## 2 System Description
### Synchronization Signal  
Let us consider the sidelink of an LTE system. As depicted in Figure 1, one LTE radio frame has a duration 10 ms and consists of equally sized 20 slots each of 0.5 ms. Each slot in turn is made up of a number of SC-FDMA symbols which can be either seven or six based on the CP mode. In contrast to LTE DL, the D2D sidelink uses a pair of PSSS and SSSS, which are periodically transmitted in two consecutive OFDM symbols every 40 ms. Physical sidelink broadcast channel (PSBCH) can be used as synchronization measurement for determining whether or not the out-of-coverage UE becomes a synchronization source, while demodulation reference signal (DMRS) can be used to produce channelestimates for demodulation of the associated physical channel. An LTE sidelink network can support 336 different sidelink identities indexed by \(N_{ID}\in\{0,1,2,\cdots,335\}\), which is retrieved from the combination of the SI embedded in the PSSS and the sidelink ID embedded in the SSSS. The SI present in the PSSS informs whether the synchronization source UE is in-network or out-of-network. The in-network-coverage case is numbered as \(\{0,1,2,\cdots,167\}\), while the out-of-network scenario is numbered as \(\{168,169,\cdots,335\}\). The PSSS is generated based on Zadoff-Chu (ZC) sequence that has the property of constant amplitude zero autocorrelation. The ZC root sequence index \(u_{i}\ (i=0,1)\) is used to represent two sets corresponding to in-network and out-of-network cases. A UE within the network coverage transmits PSSS symbols with \(u_{0}=26\) for \(N_{ID}\leq 167\) and transmits PSSS symbols with \(u_{1}=37\) otherwise. The PSSS is generated from a frequency-domain ZC sequence of length 63, with the middle part zero-padded to avoid modulating the DC subcarrier. Thus, the DC subcarrier and the rest of the subcarriers in the symbols where the PSSS is located are nulled and 31 subcarriers are mapped on each side of the DC subcarrier. Two PSSS sequences are constructed to have good periodic autocorrelation and cross-correlation properties [28]  
\[P_{i}(k)= \begin{cases}e^{-j\pi u_{i}(k^{2}+992)/63-j\pi u_{i}k},\ k\in\mathcal{ P}\\ 0,\ \text{otherwise}\end{cases}  \tag{4}\]  
where \(\mathcal{P}=\{k|-N_{p}/2\leq k\leq N_{p}/2,k\
eq 0\}\) is the set of PSSS subcarriers with \(N_{p}\) being the number of non-zero PSSS subcarriers. The primary usage of the SSSS is to retrieve the sidelink identity and to acquire the beginning point of an LTE frame. The SSSS is a frequency-domain binary phase-shift keying (BPSK) sequence, which is an interleaved combination of two length-31 scrambled maximum length sequences \(s_{w_{0}}(q)\) and \(s_{w_{1}}(q)\), with \(q=0,1,\cdots,N_{p}/2-1\). In the case of the normal CP, the SSSS is generated as  
\[H(2q)= \begin{cases}s_{w_{0}}(q)h_{0}(q)\ \ \text{in SC-FDMA symbol 11}\\ s_{w_{1}}(q)h_{0}(q)\ \ \text{in SC-FDMA symbol 12}\end{cases}  \tag{5}\]  
Figure 1: Frame structure in the long term evolution (LTE)-device-to-device (D2D) system. \[H(2q+1)= \begin{cases}s_{w_{1}}(q)h_{1}(q)d_{w_{0}}(q)\ \ \text{in SC-FDMA symbol 11}\\ s_{w_{0}}(q)h_{1}(q)d_{w_{1}}(q)\ \ \text{in SC-FDMA symbol 12}\end{cases}  \tag{6}\]  
where \(h_{i}(q)\) and \(d_{w_{i}}(q)\) represent the scrambling sequences. The subscripts \(w_{0}\) and \(w_{1}\) are directly related with the value of the sidelink identity. The sequence \(H(q)\) is mapped onto central \(N_{p}\) SSSS subcarriers with indices \(k\in\mathcal{P}\), yielding the SSSS sequence \(S_{j}(k)\) with sidelink ID \(j\). The arrangement of the PSSS and SSSS sequences differs depending on the CP type. For simple notation, we consider the situation where the first PSSS is transmitted at the \(l\)-th symbol, i.e., \(X_{l}(k)=P_{i}(k)\). According to the CP type, the first SSSS is denoted as  
\[S_{j}(k)= \begin{cases}X_{l+10}(k),&\text{for the normal CP}\\ X_{l+9}(k),&\text{for the extended CP}\end{cases}  \tag{7}\]  
where \(j\in\{0,1,2,\cdots,167\}\).