# Effective Blind Frequency Offset Estimation Scheme for BST-OFDM Based HDTV Broadcast Systems
## 4 Proposed Estimation Scheme
### MSE Analysis  
As a performance measure of the estimation scheme, we drive a closed-form expression of MSE of joint LS estimation in the frequency-selective fading channel. In our analysis, we assume that the channel is sufficiently frequency-selective such that the channel fading coefficients between neighboring subcarriers become uncorrelated. Having in mind that \(E\{\tilde{C}_{l}^{Q}(k_{i})\}=E\{\tilde{Z}_{l}^{Q}(k_{i})\}=0\) and \(\tilde{C}_{l}^{Q}(k_{i})\) is statistically independent of \(\tilde{Z}_{l}^{Q}(k_{i})\), we obtain from (24)  
\[ \begin{split} E\left\{\left|\hat{\Delta}_{c}-\Delta_{c}\right|^{2 }\right\}&=\left(\frac{1}{4\pi\rho(N_{p}-1)E_{X}^{2}}\right)^{2} \sum_{i=1}^{N_{p}-1}E\left\{\frac{\sum_{l=1}^{N_{a}}|\tilde{C}_{l}^{Q}(k_{i}) |^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\} \\ &+\left(\frac{1}{4\pi\rho(N_{p}-1)E_{X}^{2}}\right)^{2}\sum_{i=1} ^{N_{p}-1}E\left\{\frac{\sum_{l=1}^{N_{a}}|\tilde{Z}_{l}^{Q}(k_{i})|^{2}}{( \sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\}+l^{2}( \Delta_{s}).\end{split}  \tag{28}\]  
After substituting (15) to (28), we immediately establish  
\[ \begin{split} E\left\{\frac{\sum_{l=1}^{N_{a}}|\tilde{C}_{l}^{Q} (k_{i})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}} \right\}&=E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i}) |^{4}|\tilde{C}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_ {l}(k_{i+1})|^{2})^{2}}\right\}\\ &+E_{X}^{2}E\left\{\frac{\sum_{l=1}^{N_{a}}|H_{l}(k_{i+1})|^{4}| \tilde{C}_{l}^{Q}(k_{i})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{ i+1})|^{2})^{2}}\right\}\\ &+E\left\{\frac{\sum_{l=1}^{N_{a}}|\tilde{C}_{l}^{Q}(k_{i})|^{2}| \tilde{C}_{l}^{Q}(k_{i+1})|^{2}}{(\sum_{l=1}^{N_{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_ {i+1})|^{2})^{2}}\right\}.\end{split}  \tag{29}\]  
Since the variance of \(\hat{C}_{l}(k)\) in (15) is \(\sigma_{\mathcal{C}}^{2}=2E_{X}\sigma_{H}^{2}\sigma_{\mathcal{C}}^{2}+\sigma_ {\mathcal{C}}^{4}\), one can readily find that \(E\{\hat{C}_{l}^{Q}(k_{i})|^{2}|\tilde{C}_{l}^{Q}(k_{i+1})|^{2}\}=(E_{X}\sigma_ {H}^{2}\sigma_{\mathcal{C}}^{2}+\sigma_{\mathcal{C}}^{4}/2)^{2}\). Recalling from (5) that \(\sigma_{\mathcal{C}}^{2}\approx E_{X}\sigma_{H}^{2}\Delta_{c}^{2}\pi^{2}/3\) and pilot symbol is booted at \(E_{X}=16/9\), the power of the last term of the RHS in (29) is very small compared to the powers of other terms; thereby, we can omit this term. Considering that the channel is time-invariant during \(N_{a}\) symbol durations, after substituting (7) to (29), it can be calculated as  
\[E\left\{\frac{\sum_{l=1}^{N_{a}}|\tilde{C}_{l}^{Q}(k_{i})|^{2}}{(\sum_{l=1}^{N _{a}}|H_{l}(k_{i})|^{2}|H_{l}(k_{i+1})|^{2})^{2}}\right\}=\frac{E_{X}^{3} \sigma_{\mathcal{C}}^{2}}{N_{a}}\sum_{m=0}^{1}E\left\{\frac{1}{|H_{l}(k_{i+m})| ^{2}}\right\}+\frac{E_{X}^{2}\sigma_{\mathcal{C}}^{4}}{2N_{a}}\sum_{m=0}^{1}E \left\{\frac{1}{|H_{l}(k_{i+m})|^{4}}\right\}. \tag{30}\]It is obvious from (16) that the variance of \(\hat{Z}_{l}(k)\) is \(\sigma_{\hat{Z}}^{2}=2E_{X}\sigma_{H}^{2}\sigma_{Z}^{2}+2\sigma_{\mathbb{C}}^{2} \sigma_{Z}^{2}+\sigma_{Z}^{4}\). Therefore, the variances of \(\hat{\mathcal{C}}_{l}(k_{i})\hat{Z}_{l}(k_{i+1})\) and \(\hat{Z}_{l}(k_{i})\hat{Z}_{l}(k_{i+1})\) are very small compared to the powers of other terms in (16).