## 2 Proposed Framework
### NN Design via Hankelization-Based Preprocessing
The dataset configuration, including channel model parameters and signal generation settings, is summarized in Table 1. Additionally, Table 2 presents the default experimental setup, including the NN architecture, hyperparameters, and computational environment. The performance metric employed in this paper is the detection rate, defined as the ratio of correctly classified samples to the total number of test samples \(N\). The detection rate is computed as

\[\text{detection rate}=\frac{\text{card}\{n\mid\hat{c}^{(n)}=c^{(n)}\}}{N}, \forall n\in\{1,\cdots,N\}, \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Channel model & Rician fading \\ \hline \(K\)-factor & 4 dB \\ \hline Sampling rate \(T_{s}\) & 200 kHz \\ \hline Number of multipaths \(L\) & 3 \\ \hline Doppler frequency & max = 1 Hz \\ \hline Initial phase error \(\theta_{err}\) & \(U(0,2\pi)\) \\ \hline Frequency offset \(f_{err}\) & Standard deviation per sample = 0.01, \\ Maximum deviation: 500 Hz \\ \hline Timing offset \(\zeta_{err}\) & Standard deviation per sample = 0.01, \\ Maximum deviation: 500 Hz \\ \hline Number of samples per modulation per SNR & 1000 \\ \hline Length of samples & 128 \\ \hline SNR & \(-20\) to 20 dB in steps of 2 dB \\ \hline \end{tabular}
\end{table} 
Table 1: Settings of the RadioML2016.10a dataset [20,23]. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Data split (per modulation class) & 70\% training/10\% validation/20\% testing \\ \hline NN depth \(d\) & 3 \\ \hline NN width \(P_{W}\) & 24 \\ \hline Epochs & 120 \\ \hline Mini-batch size & 64 \\ \hline NN connection type & FC \\ \hline Learning rate & \(10^{-2}\) \\ \hline Activation function & Hyperbolic tangent function \\ \hline Loss function & Cross-entropy \\ \hline Optimizer & Adam \\ \hline \(\beta_{1}\), \(\beta_{2}\) & 0.9, 0.99 \\ \hline \(\epsilon\) & \(10^{-8}\) \\ \hline Experimental setup & Python 3.10.16, PyTorch 2.5.1, CUDA 12.1, cuDNN 9.1.0, \\  & NVDIA RTX A4000 GPU, Intel Core i9-11900K CPU \\ \hline \end{tabular}
\end{table} 
Table 2: Default configuration in the experiments. where \(\text{card}\{\cdot\}\) denotes the cardinality of the set. This metric reflects the classification accuracy across all test samples. The effectiveness of the proposed method was assessed through comparison with four baseline methods:

* Deep learning (CNN with real and imaginary values of the signal: The NN-based method that utilizes a CNN architecture, in which the real and imaginary parts of the input signal are treated as two separate input channels. This structure follows the design presented in [24]. * SCNN2: The NN-based method that transforms raw complex signals into spectrogram images via discrete short-time Fourier transform (STFT), applies Gaussian filtering for noise reduction, and performs classification using a dedicated CNN architecture optimized for time-frequency representations, as described in [14]. * CLDNN: The NN-based method that combines convolutional layers for local feature extraction, long short-term memory (LSTM) layers for modeling temporal dependencies, and FC layers for classification. This hybrid architecture leverages both spatial and temporal information embedded in the received signal, following the design principles in [16]. * LSTMDAE: The NN-based method that employs a denoising autoencoder (DAE) based on LSTM networks, which learns robust latent representations of noisy signals through temporal masking and reconstruction. The decoder output is jointly optimized with a modulation classification objective, thereby improving classification performance under noisy conditions, as introduced in [25]. Among the 11 modulation types in the RadioML2016.10a dataset, we select BPSK, QPSK, 16 QAM, and 64 QAM for evaluation. These schemes are chosen to cover both phase modulation and amplitude-phase combined modulation, allowing us to assess classification performance using a simple NN and minimal preprocessing.