## 2 System Description  
Let us consider an OFDM system which uses \(N\)-point FFT and \(N_{u}\) non-zero subcarriers. In the frequency domain, \(N-N_{u}\) NSCs are zero-padded. The frequency-domain signal is transformed by an \(N\)-point inverse FFT (IFFT) unit to create the time-domain signal, and a guard interval (GI) with a length of \(N_{g}\) samples is copied at the front of the OFDM symbol to remove both the ISI and ICI. Then, the \(n\)-th sample of the \(l\)-th OFDM symbol can be written by  
\[x_{l}(n)=\sum_{k=0}^{N-1}X_{l}(k)e^{j2\pi kn/N},\ -N_{g}\leq n<N, \tag{1}\]  
where \(X_{l}(k)\) is the symbol at subcarrier \(k\) with transmitted power \(E_{X}=|X_{l}(k)|^{2}\). Correspondingly, the effective duration of each OFDM symbol is equal to \(T_{s}=N_{s}T\), where \(T\) is the sampling period satisfying the Nyquist rate and \(N_{s}=N+N_{g}\). At the transmitter, the equivalent baseband signal takes the expression  
\[x(t)=\sum_{l=-\infty}^{\infty}\sum_{k=-N/2}^{N/2-1}X_{l}(k)e^{j2\pi(k/NT)(t-T _{g}-IT_{s})}p(t-IT_{s}), \tag{2}\]  
where \(p(t)=1\) for \(0\leq t<T_{s}\), and \(p(t)=0\) elsewhere. The radio signal is transmitted through a multipath fading channel corrupted by additive white Gaussian noise (AWGN). Due to the mismatch between the receiver and transmitter oscillators, the local oscillator's frequency \(f_{c}^{\prime}\) is different from the received carrier frequency \(f_{c}\), so that the frequency offset \(f_{c}^{\prime}-f_{c}\) remains after down conversion. Since the performance of IFO estimators is only marginally influenced by the STO [15; 16; 17], we restrict our attention to the detection of the IFO, considering the situation where the STO has been perfectly known at the receiver. Meanwhile, let us assume that the FFO has been estimated irrespective of the existence of the IFO so that there will remain a small FFO. As such, the time-domain signal at the receiving side is written by  
\[r(t)=e^{j2\pi(f_{c}^{\prime}-f_{c})t}x(t)\otimes h(t)+z(t), \tag{3}\]where \(\otimes\) is the linear convolution, \(h(t)\) denotes the channel impulse response (CIR), and \(z(t)\) denotes a complex zero-mean AWGN process. Considering perfect knowledge of the STO, the time-domain signal sampled at the time instant \(t=lT_{s}+nT\) is given by [16; 17]  
\[r_{l}(n)=e^{j2\pi l\epsilon\rho}x_{l}(n)\otimes h_{l}(n)+z_{l}(n), \tag{4}\]  
where \(\rho=N_{g}/N\), \(\epsilon\) is the normalized frequency offset by subcarrier spacing \(1/NT\), \(h_{l}(n)\) is the equivalent discrete-time CIR, and \(z_{l}(n)\) is a complex zero-mean AWGN having variance \(\sigma_{Z}^{2}\). According to the amount of the frequency offset between the receiver and transmitter oscillators, the frequency offset can be divided into an integer part \(\epsilon_{i}\) and a fractional part \(\epsilon_{f}\), leading to \(\epsilon=\epsilon_{i}+\epsilon_{f}\). After extracting the \(N_{g}\) GI samples and taking the FFT operation at the receiver, the frequency-domain signal at subcarrier \(k\) over the \(l\)-th symbol duration appears as  
\[R_{l}(k)=\alpha(\epsilon_{f})e^{j2\pi\epsilon(lN_{s}+N_{g})/N}H_{l}(k-\epsilon _{i})X_{l}(k-\epsilon_{i})+I_{l}(k)+Z_{l}(k), \tag{5}\]  
where \(\alpha(\epsilon_{f})=\sin(\pi\epsilon_{f})/(N\sin(\pi\epsilon_{f}/N))\) is the power leakage due to the FFO, \(H_{l}(k)\) is the FFT of \(h_{l}(n)\) with variance \(\sigma_{H}^{2}\), \(I_{l}(k)\) is the contribution to ICI with variance \(\sigma_{I}^{2}\), and \(Z_{l}(k)\) is the FFT of \(z_{l}(n)\) with variance \(\sigma_{Z}^{2}\). For small values of FFO, \(\alpha(\epsilon_{f})\approx 1\) and the variance of ICI can be easily obtained as \(\sigma_{I}^{2}\approx E_{X}\sigma_{H}^{2}\epsilon_{f}^{2}\pi^{2}/3\)[19]. In the ISDB-T system, a flexible and high-efficiency video/audio coding system, such as MPEG-4 advanced video coding (MPEG-4 AVC) is adopted.