# Effective Blind Frequency Offset Estimation Scheme for BST-OFDM Based HDTV Broadcast Systems
## 4 Proposed Estimation Scheme
### MSE Analysis
Therefore, plugging (30) and (32) into (28), we have  
\[E\left\{\left|\hat{\Delta}_{c}-\Delta_{c}\right|^{2}\right\} \tag{34}\] \[=\frac{1}{16\pi^{2}\rho^{2}N_{a}(N_{p}-1)}\left(\frac{\sigma_{Z}^ {2}}{E_{X}}\sum_{m=0}^{1}E\left\{\frac{1}{|H_{l}(k_{i+m})|^{2}}\right\}+\frac{ 2\sigma_{\mathbb{C}}^{2}\sigma_{Z}^{2}+\sigma_{Z}^{4}}{2E_{X}^{2}}\sum_{m=0}^{ 1}E\left\{\frac{1}{|H_{l}(k_{i+m})|^{4}}\right\}\right)\] \[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]0  
Using (33), after some manipulations, the final MSE of the LS CFO estimate can be obtained by  
\[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]1 \[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]2  
where \(\lambda_{1}=-E_{i}(-p_{min}/\sigma_{H}^{2})\), \(\lambda_{2}=E_{i}(-p_{min}/\sigma_{H}^{2})+e^{-p_{min}/\sigma_{H}^{2}}/(p_{min}/ \sigma_{H}^{2})\), \(\gamma_{z}=\sigma_{H}^{2}E_{X}/\sigma_{Z}^{2}\) is the average SNR, and \(\gamma_{c}=\sigma_{H}^{2}  
\[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]3 \[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]4  
Substituting (30) and (32) to (36) yields the final MSE of the SFO estimate as  
\[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]5 \[E\left\{\frac{\sum_{l=1}^{N_{a}}|\hat{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} =E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{4}|\hat{ Z}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{ 2})^{2}}\right\} \tag{31}\]6  
where the first term of the RHS disappears as SNR increases and the other terms form an irreducible error floor caused by frequency offsets.