## 2 Proposed Framework  
### System Model and Problem Formulation  
Consider the received signal \(\mathbf{r}\in\mathbb{C}^{P}\) for a single-input single-output (SISO) system, represented as  
\[\mathbf{r}[n]=e^{i(2\pi f_{err}nT_{s}+\theta_{err})}\sum_{l=0}^{L-1}\mathbf{h}[l ]\;\mathbf{s}[n-l-\zeta_{err}]+\mathbf{w}[n],\forall n\in\{0,1,\dots,P-1\}, \tag{1}\]  
where \(L\) denotes the number of multipaths, \(\mathbf{s}\) is the transmitted modulation symbol sequence, \(f_{err}\) is the frequency offset, \(\theta_{err}\) represents the initial phase error, \(T_{s}\) is the sampling rate, \(\mathbf{h}\) is the channel impulse response (CIR), \(\zeta_{err}\) is the timing offset, and \(\mathbf{w}\) is additive white Gaussian noise (AWGN) [20]. The objective is to reliably classify the modulation types of received signals embedded in noisy environments. To address this, we first define a Hankelization operator \(\hbar:\mathbb{C}^{P\times 1}\mapsto\mathbb{C}^{P_{T}\times P_{T}}\), which transforms a vector \(\mathbf{x}\) into a Hankelized matrix \(\mathbf{X}\) as follows:  
\[\mathbf{X}=\hbar(\mathbf{x})\;\text{s.t.}\;\mathbf{X}[i,j]=\mathbf{x}[i+j-1], \forall i,j\in\{1,2,\cdots\}, \tag{2}\]  
where \(P_{T}=\lfloor(P+1/2)\rfloor\) and \(P_{T}^{\prime}=P-P_{T}+1\) denote the row and column dimensions of \(\mathbf{X}\), respectively, with \(\lfloor\cdot\rfloor\) indicating the floor operation. We adopt a Hankelized matrix structure that closely approximates a square matrix. In our proposed framework, we consider two choices for the input vector \(\mathbf{x}\) to the Hankelization operator \(\hbar\). First, we apply \(\hbar\) directly to the received time-domain signal \(\mathbf{r}\) to obtain \(\mathbf{R}_{t}=\hbar(\mathbf{r})\). Second, we apply the fast Fourier transform (FFT) operator \(\mathcal{F}\) to \(\mathbf{r}\), resulting in \(\tilde{\mathbf{r}}=\mathcal{F}(\mathbf{r})\), and construct \(\mathbf{R}_{f}=\hbar(\tilde{\mathbf{r}})\). This frequency-domain representation is expected to highlight modulation-dependent spectral characteristics, thereby enhancing the separability of different modulation types, especially under noisy conditions. For notational convenience, we denote the resulting Hankelized matrix as \(\mathbf{R}_{d}\), where \(d\in\{t,f\}\) represents the time-domain \(t\) and frequency-domain \(f\) variants, respectively. We then apply SVD to \(\mathbf{R}_{d}\) as  
\[\mathbf{R}_{d}=\mathbf{U}_{d}\boldsymbol{\Sigma}_{d}\mathbf{V}_{d}{}^{H}, \tag{3}\]  
where \(\mathbf{U}_{d}\in\mathbb{C}^{P_{T}\times P_{T}}\) and \(\mathbf{V}_{d}\in\mathbb{C}^{P_{T}^{\prime}\times P_{T}^{\prime}}\) are the matrices consisting of \(P_{T}\) left and \(P_{T}^{\prime}\) right singular vectors of \(\mathbf{R}_{d}\), respectively. Also, \(\boldsymbol{\Sigma}_{d}\in\mathbb{R}^{P_{T}\times P_{T}^{\prime}}\) is a diagonal matrix whose diagonal elements are the descending ordered singular values, i.e., \(\boldsymbol{\sigma^{(i)}}_{d}\), \(\forall i\in\{1,\cdots,P_{T}\}\). We assume that the extracted singular values \(\boldsymbol{\sigma^{(i)}}_{d}\) can effectively form a discriminative latent space suitable for the modulation classification task. To empirically support this assumption, we generate noise-free samples of various modulation types in the MATLAB R2024b environment and visualize their feature distributions using t-distributed stochastic neighbor embedding (t-SNE), a nonlinear dimensionality reduction technique that transforms high-dimensional data into a low-dimensional representation [21]. As shown in Figure 1a, the absolute values of the signal show relatively dispersed and overlapping patterns across different modulation types. In contrast, the singular values derived from the Hankelized matrix in Figure 1b,c exhibit more compact distributions with improved separation between modulation categories, particularly in Figure 1c. These results suggest that the singular values better capture the intrinsic structure of the signal representations.