Improving the signal subtle feature extraction performance based on dual improved fractal box dimension eigenvectors

Because of the limitations of the traditional fractal box-counting dimension algorithm in subtle feature extraction of radiation source signals, a dual improved generalized fractal box-counting dimension eigenvector algorithm is proposed. First, the radiation source signal was preprocessed, and a Hilbert transform was performed to obtain the instantaneous amplitude of the signal. Then, the improved fractal box-counting dimension of the signal instantaneous amplitude was extracted as the first eigenvector. At the same time, the improved fractal box-counting dimension of the signal without the Hilbert transform was extracted as the second eigenvector. Finally, the dual improved fractal box-counting dimension eigenvectors formed the multi-dimensional eigenvectors as signal subtle features, which were used for radiation source signal recognition by the grey relation algorithm. The experimental results show that, compared with the traditional fractal box-counting dimension algorithm and the single improved fractal box-counting dimension algorithm, the proposed dual improved fractal box-counting dimension algorithm can better extract the signal subtle distribution characteristics under different reconstruction phase space, and has a better recognition effect with good real-time performance.

YY, 0000-0002-3867-5893 Because of the limitations of the traditional fractal boxcounting dimension algorithm in subtle feature extraction of radiation source signals, a dual improved generalized fractal box-counting dimension eigenvector algorithm is proposed. First, the radiation source signal was preprocessed, and a Hilbert transform was performed to obtain the instantaneous amplitude of the signal. Then, the improved fractal boxcounting dimension of the signal instantaneous amplitude was extracted as the first eigenvector. At the same time, the improved fractal box-counting dimension of the signal without the Hilbert transform was extracted as the second eigenvector. Finally, the dual improved fractal box-counting dimension eigenvectors formed the multi-dimensional eigenvectors as signal subtle features, which were used for radiation source signal recognition by the grey relation algorithm. The experimental results show that, compared with the traditional fractal box-counting dimension algorithm and the single improved fractal box-counting dimension algorithm, the proposed dual improved fractal box-counting dimension algorithm can better extract the signal subtle distribution characteristics under different reconstruction phase space, and has a better recognition effect with good real-time performance.

Introduction
Radiation source signal recognition technology is an emerging field in information warfare. Exploring effective signal feature extraction technology under a lower signal-to-noise ratio and achieving identification of different radiation source signals is the technical basis for the development of a new generation of electronic counterintelligence reconnaissance equipment. Owing to the complexity and urgency of identifying individual radiation sources, this has been a key topic in the field of information warfare. Therefore, it is necessary to develop a reliable and effective signal subtle feature extraction algorithm for radiation source signal recognition. Currently, many signal processing techniques have been applied to signal subtle feature extraction. However, the radiation source signal exhibits nonlinear and unsteady characteristics as a result of many nonlinear factors [1]. Slight radiation source signal information is easily submerged in the inevitable background noise and becomes difficult to extract. Therefore, conventional time-domain and frequency-domain methods may not make an accurate assessment of the radiation source signal subtle feature [2]. With the development of nonlinear dynamics, many nonlinear analytical techniques have been applied to identifying and predicting the complex dynamic nonlinearity of the radiation source signal [3]. Among them, the most typical technique is to extract the characteristic frequency from the radiation source signal through the combined usage of some advanced signal processing techniques (higher order spectra [4], wavelet package transform [5], the Hilbert transform [6], empirical mode decomposition, etc.) and further evaluating the radiation source signal by comparing it with the theoretical characteristic frequency value with the aid of expert judgement. With the development of artificial intelligence [7], radiation source signal recognition technology has been gradually introduced into the category of pattern recognition. And the validity and reliability of radiation source signal recognition technology are mainly determined by the selection of the dominant eigenvectors which characterize the signal subtle features. Recently, some entropy-based methods (hierarchical entropy [8], fuzzy entropy [9], sample entropy [10], approximate entropy [11], hierarchical fuzzy entropy [12], etc.) have been proposed for extracting dominant eigenvectors from a radiation source signal and have achieved some effect.
After extracting signal subtle features, a pattern recognition technique is also needed to achieve the radiation source signal recognition. Nowadays, a variety of pattern recognition methods have been used in radiation source signal recognition, of which the most widely used are support vector machines (SVMs) [13] and artificial neural networks (ANNs) [14][15][16]. Among them, ANN training requires a large number of samples, which is difficult or even impossible in practical applications. SVMs are based on statistical learning theory (which is especially suitable for training with a small number of samples), and have better generalization ability than ANNs and can ensure that local optimal solutions and global optimal solutions are consistent [17]. However, the accuracy of SVM classifiers depends on the choice of their optimal parameters [18,19]. To ensure the accuracy of radiation source signal recognition, some optimization algorithms and the design of complex multi-class structures [20] are often integrated into SVMs. Here, in order to solve the issue of generality versus accuracy, the grey relation algorithm (GRA) was developed to achieve accurate pattern recognition based on a small number of samples.
Aimed at solving the above problems, a dual improved generalized fractal box-counting dimension eigenvector algorithm is proposed in this paper. To achieve this, first the radiation source signal was preprocessed, and a Hilbert transform was performed to obtain the instantaneous amplitude of the signal. Then the improved fractal box-counting dimension of the signal instantaneous amplitude was extracted as the first eigenvector. At the same time, the improved fractal box-counting dimension of the signal without the Hilbert transform was extracted as the second eigenvector. Finally, the dual improved fractal box-counting dimension eigenvectors formed the multi-dimensional eigenvectors as signal subtle features, which were used for radiation source signal recognition by the GRA.
The rest of the paper is organized as follows. First, the proposed method is introduced in §2. The application and analysis of the proposed method are illustrated in §3. The conclusion is presented in §4.

Hilbert transform
The Hilbert transform is widely used in signal processing to obtain an analytic representation of the signal, which allows instantaneous calculation of the amplitude, phase and frequency. Given an actual radiation source signal f (t), its Hilbert transform is defined as follows: where P is a Cauchy Lord value. It can be seen from equation (2.1) that h(t) is a time-dependent linear function. The following relationship can be obtained from f (t) by applying convolution with (π t) −1 : Then the Fourier transform can be used to obtain the following equation: Then the following equation can be obtained: Then the time-domain result h(t) can be obtained by performing an inverse Fourier transform. The Hilbert transform h(t) of the radiation source signal f (t) represents its harmonic conjugate. The Hilbert transform can be used to obtain an analytic signal z(t) from an actual radiation source signal f (t), and the analytic signal z(t) can be expressed as follows: (2.6) Therefore, the instantaneous amplitude of the analytic signal z(t) is (2.7)

Traditional fractal box-counting dimension
Fractal theory is one of the most important branches of contemporary nonlinear sciences [21], and is suitable for processing all types of nonlinear and non-stationary phenomena; it may also be suitable for subtle feature extraction from radiation source signals. The fractal box-counting dimension algorithm has the advantage of simple calculation compared with other fractal dimension algorithms. The conventional algorithm of the fractal box-counting dimension has been widely used in the fields of image analysis [22], electromagnetic fault diagnosis and biomedicine [23], in which signals are strictly self-similar. Suppose A is a non-empty bounded subset of Euclidean space R n to be calculated, and N(A, ε) is the least number of boxes with the side length ε covering A. Then the fractal box-counting dimension can be defined as log(1/ε) . (2.8) For the actual radiation source signal sequence f (i), i = 1, 2, . . . , N 0 , there is no meaning for ε → 0 to calculate the box-counting dimension as the sampling interval σ is the highest resolution of the signal because of the existence of the sampling frequency. Therefore, the minimum side length ε of the box is often said to be equal to σ . Consider the actual sampled radiation source signal sequence f (i) as the closed set of Euclidean space R n , and the calculation process of the fractal box-counting dimension is described as follows.
Use the approximate method to make the minimum side length ε of the box covering the radiation source signal sequence f (i) equal to the sampling interval σ . And calculate the least number of boxes N(kε) with side length kε covering the signal sequence f (i), thus and where i = 1, 2, . . . , N 0 /k, k = 1, 2, . . . , K. N 0 is the number of sampling points, K < N 0 . p(kε) is the longitudinal coordinate scale of the actual sampled radiation source signal sequence f (i). Thus N(kε) can be defined as Select a fitting curve log kε ∼ log N(kε) with good linearity as a scale-free zone, and the fitting curve can be defined as log N(kε) = a log kε + b, (2.13) where k 1 ≤ k ≤ k 2 and k 1 and k 2 are the beginning and end of the scale-free zone, respectively. Generally, a least squares method is used to calculate the slope of the fitting curve, which is the fractal box-counting dimension D of the actual sampled radiation source signal sequence f (i), (2.14)

Improved generalized fractal box-counting dimension
However, for the actual radiation source signals, they do not satisfy the self-similar structure of fractal theory to some degree. Therefore, when using the traditional fractal box-counting dimension algorithm to calculate the box-counting dimension of the radiation source signals, the fitting curve often does not have a good linear structure. Because of this problem, an improved generalized fractal box-counting dimension algorithm was developed to overcome the defect of the conventional fractal box-counting dimension algorithm. The specific calculation procedure is as follows.
(1) Resample the actual radiation source signal sequence f (i), i = 1, 2, . . . , N 0 , and properly increase the sampling points to reduce the minimum side length ε to improve the calculation accuracy of the fractal box-counting dimension of the signal sequence f (i). The phase space of the signal sequence f (i) is reconstructed, and the number of iterated dimensions of the reconstructed phase space is determined according to the number of sampling points. (2) Suppose the number of sampling points of the signal sequence f (i) is N 0 = 2 n . To improve the calculation accuracy, resample the actual radiation source signal sequence f (i), and suppose the number of sampling points of the signal sequence The derivate process of the number of boxes covering the actual radiation source signal sequence f (i) can be described as follows.
When k = 1: In this case, the reconstructed phase space dimension is 2. When k = 2: In this case, the reconstructed phase space dimension is 3. When k = 3: . . , N/k. In this case, the reconstructed phase space dimension is 4. When k = K: (4) It can be seen from the above deduction that, during the reconstruction of the phase space of the radiation source signal sequence f (i) K times, the corresponding log N(kε) can be obtained at each time. And then the relation curve of log kε ∼ log N(kε) can be drawn. Since the fitting curve does not have a strict linear relationship, take the derivation of the relation curve at these K points over the improved generalized fractal box-counting dimension algorithm. The slopes D 1 , D 2 , D 3 , . . . , D K at these K points from the relation curve are the fractal box-counting dimensions in the different reconstructed phase space. Take the slopes D 1 , D 2 , D 3 , . . . , D K obtained as the K characteristic parameters for the feature vector extracted from the signal sequence f (i), which characterizes the signal subtle features.

Grey relation theory
The study of grey relation theory is the foundation of grey system theory, which is mainly based on the basic theory of space mathematics, which is used to calculate the relation coefficient and relation degree between the reference characteristic vector and each comparative characteristic vector. Grey relation theory has a good potential to be used in radiation source signal recognition for the following reasons [24]: (i) it has good tolerance to measurement noise; (ii) its algorithm is simple and can solve the issue of generality versus accuracy; (iii) it can solve the learning problem with a small number of samples; and (iv) it has the ability to assist the selection of characteristic parameters for classification. Suppose the feature vectors [D 1 , D 2 , . . . , D K , D 1 , D 2 , . . . , D K ] (i.e. the dual improved generalized fractal box-counting dimension eigenvectors) extracted from the radiation source signal to be identified are as follows: is a certain radiation source signal type to be recognized. Assume that the knowledge base between the radiation source signal type and the signal subtle features based on part of the samples is as follows: where C j (j = 1,2, . . . ) is a known radiation source signal type and c j (j = 1,2, . . . ) is a certain feature parameter.
For ρ ∈ (0, 1), is the grey relation coefficient of the kth feature parameter for B i and C j ; and ξ (B i , C j ) is the grey relation degree for B i and C j . Hereafter, B i is categorized to a certain radiation source signal type where the maximal ξ (B i , C j ) (j = 1,2, . . . ,) is calculated.

Proposed approach
In summary, the process of the proposed method for radiation source signal recognition is as follows, and the flow chart is illustrated in figure 1.
Step 1: A variety of target radiation source signals are sampled, to establish the knowledge base.
Step 2: The subtle feature vectors are extracted from the sample knowledge base through the dual improved generalized fractal box-counting dimension eigenvectors.   Step 3: The sample knowledge base for GRA is established based on the signal symptom (i.e. the extracted subtle feature vectors [D 1 , D 2 , . . . , D K , D 1 , D 2 , . . . , D K ]) and the signal pattern (i.e. the known radiation source signal type).
Step 4: The subtle feature vectors extracted from the radiation source signals to be identified are input into the GRA, and the recognition results are output.

Application and analysis
In order to verify the performance of the proposed dual improved generalized fractal box-counting dimension eigenvector algorithm, 11 different radiation source signals were used to test the recognition effect, compared with the traditional fractal box-counting dimension and the single improved generalized fractal box-counting dimension. In this paper, the radiation source signals for testing are from the Case Western Reserve University Bearing Data Center. The database has been a standard dataset for testing the effectiveness of the feature extraction algorithm and the pattern recognition algorithm. In addition, the sampled signals of the database are full of random mechanical noise, which makes the test closer to the real situation.             signal subtle feature extraction compared with the methods from [23] and [25]. The time cost of the proposed approach using a laptop computer with a 4.0 GHz dual processor for one test case is only 0.027 s and the time consumption of the proposed approach is encouraging, which shows good real-time performance.

Conclusion
Fractal dimension is an important tool to describe the complexity of a fractal body. In this paper, the fractal box-counting dimension algorithm in fractal theory is improved, and a dual improved generalized fractal box-counting dimension eigenvector algorithm is proposed. The proposed algorithm can effectively characterize the fractal box-counting dimensionality of radiation source signals under  Table 6. The recognition results by the proposed method compared with results from [23] and [25]. Note: The approach of [23] is based on multifractal theory for extracting feature vectors and a GRA for achieving pattern recognition intelligently using the extracted feature vectors. The approach of [25] is based on a four-dimensional feature extraction algorithm using entropy and Holder coefficient theories for extracting feature vectors and a GRA for achieving pattern recognition intelligently using the extracted feature vectors. In our previous works, such as [23,25], we have fully compared the recognition results with the existing feature extraction algorithm (such as entropy theory, Holder coefficient theory and multifractal theory) and a pattern recognition algorithm (such as the feed-forward backpropagation neural network, the SVM, and the adaptive GRA) in the same topic. And at this stage, we propose the improved algorithm based on our previous works [23,25] to improve the signal subtle feature extraction performance based on the dual improved fractal box-counting dimension eigenvectors.