Radar emitter individual identification method based on cca multi-domain feature fusion

Abstract

The application discloses a radar radiation source individual identification method based on CCA multi-domain feature fusion and relates to the technical field of signal processing methods. The method comprises the following steps: taking the rising edge of a radar radiation source signal envelope as an object, considering the alignment and screening of multi-domain features of time-space structure differences, and performing reference alignment of all signals; adopting a CCA algorithm to perform multi-domain feature fusion; and performing adaptive DBSCAN clustering based on the multi-domain feature fusion to identify radar radiation source individuals. The method can deeply represent the features of radar radiation source individuals under full-blind conditions, improve the clustering accuracy, and further improve the accuracy of radar radiation source individual identification.

Description

Technical Field

[0001] This invention relates to the field of signal processing methods, and in particular to a method for individual radar radiation source identification based on CCA multi-domain feature fusion. Background Technology

[0002] Individual radar emitter identification (SEI) is both a cutting-edge and challenging area of ​​radar research. In an increasingly complex electromagnetic environment, individual radar emitter identification can link individual emitters with their associated platforms, weapon systems, and strategic and tactical targets, which is of great significance for early warning analysis and intelligence reconnaissance in the field of electronic warfare.

[0003] Individual identification of radar radiation sources is an advanced task, and has been studied extensively both domestically and internationally. Achieving individual identification without being affected by transmitted signal waveforms, intentional modulation types, receiver performance, or complex electromagnetic environments places demands on the uniqueness, independence, and stability of individual feature extraction. Wang Hongwei et al. extracted high-order moment features of the pulse envelope leading edge for individual identification of unknown radiation sources, achieving good discrimination results with a signal-to-noise ratio greater than 15dB. However, the problem of poor noise resistance of time-domain features remains. Wang Qiebo et al. established a standard waveform of the pulse envelope and achieved individual identification by calculating similarity, but the standard waveform needs continuous correction. Liu Yilin et al. extracted the carrier frequency deviation of individual radiation sources using the piecewise FFT phase difference method, improving frequency resolution and solving the problem of insufficient accuracy of the FFT algorithm in estimating the center frequency of short-duration pulse signals, thus enhancing individual discrimination ability. Zhang Jiao extracted time-frequency distribution features and frequency drift characteristic curve features based on the adaptive fractional-order spectrum method. The classification results obtained are independent of the radiation source's radiated power, improving the robustness of feature extraction. Based on the strong individual representation capabilities of features, numerous classifiers (KNN, SVM, ELM) can achieve high accuracy in individual classification. However, in recent years, deep neural networks have made it possible to automatically extract features and reduce the dependence on the mechanisms and extraction methods of individual feature generation.

[0004] Although research on individual radar radiation source identification has been conducted for many years, there is currently very little research on blind clustering for individual radar radiation source identification. The main reasons are: firstly, in real-world research, some samples can always be manually labeled, and supervised algorithms trained using labeled data generally achieve high accuracy levels. However, due to the specific nature of the field, most training data comes from computer simulations and laboratory collections, with a small portion from field measurements. In the complex electromagnetic environment of actual battlefields, small sample sizes, imbalanced conditions, and the time-sensitive nature of combat place higher demands on individual radiation source identification. Secondly, the differences in the "fingerprint features" that distinguish individual radiation sources mainly exist in the radar transmitter. However, with advancements in technology, the "fingerprint features" of phase noise and key components in radar transmitters are becoming increasingly difficult to extract. Therefore, achieving clustering under completely blind conditions is extremely difficult, but it is also highly advantageous for gaining the initiative in the electromagnetic battlefield. Summary of the Invention

[0005] The technical problem to be solved by this invention is how to provide a radar radiation source individual identification method based on CCA multi-domain feature fusion that can deeply represent the characteristics of individual radiation sources under completely blind conditions and improve the accuracy of clustering.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a radar radiation source individual identification method based on CCA multi-domain feature fusion, comprising the following steps:

[0007] S1: Taking the rising edge of the radar radiation source signal envelope as the object, the alignment and filtering of multi-domain features considering the differences in spatiotemporal structure are performed to align all signals to the reference.

[0008] S2: Multi-domain feature fusion is performed using the CCA algorithm;

[0009] S3: Adaptive DBSCAN clustering based on multi-domain feature fusion is used to identify individual radar radiation sources.

[0010] The beneficial effects of adopting the above technical solution are as follows:

[0011] The proposed method takes into account the temporal structure differences of different features, using the rising edge of the radar signal envelope as the target for benchmark alignment of all signals. Secondly, considering the redundancy and concealment of individual fingerprint features in multiple domains due to structural differences, a feature selection strategy based on the correlation coefficient matrix is ​​adopted, taking the time and time-frequency domains as examples, and CCA is used for multi-domain feature fusion. Furthermore, the model derivation illustrates the role of the CCA algorithm in multi-domain complementary feature fusion and demonstrates the adaptability of multi-domain fused features to data density-based clustering algorithms. Therefore, the DBSCAN algorithm, which adaptively determines clustering parameters, is introduced in the backend, ultimately realizing the DBSCAN clustering method based on CCA multi-domain feature fusion. This method can deeply represent the features of individual radiation sources under fully blind conditions, improve clustering accuracy, and thus enhance the accuracy of radar radiation source individual identification. Attached Figure Description

[0012] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0013] Figure 1 This is a flowchart of the method described in the embodiments of the present invention;

[0014] Figure 2 This is a flowchart of the time-domain alignment process in the method described in the embodiments of the present invention;

[0015] Figure 3 These are pseudo-color images before and after temporal alignment in the method described in this embodiment of the invention;

[0016] Figures 4a-4b This is a waveform diagram of the envelope front in the method described in the embodiments of the present invention;

[0017] Figures 5a-5b This refers to the envelope front phase map in the method described in the embodiments of the present invention;

[0018] Figures 6a-6c This is the 3D-Hilbert spectrum of the signal in the method described in the embodiments of the present invention;

[0019] Figure 7 This is a schematic diagram of the CCA principle in the method described in the embodiments of the present invention;

[0020] Figure 8 This is a flowchart of the adaptive DBSCAN clustering algorithm in the method described in the embodiments of the present invention;

[0021] Figure 9a This is a clustering effect diagram (feature scatter plot) of fused features in the method described in the embodiments of the present invention;

[0022] Figure 9b This is a clustering effect diagram (clustering visualization) of the fused features in the method described in the embodiments of the present invention;

[0023] Figure 10a This is an example of an adaptive DBSCAN clustering graph (feature scatter plot) in an embodiment of the present invention.

[0024] Figure 10b This is an example of an adaptive DBSCAN clustering graph (clustering visualization) in an embodiment of the present invention.

[0025] Figure 11a This is an example of an adaptive DBSCAN clustering graph (feature scatter plot) in an embodiment of the present invention.

[0026] Figure 11b This is an example of an adaptive DBSCAN clustering graph (clustering visualization) in an embodiment of the present invention.

[0027] Figure 12a This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=6, F-score=58.82%).

[0028] Figure 12b This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=4, F-score=35.62%).

[0029] Figure 12c This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=6, F-score=74.35%).

[0030] Figure 12d This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=6, F-score=57.99%).

[0031] Figure 12e This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=3, F-score=48.10%).

[0032] Figure 12f This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=5, F-score=69.57%).

[0033] Figure 13a This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=8, F-score=48.11%).

[0034] Figure 13b This is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=8, F-score=42.03%).

[0035] Figure 13cThis is a comparison chart of the clustering effect of temporal feature combination in the embodiments of the present invention (Cluster=4, F-score=21.06%).

[0036] Figure 14a This is a comparison chart of clustering effects before and after fusion in an embodiment of the present invention (Cluster=6, F_score=74.35%);

[0037] Figure 14b This is a comparison chart of clustering effects before and after fusion in an embodiment of the present invention (Cluster=8, F_score=48.11%);

[0038] Figure 14c This is a comparison chart of clustering effects before and after fusion in an embodiment of the present invention (Cluster=6, F_score=94.74%). Detailed Implementation

[0039] 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 a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0040] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0041] like Figure 1 As shown in the figure, this invention discloses a method for individual radar radiation source identification based on CCA multi-domain feature fusion, including the following steps:

[0042] S1: Taking the rising edge of the radar radiation source signal envelope as the object, the alignment and filtering of multi-domain features considering the differences in spatiotemporal structure are performed to align all signals to the reference.

[0043] S2: Multi-domain feature fusion is performed using the CCA algorithm;

[0044] S3: Adaptive DBSCAN clustering based on multi-domain feature fusion is used to identify individual radar radiation sources.

[0045] This application explores the identification of unknown radar radiation sources from the perspective of multi-domain feature fusion, aiming to extract features that can deeply represent individual radiation sources under completely blind conditions and improve the accuracy of clustering. First, from the temporal domain perspective, starting with the rising edge of the classic radar signal envelope, which reflects individual characteristics, its rise time, phase change features, and fractal features are extracted. Second, from the time-frequency domain perspective, multiple features, such as time-frequency fractals, are calculated based on the Hilbert-Huang transform of the signal. Considering the differences in information expression and structural expression of features across different domains, a correlation coefficient screening and CCA fusion method for multi-domain feature alignment is proposed. Specifically, considering the temporal structure differences of different features, the rising edge of the radar signal envelope is used as the object for benchmark alignment of all signals. Furthermore, considering the redundancy and concealment of individual fingerprint features in multiple domains due to structural differences, a feature selection strategy based on the correlation coefficient matrix is ​​studied, taking the time and time-frequency domains as examples, and CCA is used for multi-domain feature fusion. Furthermore, a theoretical model was established to illustrate the role of the CCA algorithm in multi-domain complementary feature fusion, and the adaptability of multi-domain fused features to data density-based clustering algorithms was demonstrated. To this end, the DBSCAN algorithm, which adaptively determines clustering parameters, was introduced in the backend, ultimately realizing a DBSCAN clustering method based on CCA multi-domain feature fusion. Related experiments proved the effectiveness of this method.

[0046] Alignment and filtering of multi-domain features considering spatiotemporal structural differences

[0047] Signal preprocessing

[0048] Radar source signals are affected by environmental factors and the receiving equipment itself during propagation and reception, resulting in various noise interferences that negatively impact feature extraction and individual identification. Therefore, signal denoising plays a crucial role in this paper. Traditional denoising methods typically involve performing a Fourier transform on the received signal, followed by low-pass filtering and then an inverse Fourier transform to obtain the denoised signal. However, while low-pass filtering reduces noise, it can obscure important information, leading to signal distortion. Given the advantages of wavelet transform, such as multi-resolution, decorrelation, and flexible basis functions, wavelet transform-based signal-noise separation methods have been widely adopted. This section employs a wavelet threshold denoising algorithm to denoise the received radar source signal.

[0049] Envelope signal alignment and extraction considering temporal structure differences

[0050] 1) Normalization

[0051] Different radar sources emit pulses with varying peak amplitudes, and even radar sources from the same entity can exhibit fluctuations. Therefore, it is necessary to normalize the acquired signal data based on the maximum and minimum pulse amplitudes to eliminate the effects of these fluctuations. This normalization only involves scaling the radiation source signal and does not lose any important information.

[0052] In addition to normalizing the pulse signal amplitude, time delay also needs to be normalized. Time normalization aligns the pulse waveform, aiming to effectively utilize time-domain features for individual radiation source identification. This application employs correlation processing to align the pulse waveform.

[0053] The relevant processing steps are as follows:

[0054] Define two pulse signals x(t) and y(t) with signal lengths n1 and n2 respectively, and let m = min{n1, n2}.

[0055] Calculate the correlation function R between two pulse signals x，y (τ), τ∈[-m, m] and τ∈Z, is the time offset.

[0056]

[0057] Based on the relationship between the obtained relevant function values ​​and the time offset, find the way to make R... x，y The largest τ value represents the time delay of signal y(t) relative to x(t).

[0058] like Figure 2 As shown, the alignment process is as follows:

[0059] Input pulse signal S n = [S1, S2, ..., S n After denoising and normalizing, a new pulse signal S' is obtained. n =[S′1,S′2,…,S′ n Calculate and sort the signal-to-noise ratios (SNRs) of all N pulses, and then select the pulse with the highest SNR. As a reference pulse x(t).

[0060] For each pulse signal that needs alignment (i.e., y(t) mentioned above), (j = 1, 2, ..., n-1), calculate the correlation function R between it and the reference pulse signal x(t). x，y (τ), calculate the time delay τ between the two signals. j .

[0061] Based on the obtained time offset, the pulse signals that need to be aligned are shifted in time, so that... Align it with the reference pulse signal.

[0062] Figure 3 The top left image shows a pseudo-color map of measured radiation source data without any preprocessing, where the signal front-end positions are chaotically distributed, which is detrimental to subsequent signal analysis and feature extraction. Figure 3 The lower left image shows the pseudo-color image after alignment. As can be clearly seen from the enlarged image on the right, the pulse start positions are consistent after alignment, exhibiting a neat arrangement. Overall, the aligned data is more ordered, facilitating subsequent extraction of the envelope front and individual pulse identification.

[0063] 2) Extract the rising edge of the envelope

[0064] Different radar systems exhibit rich nonlinear characteristics in their pulse envelopes due to subtle differences in their internal mechanical and electronic components. Therefore, from the perspective of equipment differences, it is feasible to identify individual radar systems using the characteristics of their pulse envelopes.

[0065] The steps for extracting the rising edge of the envelope are as follows:

[0066] Input pulse signal S n = [S1, S2, ..., S n After normalization, a new pulse signal S is obtained. n =[S′1,S′2,…,S′ n ].

[0067] Performing a Hilbert transform on the normalized pulse signal yields the envelope signal T = [T1, T2, ..., T]. n After normalization, a new envelope signal T = T′1, T′2, ..., T′ is obtained. n ].

[0068] Find the x-coordinate index1 corresponding to the first maximum point t1 in the envelope signal T′n that is greater than 0.5.

[0069] Find the x-coordinate index2 of the nearest minimum point t2 less than 0.1 in the envelope signal T′[1:index2].

[0070] Output envelope rising edge signal T′[index1:index2].

[0071] Figures 4a-4b He Ru Figures 5a-5bThe envelope leading-edge waveforms and corresponding phase diagrams of different radar individuals are presented. Initial observation of the envelope leading-edge waveforms and corresponding phase diagrams of six radar individuals revealed certain clustering characteristics in the pulse envelope leading-edge waveforms. However, due to the continued volatility of the pulse envelope leading-edge waveforms and the insufficient distance between the curve clusters formed by different individuals, the shape of the pulse leading-edge waveforms alone is insufficient to accurately identify individual radiation sources. Therefore, further analysis was conducted, extracting other time-domain features.

[0072] 3) Temporal feature extraction based on rising edge of envelope

[0073] In the time domain, based on the rising edge of the envelope, the calculated features include fractal dimension, LZC (Lempel Ziv complexity), pulse envelope rising edge length, rising slope, information dimension, fuzzy entropy, and origin moments and central moments of various orders. Features with relatively outstanding convergence performance include fractal dimension, information dimension, LZC complexity, and pulse envelope rising time. The calculation of the pulse envelope rising edge length is relatively simple and will not be introduced here. The following sections will briefly introduce fractal dimension, LZC complexity, and information dimension.

[0074] Fractal dimension: The fractal dimension is a core parameter in fractal theory, but its calculation is quite complex and difficult in practice. In contrast, the box dimension plays a crucial role in fractal theory, reflecting the geometric scale of a fractal set and proving highly effective for signal classification and recognition. Therefore, the box dimension will be used as the primary means of describing the complexity and irregularity of fractal signals. The signal sequence s(i) is placed within a unit square, with the minimum interval of the horizontal coordinates being d = 1 / N, where N is the length of the signal sequence. Let:

[0075]

[0076] Box dimension:

[0077]

[0078] Information Dimension: While the fractal box dimension can reveal the geometric scale characteristics of fractals, it falls short when seeking a deeper understanding of the distribution of fractal sets within a spatial region. To compensate for this deficiency, the information dimension can be used. Reconstructing the signal sequence s(i), let:

[0079]

[0080]

[0081] Information dimension:

[0082]

[0083] LZC Complexity: Lempel-Ziv complexity (LZC) is a unique method for revealing the intrinsic patterns in signal waveform variations. The core of this method lies in analyzing the characteristics of signal sequences through two basic operations: copying and appending. Specifically, it calculates the number of appending operations required to construct the signal sequence and uses this number as a metric for sequence complexity. Therefore, LZC not only provides a method for quantifying signal complexity but also plays a crucial role in signal feature extraction research.

[0084] Starting from the beginning of the signal sequence, the signal is scanned sequentially, each time finding the shortest subsequence that has not appeared in previous parts and recording its length. This shortest subsequence is added to the set of known patterns, and the process continues to scan the remaining signal, repeating the above process until the signal ends. LZC complexity is the sum of the lengths of all shortest subsequences; it reflects the frequency of occurrence of novel patterns in the signal and can therefore be used as a measure of signal complexity.

[0085] 4) Extracting time-frequency domain features based on Hilbert spectrum

[0086] EMD decomposition and HHT transformation are performed on the radar radiation source signal to obtain the 3D-Hilbet spectrum, such as... Figures 6a-6c As shown, the 3D-Hilbert spectrum exhibits uneven high and low fluctuations, making fractal geometry more suitable than Euclidean geometry for describing such images. The difference box dimension, multifractal dimension, and sparsity were calculated based on the 3D-Hilbert spectrum. Related noise resistance analysis and experimental results indicate that the fractal features of the image have higher stability than the temporal features. The calculation steps are briefly described below:

[0087] Difference box dimension: Divide the N×N time-frequency domain into uniform L×L blocks. Let r = L / N, and each block have a height h = [255r], forming an L×L×h box. In the (i, j)th L×L grid, the maximum and minimum values ​​are in the l-th and k-th boxes respectively. The number of current L×L×h boxes required to cover this network is n. r If (i, j) = l - k + 1, then the number of boxes required to cover the entire surface is N. r =∑n r (i, j). Find N for all possible values ​​of L. r Linear fitting using the least squares method The resulting slope is the difference box dimension FD, which is determined by the limit of equation (7):

[0088]

[0089] To gain a more detailed understanding of the hierarchical structure of geometric objects, the multifractal dimension is introduced to describe the mass and density distribution characteristics on irregular fractal spaces. Using the results from calculating the difference box dimension, the multifractal dimension is given by the limit of equation (8):

[0090]

[0091] In equation (8), q is a parameter, and in practice, q should be selected to make MD tend to be stable.

[0092] 5) Feature selection considering differences in multi-domain feature space structure

[0093] The signals from radiation sources are multi-domain and complex. Multiple features can reflect the characteristics of an individual radiation source to varying degrees and in different aspects. However, a single feature is insufficient to describe all the attributes of an individual radiation source and is easily affected by various interferences, leading to unstable clustering results. Because there are mutually exclusive and similar relationships between features, directly using them for subsequent fusion and clustering may result in poor performance. Therefore, it is necessary to first select multi-domain features under fully blinded conditions.

[0094] In the field of pattern recognition, Fisher discriminant analysis has a significant impact on evaluating the quality of vector discrimination directions. The effectiveness of feature selection methods based on correlation coefficients is discussed below.

[0095] The feature set is represented as Assuming there are only two classes of data and the variances of the same feature in both classes are equal and similar, then measuring the distinguishing effect of a feature on the two classes only requires considering the difference in their means. ψ1=[ψ 11 , ψ 21 ,...ψ n1 ] T It can be regarded as ψ1=[μ 11 μ 21 ] T μ 11 μ 21 Let ψ2 be the mean of the two types of data features. After feature set normalization, ψ2 = [μ 12 μ 22 ] T Correlation coefficient r with ψ1:

[0096] r = <ψ1, ψ2> (9)

[0097] Consider μ 11 =μ 21 That is, feature ψ1 has no effect on distinguishing data, and at this time ψ2 satisfies μ 12 +μ 22 =r / μ 11 Let's assume If we select ψ1 and ψ2 as features, only ψ2 carries class information, and the mean distance between classes d = (μ 12 -μ 22 ) 2 Even assuming that the variances of the two classes of data with the same feature are equal, since the probability density function of the data is unknown, the only way to improve inter-class discrimination is to maximize the inter-class mean distance.

[0098]

[0099] The inter-class mean distance reaches its maximum value of 2 when r = 0, i.e., the correlation coefficients of ψ1 and ψ2 are 0. This result is obvious in the two-dimensional plane, because ψ2 may contain inter-class differences only when the correlation coefficient is very small or even 0, given that ψ1 does not have any inter-class differences.

[0100] In 3-dimensional and higher dimensions, where ψ1 still does not contain any class difference information, using the sum of squared distances between means as the objective function is inappropriate. Define matrix D:

[0101]

[0102] At this point, we should maximize the minimum value of the distances between the three classes, that is:

[0103]

[0104] The optimization problem defined by equation (11) is non-convex, and according to the cyclic symmetry of the variables, ψ is not unique when taking the maximum or minimum value. Numerical method results show that in 1 T When ψ = 0, it does not take the extreme value, but the extreme value points almost always satisfy |1. T ψ|<0.05.

[0105] Based on the above analysis, under unsupervised conditions, if ψ1 provides no class difference information for clustering, a feature vector that is as orthogonal to ψ1 as possible should be selected. Although there are countless such vectors (all in the plane with ψ1 as the normal vector), it at least provides a direction for feature selection. In other words, a feature vector that is highly correlated with ψ1 is not better than a feature vector that is weakly correlated with ψ1.

[0106] In fact, due to the inherent effectiveness of feature extraction methods, the eigenvectors in the feature matrix cannot be entirely devoid of any class difference information. Therefore, selecting eigenvectors with higher correlation coefficients is unnecessary. While a single correlation coefficient cannot fully characterize the relationship between two eigenvectors, when the feature extraction method is effective, weakly correlated eigenvectors should be more beneficial for clustering than strongly correlated ones. In pattern recognition theory, extracting statistically irrelevant discriminant vectors is an enhancement under the conditions of maximizing the inter-class scatter matrix and minimizing the intra-class scatter matrix. The discriminant vectors proposed by Jin Zhong et al., which are conjugate orthogonal with respect to the overall scatter matrix, even achieved better results than the FS vector set.

[0107] Research on unsupervised clustering feature methods mainly focuses on high-dimensional features of images, which are no less complex than feature extraction. While feature selection based on correlation coefficients is simple, subsequent experiments show that it remains effective when the number of features is small.

[0108] Table 1. Correlation coefficients of time-domain features

[0109]

[0110] Table 2 Correlation coefficients of time-frequency domain characteristics

[0111]

[0112] Based on the correlation coefficients listed in the table above, the fractal dimension and information dimension are selected for time-domain features, and the difference box dimension and multifractal dimension are selected for time-frequency domain features for subsequent feature fusion and clustering.

[0113] Fusion of multi-domain features and theoretical proof

[0114] 1) Fusion of multi-domain features

[0115] The features extracted in the time domain and time-frequency domain are centered and standardized to form a feature matrix. and n is the total number of samples, and p and q are the feature dimensions of the samples. C xx =X T X / (n-1), C yy =Y T Y / (n-1) represents the covariance matrix of X and Y, C xy =X T The Y / (n-1) cross-covariance matrix is ​​obtained from the sample estimation.

[0116] Figure 7 This is a schematic diagram of the CCA principle. CCA projects the original feature vectors to obtain canonical variables to extract the linear correlation between different feature matrices. The degree of linear correlation is as follows:

[0117]

[0118] z x =Xw x , z y =Yw y , Let z be the typical vector to be determined. x , z y For typical variables, ||z is generally restricted. x ||2=||z y ||2=1.

[0119] At this point, maximizing the correlation is equivalent to maximizing the cosine of the angle between the canonical variables:

[0120]

[0121] In equation (13), <·> represents the vector inner product operation, and δ ij Let Kronecker notation be used. Using the Lagrange multiplier method and singular value decomposition, let... where S=diag(λ1,λ2,...,λ m (0, ..., 0). The i-th pair of typical vectors The features after classical vector projection transformation in the time and time-frequency domains are Z. x =XW x and Z y =YW y . λ i >0 is The correlation coefficient. The fused feature is Z. x +Z y Equation (13) means that Z X Z Y Within the matrix, the columns are linearly independent, while the correlation between the corresponding columns reaches its maximum and decreases sequentially. Typical variables are a dimensionality-reduced representation of the original feature set, while also concisely reflecting the correlation between the variable sets.

[0122] 2) CCA Multi-Domain Fusion Feature Statistics Analysis and Theoretical Proof

[0123] E(Z) is a linear combination of E(X) and E(Y), so E(Z) = 0. Calculate the covariance matrix C of the feature fusion matrix. zz =Z T Z / (n-1)=(XW x +YW y ) T (XW x +YW y ) / (n-1). and Substituting the terms, we get the sum of the four terms:

[0124]

[0125] C zz = (2I+2S) / (n-1), where I is an m-order identity matrix and S is an m-order singular value matrix obtained by SVD decomposition of H. C zz The diagonal matrix result also shows that the columns of Z are uncorrelated. The i-th column of the feature matrix is ​​the result of fusing the i-th pair of canonical variables, and its variance is estimated using samples. This indicates that, compared to before fusion, the dispersion of any feature in Z is increased by n. 1 / 2 The reason for the decrease in the factor is essentially... The scaling effect.

[0126] P and Q are both orthogonal matrices. In the singular value decomposition result of H, U and V... T They are also orthogonal matrices, and all correspond to rotation transformations; and Diagonal matrices correspond to scaling transformations. Therefore, the role of a typical vector is to perform multiple rotations and scaling transformations on the column space of the original characteristic matrices X and Y. The scale of the scaling transformation depends on X. T X, Y T The magnitude of the eigenvalues ​​of Y. Therefore, from a transformation perspective, CCA analysis essentially involves rotating and scaling the column vectors of the original eigenmatrix to extract the correlation features between the two eigenmatrixes. Consider the scale of the scaling transformation:

[0127]

[0128] Limit the correlation coefficient when selecting features. That is, given that the features are weakly correlated, it is acceptable to make approximations. Define Rayleigh quotient R(X) T X, a):

[0129]

[0130] In equation (16), a is any non-zero vector and ||a||² = 1. T The eigenvalues ​​of X satisfy λ min ≤R(X T X, a)≤λ max Furthermore, through computer simulation, it can be estimated that the following relationship holds for its n eigenvalues:

[0131]

[0132] Expressions combining typical vectors As can be seen from (17), under the condition of feature fusion with a large number of samples, the transformation determined by CCA makes Zx Z y The degree of dispersion of Y is reduced to its original value compared to X. The dispersion of Z after fusion is also affected by the correlation coefficient between canonical variables. It should be noted that, unlike feature selection, the larger the correlation coefficient between the canonical variables used for fusion, the stronger the canonical correlation, and the more suitable it is to become a new feature vector characterizing individual differences.

[0133] The canonical variable z determined by CCA x , z y These are linear combinations of column vectors X and Y, respectively. Therefore, whether considering the overall sample or a subset of samples from a specific class, the mean remains unchanged after the linear combination, i.e., E(Z). x )=E(Z y )=E(X)=E(Y)=0, E(AX)=E(AZ)=E(AY), Select a matrix for a row, n x It represents the number of samples for a given individual. AX, AY, and AZ are the time-domain, time-frequency domain, and fused feature matrices of the corresponding radiation source individuals.

[0134] feature matrix Consider them as coordinates in p-dimensional and q-dimensional spaces, respectively, and select matrices A1 and A2 for the rows. Select different individual samples of unknown radar radiation sources [X] from the feature populations in the time domain and time-frequency domain. iq1 X iq2 ] = [A1, A2]X, [Y iq1 Y iq2 [A1, A2]Y. The "distance" between unknown radiation source individuals iq1 and iq2 in the time and time-frequency domains can be measured as d. x =||μ iq1 -μ iq2 ||2,d y =||μ iq1 -μ iq2 2. Ignoring the scaling effect and considering the linearity of the mean after a linear combination of random variables, the distance between the sample means of the radiation source individuals iq1 and iq2 after feature fusion satisfies:

[0135] min{d x d y}<d z <max{d x d y} (18)

[0136] Equation (18) shows that the relative distance between classes after fusion is between the relative distances of individual domains before fusion. Using this property to fuse complementary features of multiple domains can make it possible for samples that are clustered and mixed in a certain domain due to the small distance between classes to be successfully clustered after fusion.

[0137] To improve the accuracy of blind clustering of unknown radar radiation sources, a sufficiently large inter-class distance is fundamental to ensuring accurate clustering; that is, there should be no large overlap of features between different classes. Simultaneously, a sufficiently small intra-class variance is crucial for ensuring stable and reliable clustering. The above analysis considers the impact of the linear transformation determined by CCA on clustering from the perspectives of inter-class mean distance and intra-class variance. Specifically, the mean and variance before and after fusion show that CCA can effectively utilize complementary features from multiple domains to improve inter-class distance, while using n... 1 / 2 This significantly reduces the dispersion of features. From a clustering algorithm perspective, this essentially increases the "density" of data within each cluster based on the original data. Therefore, considering the fusion mechanism of CCA, density-based clustering algorithms are the most suitable for the fused feature matrix, which is why DBSCAN clustering is introduced in the backend.

[0138] The above analysis examined the feasibility of CCA (Clustered Classification Association) in fusing complementary features from the perspectives of inter-class distance and intra-class variance. Now, we will evaluate the effectiveness of CCA clustering from the perspectives of increasing inter-class distance and decreasing intra-class variance. The supervised classical linear classification algorithm LDA is also based on this simple idea, and its goal is to maximize the Fisher criterion function:

[0139]

[0140] Equation (19) is S b S w The generalized Rayleigh quotient, w is the projection vector to be determined, S b S w Let these be the inter-class scatter matrix and the intra-class scatter matrix. When considering only the quotients of corresponding elements in these two matrices without considering the projection vectors:

[0141]

[0142] This symmetric matrix characterizes the theoretical clustering performance of features within the domain, and is defined as the domain separation matrix. The values ​​on its diagonal are the separation scores of individual features, while the others are the separation scores of collaborative features. To compare the difference in separation scores of the feature matrices before and after fusion, we can define:

[0143]

[0144] Let J be the domain separation degree. r As a positive indicator, it theoretically evaluates the clustering effect of the feature matrix. It should be noted that CCA deals with blind signals without labels, while the intra-domain separation J... rThe size comparison should be performed after clustering is completed, so the LDA supervised classification algorithm is not actually used here, even though LDA is equivalent to CCA when encoding specific sample class labels.

[0145] Adaptive DBSCAN clustering

[0146] To adapt to complex battlefield environments and identify different radar radiation sources under label-less conditions, an unsupervised clustering algorithm is needed. Furthermore, the above demonstrates the adaptability of density-based clustering algorithms to the fused feature matrix; therefore, this application uses the DBSCAN algorithm in the CCA backend. DBSCAN is a typical density-based spatial clustering algorithm used to divide high-density point regions into clusters and effectively filter low-density point regions. However, DBSCAN is sensitive to two parameters, Eps and MinPts. These two parameters describe the density of data point distribution, dividing regions with sufficient density into clusters to form the largest set of density-connected points. Traditional DBSCAN algorithms require manual determination of Eps and MinPts parameters; improper selection leads to decreased clustering performance. Therefore, an adaptive algorithm for determining DBSCAN parameters is adopted. A parameter optimization strategy generates candidate Eps and MinPts parameters, automatically finding stable cluster number variation ranges and selecting the parameters with the lowest density threshold as the optimal parameters, effectively solving the problems of traditional DBSCAN algorithms.

[0147] 1) Definitions and steps of the adaptive DBSCAN algorithm

[0148] 1-1) Relevant Definitions

[0149] To accurately describe the algorithm, the following definitions are given:

[0150] Definition 1 (Eps neighborhood) is the region centered at a data point P with a radius of Eps.

[0151] Definition 2 (Core Point): If the number of data points in the neighborhood of a data point reaches or exceeds a preset minimum value MinPts, then this data point is called a core point.

[0152] Definition 3 (Density Direct Reach): If a data point P is located within the Eps neighborhood of a data point Q, and Q is a core object (i.e., the Eps neighborhood of Q contains at least MinPts sample points), then point P is density directly reachable from point Q.

[0153] Definition 4 (Density Reachability): If the density of core points P and Q is directly reachable, and the density of Q and U is directly reachable, then the density of P and U is reachable.

[0154] Definition 5 (Density Connectivity): If core points P and Q are density reachable, and Q and U are density reachable, then P and U are density connected.

[0155] Definition 6 (cluster) is the set of all density-connected points. This set is "maximum", meaning it cannot be expanded by adding more density-connected points, nor can it be divided into two or more density-connected subsets.

[0156] Definition 7 (Noise Points): Points that do not belong to any cluster are considered noise points or outliers. These points are neither core objects nor connected to the point density in any cluster.

[0157] Definition 8 (F-score) is a comprehensive metric for evaluating clustering results. Let precision be the percentage of correctly identified data out of the total number of identified data; and recall be the percentage of correctly identified data out of the total number of data points. Then, the clustering precision metric is defined by the F-score:

[0158]

[0159] 1-2) The steps of the adaptive DBSCAN algorithm are as follows: Figure 8 As shown.

[0160] 2) Perform adaptive DBSCAN clustering and present the results.

[0161] The canonical variables identified by the CCA algorithm passed the canonical correlation test, ensuring the reliability of the final fused features. Clustering was then performed using the adaptive DBSCAN algorithm, such as... Figures 9a-9b As shown.

[0162] Table 3. Variance of radiation source characteristics before and after fusion

[0163]

[0164] Table 4 DBSCAN Algorithm Parameters and Clustering Performance Evaluation

[0165]

[0166] Table 3 shows the parameters, clustering accuracy, and intra-domain separation defined above, determined by the adaptive DBSCAN algorithm before and after feature selection. Looking at the DBSCAN parameters, the Eps of the algorithm after feature selection are significantly lower than before, while the Minpts are the opposite. Considering the concepts of density accessibility, density reachability, and density connectivity in the DBSCAN algorithm, the changes in Eps and Minpts demonstrate that after feature selection and fusion, the core points of similar data points are closer together and more easily form density connections. Therefore, the clustering accuracy is significantly improved. This indicates that the CCA feature fusion mechanism is suitable for density-based clustering algorithms like DBSCAN, and its combined use is beneficial for blind clustering of radar radiation sources, such as... Figures 10a-10b As shown.

[0167] Taking the temporal characteristics of the individual IQ5 of a radiation source as an example, a schematic diagram is drawn based on the core points, density reachability, and density accessibility defined in the DBSCAN algorithm described above. Before fusion, although data points within a class can form multiple core points, it is impossible to determine all data points within the same class as a cluster through density connections, thus failing to achieve clustering. Figures 11a-11b As shown.

[0168] After feature fusion, even with a decrease in Eps, enough data points can still be found within the neighborhood to be identified as core points. Simultaneously, data within a class more easily form density-directed and density-reachable relationships, ultimately resulting in the majority of data points within a class being density-connected. This density-connected relationship among data points within a class, leading to clustering, is the core of the DBSCAN algorithm. CCA further densifies the data points, perfectly complementing the DBSCAN clustering process.

[0169] Comparative experiment

[0170] 1) Experimental proof of feature selection based on correlation coefficient matrix

[0171] 1-1) Intra-temporal combination

[0172] The rationale for selecting time-domain features based on correlation coefficients has been theoretically explained. Next, six sets of clustering results are compared by combining the four-dimensional time-domain features pairwise. From the distribution of time-domain features and the clustering results, the dispersion of intra-cluster time-domain features for all radiation source individuals is very high, leading to many noise points and overlapping clusters. According to the definition of the F-score, among the combinations of time-domain features, the combination of fractal dimension and information dimension has the highest clustering accuracy, reaching 74.35%, corresponding to a minimum correlation coefficient of 0.0641; while the combination of LZC complexity and information dimension has the lowest clustering accuracy, only 35.62%, corresponding to a maximum correlation coefficient of 0.5679. Figures 12a-12fAs shown, the ascending order of the correlation coefficients of the time-domain feature combinations corresponds one-to-one with the descending order of the clustering accuracy.

[0173] Table 5 Results of Temporal Feature Combination

[0174]

[0175]

[0176] 1-2) Combination within the time-frequency domain

[0177] Similarly, considering the time-frequency domain features, three sets of clustering performance comparison charts were obtained by pairwise combining the three-dimensional features in the time-frequency domain. According to the definition of the F-score, among the combinations of time-frequency domain features, the combination of difference box dimension and multifractal dimension achieved the highest clustering accuracy, reaching 48.11%, with a corresponding minimum correlation coefficient of 0.5872. Conversely, the combination of multifractal dimension and sparsity achieved the lowest clustering accuracy, at 41.06%, with a corresponding maximum correlation coefficient of 0.8977. Figures 13a-13c As shown, the results are consistent with those in the time domain. The ascending order of the correlation coefficients of the time-frequency domain feature combinations corresponds one-to-one with the descending order of the clustering accuracy. It can also be noted that, compared to the time domain, the features extracted in the time-frequency domain have higher correlations, and there are cases where the features of two different radiation sources highly overlap in the time-frequency domain. Therefore, the accuracy of combined clustering is generally lower than that of the time domain results. This indicates that overlapping radiation source individuals are difficult to distinguish using their time-frequency domain "fingerprint features." However, the advantages of the time-frequency domain are also obvious: for any radiation source individual, time-frequency domain features are generally more concentrated and less dispersed than time domain features. Combining the correlation coefficients of the time-domain and time-frequency domain feature combinations with the clustering accuracy, the experiment experimentally proves the correctness of selecting features based on the correlation coefficient matrix. That is, features with low correlation should be selected for combination as much as possible to improve the clustering accuracy.

[0178] 2) Experimental demonstration of feature fusion based on CCA

[0179] The preceding sections have theoretically and experimentally proven the correctness of feature selection based on the correlation coefficient matrix. Reliable feature selection requires reliable fusion. The following comparison examines the feature variance, intra-domain separation, and clustering accuracy of three features: the combination of best-performing clustering in the time domain (fractal dimension and information dimension), the combination of best-performing clustering in the time-frequency domain (difference box dimension and multifractal dimension), and the fused features obtained by fusing the former two based on CCA. This demonstrates the correctness of feature fusion based on CCA. In the above results of combined clustering in the time and time-frequency domains:

[0180] 1) In the time domain features, the features of different radiation sources are generally highly discrete, and the variance of features within a class is large.

[0181] 2) In the frequency domain, the characteristics of different radiation sources generally have low dispersion, and the within-class variance is relatively small compared to the time domain.

[0182] 3) In the time domain, the aliasing of individual features of different radiation sources is more severe than in the time-frequency domain, but the relative distance between classes is greater than that in the time-frequency domain. For example, iq1, iq3 and iq2, iq5 have severe aliasing in the time-frequency domain, but there are obvious differences in inter-class distance in the time domain.

[0183] Characteristics exhibited by CCA fusion features:

[0184] 1) The CCA algorithm can play a compromise role in the inter-class distance, using the separability in the time domain to change the non-separability in the time-frequency domain.

[0185] 2) The CCA algorithm significantly reduces the variance of the fused features, making the intra-class features more concentrated, and the small variance in the time-frequency domain can improve this concentration.

[0186] like Figures 14a-14c As shown, considering the DBSCAN algorithm, the CCA algorithm actually reduces Eps while increasing Minpts, thus increasing the possibility of density-connected relationships between individual data points from the same type of radiation source. Therefore, the characteristic properties obtained after CCA fusion are precisely what DBSCAN requires.

Claims

1. A method for individual radar radiation source identification based on CCA multi-domain feature fusion, characterized in that... Includes the following steps: S1: Taking the rising edge of the radar radiation source signal envelope as the object, the alignment and filtering of multi-domain features considering the differences in spatiotemporal structure are performed to align all signals to the reference. S2: Multi-domain feature fusion is performed using the CCA algorithm; S3: Adaptive DBSCAN clustering based on multi-domain feature fusion is used to identify individual radar radiation sources; Step S1 specifically includes the following methods: 1) Signal preprocessing; 2) Envelope signal alignment and extraction considering temporal structure differences; 3) Temporal feature extraction based on the rising edge of the envelope; 4) Extracting time-frequency domain features based on Hilbert spectrum; 5) Feature selection considering differences in multi-domain feature space structure; The feature selection method that considers the differences in the multi-domain feature space structure includes the following steps: The feature set is represented as Suppose there are only two classes of data and the variances of the same feature in the two classes are equal and close, so that the difference between the means is the only factor to measure the effectiveness of a feature in distinguishing between the two classes. Can be regarded as , The mean of the two types of data features; after feature set normalization, and correlation coefficient : （9） consider That is, features It has no effect on distinguishing data at this time. satisfy ,set up If selected As a characteristic, only It carries class information, and the mean distance between classes. Even assuming that the variances of the two classes of data with the same feature are equal, since the probability density function of the data is unknown, the only way to improve the inter-class discrimination is to maximize the inter-class mean distance. （10） exist Right now When the correlation coefficient is 0, the inter-class mean distance reaches its maximum value of 2; this result is obvious in a two-dimensional plane because... Under the condition of no inter-class differences, only when the correlation coefficient is very small or even 0, Only then can inter-class differences be contained; In the case of 3 dimensions and higher dimensions, and When there is still no information about class differences, using the sum of squared distances between means as the objective function is inappropriate. Therefore, a matrix should be defined. At this point, it should be maximized. The minimum distance between classes, i.e.: （12） The optimization problem defined by equation (12) is non-convex. Based on the cyclic symmetry of the variables, it can also be known that when taking the maximum / minimum value… Not unique; numerical method results show that in The time does not take the maximum or minimum value, but the maximum and minimum values ​​are almost always satisfied. ; Under unsupervised conditions, if When there is no class difference information in clustering, a single cluster should be selected. The feature vectors should be as orthogonal as possible. Although there are countless such vectors, they at least provide a direction for feature selection.

2. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 1, characterized in that... The signal preprocessing includes the following steps: The wavelet threshold denoising algorithm is used to denoise the received radar radiation source signal.

3. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 1, characterized in that... The method for aligning and extracting envelope signals that considers temporal structure differences includes the following steps: 1) Normalization processing Define two pulse signals and The signal lengths are respectively and ,make ; Calculate the correlation function between two pulse signals , and , is the time offset: （1） Based on the relationship between the obtained relevant function values ​​and the time offset, find the way to make The largest Value, this Value represents signal Compared to Time delay; The alignment process is as follows: The input pulse signal is After denoising and normalizing, a new pulse signal is obtained. ; Calculate all The signal-to-noise ratios (SNRs) of each pulse are ranked and sorted, with the pulse having the highest SNR being selected. as a reference pulse ; For each pulse signal that needs alignment , Calculate its relationship with the reference pulse signal. Correlation function between Find the time delay between the two signals. ; Based on the obtained time offset, the pulse signals that need to be aligned are shifted in time, so that... This aligns it with the reference pulse signal; 2) Extract the rising edge of the envelope The steps for extracting the rising edge of the envelope are as follows: Input pulse signal After normalization, a new pulse signal is obtained. ; The envelope signal is obtained by performing a Hilbert transform on the normalized pulse signal. After normalization, a new envelope signal is obtained. ; Finding envelope signals The first maximum point greater than 0.5 in the middle. The corresponding x-coordinate ; Searching for envelope signals Neutral The most recent local minimum value less than 0.1 corresponding x-coordinate ; Output envelope rising edge signal .

4. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 1, characterized in that... The method for extracting temporal features based on the rising edge of the envelope includes the following steps: In the time domain, based on the rising edge of the envelope, the calculated features include fractal dimension, LZC complexity, pulse envelope rising edge length, rising slope, information dimension, fuzzy entropy value, and origin moments and central moments of each order, among which: 1) Fractal dimension: Box dimension is used as the main means to describe the complexity and irregularity of fractal signals, and the signal sequence is... Placed within a unit square, the minimum interval of the x-coordinates is Let N be the length of the signal sequence, and let: （2） Box dimension: （3） 2) Information dimension: for signal sequences To refactor, let: （4） （5） The information dimension is then: （6） 3) LZC complexity: Starting from the beginning of the signal sequence, scan the signal sequentially, each time finding the shortest subsequence that has not appeared in the previous part, recording its length, adding this shortest subsequence to the known pattern set, and continuing to scan the remaining signal, repeating the above process until the signal ends.

5. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 1, characterized in that... The method for extracting time-frequency domain features based on Hilbert spectrum includes the following steps: 1) Difference box dimension: The time-frequency domain range is uniform. Blocking, ; make Each piece is high ,constitute The box; in the indivual In the grid, the maximum and minimum values ​​in the grid are respectively at the th order. and the Within each box, the current coverage required for this network is... The number of boxes is The number of boxes required to cover the entire surface is Seek all Possible values Linear fitting using the least squares method The resulting slope is the difference box dimension. , Determined by the limit of equation (7): （7） 2) Introducing the multifractal dimension to describe the mass and density distribution characteristics on irregular fractal spaces, the multifractal dimension is given by the limit of equation (8) using the results from the calculation of the difference box dimension: （8） In equation (8), For parameters, the actual selection should be... make It is trending towards stability.

6. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 1, characterized in that... The method for multi-domain feature fusion using the CCA algorithm includes the following steps: Fusion of CCA multi-domain features; CCA Multi-Domain Fusion Feature Statistics Analysis.

7. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 6, characterized in that... The fusion of CCA multi-domain features includes the following steps: The features extracted in the time domain and time-frequency domain are centered and standardized to form a feature matrix. and ; The total number of samples, The feature dimension of the sample; , express and The covariance matrix, The cross-covariance matrix is ​​estimated from the sample. CCA projects the original eigenvectors to obtain canonical variables to extract the linear correlation between different feature matrices. The degree of linear correlation is as follows: （12） in, , , , For the typical vector to be found, As a typical variable, restrictions ; At this point, maximizing the correlation is equivalent to maximizing the cosine of the angle between the canonical variables: （13） In formula (13) For vector dot product operation, Let Kronecker notation be used; using the Lagrange multiplier method and singular value decomposition, let... ,in ;No. For typical vectors , The characteristics after classical vector projection transformation in the time and time-frequency domains are: and ; for The correlation coefficient, and the fused features are The meaning of equation (13) is expressed as follows: The columns within the matrix are linearly independent, while the correlation between the corresponding columns reaches its maximum and decreases sequentially. Typical variables are a dimensionality-reduced representation of the original feature set, while also concisely reflecting the correlation between the variable sets.

8. The radar radiation source individual identification method based on CCA multi-domain feature fusion as described in claim 6, characterized in that... The method for CCA multi-domain fusion feature statistics analysis specifically includes the following steps: yes and A linear combination, therefore ; Calculate the covariance matrix of the feature fusion matrix ;Will and Substituting the terms, we get the sum of the four terms: （14） In equation (14), It is An identity matrix of order 1 yes SVD decomposition yielded A singular value matrix of order 1; The results of the diagonal matrix also illustrate The columns are uncorrelated; the first column in the feature matrix... The column is the first The variance of the fusion of canonical variables is estimated using samples. ; This indicates that compared to before fusion, The degree of dispersion of any feature in the middle is The reason for the decrease in the factor is essentially... The scaling effect; , They are all orthogonal matrices. In the results of singular value decomposition , They are also orthogonal matrices, and all correspond to rotation transformations; and A diagonal matrix corresponds to a scaling transformation; the scale of the scaling transformation depends on... , The magnitude of the eigenvalues; considering the scale of scaling: （15） Limit the correlation coefficient when selecting features. That is, given that the features are weakly correlated, it is acceptable to make approximations. Define Rayleigh commerce : （16） In equation (16) Let be any non-zero vector and ; The eigenvalues ​​satisfy Furthermore, its value can be estimated through computer simulation. For each eigenvalue, the following relationship holds: （17） Expressions combining typical vectors , As shown in equation (17), under the condition of feature fusion with a large number of samples, the transformation determined by CCA makes... The degree of dispersion is relatively Shrink back to the original size After fusion The degree of dispersion is also affected by the correlation coefficients between typical variables; Canonical variables determined by CCA , They are , A linear combination of column vectors will not change the mean. = = = = , = = , , is a matrix selected for a row. It represents the number of samples for a particular individual. That is, the time domain, time-frequency domain, and fused feature matrix of the corresponding individual radiation source; feature matrix Each is considered as in , Coordinates in 3D space, row selection matrix , Different individual samples of unknown radar radiation sources are selected from the overall characteristics in the time domain and time-frequency domain. , Unknown radiation source individuals In the time and time-frequency domains, distance can be measured as... Ignoring the scaling effects and considering the linearity of the mean after a linear combination of random variables, the individual radiation source after feature fusion... The distance between the sample means satisfies: （18） Equation (18) shows that the relative distance between classes after fusion is between the relative distance between single domains before fusion. By using this property to fuse complementary features of multiple domains, samples that are clustered and mixed due to the small distance between classes in a certain domain may be successfully clustered after fusion.

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