# Effective Blind Frequency Offset Estimation Scheme for BST-OFDM Based HDTV Broadcast Systems
## 4 Proposed Estimation Scheme
### Algorithm Description
Recalling from (18) that \(\bar{\varphi}(k)=2\Delta_{c}+(k_{i}+k_{i+1})\Delta_{s}\) is a linear function of \(\Delta_{c}\) and \(\Delta_{s}\), the proposed joint estimation of CFO and SFO can be constructed in the LS manner as follows:  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]0  
and  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]1  
where  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]2  
It is worthwhile to mention that the number of common TMCC signals \(N_{p}=13\) for 2k mode, \(N_{p}=26\) for 4k mode, and \(N_{p}=52\) for 8k mode [4]. Since the number of the common TMCC subcarriers may be insufficient in the case of 2k mode, we adopt the averaged estimation over \(N_{a}\) symbol intervals as in (14). Of course, the cost to be paid for using this averaging approach is an increase in computational complexity. For notational convenience, (17) is rearranged into  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]3  
Under the high SNR condition and pilot boosting, the argument of \(\tilde{Y}_{l}(k_{i})\) can be well approximated by  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]4  
where \(X^{Q}\) be the quadrature component of \(X\), and \(\tilde{C}_{l}(k)=\tilde{C}_{l}(k)e^{-j2\pi\bar{\varphi}(k)\rho}\) and \(\tilde{Z}_{l}(k)=\tilde{Z}_{l}(k)e^{-j2\pi\bar{\varphi}(k)\rho}\) are statistically equivalent to \(\tilde{C}_{l}(k)\) and \(\tilde{Z}_{l}(k)\), respectively. From (19)-(23), it is easy to find that the estimation error of CFO and SFO is calculated by  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]5  
and  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]6where \(I(\Delta_{s})\) and \(I(\Delta_{c})\) are the biases incurred from non-equidistant TMCC subcarriers given by  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]7  
and  
\[=\sum_{l=1}^{N_{p}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2}\bar{X}_ {l}(k_{i})\bar{X}_{l}(k_{i+1})e^{j2\pi\varphi(k_{i})\rho}+\sum_{l=1}^{N_{p}} \bar{C}_{l}(k_{i})+\sum_{l=1}^{N_{p}}\bar{Z}_{l}(k_{i}),\;i=1,2,\cdots,N_{p}-1,\]8  
As shown in (26) and (27), \(I(\Delta_{s})\) and \(I(\Delta_{c})\) are functions of frequency offsets and pilot indices so that they are constant as the SNR increases. Since frequency offsets are sufficiently small during the post-FFT tracking mode and TMCC subcarriers are evenly distributed over the entire transmission spectrum, \(I^{2}(\Delta_{s})\) and \(I^{2}(\Delta_{c})\) are insignificant as compared to the powers of their respective first terms in (24) and (25).