# Effective Blind Frequency Offset Estimation Scheme for BST-OFDM Based HDTV Broadcast Systems
## 3 Conventional Estimation Scheme  
The Post-FFT synchronization is practically performed in the frequency domain by the use of uniformly distributed pilot symbols as studied in [20; 21; 22; 23]. Although the ISDB-T system provides periodically distributed SPs for receiver synchronization, its location will be known after decoding TMCC signals. On the other hand, the presence of CPs in many broadcast systems [2; 3; 4] makes it possible to correlate two consecutive OFDM symbols. Considering non-periodically distributed property of CPs in the frequency direction, the frequency-offset estimation scheme performed on a per-subcarrier basis is appropriate for the ISDB-T system. Therefore, this paper considers a carrier-by-carrier estimation scheme [17; 18; 19] that relies on the continuous-type pilot such as CP. By assuming that \(H_{l}(k)=H_{l+1}(k)\) and \(\alpha(k)\approx 1\), one obtains a conjugate product across consecutive symbols as [20]  
\[ \begin{split} Y_{l}(k)&=R_{l}^{*}(k)R_{l+1}(k)\\ &=|H_{l}(k)|^{2}X_{l}^{*}(k)X_{l+1}(k)e^{\beta\pi\varphi(k)\rho}+ \hat{C}_{l}(k)+\hat{Z}_{l}(k),\end{split}  \tag{6}\]  
where \(\rho=N_{s}/N\), \(\hat{C}_{l}(k)\) is the combined ICI given by \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Mode 1** & **Mode 2** & **Mode 3** \\ \hline FFT size & 2k & 4k & 8k \\ Bandwidth & 5.575 MHz & 5.573 MHz & 5.572 MHz \\ Subcarrier spacing & 3.968 kHz & 1.984 kHz & 0.992 kHz \\ Effective symbol duration & 252 μs & 504 μs & 1008 μs \\ Number of segments & 13 & 13 & 13 \\ Number of carriers per segments & 108 & 216 & 432 \\ Number of usable carriers & 1405 & 2809 & 5617 \\ Number of SP carriers & \(9\times N_{c}\) & \(18\times N_{c}\) & \(36\times N_{c}\) \\ Number of CP carriers & \(N_{d}+1\) & \(N_{d}+1\) & \(N_{d}+1\) \\ Number of TMCC carriers & \(N_{c}+5\times N_{d}\) & \(2\times N_{c}+10\times N_{d}\) & \(4\times N_{c}+20\times N_{d}\) \\ Number of AC1 carriers & 26 & 52 & 104 \\ Number of AC2 carriers & \(4\times N_{d}\) & \(9\times N_{d}\) & \(19\times N_{d}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Transmission parameters for ISDB-T [4]. Figure 1: ISDB-T segment and program allocations [4]. \[\hat{\mathcal{L}}_{l}(k) =H_{l}^{*}(k)X_{l}^{*}(k)C_{l+1}(k)e^{j2\pi k\Delta_{\tau}/N}e^{-j2 \pi\varphi(k)(lN_{k}+N_{g})/N} \tag{7}\] \[+H_{l+1}(k)X_{l+1}(k)C_{l}^{*}(k)e^{-j2\pi k\Delta_{\tau}/N}e^{j2 \pi\varphi(k)((l+1)N_{k}+N_{g})/N}+C_{l}^{*}(k)C_{l+1}(k)\]  
and \(\hat{Z}_{l}(k)\) is the combined AWGN given by  
\[\hat{Z}_{l}(k) =H_{l}^{*}(k)X_{l}^{*}(k)Z_{l+1}(k)e^{j2\pi k\Delta_{\tau}/N}e^{-j 2\pi\varphi(k)(lN_{k}+N_{g})/N} \tag{8}\] \[+H_{l+1}(k)X_{l+1}(k)Z_{l}^{*}(k)e^{-j2\pi k\Delta_{\tau}/N}e^{j2 \pi\varphi(k)((l+1)N_{k}+N_{g})/N}\] \[+C_{l+1}(k)Z_{l}^{*}(k)+C_{l}^{*}(k)Z_{l+1}(k)+Z_{l}^{*}(k)Z_{l+1 }(k).\]  
From [20], \(C_{l}(k)\) behaves like zero-mean Gaussian noise with variance \(\sigma_{C}^{2}\). Since \(E\{C_{l}(k)\}=E\{Z_{l}(k)\}=0\), it immediately follows from (7) and (8) that \(E\{\hat{C}_{l}(k)\}=E\{\hat{Z}_{l}(k)\}=0\). If \(X_{l}(k)\) and \(X_{l+1}(k)\) are known pilots at the receiver, by averaging the conjugate product across the multiple pilot pairs, the pilot-compensated signal is obtained as  
\[\tilde{Y}_{l}(k) =\sum_{l=1}^{N_{k}}Y_{l}(k)X_{l}(k)X_{l+1}^{*}(k) \tag{9}\] \[=\sum_{l=1}^{N_{k}}|H_{l}(k)|^{2}E_{X}^{2}e^{j2\pi\varphi(k) \rho}+\sum_{l=1}^{N_{k}}\hat{\mathcal{L}}_{l}(k)X_{l}(k)X_{l+1}^{*}(k)+\sum_{ l=1}^{N_{k}}\hat{Z}_{l}(k)X_{l}(k)X_{l+1}^{*}(k)\]  
and its argument can be expressed as  
\[\arg\left\{\tilde{Y}_{l}(k)\right\} =2\pi\varphi(k)\rho+I_{l}(k) \tag{10}\] Figure 1:0  
where \(\arg\{x\}\) is the radian phase angle of the complex number \(x\), \(I_{l}(k)\) is the appropriate AWGN plus ICI contribution after taking an argument, and \(N_{a}\) is the number of averaging used for mitigating interferences.