## 3 Robust Low-Complexity CFO Estimator
### Iterative CFO Estimator  
Even though the proposed technique is shown to provide a reliable CFO estimate effectively, how to reduce the implementation complexity is a real challenge, especially for an OFDM system that has a large number of subcarriers. Here, we propose an efficient method to further improve the computational complexity. In comparison with the traditional approaches, the presented scheme applies the Taylor series expansion technique to simplify the optimum problem instead of using exhaustive search for finding the minimum value in Equation (14). For a small \(\epsilon\), the CFO-driven diagonal matrix \(\mathbf{D}_{\mathbf{a}(\epsilon)}\) can be approximated to a first-order Taylor series expansion expressed by:  
\[\mathbf{D}_{\mathbf{a}(\epsilon)}\approx\mathbf{I}_{N}+\epsilon\mathbf{D}_{ \mathbf{a}_{0}} \tag{30}\]  
Substituting Equation (30) into Equation (13), we have:  
\[\hat{\mathbf{h}}(\epsilon)\approx\mathbf{F}^{H}\mathbf{Q}\mathbf{D}_{\mathbf{ X}_{P}}^{-1}\mathbf{S}\left(\mathbf{I}_{N}+\epsilon\mathbf{F}\mathbf{D}_{ \mathbf{a}_{0}}\mathbf{F}^{H}\right)\mathbf{R} \tag{31}\]  
According to Equation (31), we can approximate the CRE in Equation (14) as:  
\[\begin{split} f(\epsilon)&\approx\left\|\tilde{ \mathbf{J}}\mathbf{F}^{H}\mathbf{Q}\mathbf{D}_{\mathbf{X}_{P}}^{-1}\mathbf{S} \left(\mathbf{I}_{N}+\epsilon\mathbf{F}\mathbf{D}_{\mathbf{a}_{0}}\mathbf{F}^ {H}\right)\mathbf{R}\right\|^{2}\\ &=\mathbf{r}^{H}\text{Tr}+\epsilon\mathbf{r}^{H}\mathbf{A} \mathbf{r}-\epsilon^{2}\mathbf{r}^{H}\mathbf{D}_{\mathbf{a}_{0}}\text{Tr} \mathbf{D}_{\mathbf{a}_{0}}\text{Tr} \tag{32}\]The approximation in Equation (32) reveals that the optimum problem is simplified to a one-dimension optimization by minimizing the CRE value. Clearly, it can be easily shown that the optimal selection of \(\hat{\epsilon}_{d}\) is derived as:  
\[\hat{\epsilon}_{d}=\frac{\mathbf{r}^{H}\mathbf{A}\mathbf{r}}{2\times\mathbf{r}^ {H}\mathbf{D}_{\hat{\mathbf{a}}_{0}}\mathbf{T}\mathbf{D}_{\hat{\mathbf{a}}_{0}} \mathbf{r}} \tag{33}\]  
Because the \(N\times N\) matrix \(\mathbf{T}\) is independent of the received data, it can be pre-determined while performing the CFO estimation. The above-mentioned method only works on the premise that \(\mathbf{D}_{\mathbf{a}(\epsilon)}\) is well approximated to \((\mathbf{I}_{N}+\epsilon\mathbf{D}_{\hat{\mathbf{a}}_{0}})\) if CFO is small enough, _i.e._, \(\epsilon_{d}\ll 1\). However, when \(\epsilon_{d}\) does not satisfy the condition of \(\epsilon_{d}\ll 1\), Equation (30) does not hold any more, and thus, we cannot obtain a reliable CFO estimate by means of Equation (33). For this reason, it is important to find an alternative approach for enhancing the robustness against CFO when \(\epsilon_{d}\) is not small. Next, an iteration scheme is proposed instead. Specifically, we use the current CFO estimate in Equation (33) to compensate for the received data in the next iteration. Repeating the iterations and using the \((j-1)\)-th iteration result, denoted as \(\hat{\epsilon}_{d}^{(j-1)}\), along with linear algebraic manipulations, the CFO estimate at the \(j\)-th iteration is expressed as:  
\[ \begin{array}{l}\hat{\epsilon}_{d}^{(j)}=\frac{\mathbf{r}^{(j-1)H}\mathbf{A }\mathbf{r}^{(j-1)}}{2\times\mathbf{r}^{(j-1)H}\mathbf{D}_{\mathbf{a}_{0}} \mathbf{T}\mathbf{D}_{\mathbf{a}_{0}}\mathbf{r}^{(j-1)}}\\ =\frac{\mathbf{r}^{H}\mathbf{\Lambda}(\epsilon_{d}^{(j-1)}\mathbf{A}\mathbf{A}^ {H}(\epsilon_{d}^{(j-1)})\mathbf{r}}{2\times\mathbf{r}^{H}\mathbf{\Lambda}( \epsilon_{d}^{(j-1)})\mathbf{D}_{\mathbf{a}_{0}}\mathbf{T}\mathbf{D}_{\hat{ \mathbf{a}}_{0}}\mathbf{\Lambda}^{H}(\epsilon_{d}^{(j-1)})\mathbf{r}}\end{array}  \tag{34}\]  
where, by definition,  
\[\mathbf{\Lambda}(\epsilon_{d}^{(j)})=\left.\mathbf{D}_{\mathbf{a}( \epsilon)}\right|_{\epsilon=\hat{\epsilon}_{d}^{(j)}} \tag{35}\] \[\mathbf{r}^{(j)}=\mathbf{\Lambda}^{H}(\epsilon_{d}^{(j)})\mathbf{r}\]  
In general, the iterative approach ultimately achieves a precision CFO estimate.