## 3 Reduced-Complexity ICFO and CIDS Detection Method
### Detection Algorithm  
For a simple presentation, denote \(q_{l}\) as the transmitted symbol index of the \(l\)-th NR-PSS with CIDS \(w\) located in OFDM blocks. Therefore, we have \(X_{q_{l}}(m)=P_{w}(m)\) and \(X_{q_{l}+2}(m)=S_{g}(m)\) within each SSB. As shown in (9), we perform hypothesis testing for potential ICFO and CIDS candidates, which involves significant computational demands. To address such complexity issues, a computationally efficient ICFO and CIDS synchronization scheme is proposed, without sacrificing the detection performance of (9). Figure 1 depicts the receiver structure outlining the initial synchronization process for the 5G NR communication system. Utilizing the property that the differential relation of an m-sequence does not alter the characteristics of the m-sequence, we observe that three m-sequences \(D_{\oplus}(m)\)'s (\(\widehat{w}=0,1,2\)) exhibit cyclically shifted replicas of one another. Consequently, when the number of subcarriers is sufficiently large, i.e., \(M_{\mathrm{s}}\gg 1\), the cross-correlation between any two of three m-sequences becomes essentially zero. More importantly, it can be observed that three m-sequences are interrelated as \(D_{\oplus}(m)=D_{0}(m)B_{\oplus}(m)\), where \(B_{\oplus}(m)=D_{\oplus}(m)/D_{0}(m)\in\{-1,1\}\) for \(\widehat{w}=0,1,2\). The phase rotation indicator \(B_{\widehat{w}}(m)\) captures the cyclic shift between \(D_{0}(m)\) and its cyclic-shifted versions \(D_{\oplus}(m)\) for \(\widehat{w}=1,2\). In order to mitigate the impact of interference on the estimated parameter, adopting the average estimate yields the following cost function given by  
\[\Omega_{red}(\vartheta,\widehat{w})=\sum_{l=1}^{M_{\mathrm{s}}}\sum_{m\in \mathcal{D}}\bar{Y}_{q_{l}}(m+\vartheta)D_{0}^{*}(m)B_{\oplus}^{*}(m) \tag{12}\]  
with  
\[\bar{Y}_{q_{l}}(m)\approx|G_{q_{l}}(m-v)|^{2}D_{w}(m-v)+\bar{I}_{q_{l}}(m)+\bar {Z}_{q_{l}}(m), \tag{13}\]  
where \(M_{\oplus}\) is the number of average estimates,  
\[ \begin{split}\bar{I}_{q_{l}}(m)&=G_{q_{l}}(m-v)X_{q _{l}}(m-v)I_{q_{l}}^{*}(m-1)e^{2\pi\epsilon(q_{l}M_{u}+M_{g})/M}\\ &+G_{q_{l}}^{*}(m-v-1)X_{q_{l}}^{*}(m-v-1)I_{q_{l}}(m)e^{-j2\pi \epsilon(q_{l}M_{u}+M_{g})/M}+I_{q_{l}}(m)Z_{q_{l}}^{*}(m-1)\\ &+I_{q_{l}}^{*}(m-1)Z_{q_{l}}(m)+I_{q_{l}}(m)I_{q_{l}}^{*}(m-1), \end{split}  \tag{14}\]  
and  
\[ \begin{split}\bar{Z}_{q_{l}}(m)&=G_{q_{l}}(m-v)X_{q _{l}}(m-v)Z_{q_{l}}^{*}(m-1)e^{j2\pi\epsilon(q_{l}M_{u}+M_{g})/M}\\ &+G_{q_{l}}^{*}(m-v-1)X_{q_{l}}^{*}(m-v)Z_{q_{l}}(m)e^{-j2\pi \epsilon(q_{l}M_{u}+M_{g})/M}+Z_{q_{l}}(m)Z_{q_{l}}^{*}(m-1).\end{split}  \tag{15}\]  
If we replace \(\Omega_{pss}(\vartheta,\widehat{w})\) with \(\Omega_{red}(\vartheta,\widehat{w})\) in (9), the estimates \((\vartheta,\widehat{w})\) are similarly obtained. Utilizing the cyclic-shifted feature of m-sequences, we can implement autocorrelation with the local template \(D_{\oplus}(m)\) for the remaining hypothetical candidates \(\widehat{w}\in\{1,2\}\) and  
Figure 1: Block diagram of the synchronization receiver for 5G NR systems. \(\vartheta\in\{-U,-U+1,\cdots,U\}\) by compensating the polarity of the pre-calculated autocorrelation for \(\tilde{w}=0\) on a subcarrier-wise basis. This compensation is performed based on the phase rotation indicator \(B_{\oplus}(m)\), eliminating the need for hardware multipliers, and all operations involve only shift accumulation. As a result, the completion of (12) necessitates the computation of only one autocorrelation for each trial value of \(\tilde{w}\). This formulation results in a decrease in the arithmetic complexity, as it involves performing fewer multiplication operations. Upon substituting (3) into (12), one obtains  
\[ \begin{split}\Omega_{red}(\vartheta,\tilde{w})&= \sum_{l=1}^{M_{\oplus}}\sum_{m\in\mathcal{D}}|G_{q_{l}}(m+\vartheta-v)|^{2}D_{ w}(m+\vartheta-v)D_{0}^{*}(m)B_{\oplus}^{*}(m)\\ &+\sum_{l=1}^{M_{\oplus}}\sum_{m\in\mathcal{D}}\bar{I}_{q_{l}}(m) D_{0}^{*}(m)B_{\oplus}^{*}(m)+\sum_{l=1}^{M_{\oplus}}\sum_{m\in\mathcal{D}}\bar{Z}_{q_{ l}}(m)D_{0}^{*}(m)B_{\oplus}^{*}(m).\end{split}