## 5 NB-IoT Cell Search
### NPSS Detection and Time Domain Synchronization  
Constructing a proper TM is the most efficient way to detect NPSS. However, because the traditional NF in TM will be invalid in the context of low SNR [30] and the SNR of extended coverage NB-IoT can be \(-15\) dB, solutions in [7, 8] abandon the use of NF. Nevertheless, this compromise reduces the consumption of some hardware resources, and the lengthened synchronization time consumes much power which is vital for NB-IoT terminals. Here, a noise-eliminated NF with an FFO pre-estimator is proposed to deal with low SNR conditions. As the NB-IoT samples are preceded by noise samples and NB-IoT terminals work with low mobility, the noise power \(\sigma_{\omega}^{2}\) can be estimated by these noise-only samples. The calculation process of autocorrelation summation can apply the sliding  
Figure 3: Illustration of the UTG inserted during the uplink transmission of NPRACH. method in [10]. The application of code cover in [11] avoids the plateau in the presence of NPSS, and a steep roll-off trajectory will present. As for the IFO and fine timing error, the cross-correlation between coarse detected NPSS and ideal NPSS is used as an estimator. In the following, the detailed procedures are discussed. As shown in Equation (5), the correct time of NPSS is given by  
\[\tau_{0}=\operatorname*{arg\,max}_{\tau}|\gamma(\tau)|, \tag{9}\]  
and the TM proposed in this paper is defined as \(\gamma_{prop}(\tau)\). The symbol-wise autocorrelation part and the noise-eliminated NF are represented by \(\alpha_{SW}\) and \(\beta_{NE}\), respectively. To reduce detection complexity, the authors of [7] accomplish the NPSS detection process at quite low frequency. However, not only do more radio frames have to be used, but also the synchronization precision is decreased. To accelerate this procedure, the authors of [8] adopt the octuple working frequency of [7]. The autocorrelation distance in this method is only one OFDM symbol. Our scheme is implemented with a simplified symbol-wise method compared with [7] in order to adapt to the higher working frequency. Thus, \(\alpha_{SW}\) has the following form. \[\alpha_{SW}(\tau)=\left|\sum_{r^{\prime}_{f}=1}^{R^{\prime}_{f}}\sum_{v=1}^{V} \sum_{m^{\prime}=3}^{M^{\prime}-v}\sum_{n=1}^{N+N_{g}}c(m^{\prime}-3)r_{p,r^{ \prime}_{f}}^{(m^{\prime})*}(n)c(m^{\prime}-3+v)r_{p,r^{\prime}_{f}}^{(m^{ \prime}+v)}(n)\right|^{2}, \tag{10}\]  
where \(\tau\) denotes the start sample of the subframe \(p\) in radio frame \(r^{{}^{\prime}}_{f}\), and \((\cdot)^{{}^{\prime}}\) is the hypothetical value of each parameter. \(v\) represents the number of OFDM symbols between the two parts of one correlation pair. Furthermore, the calculation of \(\alpha_{SW}(\tau+1)\) can be conducted in a convenient way,  
\[\alpha_{SW}(\tau+1)= \Bigg{|}\sum_{r^{\prime}_{f}=1}^{R^{\prime}_{f}}\sum_{v=1}^{V} \sum_{m^{\prime}=3}^{M^{\prime}-v}\bigg{(}\sum_{n=1}^{N+N_{g}}c(m^{\prime}-3) r_{p,r^{\prime}_{f}}^{(m^{\prime})*}(n)c(m^{\prime}-3+v)r_{p,r^{\prime}_{f}}^{(m^{ \prime}+v)}(n) \tag{11}\] \[\quad-c(m^{{}^{\prime}}-3)r_{p,r^{\prime}_{f}}^{(m^{\prime})*}( 1)c(m^{{}^{\prime}}-3+v)r_{p,r^{\prime}_{f}}^{(m^{\prime}+v)}(1)\] \[\quad+c(m^{{}^{\prime}}-3)r_{p,r^{\prime}_{f}}^{(m^{\prime})*}(N +N_{g}+1)c(m^{{}^{\prime}}-3+v)r_{p,r^{\prime}_{f}}^{(m^{\prime}+1)}(N+N_{g}+ v)\bigg{)}\Bigg{|}^{2}.\]  
As the NB-IoT terminals are considered to be working with low mobility, the number of used radio frames \(R^{{}^{\prime}}_{f}\) and the maximum symbol spacing \(V\) can be adjusted according to the SNR. All the NPSS detection methods implemented with TM no longer use NF. That is because NF is almost ineffective in low SNR conditions, as shown in Figure 4, where we use the Magnitude-of-Difference (MoD) NF proposed in [12] which has good performance in the AWGN channel. The result shows that although the NF still contributes a valley to add to the resolution of TM at the correct time, the valley decreases fast with the SNR. One feasible solution is to weaken the influence of noise on NF.