## 4 Timing and Carrier Frequency Synchronization through Preamble Detection
### Integer CFO Estimation and Signaling Detection  
The integer CFO can be estimated and the signaling conveyed through the frequency domain preamble can be detected when the receiver finishes the timing synchronization. The received OFDM symbol is first compensated by the estimated fractional CFO \(f_{\text{rc}}\), and then transformed to the frequency domain through \(N\)-point FFT, which yields the active carriers at the receiver given by  
\[\hat{Y}_{k}^{(d)}=Y_{k-k_{0}}^{(d)}H_{k-k_{0}}e^{-j\frac{2\pi}{N}\Delta n\cdot k }+N\cdot I_{0}\cdot\delta_{k-k_{\text{NB}}}+V_{k}, \tag{15}\]  
where \(Y_{k}^{(d)}\), \(H_{k}\), and \(\Delta n\) denote the transmitted active carrier after differential encoding operation, the channel frequency response, and the timing error which causes a phase rotation of the active carriers, respectively. \(\delta_{k}\) and \(k_{\text{NB}}\) denote the Kronecker delta function and the sub-carrier index at the frequency point \(f_{\text{NB}}\), respectively. The frequency domain noise term is denoted by \(V_{k}\). It is noted from (15) that a shift of all carriers would be caused by the integer part of CFO \(k_{0}\).  
The NBI's impact on determining the integer CFO and signaling can be eliminated as well. We only need to exclude the NBI contaminated sub-carrier through nulling the sub-carriers with excessively large power to zero. Afterward, the received active carries are applied by a differential decoding operation,  
\[\begin{split} Z_{k}&=\ \hat{Y}_{k}^{(d)}\cdot\hat{Y}_{k-2}^{(d) *}\\ &=H_{k-k_{0}}^{H}{}_{k-k_{0}-2}\cdot Y_{k-k_{0}}e^{j\frac{2\pi}{N }2\Delta n}+\hat{V}_{k}\\ &\approx\left|H_{k-k_{0}}\right|^{2}Y_{k-k_{0}}e^{j\frac{2\pi}{N }2\Delta n}+\hat{V}_{k}\end{split}, \tag{16}\]  
where \(\hat{V}_{k}\) denotes the remaining noise. The approximation in (16) holds as long as the channel frequency response at the adjacent sub-carriers are closely similar to each other, which is practical for the channel whose frequency selectively is not too severe between adjacent sub-carriers.  
Comparing (16) with (15), one can observe that the phase rotation term \(e^{-j\frac{2\pi}{N}\Delta n\cdot k}\) associated with the sub-carrier index \(k\) caused by the timing error \(\Delta n\) is eliminated through the receiver-side differential decoding, and only a fixed phase offset that has no influence on the detection of integer CFO and signaling, is left, making the proposed method immune to timing errors.  
Through the cross correlation between the local TS and the received differentially decoded sub-carriers, the signaling information and the integer CFO could be simultaneously acquired, i.e.,  
\[R_{d,k}=\frac{\sum\limits_{l=0}^{L-1}Z_{(k+4l)\bmod 4L}\cdot c_{l}^{*}}{\sum \limits_{l=0}^{L-1}\left|\hat{Y}_{(k+4l)\bmod 4L}\right|^{2}},\ \ 0\leq k<N. \tag{17}\]  
Theoretically, as shown in Figure 5, two pairs of peaks are expected to be generated by the correlation in (17) due to the integer CFO and the cyclically shifting between the two groups of TSs allocated at the \(\{4k+1\}\) and \(\{4k+3\}\) sub-carriers. Hence, the integer CFO \(k_{0}\) could be estimated through the distance between the first peak and its reference position when CFO=0. And meanwhile, the signaling parameter \(\Delta L\) could be acquired from the shifting distance between the two corresponding pairs of peaks, which is \((4\Delta L+2)\) as shown in Figure 5.