A fractal trajectory-based radio frequency fingerprinting method

Abstract

The application discloses a radio frequency fingerprint identification method based on a fractal trajectory, which comprises the following steps: obtaining original radio frequency signals of N radio frequency devices to be identified, and pre-processing the original radio frequency signals; performing transient detection based on the pre-processed original radio frequency signals, obtaining target signals, and obtaining fractal trajectories corresponding to the target signals by using a dynamic sliding window; extracting features of the N radio frequency devices based on the fractal trajectories, constructing radio frequency fingerprint feature vectors based on the features of the N radio frequency devices; and inputting the radio frequency fingerprint feature vectors into a support vector machine to identify the identities of the N radio frequency devices. Through the method, the fractal trajectory characteristics of the transient signals are analyzed in depth, a radio frequency fingerprint identification method based on the fractal trajectory is designed, the best target signal interval can be selected for radio frequency fingerprint extraction and identification regardless of the modulation mode of the received signals, and no other prior knowledge is needed.

Description

Technical Field

[0001] This invention belongs to the field of signal analysis and relates to a radio frequency fingerprint recognition method based on fractal trajectories. Background Technology

[0002] With the rapid evolution of wireless communication network technology, many constraints of traditional wired networks have been overcome, enabling wireless information transmission and greatly eliminating spatial and temporal barriers. While wireless communication networks have improved the portability of communication, they also contain various vulnerabilities, seriously threatening their security. Wireless communication networks primarily replace cables with electromagnetic waves as the carrier of information transmission, providing opportunities for malicious attackers. Wireless networks are susceptible to various attack methods, among which spoofing attacks are the most important and threatening. Attackers can deceive devices by copying a large portion of the information, thus requiring innovative mechanisms to prevent external attacks and threats. Since the tolerance effect of electronic components in wireless devices is the main cause of radio frequency fingerprints, these fingerprints are difficult to clone and are unique, thus becoming a new paradigm for wireless device authentication.

[0003] Based on the differences in the selected target signal types, radio frequency fingerprinting technology can be divided into two main research paths: based on transient signals and based on steady-state signals. The steady-state signal segment refers to the signal generated when the transmitting device starts up and enters a stable operating state. It is represented in the form of a modulated waveform, carrying the actual communication content and ensuring the full preservation of subsequent signal information to be identified. However, it requires more prior knowledge based on the modulation method and code design of the target signal, resulting in poor versatility. The transient signal segment refers to the signal generated instantaneously when the transmitting device starts up or shuts down. Although it does not carry effective data content, it contains the unique properties of the transmitting source hardware circuit and does not rely on the modulation domain information of the signal, making it more widely applicable and thus suitable for effectively constructing radio frequency fingerprint feature sets. Summary of the Invention

[0004] To address the above problems, the technical solution adopted by this invention is: a radio frequency fingerprint recognition method based on fractal trajectories, comprising the following steps:

[0005] Acquire the raw radio frequency signals of N radio frequency devices to be identified, and preprocess the raw radio frequency signals;

[0006] Transient detection is performed based on the preprocessed original radio frequency signal to obtain the target signal, and the fractal trajectory corresponding to the target signal is obtained by using a dynamic sliding window.

[0007] Features of N radio frequency devices are extracted based on fractal trajectories, and radio frequency fingerprint feature vectors are constructed based on the features of N radio frequency devices;

[0008] The RF fingerprint feature vector is input into a support vector machine to identify the identities of N RF devices.

[0009] Furthermore, the process of obtaining the fractal trajectory corresponding to the target signal using a dynamic sliding window is as follows:

[0010] A sliding window of length L is used to segment a discrete target signal x(n) of total length N. When the starting index of the sliding window moves to i, the corresponding truncated signal is defined as x. d (i);

[0011] The Higuchi algorithm is used to calculate the fractal dimension of the truncated time series. As the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained.

[0012] Furthermore: the feature extraction of N radio frequency devices based on fractal trajectories includes:

[0013] The skewness of the fractal trajectory, the spectral energy of the fractal trajectory, the constant term fitting coefficient of the fractal trajectory, the coefficient of variation of the fractal trajectory, and the Hurst exponent of the fractal trajectory.

[0014] Furthermore: the method of segmenting the discrete target signal x(n) of total length N using a sliding window of length L, when the starting index of the sliding window moves to i, yields the corresponding truncated signal defined as x. d (i);

[0015] The Higuchi algorithm is used to calculate the fractal dimension of the truncated time series. As the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained as follows:

[0016] Create a sample subset of the original signal:

[0017]

[0018] Where m is the initial time and start time of each subset, k is the time interval and determines the number of subsets, and m and k are integers.

[0019] The curve length for each subset X(m,k) defined by Higuchi is as follows:

[0020]

[0021] In the above formula, (N-1) / [(Nm) / k]k is the normalization factor for the curve length;

[0022] Plot L on a log-log scale with x-axis k. m (k) curve, L as k changes from N to 0m (k) The data should fall on one axis;

[0023] Use the least squares method to calculate the point (L) obtained in the previous step. m (k) is fitted, and the slope of the fitted curve can be used as an estimate of the fractal dimension;

[0024] Finally, as the starting point of the sliding window moves, the fractal trajectory corresponding to the target signal is obtained.

[0025] Furthermore, the expression for the skewness γ of the fractal trajectory is as follows:

[0026]

[0027] Where, σ d Let N be the standard deviation of the fractal trajectory d. d This represents the length of the fractal trajectory d. This represents the mean of the fractal locus d;

[0028] The expression for the spectral energy E of the fractal trajectory is as follows:

[0029]

[0030] Where, |D(f) 2 It is the square of the spectral density modulus of the fractal trajectory;

[0031] The expression for the fractal trajectory using the polynomial function p(x) is as follows:

[0032]

[0033] Where j represents the order of the polynomial function, a j The coefficients are those of a polynomial function of order j. When N=2, a binomial fit is performed on the fractal trajectory, and the constant term fitting coefficients are selected as features.

[0034] The expression for the coefficient of variation cv of a fractal trajectory is as follows:

[0035]

[0036] Where, σ d The standard deviation of the fractal locus The mean and standard deviation σ of the fractal locus d and mean The calculation formula is as follows.

[0037]

[0038] Furthermore, the calculation steps for the Hurst exponent of the fractal trajectory are as follows:

[0039] Divide the fractal trajectory sequence d = {d(i), i = 1, 2, 3, ..., N} into K data groups of length m. For each subgroup, define its mean as... The maximum cumulative absolute value of the deviation from the mean within this subgroup is denoted as Z. max :

[0040]

[0041] Let Z be the minimum cumulative absolute value of the deviation from the mean within the subgroup. min :

[0042]

[0043] The range within each group is denoted as R(m):

[0044] R(m)=Z max -Z min

[0045] The sample standard deviation for each group is denoted as:

[0046]

[0047] R, S, and m satisfy the following general relation:

[0048] R(m) / S(m)=cm H

[0049] Where c is a constant, R(m) / S(m) is the rescaled range, and H is the Hurst exponent, according to the equation:

[0050] log(R / S) m =log(c) + Hlog(m)

[0051] With the log(m) sequence as the independent variable, the corresponding log(R / S) m With the sequence as the dependent variable, the least squares regression method is introduced to perform linear fitting analysis on the data. The slope satisfies k = H, and the value of H is the estimated result of Hurst's exponent.

[0052] Furthermore: The RF fingerprint feature vector is input into the support vector machine to identify the identities of N RF devices. Principal component analysis is used to reduce the dimensionality of the obtained feature vector. The dimensionality-reduced features are randomly divided into training and test sets at a certain ratio and input into the support vector machine (SVM) for classification. The label results output by the trained model are compared with the actual labels to obtain the classification accuracy.

[0053] Furthermore, the process of using principal component analysis to reduce the dimensionality of the obtained eigenvectors is as follows:

[0054] 1) The extracted radio frequency fingerprint feature vector feature set X∈R n*m In this process, the sample features are demeaned by subtracting the mean of that feature from the value of the current feature in the sample set.

[0055] 2) Calculate the covariance matrix XX of the sample. T ;

[0056] 3) Perform eigenvalue decomposition on the covariance matrix to find the eigenvalues ​​and eigenvectors of the covariance matrix;

[0057] 4) Select the eigenvectors corresponding to the k largest eigenvalues ​​and transform them into P;

[0058] 5) Y = PX is the k-dimensional feature matrix obtained after dimensionality reduction.

[0059] This invention provides a radio frequency fingerprinting method based on fractal trajectories, which is a wireless device authentication method based on transient radio frequency fingerprints. This method proposes a fractal trajectory-based radio frequency fingerprinting approach based on energy-constrained signals. The radio frequency fingerprinting method utilizes the temporal self-similarity and complexity differences of the fractal trajectories of different devices to effectively extract multi-dimensional features, which are then applied to the device authentication process. Considering the impact of algorithm time complexity, feature vectors are constructed by extracting five different statistical quantities from the fractal trajectory: skewness, spectral energy, coefficient of variation, constant term fitting coefficient, and Hurst exponent, thereby improving the recognition accuracy of different devices. The authentication method of this application can accurately authenticate devices by utilizing subtle hardware differences between them.

[0060] Compared with existing technologies, the significant advantages of this invention are: Through in-depth analysis of the fractal trajectory characteristics of transient signals, this invention designs a radio frequency fingerprinting method based on fractal trajectories. This method can select the optimal target signal range for radio frequency fingerprint extraction and identification regardless of the modulation scheme of the received signal, without requiring any prior knowledge. Experiments show that this invention can effectively identify different devices with a recognition accuracy of 98.7%. Attached Figure Description

[0061] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0062] Figure 1 This is a flowchart of the method;

[0063] Figure 2 It is the original feature classification confusion matrix;

[0064] Figure 3 It is a PCA-3D feature visualization diagram. Detailed Implementation

[0065] It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0066] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] Figure 1 This is a flowchart of the method;

[0068] A radio frequency fingerprint recognition method based on fractal trajectories includes the following steps:

[0069] S1: Obtain the raw radio frequency signals of N radio frequency devices to be identified, and preprocess the raw radio frequency signals;

[0070] S2: Based on the preprocessed original radio frequency signal, transient detection is performed to obtain the transient signal segment, the target signal is acquired, and the fractal trajectory corresponding to the target signal is obtained by using a dynamic sliding window;

[0071] S3: Extract features of N radio frequency devices based on fractal trajectories, and construct a radio frequency fingerprint feature vector based on the features of N radio frequency devices;

[0072] S4: Input the RF fingerprint feature vector into the support vector machine to identify the identities of N RF devices.

[0073] Steps S1 / S2 / S3 / S4 are executed sequentially;

[0074] The preprocessing of the original radio frequency signal includes filtering and normalization.

[0075] The process of obtaining the fractal trajectory corresponding to the target signal using a dynamic sliding window is as follows:

[0076] A sliding window of length L is used to segment a discrete target signal x(n) of total length N. When the starting index of the sliding window moves to i, the corresponding truncated signal is defined as x. d (i);

[0077] The Higuchi algorithm is used to calculate the fractal dimension of the truncated time series. As the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained.

[0078] The features extracted from N radio frequency devices based on fractal trajectories include:

[0079] The skewness of the fractal trajectory, the spectral energy of the fractal trajectory, the constant term fitting coefficient of the fractal trajectory, the coefficient of variation of the fractal trajectory, and the Hurst exponent of the fractal trajectory.

[0080] The method involves using a sliding window of length L to segment a discrete target signal x(n) of total length N. When the starting index of the sliding window moves to i, the corresponding truncated signal is defined as x. d (i);

[0081] The Higuchi algorithm is used to calculate the fractal dimension of the truncated time series. As the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained as follows:

[0082] Create a sample subset of the original signal:

[0083]

[0084] Where m is the initial time and start time of each subset, k is the time interval and determines the number of subsets, and m and k are integers.

[0085] The curve length for each subset X(m,k) defined by Higuchi is as follows:

[0086]

[0087] In the above formula, (N-1) / [(Nm) / k]k is the normalization factor for the curve length;

[0088] The second step is to plot L with the horizontal axis k on a log-log scale. m (k) curve, L as k changes from N to 0 m (k) The data should fall on one axis;

[0089] Use the least squares method to calculate the point (L) obtained in the previous step. m (k) is fitted, and the slope of the fitted curve can be used as an estimate of the fractal dimension;

[0090] Finally, as the starting point of the sliding window moves, the fractal trajectory corresponding to the target signal is obtained. The expression for the skewness γ of the fractal trajectory is as follows:

[0091]

[0092] Where, σ d Let N be the standard deviation of the fractal trajectory d. d This represents the length of the fractal trajectory d. This represents the mean of the fractal locus d;

[0093] The expression for the spectral energy E of the fractal trajectory is as follows:

[0094]

[0095] Where, |D(f) 2 It is the square of the spectral density modulus of the fractal trajectory;

[0096] The expression for the fractal trajectory using the polynomial function p(x) is as follows:

[0097]

[0098] Where j represents the order of the polynomial function, a j The coefficients are those of a polynomial function of order j. When N=2, a binomial fit is performed on the fractal trajectory, and the constant term fitting coefficients are selected as features.

[0099] The expression for the coefficient of variation cv of a fractal trajectory is as follows:

[0100]

[0101] Where, σ d The standard deviation of the fractal locus The mean and standard deviation σ of the fractal locus d and mean The calculation formula is as follows.

[0102]

[0103] The calculation steps for the Hurst exponent of the fractal trajectory are as follows:

[0104] Divide the fractal trajectory sequence d = {d(i), i = 1, 2, 3, ..., N} into K data groups of length m. For each subgroup, define its mean as... The maximum cumulative absolute value of the deviation from the mean within this subgroup is denoted as Z. max :

[0105]

[0106] Let Z be the minimum cumulative absolute value of the deviation from the mean within the subgroup. min :

[0107]

[0108] The range within each group is denoted as R(m):

[0109] R(m)=Z max -Z min

[0110] The sample standard deviation for each group is denoted as:

[0111]

[0112] R, S, and m satisfy the following general relation:

[0113] R(m) / S(m)=cm H

[0114] Where c is a constant, R(m) / S(m) is the rescaled range, and H is the Hurst exponent, according to the equation:

[0115] log(R / S) m =log(c) + Hlog(m)

[0116] With the log(m) sequence as the independent variable, the corresponding log(R / S) m With the series as the dependent variable, the least squares regression method is introduced to perform linear fitting analysis on the data. The slope satisfies k = H, and these H values ​​are the estimated results of the Hurst exponent.

[0117] The RF fingerprint feature vector is input into a support vector machine to identify the identities of N RF devices. Principal component analysis is used to reduce the dimensionality of the obtained feature vector. The dimensionality-reduced features are randomly divided into training and test sets in a certain ratio, such as 7:3, and then input into the support vector machine (SVM) for classification. The label results output by the trained model are compared with the actual labels to obtain the classification accuracy.

[0118] The process of using principal component analysis to reduce the dimensionality of the obtained eigenvectors is as follows:

[0119] 1) The extracted radio frequency fingerprint feature vector feature set X∈R n*m In this process, the sample features are demeaned by subtracting the mean of that feature from the value of the current feature in the sample set.

[0120] 2) Calculate the covariance matrix XX of the sample. T ;

[0121] 3) Perform eigenvalue decomposition on the covariance matrix to find the eigenvalues ​​and eigenvectors of the covariance matrix;

[0122] 4) Select the eigenvectors corresponding to the k largest eigenvalues ​​and transform them into P;

[0123] 5) Y = PX is the k-dimensional feature matrix obtained after dimensionality reduction.

[0124] Example 1:

[0125] (1) Obtain the original radio frequency signal of the radio frequency device to be identified, and perform preprocessing and normalization;

[0126] In this embodiment, Bluetooth wireless signals from nine different devices were selected as targets and numbered 1-9. Each device collected 140 data points. Since Bluetooth signals operate in the ISM2400 band, according to the Nyquist sampling theorem, the sampling frequency must be at least 4.8 Gsps. The selected data were sampled at 5 Gsps, 10 Gsps, and 20 Gsps, with a signal duration of approximately 3 μs. The actual signals were directly captured using a high-sampling-rate oscilloscope and a low-resolution analog-to-digital converter. During signal acquisition, some unwanted spurious signals were generated by the oscilloscope; therefore, a bandpass filter was added in the preprocessing stage to remove these spurious signals.

[0127] (2) Based on the preprocessed original radio frequency signal, transient detection is performed to obtain the target signal, and the fractal trajectory corresponding to the target signal is obtained by using a dynamic sliding window.

[0128] First, transient detection is performed to obtain the transient signal segment, i.e., the target signal. Then, a sliding window is used to obtain the fractal trajectory corresponding to the transient signal segment.

[0129] A sliding window of length L is used to segment a discrete target signal x(n) of total length N. When the starting index of the sliding window moves to i, the corresponding truncated signal is defined as x. d(i). Subsequently, the Higuchi algorithm is used to accurately calculate the fractal dimension of the obtained truncated time series. The specific principle is as follows: First, a sample subset of the original signal is created according to the following method.

[0130]

[0131] Where m is the initial time and start time of each subset, and k is the time interval, which determines the number of subsets. m and k are integers. Therefore, for example, setting k=3, N=100, and m=1,2,3 will generate the following three subsets:

[0132] X(1,3):X(1),X(4),...,X(100),

[0133] X(2,3):X(2),X(5),...,X(98)

[0134] X(3,3):X(3),X(6),...,X(99).

[0135] The curve length for each subset X(m,k) defined by Higuchi is as follows:

[0136]

[0137] In the above formula, (N-1) / [(Nm) / k]k is the normalization factor for the curve length.

[0138] The second step is to plot L with the horizontal axis k on a log-log scale. m (k) curve, where the data should fall on one axis as k varies from N to 0.

[0139] Next, the least squares method is used to calculate the point (L) obtained in the previous step. m (k) is fitted, and the slope of the fitted curve can be used as an estimate of the fractal dimension. Finally, as the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained.

[0140] (3) Extract the features of N radio frequency devices based on fractal trajectories, and construct the radio frequency fingerprint feature vector based on the features of N radio frequency devices;

[0141] (3.1) Calculate the skewness of the fractal trajectory for each device as a feature;

[0142]

[0143] Where, σ d Let N be the standard deviation of the fractal trajectory d. d This represents the length of the fractal trajectory d. This represents the mean of the fractal locus d;

[0144] (3.2) Calculate the spectral energy of the fractal trajectory as a feature

[0145]

[0146] Where, |D(f)| 2 It is the square of the spectral density modulus of the fractal trajectory;

[0147] (3.3) Calculate the constant term fitting coefficient of the fractal trajectory as a feature.

[0148] The fractal trajectory is fitted using a polynomial of order N by the least squares method, i.e.:

[0149] X(t) = b1 + b2t + b3t 2 +…b N+1 t N

[0150] When N=2, the fractal trajectory is fitted with a binomial formula, and the constant term fitting coefficient is selected as a feature.

[0151] (3.4) Calculate the coefficient of variation of the fractal trajectory as a characteristic

[0152]

[0153] Where, σ d The standard deviation of the fractal locus The mean of the fractal locus. Standard deviation σ. d and mean The calculation formula is as follows:

[0154]

[0155] (3.5) Calculate the Hurst exponent of the fractal trajectory as a feature;

[0156] The rescaled range method is commonly used to calculate the Hearst exponent, and its calculation steps are as follows:

[0157] Divide the fractal trajectory sequence d = {d(i), i = 1, 2, 3, ..., N} into K data groups of length m. For each subgroup, define its mean as... The maximum cumulative absolute value of the deviation from the mean within this subgroup is denoted as Z. max :

[0158]

[0159] Let Z be the minimum cumulative absolute value of the deviation from the mean within the subgroup. min :

[0160]

[0161] The range within each group is denoted as R(m):

[0162] R(m)=Z max -Z min

[0163] The sample standard deviation for each group is denoted as:

[0164]

[0165] Hurst found that R, S, and m satisfy the following general relation:

[0166] R(m) / S(m)=cm H

[0167] Where c is a constant, R(m) / S(m) is the rescaled range, and H is the Hurst exponent. According to the equation:

[0168] log(R / S) m =log(c) + Hlog(m)

[0169] With the log(m) sequence as the independent variable, the corresponding log(R / S) m With the series as the dependent variable, the least squares regression method is introduced to perform linear fitting analysis on the data. The slope satisfies k = H, and these values ​​are the estimated results of the Hurst exponent.

[0170] (3.6) Construct the radio frequency fingerprint feature vector using the five statistical measures of the fractal trajectory calculated above: skewness, spectral energy, coefficient of variation, constant term fitting coefficient, and Hurst exponent.

[0171] (4) Input the radio frequency fingerprint feature vector into the support vector machine to identify the identities of N radio frequency devices;

[0172] The 140 samples from each device were divided into training and testing sets in a 7:3 ratio. Principal component analysis (PCA) was used to reduce the dimensionality of the resulting five-dimensional vectors. The PCA process is as follows:

[0173] 1) For the feature set X∈R n*m The sample features are demeaned. For each feature of a sample, the value of the current feature is subtracted from the mean of that feature in the sample set, i.e.

[0174] 2) Calculate the covariance matrix XX of the sample. T .

[0175] 3) Perform eigenvalue decomposition on the covariance matrix to find the eigenvalues ​​and eigenvectors of the covariance matrix.

[0176] 4) Select the eigenvectors corresponding to the top k largest eigenvalues ​​and transform them into P.

[0177] 5) Y = PX is the k-dimensional feature matrix obtained after dimensionality reduction.

[0178] Using the dimensionality-reduced feature vectors as input to the Support Vector Machine (SVM), and employing a linear kernel function for model learning and training, the model achieved classification accuracies of 98.68% and 97.39% on the training and test sets, respectively.

[0179] Figure 2 It is the original feature classification confusion matrix;

[0180] Figure 3 It is a PCA-3D feature visualization diagram.

[0181] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A radio frequency fingerprint recognition method based on fractal trajectories, characterized in that: Includes the following steps: Acquire the raw radio frequency signals of N radio frequency devices to be identified, and preprocess the raw radio frequency signals; Transient detection is performed based on the preprocessed original radio frequency signal to obtain the target signal, and the fractal trajectory corresponding to the target signal is obtained by using a dynamic sliding window. Features of N radio frequency devices are extracted based on fractal trajectories, and radio frequency fingerprint feature vectors are constructed based on the features of N radio frequency devices; The RF fingerprint feature vector is input into a support vector machine to identify the identities of N RF devices; The features extracted from N radio frequency devices based on fractal trajectories include: The skewness of the fractal trajectory, the spectral energy of the fractal trajectory, the constant term fitting coefficient of the fractal trajectory, the coefficient of variation of the fractal trajectory, and the Hurst exponent of the fractal trajectory. The skewness of the fractal trajectory The expression is as follows: in, fractal locus standard deviation Representing fractal loci Length, Representing fractal loci The average; The expression for the spectral energy E of the fractal trajectory is as follows: in, It is the square of the spectral density modulus of the fractal trajectory; Fractal trajectories using polynomial functions The expression is as follows: in Represents the order of the polynomial function. For order The coefficients of the polynomial function are selected. When the fractal trajectory is fitted with a binomial, the constant term fitting coefficient is selected as a feature; coefficient of variation of fractal trajectories The expression is as follows: in, The standard deviation of the fractal locus The mean and standard deviation of the fractal locus and mean The calculation formula is as follows: 。 2. The radio frequency fingerprint recognition method based on fractal trajectories according to claim 1, characterized in that: The process of obtaining the fractal trajectory corresponding to the target signal using a dynamic sliding window is as follows: Using a length of The sliding window has a total length of Discrete target signal To perform the segmentation, when the starting index of the sliding window moves to... When this happens, the corresponding truncated signal is defined as follows: ; The Higuchi algorithm is used to calculate the fractal dimension of the truncated time series. As the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained.

3. The radio frequency fingerprint recognition method based on fractal trajectories according to claim 2, characterized in that: The utilization length is The sliding window has a total length of Discrete target signal To perform the segmentation, when the starting index of the sliding window moves to... When this happens, the corresponding truncated signal is defined as follows: ; The Higuchi algorithm is used to calculate the fractal dimension of the truncated time series. As the starting point of the sliding window moves, the fractal trajectory corresponding to the signal is obtained as follows: Create a sample subset of the original signal: in, These are the initial time and start time of each subset. It is the time interval, and it determines the number of subsets. and It is an integer. Each subset defined by Higuchi The curve lengths are as follows: In the above formula, It is the normalization factor for the curve length; Plot the x-axis on a log-log scale. of Curve, when from When it changes to 0, The data should fall on one axis; Use the least squares method to calculate the points obtained in the previous step. By fitting the curve, the slope of the fitted curve can be used as an estimate of the fractal dimension. Finally, as the starting point of the sliding window moves, the fractal trajectory corresponding to the target signal is obtained.

4. The radio frequency fingerprint recognition method based on fractal trajectories according to claim 3, characterized in that: The calculation steps for the Hurst exponent of the fractal trajectory are as follows: fractal trajectory sequence Divided into A length of The data is grouped, and for each subgroup, its mean is defined as... And the maximum cumulative absolute value of the deviation from the mean within this subgroup is denoted as : The minimum cumulative absolute value of the deviation from the mean within this subgroup is denoted as : The range within each group is denoted as : The sample standard deviation for each group is denoted as: , , The following general relation is satisfied: in, It is a constant. For recalibrated range, That is, the Hurst exponent, according to the equation: by The sequence is the independent variable, and the corresponding With the sequence as the dependent variable, the least squares regression method is introduced to perform linear fitting analysis on the data, and its slope satisfies... The H value is the estimated result of the Hurst exponent.

5. The radio frequency fingerprint recognition method based on fractal trajectories according to claim 4, characterized in that: The RF fingerprint feature vector is input into a support vector machine to identify the identities of N RF devices. Principal component analysis is used to reduce the dimensionality of the obtained feature vector. The dimensionality-reduced features are randomly divided into training and test sets according to a certain ratio and then input into the support vector machine (SVM) for classification. The label results output by the trained model are compared with the actual labels to obtain the classification accuracy.

6. The radio frequency fingerprint recognition method based on fractal trajectories according to claim 5, characterized in that: The process of using principal component analysis to reduce the dimensionality of the obtained eigenvectors is as follows: 1) The extracted radio frequency fingerprint feature vector feature set In this process, the sample features are demeaned by subtracting the mean of that feature from the value of the current feature in the sample set. ; 2) Calculate the covariance matrix of the sample. ; 3) Perform eigenvalue decomposition on the covariance matrix to find the eigenvalues ​​and eigenvectors of the covariance matrix; 4) Select the largest front The eigenvectors corresponding to each eigenvalue are transformed into ; 5) That is, the result after dimensionality reduction. 3D feature matrix.

Images