# Effective Blind Frequency Offset Estimation Scheme for BST-OFDM Based HDTV Broadcast Systems
## 4 Proposed Estimation Scheme
### Algorithm Description  
The number and position of TMCC subcarriers are changed in accordance with which modulation is being used in ISDB-T segments. Until the segment format is identified, which is completed after TMCC decoding, some TMCC subcarriers are unknown. Fortunately, there are common TMCC signals regardless of the segment format in the ISDB-T system, which is transmitted by means of the differential BPSK (DBPSK) modulation. Hence, the differential relation \(\bar{X}_{l}(k)=X_{l}^{*}(k)X_{l+1}(k)=\pm|X_{l}(k)|^{2}\) is the same for all TMCC subcarrier positions. In order to remove its corresponding phase ambiguity, the product across adjacent pilot subcarriers \(k_{i}\) and \(k_{i+1}\) can be used instead of multiplying known pilots to \(Y_{l}(k_{i})\) as done in the conventional approach (9). With the above properties in mind, the phase-compensated signal is defined as follows:  
\[\bar{Y}_{l}(k_{i}) =\sum_{l=1}^{N_{p}}Y_{l}(k_{i})Y_{l}(k_{i+1}) \tag{14}\] \[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]  
where \(N_{p}\) is the number of common TMCC signals, \(\tilde{\varphi}(k)=2\Delta_{c}+(k_{i}+k_{i+1})\Delta_{s}\), \(\bar{C}_{l}(k_{i})\) is the combined zero-mean ICI given by  
\[\bar{C}_{l}(k_{i}) =|H_{l}(k_{i})|^{2}X_{l}^{*}(k_{i})X_{l+1}(k_{i})\hat{C}_{l}(k_{i +1})e^{j2\pi\varphi(k_{i})\rho} \tag{15}\] \[+|H_{l}(k_{i+1})|^{2}X_{l}^{*}(k_{i+1})X_{l+1}(k_{i+1})\hat{C}_{l }(k_{i})e^{j2\pi\varphi(k_{i+1})\rho}+\hat{C}_{l}(k_{i})\hat{C}_{l}(k_{i+1})\]  
and \(\bar{Z}_{l}(k_{i})\) is the zero-mean AWGN contribution given by  
\[\bar{Z}_{l}(k_{i}) =|H_{l}(k_{i})|^{2}X_{l}^{*}(k_{i})X_{l+1}(k_{i})\hat{Z}_{l}(k_{i +1})e^{j2\pi\varphi(k_{i})\rho} \tag{16}\] \[+|H_{l}(k_{i+1})|^{2}X_{l}^{*}(k_{i+1})X_{l+1}(k_{i+1})\hat{Z}_{l }(k_{i})e^{j2\pi\varphi(k_{i+1})\rho}\] \[+\hat{C}_{l}(k_{i})\hat{Z}_{l}(k_{i+1})+\hat{C}_{l}(k_{i+1})\hat{Z }_{l}(k_{i})+\hat{Z}_{l}(k_{i})\hat{Z}_{l}(k_{i+1}).\]  
Recognizing the fact that the TMCC information is common to all TMCC subcarriers and is DBPSK modulated, \(\arg\{\bar{X}_{l}(k_{i})\}=\arg\{\bar{X}_{l}(k_{i+1})\}\) for all pilot subcarrier indices \(i\)'s during the \(l\)-th period. Therefore, \(\bar{X}_{l}(k_{i})\bar{X}_{l}(k_{i+1})=E_{X}^{2}\) such that  
\[\bar{Y}_{l}(k_{i})=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}E_{ X}^{2}e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}}\bar{C}_{l}(k_{i})+ \sum_{l=1}^{N_{p}}Z_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1, \tag{17}\]  
where \(E_{X}=|X_{l}(k)|^{2}\) is the TMCC symbol energy. The property (17) is the basis of the CFO and SFO estimation in the proposed blind method, which is only valid for binary-modulated signals. Since \(E\{\tilde{C}_{l}(k_{i})\}=E\{\tilde{Z}_{l}(k_{i})\}=0\), it follows that  
\[\arg\{E\{\tilde{Y}_{l}(k_{i})\}\} =2\pi\bar{\varphi}(k)\rho \tag{18}\] \[=4\pi\Delta_{c}\rho+2\pi(k_{i}+k_{i+1})\Delta_{s}\rho.\]  
When compared to (10), it is obvious from (18) that the amount of phase rotation of \(\tilde{Y}_{l}(k_{i})\) due to frequency offsets is approximately twice, which leads to the robustness to the noise. On the contrary, one can see from (9) and (17) that the noise contribution in (17) becomes approximately four times as large as that in (9).