# Neural-Network-Based Synchronization Acquisition with Hankelization Preprocessing
## 4 Simulation Results  
### Simulation Configurations  
The parameters configured for the simulations in this paper are summarized in Table 1. The ZC sequence utilized requires the sequence length and root index as key parameters. For the simulations, we set the sequence lengths to 37 and 41 and observed the resulting performance. The configurations of the NN used in the simulations are also detailed in the table above. Instead of focusing on the complexity of the NN structure, the emphasis of this study is on extracting and utilizing key information from the data to enhance detection performance. Therefore, a relatively simple NN architecture was adopted, ensuring that the model remains efficient while effectively leveraging the preprocessed features from the proposed method.  
The metric for the simulations is the detection error rate, which quantifies the ratio of incorrectly classified samples to the total number of samples in the test dataset of size \(N\). For each test sample \(j\), the true label vector \(\mathbf{e}^{(j)}\) is represented as a one-hot encoded vector. The predicted label vector for the same sample is denoted by \(\hat{\mathbf{e}}^{(j)}\). The detection error rate is defined as the complement of the classification accuracy:  
\[\text{detection error rate}=1-\frac{\text{card}\{j\mid\hat{\mathbf{e}}^{(j)}_{ H}=\mathbf{e}^{(j)}\}}{N}, \tag{11}\]  
where \(\text{card}\{\cdot\}\) represents the cardinality of the set. Additionally, \(\hat{\mathbf{e}}_{H}\) is \([H(\hat{\mathbf{e}}[0]-\hat{\mathbf{e}}[1])\;\;H(\hat{\mathbf{e}}[1]-\hat{ \mathbf{e}}[0])]^{T}\) for hard decisions via the Heaviside function \(H(\cdot)\). This metric provides a clear and quantitative measure of detection performance across different configurations.  
To evaluate the performance of the proposed method, two baseline schemes were used for comparison:  
* Baseline 1 (Deep learning method based on cross-correlation): The NN-based method employing absolute correlation values as in conventional methods.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Length of sequence (\(P\)) & 37, 41 \\ \hline Root index (\(K\)) & A randomly selected prime number under \(P\) \\ \hline Size of training dataset & \(10^{4}\) \\ \hline Size of test dataset & \(10^{4}\) \\ \hline NN depth \(d\) & 2 \\ \hline NN width & 24 \\ \hline Max. epochs & 30 \\ \hline NN connection type & Fully-connected \\ \hline Learning rate & \(10^{-3}\) \\ \hline Activation function & Leaky ReLU \\ \hline Loss function & MSE \\ \hline Optimizer & SGDM \\ \hline Momentum & 0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Default configuration in the experiments.  
* Baseline 2 (Deep learning method based on IFFT-processed cross-correlation): The NN-based method employing absolute values of IFFT-applied correlation results as in conventional methods.
* Baseline 3 (Random forest method): The ML-based method that classifies signals using statistical features extracted from the absolute values of IFFT-processed cross-correlation results, following the approach in [16].
* Baseline 4 (Adaptive threshold method): A non-ML-based approach that detects signals by applying an adaptive threshold to the absolute values of IFFT-processed cross-correlation results based on estimated noise levels, following the approach in [25].