# A Survey of Blind Modulation Classification Techniques for OFDM Signals
## 3 Statistics-Based Approach to MC
### FB Approach  
In the FB algorithm, the expert domain feature needs to be extracted first and then decisions are made for the classification. Some of the expert domain features are the variance of the normalized signal amplitude, phase, and frequency [101], the variance of the zero-crossing interval [102], moments, cumulants [63], cyclic cumulants [5], cyclostationarity [103], Fourier transform [63], wavelet transform (WT) [17], and constellation shape [104] of the received signal. The fuzzy logic [105], entropy [106], and constellation shape recovery technique also have been used for MC. Various decision-making approaches have been employed, including maximum-likelihood detector [63], Hellinger distance [107], Euclidean distance [108], and unsupervised clustering techniques [109]. MC with Higher Order Statistics (HOS)  
Here, we provide a framework of the MC method with HOS [110]. The moment with the \(k\)th order and \(p\)th conjugations for \(x_{m}\) associated with \(x_{m}[n]\) is defined as  
\[M_{kp,x_{m}}=E\left[x_{m}^{k-p}(x_{m}^{*})^{p}\right]. \tag{13}\]  
where \(()^{*}\) represents a complex conjugate. The corresponding cumulant with \(k\)th order and \(p\)th conjugations is defined as  
\[C_{kp,x_{m}}=cum(\underbrace{x_{m},x_{m},...,x_{m}}_{k-p},\underbrace{x_{m}^{* },x_{m}^{*},...,x_{m}^{*}}_{p}), \tag{14}\]  
where \(cum()\) represents the joint cumulant function. HOS provides an integrated technique as well as a nonlinear signal processing viewpoint. Nevertheless, the information in the power spectrum of the second-order statistics is only appropriate for describing Gaussian processes statistically. In MC applications [111], a general fourth-order statistics \(C_{42,x_{m}}\) is frequently used. According to (13) and the fourth-order cumulant formula for four random variables, \(X\), \(Y\), \(Z\), and \(W\) can be expressed as  
\[cum(X,Y,Z,W)=E[XYZ]-E[XY]E[ZW]-E[XZ]E[YW]-E[XW]E[YZ], \tag{15}\]  
and  
\[ \begin{split} C_{42,x_{m}}&=cum(x_{m},x_{m},x_{m} ^{*},x_{m}^{*})\\ &=E\Big{(}|x_{m}|^{4}\Big{)}-\Big{(}\Big{|}E(x_{m}^{2})\Big{|} \Big{)}^{2}-2E^{2}\Big{(}|x_{m}|^{2}\Big{)},\end{split}  \tag{16}\]  
In a similar fashion, a typical second-order cumulant can be written as  
\[C_{21,x_{m}}=E\Big{(}|x_{m}|^{2}\Big{)}. \tag{17}\]  
The normalized fourth-order cumulant [12] is typically used to calculate MC, defined as  
\[\hat{C}_{42,x_{m}}=\frac{C_{42,x_{m}}}{C_{21,x_{m}}^{2}}. \tag{18}\]The FB-MC approaches are listed in Table 4, which highlights selected features, modulation types, channels, and undefined parameters. Multicarrier modulation given by the OFDM signal generator using an IEEE 802.16e standard is studied in [61]. Based on the standard of IEEE 802.16e, three possible modulation formats can be used, such as QPSK, 16-QAM, and 64-QAM. The mean, variance, skewness, kurtosis index, and moment order of the received signal are all considered and compared in order to determine the modulation scheme non-line-of-sight (NLOS) with six multipath components. The dominant statistic features capable of separating the QPSK modulation scheme from 16-QAM and 64-QAM are skewness, kurtosis, and variance, as determined by the statistical properties of the received signal. Furthermore, the high order moment is one of the most important statistical features that distinguish the 16-QAM modulation scheme from the 64-QAM modulation scheme. However, in the context of timing and frequency synchronization issues, this approach does not perform well. In another FB-MC [62], the amplitude moments and correlation properties are used to classify the modulation scheme for OFDM systems. This technique considers the presence of CFO, which is the cause of intercarrier interference (ICI) in the amplitude moments of the received signal. Therefore, the ICI component is estimated by using the correlation between the subcarriers. To determine the influence of ICI components in the amplitude moments, the authors derive the amplitude moment in the form of infinite series of elementary functions.