# A Survey of Blind Modulation Classification Techniques for OFDM Signals
## 3 Statistics-Based Approach to MC  
### LB Approach  
In the LB system, MC refers to numerous composite hypothesis problems. The LB-MC is based on the assumption that the probability density function (PDF) of the analyzed waveform includes all classification information, depending on the embedded modulated signal. The average likelihood ratio test (ALRT) [47], generalized likelihood ratio test (GLRT) [46], and hybrid likelihood ratio test (HLRT) [47] are the main three LB-MC techniques studied in the literature, based on the model selected for the unknown parameter. In some works in the literature, quasi-ALRT [46] and quasi-HLRT [45; 46] are also described. Figure 2: Block diagram of blind modulation classification for OFDM system. ALRT: In this method, unknown parameters are considered random variables with specific PDFs. For the hypothesis \(H_{j}\), which represents the \(j\)th modulation, \(j=1,2,...,M\), the likelihood function (LF) is as follows  
\[\Lambda_{ALRT}^{j}=\sum_{v_{j}}\Lambda[x_{m}[n]|v_{j},H_{j}]p(v_{j}|H_{j}), \tag{5}\]  
where \(\Lambda[x_{m}[n]|v_{j},H_{j}]\) denotes the conditional LF of the received signal \(x_{m}[n]\) associated with noise over \(H_{j}\), conditioned on the undefined vector \(v_{j}\) under \(H_{j}\). By integrating over \(v_{j}\) and using its known PDF, the problem is reduced to a basic hypothesis-testing problem. The conditional LF for a baseband complex AWGN is provided by  
\[\Lambda[x_{m}[n]|v_{j},H_{j}]=\frac{1}{\pi N_{0}}exp\left(-\frac{\frac{1}{\eta _{0}}\sum_{n=0}^{N-1}|x_{m}(n)-s_{m}[n]|^{2}}{N_{0}}\right) \tag{6}\]  
where \(N_{0}\) represents the power spectral density (PSD) of AWGN in W/Hz, with the autocorrelation \(E\{\omega[n],\omega^{*}[n+\tau]\}=N_{0}\delta[n]\), with \(E\{.\}\) denoting the expectation and * representing the complex conjugate. Furthermore, here \(s_{m}[n]=e^{(j2\pi\epsilon n/N+\phi)}\sum_{l=0}^{L-1}g[l]\tilde{d}_{m}[n-l- \tau]\). ALRT produces an optimal classifier in the Bayesian context when the chosen \(p(v_{j}|H_{j})\) is the same as the true PDF. GLRT: This approach considers the unknown parameters to be unknown deterministic. The best result is obtained by carrying out the so-called uniformly most powerful (UMP) test [100]. If UMP test does not exist or is difficult to obtain, a rational technique is used to estimate the unknown parameters based on the assumption that \(H_{j}\) is true, and then utilize these estimations in ALRT as if they were accurate. When maximum likelihood is applied for estimations, the hypothesis test is known as GLRT. The unknown parameters of GLRT are, of course, considered deterministic unknowns, and LF under \(H_{j}\) is provided by  
\[\Lambda_{GLRT}^{j}[x_{m}[n]]=\max_{v_{j}}\Lambda\big{[}x_{m}[n]|v_{j},H_{j} \big{]}. \tag{7}\]  
HLRT: This is the combined approaches of the above techniques, where the LF under \(H_{j}\) is defined by  
\[\Lambda_{HLRT}^{j}=\max_{v_{j_{1}}}\sum_{v_{j_{2}}}\Lambda\big{[}x_{m}[n]|v_{j_ {1}},v_{j_{2}},H_{j}\big{]}p(v_{j_{2}}|H_{j})dv_{j_{2}}, \tag{8}\]  
where \(v_{j}=\left[v_{j_{1}}^{\dagger}v_{j_{2}}^{\dagger}\right]^{\dagger}\) with \(\dagger\) as the transpose and \(v_{j_{1}}\) and \(v_{j_{2}}\) are vectors of unknown parameters treated as unknown deterministic and random variables, respectively. Generally, \(v_{j_{1}}\) and \(v_{j_{2}}\) are made up of parameters and data symbols, respectively. It is to be noted that ALRT necessitates multidimensional convergence, while GLRT necessitates multidimensional maximization. ALRT could be unrealistic due to the difficulties of performing multidimensional integration in the presence of a large number of unknown parameters and the requirement to know the PDFs. Furthermore, maximization over unknown parameters in GLRT results yield the same LF value for nested signal constellations, such as BPSK and QPSK, 16-QAM, and 64-QAM, resulting in inaccurate classification. However, with HLRT, averaging over unknown data symbols eliminates the GLRT problem of nested constellations.