A terahertz near-field covert communication method based on TTD

Abstract

A terahertz near-field covert communication method based on TTD belongs to the field of wireless communication technology. The implementation method of this invention is as follows: It utilizes near-field channel characteristics and unbounded noise uncertainty to enhance the covertness of the communication system; it analyzes the system performance indicators under the unbounded noise uncertainty model, including average covert probability, covert interruption probability, and covert rate; it adopts a TTD-based hybrid beamforming architecture, aiming to maximize the covert rate, and constructs a hybrid beamforming optimization problem under covert constraints, power constraints, and hardware constraints; in the analog domain, it uses the piecewise far-field array response to approximate the near-field array response to obtain the optimal solution of the analog beamformer, thereby mitigating the adverse effects of beam splitting and maximizing covertness; in the digital domain, it uses the SCA algorithm to transform the non-convex optimization problem into a convex form, and uses the CVX solver to solve the optimal solution of the digital beamformer to maximize the system covert rate.

Description

Technical Field

[0001] This invention relates to a terahertz near-field covert communication method based on TTD, belonging to the field of wireless communication technology. Background Technology

[0002] With the development of wireless communication technology, an increasing amount of private data and sensitive information is transmitted through wireless channels, posing a significant challenge to the security and reliability of wireless communication. Traditional encryption-based methods primarily rely on application-layer key sharing to ensure signal security, which depends mainly on the computational complexity of cracking the generated key. However, with the rapid development of computer technology, many traditional encryption techniques are becoming increasingly unreliable. Furthermore, physical layer security technologies leverage the dynamic characteristics of wireless media to minimize the information obtainable by eavesdroppers. However, all these technologies aim to ensure the security of communication content while neglecting the fact that the exposure of user communication behavior can also lead to information leakage, such as node location and transmission mode. To address this issue, scholars have proposed covert communication techniques. Covert communication techniques not only protect communication content from eavesdropping but also ensure that the communication behavior between the two parties is not 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 increasingly directional, which is beneficial for the concealment of wireless communication. THz communication, with a frequency range from 0.1THz to 10THz and capable of ultra-high transmission rates in the Tbps range, has attracted widespread attention. In addition to providing higher transmission rates, THz communication also has a natural security advantage: the high directionality of the THz band. The highly directional transmission of THz communication not only counteracts the high path loss caused by propagation, atmospheric attenuation, and scattering effects, but also confines the transmission power to a narrow beam area, naturally preventing communication activities 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] Despite the advantages of the THz band in ensuring covert communication, several challenges remain. The high directivity of the THz band effectively confines transmission power within a beam area. If eavesdroppers are located outside the beam area, covert communication is possible. However, if eavesdroppers are positioned within a fan-shaped area, the high transmission power significantly reduces the stealth of the communication. While some covert communication schemes have been proposed to combat eavesdroppers at greater distances within the beam area, they still cannot address the threat posed by eavesdroppers at closer distances. Therefore, there is an urgent need to research covert communication methods capable of countering close-range eavesdroppers. Summary of the Invention

[0005] To address the challenge of countering close-range eavesdroppers in terahertz covert communication scenarios, this invention primarily aims to provide a TTD-based terahertz near-field covert communication method. This method leverages near-field channel characteristics and unbounded noise uncertainty to enhance the covertness of the communication system. The invention analyzes system performance indicators under an unbounded noise uncertainty model, including average covert probability, covert interruption probability, and covert rate. A TTD-based hybrid beamforming architecture is employed, aiming to maximize the covert rate. Under covert constraints, power constraints, and hardware constraints, a hybrid beamforming optimization problem is constructed. In the analog domain, the optimal solution for the analog beamformer is obtained by approximating the near-field array response using a piecewise far-field array response, mitigating the adverse effects of beam splitting and maximizing covertness. In the digital domain, the non-convex optimization problem is transformed into a convex form using the SCA algorithm, and the optimal solution for the digital beamformer is solved using the CVX solver to maximize the system covert rate.

[0006] The objective of this invention is achieved through the following technical solution.

[0007] The near-field covert communication method based on TTD disclosed in this invention includes the following steps:

[0008] Step 1: Construct a covert communication system consisting of Alice (the transmitter), Bob (the legitimate user), and Willie (the eavesdropper). The covert communication system uses a simulated beamformer. and digital beamformer The constraints include power constraints, constant modulus constraints, and maximum time delay constraints; based on these constraints, the near-field channel characteristics are applied to the covert communication system to determine Willie's statistical test quantity and judgment criteria; and the expressions for the average covert probability and covert interruption probability of the covert communication system are derived.

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

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

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

[0012]

[0013] in, For a PSs-based partially analog beamformer connected to the k-th RF chain, specifically denoted as

[0014]

[0015] in, For a PS-based partially analog beamformer connected to the s-th TTD of the k-th RF chain, The phase shift implemented for the s-th TTD connected to the p-th antenna in the k-th RF chain. The number of antennas connected to each TTD; T m Specifically represented as

[0016]

[0017] in, For the delay vector implemented by TTDs connected to the k-th RF chain, t k,s The time delay implemented for the s-th TTD connected to the k-th RF chain. Additionally, the simulated beamformer... and digital beamformer It needs to meet power constraints and hardware constraints, including constant modulus constraints and maximum time delay constraints. The power constraint is as follows:

[0018]

[0019] P max Maximum transmit power; constant mode constraint is

[0020]

[0021] Maximum delay constraint is

[0022]

[0023] t max This represents the maximum latency that TTD can achieve.

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

[0025] With Alice as the origin of the reference coordinate system, r b ,θ b ∈[-1,1] represents the distance and physical direction from Bob to Alice, respectively, r w ,θ w ∈[-1,1] represent 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 m-th subcarrier at time z∈[Z] are expressed as follows:

[0026]

[0027] in, Let n be the signal transmitted by Alice at the m-th subcarrier. b,m (z) and n w,m (z) represents the noise at Bob and Willie, respectively. For frequency domain near-field legal channels, This is a frequency domain near-field eavesdropping channel;

[0028] For simplicity, let R = {b, w} represent the receiver as Bob or Willie, and the near-field channel h R,m A unified representation is used for frequency-domain near-field legitimate channels and frequency-domain near-field eavesdropping channels. The near-field channel is specifically represented as follows:

[0029]

[0030] L is the number of paths, α l Let L be the complex gain of the l-th path (l = 1, 2, ..., L). This 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 The distance between the transmitter and receiver or the scattering point; The distance between the q-th transmitting antenna and the receiver in the l-th path component is specifically expressed as:

[0033]

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

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

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

[0037]

[0038] in, This indicates that Alice did not send a signal. This indicates that Alice is sending a signal; therefore, the signal vector received by Willie is:

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

[0040] This leads to Willie's statistical test S(y). w The judgment criteria are as follows:

[0041]

[0042] in, and These are the criteria for determining whether the received signal is noise and whether the received signal is a signal plus noise, respectively, where η is Willie's detection threshold.

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

[0044] Based on equations (12) and (14) in step 1.C, Willie's false detection probability is obtained as follows:

[0045]

[0046] in, This represents the total power of the signal Willie received. Let m be the power of the signal received by Willie at the m-th subcarrier. Let m be the noise power of Willie at the m-th subcarrier;

[0047] Since the actual noise has uncertainty, according to equation (15), the average concealment probability is obtained as:

[0048]

[0049] in, Let be the probability density function of uncertain noise; from this, the probability of covert interruption is:

[0050]

[0051] To ensure concealment, it is required and P o <δ, where ε∈(0,1) is the precision, which is a sufficiently small positive number, and δ represents the acceptable maximum probability of covert interruption;

[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 at Willie, and derive the expressions for the average covert probability, covert interruption probability, and covert rate.

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

[0054] Assume Willie's noise power is the same at each subcarrier, i.e. Bob's noise power at each subcarrier is the standard noise power, i.e. Given the standard noise power; the unbounded noise uncertainty model at Willie is constructed as follows:

[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; where, Here is the dB value of the noise power at Willie. This represents the standard noise power in dB. This is a parameter used to quantify the magnitude of noise uncertainty;

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

[0058] According to equation (18), the probability density of the noise power at Willie is obtained as follows:

[0059]

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

[0061]

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

[0063] Equation (20) can be approximated as a Gaussian function, i.e.

[0064]

[0065] in,

[0066]

[0067] Step 2.C: Determine Willie's optimal detection threshold;

[0068] Based on equations (16) and (17) in step one and equation (21) in step 2.B, Willie's optimal detection threshold is obtained as follows:

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

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

[0071] Based on equations (16) and (17) in step one and equation (23) in step two, the expressions for the average concealment probability and the concealment interruption probability are obtained as follows:

[0072]

[0073] Under concealment constraints, the concealment rate is

[0074]

[0075] And satisfy Where R m Let m be the concealment rate at the m-th subcarrier;

[0076] Step 3: Based on the covert communication system constructed in Step 1 and the unbounded noise uncertainty model constructed in Step 2, and according to the power constraint, constant mode constraint, maximum time delay constraint in Step 1 and the covert rate expression in Step 2, construct the transmitter simulated beamformer F. RF,m and digital beamformer F BB,mThe optimization problem;

[0077] The optimization problem is:

[0078]

[0079] in, The constant mode constraint that a PS-based partially analog beamformer must satisfy. The latency that TTD must achieve must satisfy

[0080] Step 4: Based on the optimization problem constructed in Step 3, the analog beamformer connected to the k-th RF chain is derived. Ideal optimal solution Based on the ideal optimal solution By approximating the near-field array response vector as a piecewise far-field array response vector, we obtain... The approximate optimal solution is obtained; based on the approximate optimal solution, the partially simulated beamformer F based on PSs is derived. PS And derived a partially simulated beamformer T based on TTDs. m Finally, the optimal simulated beamformer was determined.

[0081] Step 4.A: Solve the optimization problem (27) to obtain the simulated beamformer connected to the k-th RF chain. Ideal optimal solution

[0082] definition

[0083]

[0084] in, For an analog beamformer connected to the k-th RF chain, specifically:

[0085]

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

[0087]

[0088] Among them, ψ m For any phase, Let Bob's spatial direction be at the m-th subcarrier; due to hardware limitations, this is generally not possible. Ideal optimal solution

[0089] Step 4.B: Approximate the near-field array response vector as a piecewise far-field array response vector, and obtain... The approximate optimal solution;

[0090] As explained in step 4.A, it is generally impossible to obtain the ideal optimal solution. By using a compensation method that approximates the near-field array response vector as a piecewise far-field array response vector, the near-field array response vector in step 4.A (30) is approximated as a vector divided into N segments. T The segmented far-field array response vector of the segment, i.e.

[0091]

[0092] Where, k m =2πf m / c is the wave number of the m-th subcarrier. Let represent Bob's distance and spatial direction relative to the center of the s-th subarray, respectively.

[0093]

[0094]

[0095] in, β is an intermediate variable. s Specifically, it is expressed as follows:

[0096]

[0097] Where, k c =2πf c / c represents the wavenumber of the subcarrier at the center frequency. As an intermediate variable;

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

[0099]

[0100] Step 4.C: Obtain the partially simulated beamformer F based on PSs. PS ;

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

[0102]

[0103] Based on equations (1) and (2) in step one, the partially simulated beamformer F based on PSs is obtained. PS .

[0104] Step 4.D: Obtain the partially simulated beamformer T based on TTDs. m ;

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

[0106]

[0107] in, It is an arbitrary scalar; thus, the unconstrained optimal time delay is obtained. for

[0108]

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

[0110]

[0111] in,

[0112]

[0113] According to equation (3) in step one, the partially simulated beamformer T based on TTDs is obtained. m ;

[0114] Step 4.E: Based on equation (36) in Step 4.C and equation (41) in Step 4.D, the optimal simulated beamformer is obtained as follows:

[0115]

[0116] in,

[0117]

[0118] The optimal analog beamformer for connection to the k-th RF chain;

[0119] Step 5: Based on the optimal simulated beamformer obtained in Step 4 The optimization problem in step three is redefined; then, the hidden constraints in the optimization problem are transformed into power constraints 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, transforming it from a non-convex problem into a convex problem; finally, the CVX solver is used to iteratively solve the convex optimization problem, yielding the optimal digital beamformer. To achieve terahertz near-field covert communication.

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

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

[0122]

[0123] Wherein, the concealment rate R at the m-th subcarrier m Specifically, it is expressed as follows:

[0124]

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

[0126] Step 5.B: Transform the concealment constraint into a power constraint on the received signal at Willie's location, and update the optimization problem;

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

[0128]

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

[0130]

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

[0132]

[0133] This problem is non-convex;

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

[0135]

[0136] For γ m (F BB,m Performing a first-order Taylor expansion, we obtain...

[0137]

[0138] in, As an auxiliary variable;

[0139] The auxiliary variable r obtained from equation (50) m , The optimization problem in equation (48) is transformed into:

[0140]

[0141] This problem is a convex problem;

[0142] Step 5.D: Iteratively solve the optimization problem using the CVX solver to obtain the optimal digital beamformer. To achieve terahertz near-field covert communication.

[0143] Beneficial effects:

[0144] 1. This invention discloses a near-field covert communication method based on TTD, which addresses the problem that THz communication is difficult to counter eavesdroppers at close range. It utilizes near-field channel characteristics and unbounded noise uncertainty to enhance the covertness of the communication system. By establishing a near-field covert communication model, designing and optimizing analog beamformers and digital beamformers, and realizing near-field covert communication based on TTD using digital beamformers.

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

[0146] 3. The present invention discloses a near-field covert communication method based on TTD. Based on the near-field covert communication model, the non-convex optimization problem is transformed into a convex form by using the SCA algorithm, and the optimal solution of the digital beamformer of the CVX solver is used to maximize the system concealment rate. Attached Figure Description

[0147] Figure 1 This is an overall flowchart of the near-field covert communication method based on TTD and its embodiment 1 of the present invention;

[0148] Figure 2 The average concealment probability and concealment interruption probability after implementing the near-field covert communication method based on TTD in this invention and in Embodiment 1 are related to r. w The simulation results show the changes;

[0149] Figure 3 The concealment rate of the near-field covert communication method based on TTD of the present invention and the method implemented in Embodiment 1 varies with r. w The simulation results show the changes. Detailed Implementation

[0150] To better illustrate the purpose and advantages of the present invention, the near-field covert communication method based on TTD described in the present invention will be further explained below with reference to the accompanying drawings and examples.

[0151] Example 1:

[0152] This embodiment details the specific steps of implementing the TTD-based near-field covert communication method of the present invention.

[0153] This case study considers a THz covert communication system involving a transmitter Alice, a legitimate user Bob, and an eavesdropper Willie. Alice secretly sends signals to Bob while Willie is continuously eavesdropping. Due to the high directionality of the THz band, THz communication can confine the signal transmission power to the beam area, effectively preventing the communication from being detected by eavesdroppers outside the beam area, thus possessing a natural security advantage. However, if the eavesdroppers are distributed within the beam area, the high transmission power will severely reduce the covertness of the communication. Although some covert communication schemes have been proposed to combat eavesdroppers at a greater distance within the beam area, they still cannot solve the threat of eavesdroppers at a closer distance within the beam area. Therefore, a near-field covert communication method based on TTD is adopted, which can counter eavesdroppers at a closer distance while improving the system's covertness rate.

[0154] like Figure 1 As shown in the figure, the specific implementation steps of the near-field covert communication method based on TTD disclosed in this embodiment are as follows:

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

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

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

[0158]

[0159] in, For a PSs-based partially analog beamformer connected to the k-th RF chain, specifically denoted as

[0160]

[0161] in, For a PS-based partially analog beamformer connected to the s-th TTD of the k-th RF chain, The phase shift implemented for the s-th TTD connected to the p-th antenna in the k-th RF chain. The number of antennas connected to each TTD; T m Specifically represented as

[0162]

[0163] in, For the delay vector implemented by TTDs connected to the k-th RF chain, t k,s The time delay implemented for the s-th TTD connected to the k-th RF chain. Additionally, the simulated beamformer... and digital beamformer It needs to meet power constraints and hardware constraints, including constant modulus constraints and maximum time delay constraints. The power constraint is as follows:

[0164]

[0165] P max Maximum transmit power; constant mode constraint is

[0166]

[0167] Maximum delay constraint is

[0168]

[0169] t max This represents the maximum latency that TTD can achieve.

[0170] Specifically, in this embodiment, M = 128, B = 10GHz, 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] With Alice as the origin of the reference coordinate system, r b ,θ b ∈[-1,1] represents the distance and physical direction from Bob to Alice, respectively, r w ,θ w ∈[-1,1] represent 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 m-th subcarrier at time z∈[Z] are expressed as follows:

[0173]

[0174] in, Let n be the signal transmitted by Alice at the m-th subcarrier. b,m (z) and n w,m (z) represents the noise at Bob and Willie, respectively. For frequency domain near-field legal channels, This is a frequency domain near-field eavesdropping channel;

[0175] For simplicity, let R = {b, w} represent the receiver as Bob or Willie, and the near-field channel h R,m A unified representation is used for frequency-domain near-field legitimate channels and frequency-domain near-field eavesdropping channels. The near-field channel is specifically represented as follows:

[0176]

[0177] L is the number of paths, α l Let L be the complex gain of the l-th path (l = 1, 2, ..., L). This 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 The distance between the transmitter and receiver or the scattering point; The distance between the q-th transmitting antenna and the receiver in the l-th path component is specifically expressed as:

[0181]

[0182] in, Let θ be the spatial direction of the l-th path component at the m-th subcarrier. R,l ∈[-1,1] represents the physical direction of the l-th path component, d=c / (2f c ) represents the antenna spacing, δ q =(2q-N) t -1) / 2, q=1,2,...,N t As 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 and judgment criteria;

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

[0186]

[0187] in, This indicates that Alice did not send a signal. This 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 leads to Willie's statistical test S(y). w The judgment criteria are as follows:

[0190]

[0191] in, and These are the criteria for determining whether the received signal is noise and whether the received signal is a signal plus noise, respectively, where η is Willie's detection threshold.

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

[0193]

[0194] in, This represents the total power of the signal Willie received. Let m be the power of the signal received by Willie at the m-th subcarrier. Let m be the noise power of Willie at the m-th subcarrier;

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

[0196] The probability of error detection is defined as follows:

[0197] P e =P FA +P MD >1-ε (16)

[0198] Among them, P FA and P MD Let these be the false alarm probability and the false negative probability, respectively, denoted as .

[0199]

[0200] Based on equation (16) and equations (12) and (14) in step 1.C, Willie's false detection probability is obtained as follows:

[0201]

[0202] Since the actual noise has uncertainty, according to equation (19), the average concealment probability is obtained as:

[0203]

[0204] in, Let be the probability density function of uncertain noise; from this, the probability of covert interruption is:

[0205]

[0206] To ensure concealment, it is required and P o <δ, where ε∈(0,1) is the precision, which is a sufficiently small positive number, and δ represents the acceptable maximum probability of covert interruption;

[0207] In this specific embodiment, ε = 0.01, δ = 0.01;

[0208] Thus far, 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 and judgment criteria have been determined; and finally, the expressions for the average covert probability and covert interruption probability of the covert communication system have been derived.

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

[0210] Assume Willie's noise power is the same at each subcarrier, i.e. Bob's noise power at each subcarrier is the standard noise power, i.e. Given the standard noise power; the unbounded noise uncertainty model at Willie is constructed as follows:

[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; where, Here is the dB value of the noise power at Willie. This represents the standard noise power in dB. This is a parameter used to quantify the magnitude of noise uncertainty;

[0213] Specifically, in this embodiment,

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

[0215] According to equation (22), the probability density of the noise power at Willie is obtained as follows:

[0216]

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

[0218]

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

[0220] Equation (24) can be approximated as a Gaussian function, i.e.

[0221]

[0222] in,

[0223]

[0224] Step 2.C: Determine Willie's optimal detection threshold;

[0225] Based on equations (20) and (21) in step one and equation (25) in step 2.B, Willie's optimal detection threshold is obtained as follows:

[0226] η * =max{φ1+P w / 2,P w} (27)

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

[0228] Based on equations (20) and (21) in step one and equations (25) and (27) in step two, the expressions for the average concealment probability and the concealment interruption probability are obtained as follows:

[0229]

[0230] Under concealment constraints, the concealment rate is

[0231]

[0232] And satisfy Where R m Let m be the concealment rate at the m-th subcarrier;

[0233] Thus far, from step 2.A to step 2.D, the construction of the unbounded noise uncertainty model 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 for Willie has been determined; and the expressions for 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, construct the transmitter simulated beamformer F according to equations (4), (5), and (6) in Step 1 and equation (30) in Step 2. RF,m and digital beamformer F BB,m The optimization problem is:

[0235]

[0236] in, The constant mode constraint that a PS-based partially analog beamformer must satisfy. The maximum delay constraint that the delay achieved by TTD must satisfy;

[0237] At this point, step three is complete, constructing the transmitter-side simulated beamformer F. RF,mand digital beamformer F BB,m The optimization problem.

[0238] Step 4.A: Solve the optimization problem (31) to obtain the simulated beamformer connected to the k-th RF chain. Ideal optimal solution

[0239] definition

[0240]

[0241] in, For an analog beamformer connected to the k-th RF chain, specifically:

[0242]

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

[0244]

[0245] Among them, ψ m For any phase, Let Bob's spatial direction be at the m-th subcarrier; due to hardware limitations, this is generally not possible. Ideal optimal solution

[0246] Step 4.B: Approximate the near-field array response vector as a piecewise far-field array response vector, and obtain... The approximate optimal solution;

[0247] As explained in step 4.A, it is generally impossible to obtain the ideal optimal solution. By using a compensation method that approximates the near-field array response vector as a piecewise far-field array response vector, the near-field array response vector in step 4.A (34) is approximated as a vector divided into N segments. T The segmented far-field array response vector of the segment, i.e.

[0248]

[0249] Where, k m =2πf m / c is the wave number of the m-th subcarrier. Let represent Bob's distance and spatial direction relative to the center of the s-th subarray, respectively.

[0250]

[0251] in, β is an intermediate variable.s Specifically, it is expressed as follows:

[0252]

[0253] Where, k c =2πf c / c represents the wavenumber of the subcarrier at the center frequency. As an intermediate variable;

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

[0255]

[0256] Step 4.C: Obtain the partially simulated beamformer F based on PSs. PS ;

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

[0258]

[0259] Based on equations (1) and (2) in step one, the partially simulated beamformer F based on PSs is obtained. PS .

[0260] Step 4.D: Obtain the partially simulated beamformer T based on TTDs. m ;

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

[0262]

[0263] in, It is an arbitrary scalar; thus, the unconstrained optimal time delay is obtained. for

[0264]

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

[0266]

[0267] in,

[0268]

[0269] According to equation (3) in step one, the partially simulated beamformer T based on TTDs is obtained. m ;

[0270] Step 4.E: Based on equation (36) in Step 4.C and equation (41) in Step 4.D, the optimal simulated beamformer is obtained as follows:

[0271]

[0272] in,

[0273]

[0274] The optimal analog beamformer is connected to the k-th RF chain.

[0275] Thus, from steps 4.A to 4.E, the analog beamformer connected to the k-th RF chain is obtained. Ideal optimal solution However, due to hardware limitations, it is generally impossible to obtain the ideal optimal solution; and based on the ideal optimal solution... By approximating the near-field array response vector as a piecewise far-field array response vector, we obtain... The approximate optimal solution is obtained; and then the partially simulated beamformer F based on PSs is derived. PS Furthermore, a partially simulated beamformer T based on TTDs is derived. m Finally, the optimal simulated beamformer was determined.

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

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

[0278]

[0279] Wherein, the concealment rate R at the m-th subcarrier m Specifically, it is expressed as follows:

[0280]

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

[0282] Step 5.B: Transform the concealment constraint into a power constraint on the received signal at Willie's location, and update the optimization problem;

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

[0284]

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

[0286]

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

[0288]

[0289] This problem is non-convex;

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

[0291]

[0292] For γ m (F BB,m Performing a first-order Taylor expansion, we obtain...

[0293]

[0294] in, As an auxiliary variable;

[0295] The auxiliary variable r obtained from equation (54) m , The optimization problem in equation (48) is transformed into:

[0296]

[0297] This problem is a convex problem;

[0298] Step 5.D: Iteratively solve the optimization problem using the CVX solver to obtain the optimal digital beamformer.

[0299] Thus, from step 5.A to step 4.D, the optimization problem in step three is redefined; then, the hidden constraints in the optimization problem are transformed into power constraints 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, transforming it from a non-convex problem into a convex problem; finally, the CVX solver is used to iteratively solve the optimization problem, yielding the optimal digital beamformer.

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

[0301] Figure 2 The average concealment probability and concealment interruption probability after implementing the near-field covert communication method based on TTD in this invention and in Embodiment 1 are related to r. w The simulation results show the changes.

[0302] Figure 2 The x-coordinate is r w The value range is 5-15m, and the vertical axis represents the average concealment probability and concealment interruption probability. Simulation experiments provide numerical analysis and Monte Carlo simulation of the method proposed in this invention. Figure 2 It can be seen that the numerical analysis values ​​and simulation values ​​of the method proposed in this invention are basically consistent, which verifies the accuracy of the theoretical analysis of this invention.

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

[0304] Figure 3 The x-coordinate is r w The value ranges from 5-15m, with the vertical axis representing the concealment rate. Simulation experiments compared and analyzed the near-field and far-field concealed communication methods based on TTD proposed in this invention. Figure 3 It can be seen that when Willie is closer to Alice than Bob, the concealment rate of the near-field covert communication method proposed in this invention is higher than that of the far-field covert communication method. That is, the near-field covert communication method proposed in this invention can effectively counter eavesdroppers at close range.

[0305] The above specific description further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above description 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: Includes the following steps, Step 1: Construct a covert communication system consisting of Alice (the transmitter), Bob (the legitimate user), and Willie (the eavesdropper). The covert communication system uses a simulated beamformer. and digital beamformer The constraints include power constraints, constant modulus constraints, and maximum time delay constraints; based on these constraints, the near-field channel characteristics are applied to the covert communication system to determine Willie's statistical test quantity and judgment criteria; and the expressions for the average covert probability and covert interruption probability of the covert communication system are derived. Let R = {b, w} represent the receiver as Bob or Willie, and the near-field channel h R,m The terms "frequency domain near-field legitimate channel" and "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 Let L be the complex gain of the l-th path (l = 1, 2, ..., L). This 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 f is the distance between the transmitter and receiver or the scattering point; m Let m be the frequency of the m-th subcarrier; The distance between the q-th transmitting antenna and the receiver in the l-th path component is specifically expressed as: in, Let θ be the spatial direction of the l-th path component at the m-th subcarrier. R,l ∈[-1,1] represents the physical direction of the l-th path component, d=c / (2f c ) represents the antenna spacing, f c For the center carrier frequency, δ q =(2q-N) t -1) / 2, q=1,2,...,N t As an intermediate variable; 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 at Willie; finally, derive the expressions for the average covert probability, covert interruption probability, and covert rate. Assume Willie's noise power is the same at each subcarrier, i.e. Bob's noise power at each subcarrier is the standard noise power, i.e. Standard noise power; The unbounded noise uncertainty model at Willie is constructed as follows: That is, the difference between the noise power at Willie and the standard noise power follows a normal distribution in the logarithmic domain; where, Here is the dB value of the noise power at Willie. This represents the standard noise power in dB. This is a parameter used to quantify the magnitude of noise uncertainty; Step 3: Based on the covert communication system constructed in Step 1 and the unbounded noise uncertainty model constructed in Step 2, and according to the power constraint, constant mode constraint, maximum time delay constraint in Step 1 and the covert rate expression in Step 2, construct the transmitter simulated beamformer F. RF,m and digital beamformer F BB,m Optimization problem; The optimization problem is: in, The constant mode constraint that a PS-based partially analog beamformer must satisfy. The maximum delay constraint that R must satisfy for the delay achieved by TTD is... m Let m be the concealment rate at the m-th subcarrier. Let R be the average concealment probability, R be the concealment rate, and t be the concealment rate. max Let ε be the maximum latency achievable by TTD, and ε be the precision. For the phase shift implemented by the s-th TTD connected to the k-th RF chain and the p-th antenna, t k,s For the delay of connecting to the s-th TTD implementation of the k-th RF chain, P max For maximum transmission power, t max M represents the maximum delay achievable by TTD, and M represents the number of subcarrier orthogonal frequency division multiplexing systems. Step 4: Based on the optimization problem constructed in Step 3, the analog beamformer connected to the k-th RF chain is derived. Ideal optimal solution Based on the ideal optimal solution By approximating the near-field array response vector as a piecewise far-field array response vector, we obtain... The approximate optimal solution is obtained; based on the approximate optimal solution, the partially simulated beamformer F based on PSs is derived. PS And derived a partially simulated beamformer T based on TTDs. m Finally, the optimal simulated beamformer was determined. Step 5: Based on the optimal simulated beamformer obtained in Step 4 The optimization problem in step three is redefined; the hidden constraints in the optimization problem are transformed into power constraints of the received signal at Willie, and the optimization problem is updated; the SCA algorithm is used to perform convex approximation on the optimization problem, transforming it from a non-convex problem into a convex problem; the CVX solver is used to iteratively solve the convex optimization problem, yielding the optimal digital beamformer. To achieve terahertz near-field covert communication.

2. The terahertz near-field covert communication method based on TTD as described in claim 1, characterized in that: The specific implementation method of step one is as follows: Step 1.A: Construct a covert communication system consisting of Alice (the transmitter), Bob (the legitimate user), and Willie (the eavesdropper); In the covert communication system, Alice uses N t ULA and N are uniform linear arrays composed of antenna elements. RF A radio frequency chain, or RF chain, is used to transmit N. s There are multiple data streams, while Bob and Willie are each equipped with a single antenna; the covert communication system uses an M-subcarrier orthogonal frequency division multiplexing system, with the frequency of the m-th subcarrier being f. m =f c +B(2m-1-M) / (2M), m∈{1,2,...,M}, where f c B is the center carrier frequency, and B is the bandwidth; Ailce uses an analog beamformer. and digital beamformer A hybrid beamforming architecture based on TTD; In a TTD-based hybrid beamforming architecture, the simulated beamformer F RF,m =F PS T m Composed of TTDs and PSs, This is a frequency-independent partially analog beamformer implemented by PSs. For a frequency-dependent partially analog beamformer implemented by TTDs, N T The number of TTDs connected to each RF chain; F PS Specifically represented as in, For a PSs-based partially analog beamformer connected to the k-th RF chain, specifically denoted as in, For a PS-based partially analog beamformer connected to the s-th TTD of the k-th RF chain, The phase shift implemented for the s-th TTD connected to the p-th antenna in the k-th RF chain. The number of antennas connected to each TTD; T m Specifically represented as in, For the delay vector implemented by TTDs connected to the k-th RF chain, t k,s The time delay implemented for connecting to the s-th TTD of the k-th RF chain; in addition, the analog beamformer and digital beamformer The requirements are to satisfy power constraints and hardware constraints, including constant modulus constraints and maximum time delay constraints; among them, the power constraint is as follows: P max The maximum transmit power is; the constant mode constraint is... Maximum delay constraint is t max This represents the maximum latency that TTD can achieve. Step 1.B: Apply the near-field channel characteristics to the covert communication system of 1.A; With Alice as the origin of the reference coordinate system, r b θ b ∈[-1,1] represents the distance and physical direction from Bob to Alice, respectively, r w θ w ∈[-1,1] represent 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 m-th subcarrier at time z∈[Z] are expressed as follows: in, Let n be the signal transmitted by Alice at the m-th subcarrier. b,m (z) and n w,m (z) represents the noise at Bob and Willie, respectively. For frequency domain near-field legal channels, This is a frequency domain near-field eavesdropping channel; Step 1.C: Determine Willie's statistical test and judgment criteria; According to equation (8) in step 1.B, the signals received by Willie at all subcarriers are: in, This indicates that Alice did not send a signal. This indicates that Alice is sending a signal; therefore, the signal vector received by Willie is: and w =[and w (1),...,and w (z),...,and w (Z)] (13) We obtain Willie's statistical test S(y) w The judgment criteria are as follows: in, and These are the criteria for determining whether the received signal is noise and whether the received signal is a signal plus noise, respectively, where η is Willie's detection threshold. Step 1.D: Derive the expressions for the average concealment probability and concealment interruption probability of the covert communication system; Based on equations (12) and (14) in step 1.C, Willie's false detection probability is obtained as follows: in, This represents the total power of the signal Willie received. Let m be the power of the signal received by Willie at the m-th subcarrier. Let be Willie's noise power at the m-th subcarrier; According to equation (15), the average concealment probability is obtained as: in, Let be the probability density function of uncertain noise; from this, the probability of covert interruption is: To ensure concealment, it is required and P o <δ, where ε∈(0,1) is the precision, which is a sufficiently small positive number, and δ represents the acceptable maximum probability of covert interruption.

3. The terahertz near-field covert communication method based on TTD as described in claim 2, characterized in that: The specific implementation method of step two is as follows: Step 2.A: Construct the unbounded noise uncertainty model at Willie; Step 2.B: Derive the probability density of the total noise power at Willie; According to equation (18), the probability density of the noise power at Willie is obtained as follows: Where g = ln(10) / 10; thus, the probability density of the total noise power at Willie is: Among them, M dB =10lg(M); Equation (20) can be approximated as a Gaussian function, i.e. in, Step 2.C: Determine Willie's optimal detection threshold; Based on equations (16) and (17) in step one and equation (21) in step 2.B, Willie's optimal detection threshold is obtained as follows: or * =max{φ1+P w / 2,P w } (23) Step 2.D: Derive the expressions for the average concealment probability, concealment interruption probability, and concealment rate; Based on equations (16) and (17) in step one and equation (23) in step two, the expressions for the average concealment probability and the concealment interruption probability are obtained as follows: Under concealment constraints, the concealment rate is And satisfy Where R m Let be the concealment rate at the m-th subcarrier.

4. The terahertz near-field covert communication method based on TTD as described in claim 1, characterized in that: The specific implementation method for step four is as follows: Step 4.A: Solve the optimization problem (27) to obtain the simulated beamformer connected to the k-th RF chain. Ideal optimal solution definition in, For an analog beamformer connected to the k-th RF chain, specifically: To maximize the array gain at Bob, we obtain The ideal optimal solution is: Where, ψ m For any phase, Let Bob's spatial direction be at the m-th subcarrier; Step 4.B: Approximate the near-field array response vector as a piecewise far-field array response vector, and obtain... The approximate optimal solution; By using a compensation method that approximates the near-field array response vector as a piecewise far-field array response vector, the near-field array response vector in step 4.A (30) is approximated as a vector divided into N segments. T The segmented far-field array response vector of the segment, i.e. Where, k m =2πf m / c is the wave number of the m-th subcarrier. Let represent Bob's distance and spatial direction relative to the center of the s-th subarray, respectively. in, β is an intermediate variable. s Specifically, it is expressed as follows: Where, k c =2πf c / c represents the wavenumber of the subcarrier at the center frequency. As an intermediate variable; According to equation (31), we get The approximate optimal solution is Right now Step 4.C: Obtain the partially simulated beamformer F based on PSs. PS ; According to equation (35) in step 4.B, we get Based on equations (1) and (2) in step one, the partially simulated beamformer F based on PSs is obtained. PS ; Step 4.D: Obtain the partially simulated beamformer T based on TTDs. m ; set up According to equation (37) in step 4.C, we obtain in, It is an arbitrary scalar; thus, the unconstrained optimal time delay is obtained. for By definition The optimal delay for each TTD under the maximum delay constraint is obtained as follows: in, According to equation (3) in step one, the partially simulated beamformer T based on TTDs is obtained. m ; Step 4.E: Based on equation (36) in Step 4.C and equation (41) in Step 4.D, the optimal simulated beamformer is obtained as follows: in, The optimal analog beamformer is connected to the k-th RF chain.

5. The terahertz near-field covert communication method based on TTD as described in claim 4, characterized in that: The specific implementation method of step five is as follows: Step 5.A: Based on the optimal simulated beamformer obtained in Step 4 Redefine the optimization problem; According to equation (42) in step five, the optimization problem in equation (27) is redefined as: Wherein, the concealment rate R at the m-th subcarrier m Specifically, it is expressed as follows: in, This is a low-dimensional equivalent channel; Step 5.B: Transform the concealment constraint into a power constraint on the received signal at Willie's location, and update the optimization problem; Based on the concealment constraint expression in equation (44) of step 5.A, the power constraint of the received signal at Willie is obtained as follows: Among them, P th The power threshold is specifically expressed as: According to equation (46), the optimization problem in equation (44) is transformed into: This problem is non-convex; Step 5.C: Use the SCA algorithm to approximate the optimization problem convexly, transforming the optimization problem from a non-convex problem into a convex problem; Define auxiliary variable r m ≤R m Using the SCA algorithm to perform convex approximation on equation (48) in step 5.B, we obtain... For γ m (F BB,m Performing a first-order Taylor expansion, we obtain... in, As an auxiliary variable; The auxiliary variable r obtained from equation (50) m , The optimization problem in equation (48) is transformed into: This problem is a convex problem; Step 5.D: Iteratively solve the optimization problem using the CVX solver to obtain the optimal digital beamformer. To achieve terahertz near-field covert communication.

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