## 4 Proposed Detection Method
### Sequential Detection Algorithm  
To remove the effect of the channel fading and residual STO, a conjugate product from the received PSSS and SSSS sequences is defined as  
\[ \begin{split} R_{l}(k)&=Y_{l}^{*}(k)Y_{l+D_{l}}(k) \\ &\approx|C_{l}(k-\epsilon)|^{2}P_{i}^{*}(k-\epsilon)S_{j}(k- \epsilon)e^{-jD_{t}\epsilon\rho}+W_{l}(k)\end{split}  \tag{16}\]  
with  
\[ \begin{split} W_{l}(k)&=C_{l+D_{l}}(k-\epsilon)S_{j} (k-\epsilon)Z_{l}^{*}(k)e^{-j2\pi(k-\epsilon)\tau/N}e^{j(l+D_{l})\epsilon\rho} \\ &+C_{l}^{*}(k-\epsilon)P_{i}^{*}(k-\epsilon)Z_{l+D_{l}}(k)e^{j2 \pi(k-\epsilon)\tau/N}e^{-j|\epsilon\rho}+Z_{l+D_{l}}(k)Z_{l}^{*}(k)\end{split}  \tag{17}\]  
where \(D_{t}\) is the distance between the PSSS and SSSS sequences. Note that \(D_{t}=9\) indicates the extended CP mode, while \(D_{t}=10\) corresponds to the normal CP mode. For notational convenience, it is assumed in (16) that channel remains the same during one subframe period. However, the impact of time selectivity of the channel on the detection performance will be evaluated later by means of simulations. We observe from (16) that it requires knowledge of the transmitted two synchronization signals. To enable sequential estimation of the IFO and SI, the intrinsic property of the PSSS and SSSS sequences is exploited. The sequential detection of the IFO and SI is based on observation of the correlation between the \(v\)-shifted version of \(R_{l}(k)\) and local PSSS  
\[ \begin{split}\bar{R}_{n}(k+v)&=R_{l}(k+v)P_{n}(k) e^{jD_{t}v\rho}\\ &=|C_{l}(k+v-\epsilon)|^{2}S_{j}(k+v-\epsilon)P_{i}^{*}(k+v- \epsilon)P_{n}(k)e^{jD_{l}(v-\epsilon)\rho}\\ &+W_{l}(k)P_{n}(k)e^{jD_{l}v\rho},\ D_{t}=9,10.\end{split}  \tag{18}\]  
The conjugate symmetric property of the SSSS in the time domain ensures that \(S_{j}(k)\) is binary in the frequency domain. To remove the dependency on the SSSS, an objective function is defined as  
\[\Omega_{s}(v,n)=\sum_{k\in\mathcal{P}}\left[\bar{R}_{n}(k+v)\right]^{2}, \tag{19}\]  
where \([\cdot]^{2}\) denotes the square operator. Using the fact that \(P_{0}(k)=P_{1}^{*}(k)\), the real part of (19) can be expressed in terms of \(P_{0}(k)\)  
\[ \begin{split}\Re\{\Omega_{s}(v,n)\}&=\sum_{k\in \mathcal{P}}\Re\left\{\left[R_{l}(k+v)\right]^{2}\right\}\Re\left\{\left[\bar{ P}_{0}(k)\right]^{2}\right\}\\ &+(-1)^{n+1}\sum_{k\in\mathcal{P}}\Im\left\{\left[R_{l}(k+v) \right]^{2}\right\}\Im\left\{\left[\bar{P}_{0}(k)\right]^{2}\right\},\ n=0,1, \end{split}  \tag{20}\]  
where \(\bar{P}_{n}(k)=P_{n}(k)e^{jD_{t}v\rho}\), \(\Im\{\cdot\}\) is the imaginary part of the enclosed quantity, and  
\[ \begin{split}[R_{l}(k+v)]^{2}&=|C_{l}(k+v-\epsilon )|^{4}[S_{j}(k+v-\epsilon)]^{2}[P_{i}^{*}(k+v-\epsilon)]^{2}e^{-j2D_{t}\epsilon\rho }\\ &+2|C_{l}(k+v-\epsilon)|^{2}S_{j}(k+v-\epsilon)P_{i}^{*}(k+v- \epsilon)e^{-jD_{t}\epsilon\rho}+[W_{l}(k)]^{2}.\end{split}  \tag{21}\]It is interesting to note that the uncertainty related with the PSSS is only the polarity in the second term on the right-hand side (RHS) of (20). Based on this observation, the complex conjugate relation between two PSSS sequences is exploited to estimate the IFO without relying on the transmitted PSSS information. For this purpose, we propose a novel metric as  
\[ \begin{split}\Omega_{f}(v)&=\sum_{k\in\mathcal{P}} \Re\left\{\left[R_{l}(k+v)\right]^{2}\right\}\Re\left\{\left[\bar{P}_{0}(k) \right]^{2}\right\}\\ &+\left|(-1)^{n+1}\sum_{k\in\mathcal{P}}\Im\left\{\left[R_{l}(k+v )\right]^{2}\right\}\Im\left\{\left[\bar{P}_{0}(k)\right]^{2}\right\}\right|,\ n=0,1,\end{split}  \tag{22}\]  
which indicates that \(\Omega_{f}(v)\) is constant to the index \(n\). Under the hypothesis that \(v=\epsilon\) and \(n=i\), using \(\left[S_{j}(k+v-\epsilon)\right]^{2}=E_{X}\), (22) can be expressed as  
\[