## 4 Timing and Carrier Frequency Synchronization through Preamble Detection
### Timing and Fractional CFO Estimation  
The cyclic property of the preamble can be exploited in the algorithm of timing at the receiver. Specifically, the following three pairs of cyclic parts are exploited for the calculation of SACs,  
\[R_{1,n}=\sum_{l=0}^{N/2-1}{(-1)^{(n+l)}r_{n+l}^{*}\cdot r_{n+l-N/2}}, \tag{6}\]  
\[R_{2,n}=\sum_{l=0}^{N/2-1}{(-1)^{(n+l)}r_{n+l}^{*}\cdot r_{n+l-N}}, \tag{7}\]  
\[R_{3,n}=\sum_{l=0}^{N/2-1}{(-1)^{(n+l)}r_{n+l}^{*}\cdot r_{n+l-3N/2}}. \tag{8}\]  
As shown in Figure 2, the de-scrambling operations between the three front parts '**A**', '**-**A**', '**A**' and the last part '**S(A)**' are implemented through multiplying by the descrambling coefficient \((-1)^{n}\) in the summation of (6)-(8). After that, in order to further sharpen the  
Figure 2: Proposed OFDM preamble design. correlation peak, these three SAC results are multiplied together to generate the final positive peak as given by  
\[R_{c,n}=-R_{1,n}^{*}\cdot R_{2,n}\cdot R_{3,n}. \tag{9}\]  
The proposed SAC block diagram for timing and carrier frequency synchronization is depicted in Figure 3. From the diagram and the above SAC calculation method (6)-(9), it can be found that the theoretical desired timing position \(n_{0}\) is at the first sample of the part '**S(A)**', as illustrated in Figure 2. Before calculating the correlation, a descrambling operation, i.e., multiplying by the descrambling coefficient \((-1)^{n}\), is prior implemented toward the received preamble, which is different from existing methods. It can be noticed that with the same preamble length, the computational complexity of the proposed method, mainly reflected by the number of additions and multiplications in calculating correlations, is about 50% more than that of [PERSON]'s method. This is because three groups of SAC are required to be calculated in the proposed method, whereas only two groups are calculated in [PERSON]'s method. However, the complexity of the proposed algorithm is still low, which is analyzed in detail in Section 4.3 since the summations in (6)-(8) could be implemented with recursive method [16]. At the receiver, the estimated timing position \(\hat{n}_{0}\) in the preamble can be estimated from the correlation peak of \(R_{c,n}\), and the fractional CFO can be calculated by the phase of the correlation peak as well,  
\[\hat{n}_{0}=\operatorname*{arg\,max}_{n}\{|R_{c,n}|\}, \tag{10}\]  
\[\hat{f}_{\text{frc}}=\left.\frac{\arg(R_{c,n})}{2\pi N}\right|_{n=\hat{n}_{0}}, \tag{11}\]  
where \(\arg(\cdot)\) is the phase calculation operation for a complex number, and \(\arg\max\{\cdot\}\) derives the variable that maximizes the interior objective function. To demonstrate the advantage of the proposed method in SAC peak performance over [PERSON] and [PERSON]'s methods, as illustrated in Figure 4, the simulation results are presented, where the channel noise is absent. It is noted from Figure 4 that a broad plateau whose length is the same with that of the CP is generated for [PERSON]'s method. Meanwhile, [PERSON]'s method produces four sub-peaks, with each having approximately 1/4 the amplitude of the main peak. Therefore, it can be observed that both the plateau and the sub-peaks are avoided in the proposed method, and a much sharper main peak is derived compared with that of [PERSON]'s method. Now we look at the capability of the proposed method against the NBI impact. Let us consider an NBI with the frequency \(f_{\text{NB}}\), and without loss of generality, we investigate the  
Figure 3: Proposed time–domain sliding auto-correlation module for timing and carrier frequency synchronization. first SAC result \(R_{1,n}\) in (6) at the desired correct timing position \(n_{0}\), which is applied by a descrambling operation,  
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