## 4 Proposed Detection Scheme
### Algorithm Description  
Since the TMCC signal is information-bearing, its content is available after TMCC decoding and the pilot-compensated signal is not obtained in the presence of the IFO, as performed in (10). On the other hand, the common TMCC information is present in fixed subcarrier positions regardless of modulation mode, which is transmitted by ways of differential binary phase-shift keying (DBPSK) modulation. Using this property of the continuous-type TMCC signal, the correlation from two received OFDM symbols is in the form:  
\[\hat{R}_{l}(k)=e^{j2\pi\epsilon_{i}\rho}e^{j2\pi\epsilon_{j}(1+\rho)}|H_{l}(k- \epsilon_{i})|^{2}\hat{X}_{l}(k-\epsilon_{i})+\hat{I}_{l}(k)+\hat{Z}_{l}(k), \tag{11}\]  
where \(\hat{I}_{l}(k)\) is the contribution to combined ICI given by \[ \begin{split}\hat{I}_{l}(k)&=e^{-2\pi e(l(1+\rho)+\rho)}H _{l}^{*}(k-\epsilon_{i})X_{l}^{*}(k-\epsilon_{i})I_{l+1}(k)\\ &+e^{2\pi e((l+1)(1+\rho)+\rho)}H_{l+1}(k-\epsilon_{i})X_{l+1}(k -\epsilon_{i})I_{l}^{*}(k)\\ &+I_{l+1}(k)Z_{l}^{*}(k)+I_{l}^{*}(k)Z_{l+1}(k)+I_{l}^{*}(k)I_{l +1}(k),\end{split}  \tag{12}\]  
and \(\hat{Z}_{l}(k)\) is the contribution to the combined AWGN, equal to  
\[ \begin{split}\mathcal{Z}_{l}(k)&=e^{-\beta\pi e(l (1+\rho)+\rho)}H_{l}^{*}(k-\epsilon_{i})X_{l}^{*}(k-\epsilon_{i})Z_{l+1}(k)\\ &+e^{\beta 2\pi e((1+\rho)+\rho)}H_{l+1}(k-\epsilon_{i})X_{l+1}(k- \epsilon_{i})Z_{l}^{*}(k)\\ &+Z_{l}^{*}(k)Z_{l+1}(k).\end{split}  \tag{13}\]  
As can be seen from (11), the residual FFO affects the objective metric in two ways--one is to rotate the phases of all sub-carriers by the same amount, and the other is to cause the ICI. Using the differential relation between two consecutive symbols, the TMCC-based objective metric can be defined as  
\[ \begin{split}\Omega_{d}(m)&=\sum_{k\in\mathcal{T}} \hat{R}_{l}(k+m)\\ &=e^{2\pi e\rho}e^{\beta 2\pi e_{f}(1+\rho)}\sum_{k\in\mathcal{T}}|H_{l }(k+m-\epsilon_{i})|^{2}\hat{X}_{l}(k+m-\epsilon_{i})\\ &+\sum_{k\in\mathcal{T}}\hat{I}_{l}(k+m)+\sum_{k\in\mathcal{T}} \hat{Z}_{l}(k+m),\end{split}  \tag{14}\]  
where \(\mathcal{T}\) is the set of TMCC subcarrier indices having \(N_{t}\) members, and the condition \(k+m-\epsilon_{i}\in\mathcal{T}\) informs us that a true IFO exists at the \(m\)-th shift, or rather, \(m=\epsilon_{i}\). In this case, recognizing that \(\hat{X}_{l}(k)=\pm|X_{l}(k)|^{2}\) for \(k\in\mathcal{T}\), the phase-rotated TMCC-based objective metric takes the form  
\[ \begin{split}\tilde{\Omega}_{d}(m)=B_{l}E_{X}\sum_{k\in\mathcal{T }}|H_{l}(k+m-\epsilon_{i})|^{2}+\sum_{k\in\mathcal{T}}\tilde{I}_{l}(k+m)+\sum _{k\in\mathcal{T}}\tilde{Z}_{l}(k+m),\end{split}  \tag{15}\]  
where \(\tilde{Z}_{l}(k+m)=\hat{Z}_{l}(k+m)e^{-\beta 2\pi m\rho}\) and \(B_{l}\in\{1,-1\}\) denotes the differential relation from two consecutive TMCC symbols. For notational brevity, the phase rotation term \(e^{\beta 2\pi e_{f}(1+\rho)}\) was omitted in the first quantity on the right-hand side (RHS) in (15), since its effect is removed by taking the absolute value in the proposed approach. Note that \(\tilde{Z}_{l}(k)\) is statistically equivalent to \(\hat{Z}_{l}(k)\). Under the assumption of a relatively large \(N_{t}\), (15) can be safely approximated as  
\[ \begin{split}\tilde{\Omega}_{d}(m)\approx B_{l}E_{X}N_{t}\sigma_ {H}^{2}+\sum_{k\in\mathcal{T}}\tilde{I}_{l}(k+m)+\sum_{k\in\mathcal{T}}\tilde {Z}_{l}(k+m).\end{split}  \tag{16}\]  
From the central limit theorem, (16) can be modeled as a Gaussian random variable having mean \(B_{l}E_{X}N_{l}\sigma_{H}^{2}\) and variance \(N_{t}\sigma_{I}^{2}+N_{t}\sigma_{Z}^{2}\), where \(\sigma_{I}^{2}=2E_{X}\sigma_{H}^{2}\sigma_{I}^{2}+2\sigma_{I}^{2}\sigma_{Z}^{ 2}+\sigma_{I}^{4}\) is the variance of \(\hat{I}_{l}(k)\), and \(\sigma_{Z}^{2}=2E_{X}\sigma_{H}^{2}\sigma_{Z}^{2}+\sigma_{Z}^{4}\) is the variance of \(\hat{Z}_{l}(k)\), if \(m\
eq\epsilon_{i}\), \(\hat{X}_{l}(k)\) is an unknown random signal whose mean is zero. In this hypothesis, \(\tilde{\Omega}_{d}(m)\) is expressed as  
\[