## 2 Signal Model  
The system of interest uses OFDM symbols with \(N_{z}\) non-zero subcarriers and \(N_{g}\) guard interval (GI) samples. After \(N\)-point inverse fast Fourier transform (IFFT), a time-domain signal is created, and GI with a duration of \(T_{g}\) is added to the front of the OFDM symbol to eliminate the effect of ISI. As a consequence, the period of one OFDM symbol is \(T_{u}=N_{u}T_{s}\), where \(T_{s}\) is the sampling time and \(N_{u}=N+N_{g}\). At the transmitter, the \(n\)-th time-domain sample during the \(l\)-th period can be written by  
\[x_{l}(n)=\sum_{k=-N/2}^{N/2-1}X_{l}(k)e^{j2\pi kn/N},\ n=-N_{g},-N_{g}+1, \cdots,N-1 \tag{1}\]  
where \(X_{l}(k)\) is the symbol transmitted with symbol energy \(E_{X}=|X_{l}(k)|^{2}\). In the frequency-domain, the DVB-C2 system provides three kinds of pilot symbols: scattered pilots (SPs), edge pilots (EPs) and CPs [10]. The SPs and EPs are primarily used to estimate the channel, whereas the CPs are constantly inserted into the fixed carrier position, which are dedicated to perform frequency synchronization [11; 12]. The continuous-time baseband signal transmitted is expressed as:  
\[x(t)=\sum_{l=0}^{\infty}\sum_{k=-N/2}^{N/2-1}X_{l}(k)e^{l2\pi(k/NT_{s})(t-T_{g} -IT_{u})}p(t-IT_{u}) \tag{2}\]  
where \(p(t)=1\) for \(0\leq t<T_{u}\) and \(p(t)=0\) otherwise. The transmitted signal is passed through a multipath fading channel with additive white Gaussian noise (AWGN). Since the frequency of the local oscillator \(f_{c}^{{}^{\prime}}\) is not exactly equal to the received carrier frequency \(f_{c}\), there will be a CFO \(f_{c}^{{}^{\prime}}-f_{c}\) in Hz. Since this paper focuses on the post-FFT estimation strategy, we assume that symbol timing offset (STO) has been perfectly compensated. Furthermore, we consider the situation where coarse CFO estimation has been performed before the FFT processing. In this case, small CFO and SFO will remain during the transmission of the payload. In this work, we give our attention to the frequency-offset estimation scheme using CPs. Then, the time-domain received signal can be given by:  
\[r(t)=e^{l2\pi(f_{c}^{{}^{\prime}}-f_{c})t}x(t)\otimes h(t)+z(t) \tag{3}\]  
where \(\otimes\) denotes the linear convolution, \(h(t)\) is the channel impulse response (CIR) and \(z(t)\) is a zero-mean AWGN process. The received signal is sampled at time instants \(t=IT_{u}+nT_{s}^{{}^{\prime}}\), which produces an SFO \(\xi=(T_{s}^{{}^{\prime}}-T_{s})/T_{s}\). Taking perfect STO estimation into account, the time-domain sampled signal at time instants \(t=IT_{u}+nT_{s}^{{}^{\prime}}\) can be written by:  
\[ \begin{split} r_{l}(n)&=e^{l2\pi(f_{c}^{{}^{\prime} }-f_{c})nT_{s}^{{}^{\prime}}}x_{l}(n)\otimes h_{l}(n)+z_{l}(n)\\ &=e^{l2\pi\alpha\Delta_{f}nT_{s}^{{}^{\prime}}}x_{l}(n)\otimes h _{l}(n)+z_{l}(n),\ n=-N_{g},-N_{g}+1,\cdots,N-1\end{split}  \tag{4}\]  
where \(h_{l}(n)\) is the discrete CIR, \(z_{l}(n)\) is a zero-mean AWGN contribution, \(\Delta_{f}=1/NT_{s}\) is the subcarrier spacing and \(\varepsilon\) is the CFO normalized by subcarrier spacing \(1/NT_{s}\). The channel is assumed to be constant over several OFDM blocks. After extracting the GI and taking \(N\)-point FFT for the \(l\)-th OFDM symbol, the frequency-domain received signal at subcarrier \(k\) takes the form [29; 30]:  
\[ \begin{split} R_{l}(k)&=\frac{1}{N}\sum_{n=0}^{N-1 }r_{l}((n+N_{g})T_{s}^{{}^{\prime}})e^{-l2\pi kn/N}\\ &=\alpha(k)H_{l}(k)X_{l}(k)e^{l2\pi\varphi(k)(lN_{u}+N_{g})/N}e^ {i\pi(N-1)\varphi(k)/N}+C_{l}(k)\\ &+Z_{l}(k),\ -N/2\leq k<N/2\end{split}  \tag{5}\]  
where \(\varphi(k)=\varepsilon+k\xi\), \(\alpha(k)=\sin(\pi\varphi(k))/(N\sin(\pi\varphi(k)/N))\), \(H_{l}(k)=(1/N)\sum_{n=0}^{N-1}h_{l}(n)e^{-2\pi kn/N}\) is the channel frequency response (CFR) at subcarrier \(k\) with variance \(\sigma_{H}^{2}\), \(C_{l}(k)\) is a zero-mean complex ICI contribution with variance \(\sigma_{C}^{2}\) and \(Z_{l}(k)\) is a zero-mean complex AWGN with variance \(\sigma_{Z}^{2}\).