## 3 Reduced-Complexity ICFO and CIDS Detection Method
### Performance Analysis  
This subsection presents a closed-form solution for the probability of detection failure of the proposed ICFO and CIDS synchronization scheme. For ease of derivation, we assume a slowly time-variant flat fading channel, i.e., \(G_{q_{l}}(m)\approx G_{q_{l}+1}(m)\). Let \(a\sim\mathcal{N}(b,c)\) represent that \(a\) is a Gaussian RV with mean \(b\) and variance \(c\). The probability of detection failure denoted by \(P_{f}=\text{Prob}\{(\vartheta,\tilde{w})\
eq(v,w)\}\) refers to the probability of erroneously deciding both the ICFO and CIDS. Conditioned on \(\mathcal{G}=|G_{q_{l}}(m)|^{2}\), if we assume that \((\vartheta,\tilde{w})=(v,w)\) (hypothesis \(H_{1}\)) in (17), then \(\Re\{\Omega_{red}(\tilde{v},\tilde{w})\}\sim\mathcal{N}(\mu,\sigma_{1}^{2})\) because of near-zero autocorrelation feature of m-sequence and \(M_{s}\gg 1\), where \(\mu=M_{v}(M_{s}-1)E_{X}^{2}\mathcal{G}\) and \(\sigma_{1}^{2}\approx M_{v}(M_{s}-1)E_{X}^{2}(\sigma_{I}^{2}+\sigma_{Z}^{2})/2\). Here, \(\sigma_{I}^{2}=2E_{X}\sigma_{I}^{2}\mathcal{G}+2\sigma_{I}^{2}\sigma_{Z}^{2}+ \sigma_{I}^{4}\) is the variance of (14) and \(\sigma_{Z}^{2}=2E_{X}\sigma_{Z}^{2}\mathcal{G}+\sigma_{Z}^{4}\) is the variance of (15). Under hypothesis \(H_{0}\) that \((\hat{v},\hat{w})\
eq(v,w)\) in (19), \(\Re\{\Omega_{rad}(\hat{v},\hat{w})\}\sim\mathcal{N}(0,\sigma_{0}^{2})\) with \(\sigma_{0}^{2}\approx\sigma_{1}^{2}\) when \(M_{s}\gg 1\). Let \(y=\Re\{\Omega_{red}(\hat{v},\hat{w})\}\), and assuming that ICFOs have an equal probability of occurrence, the probability of detection failure of the proposed NR-PSS synchronization method is the probability that the RV \(y\) under hypothesis \(H_{0}\) exceeds all other \(3(2U+1)-1\) RVs under hypothesis \(H_{0}\) and hypothesis \(H_{1}\). Conditioned on \(\mathcal{G}\), the probability of incorrectly detecting both the ICFO and CIDS can be expressed as  
\[P_{f}(\mathcal{G})=1-\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma_{1}^{2} }}e^{-\frac{(y-y)^{2}}{2\sigma_{1}^{2}}}\left[\int_{-\infty}^{y}\frac{1}{ \sqrt{2\pi\sigma_{0}^{2}}}e^{-\frac{y^{2}}{2\sigma_{0}^{2}}}dx\right]^{6U+2}dy. \tag{20}\]  
Thus, \(P_{f}(\mathcal{G})\) can be obtained as  
\[P_{f}(\mathcal{G})=1-\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^ {2}}{2}}[1-Q(y+\lambda)]^{6U+2}dy, \tag{21}\]  
where \(Q(\cdot)\) is the \(Q\)-function and \(\lambda=\mu/\sigma_{1}\). In (21), we have  
\[\lambda=\sqrt{\frac{2M_{v}(M_{s}-1)}{2(1/\gamma_{z}+/\gamma_{i})+(1/\gamma_{z} +/\gamma_{i})^{2}}}, \tag{22}\]  
where \(\gamma_{z}=E_{X}\mathcal{G}/\sigma_{Z}^{2}\) is the received signal-to-noise ratio (SNR) and \(\gamma_{i}=E_{X}\mathcal{G}/\sigma_{I}^{2}\) is the received signal-to-ICI ratio (SIR). To simplify the derivation of (21), the following upper-bound presented in [27; 28] is adopted  
\[Q(y+\lambda)\leq\sum_{i=1}^{M_{e}}a_{i}e^{-\frac{b_{i}(y+\lambda)^{2}}{2}}, \tag{23}\]  
where \(a_{i}\) and \(b_{i}\) represent suitable coefficients for deciding the accuracy of the approximation. It is important to note that the bound approaches the precise value as \(M_{e}\) increases. By substituting (23) into (21) and applying the multinomial theorem, we obtain the lower bound for \(P_{f}(\mathcal{G})\)  
\[ \begin{split} P_{f}(\mathcal{G})&\geq 1-\sum_{u_{1}+ \cdots+u_{M_{\text{c}}+1}=6U+2}\ \prod_{i=1}^{M_{e}}(-a_{i})^{u_{i}}\binom{6U+2}{u_{1},u_{2},\cdots,u_{M_{ \text{c}}+1}}\\ &\times\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{y^ {2}}{2}}e^{-\frac{(y+\lambda)^{2}}{2}\sum_{i=1}^{M_{e}}b_{i}u_{i}}dy,\end{split}  \tag{24}\]  
which can be further derived as  
\[ \begin{split} P_{f}(\mathcal{G})&\geq 1-\sum_{u_{1}+ \cdots+u_{M_{\text{c}}+1}=6U+2}\ \prod_{i=1}^{M_{e}}(-a_{i})^{u_{i}}\binom{6U+2}{u_{1},u_{2},\cdots,u_{M_{ \text{c}}+1}}\\ &\times\left(1+\sum_{i=1}^{M_{e}}b_{i}u_{i}\right)^{-1/2}e^{- \frac{\lambda^{2}}{2}\left\{1-\left(1+\sum_{i=1}^{M_{e}}b_{i}u_{i}\right)^{-1} \right\}},\end{split}