## 2 The Principle of SFO Estimation and Compensation  
The OFDM frame consists of one frame head and multiple data-carrying OFDM symbols. The frame head is composed of several training sequences. As illustrated in Figure 1, we assume that the phase-conjugated pilots are inserted after the \(m\)-th symbol and the \(n\)-th symbol on the \(k\)-th subcarrier. On the receiver side, the training sequences are used for symbol synchronization and channel estimation. When the system works in an asynchronous way, the \(m\)-th OFDM symbol of the \(k\)-th subcarrier in the frequency domain can be approximately written as:  
\[R_{m,k}=S_{m,k}\times H_{k}e^{i\phi_{m,k}}+W_{m,k},\qquad k=0,1,2,\ldots,N-1 \tag{1}\]  
where \(S_{m,k}\) and \(R_{m,k}\) are the transmitted and received frequency domain signal on the \(m\)-th OFDM symbol of the \(k\)-th subcarrier, \(H_{k}\) is the channel response on the \(k\)-th subcarrier, \(W_{m,k}\) represents the noise, \(\phi_{m,k}=2\pi mkN_{T}\Delta/N\) is the SFO-induced phase rotation for the \(m\)-thOFDM symbol of the \(k\)-th subcarrier, \(N\) is the FFT size, \(N_{T}\) is the length of the OFDM symbol including CP, the SFO between transmitter and receiver is denoted by \(\Delta=(f_{t}-f_{r})/f_{r}\), \(f_{t}\) and \(f_{r}\) represent the sampling frequency in the DAC and ADC, respectively.  
After zero-forcing equalization, the equalized signal is expressed as:  
\[\hat{R}_{m,k}=S_{m,k}e^{i(\phi_{m,k}+\theta_{k})}+W^{\prime}{}_{m,k} \tag{2}\]  
where \(\theta_{k}\) is the common phase error caused by inaccurate channel estimation. The OFDM symbols from the same subcarrier index are generally considered to have the same common phase error \(\theta_{k}\)[22]. Thus, the product of a pair of phase-conjugated OFDM symbols is followed as:  
\[\hat{R}_{m,k}\times\hat{R}_{m+1,k}=\left|S_{m,k}\right|^{2}e^{i(\phi_{m,k}+ \phi_{m+1,k}+2\theta_{k})}+N_{m,k} \tag{3}\]  
where \(N_{m,k}\) can be considered as additive noise.  
Then, the superposition value of phase rotation on the phase-conjugated symbols can be given by:  
\[\hat{\phi}_{m,k}=arg\{\hat{R}_{m,k}\times\hat{R}_{m+1,k}\}=2\pi N_{T}k\Delta(2 m+1)/N+2\theta_{k}+\omega_{k} \tag{4}\]  
where \(arg\{\cdot\}\) stands for the angle operation, \(\omega_{k}\) represents the phase noise. As shown in Figure 1, a subtraction operation is performed to suppress the common phase error \(\theta_{k}\), the difference of superposition values between two groups of phase-conjugated symbols in the \(i\)-th OFDM frame can be expressed as:  
\[\hat{\phi}_{i,k}=\hat{\phi}_{n,k}-\hat{\phi}_{m,k}=4\pi N_{T}k\times(n-m) \Delta/N+\omega_{k}^{\prime} \tag{5}\]  
To further reduce the influence of the enhanced phase noise \(\omega_{k}^{\prime}\) and improve the estimation precision, the phase difference \(\hat{\phi}_{k}\) is obtained by the averaging processor of \(N_{f}\) OFDM frames and can be expressed by:  
\[\hat{\phi}_{k}=\frac{1}{N_{f}}{\sum_{i=1}^{N_{f}}\hat{\phi}_{i,k}} \tag{6}\]  
In this way, the SFO can be accurately estimated as follows:  
\[\hat{\Delta}=\hat{\phi}_{k}\times N/\left(4\pi N_{T}k\times(n-m)\right) \tag{7}\]  
Figure 1: Principle of the SFO estimation based on the phase-conjugated pilots.  
Finally, the \(m\)-th OFDM symbol on the \(k\)-th subcarrier should be multiplied by a compensation factor \(exp(-j2\pi kmN_{T}\hat{\Delta}/N)\) to correct the SFO-induced subcarrier phase rotation.