# Neural-Network-Based Synchronization Acquisition with Hankelization Preprocessing
## 3 Proposed Method
### NN Design via Hankelization-Based Preprocessing  
As previously discussed, although the rank-1 property of the Hankel matrix \(\mathbf{Z}\) is no longer satisfied due to noise-induced variations and the intrinsic correlation of the ZCsequence, \(\mathbf{Z}\) retains useful information related to the synchronization signal. Therefore, we leverage its singular values to construct an efficient latent space.  
Now, we propose an integrated framework that combines Hankelization and NN utilization to identify the low-dimensional representation of complex data. To achieve this, we define the required terms as follows:  
* \(\overline{\boldsymbol{\sigma}}\in\mathbb{R}_{+}^{P}\): The vector of singular values of the Hankel matrix \(\mathbf{Z}\), which is sorted in descending order.
* \(\mathbf{e}\in\{[1,0]^{T},[0,1]^{T}\}\): The one-hot encoding vector representing whether the synchronization signal is present or absent, where \([\cdot]^{T}\) denotes the transpose operator. We set the presence and absence of the synchronization signal as \([1,0]^{T}\) and \([0,1]^{T}\), respectively.
* \(\mathcal{L}\): The loss function, e.g., the mean squared error (MSE).
* \(\Theta\): The model parameter, i.e., weight matrices and bias vectors.
* \(\
abla_{\Theta}\): The gradient operator with respect to \(\Theta\).
* \(P_{L}\): The latent space dimension.
* \(d\): The NN model depth.
* \(M\): The size of the training dataset.
* \(N\): The size of the test dataset.  
With these terms, we define the fully-connected (FC) NN model denoted by \(\mathcal{N}\). Finally, we can define the optimized NN model, denoted by \(\mathcal{N}^{*}\), as follows:  
\[\mathcal{N}^{*}=\operatorname*{arg\,min}_{\mathcal{N}}\sum_{i=1}^{M}\mathcal{ L}(\mathbf{e}^{(i)},\mathcal{N}(\overline{\boldsymbol{\sigma}}^{(i)})), \tag{8}\]  
where \(i\) is the index of the training dataset. In addition, we use the MSE as the loss function \(\mathcal{L}\). We choose the stochastic gradient descent with momentum (SGDM) as the optimizer for the update rule for the model; thus, the parameter \(\Theta_{t}\) at iteration \(t\) is computed as follows:  
\[\Theta_{t+1}\leftarrow\Theta_{t}-\eta\sum_{i\in\mathcal{B}_{t}}\
abla_{\Theta }\mathcal{L}(\mathbf{e}^{(i)},\mathcal{N}(\overline{\boldsymbol{\sigma}}^{(i )}))+\gamma(\Theta_{t}-\Theta_{t-1}), \tag{9}\]  
where \(\eta\) is the learning rate, \(\mathcal{B}_{t}\) represents the set containing the indices of the input/output pairs in the current mini-batch, and \(\gamma\) is the momentum coefficient that determines the contribution of the previous gradient step to the current update. Through the procedures of (8) and (9), we can optimize the NN model. Now, we can use the optimized NN model for the hypothesis testing of binary classification as follows:  
\[\hat{\mathbf{e}}[0]\overset{Absent}{\underset{Present}{\overset{\eqref{eq:NN}}{ \underset{Present}{\Leftrightarrow}}}}\hat{\mathbf{e}}[1],\text{ where }\hat{\mathbf{e}}=\mathcal{N}^{*}(\overline{ \boldsymbol{\sigma}}). \tag{10}\]  
Based on (10), we now evaluate the performance of the optimized NN model \(\mathcal{N}^{*}\) through numerical simulations in the next section.  
To analyze the computational efficiency of the proposed method, considering the Hankel matrix dimensions \(P\), the latent space dimension \(P_{L}\), and the NN model depth \(d\), the proposed method requires a floating-point operation (FLOP) complexity \(\mathcal{O}(dP_{L})\) in terms of online complexity for an FC model's operation. This complexity is comparable to, or even lower than, that of conventional NN models. However, the proposed method incurs an additional online complexity of \(\mathcal{O}(P^{3})\) FLOP for singular value extraction. Although this additional cost increases computational demand, it improves overall performance.