## 4 Artificial Neural Network-Based Classification of Hierarchical Symbols  
In this section, we briefly outline a generic approach, which is later improved and forms our main contribution. The main task is to infer the value of hierarchical symbol \(b\), based on the observation \(x\). Obviously, knowledge of the parameter \(h\) in (1) is crucial for this task. The expected channel phase drift over time opens a problem of ambiguity of deciding the hierarchical symbol \(b\).  
A basic background idea described in this paper is to utilize a classical ML approach for this problem, such that a training dataset \(\mathcal{D}=\{(\mathbf{x}_{d},\mathbf{t}_{d})\}_{d=1}^{D}\) of size \(D\) is utilized to train an ANN classifier for this task. Vector \(\mathbf{x}_{d}=[\Re\{x\},\Im\{x\}]^{T}\) corresponds to the coordinates in the constellation space (an alternative would be to take \([|x|,\measuredangle x]^{T}\) as the input vector, but it would introduce the problem of phase discontinuity). Vector \(\mathbf{t}_{d}\) with dimensionality 2 represents the desired output value of the ANN. It should be clear that even though we do not apply any explicit feature extraction, such preprocessing takes place in the form of a projection of the received signal onto a set of basis functions, thus giving the coordinates of \(\mathbf{x}_{d}\) in the constellation space. For the purpose of simulations, however, we work directly in the constellation space.  
Let us now describe how to obtain the training dataset. Consider that initially, nodes \(S_{A}\) and \(S_{B}\) simultaneously transmit a predetermined, pseudorandom sequence of symbols \(\left\{b_{A,d}\right\}_{d=1}^{D},\)\(\left\{b_{B,d}\right\}_{d=1}^{D}.\) Furthermore, let node \(R\) be aware of the resulting hierarchical target sequence \(b_{d}=\left\{b_{A,d}\oplus b_{B,d}\right\}\). For the purpose of training the ANN using the backpropagation algorithm, this sequence is transformed to the expected ANN output vector \(\mathbf{t}_{d}\) as \(\mathbf{t}_{d}=\left[b_{d},\,1-b_{d}\right].\) Thus, the observation \(\mathbf{x}_{d}(x)\) together with the corresponding value \(\mathbf{t}_{d}\) form the required training dataset for the supervised ML hypothesis class.  
In the case of the considered BPSK modulation, the hierarchical symbol is binary. See Figure 1 for a demonstrative example. Therein, two colors distinguish the classes corresponding to the two values of the hierarchical target symbol \(b\). The phase of \(h\), denoted \(\measuredangle h\), dictates the rotation of the superimposed constellation.  
It is natural to visualize the trained classifierin terms of its decision regions, as shown in [13]. An example of trained decision regions is shown in Figure 2. Therein, two reference solutions are provided. The trained decision regions are shown in the central subfigure. The true metric map is based on the analytically optimal metric; see [13] for details. The distance-based map is an approximation of the true metric map, based on the Euclidean distance between the neighboring points only, also derived in [13].  
With the above description of the nature of the training dataset, it is straightforward to design an ANN and train it using the backpropagation algorithm and perform the classification of the hierarchical symbols; see [1]. The specific implementation is not the main objective of this paper. For more details; see [15].  
Figure 1: Demonstration of the training dataset.