# Effective Blind Frequency Offset Estimation Scheme for BST-OFDM Based HDTV Broadcast Systems
## 2 System Model  
In this paper, we consider an OFDM system using \(N\)-point inverse fast Fourier transform (IFFT) of size \(N\). One OFDM symbol comprises \(N_{u}<N\) non-zero subcarriers, and remnant \(N-N_{u}\) subcarriers are assumed to be zero-inserted. After IFFT operation, the time-domain signal can be generated and a guard interval (GI) with a length of \(N_{g}\) is inserted at the front of OFDM symbols in order to remove both inter-symbol interference (ISI) and inter-carrier interference (ICI). The \(l\)-th transmitted time domain signal can be given by  
\[x_{l}(n)=\sum_{k=0}^{N-1}X_{l}(k)e^{j2\pi k\alpha n/N},\ n=-N_{g},N_{g}+1,\cdots, N-1, \tag{1}\]  
where \(X_{l}(k)\) is the symbol transmitted at the \(k\)-th subcarrier. Consequently, the effective duration of one OFDM symbol is \(N_{s}T_{s}\), where \(T_{s}\) is the sample time interval and \(N_{s}=N+N_{g}\). Since our focus is on the post-FFT frequency synchronization, it is assumed that the coarse symbol timing offset (STO) and frequency offset estimation has been performed at the pre-FFT stage. Since many accurate coarse estimation schemes were studied in the literature [24; 25], it is reasonable to assume that residual STO, CFO, and SFO are small enough after pre-FFT synchronization stage. Let \(\Delta_{c}\) be the normalized CFO by the subcarrier spacing \(\Delta_{f}=1/NT_{s}\) and \(\Delta_{s}\) be the normalized SFO by the sampling frequency interval \(1/T_{s}\)  
After IFFT, the received frequency-domain signal can be represented as [20; 21; 22; 23]  
\[R_{l}(k)=\alpha(k)H_{l}(k)X_{l}(k)e^{-j2\pi k\Delta_{\tau}/N}e^{j2\pi\varphi(k )(lN_{s}+N_{g})/N}+C_{l}(k)+Z_{l}(k), \tag{2}\]  
where \(\varphi(k)=\Delta_{c}+k\Delta_{s}\), \(\alpha(k)\) is an amplitude attenuation incurred by \(\varphi(k)\), \(H_{l}(k)\) is the CFR with variance \(\sigma_{H}^{2}\), \(\Delta_{\tau}\) is the residual STO, \(Z_{l}(k)\) is the zero-mean additive white Gaussian noise (AWGN) with variance \(\sigma_{Z}^{2}\), and \(C_{l}(k)\) is the frequency-offset-induced ICI with variance \(\sigma_{\mathbb{C}}^{2}\). In (2), \(\alpha(k)\) and \(C_{l}(k)\) can be expressed as  
\[\alpha(k)=\frac{\sin(\pi(\varphi(k))}{N\sin(\pi\varphi(k)/N)} \tag{3}\]  
and  
\[C_{l}(k)=\sum_{n=0,n\
eq k}^{N-1}H_{l}(n)X_{l}(n)e^{j2\pi\varphi(n)(lN_{s}+N_ {g})/N}\frac{\sin(\pi(\varphi(n)+n-k))}{N\sin(\pi(\varphi(n)+n-k)/N)}e^{j\pi( N-1)(\varphi(n)+n-k)N}. \tag{4}\]  
Assuming that \(X_{l}(n)\) are statistically independent for different \(n\)'s and \(l\)'s, from the central limit theorem \(C_{l}(k)\) is treated as a zero-mean Gaussian random variable with variance [20]  
\[\sigma_{\mathbb{C}}^{2}=E_{X}\sigma_{H}^{2}\sum_{n=0,n\
eq k}^{N-1}\frac{\sin ^{2}(\pi(\varphi(n)+n-k))}{N^{2}\sin^{2}(\pi(\varphi(n)+n-k)/N)}, \tag{5}\]  
where \(E_{X}=|X_{l}(k)|^{2}\). Note that \(\alpha(k)\approx 1\) and \(\sigma_{\mathbb{C}}^{2}\approx E_{X}\sigma_{H}^{2}\Delta_{c}^{2}\pi^{2}/3\) for typical values of \(\Delta_{c}\) and \(\Delta_{s}\)[9]. Figure 1 shows the assignment of an OFDM segment and program in the ISDB-T system. As shown in Figure 1, one ISDB-T channel consists of 13 OFDM segments and up to three hierarchical layers can be supported with respect to these segments. The audio program segment is positioned in the middle of the frequency band, which is called narrowband ISDB-T. The narrowband ISDB-T mode only uses single or triple OFDM segments, whereas the wide-band ISDB-T system contains up to 13 segments and supports HDTV service. Table 1 summarizes the basic transmission parameters of each mode for ISDB-T, where \(N_{d}\) and \(N_{c}\) are the numbers in differential-modulation and coherent-modulation segments, respectively. One OFDM frame includes 204 OFDM symbols and pilot carriers. The pilot signals include SP, CP, TMCC, and AC. The SP is present only in coherent-modulation segments in order to estimate the channel, whereas the other pilots can be mainly used to acquire time and frequency synchronization.