## 2 Materials and Methods
### System Design  
As shown in Equation (1), RF impairments, such as CFO, phase offset, and AWGN, affect the transmitted signal, causing deviations in the received signal. Addressing and compensating for these impairments enables accurate estimation of the transmitted signal from the received one. We propose a multi-stage DL system that combines explicit and implicit estimation approaches to address these impairments. A dedicated neural network (NN) model estimates the CFO, and this estimation is used with signal processing techniques to mitigate its effect on the received signal. For the remaining impairments, such as phase offset and AWGN, an implicit estimation approach is adopted. Another dedicated neural network jointly estimates and corrects these impairments by capturing complex patterns and dependencies in the data. Training on a moderately diverse dataset, the network learns to implicitly estimate and compensate for the combined effect of phase offset and noise. By referring to Figure 1 and following the sequential numbering assigned to each line, readers can effortlessly navigate and comprehend the system design. The system begins by estimating the CFO using a dedicated fully convolutional network (FCN). The estimated CFO is then utilized in the CFO correction block, where the received signal \(\mathbf{y}\) is multiplied by \(e^{-j2\pi/\hat{\mathbf{z}}n}\) to obtain an intermediate signal \(\hat{\mathbf{z}}\), which undergoes a certain level of CFO correction. This stage is known as coarse CFO estimation and correction. Subsequently, the same FCN is employed, with the previous network's output, i.e., the CFO-corrected intermediate signal \(\hat{\mathbf{z}}\), serving as the input. This stage, referred to as fine CFO estimation and correction, further refines the CFO estimation and yields the signal \(\hat{\mathbf{z}}\), which represents a well-CFO-corrected signal. The coarse estimation result is denoted as \(\hat{f}_{\Delta}\), while the fine estimation result is denoted as \(\hat{f}_{\Delta}\). Afterwards, a U-Net [36]-based autoencoder, known for its effectiveness in noise mitigation applications [37; 38; 39], is employed to compensate for phase offset and denoise the signal. Next, the energies of both the received signal \(\mathbf{y}\) and the estimated transmitted signal \(\hat{\mathbf{x}}\) are computed, as depicted in Figure 1. These energy values are utilized to determine the energy of the noise \(E_{\hat{\mathbf{w}}}\). Finally, the estimated noise energy is multiplied by a factor \(k\), and together with \(E_{y}\), it is employed in the binary hypothesis testing to make a decision between \(H_{0}\) and \(H_{1}\). Determining the value of parameter \(k\) through an exhaustive search, covering all real values from one to infinity, and selecting the optimal \(k\) where the probabilities of false alarm and missed detection intersect is theoretically valid. However, implementing such an approach is computationally infeasible. Therefore, we employ a practical and empirical method to determine an appropriate value for \(k\). Specifically, we create a vector of 200 evenly spaced numbers ranging from 1 to 1.5, with the upper limit of 1.5 chosen based on empirical analysis. This approach allows for an efficient and effective selection of the parameter \(k\). Each element of this vector is substituted for \(k\), and the corresponding  
Figure 1: System design diagram. false alarm and missed detection rates are evaluated. The values of \(k\) that resulted in the intersection of the false alarm and missed detection rates are identified as the most favorable choice, as demonstrated in Figure 2. The average central processing unit time required to perform this process is approximately 800 ms. While recurrent neural networks (RNNs) are commonly used for time series problems, CNNs and FCNs have also proven effective [40; 41; 42; 43]. In our system, we employ FCNs for CFO estimation due to their ability to detect and combine complex patterns, making them suitable for this task.