# A Survey of Blind Modulation Classification Techniques for OFDM Signals
## 3 Statistics-Based Approach to MC
### LB Approach
In the case of a two-hypothesis classification problem, a decision is made on the basis of  
\[\Lambda_{H}^{(1)}[x_{m}[n]]/\Lambda_{H}^{(2)}[x_{m}[n]]\overset{H_{1}}{\underset {H_{2}}{\overset{H_{2}}{\overset{H_{1}}{\geq}}}}\eta_{l},l=A(\text{ALRT}),G( \text{GLRT}),H(\text{HLRT}), \tag{9}\]where \(\eta_{l}\) represents the threshold. The left-hand side of (9) represents the likelihood ratio, and the test is referred to as the ALRT, GLRT, and the HLRT, respectively, depending on the approach used to estimate the LF. The extension of (8) to multiple classes is simple. Likewise, the log function can be extended to the two members of the inequality (9). Table 2 lists multiple LB-MC algorithms proposed in the literature, outlining the modulation types, uncertain parameters, and the channel employed. An LB-MC for the OFDM system is studied in [45]. The aim of this work is limited to reliable blind MC schemes. A maximum likelihood that provides optimal performance in the presence of AWGN is introduced. A sub-optimal classifier is obtained based on the optimal maximum-likelihood classifier to minimize the computational complexity. The accuracy of such classifiers is evaluated through Monte Carlo simulations. In the simulation, an OFDM system with 64 subcarriers is considered. The subcarriers are divided into 4 bands, each of which has 16 subcarriers. In each sub-band, four distinct modulation formats, namely BPSK, QPSK, 16-QAM, and 64-QAM, are used to transmit the signal according to the channel conditions. Perfect CSI is considered for the simulation. It is observed that the proposed sub-optimal algorithm achieves near to optimal performance with significantly less complexity. As a result, it can be used rather than signaling in realistic systems to improve spectral efficiency. In the proposed method [46], a modified quasi-log-likelihood ratio (QLLR) based MC for the OFDM system is studied. The ALRT- and GLRT-based classifiers need few symbols to achieve acceptable classification performance in the presence of appropriate channel estimation with relatively high SNR. To achieve acceptable performance, a modified QLLR-based classifier needs high SNR and more symbols but their computational complexity remains lower compared to the ALRT- and GLRT-based classifiers. In order to classify QPSK, 16-QAM, and 64-QAM, the modified QLLR test is applied on received symbol sets. This method seems to be feasible if the operating point of SNR is comparatively high as compared to ALRT- and GLRT-based classifiers. Another LB-MC for OFDM with index modulation (OFDM-IM) is analyzed in [47]. The modulation parameters in OFDM-IM often include the number of active subcarriers in addition to the constellation of signals, which distinguishes them from traditional modulations. Specifically, two MC cases are assumed. One is the MC with known CSI, and another is the MC with unknown CSI. ALRT, HLRT-LLR, and HLRT-energy-based classifiers are considered for the case of known CSI. When compared to ALRT, both HLRT-LLR and HLRT-energy have lower computational complexity, but show degradation in classification performance. In the case of unknown CSI, the energy-based detector is first used to recognize the active subcarriers, then the expectation-maximization (EM) algorithm is employed to estimate the CSI for each hypothesis. The number of subcarrier \(N=128\)