# A Frequency-Domain Estimation Scheme for Frequency Offset with Large Range in OFDM Systems
## 3 Proposed Frequency Offset Estimation Scheme
### The Low-Complexity Implementation of the Proposed Scheme  
In this subsection, we provide a low-complexity implementation of the proposed scheme by simplifying the computation of frequency offset compensation in Algorithm 1. The basic idea for frequency offset compensation is firstly to obtain the frequency-domain compensation coefficients \(\mathbf{Z}_{l}\) with the candidate frequency offset \(\hat{\epsilon}(l)\), and then calculate the cyclic convolution between the compensation coefficients \(\mathbf{Z}_{l}\) and the received pilot sequence \(\mathbf{Y}\) to obtain the compensated pilot sequence \(\tilde{\mathbf{Y}}_{l}(k),k\in\mathcal{S}\). It is noted that the compensation complexity can be reduced if the compensation coefficients \(\mathbf{Z}_{l}\) contain a large amount of zero elements. Based on this idea, we consider simplifying the compensation coefficients \(\mathbf{Z}_{l}\) as follows. The frequency-domain compensation coefficients \(\mathbf{Z}_{l}\) can be obtained by performing the FFT on the time-domain compensation coefficients \(\mathbf{z}_{l}=(e^{\frac{-|2\pi\cdot\hat{\varepsilon}(l)|}{N}},e^{\frac{-|2\pi \cdot\hat{\varepsilon}(l)|}{N}},\ldots,\)\(e^{\frac{-|2\pi\cdot(N-1)\cdot\hat{\varepsilon}(l)|}{N}})\); that is  
\[\mathbf{Z}_{l}(k)=\sum_{n=0}^{N-1}e^{\frac{-|2\pi\cdot n\hat{\varepsilon}(l)|}{N} }\cdot e^{\frac{-|2\pi\cdot\hat{\varepsilon}(l)|}{N}} \tag{14}\]  
Let us consider \(\hat{\varepsilon}(l)\in(-1/2,1/2]\) and the general case \(\hat{\varepsilon}(l)\
eq 0\), where the coefficient \(\mathbf{Z}_{l}(k)\) is  
\[\mathbf{Z}_{l}(k)=\frac{1-e^{-\frac{|2\pi\cdot\hat{\varepsilon}(l)|}{N}}}{1-e^{ \frac{-|2\pi\cdot\hat{\varepsilon}(l)|+k}{N}}},k=0,1,2,\ldots,N-1. \tag{15}\]  
and  
\[||\mathbf{Z}_{l}(k)||^{2}=\frac{2-2\cos\left[2\pi\cdot\hat{\varepsilon}(l)\right] }{2-2\cos\left[\frac{2\pi\cdot\hat{\varepsilon}(l)+k}{N}\right]}. \tag{16}\]  
According to (16), \(||\mathbf{Z}_{l}(k)||^{2}\) is monotonically decreasing with \(\{k|-\hat{\varepsilon}(l)<k\leq N/2-\hat{\varepsilon}(l),k\in\mathcal{Z}\}\), and monotonically increasing with \(\{k|N/2-\hat{\varepsilon}(l)\leq k<N-\hat{\varepsilon}(l),k\in\mathcal{Z}\}\), where \(\mathcal{Z}\) is the integer set. Therefore, the maximum value of \(||\mathbf{Z}_{l}(k)||^{2}\) is obtained at \(k=0\) or \(k=N-1\). Figure 3 shows the modulus of \(\mathbf{Z}_{l}(k)\), where \(N=1024\), \(\hat{\varepsilon}(l)=-0.3\), \(-0.1\), \(0.2\) and \(0.4\). It is verified that the maximum value of \(|\mathbf{Z}_{l}(k)|\) is obtained at \(k=0\) or \(k=N-1\), which means that the compensation coefficients \(\mathbf{Z}_{l}(k)\) around \(k=0\) or \(k=N-1\) would have a greater effect on the compensation results than those around \(k=N/2\). In order to reduce the calculation complexity, we only keep \(M\) compensation coefficients with the larger modulus and set the other \(N-M\) coefficients to zeros in \(\mathbf{Z}_{l}\). In  
Figure 3: The modulus of \(\mathbf{Z}_{l}(k)\) with different values of \(\hat{\varepsilon}(l)\). this way, both the compensation coefficients calculation and cyclic convolution can be simplified. The simplified compensation coefficients \(\bar{\mathbf{Z}}_{l}\) are given by  
\[\bar{\mathbf{Z}}_{l}(k)=\left\{ \begin{array}{cc}0,&k=\frac{M}{2},\frac{M}{2}+1, \ldots,N-\frac{M}{2}-1\\ \mathbf{Z}_{l}(k),&\text{otherwise}\end{array} \right. \tag{17}\]  
After obtaining the simplified compensation coefficients \(\mathbf{Z}_{l}\), we calculate the cyclic convolution between \(\bar{\mathbf{Z}}_{l}\) and the received pilot sequence \(\mathbf{Y}\) to obtain the compensated pilot sequence \(\bar{\mathbf{Y}}_{l}(k),k\in\mathcal{S}\).