## 4 RFF Extraction and Identification Methods
### Preamble Field-Based RFF Extraction Method
It is assumed that the multipath channel characteristics \(h(n)\) experienced by each of the four sequences in a frame signal are the same. In general, the same receiver is used for signal reception in the system, so the influence of the receiver RFF \(f^{r}(n)\) on the received signal is also regarded as the same. Additive white Gaussian noise can be proved to be a stationary random process with zero mean and constant variance, hence all four \(v(n)\) sequences are random sequences. Additive noise \(v(n)\) can be attenuated by signal superposition in time domain. Since the mean value of additive white Gaussian noise is zero, when enough noise sequences are added, the value of the stacked sequence is in theory zero, which proves that the method of superposition denoising method is feasible. In a frame of PPDU signal, the received two STF sequences \(y_{STF1}(n)\) and \(y_{STF2}(n)\), and two LTF sequences \(y_{LTF1}(n)\) and \(y_{LTF2}(n)\), can be added separately to enhance the fingerprint features and ease the influence of noise to a certain extent. Although only two sequences are superimposed each time, the fingerprint part \(f^{t}(n)\) is enhanced twice, and the amplitude of noise \(v(n)\) is reduced compared to the original. In the V2X system using DSRC technology, the vehicle is moving at a high speed in most cases, and the relative position between the receiver and the sender is often changing, so that each frame of the received signal has a variable channel factor, which adversely affects the extraction of stable RFFs. By using the repeated preamble structure defined in the PPDU frame format of IEEE 802.11p standard, transforming the time domain signal into the frequency domain and applying the division operation can remove the interference of channel condition on the fingerprint. After the above time domain superposition operations of STF and LTF, it can be approximately considered that the noise \(v(n)\) has been effectively eliminated. The expressions of the superimposed sequences transformed into frequency domain are  
\[Y_{STF1}(k)+Y_{STF2}(k) =X_{STF1}(k)F^{t}_{STF1}(k)H(k)F^{r}(k)+X_{STF2}(k)F^{t}_{STF2}(k )H(k)F^{r}(k) \tag{19}\] \[Y_{LTF1}(k)+Y_{LTF2}(k) =X_{LTF1}(k)F^{t}_{LTF1}(k)H(k)F^{r}(k)+X_{LTF2}(k)F^{t}_{LTF2}(k )H(k)F^{r}(k) \tag{20}\]  
where \(Y(k)\), \(X(k)\), \(F^{t}(k)\), \(H(k)\), \(F^{r}(k)\) are the frequency domain of \(y(n)\), \(x(n)\), \(f^{t}(n)\), \(h(n)\), \(f^{r}(n)\), respectively. According to Equations (1) and (2), the amplitudes of the non-zero spectrum of short training symbol and long training symbol are different in the frequency domain. When the signals are transmitted with different amplitudes, the nonlinearity of the power amplifier in the transmitter will cause it to work in different states [30]. The difference in signal transmission power and energy will result in different nonlinear fingerprints generated by nonlinear electronic components such as power amplifiers. When the two identical STF sequences and two identical LTF sequences are transmitted, the frequency domain of transmitter RFF can be uniformly expressed as \(F^{t}_{STF}(k)\) and \(F^{t}_{LTF}(k)\), while these two fingerprint expressions are different. When the two parts receiving signals are divided in the following steps, even if the consistent linear part is removed in the division operation, the nonlinear RFF components can still be preserved. Next, the frequency domain expressions of the two superimposed sequences are divided to remove the multipath channel, which can be shown as  
\[ \begin{array}{ll}RFF(k)&=\frac{Y_{STF1}(k)+Y_{STF2}(k)}{Y_{ITF1}(k)+Y_{ITF2}(k)} \\ &=\frac{X_{STF1}(k)F_{STF2}(k)(H(k)F^{\prime}(k)+X_{STF2}(k)F_{STE}^{\dagger}(k )H(k)F^{\prime}(k)}{X_{ITF1}(k)F_{ITF}^{\dagger}(k)H(k)F^{\prime}(k)+X_{STF2}(k )F_{ITF}^{\dagger}(k)H(k)F^{\prime}(k)}\\ &=\frac{X_{STF1}(k)F_{STF}(k)+X_{STF2}(k)F_{STE}^{\dagger}(k)}{X_{ITF1}(k)F_{ ITF}^{\dagger}(k)+X_{STF2}(k)F_{ITF}^{\dagger}(k)}\end{array}