Research on multifractal dimension and improved gray relation theory for intelligent satellite signal recognition

In the wake of the development and advancement of signal processing technology for communication radiation source individual, Signal ﬁngerprint feature extraction and analysis technology for communication radiation source individual has broad application prospects in many ﬁelds. To effectively extract the individual characteristics of different modulated signals under low SNR environment, and recognize the subtle features for communication radiation source individuals has been a hot spot. Aiming at the problem of signal feature extraction and classiﬁer design under low SNR environment, in the paper, a multifractal dimension and improved gray relation theory based classiﬁer design algorithm is proposed. Firstly, the multifractal dimension feature extraction of nine modulated communication signals is realized. Then multifractal dimension features of these modulated signals under different SNR are compared. An improved gray relation algorithm is used to recognize the extracted subtle characteristics. Meanwhile, FSK signal is used to simulate radio subtle features by adding different distribution of noise. Subtle feature extraction by means of multifractal dimension algorithm and pattern recognition by means of improved gray relation algorithm are used to test the effectiveness of the proposed method for identiﬁcation of modulated signals and radio subtle features. The simulation results show that the recognition success rate of nine different communication modulated signals can reach 93% even under the SNR of 2 dB, and the recognition success rate of the subtle features of distributed noise can reach 100%. The proposed method provides an effective theoretical basis for identifying of radio modulated signals and communication radiation source individual subtle features.


Introduction
Under the background of the increasingly widespread application of communication technology disciplines and the rapid development of signal processing disciplines, the identification of communication radiation source individuals has gradually attracted the researchers attention [1,2]. With the continuous improvement of anti-reconnaissance and anti-jamming technologies, traditional intelligence information methods are difficult to achieve the information acquisition. With the advance in technologies such as computer technology, fusion algorithms, and electronic information science, signal feature analysis [3] and signal information extraction techniques [4] also develop rapidly, thus achieving the full use of information that was not available in the past. The contained target information from the signal characteristics and subtle features of communication radiation source individuals has good mining potential [5,6]. How to use the subtle features of the radiation source individuals to achieve reconnaissance and surveillance of targets on the battlefield in today's complex electromagnetic environment is a new and valuable research field [7,8]. This technology is deeply welcomed by various combat troops and has great application value. The fractal theory [9][10][11][12] is often referred to as the geometry that exists in nature and complements the chaos theory of the dynamic system. In 1975, the American French mathematician, B.B. Mandelbrot, first proposed the geometric concept of fractals. With the continuous development of fractal theory, he pointed out that parts of anything in the world may be in a certain process, or under certain conditions, certain aspects of things (such as energy, function, information, time, structure, morphology, etc.) show similarities with the whole. And the change in spatial dimension can be continuous or discrete, which expands people's horizons.
Fractal dimension theory has been widely used in fault diagnosis [10,13], image information processing [14,15] and other fields, which can effectively describe the waveform characteristics of signals. For instance, the fractal-box dimension algorithm uses the boxes with different side lengths to depict the signal waveform change [16,17]. Literature [18] uses the box dimension algorithm to depict the signal features, where the smaller the box side length, the longer the computational time, the higher the recognition success rate of signals. The multifractal dimension algorithm can describe the complexity and irregularity of the signal compared with fractal-box dimension algorithm [19]. It can calculate different dimension characteristics of signals, describe signal features from different dimensions, and reflect the distribution density of signal in the area space, thereby achieving recognition of the signal. Ding Kai et al. used the multifractal dimension algorithm to identify different signals, and the high recognition efficiency was achieved in the lower SNR environment, but the proposed algorithm has high computational complexity and costs long simulation time [20]. Yang et al. applied the fractal theory to the analysis and processing of noise, which has a wide range of application effects. It mainly discusses the root cause, model and analysis method of 1/f noise, and the application of fractal estimation in signal processing is well summarized [21]. Wu et al. studied the fractal growth model and summarized its research in materials science [22]. The literature [23][24][25] summarizes the application of fractals in heat transfer and mass transfer of media, the application in geomorphology, and the application in manufacturing systems.
It can be seen from the literatures in recent years that fractal dimension has achieved good development in various fields. Therefore, the research on fractal signal recognition [26][27][28] has also become the focus of this paper. The main contributions of this paper are as follows: first, extract the multifractal dimension features of nine modulated signals under different SNR environment, then the features of FSK signals with different distributed noise are extracted, at last, an improved adaptive interval gray relation algorithm [29,30] is used to classify the extracted characteristics, and finally the purpose of accurately identifying signals is achieved.

Basic definition of multifractal dimension
Multifractal dimension algorithm can depict the signals features from different levels and describe the signal waveform probability distribution characteristics. Multifractal dimension algorithm splits the research object into N small regions. Set the linearity of ith region as e i , and the density distribution function P i of ith region can be described by different scale indices a i as following: The non-integer a i is generally called a singularity index, and its value corresponds to its region.
To obtain the distributed characteristics of a succession of subsets, define the function X q ðeÞ, which is the probability-weighted summation of the regions: Further the generalized fractal dimension D q can be obtained as follows: X q ðeÞ shows the role of various sizes P i , and it can be seen from the Eq. (3) that, the area with high probability plays a major role of the sum P P q i when q ) 1 , and at this time, X q ðeÞ and D q reflect the nature of the high probability region (i.e., dense region)When q ! 1, the small probability can be ignored, and only the large probability P i is considered, so that the calculation of D q is simplified. On the contrary, X q ðeÞ and D q reflect the nature of a small probability region (i.e., sparse region) when q ( 1. Thus, the properties of different probability characteristic regions are represented by different q values. After the weighted summation process, a signal is divided into a number of regions, which have different degrees of singularity. Therefore, it is possible to understand the fine structure inside the signal hierarchically. Define D q as capacity dimension (i.e., box dimension) D 0 , information dimension D 1 , and correlation dimension D 2 respectively when q ¼ 0; 1; 2.
3 Design method of classifier based on gray relation

Ordinary gray relation algorithm
The basic idea of gray relation theory is to quantitatively describe and compare the changes and development trends of a system. Suppose the behavior sequence of the system is: where X 0 represents the reference sequence and X 1 ; X 2 ; . . .; X m represents the comparative sequence. Set: Here, define q 2 ð0; 1Þ as the resolution coefficient, which is usually set as 0.5. And cðX 0 ; X i Þ is called the gray relation degree between X 0 and X i , often abbreviated as c 0i . The k-point relation coefficient c x 0 ðkÞ; x i ðkÞ ð Þis abbreviated as c 0i ðkÞ.
The specific calculation process of the gray relation degree is listed as follows: Step 1: Calculate the initial phase value (or mean value phase) for each sequence, that is: Step 2: Calculate the difference sequences, that is: Step 3: Calculate the maximum difference and the minimum difference, namely: Step 4: Calculate the relation coefficient value as follows: . . .; n; Here, q is named as resolution coefficient, which values between 0 and 1.
Step 5: Finally, calculate the relation value between sequences, namely: Then c 0i represents the relation degree between the sequences, that is, the similarity degree between the sequences, n is the number of features.

Interval gray relation algorithm
The interval gray relation algorithm uses the feature interval of the signals to classify the extracted features, and has better recognition effect for overlapping features. The calculation process is depicted as follows: Define the feature interval matrix first as: where m represents the signal type; n represents the number of characteristic parameters, and s min mn represents the minimum value of the fluctuation range of the nth eigenvalue of the modulated signal m; s max mn represents the maximum value of the fluctuation range of the nth eigenvalue of the modulation signal m.
Suppose the feature interval of the nth feature of the signal to be identified is ½S min 0n S max 0n , then the interval dissociation degree of the known signal feature interval is defined as: Then the interval gray relation coefficient can be obtained as follows: where q ¼ 0:5 By calculating the value of n mn in order, the interval relation coefficient matrix can be constructed: n m1 n m2 Á Á Á n mn where i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n.

Wireless Networks
Therefore the gray relation degree can be calculated as follows: The value of i corresponding to the largest n i value is the category to which the signal to be identified belongs.

Improved adaptive mean gray relation algorithm
Suppose there is a certain feature parameter x from feature parameter set X of a certain signal and the value of its possible eigenvalue is fx i g ¼ fx 1 ; x 2 ; . . .; x n g. And for any i, j, x i 6 ¼ x j , take the mean value of the n number of eigenvalues as the standard reference value: Theoretical analysis shows that the mean value of eigenvalues is selected as the reference sequence, which has better stability compared with the randomly selected reference samples. When the mean value samples are close, that is, the parameter values of different signals overlap too much, the recognition success rate will drop rapidly, so the inter-class distances between samples are required by the improved algorithm to some extent. For the improved adaptive mean value sample relation algorithm, considering with the concept of residual in information theory, the feature distance Dx i ðkÞ is processed as follows: where m is the type of signal and the expression of the entropy is: Then the maximum entropy expression is: Then the relative entropy value is: In view of the concept of residual in information theory, the residual of the kth eigenvalue is: The substantial meaning of the residual is the removal of the difference between the kth characteristic entropy value and the optimal feature entropy value. The larger D k is, the more important the feature is, and the greater the weight should be given. Finally, the weight a ik of the kth eigenvalue is calculated as: where P n k¼1 a ik ¼ 1; a ik ! 0 Multiply the weight coefficient by the corresponding relation coefficient to obtain the relation degree value, namely: From the statistical perspective, the more the deviation, the more the characteristics can reflect the differences between the categories. Therefore, it can be considered that the greater the degree of difference in the features, the more important the feature is. Therefore, Compared with other classifier algorithm [3,31,32], the improved gray relation algorithm has a certain adaptive effect and easy to calculate the recognition results.

Improved adaptive interval gray relation algorithm
The improved interval gray relation value algorithm first needs to calculate the relation coefficient matrix ½n ij ½n ij ¼ The algorithm of the weight value a ij is the same as the calculation method in Sect. 3.3. Finally, the weight value assigned to the common interval relation algorithm correspondingly is as follows: The obtained interval gray relation degree n i at this time has certain adaptive ability, which has better recognition effect compared with ordinary interval gray relation algorithm. According to the above discussion, the improved adaptive interval gray relation algorithm was used as a classifier to achieve accurate identification.

Basic steps of algorithm implementation
In this section, a new modulation signal feature extraction algorithm based on complexity features is proposed. And multifractal dimension theory is applied to the modulation signal feature extraction. The specific steps of algorithm implementation are as follows: Step 1: the received unknown communication signal is pre-processed, that is, discretized it as: suppose the received radiation source individual signal is S , and the preprocessed discrete signal sequence be fsðiÞg, where i ¼ 1; 2; . . .; N 0 is the number of signal sampling points and N 0 is the signal sequence length.
Step 2: reorganize the discretized signal sequence: First, for the preprocessed discrete communication signal sequence fsðiÞg; i ¼ 1; 2; . . .; N 0 , define the characteristic parameters as follows: Define: n ¼ log N 0 2 , which indicate the times of the number of different vectors of the signal that to be reorganized.
Define: tðjÞ ¼ 2 j , which indicates the number of discrete signal points in each recombination signal, where j ¼ 1; 2; . . .; n represent the values of the number of different vectors of the recombination signal.
Step 3: the multifractal dimension operations are performed on the reconstructed eigenvectors, and different dimensions are selected to extract the signals multifractal dimension features: The multifractal dimension feature describes the different level characteristics of things. A multifractal dimension can be regarded as a union of fractal subsets of different dimensions. The research object is divided into M small regions, and set the linearity of the ith region as e i and the density distribution function of the ith region as P i , and then the scale index a 1 of the ith region can be described as following.
The non-integer a i is called the singular exponent, which represents the fractal dimension of a certain region. Since a signal can be divided into many different small regions, a variable f ðaÞ composed of a series of different a i can be obtained, and f ðaÞ becomes a signal multifractal spectrum. Define the function X q ðeÞ as the weighted summation of the probabilities for each region, and e is the magnitude of the linearity, and q is power for density distribution function P i , i.e.: Define generalized fractal dimension D q as follows: Thus each of the recombination signals S(j) in step 2 is summarized, and S(j) represents the jth recombination signal, namely: where J 1 ; 2; . . .J 0 ; j ¼ 1; 2; . . .n, S J is the sum of Jth recombination signal, and J 0 is the number of recombination signals. Then sum the entire discrete signal sequence, and the sum is S , ie: where i ¼ 0; 1; 2; . . .N 0 S i is ith sample point value for the discrete signal sequence, and then the Jth probability measure P j is defined as: where J ¼ 1; 2; . . .J 0 The multifractal dimension feature of the signal can be obtained by taking P J into the calculation formula of multifractal dimension D q .
Finally, the adaptive interval gray relation algorithm is used to recognize the obtained features, and then the recognition rate is calculated to test the effectiveness of the proposed algorithm. Take nine commonly used communication modulation  signals, such as 16QAM, 2ASK, 2FSK, 4FSK, QPSK,  8FSK, 4ASK, BPSK, 32QAM, calculate the multifractal dimension features under three different SNR environments of -10 dB, 0 dB and 10 dB. And the simulation results are shown from Figs. 1, 2 and 3, where the abscissa represents Ine and the ordinate represents Inxq respectively, corresponding to two parameters in the definition of multifractal dimension, and each point on the curve represents a set of multifractal feature values for different signals.

Simulation results and analysis
It can be seen from the simulation diagram that the multifractal dimension of different modulation signals is obviously different. The signal can be distinguished by the fractal dimension features at different dimensions even at a lower SNR. The gray relation classifier algorithm and its proposed improved algorithm are used to classify and identify the extracted features by using the multifractal features of the signal as feature database. At the same time, the recognition results of the traditional neural network classifier are used to compare and contrast. And the recognition rate under different SNRs is seen in Table 1.
It can be seen from the recognition results in Table 1 that under the SNR of 10 dB and 20 dB, the four classifiers all have good recognition performance. According to the analysis of Fig. 1, these classifiers can obtain a recognition success rate of more than 95% when these characteristic parameters have better separation characteristics. And the recognition success rates of the common gray relation algorithm, the improved mean sample relation algorithm, and the improved adaptive mean sample relation algorithm all will decrease when the SNR decreases. Only the neural network classifier and the gray relation classifier with improved adaptive entropy weight can obtain more than 90% recognition rate when the SNR is 2 dB. when the SNR is 0 dB, the performance of classifiers obviously decreases. It can be seen from the analysis of Fig. 2 that the neural network based classifier and the entropy weight based interval gray relation algorithm have the best recognition effect when the signal characteristic parameters overlap partially.
In order to measure the complex characteristics for different algorithms, the computational time for different classifiers is shown in Table 2.
It can be seen from the simulation results in Table 2 that the computational time of the gray relation algorithm based classifiers is basically no difference, i.e., the complexity of these classifiers is equivalent. Compared with the improved   adaptive interval gray relation algorithm, the neural network classifier has a better classification effect. However, due to the long time required to train test samples, the improved gray correlation algorithm has better application value than neural network classifier in real-time performance.
Similarly, the FSK signal transmitted by the communication station is taken as an example, and five different distributed noise sequences are generated and added on the FSK signal. Each distribution randomly generates 1000 samples. The Monte Carlo simulation experiment is carried out to explore the recognition effect for different distributed noise. The multifractal characteristic diagram of the FSK signal with different distributed noise sequences and the simulation results are shown in Fig. 4 (where the ordinate A represents the amplitude value; the abscissa represents the time on the left side; the abscissa represents Ine and the ordinate represents InXq on the right side).
Different distributed noises represent different subtle features carried by radios transmitting of the same signal to test the recognition effect of the proposed method on the subtle ''fingerprint'' features. It can be seen from the simulation results that the multifractal dimension characteristic curves of FSK signal with different distributed noise sequences are slightly different. Using the improved gray relation theory, the relation degree of the fractal results obtained in each reconstruction space are calculated. And the recognition success rate can reach 100%, which verifies the effectiveness of the improved algorithm proposed in the paper.

Conclusion
After years of development, the individual identification of radiation sources has achieved certain results. But it has not yet formed a complete scientific system, which involves many complicated factors, including the increasing number and complexity of radio signals, the increasing electromagnetic environment, the gradually increasing noise interference, and the application of a large number of new system communication devices, resulting in traditional algorithms in many cases can not accurately achieve signal recognition in complex electromagnetic environments.
Aiming at these problems, this paper proposes a multifractal dimension and improved gray relation theory based classifier design algorithm. Firstly, the effectiveness of the proposed algorithm for different types of modulated signals is validated. Then, different radio noises are simulated to validate the effectiveness of subtle feature recognition of signals by the proposed algorithm. Simulation results illustrate that the recognition success rate for nine different communication modulated signals can achieve 93% recognition rate even under the SNR of 2 dB. And the identification of subtle features of distributed noise can reach 100% recognition rate. This provides an effective theoretical basis for the identification of radio modulated signals and subtle features of signals. Although the simulation results show the effectiveness of the proposed algorithm, for increasingly complex electromagnetic environment, the radio subtle features are not stable due to many external factors. How to extract robust radio features for complex electromagnetic environments is the hotspot that scholars still need to continue to study in the future.