## 5 Simulation Results
### Simulation Analysis of Fractal Features
Calculating a feature parameter \(C\) for the \(i\)th modulation method, the zero-center normalized statistical variance \(Var(C_{i})\) of the corresponding parameter values at different SNR is shown as follows:  
\[Var(C_{i})=\frac{1}{K}\sum\limits_{k=1}^{K}{[C_{i}(k)-E(C_{i}(k))]^{2}}. \tag{16}\]  
\(K\) is the number of samples for each feature. \(C_{i}(k)\) is the zero-center normalized statistical value of parameter value \(C\), which corresponds to the feature value \(c_{i}(k)\) for the \(i\)th modulation under different SNRs. \(C_{i}(k)\) can be calculated by the following equation:  
\[C_{i}(k)=\frac{c_{i}(k)}{\sum\limits_{k=1}^{K}{c_{i}(k)}}-1,k=0,1,2,\ldots K \tag{17}\]  
Then, the sum of the zero-center normalized statistical variance \(Var(C_{i})\) is calculated with \(n\) different modulation modes as \(SVar(C)\):  
\[SVar(C)=\sum\limits_{i=1}^{n}{Var(C_{i})} \tag{18}\]  
Table 1 shows the anti-noise evaluation results of the five different fractal features. The \(SVar(C)\) of the five different fractal features are all within \(\times 10^{-3}\), which reflects the superior noise immunity of fractal features. Therefore, fractal features have an advantage in overcoming the problem of poor robustness in traditional automatic modulation recognition. Furthermore, in order to show the data distribution of the five different features more clearly, the box diagram distribution of these features is given at SNR = 0 dB. A box diagram is a statistical method to reflect the discrete data distribution. The box diagram consists of five data points: minimum, lower quartile, median, upper quartile, and maximum. The lower quartile, median, and upper quartile make up a \"box with compartments\". An extension line is established between the quartiles and the maximum value. There is always a wide variety of \"dirty data\" in real data, which are also called \"outliers\". Outliers are picked up individually. Therefore, the outliers of a dataset can be recognized intuitively, and the data dispersion and bias can be judged. The box diagram distribution is shown in Figure 4.  
According to Figure 4, the data distribution of the Sevcik dimension had the biggest difference, which means it had the best ability to distinguish the eight different kinds of communication modulation signals. The data distribution of the box dimension and Katz dimension were also ideal, where the intra-class distribution of the Katz dimension was the most concentrated and the aggregation degree was the best. The data ranges of Higuchi dimension and Petrosian dimension were small, and the data distribution between different signals was similar. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Box** & **Katz** & **Higuchi** & **Petrosian** & **Sevcik** \\ \hline SVar/\(\times 10^{-3}\) & 2.5 & 1.88 & 9 & 0.03 & 9.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Robustness evaluation (SVar, zero-center normalized statistical variance). Computational complexity is also very important for evaluating features. Therefore, the running times of the five fractal features were calculated with a DELL Inspiron 14 laptop. The CPU was an Intel(R) Core(TM) i7-4510U, and the memory was 8 GB. The results are shown in Table 2.  
The Table 2 indicates that the running time of the Sevcik dimension was the shortest, meaning that the Sevcik dimension had the lowest computational complexity. The running time of the Higuchi dimension was more than 30 times longer than the others. Therefore, the computational complexity of the Higuchi dimension was much higher than that of the others. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Box** & **Katz** & **Higuchi** & **Petrosian** & **Sevcik** \\ \hline Runtime/s & 1.92 & 1.66 & 82.199 & 2.83 & 1.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristic complexity evaluation. Figure 4: Characteristic box diagram evaluation of five different fractal dimensions.