## 4 Proposed Detection Scheme
### Algorithm Description  
From the fact that \(P_{1}(k)=P_{2}^{*}(k)\), it follows that \(D_{1}(k)=D_{2}^{*}(k)\). This property yields Equation (9) in the form  
\[\Phi_{a}^{I}(m,n)=\sum_{k\in\mathcal{D}^{+}}\hat{Y}_{l}^{I}(k+m)D _{1}^{I}(k)+(-1)^{n+1}\sum_{k\in\mathcal{D}^{+}}\hat{Y}_{l}^{Q}(k+m)D_{1}^{Q}( k),\ n=1,2, \tag{14}\]which indicates that the ambiguity associated with two ZC root indices \(z_{1}\) and \(z_{2}\) is only the polarity in the second term of the right-hand side (RHS). Therefore, the metrics for \(z_{1}\) and \(z_{2}\) are based on the same observation. Leveraging the above property, a reduced search space based metric is given by  
\[\Phi_{d}(m,n)=\sum_{k\in\mathcal{D}^{+}}\hat{\Upsilon}_{I}^{I}(k+m)D_{n}^{I}(k )+\left|\sum_{k\in\mathcal{D}^{+}}\hat{\Upsilon}_{I}^{Q}(k+m)D_{n}^{Q}(k) \right|,\ n=0,1, \tag{15}\]  
where the polarity depending on \(n\) is removed by taking the absolute value, and \(\Phi_{d}(m,n)\) represents the joint metric of the proposed scheme. After substituting Equations (5) to (15), one gets for \(n=0,1\)  
\[\Phi_{d}(m,n)=\sum_{k\in\mathcal{D}^{+}}\mathcal{H}_{i}^{I}(k+m-v)D_{n}^{I}(k )+\sum_{k\in\mathcal{D}^{+}}\hat{W}_{I}^{I}(k)+\left|\sum_{k\in\mathcal{D}^{+ }}\mathcal{H}_{i}^{Q}(k+m-v)D_{n}^{Q}(k)+\sum_{k\in\mathcal{D}^{+}}\hat{W}_{I }^{Q}(k)\right|, \tag{16}\]  
with  
\[ \begin{split}\mathcal{H}_{i}(k+m-v)&=|H_{I}(k+m-v )|^{2}D_{i}(k+m-v)\\ &+|H_{I}(-k+m-v+1)|^{2}D_{i}^{*}(-k+m-v+1)\end{split}  \tag{17}\]  
and  
\[\hat{W}_{I}^{I/Q}(k)=\{\hat{W}_{I}^{I/Q}(k)+\hat{W}_{I}^{I/Q}(-k+1)\}D_{n}^{I /Q}(k), \tag{18}\]  
where \(I/Q\) represents the corresponding term-wise pair. When the local PSS with ZC root index \(z_{n}\) is synchronized with the \(m\)-th shifted version of the received PSS with ZC root index \(z_{i}\), or, equivalently, when \(m=v\) and \(n=i\), Equation (16) becomes  
\[\Phi_{d}(m,n)=\sum_{k\in\mathcal{D}^{+}}|\hat{H}_{I}(k)|^{2}|D_{n}^{I}(k)|^{2} +\sum_{k\in\mathcal{D}^{+}}\hat{W}_{I}^{I}(k)\ +\left|\sum_{k\in\mathcal{D}^{+}}|\hat{H}_{I}(k)|^{2}|D_{n}^{Q}(k)|^{2}+\sum_ {k\in\mathcal{D}^{+}}\hat{W}_{I}^{Q}(k)\right|, \tag{19}\]  
where \(|\hat{H}_{I}(k+m-v)|^{2}=|H_{I}(k+m-v)|^{2}+|H_{I}(-k+m-v+1)|^{2}\). If \(D_{i}(k)\)'s are statistically independent for different \(i\)'s and \(k\)'s, using the central limit theorem, the first term on the RHS of Equation (15) converges in distribution to a zero-mean Gaussian RV under the hypothesis that the PSS signal is not synchronized (i.e., \(m\
eq v\)). Thus, the second term on the RHS of Equation (15) is a folded Gaussian RV when \(m\
eq v\). The IFO and partial SID are firstly recovered by finding the peak of \(\Phi_{d}(m,n)\) over \((m,n)\), which is expressed as  
\[(\hat{v},\hat{n})=\operatorname*{arg\,max}_{|m|\leq M,n\in\{0,1\}}\Phi_{d}(m, n). \tag{20}\]  
Inspection of Equation (20) reveals that the joint search space can be reduced to \(4M+2\) hypotheses, leading to substantial complexity reduction. If \(\hat{n}=0\) in Equation (20), the MTC UE decides that the transmitted PSS was associated with ZC root index \(z_{0}\). In this case, the joint detection of IFO and SID is finished. Otherwise, it is assumed that the PSS with ZC root index \(z_{1}\) or \(z_{2}\) was transmitted from the eNodeB. To fully complete the PSS detection task, therefore, it is necessary to determine which PSS was actually transmitted and this is accomplished by deciding  
\[\Phi_{a}^{I}(\hat{v},1)\underset{z_{2}}{\gtrless},\Phi_{a}^{I}(\hat{v},2), \tag{21}\]  
where  
\[\Phi_{a}^{I}(\hat{v},1)-\Phi_{a}^{I}(\hat{v},2)=2\sum_{k\in\mathcal{D}^{+}} \hat{\Upsilon}_{I}^{Q}(k+\hat{v})D_{1}^{Q}(k). \tag{22}\]Using Equation (14) and recalling that \(D_{1}(k)=D_{2}^{*}(k)\), Equation (21) can be put in an equivalent form as  
\[\hat{n}=\frac{3-s(\Phi_{a}^{I}(\vartheta,1)-\Phi_{a}^{I}(\vartheta,2))}{2}, \tag{23}\]  
where \(s(x)=1\) for \(x\geq 0\) and \(s(x)=-1\), otherwise. The joint IFO and SID detection algorithm is summarized below. 1.