## 2 Signal Model and Related Work  
The modulated symbols \(\{X_{k}\}_{k=0}^{N-1}\) at \(N\) sub-carriers are processed with an \(N\)-point inverse fast Fourier transform (IFFT) to produce the OFDM symbol as given by  
\[x_{n}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X_{k}\cdot e^{i\frac{2\pi nf}{N}nk}. \tag{1}\]  
Taking the additive white Gaussian noise (AWGN), frequency-selective multi-path channel, the carrier frequency offset (CFO) and the NBI into consideration, the received signal can be represented as,  
\[r_{n}=\sum_{l=0}^{L_{h}-1}h_{l}\cdot x_{n-n_{0}-l}\cdot e^{i2\pi nf_{\rm c}}+l_{ 0}\cdot e^{i2\pi nf_{\rm NB}}+\
u_{n}, \tag{2}\]  
where \(n_{0}\), \(f_{\rm c}\) and \(\
u_{n}\) are the unknown time of symbol arrival (which is the desired correct timing position to be estimated by the receiver), CFO and AWGN, respectively [21]. The multi-path channel impulse response (CIR) is denoted by \(\{h_{l}\}_{l=0}^{L_{h}-1}\), which is modeled by \(L_{h}\) delay taps. The NBI amplitude at the frequency point \(f_{\rm NB}\) is denoted by \(I_{0}\). Normally, the CFO is normalized by the sub-carrier spacing \(1/N\), i.e., \(f_{\rm c}=k_{0}/N+f_{\rm frc}\), where \(k_{0}\) is an integer representing the integer part of the CFO, and the remaining fractional part of the CFO is denoted by \(f_{\rm frc}\). The target of the synchronization process at the receiver is estimating the correct timing position and determining the CFO without knowledge of prior channel state information. There have been some conventional methods proposed by [PERSON] et al. [16] and [PERSON] et al. [17]. Specifically, to achieve this goal, as illustrated in Figure 1a, [PERSON] proposed a preamble design whose time-domain structure consists of the two same components [16]. The two cyclic parts 'B' are used to calculate the SAC result for synchronization. However, because of the CP part, a wide plateau around the correct timing position would be generated in the SAC result, which would lead to ambiguity and inaccuracy in estimating the timing position. In order to cut down the plateau, a novel time-domain preamble as denoted by \([\mathbf{A}~{}~{}\mathbf{A}~{}-~{}\mathbf{A}~{}-~{}\mathbf{A}]\) was proposed by [PERSON], which is depicted in Figure 1b [17]. This deliberate design with two opposite components is capable of sharpening the SAC peak with respect to that of [PERSON]'s. Nevertheless, some sub-peaks are inevitably generated in the SAC result, leading to an increase in the probability of false detection, especially in the case of small SNR. Taking the NBI at the frequency point \(f_{\rm NB}\) into consideration, the SAC result of the conventional preamble based on two identical parts at the desired timing position \(n_{0}^{\prime}\) is  
Figure 1: Conventional state-of-the-art designs of the OFDM preamble for synchronization. given as (we first omit the noise and multipath channel terms for simplicity, and then the analysis can be easily extended to multi-path channels as described in Appendix A),  
\[ \begin{array}{l}R_{c,\mu_{0}^{\prime}}=\sum\limits_{l=0}^{L_{c}-1}r_{\mu_{0}^{ \prime}+l}\cdot\tau_{\mu_{0}+l+N_{c}}^{*}\\ =\sum\limits_{l=0}^{L_{c}-1}\left(x_{l}+I_{0}e^{j2\pi f_{\rm NB}l}\right)\ \cdot \left(x_{l}+I_{0}e^{j2\pi f_{\rm NB}(l+N_{c})}\right)^{*}\\ =\sum\limits_{l=0}^{L_{c}-1}\left(|x_{l}|^{2}+I_{0}x_{l}e^{-j2\pi f_{\rm NB}(l+ N_{c})}\right.\\ \left.+\,I_{0}x_{l}^{*}e^{2\pi f_{\rm NB}l}+I_{0}^{2}e^{-j2\pi f_{\rm NB}N_{c}} \right)\\ \approx\sum\limits_{l=0}^{L_{c}-1}|x_{l}|^{2}+L_{c}\cdot I_{0}^{2}e^{-j2\pi f_ {\rm NB}N_{c}}\end{array} , \tag{3}\]  
where \(N_{c}\) and \(L_{c}\) denote the length of the correlation lag and the identical components, and \((\cdot)^{*}\) represents the complex conjugation operation. Since the transmitted preamble \(\{x_{l}\}\) and the NBI \(\{e^{j2\pi f_{\rm NB}l}\}\) are non-coherent signals, the cross terms in (3) are eliminated after sum averaging and can be neglected with respect to the preamble SAC component \(|x_{l}|^{2}\) and the significant NBI component \(L_{c}I_{0}^{2}e^{-j2\pi f_{\rm NB}N_{c}}\).