## 2 System Description  
### Signal Model  
We consider an OFDM system consisting of \(M\) equi-spaced subcarriers [15; 16; 17; 18]. To create a time-domain OFDM symbol with a duration of \(T\), we apply an \(M\)-point inverse DFT (IDFT) to the complex-valued data symbol \(X_{q}(m)\) with symbol energy \(E_{X}=\mathbb{E}\{|X_{q}(m)|^{2}\}\), where \(m=0,1,\cdots,M-1\) is the subcarrier index and \(\mathbb{E}\{\cdot\}\) denotes the expectation operator. In order to ensure interference-free transmission, we append a CP of a duration \(T_{g}\) to the effective part of the OFDM signal, producing one OFDM symbol of a duration \(T_{u}=T+T_{g}\). With a duration of \(T_{u}=(M+M_{g})/f_{s}\), the transmitted signal in the time domain for the \(q\)-th symbol period can be written as  
\[x_{q}(n)=\sum_{m=0}^{M-1}X_{q}(m)e^{j2\pi mn/M},\ n=-M_{g},-M_{g}+1,\cdots,M-1, \tag{1}\]  
where \(f_{s}\) represents the sampling frequency, \(j=\sqrt{-1}\) is the square root of minus one, and \(M_{g}\) is the CP length.  
The transmitted signal undergoes multipath fading and is distorted by additive white Gaussian noise (AWGN). Moreover, the received signal frequently experiences the CFO arising from the discrepancy between the local oscillators of the transmitter and receiver. This paper considers a scenario where uncompensated ICFO, residual CFO (RCFO), and residual sampling time offset (RSTO) persist after the initial synchronization process has been performed in the pre-DFT phase [24; 25]. As studied in [23], the impact of RSTO on the synchronization performance during the post-DFT detection phase is minimal. Assuming the receiver perfectly knows the RSTO, the transmitted signal \(x_{q}(n)\) undergoes linear convolution with the multi-path fading channel. Hence, the received signal in the time domain for the \(q\)-th symbol period can be given by  
\[\begin{split} y_{q}(n)&=e^{j2\pi q(v+\eta)M_{u}/M}e ^{j2\pi(v+\eta)n/M}x_{q}(n)\otimes h_{q}(n)\\ &+z_{q}(n),\ n=-M_{g},-M_{g}+1,\cdots,M-1,\end{split} \tag{2}\]  
where \(v\) represents the ICFO normalized by the subcarrier spacing denoted by \(\Delta_{f}\), \(\eta\) is the RCFO normalized by \(\Delta_{f}\), \(M_{u}=M+M_{g}\), \(\otimes\) denotes the linear convolution operator, \(h_{q}(n)\) is the discrete channel impulse response with \(L\) resolvable multipaths, and \(z_{q}(n)\) is the zero-mean AWGN with variance \(\sigma_{z}^{2}\). If we assume that the CP duration is substantially longer to mitigate the impact of the multipath channel, it can be ideally removed from the received time-domain signal. By conducting the DFT process on the CP-discarded signal, the frequency domain received signal can be expressed as [26]  
\[Y_{q}(m)\approx G_{q}(m-v)X_{q}(m-v)e^{j2\pi e(qM_{a}+M_{g})/M}+I_{q}(m)+Z_{q}(m ),\ |v|\leq U, \tag{3}\]  
where \(Y_{q}(m)\) denotes the demodulated OFDM symbol for the symbol index \(q,\epsilon=v+\eta\), \(G_{q}(m)\) represents the channel state information (CSI) following a zero-mean Gaussian distribution with variance \(\sigma_{G}^{2}\), \(I_{q}(m)\) represents the inter-carrier interference (ICI) term with variance \(\sigma_{I}^{2}\approx E_{X}\sigma_{G}^{2}\eta^{2}\pi^{2}/3\) for relatively small values of \(\eta\), \(Z_{q}(m)\) represents the zero-mean AWGN with variance \(\sigma_{Z}^{2}\), and \(U\) corresponds to the largest value of ICFO.