## 2 Signal Model  
For a classical OFDM system, we consider that it has a total bandwidth composed of \(N\) subcarriers and employs the technique of pilot-based signal correction, in which quadrature amplitude modulation (QAM) transmission is utilized over multipath fading channels. Let the total \(K\) pilot symbols and \((N-K)\) data symbols be modulated each by an individual subcarrier, such that the inter-carrier interference is absent. The modulated symbols \(\mathbf{X}\) are processed by the inverse fast Fourier transformer (IFFT), and then transformed into the time-domain signals obtained by \(\mathbf{x}=\mathbf{F}^{H}\mathbf{X}\), where \(\mathbf{F}\in C^{N\times N}\) denotes the FFT matrix with the \((k,l)\)-th entry being \(e^{j2\pi(k-1)(l-1)/N}/\sqrt{N}\). For a channel of maximum delay spread \(L\), the OFDM symbol requires a cyclic prefix (CP) of length \(N_{g}\geq L\) so as to eliminate the inter-symbol interference (ISI). After the CP is removed at the receiver end, the received OFDM symbols are then FFT processed. In the presence of CFO, the received symbol block is given by:  
\[ \begin{split}\mathbf{R}&=\mathbf{F}\mathbf{D_{a( \epsilon_{d})}}\mathbf{F}^{H}\mathbf{D_{X}}\mathbf{F}\mathbf{h}+\mathbf{W}\\ &=\mathbf{F}\mathbf{D_{a(\epsilon_{d})}}\mathbf{F}^{H}\mathbf{D_{X }}\mathbf{H}+\mathbf{W}\end{split}  \tag{1}\]  
where \(\epsilon_{d}\) is the related frequency offset normalized to subcarrier spacing and the diagonal matrix \(\mathbf{D_{a(\epsilon_{d})}}\) represents the CFO effect with \(\mathbf{a}(\epsilon_{d})=[1,e^{j2\pi\epsilon_{d}/N},\cdots,e^{j2(N-1)\pi \epsilon_{d}/N}]^{T}\). The channel impulse response is given by \(\mathbf{h}=[h(0),h(1),\cdots,h(L),\mathbf{0}_{N-L-1}^{T}]^{T}\), and the \(N\times 1\) vector \(\mathbf{H}=\mathbf{F}\mathbf{h}\) is the channel frequency response (CFR). \(\mathbf{W}\) denotes the complex white Gaussian channel noise, which is assumed to be independent of the transmitted data. The channel estimation (CE) aims at extracting \(\mathbf{H}/\mathbf{h}\) from the received data in Equation (1). We briefly review the pilot-based CE technique, which is widely used because of its simplicity. Let the \(k\)-th pilot symbol be modulated on the \(c_{k}\)-th subcarrier where \(k=0,1,...,K-1\). Stacking all of the \(K\) entries in order yields a \(K\times 1\) vector \(\mathbf{R}_{P}\), which is expressed by:  
\[ \begin{split}\mathbf{R}_{P}&\stackrel{{ \Delta}}{{=}}\left[R_{c_{0}}\ R_{c_{1}}\ \cdots\ R_{c_{K-1}}\right]^{T}=\mathbf{S}\mathbf{R}\\ &=\mathbf{S}\mathbf{F}\mathbf{D}_{\mathbf{a}(\epsilon_{d})} \mathbf{F}^{H}\mathbf{D}_{\mathbf{X}}\mathbf{H}+\mathbf{S}\mathbf{W}\end{split}  \tag{2}\]  
where the \(K\times N\) matrix \(\mathbf{S}=[\mathbf{e}_{c_{0}},\mathbf{e}_{c_{1}},\cdots,\mathbf{e}_{c_{K-1} }]^{T}\), with \(\mathbf{e}_{k}\) being the \(k\)-th column vector of an \(N\times N\) identity matrix \(\mathbf{I}_{N}\). Based on the pre-known pilot symbols, the CFR estimate associated with the pilot symbols can be written by:  
\[ \begin{split}\hat{\mathbf{H}}_{P}&=\mathbf{D}_{ \mathbf{X}_{P}}^{-1}\mathbf{R}_{P}\\ &=\mathbf{D}_{\mathbf{X}_{P}}^{-1}\mathbf{S}\mathbf{F}\mathbf{D} _{\mathbf{a}(\epsilon_{d})}\mathbf{F}^{H}\mathbf{D}_{\mathbf{X}}\mathbf{H}+ \mathbf{D}_{\mathbf{X}_{P}}^{-1}\mathbf{S}\mathbf{W}\end{split}  \tag{3}\]  
where \(\mathbf{X}_{P}=\mathbf{S}\mathbf{X}\) is the data vector associated with the corresponding pilot symbols. Clearly, when CFO is absent \((\epsilon_{d}=0)\), we have \(\mathbf{D}_{\mathbf{a}(\epsilon_{d})}=\mathbf{I}_{N}\), which allows the CFR to be successfully estimated. When the input signal-to-noise ratio (SNR) is sufficiently large, the estimated CFR reaches:  
\[ \begin{split}\hat{\mathbf{H}}_{P}&=\mathbf{S} \mathbf{H}+\mathbf{D}_{\mathbf{X}_{P}}^{-1}\mathbf{S}\mathbf{W}\\ &\approx\mathbf{S}\mathbf{H}\end{split}  \tag{4}\]  
However, in mobile environments, CFO usually exists due to Doppler shift and/or mismatch between the transmitter and receiver oscillators.