# Modulation Recognition System of Electromagnetic Interference Signal Based on SDR
## 3 High Order Cumulant Modulation Recognition Model
The \(f_{2}\) value of QPSK, 16 QAM, and 64 QAM signals is about 1, while the \(f_{2}\) value of the 8 QAM signal is about 1.7. Therefore, the 8 PSK signal and the 8 QAM signal can be recognized by setting the decision threshold to 1.4 and 0.5, respectively. Figure 2c shows the simulation results of the characteristic parameter \(f_{3}\). It can be seen that the \(f_{3}\) value of QPSK is about 16, the values of 16 QAM and 64 QAM are kept between 14 and 13.5, and the QPSK signal can be recognized by setting the decision threshold value to 15. Figure 2d shows the simulation values of the characteristic parameter \(f_{4}\). It can be seen that the \(f_{4}\) value of 16 QAM and 64 QAM is relatively close, and the value increases with the increase in signal-to-noise ratio. This is because the cumulant values of each order of 16 QAM and 64 QAM signals are relatively close, and the two signals are seriously affected by noise due to their large order. Therefore, it is difficult to distinguish the two signals by setting the decision threshold. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Characteristic Parameter** & **BPSK** & **QPSK** & **8 PSK** & **8 QAM** & **16 QAM** & **64 QAM** \\ \hline \(f_{1}\) & \(1\) & 0 & 0 & 0 & 0 & 0 \\ \(f_{2}\) & \(1\) & \(1\) & 0 & 1.72 & 1 & 1 \\ \(f_{3}\) & 21.125 & 16 & 16 & 19.151 & 13.759 & 713.615 \\ \(f_{4}\) & 4 & 2 & 1 & 1.82 & 1.36 & 1.24 \\ \hline \end{tabular}
\end{table} 
Table 2: Characteristic parameter values of MPSK and MQAM interference signals characteristic parameters. According to Equation (12), when the constellation phase offset is very small, i.e., \(\theta=0\), the MQAM signal can be expressed. \[r(k) =s(k)+n(k) \tag{21}\] \[=[s_{I}(k)+n(k)]+j[s_{Q}(k)+n_{Q}(k)]\]  
The order of \(r(k)\) is reduced once, and a new signal sequence \(r^{\prime}(k)\) is obtained. \[r^{\prime}(k) =|\text{Re}[s(k)+n(k)]|+j|\text{Im}[s(k)+n(k)]| \tag{22}\] \[=|s_{I}(k)+n_{I}(k)|+j|s_{Q}(k)+n_{Q}(k)|\] \[=Y_{I}(k)+jY_{Q}(k)\]  
Under the condition of high signal-to-noise ratio, the probability density function of \(Y_{\text{I}}(k)\) is approximately. \[f_{Y_{\text{I}}[k]}(y)\approx\frac{1}{\sqrt{\pi N_{0}}}\exp\left[-\frac{\left( y-|s_{I}(k)|\right)^{2}}{N_{0}}\right] \tag{23}\]  
where \(Y_{\text{I}}(k)\) approximately obeys the Gaussian distribution with mean value \(|\mathfrak{s}_{1}(k)|\) and variance \(N_{0}/2\). Similarly, \(Y_{\text{Q}}(k)\) also approximately obeys the Gaussian distribution with a mean value of \(|s_{\text{Q}}(k)|\) and a variance of \(N_{0}/2\). Therefore, the signal \(r^{\prime}(k)\) can be expressed. Figure 2: Simulation results of different characteristic parameter. \[r^{\prime}(k) = (|\text{Re}[s(k)]+n_{I}(k)|)+j(|\text{Im}[s(k)]|+n_{Q}(k)) \tag{24}\] \[= (|s_{I}(k)|+j|s_{Q}(k)|)+n(k)\] \[r(k) =s(k)+n(k) \tag{21}\]0  
The new signal can be calculated from Equation (24), and the order of the new signal becomes one quarter of the original signal, that is, M\({}^{\prime}\) = M/4, which realizes the order reduction processing of the signal. After the reduced order processing, the values of the characteristic parameters \(f_{4}\) of the 16 QAM signal and the 64 QAM signal are shown in \[r(k) =s(k)+n(k) \tag{21}\]1  
As can be seen from Figure 3, with the increase in signal-to-noise ratio, the \(f_{4}\) value gap between the two signals gradually becomes larger, and the \(f_{4}\) value of 64 QAM gradually stabilizes at about 2.26. Therefore, by setting the decision threshold to 2.27, respectively, the 16 QAM signal and the 64 QAM signal can be distinguished.