## 5 Slowly Varying Fading Parameter in 2 WRC
### Numerical Analysis  
The above-described procedures P1 - P3 were implemented, and we provide numeric simulation results. The angle of relative fading \(\measuredangle h\) of consequent pilot sequences used for the training process changes uniformly as \(\measuredangle h_{i+1}-\measuredangle h_{i}=4.5^{\circ}\). As justified by results presented in Figure 3, a pilot sequence of length \(D=2000\) was considered. We experimentally determined \(E_{1}=30\) and \(E_{2}=20\) to be the suitable numbers of training epochs with the corresponding learning rates \(\eta_{1}=0.05\) and \(\eta_{2}=0.07\). With respect to the block diagram in Figure 4, consider \(I=3\). To provide an in-depth look, an analysis of the ANN weights' evolution was performed. To evaluate the rates of change of all weights \(\alpha\) in each layer of the ANN, the following expression was used:  
\[a^{(j)}(e)=\frac{1}{A^{(j)}}\sum_{i}\left[\alpha_{i}^{(j)}(e)-\alpha_{i}^{(j) }(e-1)\right]^{2}, \tag{2}\]  
where the upper index \((j)\) specifies the layer, \(A^{(j)}\) is the number of weights in layer \((j)\), \(e\) is a counter of training epochs, and \(i\) identifies the specific weights of the layer. For comparison, this expression is evaluated and graphically represented in Figure 5 for procedure P3, without re-initialization, and in Figure 6 for procedure P1, where the weights are always re-initialized, and thus no prior information is utilized (graphically, the inspection of procedure P2 resulted in similar results as for P3, and the figure is therefore omitted). Note that in these figures, it is clearly observable how the training process  
Figure 4: An illustrative block diagram representing procedures P1 (green), P2 (blue), and P3 (red). The difference to be emphasized is the strategy of re-initialization of the ANN weights.  
converges. The peaks of the curves correspond to the changes of the parameter \(\measured{h_{i}}\), as marked by arrows. These results are useful to optimize parameters \(E_{1}\), \(\eta_{1},E_{2}\), \(\eta_{2}\).  
In Figure 7, we provide a comparison of the approaches in terms of the bit-error rate (BER) over different values of the SNR. Therein, two reference solutions are provided, as addressed in Section 4. Recall that these reference solutions are based on perfect knowledge of \(\measured{h_{i}}\) and therefore are naturally superior to the trained solutions. Procedures P1-P3, represented in Figure 4, were tested for two values of the parameter \(I\), namely \(I=3\) and \(I=5\). In this result, we observe that the most efficient strategy  
Figure 5: Relative squared evolution of the weights for procedure P3, where the weights are initialized only once, and for every subsequent value of \(\measured{h_{i}}\), the weights are updated via backpropagation without subsequent re-initializations.  
Figure 6: Relative squared evolution of weights for procedure P1, where the weights are initialized, and for every subsequent value of \(\measured{h_{i}}\), the weights are re-initialized prior to the further training process. The legend from Figure 5 is used.  
is procedure P3. Clearly, it is also better to keep longer training sequences more often, i.e., \(I=3\) is preferable to \(I=5\).