# A Survey of Blind Modulation Classification Techniques for OFDM Signals
## 3 Statistics-Based Approach to MC
### Maximum a Posteriori (MAP) Approach  
MC is the process of determining the modulation format of received signals from a set of \(L\) modulation formats \(\mathfrak{M}=\{M_{j},\,j=1,2,...,M\}\), based on a series of \(N\) received samples \(\mathbf{x}_{m}=[x_{m}[0],x_{m}[1],...,x_{m}[N_{s}-1]]\). The maximum a posteriori (MAP) criteria can be used to find the optimal modulation classifier by using the [PERSON] decision principle [57]. For received signal \(x_{m}\), the a posteriori probability of \(M_{j}\) is defined as \(P(M_{j}|x_{m})\), and the decision is made by  
\[\hat{M}_{j}=\arg\max_{M_{j}\in\mathfrak{M}}P(M_{j}|x_{m}), \tag{10}\]  
Another well-known classifier originating from the MAP criteria is the ML classifier. The a posteriori probability can be expressed using the [PERSON] rule as  
\[P(M_{j}|x_{m})=\frac{P(x_{m}|M_{j})P(M_{j})}{P(x_{m})}, \tag{11}\]  
where \(P(x_{m}|M_{j})\) denotes the likelihood of the received samples \(x_{m}\) when the modulation format \(M_{j}\) is given, \(P(M_{j})\) is the prior likelihood of the modulation format \(M_{j}\), and \(P(x_{m})\) is the marginal likelihood of the received samples \(x_{m}\), which is independent of \(M_{j}\). When all the candidate modulation formats are equiprobable, then the MAP classifier is identical to the ML classifier [51]. \[\hat{M}_{j}=\arg\max_{M_{j}\in\mathfrak{M}}P(x_{m}|M_{j}). \tag{12}\]  
Table 3 lists multiple MAP-based MC algorithms studied in the literature, outlining the modulation types, uncertain parameters, and the channel employed. A MAP-based MC algorithm in time division duplex (TDD) based OFDM systems with adaptive QAM modulation is studied in [50]. It takes advantage of the channel reciprocity in TDD systems and the data rate of transmission. Unlike the signaling-free adaptive modulation technique, MC and data detection are decoupled here, resulting in significantly decreased computational complexity. Moreover, this technique utilizes the fixed bit allocation table (BAT) for all transmission frames. As a result, more symbols of the same modulation scheme can be employed to make a decision. Compared to the traditional ML method, simulations have validated the superior classification performance of the modified MAP algorithm. This technique allows adaptive modulation to be applied in wireless OFDM systems without reducing the effective data rate due to the signaling of the BAT. A novel efficient MC algorithm in wireless TDD-based OFDM systems with adaptive modulation is analyzed in [51]. The frequency-selective behavior of the channel is experienced by a finite impulse response (FIR) filter model with Rayleigh fading coefficients. Jakes' spectrum with the Doppler frequency \(f_{dm}\) is used to model the time correlation of the different path coefficients. This adaptive modulation approach adapts modulation formats among BPSK, QPSK, 16-QAM, and 64-QAM to a group of two adjacent subcarriers. The conventional maximum-likelihood method is modified to a MAP classifier that uses reciprocity of the channels in TDD systems. Moreover, a less computationally complex classifier based on the MAP criteria is developed and evaluated, which is desirable for real-time implementations. The feasibility of complexity reductions is validated by simulations. The classification performance of the proposed technique is slightly reduced in terms of the packet error rate compared to perfectly known modulation schemes. \begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline  
A framework of MAP algorithms for MC in OFDM-based communication systems with adaptive modulation is studied in [52]. This work extends the achievements in MAP-based MC [50; 51] by adding a new constraint to the framework. In this paper, a metric approximation is used, whose accuracy increases with rising SNR; the reason behind using this is the high computational complexity of the optimal algorithm.