## 2 Background
### Frame Synchronization and Zadoff-Chu Sequences  
Frame detection based on the use of either pilot or preamble sequences known at the receiver is a classical strategy to achieve synchronization. Typically, these markers have proprieties that facilitate frame estimation detection [12]. Briefly, the frame beginning position is estimated by producing a correlation signal between received frame \(y\) and known \(L_{p}\) length preamble \(p_{n}\)[13] according to  
\[\mathrm{C}_{y,p_{n}}(l)=\sum_{m}y(m)p_{n}^{*}(l+m), \tag{3}\]  
where \(p_{n}^{*}(l)\) is the conjugate of the preamble sequence. Depending on the chosen preamble sequence and its proprieties, when \(y\approx p_{n}\), at least a high peak is produced. When performing autocorrelation, highest peak \(C_{max}\) is given by  
\[\mathrm{C}_{max}=\sum_{m}\lvert p_{n}(m)\rvert^{2}, \tag{4}\]  
where total correlation length is equal to \(2L_{p}-1\). When the frame's estimated starting point is obtained, it is important to decide if the marked position is indeed the beginning of the frame or, e.g., define a detection probability of the following events  
\[\left\{ \begin{aligned} \mathcal{H}_{0}&:\text{Signal absence}\\ \mathcal{H}_{1}&:\text{Signal detected}\end{aligned}  \Leftrightarrow\left\{ \begin{aligned} \mathcal{H}_{0}&:y=\eta\\ \mathcal{H}_{1}&:y=x_{n}+\eta\end{aligned} ,\right. \tag{5}\]  
where, for the absence hypothesis, the received signal is composed only by noise component \(\eta\) and the detected hypothesis is the superposition between noise and transmitted signal blocks \(x_{n}\). Based on the \(C_{max}\) and noise distribution, it is important to define a threshold, \(\delta_{decision}\), and a decision rule considering a length-\(2L_{p}\) sliding window analysis of (3) within each analysis. interval \(\mathcal{L}\)  
\[\left\{\begin{aligned} \text{select }\mathcal{H}_{1}& if \quad \begin{aligned} &\underset{l\in\mathcal{L}}{max}C_{y,p_{n}}(l)& >\delta_{decision}\\ \text{select }\mathcal{H}_{0}& if \quad\underset{l\in\mathcal{L}}{max}C_{y,p_{n}}(l)& <\delta_{decision}\end{aligned} .\right. \tag{6}\]  
The \(\delta_{decision}\) must be set in order to maximize the success frame detection probability (\(P_{D}\)) and reduce situations of either false alarms (\(P_{F}\)) or missed frame detection (\(P_{M}\)). Mathematically, the probabilities of these definitions can be expressed as  
\[P_{D}=P_{r}\{\text{Select }\mathcal{H}_{1}|\mathcal{H}_{1}\} \times P_{r}\{H_{1}\}+P_{r}\{\text{Select }\mathcal{H}_{0}|\mathcal{H}_{0}\}\times P_{r}\{H_{0}\}, \tag{7}\] \[P_{M}=P_{r}\{\text{Select }\mathcal{H}_{1}|\mathcal{H}_{0}\},\] (8) \[P_{F}=P_{r}\{\text{Select }\mathcal{H}_{0}|\mathcal{H}_{1}\}. \tag{9}\]  
An essential part for achieving reliable signal recovery is the choice of optimal preamble sequences. In the LTE standard, as part of the system acquisition process, two preamble sequences defined as PSS and secondary synchronization signals (SSS) are used in the downlink process [24]. Although these SSS sequences are based on maximum-lengthsequences, also known as M sequences, the PSS is composed by Zadoff-Chu sequences [14] of odd-length period \(N_{ZC}\), given by [15],  
\[zc_{n}[q]=e^{\frac{-|m\mu_{n}(q+1)}{N_{ZC}}},\quad q=0,1,\ldots,N_{ZC}-1, \tag{10}\]  
where \(n\) is the root index and \(n\in\{1,\ldots,N_{ZC}-1\}\). ZC sequences exhibit fundamental proprieties that aid the receiver in synchronization tasks [24; 25]:  
* constant amplitude that limits the peak-to-average-power ratio (PAPR);
* \"perfect\" cyclic autocorrelation, i.e., correlation with its circularly shifted version is zero at different samples from \(N_{ZC}\), and nonzero only at one instant, which corresponds to the \(N_{ZC}\) sample. The zero-autocorrelation ZC property can be formulated by (3), resulting in \[\mathrm{C}_{zc,zc}[l]=\sum_{m}zc[m]zc^{*}[m+l]=P_{zc}[l],\] (11) where \(P_{zc}\) corresponds to the autocorrelation peak positioned in the delayed l sample.