## Appendix A  
The proof of the validity and capability of the proposed method against the NBI in multi-path channels is presented as follows. First, we look at the SAC window \(R_{1,n}\) at the desired timing position \(n_{0}\) as given by (12). Taking the frequency-selective fading channel condition into consideration with the CIR denoted by \(\{h_{k}\}_{k=0}^{L_{k}-1}\), the SAC result is derived as given by (A3), where the cross terms between the NBI and the preamble sequence can still be eliminated by sum averaging and de-scrambling as they can be in the AWGN channel shown by (12) because they are  
Figure 8: False probability of integer CFO estimation and signaling detection. incoherent signals. The NBI component as pointed out above is also eliminated through de-scrambling and sum averaging as well. Hence, the SAC result of \(R_{1,n}\) yields the peak value as given by (A3) at the desired timing position \(n_{0}\), which is free from the impacts of the NBI. Similarly, we have  
\[R_{2,n}\approx-\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\] (A1)  
\[R_{3,n}\approx\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\] (A2)  
\[R_{1,n_{0}}=\sum\limits_{l=0}^{N/2-1}(-1)^{l}r_{n_{0}+l}^{*}\cdot r _{n_{0}+l-N/2}\] (A3) \[=\sum\limits_{l=0}^{N/2-1}(-1)^{l}\Bigg{[}\sum\limits_{k=0}^{L_{ k}-1}h_{k}\cdot(-1)^{l-k}x_{l-k}+I_{0}\exp(j2\pi f_{\text{NB}}l)\Bigg{]}^{*}.\] \[\Bigg{[}\sum\limits_{m=0}^{N/2-1}h_{m}\cdot x_{l-m}+I_{0}\exp(j2 \pi f_{\text{NB}}(l-N/2))\Bigg{]}\] \[=\sum\limits_{l=0}^{N/2-1}\Bigg{[}(-1)^{l}\Bigg{(}\sum\limits_{k =0}^{L_{k}-1}h_{k}\cdot(-1)^{l-k}x_{l-k}\Bigg{)}^{*}\Bigg{(}\sum\limits_{m=0} ^{L_{k}-1}h_{m}\cdot x_{l-m}\Bigg{)}\] \[+(-1)^{l}\Bigg{(}\sum\limits_{k=0}^{L_{k}-1}h_{k}\cdot(-1)^{l-k} x_{l-k}\Bigg{)}^{*}I_{0}\exp(j2\pi f_{\text{NB}}(l-N/2))\] \[+(-1)^{l}\Bigg{(}\sum\limits_{m=0}^{L_{k}-1}h_{m}\cdot x_{l-m} \Bigg{)}I_{0}\exp(-j2\pi f_{\text{NB}}l)+(-1)^{l}\underbrace{I_{0}^{2}\exp(-j 2\pi f_{\text{NB}}N/2)}_{\text{NBI component}}\Bigg{]}\] \[\approx\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum \limits_{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m},\]  
\[R_{2,n}\approx-\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\]0 (A4) \[R_{2,n}\approx-\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\]1 \[R_{2,n}\approx-\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\]2 \[R_{2,n}\approx-\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\]3 \[R_{2,n}\approx-\sum\limits_{l=0}^{N/2-1}\sum\limits_{k=0}^{L_{k}-1}\sum\limits _{m=0}^{L_{k}-1}(-1)^{k}h_{k}^{*}h_{m}x_{l-k}^{*}x_{l-m}\]4  
Therefore, all the three SAC results are free from the impacts of the NBI, making the final peak value \(R_{3,\hat{n}_{0}}\) immune to the NBI, so the estimated timing position \(\hat{n}_{0}\) accurately matches the desired timing position \(n_{0}\) in the presence of the NBI under the multi-path channel. The analysis for the contamination of the NBI upon the conventional methods as given by (3) can be extended to multi-path channels through similar reasoning. Considering multi-path fading channel, the SAC result \(R_{c,n}\) of the conventional methods at the desired timing position \(n_{0}^{\prime}\) is given by (A4). From (A4), it can be noted that, although the cross terms between the NBI and the preamble sequences are eliminated through sum averaging, the NBI component as pointed out above remains in the SAC result because there is no scrambling to eliminate it.