# Modulation Recognition System of Electromagnetic Interference Signal Based on SDR
## 3 High Order Cumulant Modulation Recognition Model
The theoretical values of the second, fourth, and sixth order cumulants of each modulation signal calculated are shown in \[\Phi^{k}(\omega)=\frac{d^{k}\Phi(\omega)}{d\omega^{k}}=j^{k}E\Big{\{}X^{k}\exp (j\omega X)\Big{\}} \tag{12}\]0  
It can be seen from Table 1 that the six modulation-type signals cannot be completely distinguished by high-order cumulants. Therefore, using the ratio of absolute values to construct the characteristic parameters to distinguish the six modulation types of signals in electromagnetic interference can not only eliminate the influence of signal amplitude change on the characteristic parameters, but also eliminate the influence of phase jitter \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Modulation Type** & **1C20** & **1C21** & **1C40** & **1C41** & **1C42** & **1C60** & **1C63** \\ \hline BPSK & E & E & 2E\({}^{2}\) & 2E\({}^{2}\) & 2E2 & 16E\({}^{3}\) & 13E\({}^{3}\) \\ QPSK & 0 & E & E\({}^{2}\) & 0 & E\({}^{2}\) & 0 & 4E\({}^{3}\) \\
8 PSK & 0 & E & 0 & 0 & E\({}^{2}\) & 0 & 4E\({}^{3}\) \\
8 QAM & 0 & E & 1.15E\({}^{2}\) & 0 & 0.67E\({}^{2}\) & 3.44E\({}^{3}\) & 2.4E\({}^{3}\) \\
16 QAM & 0 & E & 0.68E\({}^{2}\) & 0 & 0.68E\({}^{2}\) & 0 & 2.08E\({}^{3}\) \\
64 QAM & 0 & E & 0.619E\({}^{2}\) & 0 & 0.619E\({}^{2}\) & 0 & 1.797E\({}^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Theoretical value of cumulants of each order of MPSK and MQAM interference signals. on the characteristic parameters. The four characteristic parameters are: \(f_{1}=|\text{C}_{41}|/|\text{C}_{42}|\), \(f_{2}=|\text{C}_{40}|/|\text{C}_{42}|\), \(f_{3}=|\text{C}_{63}|^{2}/|\text{C}_{42}|^{3}\), \(f_{4}=(|\text{C}_{42}|+|\text{C}_{40}|)/|\text{C}_{21}|^{2}\). The theoretical values of characteristic parameters of each modulation signal calculated are shown in Table 2.  
It can be seen from Table 2 that the \(f_{1}\) characteristic parameter value of the BPSK signal is different from that of the other five modulation signals. Therefore, the modulation signal can be divided into BPSK and QPSK, 8 PSK, 8 QAM, 16 QAM, and 64 QAM by using \(f_{1}\) characteristic parameters, so as to identify the BPSK signal; Using \(f_{2}\) characteristic parameters, the modulation signal can be divided into three groups 8 PSK, QPSK, 16 QAM, 64 QAM, and 8 QAM, so as to identify 8 PSK signal and 8 QAM signal. Using the \(f_{3}\) characteristic parameter, the modulation signal can be divided into QPSK, 16 QAM, and 64 QAM, so as to identify the QPSK signal. Finally, the 16 QAM signal and the 64 QAM signal can be identified by \(f_{4}\) characteristic parameters. According to the above analysis, the characteristic parameters \(f_{1}\), \(f_{2}\), \(f_{3}\) and \(f_{4}\) of BPSK, QPSK, 8 PSK, 8 QAM, 16 QAM, and 64 QAM modulation signals are simulated and analyzed, respectively, so as to set the appropriate decision threshold to construct the decision tree classifier. The parameters of the simulation environment are set as follows: carrier frequency is 1 MHz; sampling frequency is 6 MHz; symbol rate is 1000 b/s; symbol length N is 3000; SNR = 0 dB\(\sim\)20 dB; noise is Gaussian white noise. Under the same SNR, 200 simulations are carried out, the average value is taken as the simulation value of the characteristic parameters of this modulated signal, and the simulation results are shown in Figure 2.  
Figure 2a shows the simulation results of characteristic parameter \(f_{1}\). It can be seen that under different signal-to-noise ratios, the \(f_{1}\) value of BPSK signals is maintained at about 1, while the \(f_{1}\) value of the QPSK, 8 PSK, 8 QAM, 16 QAM, and 64 QAM signals is maintained in the range of 0 to 0.2. With the increase in signal-to-noise ratio, the \(f_{1}\) value of five signals gradually tends to 0. Therefore, setting the decision threshold to 0.6 can identify BPSK signals. Figure 2b shows the simulation results of characteristic parameter \(f_{2}\). It can be seen that the \(f_{2}\) value of the 8 PSK signal remains in the range of 0 to 0.2 and gradually tends to 0.