## 2 Related Work  
Both cell search and frequency tracking are important issues in NB-IoT and legacy LTE, and many articles on these points have been published by scholars. As the cell search procedure can be divided into NPSS detection and NSSS detection, we will discuss the existing solutions in three parts: (1) for NPSS detection, (2) for NSSS detection, and (3) for frequency tracking separately. 1. Although NPSS detection is a problem of deterministic signal detection, the signal location is uncertain and it is actually a synchronization problem. Traditionally, NPSS detection is accomplished with symbol-wise sliding autocorrelation by using the duplicate property of NPSS [7], and this method tries to use quite several NB-IoT frames to achieve acceptable performance. However, to reduce the hardware consumption and computation complexity, this method operates at very low frequency with the decimated samples. As a result, lots of radio frames are occupied by the NPSS detector which increases the communication link setup time of the NB-IoT transceiver system and results in excessive power consumption. To shorten the detection time, [PERSON] and [PERSON]. [8] adopt a full rate autocorrelation method. As they make the autocorrelation window longer than one subframe, the detected location of NPSS will become indistinct. In [9], [PERSON] et al. present an ML NPSS detector which can jointly estimate the frequency offset and symbol timing. The authors declare that this solution can estimate the whole range frequency and timing offset simultaneously, and it is true that it can deal with small range frequency offset. However, the estimation accuracy is severely restricted by the FFT points and working frequency. If high-precision frequency offset tracking is demanded, more than 64 k points FFT is demanded, which is unrealistic in NB-IoT terminals. Another aspect is that the existence of an integer frequency offset is not considered in this solution. Additionally, frame synchronization based on duplicated synchronization signals becomes the research object of many scholars. [PERSON] and [PERSON] [10] proposed taking advantage of the preamble to construct a timing metric (TM) which consists of an autocorrelation part and a normalization factor (NF). To ensure that the TM is robust to CFO, a modified TM is proposed in [11]. To deal with different CFO and SNR scenes [12], summarizes two TMs based on two differential NFs and gives the suitable CFO and SNR ranges for each TM. The results show that these methods reduce the ability to deal with low SNR conditions. The authors of [13; 14; 15] make use of high-order statistics to accelerate the detection and synchronization procedures. Nevertheless, even though these methods bring some performance improvement, the enormous amount of calculation is difficult to achieve with the hardware of IoT terminals. 2. When NPSS detection is conducted, and timing synchronization and CFO recovery have been completed, NSSS detection will be performed in the frequency domain. Different from legacy LTE, the cell ID of NB-IoT is decided by NSSS entirely which has more diverse sequences compared with LTE [8]. However, as both SSS detection and NSSS detection are determinate signal detection problems, many SSS detection schemes can be used for reference. Furthermore, these solutions can be mainly divided into two classes: coherent detection and non-coherent detection [16]. Considering the coherent SSS detection, the key operation is resorting to the known PSS to estimate the channel frequency responses (CFR) of each subcarrier in the frequency domain, and using the CFRs to improve the performance of correct SSS cross correlation results [17; 18]. Unfortunately, despite NB-IoT operating in slow fading wireless environments [19; 20], which means that even NPSS and NSSS are separated by no less than 4 subframes, the channels can still be regarded as constant.