## 3 Results
### Transmitted Signal Estimation  
Figure 5 presents the results for randomly selected signals at various SNR levels, including the true transmitted signal (blue), the estimated transmitted signal (red), and the corresponding 95% prediction interval (gray). In the previous subsection focusing on CFO estimation, we established an inverse relationship between the SNR and MAE, indicating that higher SNR values lead to lower CFO estimation errors. Expanding on this observation, Figure 5 reveals that the combined effect of low SNR and the inferior quality of CFO estimation at lower SNR levels, in comparison with higher SNR levels, leads to a decrease in the performance of the transmitted signal estimation. On the other hand, the inverse is also true, where higher SNR levels and improved CFO estimation contribute to enhanced performance in signal estimation. Table 6 provides an overview of the RF impairments present in each of the signals presented in Figure 5.  
Figure 5: Transmitted signal estimation results. (**a**) A randomly selected estimation result of the transmitted signal at 0 dB. (**b**) A randomly selected estimation result of the transmitted signal at 10 dB. (**c**) A randomly selected estimation result of the transmitted signal at 20 dB. While the estimation performance of the transmitted signal is indeed satisfactory at high SNR levels, it is important to note that it is not solely determined by the SNR. Figure 6a provides a clear example of this, where, despite an SNR of 20 dB, the transmitted signal estimation performance is poor. This can be attributed to a phase offset of 45 degrees, as the QPSK symbol constellations become indistinguishable from the baseband signal observations at \(\pm 45\) degrees. Furthermore, it is essential to acknowledge the presence of outliers in the estimation results. Figure 6b highlights cases where the transmitted signal estimation performance is poor, particularly when there is no phase offset of \(\pm 45\) degrees. These outliers, although relatively infrequent, do not significantly impact the overall system performance. We demonstrate this by presenting the results that follow. Table 7 provides a breakdown of the RF impairments present in each of the signals featured in Figure 6.  
The assessment of the overall performance of the U-Net-based autoencoder neural network cannot be effectively conducted by visually inspecting individual pairs of true and estimated transmitted signals. To address this, we adopt a quantitative approach by computing the residuals between the true transmitted signal and the received signal, as \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Subfigure** & **True CFO** & **CFO Est.** & \
\begin{tabular}{c} **1 True CFO** \\ **– Est. CFO** \\ \end{tabular} & **SNR (dB)** &
\begin{tabular}{c} **Phase Offset** \\ **(degrees)** \\ \end{tabular} \\ \hline (a) & 0.5143 & 0.3229 & 0.1914 & 0 & 15 \\ (b) & 0.5429 & 0.4987 & 0.0442 & 10 & -25 \\ (c) & 0.4768 & 0.4827 & 0.0059 & 20 & 35 \\ \hline \hline \end{tabular} CFO values are rounded to 4 decimal places.  
\end{table} 
Table 6: Radio Frequency (RF) impairments that are present in each of the signals provided in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Subfigure** & **True CFO** & **CFO Est.** & \
\begin{tabular}{c} **1 True CFO** \\ **– Est. CFO** \\ \end{tabular} & **SNR (dB)** &
\begin{tabular}{c} **Phase Offset** \\ **(degrees)** \\ \end{tabular} \\ \hline (a) & 0.9196 & 0.9200 & 0.0004 & 20 & 45 \\ (b) & 0.1057 & 0.1639 & 0.0582 & 20 & \(-15\) \\ \hline \hline \end{tabular} CFO values are rounded to 4 decimal places.  
\end{table} 
Table 7: RF impairments that are present in each of the signals provided in Figure 6.  
Figure 6: Transmitted signal estimation results—outliers. (**a**) Example of the estimated transmitted signal at 20 dB with a phase offset of 45 degrees. (**b**) Outlier example of the estimated transmitted signal at 20 dB with a phase offset that is not \(\pm 45\) degrees.