## 2 Methodology
### CFO Estimation in OFDM System with ICI  
Assume that the OFDM system used for the analysis below has \(N\) subcarriers. When \(H(k)\) denotes the complex channel frequency response on the \(k\)-th received subcarrier, the channel fading diagonal matrix is \(\boldsymbol{\Lambda}=diag(H(0),H(1),\ldots,H(N-1))\). Received OFDM block symbol \(\boldsymbol{Y}\) with a CFO is represented as follows:  
\[\boldsymbol{Y}=\boldsymbol{X}\boldsymbol{\Lambda}\boldsymbol{S}+\boldsymbol{W}, \tag{1}\]  
where \(\boldsymbol{S}\) denotes transmitted block symbols, and \(\boldsymbol{W}\) represents the additive white Gaussian noise vector. \(\boldsymbol{S}\) represents an \(N\times N\) ICI matrix. The specific elements can be represented by the following formula :  
\[\boldsymbol{S}=\left[\begin{array}{cccc}S(0)&S(-1)&\cdots&S(1-N)\\ S(1)&S(0)&\cdots&S(2-N)\\ \vdots&\vdots&\ddots&\vdots\\ S(N-1)&S(N-2)&\cdots&S(0)\end{array}\right]_{N\times N}. \tag{2}\]  
The element in the \(p\)-th row and \(q\)-th column of the \(\boldsymbol{S}\) matrix is \(\boldsymbol{S}_{p,q}=\boldsymbol{S}(p-q)\), which is calculated in the following form:  
\[\boldsymbol{S}_{p,q}=\frac{\sin(\pi(p-q+\varepsilon))}{N\sin(\pi(p-q+ \varepsilon)/N)}\mathrm{e}^{\mathrm{i}\pi\frac{(p-q+\varepsilon)(N-1)}{N}}, \tag{3}\]  
where \(\varepsilon=\Delta F/\Delta f\) represents the normalized CFO calculated using the CFO \(\Delta F\) and subcarrier spacing \(\Delta f\).  
According to (1), the OFDM received symbol with CFO can be regarded as the MC-CDMA system [26, 27]. Therefore, the received \(k-\)th data symbol can be considered as the \(k-\)th vector to form the spreading code matrix. By performing eigenvalue decomposition on the \(\boldsymbol{S}\) matrix, the following relationship can be obtained:  
\[\boldsymbol{S}=\boldsymbol{F}^{\mathrm{H}}\boldsymbol{\Phi}(\varepsilon) \boldsymbol{F}, \tag{4}\]  
where \(\boldsymbol{\Phi}\) is a diagonal matrix containing the CFO information, with the \(n-\)th element of \(\boldsymbol{\Phi}  
\[\mathbf{R}=\mathbf{Y}\mathbf{S}^{H}\mathbf{\Lambda}^{-1}=\mathbf{X}+\mathbf{W}\mathbf{S}^{H}\mathbf{\Lambda}^{-1}. \tag{6}\]  
where \(\mathbf{W}\mathbf{S}^{H}\) and \(\mathbf{W}\) have the same covariance matrix because \(\mathbf{S}^{H}\) is also an orthogonal matrix. The estimated values of transmitted symbol vector \(\mathbf{X}\) can be obtained by performing the decision of \(\mathbf{R}\), that is \(\mathbf{\hat{X}}=D(\mathbf{R})\), where \(D(\cdot)\) represents the decision function. Subsequently, the received signal can be reconstructed as \(\hat{\mathbf{Y}}\) with \(\mathbf{\hat{X}}\) and ICI coefficient matrix \(\mathbf{S}(\hat{\varepsilon})\):  
\[\hat{\mathbf{Y}}=\mathbf{\hat{X}}\mathbf{\Lambda}\mathbf{S}(\hat{\varepsilon}). \tag{7}\]  
When the CFO is correctly estimated and compensated, the estimated values of the received signals are equal to the received signals without the influence of noise \(\hat{\mathbf{Y}}=\mathbf{Y}\). Therefore, a smaller difference between the reconstructed and received signals (\(\hat{\mathbf{Y}}\) and \(\mathbf{Y}\), respectively) indicates that the estimated CFO, \(\hat{\varepsilon}\), is closer to the actual CFO. The Euclidean distance between the received and reconstructed signals can be used to measure the difference between them:  
\[\begin{split} J(\hat{\varepsilon})=&\|\hat{\mathbf{Y}} -\mathbf{Y}\|^{2}=\|\mathbf{\hat{X}}\mathbf{\Lambda}\mathbf{S}(\hat{\varepsilon})-\mathbf{Y}\|^{2 }\\ =&\|D(\mathbf{R})\mathbf{\Lambda}\mathbf{S}(\hat{\varepsilon})- \mathbf{Y}\|^{2}\\ =&\Big{\|}D\Big{(}\mathbf{Y}\mathbf{\hat{S}}^{H}(\hat{ \varepsilon})\mathbf{\Lambda}^{-1}\Big{)}\mathbf{\Lambda}\mathbf{S}(\hat{\varepsilon})- \mathbf{Y}\Big{\|}^{2}.\end{split} \tag{8}\]  
Therefore, the optimal CFO value, \(\hat{\varepsilon}\), can be obtained by minimizing the cost function, J(\(\hat{\varepsilon}\)):  
\[\hat{\varepsilon}=\arg\min(J(\hat{\varepsilon})). \tag{9}\]  
A flowchart that describes how to execute the CFO estimation of the OFDM system with ICI is shown in Figure 1.