## 3 Signal Acquisition and Preprocessing
### Signal Preprocessing  
Before extracting the RFF features, it is necessary to preprocess the captured PPDU frames, which mainly includes time synchronization and frequency offset compensation. In the previous signal acquisition step, the starting position of each frame is only roughly determined, which may have a certain time deviation from the actual signal start point. While in the process of frequency offset compensation and fingerprint extraction, there is a high requirement for the accuracy of time synchronization. Therefore, in this section an improved synchronization method is proposed based on the existing method, in which the short preamble is utilized for coarse synchronization first, and then the long preamble is utilized for fine synchronization. Both the coarse synchronization and the fine synchronization are realized by performing complex conjugate correlation operation between the received signal and the corresponding local preamble signal, which can make synchronization more precise through two correlation operations. Due to the strong correlation of short preamble, it is first used for coarse synchronization. During the correlation operation for each PPDU frame, the complete 16 \(\upmu\)s short preamble field is used, and the range of M sampling points before and after the starting point determined by the acquisition process are contained to ensure that the actual starting point is included in this range. Assume that the local ideal short preamble signal is \(x_{short}(n)\), and the received signal is \(y_{short}(n)\), whose energy normalized complex conjugate correlation value is  
\[r_{short}(m)=\sum\limits_{n=0}^{N-1}\frac{y_{short}(n+m)x_{short}^{*}(n)}{ \sqrt{R_{xx}(0)R_{yy}(0)}} \tag{7}\]where  
\[R_{xx}(0)=\sum_{n=0}^{N-1}x_{short}(n)x_{short}^{*}(n)=\sum_{n=0}^{N-1}|x_{short} (n)|^{2} \tag{8}\]  
\[R_{yy}(0)=\sum_{n=0}^{N-1}y_{short}(n)y_{short}^{*}(n)=\sum_{n=0}^{N-1}|y_{short} (n)|^{2} \tag{9}\]  
\(m=0,1,\ldots,2M-1\), \(M\) denotes the sampling points before and after the starting point determined during signal acquisition, \(n=0,1,\ldots,N-1\), \(N\) denotes the sampling points of a short training symbol. The position \(\overset{\wedge}{m}\) is the starting point of the coarse synchronization of each frame where the maximum value \(r_{short}(m)\) is obtained by  
\[\overset{\wedge}{m}=\underset{m}{\operatorname{argmax}}\{r_{short}(m)\} \tag{10}\]  
However, in the case of severe multipath channels and mobile environment, the peak point performed by correlation calculation in the coarse synchronization process may have several points deviation. Therefore, it is necessary to use the long preamble field for fine synchronization. The fine synchronization utilizes the 16 \(\upmu\)s long preamble field for correlation operation, whose sliding searching range is within 3.2 \(\upmu\)s before and after the starting point determined by the coarse synchronization, namely a total of 6.4 \(\upmu\)s, which is the length of a long training symbol, and the correlation operation method and calculation equations are similar to the coarse synchronization process. The frequency offset is mainly caused by the inconsistency of the frequency between the wireless signal receiver and the transmitter oscillator. In the process of frequency offset compensation, the frequency offset values are estimated and then averaged through multiple periodic repetitive signals of the short preamble. In the frequency offset estimation process, the phase offsets between the corresponding points of the two adjacent short training symbols are estimated, and then their average value is calculated. The 10 short training symbols can be used to calculate the phase offset between the adjacent symbols for 9 times, and finally the phase offset can be converted to the frequency offset. The specific calculation process of frequency offset estimation is as follows.