# Neural-Network-Based Synchronization Acquisition with Hankelization Preprocessing
## 3 Proposed Method
### Effectiveness of the Proposed Method Related to the Low Rank Properties of a Hankelized Matrix  
The inverse fast Fourier transform (IFFT) of the correlation function, denoted as \(\mathbf{z}\in\mathbb{C}^{2P-1}\), is reconstructed into a Hankel matrix \(\mathbf{Z}\in\mathbb{C}^{P\times P}\), as given below:  
\[\mathbf{Z}[i,j]=\mathbf{z}[i+j-1],\quad\forall i,j\in\{1,2,\ldots\}. \tag{2}\]  
Figure 1: An illustration of an example of the proposed framework. Here, \(\mathbf{z}\) is derived from the cross-correlation between the received noisy signal \(\mathbf{y}\) and the ZC sequence \(\mathbf{x}_{K}\). The cross-correlation function is expressed as  
\[\mathcal{R}_{\mathbf{y}\mathbf{x}_{K}}[m]=\sum_{p=0}^{P-1}\mathbf{y}[p]\cdot \mathbf{x}_{K}[p+m],\quad\forall m\in\{-P+1,\ldots,P-1\}. \tag{3}\]  
and the IFFT is applied to \(R_{\mathbf{y}\mathbf{x}_{K}}[m]\), resulting in \(\mathbf{z}\) used to construct the Hankel matrix:  
\[\mathbf{z}[n]=\frac{1}{2P-1}\sum_{m=0}^{2P-2}R_{\mathbf{y}\mathbf{x}_{K}}[m] \cdot e^{i\frac{2\pi m}{2P-1}},\quad\forall n\in\{0,1,\ldots,2P-2\}. \tag{4}\]  
The sequence \(\mathbf{z}\) captures the correlation properties between the received signal \(\mathbf{y}\) and the ZC sequence \(\mathbf{x}_{K}\). When used to construct the Hankel matrix \(\mathbf{Z},\mathbf{z}\) generally results in a full-rank matrix. This is caused not only by noise-induced variations in practical scenarios but also by the intrinsic correlation characteristics of the ZC sequence, which can introduce small residual variations even under ideal noise-free conditions. Despite being full-rank, \(\mathbf{Z}\) retains significant information about the synchronization signal. Specifically, the dominant periodic patterns of the reference sequence, which are critical for synchronization, remain embedded within the structure of \(\mathbf{Z}\). To validate this, we analyze an idealized scenario where the received signal \(\mathbf{y}\) is identical to the \(\mathbf{x}_{K}\). In this ideal case, the cross-correlation function \(\mathcal{R}_{\mathbf{y}\mathbf{x}_{K}}\) simplifies to the autocorrelation of \(\mathbf{x}_{K}\), dominated by its periodic structure. The dominant periodic component is characterized by the peak index \(f_{p}\) of the autocorrelation function:  
\[f_{p}=\arg\max_{m}\Bigl{|}R_{\mathbf{y}(=\mathbf{x}_{K})\mathbf{x}_{K}}[m] \Bigr{|}. \tag{5}\]  
Based on (5), an ideal sequence \(\mathbf{s}\in\mathbb{C}^{2P-1}\) can be expressed as \(\mathbf{s}[n]=e^{j2\pi nf_{p}/(2P-1)}\) for \(n\in\{0,1,\ldots,2P-2\}\). From this sequence, a Hankel matrix \(\mathbf{S}\in\mathbb{C}^{P\times P}\) is constructed following (2), and in the ideal case, \(\mathbf{S}\) exhibits a rank-1 property. To validate this claim, we analyze the row structure of \(\mathbf{S}\). From (2), the \(u\)-th row of \(\mathbf{S}\) can be represented as follows:  
\[\mathbf{S}[u,:]=e^{j2\pi nf_{p}/(2P-1)}\cdot\mathbf{v}, \tag{6}\]  
where \(\mathbf{v}=[1,e^{j2\pi f_{p}/(2P-1)},e^{j4\pi f_{p}/(2P-1)},\ldots,e^{j2\pi(P- 1)f_{p}/(2P-1)}]\) serves as the basis vector for all rows of \(\mathbf{S}\). This indicates that all rows of \(\mathbf{S}\) are constant multiples of the basis vector \(\mathbf{v}\). Specifically, any two rows \(\mathbf{S}[u,:]\) and \(\mathbf{S}[v,:]\) satisfy the following relationship:  
\[\mathbf{S}[u,:]=e^{j2\pi f_{p}(u-v)/(2P-1)}\cdot\mathbf{S}[v,:],\quad\forall u,v\in\{1,2,\ldots,P\}. \tag{7}\]  
Since all rows are scalar multiples of the same basis vector, \(\mathbf{S}\) has a rank-1 in the ideal case. In accordance with the symmetry, the same property holds for the columns of \(\mathbf{S}\). Thus, \(\mathbf{S}\) is confirmed to exhibit a rank-1 structure under ideal conditions. This ideal rank-1 structure of \(\mathbf{S}\) ensures that the underlying synchronization signal is well represented in the matrix. As noted, in practical scenarios, the presence of noise and residual components causes the Hankel matrix \(\mathbf{Z}\), constructed from the noisy signal, to deviate from the ideal rank-1 property.