# Robust Sampling Frequency Offset Estimation for OFDM over Frequency Selective Fading Channels
## 4 Proposed Estimation Scheme for DRM with Frequency Selective Fading  
In this section, a pilot-assisted SFO estimation algorithm is proposed with an improved accuracy in the presence of frequency selectivity. With the use of GRCs, a simple way of designing an unbiased estimation scheme is designed and its estimation performance is analytically derived. As reported in [22; 23], (14) is approximately unbiased in the case of frequency selective slowly fading channelswhen a small SFO is encountered. In addition, the unbiasedness of (14) is guaranteed if the pilot subcarriers are located symmetrically and distributed uniformly around the DC carrier. Unfortunately, the preliminary condition is not valid for DRM mode C and only one GRC pattern to meet this condition is existed for other DRM modes. To solve this problem, we derive a robust unbiased SFO estimation method, which has no restriction to pilot patterns. In contrast to the conventional schemes, the proposed scheme uses two-step correlation, namely temporal correlation and frequential correlation, which are adopted to remove the need of channel state information (CSI) and get an unbiased estimate in the presence of frequency selective fading, respectively. In the first step, we consider the temporal correlation as similarly used in the conventional scheme (14) to offset the unknown channel \(H_{l}(k)\). Without the loss of generality, in the following derivation we assume that \(H_{l}(k)\approx H_{l+D_{l}}(k)\), which leads to  
\[\hat{R}_{l}(k)=|H_{l}(k)|^{2}E_{X}e^{\beta 2\pi\phi(k)D_{l}N_{u}/N}+\tilde{I}_{l}(k )+\tilde{Z}_{l}(k) \tag{21}\]  
where \(E_{X}=|X_{l}(k)|^{2}\) is the pilot symbol energy. For the simple description, we omit the parameter \(b_{l}(k)\in\{1,-1\}\), assuming that its polarity is compensated at the receiver. In the next step, the frequential correlation is adopted to obtain an unbiased SFO estimation in a severe frequency selective fading channel. The frequential correlation is performed on two pilot subcarriers spaced \(k_{m}+k_{N_{p}/2+1-m}\) (\(m=1,\,2,\,\cdots,\,N_{p}/2\)) subcarrier apart, which is defined by  
\[ \begin{split}\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m}) &=|H_{l}(k_{m})|^{2}|H_{l}(-k_{N_{p}/2+1-m})|^{2}E_{X}^{2}e^{\beta 2\pi(k_{m}+k_{N_{p}/2+1-m}) \Delta_{k}D_{l}N_{u}/N}\\ &\quad+\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\end{split}  \tag{22}\]  
where \(\mathcal{I}_{l}(k_{m})\) is the ICI contribution given by  
\[ \begin{split}\mathcal{I}_{l}(k_{m})&=|H_{l}(k_{m})| ^{2}E_{X}\tilde{I}_{l}^{*}(-k_{N_{p}/2+1-m})e^{\beta 2\pi\phi(k_{m})D_{l}N_{u}/N}+\tilde{Z}_{l}(k_{m})\tilde{I}_{l}^{*}(-k_{N_{p}/2 +1-m})\\ &\quad+|H_{l}(-k_{N_{p}/2+1-m})|^{2}E_{X}\tilde{I}_{l}(k_{m})e^{ -\beta 2\pi\phi(-k_{N_{p}/2+1-m})D_{l}N_{u}/N}\\ &\quad+\tilde{Z}_{l}^{*}(-k_{N_{p}/2+1-m})\tilde{I}_{l}(k_{m})+ \tilde{I}_{l}(k_{m})\tilde{I}_{l}^{*}(-k_{N_{p}/2+1-m})\end{split}  \tag{23}\]  
and \(\mathcal{Z}_{l}(k_{m})\) is the AWGN contribution given by  
\[ \begin{split}\mathcal{Z}_{l}(k_{m})&=|H_{l}(k_{m})| ^{2}E_{X}\tilde{Z}_{l}^{*}(-k_{N_{p}/2+1-m})e^{\beta 2\pi\phi(k_{m})D_{l}N_{u}/N}\\ &\quad+|H_{l}(-k_{N_{p}/2+1-m})|^{2}E_{X}\tilde{Z}_{l}(k_{m})e^{ -\beta 2\pi\phi(-k_{N_{p}/2+1-m})D_{l}N_{u}/N}+\tilde{Z}_{l}(k_{m})\tilde{Z}_{l}^{*}(- k_{N_{p}/2+1-m})\end{split}  \tag{24}\]  
Since \(E\left[\mathcal{Z}_{l}(k)\right]=E\left[\mathcal{I}_{l}(k)\right]=0\), we eventually establish an estimate for \(\Delta_{s}\) as  
\[\hat{\Delta}_{s}=\frac{N}{2\pi D_{l}N_{u}}\frac{\sum_{m=1}^{N_{p}/2}\arg\left\{ \hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})\right\}}{\sum_{m=1}^{N_{p}/ 2}\left\{k_{m}+k_{N_{p}/2+1-m}\right\}} \tag{25}\]  
We compute hereafter the MSE of the estimation scheme (25) when the SNR is assumed to be high. Denote \(m_{v}=2\pi\Delta_{s}D_{l}(k_{m}+k_{N_{p}/2+1-m})N_{u}/N\).