## 2 System Model  
We consider a three-terminal cooperative system as shown in Figure 1, where the source \(S\) has data for the delivery to the destination \(D\) with the aid of the relay \(R\). To simplify the mathematical notations and analytical expressions, we assume that all nodes have a single antenna element. We leave the implementation of multiple antennas at each node to future development. The relay implements an AF protocol and runs in a half-duplex configuration that it transmits and receives at two distinct times [24].  
At both the relay and the destination, we expect the additive white Gaussian noise (AWGN) variance to be \(\sigma_{n}^{2}\). We denote \(\mathbf{u}=[u(0),u(1),\cdots,u(K-1)]\) as a frame of transmission comprising \(K\) data symbols. Here, each data symbol is picked at random from a specific signal constellation \(\alpha\). The source selects \(\alpha\) from a pool of candidates for each transmission frame to match the quality of service and capacity demands. A frame transmission will normally take place over two distinct stages.  
During the first phase of the process, the frame \(\mathbf{u}\) is sent via two distinct connections in order to reach both the relay and the final destination. We call \(\mathbf{h}_{SR}=[h_{SR}(0),\cdots,h_{SR}(L-1)]\) and \(\mathbf{h}_{RD}=[h_{RD}(0),\cdots,h_{RD}(L-1)]\) the channel impulse responses between \(S\) and \(R\) and between \(R\) and \(D\), respectively. Here, \(L\) is the number of channel taps. We express the \(k\)th received samples at the relay, \(z_{R}(k)\), and the destination, \(z_{D}(k)\), analytically as  
\[z_{R}(k)=\sum_{l=0}^{L-1}h_{SR}(l)u(k-l)+n_{R}(k), \tag{1}\]  
and  
\[z_{D}^{(1)}(k)=\sum_{l=0}^{L-1}h_{SD}(l)u(k-l)+n_{D}^{(1)}(k), \tag{2}\]  
where \(n_{R}(k)\) and \(n_{D}^{(1)}(k)\) are the noise samples at the relay and destination, respectively. Here, we underline that both situations of direct connection presence and absence are extensively discussed in the literature (e.g., [19; 20; 21; 22; 23; 24; 25] and references therein). We focus on the existence of direct link in this investigation.  
Figure 1: The three-terminal cooperative system under consideration.  
Phase two of transmission entails re-sending the processed the signal that was received by the AF relay. Accordingly, the input signal at the node \(D\) is written as  
\[z_{D}^{(2)}(k)=C\sum_{l=0}^{L-1}h_{RD}(l)z_{R}(k-l)+n_{D}^{(2)}(k), \tag{3}\]  
where \(C\) is the scaling factor, and it is provided in the following form  
\[C=\frac{1}{\sqrt{\sum_{l=0}^{L-1}\left|h_{SR}(l)\right|^{2}+\sigma_{n}^{2}}}. \tag{4}\]  
We set \(C\) to ensure that the output power of the AF relay and the source are identical. Plugging (1) into (3), we obtain  
\[z_{D}^{(2)}(k)=C\sum_{l=0}^{L-1}h_{RD}(l)\sum_{l^{\prime}=0}^{L-1}h_{SR}(l^{ \prime})u\big{(}k-l-l^{\prime}\big{)}+w_{D}^{(2)}(k), \tag{5}\]  
where \(w_{D}^{(2)}(k)\) is the \(k\)th noise sample and it incorporates both \(n_{D}^{(2)}(k)\) and \(n_{R}(k)\) contributions as  
\[w_{D}^{(2)}(k)=n_{D}^{(2)}(k)+C\sum_{l=0}^{L-1}h_{RD}(l)n_{R}(k-l). \tag{6}\]  
The objective of this research is to devise a method for determining the allotted modulation type \(\alpha\) by utilizing the features extracted from the signals \(z_{D}^{(1)}(k)\) and \(z_{D}^{(2)}(k)\), \(k=0,\cdots,K-1\).