## 2 Proposed Framework
### NN Design via Hankelization-Based Preprocessing  
Now, we propose a harmonized framework with Hankelization and NN utilization. The relevant terms are defined as follows:  
* \(\mathcal{L}\): The loss function. * \(\Theta\): The collection of all trainable parameters, including weight matrices and bias vectors. * \(\
abla_{\Theta}\): The parameter gradient computed with respect to \(\Theta\). * \(C\): The number of modulation classes. * \(M\): The number of training datasets. * \(N\): The number of test datasets. Let \(c^{(m)}\in\{1,2,\dots,C\}\) denote the ground truth modulation class index for the \(m\)-th training sample. The corresponding one-hot encoding vector \(\mathbf{e}^{(m)}\in\{0,1\}^{C}\) is defined as:  
\[\mathbf{e}^{(m)}[c]=\left\{ \begin{array}{ll}1&\text{if }c=c^{(m)},\\ 0&\text{otherwise}.\end{array} \right. \tag{5}\]  
For notational convenience, we define the vector of singular values \(\sigma_{d}^{(m)}\in\mathbb{R}_{+}^{P_{T}}\) of the Hankelized matrix as follows:  
\[\sigma_{d}^{(m)}=\text{diag}(\Sigma_{d}). \tag{6}\]  
We denote the fully connected (FC) NN model by \(\mathcal{N}_{p}\), which is defined using the above terms. Then, the optimized NN model, denoted by \(\mathcal{N}_{p}^{*}\), is defined as follows:  
\[\mathcal{N}_{p}^{*}=\underset{\mathcal{N}_{p}}{\text{arg min}}\ \sum_{m=1}^{M}\ \mathcal{L}(\mathbf{e}^{(m)},\mathcal{N}_{p}(\sigma_{d}^{(m)})), \tag{7}\]  
where \(\mathcal{L}\) is the loss function, and we use the cross-entropy in this work. Figure 3: Overview of the architecture underlying the proposed framework. To optimize NN model, we adopt the adaptive moment estimation (Adam) optimizer as the update rule for the model; thus, the parameter \(\Theta_{t}\) at iteration \(t\) is computed as follows:  
\[\Theta_{t+1}\leftarrow\Theta_{t}-\eta\cdot\frac{\mathbf{m}_{t+1}}{\sqrt{\mathbf{ v}_{t+1}+\epsilon}}, \tag{8}\]  
\[\mathbf{m}_{t+1}\leftarrow\beta_{1}\mathbf{m}_{t}+(1-\beta_{1})\sum_{m\in \mathcal{B}_{t}}\
abla_{\Theta}\Big{(}\mathcal{L}(\mathbf{e}^{(m)},\mathcal{N }_{p}(\boldsymbol{\sigma}_{d}^{(m)};\Theta_{t}))\Big{)}, \tag{9}\]  
where \(\mathcal{B}_{t}\) refers to the set of indices associated with the input/output samples in the current minibatch. The learning rate is denoted by \(\eta\), while \(\mathbf{m}\) and \(\mathbf{v}\) represent the first and second moment estimates, respectively. And \(\beta_{1}\) and \(\beta_{2}\) are the exponential moving average of momentum and the exponential moving average of root mean square propagation (RMSProp), respectively. Now, in accordance with (7), (8), and (9), we can obtain the optimized \(\mathcal{N}_{p}^{*}\). Then, the result vector \(\hat{\mathbf{e}}\) from \(\boldsymbol{\sigma}_{d}\) can be obtained as follows:  
\[\hat{\mathbf{e}}=\mathcal{N}_{p}^{*}(\boldsymbol{\sigma}_{d}). \tag{10}\]  
Finally, the predicted modulation class index \(\hat{c}\) is determined as the index corresponding to the maximum value of the NN output vector:  
\[\hat{c}=\arg\max_{c}\hat{\mathbf{e}}[c]. \tag{11}\]  
Algorithm 1 shows the procedure of the proposed method. The performance of the optimized NN model \(\mathcal{N}_{p}^{*}\) is evaluated through numerical simulations in the following section. ```
1:\(\text{Collect training dataset }\{((\mathbf{r}^{(m)},\mathbf{e}^{(m)}))_{m=1}^{M}\). 2:\(\text{Select domain }d\in\{t,f\}\) for all samples: \(d=t\) (time-domain) or \(d=f\) (frequency-domain via FFT). 3:for\(m\gets 1\)to\(M\)do
4: Build Hankelized matrix: \(\mathbf{R}_{d}^{(m)}= \begin{cases}\hbar(\mathbf{r}^{(m)})&\text{if }d=t,\\ \hbar(\mathcal{F}(\mathbf{r}^{(m)}))&\text{if }d=f.\end{cases} \)
5: Compute SVD based on (3): \(\mathbf{R}_{d}^{(m)}=\mathbf{U}_{d}\boldsymbol{\Sigma}_{d}\mathbf



## 3 Simulation Results

### Simulation Configurations

The experiments in this study are conducted using the widely used RadioML2016.10a dataset introduced in [23], which is commonly adopted for evaluating AMC performance.