 1.C: Determine Willie's statistical test amount and judgment criteria;

[0036] According to equation (8) in step 1.B, the signal received by Willie at all subcarriers is:

[0037]

[0038] in, It means Alice did not send a signal. Indicates that Alice is sending a signal; thus, the signal vector received by Willie is:

[0039] y w =[y w (1),...,y w (z),...,y w (Z)] (13)

[0040] This gives the Willie statistical test S(y w ) and the judgment criteria are:

[0041]

[0042] in, and are the determination of the received signal as noise and the determination of the received signal as signal plus noise, respectively, and η is the detection threshold of Willie;

[0043] Step 1.D: Obtain the expressions of average concealment probability and concealment interruption probability of the covert communication system;

[0044] According to equations (12) and (14) in step 1.C, the error detection probability of Willie is obtained as:

[0045]

[0046] in, is the total power of the signal received by Willie, is the power of the signal received by Willie at the mth subcarrier, is the noise power of Willie at the mth subcarrier;

[0047] Since the actual noise is uncertain, according to formula (15), the average concealment probability is:

[0048]

[0049] in, is the probability density function of uncertain noise; the concealed outage probability is obtained as follows:

[0050]

[0051] To ensure confidentiality, and P o <δ, where ε∈(0,1) is the precision, which is a sufficiently small positive number, and δ represents the maximum acceptable concealed outage probability;

[0052] Step 2: Based on the near-field covert communication system constructed in step 1, construct an unbounded noise uncertainty model at Willie; derive the probability density of the total noise power at Willie under this noise model; then determine the optimal detection threshold of Willie, and derive the expressions for the average concealment probability, concealment interruption probability and concealment rate;

[0053] Step 2.A: Construct an unbounded noise uncertainty model at Willie;

[0054] Assume that the noise power of Willie at each subcarrier is the same, that is, The noise power of Bob at each subcarrier is the standard noise power, that is is the standard noise power; the unbounded noise uncertainty model at Willie is constructed as:

[0055]

[0056] That is, the difference between the noise power at Willie and the standard noise power follows a normal distribution in the logarithmic domain; is the dB value of the noise power at Willie, is the dB value of the standard noise power, is a parameter that quantifies the size of the noise uncertainty;

[0057] Step 2.B: Derive the probability density of the total noise power at Willie;

[0058] According to formula (18), the probability density of the noise power at Willie is:

[0059]

[0060] Where g = ln(10) / 10; the probability density of the total noise power at Willie is:

[0061]

[0062] Among them, M dB =10lg(M);

[0063] Approximate equation (20) to a Gaussian function, that is

[0064]

[0065] in,

[0066]

[0067] Step 2.C: Determine the optimal detection threshold of Willie;

[0068] According to equations (16) and (17) in step 1 and equation (21) in step 2.B, the optimal detection threshold of Willie is

[0069] η * =max{φ1+P w / 2,P w} (twenty three)

[0070] Step 2.D: Obtain the expressions for average concealment probability, concealment interruption probability and concealment rate;

[0071] According to equations (16) and (17) in step 1 and equation (23) in step 2, the expressions of average concealment probability and concealment interruption probability are obtained as follows:

[0072]

[0073] Under the concealment constraint, the concealment rate is

[0074]

[0075] And meet Where R m is the concealment rate at the mth subcarrier;

[0076] Step 3: Based on the covert communication system constructed in step 1 and the unbounded noise uncertainty model constructed in step 2, the transmitter analog beamformer F is constructed according to the power constraint, constant modulus constraint, maximum delay constraint in step 1 and the covert rate expression in step 2. RF,m and digital beamformer F BB,mOptimization problem;

[0077] The optimization problem is:

[0078]

[0079] in, is the constant modulus constraint that the PS-based partial analog beamformer must satisfy, The delay achieved for TTD must satisfy

[0080] Step 4: Based on the optimization problem constructed in step 3, derive the analog beamformer connected to the kth RF chain The ideal optimal solution Based on the ideal optimal solution Approximating the near-field array response vector as the segmented far-field array response vector, we obtain The approximate optimal solution of the PSs-based partial analog beamformer F is obtained based on the approximate optimal solution. PS , and derive the partial simulation beamformer T based on TTDs m ; Finally, the optimal analog beamformer is obtained

[0081] Step 4.A: Solve the optimization problem (27) to obtain the analog beamformer connected to the kth RF chain The ideal optimal solution

[0082] definition

[0083]

[0084] in, is the analog beamformer connected to the kth RF chain, which is specifically expressed as:

[0085]

[0086] To maximize the array gain at Bob, we have The ideal optimal solution is:

[0087]

[0088] Among them, ψ m For any phase, is the spatial direction of Bob at the mth subcarrier; due to hardware limitations, it is generally not possible to obtain The ideal optimal solution

[0089] Step 4.B: Approximate the near-field array response vector as the segmented far-field array response vector, yielding The approximate optimal solution of ;

[0090] According to the description in step 4.A, it is generally impossible to obtain an ideal optimal solution. By using the compensation method of approximating the near-field array response vector as a segmented far-field array response vector, the near-field array response vector of equation (30) in step 4.A is approximated as being segmented into N T The segmented far-field array response vector of the segment is

[0091]

[0092] Among them, k m =2πf m / c is the wave number of the mth subcarrier, are the distance and spatial direction of Bob relative to the center of the s-th subarray, respectively, which can be expressed as:

[0093]

[0094]

[0095] in, is the intermediate variable; s Specifically expressed as:

[0096]

[0097] Among them, k c =2πf c / c is the wave number of the subcarrier at the center frequency, is an intermediate variable;

[0098] According to formula (31), we get The approximate optimal solution is Right now

[0099]

[0100] Step 4.C: Derive the PSs-based partial analog beamformer F PS ;

[0101] According to step 4.B, equation (35), we get

[0102]

[0103] According to equations (1) and (2) in step 1, we get the PSs-based partial analog beamformer F PS .

[0104] Step 4.D: Derive the partial analog beamformer T based on TTDs m ;

[0105] set up According to step 4.C, equation (37), we get

[0106]

[0107] in, is an arbitrary scalar; thus we get the unconstrained optimal delay for

[0108]

[0109] By definition The optimal delay for each TTD under the maximum delay constraint is

[0110]

[0111] in,

[0112]

[0113] According to equation (3) in step 1, we get the partial analog beamformer T based on TTDs: m ;

[0114] Step 4.E: According to equation (36) in step 4.C and equation (41) in step 4.D, the optimal analog beamformer is obtained as:

[0115]

[0116] in,

[0117]

[0118] is the optimal analog beamformer connected to the kth RF chain;

[0119] Step 5: Optimal analog beamformer obtained in step 4 Redefine the optimization problem in step 3; then transform the concealment constraint in the optimization problem into the power constraint of the received signal at Willie, and update the optimization problem; then use the SCA algorithm to perform convex approximation on the optimization problem, transforming the optimization problem from a non-convex problem to a convex problem; finally, use the CVX solver to iteratively solve the optimization convex problem and obtain the optimal digital beamformer Realize terahertz near-field covert communication.

[0120] Step 5.A: Based on the optimal analog beamformer obtained in step 4 Redefine the optimization problem;

[0121] According to equation (42) in step 5, the optimization problem in equation (27) is redefined as:

[0122]

[0123] Among them, the concealment rate R at the mth subcarrier m Specifically expressed as:

[0124]

[0125] in, is a low-dimensional equivalent channel;

[0126] Step 5.B: Convert the concealment constraint into the power constraint of the received signal at Willie and update the optimization problem;

[0127] According to the concealment constraint expression of equation (44) in step 5.A, the power constraint of the received signal at Willie is obtained as:

[0128]

[0129] Among them, P th is the power threshold, which is specifically expressed as:

[0130]

[0131] According to formula (46), the optimization problem in formula (44) is transformed into:

[0132]

[0133] The problem is non-convex;

[0134] Step 5.C: Use the SCA algorithm to perform convex approximation on the optimization problem, transforming the optimization problem from a non-convex problem to a convex problem; define the auxiliary variable r m ≤R m , use the SCA algorithm to perform convex approximation on equation (48) in step 5.B, and get

[0135]

[0136] γ m (F BB,m ) is expanded by the first order Taylor, and we get

[0137]

[0138] in, is an auxiliary variable;

[0139] According to the auxiliary variable r obtained in formula (50) m , The optimization problem in formula (48) is transformed into:

[0140]

[0141] This problem is a convex problem;

[0142] Step 5.D: Use CVX solver to iteratively solve the optimization problem and obtain the optimal digital beamformer Realize terahertz near-field covert communication.

[0143] Beneficial effects:

[0144] 1. The present invention discloses a near-field covert communication method based on TTD. Aiming at the problem that THz communication is difficult to resist eavesdroppers at a close distance, the concealment of the communication system is enhanced by using the near-field channel characteristics and unbounded noise uncertainty. By establishing a near-field covert communication model, designing and optimizing analog beamformers and digital beamformers, near-field covert communication based on TTD is realized according to the digital beamformer.

[0145] 2. The present invention discloses a TTD-based near-field covert communication method, which is based on a near-field covert communication model and obtains the optimal solution of the analog beamformer by using the segmented far-field array response to approximate the near-field array response, thereby suppressing the adverse effects of the beam splitting effect and maximizing the concealment.

[0146] 3. The present invention discloses a TTD-based near-field covert communication method, which is based on a near-field covert communication model. The SCA algorithm is used to transform the non-convex optimization problem into a convex form, and the optimal solution of the CVX solver digital beamformer is used to maximize the system hiding rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0147] Figure 1 It is the overall flow chart of the near-field covert communication method based on TTD and embodiment 1 of the present invention;

[0148] Figure 2 is the average concealment probability and concealment interruption probability after the implementation of the TTD-based near-field covert communication method of the present invention and the method in Example 1 with r w Simulation result diagram of changes;

[0149] Figure 3 is the concealed rate of the near-field covert communication method based on TTD of the present invention and the method implemented in Example 1 with r w The simulation results of the changes. DETAILED DESCRIPTION

[0150] In order to better illustrate the purpose and advantages of the present invention, the TTD-based near-field covert communication method of the present invention is further described below with reference to the accompanying drawings and examples.

[0151] Embodiment 1:

[0152] This embodiment describes in detail the specific implementation steps of the TTD-based near-field covert communication method of the present invention.

[0153] This case considers that there are transmitter Alice, legitimate user Bob and eavesdropper Willie in the THz covert communication system. Alice sends signals to Bob covertly while Willie continues to eavesdrop. Due to the high directivity of the THz frequency band, THz communication can limit the transmission power of the signal within the beam area, effectively preventing the communication behavior from being detected by eavesdroppers outside the beam area, and has a natural security advantage. However, if the eavesdroppers are distributed in the beam area, the higher transmission power will seriously reduce the concealment of the communication behavior. Although some covert communication schemes have been proposed to combat eavesdroppers at a long distance in the beam area, they still cannot solve the threat of eavesdroppers at a close distance in the beam area. For this reason, a near-field covert communication method based on TTD is adopted, which can combat eavesdroppers at a close distance and improve the system concealment rate.

[0154] like Figure 1 As shown, the present embodiment discloses a TTD-based near-field covert communication method, and the specific implementation steps are as follows:

[0155] Step 1.A: Construct a covert communication system consisting of transmitter Alice, legitimate user Bob, and eavesdropper Willie;

[0156] In the covert communication system, Alice uses N t The uniform linear array ULA and N RF A radio frequency chain, RF chain, to transmit N s data streams, and Bob and Willie are equipped with a single antenna; the covert communication system uses an M-subcarrier orthogonal frequency division multiplexing system, and the frequency of the m-th subcarrier is f m =f c +B(2m-1-M) / (2M), m∈{1,2,...,M}, where f c is the center carrier frequency, B is the bandwidth; Ailce uses an analog beamformer and digital beamformer The TTD-based hybrid beamforming architecture is composed.

[0157] In the TTD-based hybrid beamforming architecture, the analog beamformer F RF,m =F PS T m It is composed of TTDs and PSs. is a frequency-independent partial analog beamformer implemented by PSs, is the frequency-dependent partial analog beamformer implemented by TTDs, N T The number of TTDs connected to each RF chain; F PS Specifically expressed as

[0158]

[0159] in, is a PSs-based partial analog beamformer connected to the kth RF chain, specifically expressed as

[0160]

[0161] in, is the PS-based partial analog beamformer of the s-th TTD connected to the k-th RF chain, The phase shift achieved for the pth antenna connected to the sth TTD pair connected to the kth RF chain, is the number of antennas connected to each TTD; T m Specifically expressed as

[0162]

[0163] in, The delay vector, t, is realized for the TTDs connected to the kth RF chain. k,s The delay achieved for the sth TTD connected to the kth RF chain. In addition, the simulated beamformer and digital beamformer The power constraint and hardware constraint need to be met. The hardware constraint includes constant modulus constraint and maximum delay constraint. The power constraint is

[0164]

[0165] P max is the maximum transmit power; the constant modulus constraint is

[0166]

[0167] The maximum delay constraint is

[0168]

[0169] t max is the maximum delay that TTD can achieve;

[0170] Specifically in this embodiment, M=128, B=10 GHz, f c =300GHz, N t =256, N RF =16, NT =16, N s =1,P max =10dBW, t max =10ns;

[0171] Step 1.B: Apply the near-field channel characteristics to the covert communication system of 1.A;

[0172] Alice is the origin of the reference coordinate system, r b ,θ b ∈[-1,1] are the distance and physical direction from Bob to Alice, r w ,θ w ∈[-1,1] are the distance and physical direction from Willie to Alice respectively. Substituting the near-field channel into the covert communication system constructed in step 1.A, the received signals of Bob and Willie at the mth subcarrier at time z∈[Z] are expressed as:

[0173]

[0174] in, is the signal sent by Alice at the mth subcarrier, n b,m (z) and n w,m (z) are the noises at Bob and Willie respectively, is a frequency domain near-field legal channel, It is a frequency domain near-field eavesdropping channel;

[0175] For simplicity, we use R = {b, w} to indicate that the receiver is Bob or Willie, and the near-field channel h R,m The frequency domain near-field legal channel and the frequency domain near-field eavesdropping channel are uniformly represented. The near-field channel is specifically represented as:

[0176]

[0177] L is the number of paths, α l is the complex gain of the lth (l=1,2,...,L)th path, is the near-field array response vector.

[0178] Derived from spherical waves, it is expressed as:

[0179]

[0180] Where c is the speed of light, r R,l is the distance between the transmitter and the receiver or scattering point; is the distance between the qth transmitting antenna and the receiver in the lth path component, which can be specifically expressed as:

[0181]

[0182] in, is the spatial direction of the lth path component at the mth subcarrier, θ R,l ∈[-1,1] is the physical direction of the lth path component, d=c / (2f c ) is the antenna spacing, δ q =(2q-N t -1) / 2, q=1,2,...,N t is an intermediate variable;

[0183] Specifically in this embodiment, r b =20m, r w =10m,θ b =0,θ w =0, L = 3, c = 3 × 10 8 m / s;

[0184] Step 1.C: Determine Willie's statistical test amount and judgment criteria;

[0185] According to equation (8) in step 1.B, the signal received by Willie at all subcarriers is:

[0186]

[0187] in, It means Alice did not send a signal. Indicates that Alice is sending a signal; assuming Willie uses a radiometer as a detector, the signal vector received by Willie is:

[0188] y w =[y w (1),...,y w (z),...,y w (Z)] (13)

[0189] This gives the Willie statistical test S(y w ) and the judgment criteria are:

[0190]

[0191] in, and are the determination of the received signal as noise and the determination of the received signal as signal plus noise, respectively, and η is the detection threshold of Willie;

[0192] When Z→∞ in equation (14), the Willie statistical test quantity is obtained as follows:

[0193]

[0194] in, is the total power of the signal received by Willie, is the power of the signal received by Willie at the mth subcarrier, is the noise power of Willie at the mth subcarrier;

[0195] Step 1.D: Obtain the expressions of average concealment probability and concealment interruption probability of the covert communication system;

[0196] The false detection probability is defined as

[0197] P e =P FA +P MD >1-ε (16)

[0198] Among them, P FA and P MD are the false alarm probability and missed detection probability, respectively, expressed as

[0199]

[0200] According to equation (16) and equation (12) and equation (14) in step 1.C, the error detection probability of Willie is obtained as:

[0201]

[0202] Since the actual noise is uncertain, according to formula (19), the average concealment probability is:

[0203]

[0204] in, is the probability density function of uncertain noise; the concealed outage probability is obtained as follows:

[0205]

[0206] To ensure confidentiality, and P o <δ, where ε∈(0,1) is the precision, which is a sufficiently small positive number, and δ represents the maximum acceptable concealed outage probability;

[0207] Specifically in this embodiment, ε=0.01, δ=0.01;

[0208] At this point, from step 1.A to step 1.E, the construction of the covert communication system has been completed; the near-field channel characteristics have been applied to the covert communication system; Willie's statistical test quantity and judgment criteria have been determined; and finally, the expressions for the average concealment probability and concealment interruption probability of the covert communication system have been obtained.

[0209] Step 2.A: Construct an unbounded noise uncertainty model at Willie;

[0210] Assume that the noise power of Willie at each subcarrier is the same, that is, The noise power of Bob at each subcarrier is the standard noise power, that is is the standard noise power; the unbounded noise uncertainty model at Willie is constructed as:

[0211]

[0212] That is, the difference between the noise power at Willie and the standard noise power follows a normal distribution in the logarithmic domain; is the dB value of the noise power at Willie, is the dB value of the standard noise power, is a parameter that quantifies the size of the noise uncertainty;

[0213] Specifically to this embodiment,

[0214] Step 2.B: Derive the probability density of the total noise power at Willie;

[0215] According to formula (22), the probability density of the noise power at Willie is:

[0216]

[0217] Where g = ln(10) / 10; the probability density of the total noise power at Willie is:

[0218]

[0219] Among them, M dB =10lg(M);

[0220] Approximate equation (24) to a Gaussian function form, that is,

[0221]

[0222] in,

[0223]

[0224] Step 2.C: Determine the optimal detection threshold of Willie;

[0225] According to equations (20) and (21) in step 1 and equation (25) in step 2.B, the optimal detection threshold of Willie is

[0226] η * =max{φ1+P w / 2,P w} (27)

[0227] Step 2.D: Obtain the expressions for average concealment probability, concealment interruption probability and concealment rate;

[0228] According to equations (20) and (21) in step 1 and equations (25) and (27) in step 2, the expressions of average concealment probability and concealment interruption probability are obtained respectively:

[0229]

[0230] Under the concealment constraint, the concealment rate is

[0231]

[0232] And meet Where R m is the concealment rate at the mth subcarrier;

[0233] So far, from step 2.A to step 2.D, the construction of the uncertainty model of unbounded noise at Willie has been completed; the probability density of the total noise power at Willie under this noise model has been derived; the optimal detection threshold of Willie has been determined; and the expressions of the average concealment probability, concealment interruption probability and concealment rate have been obtained;

[0234] Step 3.A: Based on the covert communication system constructed in step 1 and the unbounded noise uncertainty model constructed in step 2, according to equations (4), (5), (6) in step 1 and equation (30) in step 2, construct the transmitter analog beamformer F RF,m and digital beamformer F BB,m The optimization problem is:

[0235]

[0236] in, is the constant modulus constraint that the PS-based partial analog beamformer must satisfy, The maximum delay constraint that the delay achieved by TTD must satisfy;

[0237] At this point, step three has completed the construction of the transmitter analog beamformer F RF,mand digital beamformer F BB,m Optimization problem.

[0238] Step 4.A: Solve the optimization problem (31) to obtain the analog beamformer connected to the kth RF chain The ideal optimal solution

[0239] definition

[0240]

[0241] in, is the analog beamformer connected to the kth RF chain, which is specifically expressed as:

[0242]

[0243] To maximize the array gain at Bob, we have The ideal optimal solution is:

[0244]

[0245] Among them, ψ m For any phase, is the spatial direction of Bob at the mth subcarrier; due to hardware limitations, it is generally not possible to obtain The ideal optimal solution

[0246] Step 4.B: Approximate the near-field array response vector as the segmented far-field array response vector, yielding The approximate optimal solution of ;

[0247] According to the description in step 4.A, it is generally impossible to obtain an ideal optimal solution. By using the compensation method of approximating the near-field array response vector as a segmented far-field array response vector, the near-field array response vector of equation (34) in step 4.A is approximated as being segmented into N T The segmented far-field array response vector of the segment is

[0248]

[0249] Among them, k m =2πf m / c is the wave number of the mth subcarrier, are the distance and spatial direction of Bob relative to the center of the s-th subarray, respectively, which can be expressed as:

[0250]

[0251] in, is the intermediate variable;s Specifically expressed as:

[0252]

[0253] Among them, k c =2πf c / c is the wave number of the subcarrier at the center frequency, is an intermediate variable;

[0254] According to formula (34), we get The approximate optimal solution is Right now

[0255]

[0256] Step 4.C: Derive the PSs-based partial analog beamformer F PS ;

[0257] According to step 4.B, equation (39), we get

[0258]

[0259] According to equations (1) and (2) in step 1, we get the PSs-based partial analog beamformer F PS .

[0260] Step 4.D: Derive the partial analog beamformer T based on TTDs m ;

[0261] set up According to step 4.C, formula (41), we get

[0262]

[0263] in, is an arbitrary scalar; thus we get the unconstrained optimal delay for

[0264]

[0265] By definition The optimal delay for each TTD under the maximum delay constraint is

[0266]

[0267] in,

[0268]

[0269] According to equation (3) in step 1, we get the partial analog beamformer T based on TTDs: m ;

[0270] Step 4.E: According to equation (36) in step 4.C and equation (41) in step 4.D, the optimal analog beamformer is obtained as:

[0271]

[0272] in,

[0273]

[0274] is the optimal analog beamformer connected to the kth RF chain.

[0275] At this point, from step 4.A to step 4.E, the analog beamformer connected to the kth RF chain is obtained The ideal optimal solution However, due to hardware limitations, it is generally impossible to obtain an ideal optimal solution. Approximating the near-field array response vector as the segmented far-field array response vector, we obtain The approximate optimal solution of ; and then the partial analog beamformer F based on PSs is obtained PS ; Then we can get the partial simulation beamformer T based on TTDs m ; Finally, the optimal analog beamformer is obtained

[0276] Step 5.A: Based on the optimal analog beamformer obtained in step 4 Redefine the optimization problem;

[0277] According to equation (46) in step 5, the optimization problem in equation (31) is redefined as:

[0278]

[0279] Among them, the concealment rate R at the mth subcarrier m Specifically expressed as:

[0280]

[0281] in, is a low-dimensional equivalent channel;

[0282] Step 5.B: Convert the concealment constraint into the power constraint of the received signal at Willie and update the optimization problem;

[0283] According to the concealment constraint expression of equation (48) in step 5.A, the power constraint of the received signal at Willie is obtained as:

[0284]

[0285] Among them, P th is the power threshold, which is specifically expressed as:

[0286]

[0287] According to formula (50), the optimization problem in formula (48) is transformed into:

[0288]

[0289] The problem is non-convex;

[0290] Step 5.C: Use the SCA algorithm to perform convex approximation on the optimization problem, transforming the optimization problem from a non-convex problem to a convex problem; define the auxiliary variable r m ≤R m , use the SCA algorithm to perform convex approximation on equation (52) in step 5.B, and get

[0291]

[0292] γ m (F BB,m ) is expanded by the first order Taylor, and we get

[0293]

[0294] in, is an auxiliary variable;

[0295] According to the auxiliary variable r obtained in formula (54) m , The optimization problem in formula (48) is transformed into:

[0296]

[0297] This problem is a convex problem;

[0298] Step 5.D: Use CVX solver to iteratively solve the optimization problem and obtain the optimal digital beamformer

[0299] At this point, from step 5.A to step 4.D, the optimization problem of step 3 is redefined; then the concealment constraint in the optimization problem is converted into the power constraint of the received signal at Willie, and the optimization problem is updated; then the SCA algorithm is used to perform convex approximation on the optimization problem, and the optimization problem is converted from a non-convex problem to a convex problem; finally, the CVX solver is used to iteratively solve the optimization problem and the optimal digital beamformer is obtained.

[0300] Based on repeated iterations of step two, step three, step four, and step five, the analog beamformer and digital beamformer are designed and optimized to maximize the system concealment rate under concealment constraints, power constraints, and hardware constraints, thereby ensuring the concealment of communication behavior and improving the effectiveness of communication.

[0301] Figure 2 is the average concealment probability and concealment interruption probability after the implementation of the TTD-based near-field covert communication method of the present invention and the method in Example 1 with r w The simulation results of the changes.

[0302] Figure 2 The horizontal axis is r w , the value range is 5-15m, the ordinate is the average concealment probability and concealment interruption probability, the simulation experiment gives the numerical analysis and Monte Carlo simulation of the method proposed in the present invention, Figure 2 It can be seen that the numerical analysis value of the method proposed in the present invention is basically consistent with the simulation value, which verifies the accuracy of the theoretical analysis of the present invention.

[0303] Figure 3 is the concealed rate of the near-field covert communication method based on TTD of the present invention and the method implemented in Example 1 with r w The simulation results of the changes.

[0304] Figure 3 The horizontal axis is r w , the value range is 5-15m, the ordinate is the concealment rate, the simulation experiment compares and analyzes the near-field covert communication method based on TTD and the far-field covert communication method proposed in the present invention. Figure 3 It can be seen that when Willie is closer to Alice than Bob, the covert rate of the near-field covert communication method proposed in the present invention is higher than that of the far-field covert communication method, that is, the near-field covert communication method proposed in the present invention can effectively counteract eavesdroppers at a closer distance.

[0305] The specific description above further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A terahertz near-field covert communication method based on TTD, characterized in that: The following steps are included: Step 1: Construct a covert communication system consisting of transmitter Alice, legitimate user Bob, and eavesdropper Willie. The analog beamformer of the covert communication system and digital beamformer The constraints include power constraint, constant modulus constraint and maximum delay constraint; on this basis, the near-field channel characteristics are applied to the covert communication system, and the Willie statistical test quantity and judgment criteria are determined; the expressions of the average concealment probability and concealment interruption probability of the covert communication system are obtained; Step 2: Based on the near-field covert communication system constructed in step 1, construct an unbounded noise uncertainty model at Willie; derive the probability density of the total noise power at Willie under this noise model; then determine the optimal detection threshold of Willie; finally, derive the expressions for the average concealment probability, concealment interruption probability and concealment rate; Step 3: Based on the covert communication system constructed in step 1 and the unbounded noise uncertainty model constructed in step 2, the transmitter analog beamformer F is constructed according to the power constraint, constant modulus constraint, maximum delay constraint in step 1 and the covert rate expression in step 2. RF,m and digital beamformer F BB,m Optimization problem; Step 4: Based on the optimization problem constructed in step 3, derive the analog beamformer connected to the kth RF chain The ideal optimal solution Based on the ideal optimal solution Approximating the near-field array response vector as the segmented far-field array response vector, we obtain The approximate optimal solution of the PSs-based partial analog beamformer F is obtained based on the approximate optimal solution. PS , and derive the partial simulation beamformer T based on TTDs m ; Finally, the optimal analog beamformer is obtained Step 5: Optimal analog beamformer obtained in step 4 Redefine the optimization problem in step 3; transform the concealment constraint in the optimization problem into the power constraint of the received signal at Willie, and update the optimization problem; use the SCA algorithm to perform convex approximation on the optimization problem, transforming the optimization problem from a non-convex problem to a convex problem; use the CVX solver to iteratively solve the optimization convex problem and obtain the optimal digital beamformer Realize terahertz near-field covert communication.

2. A terahertz near-field covert communication method based on TTD as claimed in claim 1, characterized in that: The specific implementation method of step one is: Step 1.A: Construct a covert communication system consisting of transmitter Alice, legitimate user Bob, and eavesdropper Willie; In the covert communication system, Alice uses N t The uniform linear array ULA and N RF A radio frequency chain, RF chain, to transmit N s data streams, and Bob and Willie are equipped with a single antenna; the covert communication system uses an M-subcarrier orthogonal frequency division multiplexing system, and the frequency of the m-th subcarrier is f m =f c +B(2m-1-M) / (2M), m∈{1,2,...,M}, where f c is the center carrier frequency, B is the bandwidth; Ailce uses an analog beamformer and digital beamformer The TTD-based hybrid beamforming architecture is composed of: In the TTD-based hybrid beamforming architecture, the analog beamformer F RF,m =F PS T m It is composed of TTDs and PSs. is a frequency-independent partial analog beamformer implemented by PSs, is the frequency-dependent partial analog beamformer implemented by TTDs, N T The number of TTDs connected to each RF chain; F PS Specifically expressed as in, is a PSs-based partial analog beamformer connected to the kth RF chain, specifically expressed as in, is the PS-based partial analog beamformer of the s-th TTD connected to the k-th RF chain, The phase shift achieved for the pth antenna connected to the sth TTD pair connected to the kth RF chain, is the number of antennas connected to each TTD; T m Specifically expressed as in, The delay vector, t, is realized for the TTDs connected to the kth RF chain. k,s The delay achieved for the sth TTD connected to the kth RF chain; in addition, the simulated beamformer and digital beamformer The power constraint and hardware constraint need to be met. The hardware constraint includes constant modulus constraint and maximum delay constraint. The power constraint is P max is the maximum transmit power; the constant modulus constraint is The maximum delay constraint is t max is the maximum delay that TTD can achieve; Step 1.B: Apply the near-field channel characteristics to the covert communication system of 1.A; Alice is the origin of the reference coordinate system, r b ,θ b ∈[-1,1] are the distance and physical direction from Bob to Alice, r w ,θ w ∈[-1,1] are the distance and physical direction from Willie to Alice respectively. Substituting the near-field channel into the covert communication system constructed in step 1.A, the received signals of Bob and Willie at the mth subcarrier at time z∈[Z] are expressed as: in, is the signal sent by Alice at the mth subcarrier, n b,m (z) and n w,m (z) are the noises at Bob and Willie respectively, is a frequency domain near-field legal channel, It is a frequency domain near-field eavesdropping channel; R = {b, w} is used to indicate that the receiver is Bob or Willie, and the near-field channel h R,m The frequency domain near-field legal channel and the frequency domain near-field eavesdropping channel are uniformly represented; the near-field channel is specifically represented as: L is the number of paths, α l is the complex gain of the lth (l=1,2,...,L)th path, is the near-field array response vector; Derived from spherical waves, it is expressed as: Where c is the speed of light, r R,l is the distance between the transmitter and the receiver or scattering point; is the distance between the qth transmitting antenna and the receiver in the lth path component, which can be specifically expressed as: in, is the spatial direction of the lth path component at the mth subcarrier, θ R,l ∈[-1,1] is the physical direction of the lth path component, d=c / (2f c ) is the antenna spacing, δ q =(2q-N t -1) / 2, q=1,2,...,N t is an intermediate variable; Step 1.C: Determine Willie's statistical test amount and judgment criteria; According to equation (8) in step 1.B, the signal received by Willie at all subcarriers is: in, It means Alice did not send a signal. Indicates that Alice is sending a signal; thus, the signal vector received by Willie is: and w =[and w (1),...,and w (z),...,and w (Z)](13) Get Willie's statistical test S(y w ) and the judgment criteria are: in, and are the determination of the received signal as noise and the determination of the received signal as signal plus noise, respectively, and η is the detection threshold of Willie; Step 1.D: Obtain the expressions of average concealment probability and concealment interruption probability of the covert communication system; According to equations (12) and (14) in step 1.C, the error detection probability of Willie is obtained as: in, is the total power of the signal received by Willie, is the power of the signal received by Willie at the mth subcarrier, is the noise power of Willie at the mth subcarrier; According to formula (15), the average concealment probability is: in, is the probability density function of uncertain noise; the concealed outage probability is obtained as follows: To ensure confidentiality, and P o <δ, where ε∈(0,1) is the precision, which is a sufficiently small positive number, and δ represents the maximum acceptable concealed outage probability.

3. A terahertz near-field covert communication method based on TTD as claimed in claim 2, characterized in that: The specific implementation method of step 2 is: Step 2.A: Construct an unbounded noise uncertainty model at Willie; Assume that the noise power of Willie at each subcarrier is the same, that is, The noise power of Bob at each subcarrier is the standard noise power, that is is the standard noise power; the unbounded noise uncertainty model at Willie is constructed as: That is, the difference between the noise power at Willie and the standard noise power follows a normal distribution in the logarithmic domain; is the dB value of the noise power at Willie, is the dB value of the standard noise power, is a parameter that quantifies the size of the noise uncertainty; Step 2.B: Derive the probability density of the total noise power at Willie; According to formula (18), the probability density of the noise power at Willie is: Where g = ln(10) / 10; the probability density of the total noise power at Willie is: Among them, M dB =10lg(M); Approximate equation (20) to a Gaussian function, that is in, Step 2.C: Determine the optimal detection threshold of Willie; According to equations (16) and (17) in step 1 and equation (21) in step 2.B, the optimal detection threshold of Willie is or * =max{φ1+P w / 2,P w } (23) Step 2.D: Obtain the expressions for average concealment probability, concealment interruption probability and concealment rate; According to equations (16) and (17) in step 1 and equation (23) in step 2, the expressions of average concealment probability and concealment interruption probability are obtained as follows: Under the concealment constraint, the concealment rate is And meet Where R m is the concealment rate at the mth subcarrier.

4. A TTD-based terahertz near-field covert communication method as claimed in claim 3, characterized in that: In step three, The optimization problem is: in, is the constant modulus constraint that must be satisfied by the PS-based partial analog beamformer, 0≤t k,s ≤t max , The maximum delay constraint that the delay achieved by TTD must satisfy.

5. A terahertz near-field covert communication method based on TTD as claimed in claim 4, characterized in that: The specific implementation method of step 4 is: Step 4.A: Solve the optimization problem (27) to obtain the analog beamformer connected to the kth RF chain The ideal optimal solution definition in, is the analog beamformer connected to the kth RF chain, which is specifically expressed as: To maximize the array gain at Bob, we have The ideal optimal solution is: Among them, ψ m For any phase, is the spatial direction of Bob at the mth subcarrier; Step 4.B: Approximate the near-field array response vector as the segmented far-field array response vector, yielding The approximate optimal solution of ; By using the compensation method of approximating the near-field array response vector as a segmented far-field array response vector, the near-field array response vector of equation (30) in step 4.A is approximated as being segmented into N T The segmented far-field array response vector of the segment is Among them, k m =2πf m / c is the wave number of the mth subcarrier, are the distance and spatial direction of Bob relative to the center of the s-th subarray, respectively, which can be expressed as: in, is the intermediate variable; s Specifically expressed as: Among them, k c =2πf c / c is the wave number of the subcarrier at the center frequency, is an intermediate variable; According to formula (31), we get The approximate optimal solution is Right now Step 4.C: Derive the PSs-based partial analog beamformer F PS ; According to step 4.B, equation (35), we get According to equations (1) and (2) in step 1, we get the PSs-based partial analog beamformer F PS ; Step 4.D: Derive the partial analog beamformer T based on TTDs m ; set up According to step 4.C, equation (37), we get in, is an arbitrary scalar; thus we get the unconstrained optimal delay for By definition The optimal delay for each TTD under the maximum delay constraint is in, According to equation (3) in step 1, we get the partial analog beamformer T based on TTDs: m ; Step 4.E: According to equation (36) in step 4.C and equation (41) in step 4.D, the optimal analog beamformer is obtained as: in, is the optimal analog beamformer connected to the kth RF chain.

6. A terahertz near-field covert communication method based on TTD as claimed in claim 5, characterized in that: The specific implementation method of step five is: Step 5.A: Based on the optimal analog beamformer obtained in step 4 Redefine the optimization problem; According to equation (42) in step 5, the optimization problem in equation (27) is redefined as: Among them, the concealment rate R at the mth subcarrier m Specifically expressed as: in, is a low-dimensional equivalent channel; Step 5.B: Convert the concealment constraint into the power constraint of the received signal at Willie and update the optimization problem; According to the concealment constraint expression of equation (44) in step 5.A, the power constraint of the received signal at Willie is obtained as: Among them, P th is the power threshold, which is specifically expressed as: According to formula (46), the optimization problem in formula (44) is transformed into: The problem is non-convex; Step 5.C: Use the SCA algorithm to perform convex approximation on the optimization problem, transforming the optimization problem from a non-convex problem to a convex problem; Define auxiliary variable r m ≤R m , use the SCA algorithm to perform convex approximation on equation (48) in step 5.B, and get γ m (F BB,m ) is expanded by the first order Taylor, and we get in, is an auxiliary variable; According to the auxiliary variable r obtained in formula (50) m , The optimization problem in formula (48) is transformed into: This problem is a convex problem; Step 5.D: Use CVX solver to iteratively solve the optimization problem and obtain the optimal digital beamformer Realize terahertz near-field covert communication.

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