## 5 NB-IoT Cell Search
### NPSS Detection and Time Domain Synchronization
To achieve this aim, we created a noise-eliminated differential NF which is composed of a differential part and a noise elimination part. The noise power \(\sigma_{\omega}^{2}\) can be estimated with a moving average filter (MAF) before the arrival of the NB-IoT frame. The average window is set as \(T_{0}\), and the estimation is as follows,  
\[\sigma_{\omega}^{2}(n_{0})=\frac{1}{T}\sum_{n=n_{0}}^{n_{0}+T}|r_{\omega}(n)| ^{2}, \tag{12}\]  
where \(n_{0}\) denotes the window start time of MAF. Furthermore, in order to promote the accuracy of estimation, \(\Lambda\) consecutive windows are coherently averaged until the NB-IoT radio frame presents, and an infinite impulse response filter is applied. \[\overline{\sigma}_{\omega}^{2}(n_{\lambda})=\eta\overline{\sigma}_{\omega}^{2}(n_{ \lambda-1})+(1-\eta)\sigma_{\omega}^{2}(n_{\lambda}), \tag{13}\]  
with  
\[\overline{\sigma}_{\omega}^{2}(n_{0})=\sigma_{\omega}^{2}(n_{0}), \tag{14}\]  
here, \(\eta\) is set as a constant 0.8 (This value is obtained by software simulation with proper NB-IoT working environment). Conventional MoD NF has the original form of  
\[\beta_{MOD}(\tau)=\sum_{n=\tau+1}^{\tau+N+N_{g}}\left|r(n)-r(n+N+N_{g})\right|^ {2}. \tag{15}\]  
The above equation is the two identical parts version in [11]. However, to adapt to the structure of NPSS, we modify it into a symbol-wise form similar to \(\alpha_{SW}(\tau)\) as the following equation. \[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]0  
Since the NB-IoT terminals work with a slow fading channel, the multi-path signals in one detection window can be assumed as independent zero mean Gaussian random signals with the variance of \(\delta_{s}^{2}\). By using the approximation similar to [11] and considering the presence of NPSS, the below expression can be obtained. \[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]1 \[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]2  
with  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]3  
Similarly, when the NPSS is absent, the approximation is  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]4  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]5 Comparison of MOD normalization factors at different SNRs. where \(\tau_{p}\) and \(\tau_{a}\) represent the present and absent time of the NPSS signal, respectively. The ratio between \(\beta(\tau_{p})\) and \(\beta(\tau_{a})\) dictates the performance of NF. Considering the following relationship  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]6  
when an NF is effective, the parameter \(\Re>0\) is required. The precise range of \(\varepsilon_{F}\) to fulfil \(\Re>0\) is too complex to be decided, as it is determined jointly by \(V\) and \(M^{\prime}\). So, we investigate a reduced range by the following equation,  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]7  
with  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]8  
It is obvious that when the NPSS is absent, the \(\hat{\varepsilon}_{F}\) is just a random phase rotation to the NF. When the FFO estimation error fulfils the following relationship  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]9  
the \(\Re\) of \(\beta_{MMOD}^{{}^{\prime\prime}}(\tau)\) can be written as  
\[\alpha_{SW}(\tau)=\left|\sum_{r^{\prime}_{f}=1}^{R^{\prime}_{f}}\sum_{v=1}^{V} \sum_{m^{\prime}=3}^{M^{\prime}-v}\sum_{n=1}^{N+N_{g}}c(m^{\prime}-3)r_{p,r^{ \prime}_{f}}^{(m^{\prime})*}(n)c(m^{\prime}-3+v)r_{p,r^{\prime}_{f}}^{(m^{ \prime}+v)}(n)\right|^{2}, \tag{10}\]0 \[\alpha_{SW}(\tau)=\left|\sum_{r^{\prime}_{f}=1}^{R^{\prime}_{f}}\sum_{v=1}^{V} \sum_{m^{\prime}=3}^{M^{\prime}-v}\sum_{n=1}^{N+N_{g}}c(m^{\prime}-3)r_{p,r^{ \prime}_{f}}^{(m^{\prime})*}(n)c(m^{\prime}-3+v)r_{p,r^{\prime}_{f}}^{(m^{ \prime}+v)}(n)\right|^{2}, \tag{10}\]1  
The above equation shows that the SNR is larger, and the NF performance is better.