# A Frequency-Domain Estimation Scheme for Frequency Offset with Large Range in OFDM Systems
## 3 Proposed Frequency Offset Estimation Scheme
### Proposed Estimation Scheme  
The proposed scheme takes advantage of the autocorrelation of the pilot sequence and the \(2\pi\)-periodicity of the phase offset between the channels of two pilot OFDM symbols, which enables a large estimation range for frequency offset. Figure 2 shows a frame structure with two pilot OFDM symbols. Due to the frequency offset, there exists a phase offset \(\alpha\) between the channels of the two pilot symbols. The phase offset \(\alpha\) could be estimated as [9]  
\[\alpha=\text{angle}(\sum_{k\in\mathcal{S}}\mathbf{H}_{1}(k)\mathbf{H_{0}}^{*}(k)) \tag{6}\]  
where \(\mathbf{H}_{0}(k)\) and \(\mathbf{H}_{1}(k)\) denote the \(k\)-th subcarrier channels of pilot OFDM symbol 0 and symbol 1, respectively. The channels \(\mathbf{H}_{0}(k)\) and \(\mathbf{H}_{1}(k)\) could be estimated as  
\[\mathbf{H}_{0}(k) =\mathbf{Y}^{(0)}(k)\cdot\left[\mathbf{X}^{(0)}(k)\right]^{*},k\in\mathcal{S} \tag{7}\] \[\mathbf{H}_{1}(k) =\mathbf{Y}^{(1)}(k)\cdot\left[\mathbf{X}^{(1)}(k)\right]^{*},k\in\mathcal{S} \tag{8}\]  
where \(\mathbf{X}^{(0)}(k)\) and \(\mathbf{X}^{(1)}(k)\) denote the transmitted symbols on the \(k\)-th subcarriers of the pilot OFDM symbol 0 and symbol 1, respectively, and \(\mathbf{Y}^{(0)}(k)\) and \(\mathbf{Y}^{(1)}(k)\) denote the corresponding received symbols of \(\mathbf{X}^{(0)}(k)\) and \(\mathbf{X}^{(1)}(k)\), respectively. With the estimated phase offset \(\alpha\), the frequency offset could be conventionally estimated as \(\alpha/(2\pi m)\), where \(m\) denotes the number of OFDM symbols between the two pilot symbols, as shown in Figure 2. In the conventional estimation method, the estimation range of the frequency offset is \((-1/2m,1/2m]\), since \(\alpha\in(-\pi,\pi]\). In particular, if \(m>1\), the estimation range cannot cover \((-1/2,1/2]\), this would lead to an inaccurate FFO estimation when the frequency offset is close to \(\pm 1/2\). It is noted that the real phase offset between \(\mathbf{H}_{0}(k)\) and \(\mathbf{H}_{1}(k)\) exceeds \(2\pi\), when the frequency offset is large. Therefore, the real phase offset should be \(\alpha+2\pi l\), where \(l\) is an integer. With the phase offset \(\alpha+2\pi l\), the estimated frequency offset could be  
\[\hat{e}(l)=\frac{1}{2\pi m}(\alpha+2\pi l),\;l=0,\pm 1,\pm 2,\ldots \tag{9}\]  
In order to determine the final value of \(l\) corresponding to the real frequency offset, we firstly determine the range of \(l\). Since \(l\) is an integer, the range of \(l\) is denoted as  
\[l\in\mathcal{L}=\{-L_{1},-L_{1}+1,\ldots,-1,0,1,\ldots,L_{2}-1,L_{2}\} \tag{10}\]  
where both \(L_{1}\) and \(L_{2}\) are positive integers. Assuming the maximum Doppler shift in the system is known and denoted as \(\epsilon_{max}\) (\(\epsilon_{max}>0\)), we have \(-\epsilon_{max}\leq e(l)\leq\epsilon_{max}\). Then, according to (9) and (10), \(L_{1}\) and \(L_{2}\) could be derived as  
\[L_{1} =\lfloor m\cdot\epsilon_{max}+\frac{\alpha}{2\pi}\rfloor \tag{11}\] \[L_{2} =\lfloor m\cdot\epsilon_{max}-\frac{\alpha}{2\pi}\rfloor \tag{12}\]  
where \(\lfloor x\rfloor\) denotes the lower integer of \(x\). In particular, when \(\epsilon_{max}=1/2\), we have \(L_{1}=\lfloor\frac{m}{2}+\frac{\alpha}{2\pi}\rfloor\) and \(L_{2}=\lfloor\frac{m}{2}-\frac{\alpha}{2\pi}\rfloor\). Figure 2: A frame structure with two pilot OFDM symbols. With the obtained set \(\mathcal{L}\), we take advantage of the autocorrelation of the pilot sequence to determine the final value of \(l\) corresponding to the real frequency offset, which makes reference to the basic idea of IFO estimation shown in (4). Firstly, the candidate frequency offset \(\hat{\epsilon}(l)\) can be calculated by (9) for each \(l\in\mathcal{L}\). Secondly, each \(\hat{\epsilon}(l)\) is then used to compensate the received pilot sequence \(\mathbf{Y}(k),k\in\mathcal{S}\) and to obtain the compensated pilot sequence \(\tilde{\mathbf{Y}}_{l}(k),k\in\mathcal{S}\).