High-reliability device authentication method based on antenna array error characteristics and double-beam transmission

Abstract

The application belongs to the technical field of wireless communication security and signal processing, and discloses a high-reliability device authentication method based on antenna array error characteristics and double-beam transmission, which comprises the following steps: 1, constructing a double-beam, wherein signals received by a user equipment are transmitted through the double-beam; 2, actively tracking the double-beam and adjusting the direction of the double-beam in real time; 3, modeling a statistical analysis model of a radiation mode; 4, defining a radiation mode characteristic extraction target, completing characteristic extraction, and calculating gain error values, phase error values and position error values; 5, calculating signal energy, comparing the signal energy with a preset threshold as an energy detection statistical quantity, judging the legitimacy of a signal source, and completing device authentication. The authentication scheme provided by the application can effectively cope with the influence of array errors, thereby providing a reliable and efficient authentication solution for a millimeter wave communication system and helping to enhance the security of a communication link.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication security and signal processing technology, specifically a highly reliable device authentication method based on antenna array error characteristics and dual-beam transmission. Background Technology

[0002] Millimeter-wave communication systems are a crucial component of next-generation wireless communication technologies. Leveraging the wide bandwidth of their high-frequency bands, they can meet the high data rate demands of 5G and future wireless networks. Through compact antenna array design and efficient beamforming, millimeter-wave communication significantly enhances communication capacity and directionality, becoming a vital technological foundation for innovation in key areas such as industrial automation, autonomous driving, and precision agriculture. Furthermore, with their high directionality, high data rate, and high frequency, millimeter-wave communication systems are widely considered a significant driving force for realizing future intelligent interconnectivity.

[0003] However, the security of millimeter-wave communication systems faces numerous challenges in increasingly complex network environments. The high directionality and limited coverage of millimeter-wave signals make them vulnerable to identity spoofing attacks. Attackers can impersonate legitimate users or devices to gain unauthorized access, thereby disrupting communication links or even committing malicious acts. This potential harm is particularly pronounced in critical scenarios with high security requirements, such as autonomous driving and industrial automation. Therefore, designing efficient and reliable security mechanisms to address the authentication problem of millimeter-wave communication systems has become a crucial and urgent task.

[0004] Currently, Physical Layer Authentication (PLA) is an emerging technology that achieves identity verification by utilizing the unique physical characteristics of wireless channels or hardware. Compared with traditional cryptographic methods, PLA offers significant advantages such as no need for key management, adaptability to dynamic networks, and low computational overhead. However, existing PLA methods still have significant shortcomings, mainly: First, many schemes rely on single physical characteristics such as channel sparsity or hardware defects, resulting in insufficient distinguishability of authentication features and difficulty in achieving efficient and reliable identity authentication in complex dynamic scenarios; Second, single-beam transmission authentication methods are sensitive to beam blocking and device movement, leading to poor stability and reliability; Third, there is a lack of systematic statistical performance evaluation methods, making it difficult to comprehensively quantify the actual effectiveness of authentication schemes.

[0005] Therefore, how to fully utilize the multidimensional physical characteristics of millimeter-wave communication systems to improve the accuracy and reliability of equipment authentication is an important research direction at present. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a highly reliable device authentication method based on antenna array error characteristics and dual-beam transmission. By combining three random array error characteristics—gain, phase, and position—the distinguishability of radiation patterns is improved. The use of dual-beam transmission technology enhances authentication reliability. By statistically modeling the radiation patterns and introducing composite hypothesis testing theory, the authentication scheme proposed in this invention can effectively address the impact of array errors, thus providing a reliable and efficient authentication solution for millimeter-wave communication systems and contributing to enhanced communication link security.

[0007] To achieve the above objectives, the present invention is implemented through the following technical solution:

[0008] This invention is a highly reliable device authentication method based on antenna array error characteristics and dual-beam transmission. This method is applied to millimeter-wave communication systems. User equipment (UE) needs to verify whether the signal it receives originates from a legitimate base station (Alice) and not a spoofed attacker (Eve). By extracting and analyzing the gain, phase, and position error characteristics of the base station's radiation pattern, the UE determines the source of the signal. The specific authentication method includes the following steps:

[0009] Step 1: Construct dual beams. The signal received by the user equipment (UE) is transmitted through dual beams, and the parameters of the transmission path are calculated, and the dual beam weighting vector is calculated.

[0010] Step 2: Actively track the dual beams and adjust their direction in real time to ensure that the user equipment (UE) receives signals efficiently;

[0011] Step 3: Model the statistical analysis model of the radiation model, and statistically analyze the mean, variance, and correlation of the real and imaginary parts of the radiation model. Then, describe the distribution of the radiation model through the statistical characteristics.

[0012] Step 4: Define the target for radiation mode characteristic extraction, complete the characteristic extraction, and calculate the gain error value, phase error value, and position error value based on the extracted characteristic parameters;

[0013] Step 5: Calculate the signal energy and compare it with a preset threshold as an energy detection statistic to determine the legitimacy of the signal source and complete the device authentication.

[0014] A further improvement of the present invention is that step 1 specifically includes the following steps:

[0015] Step 1.1: Define the received signal as:

[0016] y(t)=hΓws(t)+ν(t)

[0017] Where Γ represents the gain-phase-position error matrix, s(t) is the transmitted signal of the legitimate base station (Alice) at time t, ν(t) represents the additive noise at the receiving antenna, and w = [W1,...,W M ] T This represents the component values ​​of the beam weighting vector, where M represents the number of antennas and the beam weighting for each path. Represented as: Indicates the path propagation angle of the antenna array. The response vector on, For the propagation angle of this path, The value can be 1 or 2;

[0018] Step 1.2: Assume the millimeter-wave propagation channel follows a geometry-based L-path model, with the channel expression as follows:

[0019]

[0020] in, Indicates the first The complex gain of the first scattering path, the channel of the second path is expressed as: h2 = h1δe jβ , β represents relative decay, β∈[0,2π] represents relative phase shift, and j is the imaginary unit;

[0021] Step 1.3, Received Signal Update and Weighted Calculation: By combining the channel information of the two beam paths, the received signal is updated as follows:

[0022] y(t)=(α1b1+α1δe jβ b1)s(t)+ν(t)

[0023] Where α1 represents the complex gain of the first scattering path, and b1 is the propagation angle dependent on that path. and radiation modes at frequency f;

[0024] Step 1.4: Calculate the dual-beam weights:

[0025]

[0026] Where θ1 and θ2 represent the propagation angles of the first path and the second path, respectively, and w1 and w2 represent the beam weighting of the first path and the second path, respectively;

[0027] Step 1.5: Since the relative amplitude δ and relative phase shift β are expressed as channel ratios: By using additional channel sensing, and setting beams w(θ1,θ2,1,0) and w(θ1,θ2,1,π / 2), the channel strength is estimated respectively:

[0028]

[0029] Based on the values ​​of P3 and P4, the ratio of h2 to h1 is obtained, which is used to calculate the precise values ​​of the relative amplitude δ and the relative phase shift β.

[0030]

[0031] in, This indicates the operation of taking the real part. This represents the imaginary part operation, P1 = ||h1|| 2 P2 = ||h2|| 2 , which represents the channel strength.

[0032] A further improvement of the present invention is that, in step 2, actively tracking the dual beams and adjusting the dual beam reflection in real time specifically includes the following steps:

[0033] Step 2.1: To capture changes in beam direction, define the power measurement formula for each beam:

[0034] P i (t)=Ω T (θ i +φ i (t))+Ω R +P T -P c i = 1, 2;

[0035] Where, θ i +φ i (t) represents the beam angle, Ω T (θ i +φ i (t) represents the beam transmit gain, which varies with the beam angle, Ω R P represents the receiver gain. T P represents transmission power. c This indicates the power loss due to channel attenuation;

[0036] Step 2.2: Construct the uniform linear array (ULA) gain model. In the uniform linear array (ULA), the transmit gain is:

[0037]

[0038] Where M represents the number of antennas, and the offset angle φ is estimated from the measured beam power using an inverse function. i (t0);

[0039] Step 2.3, Differential Calculation of Beam Power Variation: By calculating the difference between the beam power at time t = t0 and the initial time t = 0, the beam direction offset is obtained, directly reflecting the trend of beam direction change and ensuring continuous signal reception.

[0040] P i (t0)-P i (0)=Ω T (θ i +φ i (t0))-Ω T (θ i ).

[0041] A further improvement of the present invention is that step 3 specifically includes the following steps:

[0042] Step 3.1: Model the statistical analysis model of the radiation mode: Statistically describe the gain error, phase error, and position error, assuming that the gain error, phase error, and position error are independent random variables and satisfy the following distribution:

[0043]

[0044] Where, μ g μ is the mean of the gain error. ψ This represents the mean of the phase error. For gain error variance, The phase error variance, This represents the variance of the positional error.

[0045] Step 3.2, Statistical analysis of the mean of the real and imaginary parts of the radiation model: Define the following lemma: If the random variable θ follows a normal distribution with mean μ and standard deviation σ, then the expected values ​​of the cosine and sine are respectively:

[0046]

[0047] Where a is a constant, σ is the standard deviation, and b is the real part of the radiation mode. R and the imaginary part b I The definition of the mean of the real part of the radiation mode and the mean of the imaginary part They are respectively:

[0048]

[0049] Expanding using the difference formulas for cosine and sine, we further obtain:

[0050]

[0051] Let phase Θ m The mean and variance are as follows:

[0052] μ Θ =μ ψ ,

[0053] Using the lemma, we obtain:

[0054]

[0055] The mean values ​​of the real and imaginary parts of the radiation mode are:

[0056]

[0057] Where, c0 = 1 + μ g ;

[0058] Step 3.3: Statistically calculate the variance of the real and imaginary parts of the radiation model: The mean power of the radiation model consists of two parts: in,

[0059]

[0060] Based on the above formula, calculate the real part b of the radiation mode. R and the imaginary part b I The squared mean power values ​​are expressed as follows:

[0061]

[0062] in, and This indicates the power contribution from the same antenna element:

[0063]

[0064] in The comprehensive statistical components representing the gain error. This indicates the power contribution from different antenna elements:

[0065]

[0066] in, This represents the mean component of the gain error;

[0067] The formulas for the square mean of the real and imaginary parts of the radiation model are modified using the square mean formula:

[0068]

[0069] Calculate the variances of the real and imaginary parts of the radiation mode using the relationship between the power square mean and the mean:

[0070]

[0071] Step 3.4: Statistical characteristics of the correlation between the real and imaginary parts of the radiation model: Define the Pearson correlation coefficient between the real and imaginary parts of the radiation model as:

[0072]

[0073] Among them, Cov(b R ,b I ) represents covariance. Expanding the covariance formula yields:

[0074]

[0075] Step 3.5: Describe the distribution of the radiation mode through statistical properties: the real part b of the radiation mode R and the imaginary part b I All follow a Gaussian distribution, and their amplitude statistical distribution is derived based on the central limit theorem:

[0076] Beckmann distribution: The amplitude of the radiation mode is described by the Beckmann distribution, whose probability density function (PDF) is:

[0077]

[0078] in,

[0079] The cumulative distribution function (CDF) is:

[0080] Rice distribution: Under the assumption that the variances of the real and imaginary parts are equal and uncorrelated, the amplitude distribution of the radiation mode is approximately described by the Rice distribution, whose probability density function (PDF) is:

[0081]

[0082] in, This represents the ratio of direct path power to power along other paths. Indicates the total received power. This represents the mean of the amplitude. I0 represents the variance of the amplitude, and I0 represents the zero-order modified Bessel function.

[0083] A further improvement of the present invention is that step 4 specifically includes the following steps:

[0084] Based on the sample signal y(t), using the formula: Estimate the sample variance.

[0085] According to the formula Construct a tensor model χ, decompose the tensor model using COMFAC decomposition, jointly estimate the angle and error parameters to construct a matrix Q, estimate the gain-phase-position error, extract the beam direction, extract error characteristics, and output the statistical parameter μ. g ,

[0086] A further improvement of the present invention is that step 5 specifically includes the following steps:

[0087] Step 5.1: At the receiving end, the user equipment verifies whether the signal y(t) comes from a legitimate base station (Alice). The verification process is modeled as a composite hypothesis testing problem, with the following assumptions:

[0088] Assume H0: The signal originates from a legitimate base station (Alice), and its radiation pattern is as follows:

[0089] Alternative hypothesis H1: The signal originates from a spoofed attacker (Eve), and its radiation pattern is as follows:

[0090] Step 5.2: Based on hypothesis testing, construct the energy detection statistic, i.e., the received signal energy Y, and compare it with a preset threshold λ. t For comparison, the energy detection formula is:

[0091]

[0092] Where, λ t This indicates the preset detection threshold, used to distinguish whether a signal comes from a legitimate user. Let y(t) be the noise power and y(t) be the received signal. If Y > λ t If Y ≤ λ, then accept the null hypothesis H0, i.e., the signal comes from a legitimate base station; t If we accept the opposing hypothesis H1, that is, the signal comes from a fake attacker (Eve), then we can effectively determine the legitimacy of the signal source.

[0093] The beneficial effects of this invention are: (1) By establishing a statistical model of the radiation pattern, this invention uses Rice distribution to approximate the radiation pattern modulus containing gain, phase, and position errors, proving that this distribution can accurately describe the cumulative distribution characteristics of the radiation pattern. This modeling method provides a precise mathematical basis for the theoretical analysis of physical layer authentication. (2) By integrating multi-dimensional hardware features such as gain, phase, and position errors, this invention effectively improves the distinguishability of radiation patterns, making physical layer authentication more accurate and reliable. At the same time, combined with dual-beam transmission technology, it further enhances the authentication stability of the system in complex environments, especially showing higher robustness in congested scenarios. (3) In the authentication process, this invention uses single radio frequency chain and phased array technology, which reduces the complexity of hardware implementation and power consumption requirements by optimizing resource utilization. In addition, the closed-form expression of the detection probability and false alarm probability based on the statistical model provides precise theoretical support for system performance evaluation. (4) This invention uses the hardware features of the device for physical layer authentication, getting rid of the dependence of traditional encryption technology on key management. In dynamic networks and resource-constrained environments, it can effectively resist identity-based spoofing attacks, significantly improving the security and reliability of millimeter-wave communication systems. Attached Figure Description

[0094] Figure 1 This is a schematic diagram of a millimeter-wave communication system.

[0095] Figure 2 This is a schematic diagram illustrating the effects of gain error, phase error, and position error on the antenna radiation pattern.

[0096] Figure 3 This is a graph showing the experimental results of the statistical analysis model of the radiation pattern.

[0097] Figure 4 This is a graph showing the experimental results of analyzing the influence of different orientation angles (DoD) on the statistical characteristics of radiation patterns.

[0098] Figure 5 This is an experimental graph showing the impact of attacker signature parameters on authentication performance.

[0099] Figure 6 This is a graph showing the experimental results of how user equipment (UE) mobility affects the authentication performance of millimeter-wave communication systems.

[0100] Figure 7 This is an experimental result diagram showing how dual-beam transmission technology improves authentication performance.

[0101] Figure 8 This is a graph showing the experimental results comparing the authentication performance of different combinations of authentication features on millimeter-wave communication systems. Detailed Implementation

[0102] The embodiments of the present invention will be disclosed below with reference to the drawings. For clarity, many practical details will be described in the following description. However, it should be understood that these practical details are not intended to limit the invention. That is, in some embodiments of the invention, these practical details are not essential.

[0103] like Figure 1 As shown, this invention establishes a millimeter-wave wireless communication system, including a legitimate base station (Alice): a uniform linear array (ULA) with M antennas, responsible for transmitting legitimate signals to user equipment; an attacker (Eve): attempting to impersonate the legitimate base station to send forged information; and a user equipment (UE): a single-antenna device used to receive signals from the base station and verify the base station's identity through radiation mode characteristics.

[0104] The attack scenario is determined: Eve attempts to impersonate Alice to deliver false information to the UE. The UE extracts parameters such as gain error, phase error, and antenna position error from radiation pattern characteristics through the beam training phase of millimeter wave standards, such as 802.11ad, 802.11ay, and 5G-NR, and uses these parameters for base station authentication.

[0105] Establish a frequency response model for the antenna array: Define the complex frequency response of M antenna elements at carrier frequency f as: in, This indicates the nominal gain, representing the gain amplitude of the antenna element.

[0106] Represents the nominal phase, and represents the phase offset of the antenna element. Establishing the actual frequency response model: Considering errors in the actual manufacturing process, the actual frequency response can be defined as: Where, ε g,m (f) represents the gain error, describing the deviation between the actual gain and the nominal gain, 0 < (1 + ε) g,m (f))<∞。 ε ψ,m (f) represents the phase error, describing the deviation between the actual phase and the nominal phase. The influence of antenna element position error is defined, assuming the nominal position of the m-th antenna element is p. m The actual position has an offset ε p,m Then its actual position can be represented as: p m +ε p,m Define the beam weighting function: The beamforming weighting function in the far-field direction θ is defined as follows:

[0107] Among them, w m (f) represents the beamweighted amplitude of the antenna element. This represents the beam-weighted phase. When considering wavenumber... Based on the signal direction θ, the far-field radiation mode is defined as: in, c represents the product of the nominal gain and the beam weighting coefficient. m (f)=1+ε g,m (f) represents the actual gain factor, which includes the gain error.

[0108] Θ m (θ,f) represents the actual phase deviation, caused by phase error and position error: Θ m (θ,f)=ε ψ,m (f)+kε p,m sinθ. The nominal phase is defined as follows: in, θ represents the beam-weighted phase, and θ0 represents the reference direction.

[0109] The radiation mode is decomposed into real and imaginary parts according to Euler's formula:

[0110]

[0111] Among them, b R (θ,f) represents the real part of the radiation mode, f represents the cosine component of the radiation pattern, and b represents the real part of the radiation mode. I (θ,f) represents the imaginary part of the radiation mode, and f represents the sinusoidal component of the radiation pattern.

[0112] Based on the above authentication background, the authentication objectives of this invention are: 1. to verify whether the source of the signal is a legitimate base station (Alice); 2. to prevent malicious attackers (Eve) from committing identity fraud by spoofing base stations; 3. to provide an efficient and reliable authentication mechanism to enhance the security of millimeter-wave communication.

[0113] Based on the authentication objective, this invention provides a highly reliable device authentication method based on antenna array error characteristics and dual-beam transmission. The specific authentication method includes the following steps: Step 1: Constructing dual beams. The signal received by the user equipment (UE) is transmitted through dual beams, and the parameters of the transmission path are calculated, as well as the dual-beam weighting vector. In millimeter-wave communication, path loss increases due to the shorter wavelength, and sensitivity to obstacles is significantly enhanced. However, the propagation environment of millimeter-wave signals typically exhibits sparsity, with only one or two significant paths contributing to communication. Therefore, by utilizing the sparsity characteristics of multipaths, the efficiency of the communication system can be improved. For scenarios with two paths, a two-beam weighting mechanism is designed to optimize channel utilization efficiency. Specifically, it includes the following steps: Step 1.1: Define the received signal as:

[0114] y(t)=hΓws(t)+ν(t)

[0115] Where Γ represents the gain-phase-position error matrix, s(t) is the transmitted signal of the legitimate base station (Alice) at time t, and ν(t) represents the additive noise at the receiving antenna, satisfying zero mean and variance. The complex circularly symmetric Gaussian distribution, w=[W1,...,W M ] T This represents the component values ​​of the beam weighting vector, where M represents the number of antennas and the beam weighting for each path. Represented as: Indicates the path propagation angle of the antenna array. The response vector on, For the propagation angle of this path, The value can be 1 or 2;

[0116] Step 1.2: Assume the millimeter-wave propagation channel follows a geometry-based L-path model, with the channel expression as follows:

[0117]

[0118] in, Indicates the first The complex gain of the first scattering path, the channel of the second path is expressed as: h2 = h1δe jβ , β represents relative decay, β∈[0,2π] represents relative phase shift, and j is the imaginary unit;

[0119] Step 1.3, Received Signal Update and Weighted Calculation: By combining the channel information of the two beam paths, the received signal is updated as follows:

[0120] y(t)=(α1b1+α1δe jβ b1)s(t)+ν(t)

[0121] Where α1 represents the complex gain of the first scattering path, and b1 is the propagation angle dependent on that path. and radiation modes at frequency f;

[0122] Step 1.4: Calculate the dual-beam weights:

[0123]

[0124] Where θ1 and θ2 represent the propagation angles of the first path and the second path, respectively, and w1 and w2 represent the beam weighting of the first path and the second path, respectively;

[0125] Step 1.5: Since the relative amplitude δ and relative phase shift β are expressed as channel ratios: By using additional channel sensing, and setting beams w(θ1,θ2,1,0) and w(θ1,θ2,1,π / 2), the channel strength is estimated respectively:

[0126]

[0127] Based on the values ​​of P3 and P4, the ratio of h2 to h1 is obtained, which is used to calculate the precise values ​​of the relative amplitude δ and the relative phase shift β.

[0128]

[0129] in, This indicates the operation of taking the real part. This represents the imaginary part operation, P1 = ||h1|| 2 P2 = ||h2|| 2 , which represents the channel strength.

[0130] Step 2: Actively track the dual beams and adjust their direction in real time to ensure efficient signal reception by the User Equipment (UE). In scenarios where the UE is moving, the initial dual beams may shift. Even a small angular shift, such as 14°, can cause a 20dB drop in signal strength, or even complete signal loss. Therefore, an active dual-beam tracking method is proposed to adjust the beam direction in real time, specifically including the following steps:

[0131] Step 2.1: To capture changes in beam direction, define the power measurement formula for each beam:

[0132] P i (t)=Ω T (θ i +φ i (t))+Ω R +P T -P c i = 1, 2;

[0133] Where, θ i +φ i (t) represents the beam angle, Ω T (θ i +φ i (t) represents the beam transmit gain, which varies with the beam angle, Ω R P represents the receiver gain. T P represents transmission power. c This indicates power loss due to channel attenuation, such as path loss or reflection loss.

[0134] Step 2.2: Construct the uniform linear array (ULA) gain model. In the uniform linear array (ULA), the transmit gain is:

[0135]

[0136] Where M represents the number of antennas, and the offset angle φ is estimated from the measured beam power using an inverse function. i (t0);

[0137] Step 2.3, Differential Calculation of Beam Power Variation: By calculating the difference between the beam power at time t = t0 and the initial time t = 0, the beam direction offset is obtained, directly reflecting the trend of beam direction change and ensuring continuous signal reception.

[0138] P i (t0)-P i (0)=Ω T (θ i +φ i (t0))-Ω T (θ i ).

[0139] Step 3: Model the statistical analysis model of the radiation pattern, and statistically analyze the mean, variance, and correlation of the real and imaginary parts of the radiation pattern. Use these statistical characteristics to describe the distribution of the radiation pattern.

[0140] Figure 2 (a)-(d) respectively show the effects of gain error, phase error, position error and the combined effect of the three on the antenna radiation mode, including the change in beam shape.

[0141] Specifically, step 3 includes the following steps:

[0142] Step 3.1: Model the statistical analysis model of the radiation mode: Statistically describe the gain error, phase error, and position error, assuming that the gain error, phase error, and position error are independent random variables and satisfy the following distribution:

[0143]

[0144] Where, μ g μ is the mean of the gain error. ψ This represents the mean of the phase error. For gain error variance, The phase error variance, This represents the variance of the positional error.

[0145] Step 3.2, Statistical analysis of the mean of the real and imaginary parts of the radiation model: Define the following lemma: If the random variable θ follows a normal distribution with mean μ and standard deviation σ, then the expected values ​​of the cosine and sine are respectively:

[0146]

[0147] Where a is a constant, σ is the standard deviation, and b is the real part of the radiation mode. R and the imaginary part b I The definition of the mean of the real part of the radiation mode and the mean of the imaginary part They are respectively:

[0148]

[0149] Expanding using the difference formulas for cosine and sine, we further obtain:

[0150] Let phase Θ m The mean and variance are as follows:

[0151] μ Θ =μ ψ , Using the lemma, we obtain:

[0152]

[0153] The mean values ​​of the real and imaginary parts of the radiation mode are:

[0154]

[0155] Where, c0 = 1 + μ g ;

[0156] Step 3.3: Statistically calculate the variance of the real and imaginary parts of the radiation model: The mean power of the radiation model consists of two parts: in,

[0157]

[0158] Based on the above formula, calculate the real part b of the radiation mode. R and the imaginary part b I The squared mean power values ​​are expressed as follows:

[0159]

[0160] in, and This indicates the power contribution from the same antenna element:

[0161]

[0162] in The comprehensive statistical components representing the gain error. This indicates the power contribution from different antenna elements:

[0163]

[0164] in, This represents the mean component of the gain error;

[0165] The formulas for the square mean of the real and imaginary parts of the radiation model are modified using the square mean formula:

[0166]

[0167] Calculate the variances of the real and imaginary parts of the radiation mode using the relationship between the power square mean and the mean:

[0168]

[0169] Step 3.4: Statistical characteristics of the correlation between the real and imaginary parts of the radiation model: Define the Pearson correlation coefficient between the real and imaginary parts of the radiation model as:

[0170]

[0171] Among them, Cov(b R ,b I ) represents covariance. Expanding the covariance formula yields:

[0172]

[0173] Step 3.5: Describe the distribution of the radiation mode through statistical properties: the real part b of the radiation mode R and the imaginary part b I All follow a Gaussian distribution, and their amplitude statistical distribution is derived based on the central limit theorem:

[0174] Beckmann distribution: The amplitude of the radiation mode is described by the Beckmann distribution, whose probability density function (PDF) is:

[0175]

[0176] in,

[0177] The cumulative distribution function (CDF) is:

[0178] Rice distribution: Under the assumption that the variances of the real and imaginary parts are equal and uncorrelated, the amplitude distribution of the radiation mode is approximately described by the Rice distribution, whose probability density function (PDF) is:

[0179]

[0180] in, This represents the ratio of direct path power to power along other paths. Indicates the total received power. This represents the mean of the amplitude. I0 represents the variance of the amplitude, and I0 represents the zero-order modified Bessel function.

[0181] Step 4: Define the target for radiation mode characteristic extraction, complete the characteristic extraction, and calculate the gain error, phase error, and position error values ​​based on the extracted characteristic parameters. This specifically includes the following steps:

[0182] Step 4.1: Define radiation mode characteristics and extract the key parameter μ of the target, i.e., estimate the direction. b and key parameters in frequency These parameters are affected by array errors such as gain, phase, and position errors. By extracting these key characteristics, system errors can be effectively identified and corrected.

[0183] Step 4.2: Define the gain-phase-position error matrix: Γ=diag(r)

[0184] in, M c ρ represents the number of well-calibrated antenna elements. m Parameters representing the combined gain, phase, and position error;

[0185] Step 4.3: Establish the received signal model. The received signal considering gain-phase-position error is represented as follows:

[0186] y(t)=hΓws(t)+v(t)

[0187] =(ΓA) H ws(t)+ν(t)

[0188] Where, Aα=A t A t =[a(θ1),a(θ2)],α=[α1,α2] T Let represent the beam direction matrix, and ν(t) represent additive noise, which follows a Gaussian distribution;

[0189] Step 4.4: Estimate the sample variance using multiple snapshot data K:

[0190] in, Where K is the number of samples, y(t) is the signal received at time t, Γ is the error matrix including gain, phase, and position errors, A is the transmission matrix of the antenna array, and R... s Given the covariance matrix of the signal, the error level in the signal can be assessed by calculating the sample variance.

[0191] Step 4.5: Construct the tensor model Decomposing the tensor y(t) yields the gain-phase-position error estimation matrix Q: Estimate the gain-phase-position error based on matrix Q:

[0192] Step 4.6: Correct the direction matrix to complete feature extraction.

[0193]

[0194] Step 4.7: Calculate the beam direction based on the estimated matrix. in, Represents the relationship of the estimated direction matrix, A Γ,1 and A Γ,2 It is an estimate of the direction matrix;

[0195] Step 4.8: By extracting characteristic parameters, calculate the error value step by step, including: gain error calculation: estimate the gain error. in, Indicates the parameters obtained from the joint estimation; location error calculation: in, and These are the direction estimates for path 1 and path 2, respectively; phase error calculation: in, The phase estimate is represented by μ, which ultimately yields the statistical characteristics of the gain, position, and phase error: μ g ,

[0196] Step 5: Calculate the signal energy and compare it with a preset threshold as an energy detection statistic to determine the legitimacy of the signal source and complete the device authentication. This includes the following steps:

[0197] Step 5.1: At the receiving end, the user equipment verifies whether the signal y(t) comes from a legitimate base station (Alice). The verification process is modeled as a composite hypothesis testing problem, with the following assumptions:

[0198] Assume H0: The signal originates from a legitimate base station (Alice), and its radiation pattern is as follows:

[0199] Alternative hypothesis H1: The signal originates from a spoofed attacker (Eve), and its radiation pattern is as follows:

[0200] Step 5.2: Based on hypothesis testing, construct the energy detection statistic, i.e., the received signal energy Y, and compare it with a preset threshold λ. t For comparison, the energy detection formula is:

[0201]

[0202] Where, λ t This indicates the preset detection threshold, used to distinguish whether a signal comes from a legitimate user. Let y(t) be the noise power and y(t) be the received signal. If Y > λ t If Y ≤ λ, then accept the null hypothesis H0, i.e., the signal comes from a legitimate base station; t If we accept the opposing hypothesis H1, that is, the signal comes from a fake attacker (Eve), then we can effectively determine the legitimacy of the signal source.

[0203] To evaluate the reliability and effectiveness of authentication methods in practical applications, two main metrics for detection performance are defined: detection probability P. D,I The probability of correct detection is P, assuming the signal originates from attacker Eve. D,I =Pr(Y>λ) t |H1). False alarm probability P F,I The probability of false detection when the signal comes from the legitimate user Alice: P F,I =Pr(Y>λ) t |H0).

[0204] The statistic Y is approximately the sum of squares of a non-zero mean Gaussian random variable, and its probability density function (PDF) is:

[0205]

[0206] in, I K-1 (x) represents the modified Bessel function.

[0207] In the statistical distribution of energy detection, the generalized Marcum Q function Q is used. n (a,b) describes the performance of energy detection. Its definition is as follows: Where a represents the magnitude parameter of a non-zero mean Gaussian random variable, and b is the detection threshold parameter. n-1 (x) is the modified Bessel function. This function is used to quantify the probability of signal detection and false alarm in hypothesis testing.

[0208] Based on hypothesis H1, the detection probability P D,I Represented as: in, F Y (y|H1) represents the cumulative distribution function under hypothesis H1.

[0209] For hypothesis H0, the false alarm probability P F,I Represented as: in, λ t This indicates the detection threshold.

[0210] To more accurately describe the performance in a real channel, the detection probability needs to be averaged. Assuming path fading exists in the channel, the detection probability P... D The expression is: Among them, f γ1 (x) represents the channel amplitude distribution under hypothesis H1.

[0211] In millimeter-wave communication systems, the uncertainty and randomness of the channel cause changes in the statistical properties of the received signal. Modeling using the Marcum Q function can accurately describe the signal detection performance under different assumptions, and this can be achieved by adjusting the parameter λ. t Optimize the performance of the detection algorithm.

[0212] The performance analysis is further extended to random channels. In the case of path fading, the average behavior of detection performance under different assumptions needs to be solved by integration. Specifically:

[0213] Detection probability P under channel D :

[0214] False alarm probability P under channel F : By averaging the detection and false alarm probabilities, the system performance under random channels can be better described.

[0215] Define the detection probability P D And the false alarm probability P F Detection probability P D And the false alarm probability P F It is a key indicator for evaluating system performance, and its mathematical formula is as follows:

[0216]

[0217]

[0218] Among them, Q n (a,b) represents the generalized Marcum Q function, used to describe the distribution characteristics of a random variable. κ E and κ A These are the shape parameters of the attack channel and the legitimate channel, respectively, representing the power ratio of the direct path to the scattering path. and These represent the normalized signal-to-noise ratio, reflecting the impact of channel conditions on performance.

[0219] The Rice distribution is used to describe the radiation mode. To accurately describe the distribution of the actual radiation mode, we assume... Following a non-central chi-square distribution, based on the Rice distribution model, the amplitude distribution of the radiation mode can be represented by the following probability density function (PDF):

[0220]

[0221] Where I0 is the zeroth-order modified Bessel function, κ is the parameter of the non-central chi-square distribution. E It is a parameter related to error.

[0222] The integral formula for calculating the detection probability. The probability density function (PDF) is used. And the false alarm probability P F,I Detection probability P D In the process, the new detection probability P is calculated. D The expression. To simplify calculations, variable substitution is used to transform the formula into its standard form:

[0223] This integral form helps in calculating the detection probability P. D And evaluate the performance of the authentication method in a real communication system.

[0224] Define the detection probability P when K=1 D The formula. When K=1, the defined detection probability P D And the false alarm probability P F This can be evaluated by using changes in the variables. Let... The above equation then becomes:

[0225]

[0226]

[0227] When K=1, the detection probability P D It can be characterized as:

[0228]

[0229] The generalized integral formula is applied to random channel analysis. To describe the performance under random channel conditions, the detection probability P... D And the false alarm probability P F Calculated using the following integral formula: in, The probability density function (PDF) represents the attack channel gain. The probability density function represents the legitimate channel gain.

[0230] The randomness of the channel gain is modeled using the moment generating function (MGF), which is expressed as:

[0231]

[0232] in, and These represent the stochastic characteristics of the attack channel and the legitimate channel gain, respectively. The introduction of MGF simplifies integral calculations and is used to analyze the distributions of detection probability and false alarm probability.

[0233] Since many channel distributions are combined by adding or multiplying them, using the Moment Generating Function (MGF) is more convenient than calculating the average error probability over the channel PDF. The average detection probability PD is calculated using the Moment Generating Function (MGF) through the contour integral representation of the Marcum Q function with the contour radius. Therefore, PD (similar to PF) can be expressed as:

[0234] Where o represents a circular outline with radius r∈[0,1], Φ(x) represents the MGF of x, and by The calculation yielded the result.

[0235] When K > 1, combining the Marcum Q function and the MGF of the channel gain, the detection probability can be expressed as the path integral formula:

[0236]

[0237] Where o represents the integration path as a closed path around the origin, z K This indicates the impact of the number of samples on performance, and ξ E,2 =κ E ξ E,1 (1-ξ E,1 ).

[0238] To verify whether the proposed statistical model is consistent with the characteristics of actual signals, numerical simulations and theoretical models were compared. The cumulative distribution function (CDF) was calculated using Monte Carlo simulation and compared with theoretical models of the Beckmann or Rice distributions to ensure that the statistical model accurately describes the radiation characteristics of actual signals. This verification ensures the matching degree between the mathematical model of the certification process and practical applications, further guaranteeing the reliability of the certification method.

[0239] Specifically, this involves setting system parameters and identifying key parameters affecting millimeter-wave reliable communication and authentication performance. The characteristic parameters for the legitimate base station (Alice) and the attacker (Eve) are set as follows: Gain error μ A,g =0,σ A,g =0.1dB,σ A,ψ =10°,σ A,p=0.1λ, where λ represents the wavelength, the operating frequency f = 30 GHz, and the bandwidth is set to 1 GHz. The communication distance is set to between 50 and 100 meters, and the noise power spectral density is -174 dBm / Hz. According to the ITU-RP.676-9 standard and the free space loss formula, the propagation loss is taken as -124.6 dB. The base station uses a 161×161 element uniform linear array (ULA), with an antenna gain of 31.6 dB, and beam directions (DoD) set to θ1 = 60° and θ2 = 40°, respectively, to describe the LOS and NLOS link scenarios.

[0240] To verify the statistical properties of the radiation mode, the calculated CDF was compared with the Beckman and Rice distributions. A LOS link was selected as the verification example, and a 1×10⁻⁶ Ω·cm distribution was calculated using Monte Carlo simulation. 5 The radiation mode takes into account three cases: gain, phase, and position error. Figure 3 The statistical model validation is demonstrated, including comparisons with the Beckmann and Rice distributions. The results show a high degree of agreement between the Beckmann and Rice distributions and the Monte Carlo simulation results, validating the accuracy of the proposed model. (a) shows a comparison between the Beckmann distribution and the Monte Carlo simulation results, and (b) shows a comparison between the Rice distribution and the Monte Carlo simulation results. Based on the experimental results of the statistical analysis model of the radiation model, the validation results show a high degree of agreement between the theoretical models of the Beckmann and Rice distributions and the simulation data, validating the excellent fit of the proposed model in actual simulations.

[0241] To verify the effectiveness and superiority of the authentication scheme proposed in this invention, it was compared with several existing identity verification methods through numerical simulation, and multiple performance indicators were evaluated.

[0242] The variations in variance and correlation coefficient between the real and imaginary parts of the radiation model under different orientation angles (DoD) were analyzed. Figure 4 This demonstrates how the variance and correlation coefficient between the real and imaginary parts of the radiation map vary with the DoD (Angle of Arrival). Figure 4 (a) The variances of the real and imaginary parts of the radiation model are shown, and the results indicate that the variances are very small and the changes are negligible. Figure 4 (b) The correlation coefficients between the real and imaginary parts of the radiation pattern are shown, indicating that the correlation coefficients do not exceed 0.3, suggesting that the correlation between them is negligible. This result demonstrates that the CDF of the radiation pattern modulus can be effectively approximated by the Rice distribution, and that at a specific turning angle, the Rice distribution exhibits good overall accuracy and minimal deviation from the Monte Carlo simulation results.

[0243] Figure 5 This demonstrates the effect of different attacker characteristic parameters on the false alarm probability (P). F) and detection probability (P) D The effects of different thresholds are shown in (a) and (b). (a) shows the changes in detection probability at different thresholds, with larger thresholds leading to a decrease in detection probability. (b) shows the changes in the ROC curve under different feature parameters, with the mean and variance of the gain error having a significant impact on authentication performance.

[0244] By analyzing the detection probability P D And the false alarm probability P F The study found that when the threshold is low, the system can effectively identify identity spoofing attacks launched by attackers. However, if the threshold is set too high, the probability of false alarms increases, which may cause the system to mistakenly identify attackers as legitimate users, leading to information leakage or malicious information injection.

[0245] By analyzing the changes in radiation mode power of LOS and NLOS links under different UE motion angles, Figure 6 The impact of UE motion on authentication performance was analyzed. (a) shows the effect of UE motion angle on radiation mode power; as the motion angle increases, the power gradually decreases, but the decrease is less than 3 dB. (b) shows the ROC curves for different motion angles; as the angle increases, the authentication performance decreases slightly. Further analysis indicates that larger motion angles lead to a slight performance degradation, mainly due to beam misalignment or blocking caused by UE mobility. However, the performance loss is relatively limited, indicating that the authentication system is highly robust to internal hardware failures.

[0246] By comparing the receiver operating characteristic (ROC) curves and overload probability (PO) under single-beam and dual-beam configurations, such as Figure 7 As shown, the results demonstrate that the authentication system employing a dual-beam scheme exhibits a significant performance improvement compared to the single-beam scheme. Figure 7 As shown in (a), the success rate of authentication (ROC) improved by approximately 5%, and the probability of overload (PO) decreased by approximately 1.5%, as Figure 7 As shown in (b), this improvement is attributed to the fact that dual-beam systems can effectively avoid the link blocking problem that single-beam schemes may face, thereby enhancing the robustness and anti-interference capability of the system.

[0247] like Figure 8 As shown, the experiment compares the performance of the authentication method based on gain-phase-position error with that based on a single gain error. The results show that the method combining the characteristics of three radiation modes (gain, phase, and position error) exhibits the best authentication performance. Figure 8 As shown in (a). Especially in the detection probability P DCompared to methods using only gain error, this method improves detection accuracy by approximately 10%. Furthermore, utilizing multi-dimensional radiation mode characteristics provides higher authentication reliability and enables more accurate decision-making from different signal angles and dimensions. Experiments analyzed the impact of beam alignment accuracy on authentication performance. Results show that precise beam alignment significantly improves authentication performance. Conversely, as... Figure 8 As shown in (b), when beam alignment errors are large, such as Δ = 15°, authentication performance degrades significantly. At low signal-to-noise ratios (SNR), signal characteristics become difficult to identify, leading to authentication failure. Therefore, maintaining beam alignment accuracy is crucial for achieving the desired authentication performance.

[0248] The authentication method of this invention can effectively cope with the motion changes of the UE. The dual-beam scheme improves the authentication success rate (ROC) by about 5% and reduces the probability of overload (PO) by about 1.5%. Compared with the single gain error scheme, the authentication method combining gain, phase and position errors improves the detection accuracy by about 10%. Precise beam alignment significantly improves the authentication performance, especially under low signal-to-noise ratio (SNR) conditions, where beam alignment errors can cause a significant drop in authentication performance.

[0249] The above description is merely an embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of the present invention should be included within the scope of the claims of the present invention.

Claims

1. A highly reliable device authentication method based on antenna array error characteristics and dual-beam transmission, characterized in that: This device authentication method is applied to millimeter-wave communication systems. User equipment (UE) needs to verify whether the signal it receives originates from a legitimate base station (Alice) and not a spoofed attacker (Eve). By extracting and analyzing the gain, phase, and antenna element position error characteristics of the base station's radiation pattern, the UE determines the legitimacy of the signal source. The specific authentication method includes the following steps: Step 1: Construct dual beams. The signal received by the user equipment (UE) is transmitted through dual beams, and the parameters of the transmission path are calculated, and the weighted vector of the dual beams is calculated. Step 2: Actively track the dual beams and adjust their direction in real time to ensure that the user equipment (UE) receives signals efficiently; Step 3: Model the statistical analysis model of the radiation model, and statistically analyze the mean, variance, and correlation of the real and imaginary parts of the radiation model. Then, describe the distribution of the radiation model through the statistical characteristics. Step 4: Define the target for radiation mode characteristic extraction, complete the characteristic extraction, and calculate the gain error value, phase error value, and position error value based on the extracted characteristic parameters; Step 5: Calculate the signal energy and compare it with a preset threshold as an energy detection statistic to determine the legitimacy of the signal source and complete the device authentication. This includes the following steps: Step 5.1: At the receiving end, the user equipment verifies the received signal. Whether the data originates from a legitimate base station (Alice) is modeled as a composite hypothesis testing problem, with the following hypotheses: Assumption The signal originates from the legitimate base station Alice, and its radiation pattern is as follows: : ; Alternative Hypothesis The signal originates from the masquerading attacker Eve, and its radiation pattern is as follows: : , in, This represents the complex gain of the scattering path of the signal emitted by Alice. This represents the relative amplitude of the signal emitted by Alice. This indicates the relative phase shift of the signal emitted by Alice. This represents the complex gain of the scattering path of the signal emitted by Eve. This indicates the relative amplitude of the signal emitted by Eve. This indicates the relative phase shift of the signal emitted by Eve. This is the radiation pattern when the signal originates from Eve, the posing attacker. Indicates additive noise. for Step 5.2: Based on hypothesis testing, construct the energy detection statistic, i.e., the received signal energy. and compare it with a preset threshold. For comparison, the energy detection formula is: in, This indicates the preset detection threshold, used to distinguish whether a signal comes from a legitimate user. For noise power, For the sample size, For the received signal, if Then accept the null hypothesis. That is, the signal comes from a legitimate base station; if Then accept the opposing hypothesis. The signal originated from Eve, who was posing as the attacker.

2. The high-reliability device authentication method based on antenna array error characteristics and dual-beam transmission according to claim 1, characterized in that: Step 1 specifically includes the following steps: Step 1.1: Define the received signal as: in, This represents the gain-phase-position error matrix. for Alice's legitimate base station transmits signals at all times. This represents the additive noise at the receiving antenna. This represents the component values ​​of the beam weighting vector. This indicates the number of antennas and the beam weighting for each path. Represented as: , Indicates the path propagation angle of the antenna array. The response vector on, For the propagation angle of this path, The value can be 1 or 2; Step 1.2: Assume the millimeter-wave propagation channel follows a geometry-based L-path model, with the channel expression as follows: in, Indicates the first The complex gain of the first scattering path, and the channel of the second path are expressed as: , Indicates relative amplitude. Indicates relative phase shift, The imaginary unit; Step 1.3, Received Signal Update and Weighted Calculation: By combining the channel information of the two beam paths, the received signal is updated as follows: in, This represents the complex gain of the first scattering path. For propagation angle dependent on this path and frequency Radiation patterns; Step 1.4: Calculate the dual-beam weights: in, and These represent the propagation angles of the first and second paths, respectively. and These represent the beam weighting of the first and second paths, respectively. Step 1.5, due to relative amplitude and relative phase shift Expressed as channel ratio: By using additional channel probing, beams can be set. Estimate the channel strength respectively: And according to , Value, obtain and The ratio is used to calculate the relative amplitude. and relative phase shift The exact value: in, This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part. , representing channel strength.

3. The high-reliability device authentication method based on antenna array error characteristics and dual-beam transmission according to claim 2, characterized in that: In step 2, actively tracking the dual beams and adjusting their direction in real time specifically includes the following steps: Step 2.1: To capture changes in beam direction, define the power measurement formula for each beam: ； in, Indicates the beam angle. This indicates the transmit gain of the beam, which varies with the beam angle. Indicates the receive gain. Indicates transmission power. This indicates the power loss due to channel attenuation; Step 2.2: Construct the gain model of the uniform linear array ULA. In the uniform linear array ULA, the transmit gain is: in, Indicates the number of antennas. The azimuth angle is represented by an inverse function, and the offset angle is estimated from the measured beam power. ; Step 2.3, Differential Calculation of Beam Power Variation: By calculating the time... and initial time The difference in beam power yields the beam direction offset, directly reflecting the trend of beam direction change and ensuring continuous signal reception. 。 4. The high-reliability device authentication method based on antenna array error characteristics and dual-beam transmission according to claim 3, characterized in that: Step 3 specifically includes the following steps: Step 3.1: Model the statistical analysis model of the radiation mode: Statistically describe the gain error, phase error, and position error, assuming the gain error... Phase error and position error They are independent random variables and satisfy the following distribution: in, This represents the mean of the gain error. This represents the mean of the phase error. For gain error variance, The variance of the phase error. This represents the variance of the positional error. Step 3.2: Calculate the mean of the real part and the mean of the imaginary part of the radiation model: Based on the real part of the radiation model... and the virtual part The definition of the mean of the real part of the radiation mode and the mean of the imaginary part They are respectively: Expanding using the difference formulas for cosine and sine, we further obtain: ； Let phase The mean and variance are as follows: Where k is the wavenumber and cm is the actual gain factor. This represents the product of the nominal gain and the beam weighting coefficient. Indicates the actual phase deviation. Indicates the nominal phase. Let the direction angle be , then we can further obtain: The mean values ​​of the real and imaginary parts of the radiation mode are: in, ; Step 3.3: Calculate the variance of the real and imaginary parts of the radiation model. and the virtual part The squared mean power values ​​are expressed as follows: ,in, and This indicates the power contribution from the same antenna element. This indicates the power contribution from different antenna elements; in, Represents the real part of the radiation mode The power square mean, Represents the real part of the radiation mode The square of the mean, This represents the mean component of the gain error. For phase variance Indicates the nominal phase. For phase The mean; Step 3.4: Statistical characteristics of the correlation between the real and imaginary parts of the radiation model: Define the Pearson correlation coefficient between the real and imaginary parts of the radiation model as: in, Describing covariance, Expanding the covariance formula yields: ； Step 3.5: Describe the distribution of the radiation mode through statistical properties: the real part of the radiation mode. and the virtual part All follow a Gaussian distribution, and their amplitude statistical distribution is derived based on the central limit theorem: The amplitude of the radiation mode is described by the Beckmann distribution, and its probability density function PDF is: in, The cumulative distribution function (CDF) is: ; Assuming that the variances of the real and imaginary parts are equal and uncorrelated, the amplitude distribution of the radiation mode is approximately described by the Rice distribution, whose probability density function PDF is: in, This represents the ratio of direct path power to power along other paths. Indicates the total received power. This represents the mean of the amplitudes. The variance of the amplitude is represented. This represents the zeroth-order modified Bessel function.

5. The high-reliability device authentication method based on antenna array error characteristics and dual-beam transmission according to claim 4, characterized in that: Step 4 specifically includes the following steps: Step 4.1: Define radiation mode characteristics and extract key parameters for target estimation. and key parameters in frequency ; Step 4.2, Define the gain-phase-position error matrix: ; in, , This indicates the number of well-calibrated antenna elements. Parameters representing the combined gain, phase, and position error. Beam direction; Step 4.3: Establish the received signal model. The received signal considering gain-phase-position error is represented as follows: in, , , Represents the beam direction matrix. This represents additive noise that follows a Gaussian distribution. Step 4.4: Estimate the sample variance using multiple snapshot data K: in, , It is the variance term of the signal in each snapshot. For the sample size, time Received signal, The error matrix contains gain, phase, and position errors. The transmission matrix of the antenna array, Given the covariance matrix of the signal, the error level in the signal can be assessed by calculating the sample variance. Step 4.5: Construct the tensor model ,in, For the direction matrix containing gain, phase, and position error, the tensor By decomposing the matrix, we obtain the estimation matrix of gain-phase-position error. According to the matrix Estimate gain-phase-position error ; Step 4.6: Correct the direction matrix to complete feature extraction. ； in, express The first line; Step 4.7: Calculate the beam direction based on the estimated matrix. ; Step 4.8: By extracting characteristic parameters, calculate the error value step by step, where: Gain error calculation: Estimating the gain error : ,in, This represents the parameters obtained from the joint estimation; Position error calculation: ,in, and These are the direction estimates for path 1 and path 2, respectively. Phase error calculation: ,in, The phase estimate is represented, and the statistical characteristics of gain, position, and phase error are finally obtained: .

6. The high-reliability device authentication method based on antenna array error characteristics and dual-beam transmission according to any one of claims 1-5, characterized in that: The millimeter-wave communication system includes: a legitimate base station Alice: a uniform linear array ULA with M antennas, responsible for transmitting legitimate signals to user equipment; an attacker Eve: attempting to impersonate a legitimate base station to send forged information; and user equipment UE: a single-antenna device used to receive signals from the base station and verify the base station's identity through radiation mode characteristics.

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