## 2 Related Work  
In this section, we provide a description of some works related to fractal feature extraction. Fractal-based pattern recognition has been paid increasing attention, because it can effectively deal with feature information in a complex electromagnetic environment. One of the first studies of fractal theory [15; 16] was conducted by [PERSON] in 1975, as a branch of modern mathematical theory. The fractal dimension is the main parameter in fractal theory. It quantitatively describes the complexity of the fractal set. As a kind of time series, the communication modulation signal can be effectively described by the fractal dimension. There are many definitions of and measurement methods for fractal dimension [17; 18; 19; 20].  
The common one-dimensional fractal dimensions are box dimension, Hausdorff-Besicovitch dimension, Sevcik fractal dimension, Katz dimension, Higuchi dimension, and Petrosian dimension. Among them, the box dimension is easy to calculate, and a box with different side lengths is used to describe the change of the signal waveform. Smaller side lengths of the box lead to a longer calculation time, but the recognition rate of the signal will increase. In References [21; 22], the box dimension was used to characterize the signal characteristics. The Hausdorff-Besicovitch dimension [23; 24] was the basic fractal dimension. Hausdorff-Besicovitch and box dimensions were considered to minimize the antenna size, holding a high radiation efficiency in Reference [25]. This confirmed the importance of Hausdorff-Besicovitch Dimension, but also indicated that it has a high computational complexity, making it difficult to realize in practice. The Sevcik fractal dimension was proposed to calculate the fractal dimension of waveforms in Reference [26]. This method could be used to quickly estimate the Hausdorff-Besicovitch dimension of waveforms and measure the complexity and randomness of waveforms. [PERSON] proposed a fast method to estimate the fractal dimension of a finite sequence [27], which converts the data to binary sequence before estimating the fractal dimension from time series. [PERSON] compared the estimation accuracy delivered by [PERSON]'s and [PERSON]'s methods on four synthetic fractal waveforms. The results indicated that [PERSON]'s method invariably under-estimated the true dimension, but [PERSON]'s method was more accurate in estimating the fractal dimensions.  
In addition, in Reference [28], the definition and mathematical analysis of the Weierstrass-Mandelbrot fractal function were described. The application of fractal functions in engineering was discussed in Reference [29], including fractal modeling and calculation. In Reference [30], the effect of multiple fractals on a digital communication signal was studied under different noise distributions. The simulation results showed that multiple fractals could effectively extract the feature of FSK (Frequency Shift Keying) signals under different noise distributions. Additionally, the use of multiple fractals was more effective in depicting subtle features [31; 32; 33]. However, the problem of large complexity in multifractal calculation could not be avoided [34; 35].  
Therefore, how to select the proper fractal dimension algorithm according to the signal complexity, and then accurately classify the different communication modulation signals are the key problems in this paper. Therefore, we conducted a systematic empirical study on how to choose fractal dimension features for classifying communication modulation signals.