# Modulation Recognition System of Electromagnetic Interference Signal Based on SDR
## 3 High Order Cumulant Modulation Recognition Model  
Modulation signal feature extraction is designed to extract the feature information that can reflect the signal modulation type from the pre-processed signal sequence and use it for classification and recognition. Because the higher-order cumulant of the Gaussian random variable is equal to zero, the higher-order cumulant is used as the characteristic parameter in modulation recognition to remove the influence of Gaussian white noise and improve the anti-noise performance. At the same time, combined with the decision tree classifier, the classification and recognition of modulation signals other than 16 QAM and 64 QAM signals are realized by step-by-step processing. For the random variable \(X\), whose probability density function is \(f(X)\), its first characteristic function \(\Phi(\omega)\) is expressed. \[ \begin{split}\Phi\big{(}\omega\big{)}=E\big{\{}\exp(j\omega X) \big{\}}\\ =\int_{-\infty}^{\infty}f\big{(}X\big{)}\exp(j\omega X)dX\end{split}  \tag{10}\]  
The k-order moment \(m_{k}\) of random variable \(X\) is expressed. \[m_{k}=E\Big{\{}X^{k}\Big{\}}=\int_{-\infty}^{\infty}X^{k}f(X)dX \tag{11}\]  
By finding the k-order derivative of Equation (10), Equation (12) can be obtained. \[\Phi^{k}(\omega)=\frac{d^{k}\Phi(\omega)}{d\omega^{k}}=j^{k}E\Big{\{}X^{k}\exp (j\omega X)\Big{\}} \tag{12}\]  
According to Equations (11) and (12), when \(\omega\) = 0, the k-order moment formula of random variable \(X\) can be obtained. \[m_{k}=E\Big{\{}X^{k}\Big{\}}=(-j)^{k}\,\frac{d^{k}\Phi(\omega)}{d\omega^{k}} \Bigg{|}_{\omega=0}=(-j)^{k}\Phi^{(k)}(0) \tag{13}\]  
By performing natural logarithm operation on the first characteristic function \(\Phi(\omega)\), the second characteristic function can be obtained. \[\Psi(\omega)=\ln\Phi(\omega) \tag{14}\]  
Similarly, the calculation formula of the k-order moment of the second characteristic function can be deduced as follows. \[c_{k}=(-j)^{k}\frac{d^{k}\ln\Phi(\omega)}{d\omega^{k}}\Bigg{|}_{\omega=0}=(-j)^{k} \Psi^{(k)}(0) \tag{15}\]  
where \(c_{k}\) is the k-order cumulant of random variable \(X\). For a stationary continuous random process with multiple random variables, the k-order moment and the k-order cumulant of the random signal \(X(t)\) can be obtained. \[m_{kx}(\tau,\cdots,\tau_{k-1}) =\text{mom}[X(t),\cdots,X(t+\tau_{k-1})] \tag{16}\] \[c_{kx}(\tau,\cdots,\tau_{k-1}) =\text{cum}[X(t),\cdots,X(t+\tau_{k-1})] \tag{17}\]  
where mom represents the joint moment and cum represents the joint cumulant. If \(X(t)\) is a complex stationary random process, the expression of its p-order q-order mixing moment can be defined. \[M_{pq}=E\Big{[}X(t)^{p-q}\big{(}X^{*}(t)\big{)}^{q}\Big{]} \tag{18}\]  
where * represents conjugation. The \(X(t)\) higher-order cumulant is defined. \[c_{pq}=\text{cum}\Big{[}X(t),\cdots,X(t),X^{*}(t),\cdots,X^{*}(t)\Big{]} \tag{19}\]  
where the number of \(X(t)\) is \(p\)-\(q\) and the number of \(X^{*}(t)\) is \(q\). At the same time, there is a mutual conversion relationship between higher-order moment and higher-order cumulant function. \[\text{cum}\big{(}X_{1},\cdots X_{k}\big{)}=\sum_{n}(-1)^{q-1}\big{(}q-1)!\prod _{p=1}^{q}E\Big{(}\prod_{l\in U_{p}}X_{l}\Big{)} \tag{20}\]  
Equation (20) is called moment-cumulant conversion formula, i.e., M-C formula, where \(X_{l},l=1,2,\ldots k\) are k stationary random variables and \(\Sigma(\bullet)\) represents the sum of all unconnected ordered partitioned sets \(X=(X_{1},X_{2},\ldots,X_{k})\). Up is the subscript set of the elements in the \(p\)-th subset of the \(q\)-th subset. For zero mean complex random process \(X(t)\), the relationship between the second to sixth order cumulants and their higher-order moments can be obtained from the M-C formula:\(C_{20}=M_{20}\); \(C_{21}=M_{21}\), \(C_{40}=M_{40}-3M_{20}^{2}\), \(C_{41}=M_{41}-3M_{21}M_{30}\), \(C_{42}=M_{42}-|M_{20}|^{2}-2M_{21}^{2}\), \(C_{60}=M_{60}-15M_{40}M_{20}+30M_{20}^{3}\), \(C_{63}=M_{63}-9C_{42}C_{21}-6C_{21}^{3}\).