## 5 OFDM Signal Identification Based on Inherent Features
### Multidimensional Feature-Based OFDM Signal Identification  
Based on the parameter analysis results described earlier, the cyclic prefix is removed from the received low-orbit satellite OFDM baseband signal. Subsequently, the Fast Fourier Transform (FFT) is applied to isolate the subcarrier signals across individual subchannels. By leveraging the multidimensional feature vectors of these subcarrier signals in conjunction, the modulation scheme employed for each subcarrier can be identified.  
This paper introduces a multidimensional domain feature extraction technique, which yields a total of 13 types of signal features. The specific categories of these signal features are depicted in Figure 5.  
Higher-order cumulants possess the property of reflecting the higher-order statistical characteristics of modulated signals, offering robustness against fading and effective suppression of Gaussian white noise. Moreover, different digital modulation signals exhibit distinct higher-order cumulants. Consequently, these cumulants with their unique features can serve as a basis for classifying modulated signals.  
Figure 5: Intrinsic features in multiple domains of each subcarrier in OFDM.  
Based on Equations (34) and (35), it is feasible to compute the second-order, fourth-order, sixth-order, and eighth-order cumulants:  
\[\begin{split} C_{20}&=M_{20},\\ C_{21}&=M_{21},\\ C_{40}&=M_{40}-3M_{20}^{2},\\ C_{41}&=M_{41}-3M_{21}M_{20},\\ C_{42}&=M_{42}-|M_{20}|^{2}-2M_{21}^{2},\\ C_{60}&=M_{60}-15M_{40}M_{20}+30M_{21}^{2},\\ C_{61}&=M_{61}-5M_{40}M_{21}-10M_{20}M_{41}\\ &+30M_{21}M_{20}^{3},\\ C_{63}&=M_{63}-6M_{41}M_{20}-9M_{21}M_{42}\\ &+18M_{21}M_{20}^{2}+12M_{21}^{3},\\ C_{80}&=M_{80}-28M_{20}M_{60}-35M_{40}^{2}\\ &+420M_{40}M_{20}^{2}-630M_{20}^{4}.\end{split} \tag{50}\]  
Wavelet transform is a time-frequency analysis method for signals, characterized by its multi-resolution analysis capabilities, as well as its localization properties in both time and frequency domains. By employing wavelet transform to decompose the signal at different scales, it is possible to delineate the details of various modulation type signals. The use of different wavelet bases to analyze the same problem can yield distinct results. Common wavelet functions include the Haar, Daubechies, Coflet, and Symlet families, and the Morlet wavelet, among others, owing to its simplicity and ease of computation; the Haar wavelet is suitable for engineering applications. Therefore, this paper adopts the Haar wavelet. The Haar wavelet is defined as  
\[\Psi(t)=\begin{cases}1,&-0.5<t\leq 0,\\ -1,&0<t\leq 0.5,\\ 0,&others.\end{cases} \tag{51}\]  
The wavelet basis function, denoted as \(\Psi_{(a,b)}(t)\), is defined as  
\[\Psi_{(a,b)}(t)=\begin{cases}\frac{1}{\sqrt{a}},&-0.5a<t\leq 0,\\ -\frac{1}{\sqrt{a}},&0<t\leq 0.5a,\\ 0,&others.\end{cases} \tag{52}\]  
In the entropy domain, features such as the Renyi entropy, power spectrum Shannon entropy, and power spectrum exponential entropy are utilized as characteristic parameters for modulation identification. The definition of the \(\alpha\)-th order Renyi entropy is  
\[H_{\text{a}}(X)=-\frac{1}{\alpha-1}\log E_{X}[P(X)^{\alpha-1}], \tag{53}\]  
where \(X\) and \(P(X)\) represent random variables, and \(E\) denotes the expectation operator.  
When \(\alpha-1=s\) approaches 0, the Renyi entropy reduces to the Shannon entropy, whose formula is given by  
\[H=E[-\log_{2}p_{i}]=\sum_{i=1}^{n}p_{i}\log_{2}p_{i}. \tag{54}\]Here, \(E\) represents the expectation operator, and \(p_{i}\) is the probability.  
The power spectrum exponential entropy signifies the uncertainty of signal energy under the division of the power spectrum, providing a quantitative description of the complexity of energy distribution in the frequency domain. The power spectrum exponential entropy is defined as  
\[H=\mathrm{E}\bigg{[}\mathrm{e}^{1-p_{i}}\bigg{]}=\sum_{i=1}^{n}p_{i}\mathrm{e}^{ 1-p_{i}}, \tag{55}\]  
where \(H\) represents the entropy value, \(E\) denotes the expectation operator, and \(p_{i}\) is the probability.