# A Survey of Blind Modulation Classification Techniques for OFDM Signals
## 3 Statistics-Based Approach to MC
### FB Approach
It is observed that the amplitude moments increase as the frequency offset increases. Considering 4096 subcarriers in an OFDM symbol, at least 10 OFDM symbols are required to achieve the desired classification accuracy at 30 dB SNR. This approach outperforms the existing amplitude moment-based approach with the prior information about CFO. This is due to the estimation and elimination of the ICI components in the  
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amplitude moments. However, this MC algorithm is restricted to known CSI and proper synchronization circumstances. In [63], another FB blind MC approach is suggested and implemented on radio frequency (RF) testbed for OFDM signals. The authors use the combined features of DFT and the fourth-order cumulant, as shown in Figure 3. This algorithm does not need prior information about the signal parameters and CSI. It also works effectively when there are synchronization problems, such as timing, frequency, and phase errors. Before the feature extraction process, a random uniformly distributed timing offset is added in each OFDM symbol to reduce the influence of the timing offsets. The authors have listed BPSK, QPSK, OQPSK, MSK, and 16-QAM for the OFDM signal. The number of subcarrier \(N=1024\), CP length \(N_{cp}=N/4\), channel tap \(L=4\), number of OFDM symbol 50, normalized CFO \(-0.5<\epsilon<0.5\), symbol timing offset \([-N/2,N/2]\), sampling rate 50 Msamples/s, symbol rate 1 Msymbols/s, and Rayleigh channel are considered the simulation parameters for this technique. Classification is carried out in two stages. First, the received signal is transformed into the frequency domain by using the DFT operation, then the normalized fourth-order cumulant of the frequency domain is calculated. The modulation formats OQPSK, MSK, and 16-QAM can be distinguished by the normalized fourth-order cumulant, which is expressed as  
\[\tilde{C}_{42_{R}}=\frac{1}{K}\sum_{m=1}^{K}\frac{\tilde{C}42_{X_{m}}-\left| \frac{1}{K}\sum_{m=0}^{K-1}e^{-\tilde{J}4\pi\gamma/K(\tau+\theta_{u})}X_{m}^{2 }[v]\right|^{2}}{\frac{1}{K}\sum_{m=0}^{K-1}\left|X_{m}[v]\right|^{2}-C_{21,W}}, \tag{19}\]  
where \(X_{m}[v]\) represents the DFT of the received signal \(x_{m}[n]\), \(C_{21,W}=\sigma_{W}^{2}\) represents the estimated variance of AWGN, and \(K\) is the total number of OFDM symbols. The histogram of the above is given in Figure 4. The second stage performs the DFT of the square of the received signal then calculates the normalized fourth-order cumulant, which is expressed as  
\[\tilde{C}_{42_{U}}=\frac{1}{K}\sum_{m=1}^{K}\frac{\tilde{C}42_{U_{m}}-\left| \frac{1}{K}\sum_{m=0}^{K-1}e^{-\tilde{J}4\pi\gamma/K(\tau+\theta_{u})}U_{m}^{2 }[v]\right|^{2}}{\frac{1}{K}\sum_{m=0}^{K-1}\left|U_{m}[v]\right|^{2}-C_{21,W}}, \tag{20}\]  
where \(U_{m}[v]=X_{m}[v]\)\(\otimes\)\(X_{m}[v]\), \(\otimes\) denotes the linear convolution operator. For BPSK and QPSK modulation schemes, the above Equation (20) gives different values, as shown in Figure 5.  
Figure 3: Schematic diagram of blind modulation classification studied in [63]. In the paper [64], the authors use a method for applying wavelet transform (WT) to OFDM and SC signals to extract their transient characteristics and then use the transient characteristics to identify the two types of signals. Good performance can be achieved in low SNR and multipath channel conditions. In addition, the effects of the sample rate and symbol rate on the identification algorithms are analyzed and simulated. The author conducts a variety of simulation experiments to assess the performance of the proposed identification algorithms and the effects of the sample rate and symbol rate on the identification algorithms. All results are based on 100 Monte Carlo trials. The percentage of correct classification (PCC) versus the SNR plot represents the average variance of signals versus SNR. Every 8000 data samples make up of a trial source. We notice that the average variance is large in the OFDM curve and small in the SC curve.