## 1 Introduction  
To ensure realization of machine-to-machine communications, the third generation partnership project proposed enhanced long-term evolution machine-type communication (LTE-MTC), extended coverage-global system for mobile communications for the Internet of Things (EC GSM-IoT), and narrow-band IoT (NB-IoT) as cellular-based low-power wide area (LPWA) schemes for the IoT [1; 2; 3]. Among them, MTC introduces a set of physical layer functions targeting supporting the wide coverage, big network, and low power consumption of user equipment (UE). Due to the limited transmission power, however, the MTC UE cannot arbitrarily increase its power to reach the enhanced base station (eNodeB). In particular, MTC devices in low mobility are usually encountered with deep coverage holes, leading to poor operating signal-to-noise ratio (SNR) [4]. Therefore, a main challenging issue in MTC is to provide coverage enhancement (CE) to enhance battery life and improve the performance of MTC units suffering from poor coverage problems [5]. A variety of techniqUE such as symbol repetition, frequency hopping, and reference-signal boosting have been suggested to improve detection ability of MTC UE in a poor coverage area [6; 7]. Since narrow bandwidth and single-antenna configuration are adopted in LTE-MTC, either frequentist or spacial diversity cannot be used. For this reason, frequency hopping is a new added feature to improve the performance of MTC UE within the reduced bandwidth [6]. To further support CE, subframe repetition over time is introduced in LTE-MTC. Using a repetition mechanism, the effect of time diversity can favorably enhance the capacity of the LTE-MTC network. However, such a repetition increases latency and wake-up time of MTC UE [7]. To shorten the stand-by time of the MTC UE, a discontinuous reception (DRX) scheme is used in the LTE-MTC system [8]. An MTC UE equipped with a DRX cycle wakes up only during its paging occasion to monitor paging messages from eNodeB, while it sleeps at other time periods, which facilitates the UE to sleep for very long periods of time. The prolonged sleep duration between events causes the local oscillator to deviate from its original operating frequency [9]. Since the receiver may lose its synchronization while switching off parts of the circuitry, it is necessary for the MTC receiver to wake up periodically at a specific time to achieve synchronization and cell identification. Because of the vulnerability of orthogonal frequency division multiplexing (OFDM) to synchronization imperfections, a reduced-cost synchronization scheme is necessary for the MTC device to shorten the search period, thus extending sleeping duration. Before connecting with the eNodeB, the MTC UE attempts to acquire synchronization looking for the position of the maximum correlation of the cyclic prefix (CP) [10] and identifying the synchronization signals, called the primary synchronization signal (PSS) and secondary synchronization signal (SSS) [11; 12; 13]. The PSS carries the information of sector ID (SID), whereas the SSS is transmitted to deliver the information about group ID (GID) and is also used to determine 10 ms radio frame boundary. Since the MTC UE can complete SSS detection only after the SID is correctly retrieved, the accurate identification of the PSS plays an important role in overall cell search procedure [9]. There exist some studies focusing on the PSS detection [13; 14; 15; 16; 17; 18]. Some of these methods perform in the time domain [13; 14], while the others use frequency-domain samples obtained from a discrete Fourier transform (DFT) unit [15; 16; 17; 18]. In [14], the PSS detection method was proposed to be performed by finding the maximum of the cross-correlation in the time domain, wherein the integer carrier frequency offset (IFO) degrades the accuracy of detecting the PSS.