# Data-Aided Frequency Offset Estimation for CE-OFDM Broadband Satellite Systems
## 5 Pilot Symbol-Based CFO Estimator and the CRB  
The structure of the PSCE includes a phase demodulator, a phase unwrap module, and a DFT module, and it is shown in Figure??. The difference with [??] is that the CFO is estimated based on the entire pilot symbols instead of a single null subcarrier. We will prove that this can improve the estimation accuracy. According to the analysis above, the output of DFT is \(Y_{\theta}(k)\) (Equation (4)). If the pilot symbols \(S_{p}(k)|_{k=1}^{N_{p}}\) are inserted at the transmitter, then Equation (4) can be rewritten as (5), where \(\overline{Y}_{\theta}(k)=Y_{\theta}(k)-2\pi\hbar S_{p}(k)\). \[\overline{Y}_{\theta}(k)=\frac{1}{\sqrt{N}}\!\!\sum_{n=0}^{N-1} \big{(}\frac{2\pi n\Delta f}{N}+\theta\big{)}e^{\frac{-2\pi\hbar t}{N}}+ \overline{W}_{\theta}(k) \tag{5}\]  
Figure 4: Pilot symbol structure. The linear relationship between the \(\Delta f\) and the \(\theta\) as well as the \(\overline{Y}_{\theta}(k)\) can be expressed as  
\[\mathbf{V}\cdot\left[ \begin{array}{cc}\Delta f&\theta\end{array} \right]^{T}+\mathbf{W} =\mathbf{Q} \tag{6}\]  
where \(\mathbf{V}\), \(\mathbf{W}\), and \(\mathbf{Q}\) are:  
\[\mathbf{V}= \begin{bmatrix}\sum\limits_{n=0}^{N-1}\frac{2g\mu}{N}\cos(\frac{2\pi n0 }{N})&\sum\limits_{n=0}^{N-1}\cos(\frac{2\pi n0}{N})\\ \ldots&\ldots\\ \sum\limits_{n=0}^{N-1}\frac{2\pi n}{N}\cos(\frac{2\pi n(N-1)}{N})&\sum\limits _{n=0}^{N-1}\cos(\frac{2\pi n(N-1)}{N})\\ \sum\limits_{n=0}^{N-1}\frac{2\pi n}{N}\sin(\frac{2\pi n0}{N})&-\sum\limits_{n =0}^{N-1}\sin(\frac{2\pi n0}{N})\\ \ldots&\ldots\\ -\sum\limits_{n=0}^{N-1}\frac{2\pi n}{N}\sin(\frac{2\pi n(N-1)}{N})&-\sum \limits_{n=0}^{N-1}\sin(\frac{2\pi n(N-1)}{N})\\ \end{bmatrix}  \tag{7}\]  
\[\mathbf{W}= \begin{bmatrix}\Re\left\{\overline{W}_{\theta}(0)\right\},\ldots,\Re \left\{\overline{W}_{\theta}(N-1)\right\},\Im\left\{\overline{W}_{\theta}(0) \right\},\ldots,\Im\left\{\overline{W}_{\theta}(N-1)\right\}\end{bmatrix} ^{T}\]  
\[\mathbf{Q}= \begin{bmatrix}\Re\left\{\overline{Y}_{\theta}(0)\right\},\ldots,\Re \left\{\overline{Y}_{\theta}(N-1)\right\},\Im\left\{\overline{Y}_{\theta}(0) \right\},\ldots,\Im\left\{\overline{Y}_{\theta}(N-1)\right\}\end{bmatrix} ^{T}\]  
It can be seen from Equation (6) that \(\Delta f\) can be estimated by a minimum variance unbiased (MVU) estimator according to the theory in [11]. The MVU estimator of \(\Delta f\) and the corresponding CRB can be obtained as (8) and (9), respectively:  
\[\Delta\hat{f}=((\mathbf{V}^{T}\mathbf{V})^{-1}\mathbf{V}^{T}\mathbf{Q})_{1,1}, \tag{8}\]  
\[CRB=\frac{\sigma_{w}^{2}}{2\overline{A}}((\mathbf{V}^{T}\mathbf{V})^{-1})_{1,1}, \tag{9}\]  
where \(\sigma_{w}^{2}/\overline{A}\) is the variance of \(\overline{W}_{\theta}(k)\). Similar to [11], the ideal estimation range of \(\Delta f\) is \((-N/2,N/2)\), which is half the size of IDFT. After some simplifications, we obtain  
\[CRB=\frac{\sigma_{w}^{2}}{2\overline{A}\sum\limits_{i=2}^{2N}\left|\mathbf{V}_{i,1 }\right|^{2}}. \tag{10}\]  
Therefore, the purpose of the pilot symbol placement is to minimize the CRB, which is equal to the maximum value of \(\sum\limits_{i=2}^{2N}\left|\mathbf{V}_{i,1}\right|^{2}\). The relationship between the value of \(\left|\mathbf{V}_{i,1}\right|^{2}\) and the frequency position of the pilot symbol \(S_{p}(k)\) is shown in Figure 11. If a pilot symbol is inserted at the transmitter, the corresponding value of \(\mathbf{V}_{i,1}\) and \(\mathbf{V}_{N+i,1}\) can be obtained (e.g., insert \(S_{p}(1)\) to obtain \(\mathbf{V}_{2,1}\) and \(\mathbf{V}_{N+2,1}\)), which corresponds to the real and the imaginary coefficients of this pilot symbol. The value of \(\left|\mathbf{V}_{i,1}\right|^{2}\) can be obtained through mathematical analysis:  
\[\left|\mathbf{V}_{i,1}\right|^{2}\approx\frac{\pi^{2}}{N},\quad\;i\in[2,N]\]  
\[\left|\mathbf{V}_{N+i,1}\right|^{2}\;=\left|\mathbf{V}_{2N+2-i,1}\right|_{2},\;\;i\in [2,\frac{N}{2}] \tag{11}\]  
Figure 5: The structure of pilot symbol-based CFO estimator (PSCE).