## 3 Reduced-Complexity ICFO and CIDS Detection Method
### Complexity Analysis  
To show the benefits of the proposed approach, we compare it with three representative NR-PSS synchronization schemes as baselines [20; 21; 22; 23]. These benchmark methods can be easily extended to enable \(M_{\mathrm{v}}\)-average estimation. The first benchmark is the approximated MMSE (AMMSE) detection formulated as [20]  
\[(\hat{v},\hat{w})=\underset{(\hat{v},\hat{w})}{\text{arg max}}\sum_{l=1}^{M_{ \mathrm{c}}}\left\|\mathbf{U}^{H}\mathbf{\hat{Y}}_{q_{l}}(\tilde{v},\tilde{w}) \right\|^{2}, \tag{32}\]where \(\|\cdot\|^{2}\) denotes the two-norm of vector, the columns of \(\mathbf{U}\) consists of the \(M_{r}\) normalized eigenvectors of \(\mathbf{F}\mathbf{F}^{H}\) involved to the \(M_{r}\) largest eigenvalues, \(M_{r}\) stands for the rank order of the reduced channel matrix, \(\mathbf{F}\) denotes the \(M_{\mathrm{s}}\times L\) normalized DFT matrix whose \((m,n)\)-th component is \(e^{-\beta\tau\mathrm{\,arm}/M}\), and \(\bar{\mathbf{Y}}_{q_{l}}(\hat{v},\hat{w})\) is the \(M_{\mathrm{s}}\)-dimensional vector with components \(\{\bar{Y}_{q_{l}}(m+\hat{v})D_{\mathrm{s}}^{*}(m);m\in\mathcal{D}\}\). The second reference is the joint detection of ICFO and CIDS (JDICID), which can be formulated as [21; 22]  
\[(\hat{v},\hat{w})= \underset{(\hat{v},\hat{w})}{\text{arg max}}\ \Re\Bigg{\{}\sum_{l=1}^{M_{ \mathrm{s}}}\sum_{m\in\mathcal{D}}\bar{Y}_{q_{l}}(m+\hat{v})D_{\mathrm{s}}^{*} (m)\Bigg{\}}. \tag{33}\]  
Finally, the third benchmark is the sequential detection of ICFO and CIDS (SDICID) [23] and applying \(M_{\mathrm{v}}\)-times average estimation, the ICFO can initially be estimated as  
\[\hat{v}= \underset{\hat{v}}{\text{arg max}}\ \Re\Bigg{\{}\sum_{l=1}^{M_{ \mathrm{s}}}\sum_{m\in\mathcal{D}}\bar{Y}_{q_{l}}(m+\hat{v})\bar{D}^{*}(m) \Bigg{\}}, \tag{34}\]  
where \(\bar{D}(m)=\sum_{\hat{w}=0}^{2}D_{\mathrm{s}}(m)\). Using the estimate \(\hat{v}\), the CIDS is sequentially detected as  
\[\hat{w}= \underset{\hat{w}}{\text{arg max}}\ \Re\Bigg{\{}\sum_{l=1}^{M_{ \mathrm{s}}}\sum_{m\in\mathcal{D}}\bar{Y}_{q_{l}}(m+\hat{v})D_{\mathrm{s}}^{*} (m)\Bigg{\}}. \tag{35}\]  
To ensure fair comparisons, we consider the number of floating point operations (flops) required for every arithmetic operation. Specifically, we assume six flops for every complex multiplication, two flops for every complex addition, and three flops for every complex magnitude, as specified in [29]. Regarding the JDICID scheme, it requires \((5M_{\mathrm{s}}-4)M_{\mathrm{v}}-2\) flops to compute the quantity \(\Re\{\sum_{l=1}^{M_{\mathrm{s}}}\sum_{m\in\mathcal{D}}\bar{Y}_{q}(m+\hat{v} )D_{\mathrm{s}}^{*}(m)\}\) for every hypothesis. Adopting \(M_{\mathrm{v}}\)-times average estimation, therefore, the total count of flops associated with (33) is obtained as \((6U+3)(5M_{\mathrm{s}}-6)M_{\mathrm{v}}\). For the AMMSE scheme, the overall processing complexity is given by \((6U+3)(10M_{\mathrm{s}}+2)M_{\mathrm{v}}M_{r}\). Identically, the SDICID method involves computing \(\Re\{\sum_{l=1}^{M_{\mathrm{s}}}\sum_{m\in\mathcal{D}}\bar{Y}_{q_{l}}(m+\hat{ v})\bar{D}^{*}(m)\}\) and \(\Re\{\sum_{l=1}^{M_{\mathrm{s}}}\sum_{m\in\mathcal{D}}\bar{Y}_{q_{l}}(m+\hat{ v})D_{\mathrm{s}}^{*}(m)\}\) with \((5M_{\mathrm{s}}-4)M_{\mathrm{v}}-2\) and \((2M_{\mathrm{s}}-1)M_{\mathrm{v}}-2\) flops for each trial, respectively. As a consequence, (34) requires \((2U+1)(5M_{\mathrm{s}}-6)M_{\mathrm{v}}\) flops to estimate the ICFO, whereas (35) requires \(3(2M_{\mathrm{s}}-3)M_{\mathrm{v}}\) flops to identify the CIDS. To summarize, the SDICID method needs \((2U+1)([5M_{\mathrm{s}}-4)M_{\mathrm{v}}-2]+3([2M_{\mathrm{s}}-1)M_{\mathrm{v }}-2]\) flops to sequentially obtain the ICFO and CIDS. In the end, we examine the arithmetic operations associated with the proposed detection method.