# Robust Symbol and Frequency Synchronization Method for Burst OFDM Systems in UAV Communication
## 2 System Model
### Symbol and Frequency Synchronization Algorithms  
The modulation and demodulation of the OFDM are implemented using inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT). A symbol time synchronization should be performed to determine the start point of each OFDM symbol to take the FFT at the receiver end. Since the OFDM signal is sensitive to carrier frequency offset caused by the Doppler shift or local oscillator (LO) instabilities between the transmitter and the receiver, an accurate frequency synchronization estimation and compensation must be performed before demodulation. In a burst OFDM system, preamble-based symbol and frequency synchronization methods are commonly employed. Figure 2 illustrates an example of a preamble with a repetitive structure, where CP denotes the cyclic prefix. The fundamental concept is to leverage the good correlation characteristics of a training sequence. By correlating the received signal with the known training sequence at the receiver, a sharp correlation peak can be achieved. The Zadoff-Chu sequence can be defined as follows [19]:  
\[p_{k}(L,u,q)=\left\{ \begin{array}{cc}\exp\left(-j\frac{2\pi u}{L}(\frac{k^{2} }{2}+qk)\right),&\text{when $L$ is odd}\\ \exp\left(-j\frac{2\pi u}{L}(\frac{k(k+1)}{2}+qk)\right),&\text{when $L$ is even} \end{array} \right., \tag{1}\]  
where \(j\) is the imaginary unit \(j=\sqrt{-1}\), \(k\) is an integer time instant ranging from 0 to \(L-1\), \(L\) is the length of the ZC sequence, \(u\) is a root index that is prime to \(L\), \(q\) is an arbitrary integer, and \(q=0\) in this work. The auto-correlation of ZC sequence with \(L=128\) and \(u=127\) is shown in Figure 3.  
Figure 3: Autocorrelation of ZC sequence with length \(L=128\) in the (red) time and (blue) frequency domains. Figure 2: Traditional frame structure design. Let \(r(n)\) be the received signal at time \(n\), and \(s(k)\) be the known training sequence of length \(L\). The complex correlation \(V(n)\) can be expressed as [27]:  
\[V(n)=\sum_{k=0}^{L-1}|r(n+k)s^{*}(k)|^{2}, \tag{2}\]  
where \(s^{*}(k)\) is the complex conjugate of \(s(k)\). The presence of a preamble can be detected by comparing the correlation value with a pre-established threshold \(V_{th}\), and as a result, an accurate position of the synchronization training sequence can be determined. According to Equation (2), the calculation of \(V(n)\) requires \(L\) complex multiplication operations and \(L-1\) addition operations, leading to high implementation complexity. Additionally, the carrier frequency offset can be estimated from the phase angle of the product of a repetitive ZC sequence. For a given carrier frequency offset \(f_{\omega}\), the received signal can be expressed as:  
\[r(n)=x(n)e^{j2\pi f_{\omega}nT_{s}}+w(n), \tag{3}\]  
where \(x(n)\) represents the transmission signal, \(T_{s}\) is the sampling period, and \(w(n)\) denotes the noise component. The time-delayed autocorrelation of the two training symbols can be expressed as:  
\[ \begin{split} R&=\sum_{n=0}^{L-1}r(n)r^{*}(n+L)\\ &=\sum_{n=0}^{L-1}\left[|x(n)|^{2}e^{-j2\pi f_{\omega}LT_{s}}+x(n )e^{j(2\pi f_{\omega}nT_{s}+\varphi_{0})}w^{*}(n+L)\right.\,\\ &\left.+x^{*}(n+L)e^{-j(2\pi f_{\omega}(n+L)T_{s}+\varphi_{0})}w( n)+w(n)w^{*}(n+L)\right]\end{split}  \tag{4}\]  
where \(\varphi_{0}\) denotes the initial phase offset between the transmitter and receiver. In the AWGN channel, the noise component \(w(n)\) is modeled as a zero-mean independent and identically distributed Gaussian random variable. Therefore, the expectation value of product elements \(x(n)e^{j(2\pi f_{\omega}nT_{s}+\varphi_{0})}w^{*}(n+L)\), \(x^{*}(n+L)e^{-j(2\pi f_{\omega}(n+L)T_{s}+\varphi_{0})}w(n)\), and \(w(n)w^{*}(n+L)\) are all zero. As a result, Equation (4) can be approximated as [23]  
\[\hat{R}=e^{-j(2\pi f_{\omega}LT_{s})}\sum_{n=0}^{L-1}|x_{n}|^{2}. \tag{5}\]  
Therefore, the carrier frequency offset can be estimated by  
\[\hat{f}_{w}=-\frac{1}{2\pi LT_{s}}\angle\hat{R}, \tag{6}\]  
where \(\angle\hat{R}\) is the angle to \(\hat{R}\).