# A Frequency-Domain Estimation Scheme for Frequency Offset with Large Range in OFDM Systems
## 3 Proposed Frequency Offset Estimation Scheme
### Proposed Estimation Scheme
Finally, the correlation value between \(\tilde{\mathbf{Y}}_{l}(k),k\in\mathcal{S}\) and the transmitted pilot sequence \(\mathbf{X}(k),k\in\mathcal{S}\) is calculated, and the \(\hat{\epsilon}(l)\) corresponding to the maximum correlation value is the final estimation result, i.e.,  
\[\hat{\epsilon}=\operatorname*{arg\,max}_{\hat{\epsilon}(l)}\left| \left|\sum_{k\in\mathcal{S}}\tilde{\mathbf{Y}}_{l}(k)\cdot\mathbf{X}^{*}(k) \right|\right|^{2}\] \[\mathbf{H}_{0}(k) =\mathbf{Y}^{(0)}(k)\cdot\left[\mathbf{X}^{(0)}(k)\right]^{*},k\in\mathcal{S} \tag{7}\]0  
Since the phase offset \(\alpha+2\pi l\) could be more than \(2\pi\), the estimation range of (13) could completely cover \((-1/2,1/2]\), independent of the limitation of \(m\). Furthermore, the proposed estimation (13) combined with the IFO estimation (4) could achieve a large estimation range, such as more than one subcarrier spacing estimation. The proposed frequency offset estimation scheme is summarized in Algorithm 1. It is noted that the frequency offset compensation method obviously affects the computing efficiency of Algorithm 1. One direct method is firstly to compensate the received pilot signal in the time domain, and then obtain the pilot sequence by FFT. The number of complex-multiplex operations is \(N+\frac{N}{2}\log_{2}N\) for one-time compensation, if the FFT operator is implemented by a radix-2 structure [23]. In the 5G NR base station, the received signal could be separately processed in different units, such as time-domain processing in the remote radio unit (RRU) and frequency-domain processing in the baseband unit (BBU) [24]. If both the candidate frequency offset estimation and compensation of Algorithm 1 are processed in the frequency domain, the control information interaction between different units can be simplified. For this purpose, we present a low-complexity and frequency-domain compensation method in the following subsection. ```
1:Input\(\mathbf{X}^{(0)}(k),\mathbf{X}^{(1)}(k)\), \(\mathbf{Y}^{(0)}(k)\), \(\mathbf{Y}^{(1)}(k)\), \(\mathcal{S}\), \(m\), \(\varepsilon_{max}\)
2:Calculate the channels \(\mathbf{H}_{0}(k)\) and \(\mathbf{H}_{1}(k)\), \(k\in\mathcal{S}\) with (7) and (8), respectively. 3:Calculate the initial phase offset \(\alpha\) with (6). 4:Calculate \(L_{1}\) and \(L_{2}\) with (11) and (12), respectively. 5:Initial a correlation value storage \(\mathbf{R}\) with the size of \(L_{1}+L_{2}+1\), and initial \(i=0\). 6:for\(l=-L_{1}\) to \(L_{2}\)do
7: Calculate \(\hat{\epsilon}(l)\) with (9). 8: Compensate the received pilot sequence with \(\hat{\epsilon}(l)\). 9: Calculate the correlation value \(r_{i}\) between the compensated pilot sequence and the transmitted pilot sequence. 10: Assign \(\mathbf{R}(i)=r_{i}\), where \(\mathbf{R}(i)\) is the \(i\)-th element in \(\mathbf{R}\), and let \(i=i+1\). 11:endfor
12: Determine \(l^{*}\), which is corresponding to the maximum element in \(\mathbf{R}\). 13:Output: the frequency offset estimation result \(\hat{\epsilon}=\hat{\epsilon}(l^{*})\). ```  
**Algorithm 1** The proposed frequency offset estimation scheme