## 5 Numerical Results
### Experimental Settings and Performance Metric  
The dataset consists of signals modulated by BPSK, QPSK, 8 PSK, 16 QAM, 64 QAM, and 256 QAM, with the SNRs ranging within \(\{-5,0,5,10,15,20,25\}\) dB. For each case in the combination set of such modulation schemes and SNR conditions, 2000 samples are generated; therefore, the complete dataset comprises 84,000 samples in total. To ensure effective model training and evaluation, a dataset is constructed with training, validation, and test subsets with the proportion of 7:2:1, respectively. The training phase spans 80 epochs with a batch size of 400 samples per iteration, optimized by the Adam algorithm with a learning rate of 0.0035. More experiment settings are listed in Table 1.  
The experiments are conducted on a device with a single NVIDIA RTX 4090 GPU (24 GB), an Intel Xeon Gold 6430 CPU (16 cores), and 128 GB RAM. With the specified configurations, model training takes an average of approximately 7.2 min, with each epoch spanning an average of 5.4 s. With a trained model, it takes 0.054 ms for modulation classification of a signal. According to the above time costs, the proposed model can be implemented in real-time wireless operations. First, once trained, the model can be utilized for an extended period and does not need to be retrained unless the network undergoes significant changes. Second, 0.054 ms is an acceptable operation duration compared to the intervals (e.g., typically 0.1\(\sim\)10 ms in different cases) between two modulation classifications in real wireless systems.  
In the experiments, without loss of generality, the pilot values within the pilot matrix are kept identical for simplification. The values adopted for different modulation types are listed in Table 2.  
The performance of modulation classification is evaluated based on the accuracy, precision, recall, and F1-score metrics. Let the entire sample set be divided into four subsets based on the true labels and predicted outcomes, i.e., the sets of true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN); then, the four metrics are computed as follows:  
The accuracy quantifies the proportion of correctly classified samples in the entire dataset and is defined by  
\[\text{Accuracy}=\frac{\text{TP}+\text{TN}}{\text{TP}+\text{TN}+\text{FP}+ \text{FN}}. \tag{11}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Delay-Doppler grid size & N = 32, M = 64 \\ Pilot symbol dimensions & 3 \(\times\) 3 \\ Guard interval lengths & 2 \\ Sampling rate (kHz) & 100 \\ Maximum Doppler shift (Hz) & 1000 \\ Carrier frequency (GHz) & 5 \\ Channel model & Extended vehicular A model (EVA) \\ Modes of modulation & BPSK, QPSK, 8 PSK, 16 QAM, 64 QAM, 256 QAM \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Modulation Type** & **Pilot Type** & **Value** \\ \hline BPSK & Real number & 2 \\ QPSK & Complex number & \(1+j\) \\
8 PSK & Phase rotation & \(e^{j\pi/4}\) \\
16 QAM & Complex number & \(1.5+1.5j\) \\
64 QAM & Complex number & \(2+2j\) \\
256 QAM & Real number & 2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pilot signal configurations.  
The precision measures the proportion of correctly predicted positive instances among all the predicted positive instances:  
\[\text{Precision}=\frac{\text{TP}}{\text{TP}+\text{FP}}. \tag{12}\]  
The recall measures the proportion of correctly predicted positive instances among all the actual positive instances:  
\[\text{Recall}=\frac{\text{TP}}{\text{TP}+\text{FN}}. \tag{13}\]  
The F1-score is the harmonic mean of precision and recall, providing a balanced evaluation metric:  
\[\text{F1}=2\times\frac{\text{Precision}\times\text{Recall}}{\text{Precision}+ \text{Recall}}. \tag{14}\]