# Modulation Recognition System of Electromagnetic Interference Signal Based on SDR
## 2 Modulation Recognition System  
Modulation recognition refers to the classification and judgment of the modulation type of the signal according to its relevant characteristics under the condition of unknown target signal information, including signal pre-processing, feature extraction, classification, and recognition, as shown in Figure 1.  
The modulation signal pre-processing is to estimate the frequency and symbol rate of the signal, complete clock synchronization, and perform matching filtering, so that the received signal can remove some interference, and provide easier signal data for later feature extraction and classification and recognition. In the communication system, the signal \(r(t)\) received by the receiver can be expressed. \[r(t) =s(t)+n(t) \tag{1}\] \[=\sqrt{E}\sum_{n}s[nT]g(t-nT)e^{i(\omega_{c}t+\theta_{c})}+n(t)\]  
where \(s[nT]\) is the transmitted symbol sequence, \(E\) is the symbol energy, \(g(t)\) is the symbol waveform, \(\omega_{c}\) is the carrier frequency, \(\theta_{c}\) is the carrier phase, and \(n(t)\) is Gaussian white noise. Before identifying the received interference signal, first pre-process the interference signal, estimate the signal frequency and symbol rate, and complete clock synchronization and matching filtering. The position of the discrete spectral line of the modulation signal contains the frequency and symbol rate information of the signal, so the frequency and symbol rate of the signal can be estimated according to the spectral line position of each modulation signal. The modulation signal \(s(t)\) is constructed as a cyclostationary random process, and its time-varying autocorrelation function is expressed. \[R_{s}(t;\tau)=E[s(t)s^{*}(t-\tau)]\]  
Figure 1: The structure of the software radio modulation recognition system. If \(R_{s}(t;\tau)\) has the statistical characteristic with the time period \(T_{0}\), then \(s(t)\) is called a second-order cyclostationary process and Equation (3) can be obtained. \[\left\{ \begin{aligned} R_{s}(t;\tau)&=\sum_{m}R_{s}^{a} (\tau)\exp(j2\pi\alpha t)\\ R_{s}^{a}(\tau)&=\frac{1}{T_{0}}\int_{-\frac{T_{0}}{2 }}^{\frac{T_{0}}{2}}R_{s}(t;\tau)\exp(-j2\pi\alpha t)dt\end{aligned} \right. \tag{3}\]  
where, \(\alpha=m/T_{0}\), \((m=0,1,2,3\dots)\) is the cycle frequency. Since \(R_{s}(t;\tau)\) has periodic time variability, so the signal will appear discrete spectral lines in the frequency domain, which is called the spectral characteristics of the signal. When \(\tau=0\), the statistical expected value \(R_{s}(t;0)\) of the quadratic power of the signal. \[u(t)=R_{s}(t;0)=E\Big{[}s^{2}(t)\Big{]} \tag{4}\]  
The quadratic spectrum of the signal is obtained by Fourier transform: \(U(f)=F\{E[s^{2}(t)]\}\). Similarly, the fourth power spectrum \(V(f)=F\{E[s^{4}(t)]\}\) and eighth power spectrum of the signal \(W(f)=F\{E[s^{8}(t)]\}\) can be obtained. Different modulation signals have different spectral line characteristics in each order of power spectrum. Taking QPSK as an example, when the modulation order M = 4, the information sequence of QPSK signal is \(\theta=\{\pi/4,3\pi/4,5\pi/4,7\pi/4\}\), so the statistical expectation of the quadratic form of QPSK signal is equal to 0. \[ \begin{aligned} E\Big{[}s^{2}(t)\Big{]}&=E \Bigg{[}\sum_{n}\sum_{m}(a_{n}+jb_{n})(a_{m}+jb_{m})g(t-nT)g(t-mT)\Bigg{]}\\ &=\sum_{n}E\Big{[}a_{n}^{2}-b_{n}^{2}\Big{]}g(t-nT)\end{aligned}  \tag{5}\]  
According to Equation (5), \(E[s^{2}(t)]\) does not have periodic characteristics, so there is no discrete spectral line in the quadratic spectrum of QPSK signal. The quartic statistical expected value of QPSK signal can be expressed. \[ \begin{aligned} E\bigg{[}s^{4}(t)\bigg{]}&=E \Bigg{[}\sum_{n}\bigg{[}a_{n}^{4}+b_{n}^{4}-2a_{n}^{2}b_{n}^{2}\bigg{]}g^{4}(t -nT)\Bigg{]}\\ &=\bigg{(}-E_{s}^{2}\bigg{)}g^{4}(t-nT)\end{aligned}  \tag{6}\]  
where \(E_{s}\) is the symbolic average energy. The quartic spectrum of QPSK signal can be obtained by Fourier transform of Equation (6). \[