## 1 Introduction  
Orthogonal frequency division multiplexing (OFDM) has been shown to be an efficient modulation technique in wireless communication systems due to its excellent bandwidth efficiency and resistance to channel impairments. Long term evolution (LTE) is an attractive technology to be applied for a variety of wireless applications due to its ability to provide high spectral efficiency and low latency using OFDM transmission, which has been standardized by the third generation partnership project (3 GPP) [1]. Recently, cellular internet of things (IoT) solutions such as narrowband IoT and device-to-device (D2D) communications have been paid much attention as a new communication paradigm in existing and future cellular LTE networks [2; 3; 4; 5; 6; 7; 8]. D2D is considered a promising technology that provides a mechanism for reusing LTE's uplink physical layer such as single-carrier frequency division multiple access (SC-FDMA) and reference signals [5; 6]. Thus, IoT that enables ubiquitous connection among D2D devices without the essential control of the centralized supervision is a prime enabler to facilitate the evolution of routing the traffic via enhanced base station (eNodeB) from LTE-A systems to future 5G [8; 9; 10; 11]. In the LTE-D2D network, it is a challenging task to maintain the synchronization especially for out-of-network coverage scenarios, wherein the D2D devices have to synchronize with each other. To this end, the D2D device has not only to estimate symbol timing offset (STO) and carrier frequency offset (CFO) but also to capture the sidelink identity [12; 13; 14; 15; 16; 17; 18]. Toward this, a synchronization source user equipment (UE) periodically transmits two specified synchronization signals, named as primary sidelink synchronization signal (PSSS) and secondary sidelink synchronization signal (SSSS). Most of the existing synchronization schemes are designed for downlink (DL) detection and are not the best solution for sidelink synchronization, but they can be applied to the D2D scenarios [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. The sidelink identity is acquired by combining the set information (SI) transmitted in the PSSS and the sidelink ID transmitted in the SSSS. The initial cell search procedure in the LTE system generally involves three steps. During the first step of the procedure, the D2D device exploits the redundant cyclic prefix (CP) to find the fractional frequency offset (FFO) and initial STO [12; 13; 14]. In the second step, the PSSS is used to retrieve the SI and detect integer frequency offset (IFO) [20; 21; 22; 23; 24; 25; 26; 27]. After the completion of this stage, the SSSS is identified to determine the frame boundary and the sidelink ID [16; 17; 18]. Once the IFO and SI acquisition has been successfully achieved, the D2D UE can attempt to decode the SSSS and thus the correct detection of the PSSS is the most challenging task in the overall synchronization process. There are many works concentrated mainly on the detection of the PSSS in the literature [19; 20; 21; 22; 23; 24; 25; 26; 27]. Some of these methods perform in the time domain [19; 20], while others use the frequency-domain samples obtained from the discrete Fourier transform (DFT) unit [21; 22; 23; 24; 25; 26; 27]. In [19; 20], PSSS detection scheme has been presented to be performed by looking for the peak of the cross-correlation in the time domain. However, the performance of these approaches is significantly degraded in the case when the IFO is present. To solve this drawback, various differential correlation based PSSS detection schemes have been proposed in the frequency domain [21; 22; 23], wherein the IFO is estimated along with the PSSS. As for the IFO, it can be estimated either in a joint or decoupled way.