## 2 Signal Model
Even though the CFO is very small, the estimator provided by Equation (3) may not work properly. To relieve the impact on the system performance caused by CFO, a variety of approaches have been proposed in the past. Let us consider a typical example, which employs redundant pilot symbols to perform CFO estimation [19]. Recall that the desired channel response with a channel length of \(L\) can be modeled by \(\mathbf{h}=[h(0),h(1),\cdots,h(L)\ \ \mathbf{0}_{N-L-1}^{T}]^{T}\), where \(h(i)\
eq 0,\)\(i=0,1,..,L\) and the other entries are all zeros. For the estimated channel response \(\hat{\mathbf{h}}\), the last \(N-L-1\) entries of \(\hat{\mathbf{h}}\) are negligible, because they are essentially induced by the channel noise while CFO is absent. Else, the magnitudes of these entries increase with the value of CFO. That is, the minimization of the energy composed of the last \(N-L-1\) entries of \(\hat{\mathbf{h}}\) can be regarded as a useful criterion for estimating the CFO. The procedure of [PERSON]'s algorithm [19] is summarized as follows:  
**(1)**: Compute the CFO-compensated data:  
\[\tilde{\mathbf{R}}(\epsilon)=\mathbf{F}\mathbf{D}_{\mathbf{a}(\epsilon)}^{H} \mathbf{F}^{H}\mathbf{R} \tag{5}\]
**(2)**: Compute the CFR estimate of pilot symbols according to Equation (3):  
\[\hat{\mathbf{H}}_{P}(\epsilon)=\mathbf{D}_{\mathbf{X}_{P}}^{-1}\mathbf{S} \tilde{\mathbf{R}}(\epsilon) \tag{6}\]
**(3)**: By operating \(K\)-point inverse discrete Fourier transform (IDFT), the time-domain channel response estimate of size \(K\) is given by:  
\[ \begin{split}\hat{\mathbf{h}}_{[PERSON]}(\epsilon)&= \mathbf{S}\mathbf{F}^{H}\mathbf{S}^{T}\hat{\mathbf{H}}_{P}(\epsilon)\\ &=\mathbf{S}\mathbf{F}^{H}\mathbf{S}^{T}\mathbf{D}_{\mathbf{X}_{ P}}^{-1}\mathbf{S}\tilde{\mathbf{R}}(\epsilon)\end{split}  \tag{7}\]* Compute the energy of the last \(K-L-1\) entries of \(\hat{\mathbf{h}}_{[PERSON]}(\epsilon)\), which is termed the channel residual energy (CRE), defined by: \[ \begin{split} f(\epsilon)&=\sum_{j=\hat{L}+2}^{K} \left|[\hat{\mathbf{h}}_{Chung}(\epsilon)]_{j}\right|^{2}\\ &=\left|\mathbf{J}\hat{\mathbf{h}}_{Chung}(\epsilon)\right|^{2} \end{split} \] (8) where \(\mathbf{J}=[\mathbf{O}_{(K-\hat{L}-1)\times(\hat{L}+1)},\mathbf{I}_{K-\hat{L} -1}]\) and \(\hat{L}\) is the estimate of the channel order \(L\). * Determine the CFO estimate in accordance with: \[ \begin{split}\hat{\epsilon}_{d}&=\text{arg}\min_{ \epsilon}\ \ f(\epsilon)\\ &=\text{arg}\min_{\epsilon}\ \ \left\|\mathbf{J}\mathbf{F}_{K}^{H} \mathbf{D}_{\mathbf{X}_{P}}^{-1}\text{S}\tilde{\mathbf{R}}(\epsilon)\right\|^ {2}\end{split} \] (9) where \(\mathbf{F}_{K}\) denotes the first \(K\) columns of \(\mathbf{F}\). In spite of proving good performance, [PERSON]'s method deteriorates in performance while \(L\) is not correctly estimated. Under the condition of \(\hat{L}>L\), although the last few entries of \(\hat{\mathbf{h}}_{Chung}(\epsilon)\) may be very small, they result in estimation errors. On the other hand, this scheme fails to provide acceptable performance when \(\hat{L}<L\). How to correctly estimate the channel order is a critical challenge for [PERSON]'s algorithm. What is worse is that this method breaks down as the number of pilot symbols, \(K\), is less than or equal to the channel order. In addition, the searching granularity for \(\epsilon_{d}\) in Equation (9) is given by a heuristic approach, which involves the trade-off of estimate accuracy and computational complexity. For these considerations, in the next section, we will present a robust CFO estimator for a pilot-based OFDM system while avoiding these unfavorable factors.