## 2 Materials and Methods
### Bias and Variance Analysis with Cross-Correlation Function Combined with Hilbert Transform for TPE  
Another approach that has been used in this study is applying Hilbert transform [17; 23] to one of the output signals, specifically the second one:  
\[x_{2}^{\sim}(t)=\int_{0}^{T}\frac{x_{2}(\alpha)}{t-\alpha}d\alpha \tag{44}\]  
The correlation function for signals \(x_{1}(t),\ x_{2}^{\sim}(t)\), which is also the specific target function, is now constructed as:  
\[J_{CCFHT}(\tau)=R_{x_{1}x_{2}}^{\sim}(\tau)=\int_{0}^{T}x_{1}(\alpha)x_{2}^{ \sim}(\tau-\alpha)d\alpha;\forall-T\leq\tau\leq T \tag{45}\]  
Unlike the CCF method, using the CCFHT method, the time period is estimated as the zero point of the new correlation function:  
\[\tau_{0}=\arg(R_{x_{1}x_{2}}^{\sim}(\tau)=0) \tag{46}\]  
The Fourier transform of the objective function is defined as [17]:  
\[\begin{array}{c}-jsign\ (f).X_{1}(f)Y_{2}(f)=-j.\ sign\ (f).(D_{1}(f)+N_{1}(f))(D_{2}^{ \ast}(f)+N_{2}^{\ast}(f))\cdot\\ e^{2\pi f\tau_{0}}=-j.sign\ (f)M(f)\end{array} \tag{47}\]  
Then, the following equations can be proceeded:  
\[\begin{array}{c}J_{CCFHT}(\tau)=\int_{-B}^{B}-j.sign\ (f)M(f)e^{2\pi f( \tau_{0}-\tau)}df\\ J_{CCFHT}(\tau_{0})=\int_{-B}^{B}-j.sign\ (f)M(f)df\\ \mathbb{E}[J_{CCFHT}(\tau_{0})]=2.Im\{D_{1}(f_{0})D_{2}^{\ast}(f_{0})\}. \Delta f\end{array} \tag{48}\]  
\[\begin{array}{c}J_{CCFHT}^{\prime}(\tau)=\frac{d}{d\tau}J_{CCFHT}(\tau)=2 \pi\int_{-B}^{B}f.sign(f)M(f)e^{2\pi f(\tau_{0}-\tau)}df\end{array} \tag{49}\]  
\[\begin{array}{c}J_{CCFHT}^{\prime}(\tau)=2\pi\int_{-B}^{B}f.sign(f)\mathbb{ E}[M(f)]df\end{array} \tag{50}\]  
\[\begin{array}{c}\mathbb{E}\left[J_{CCFHT}^{\prime}(\tau_{0})\right]=4\pi f_ {0}Re\{D_{1}(f_{0})D_{2}^{\ast}(f_{0})\}.\Delta f\\ \mathbb{E}\left[J_{CCFHT}^{\prime}(\tau_{0})\right]^{2}=16\pi^{2}f_{0}^{2}.Re ^{2}\{D_{1}(f_{0})D_{2}^{\ast}(f_{0})\}.\Delta^{2}f\\ =16\pi^{2}f_{0}^{2}.\alpha_{N1}^{2}\alpha_{N2}^{2}.\frac{1}{\Delta^{2}f} \times SNR_{1}.SNR_{2}{\cos^{2}\Theta_{0}}\\ \mathbb{E}\left[J_{CCFHT}(\tau_{0})^{2}\right]=2\alpha_{N1}^{2}\alpha_{N2}^{2}.\Big{\{}\frac{\text{SNR}_{1}SNR_{2}}{\Delta^{2}f}.\sin^{2}\Theta_{0}+\frac{T }{2B}.(\text{SNR}_{1}+\text{SNR}_{2})\\ +\frac{T^{2}.\Delta f}{4B}\Big{\}}\end{array} \tag{52}\]  
The bias and variance of the estimation using the CCFHT method are defined:  
\[\begin{array}{c}bias(\tau_{0})_{CCFHT}=-\frac{\mathbb{E}[J_{CCFHT}(\tau_{0 })]}{\mathbb{E}\left[J_{CCFHT}^{\prime}(\tau_{0})\right]}=-\frac{1}{2\pi f_{0} }.\tan\Theta_{0}\end{array} \tag{53}\]  
\[\begin{array}{c}bias(\tau_{0})_{CCFHT}=-\frac{\mathbb{E}[J_{CCFHT}(\tau_{0 })]}{\mathbb{E}\left[J_{CCFHT}^{\prime}(\tau_{0})\right]}=-\frac{1}{2\pi f_{0} }.\tan\Theta_{0}\end{array} \tag{54}\]  
Again, in the case of the time constants of \(H_{1}(s)\) and \(H_{2}(s)\) coinciding, it appears:  
\[\begin{array}{c}bias(\tau_{0})_{CCFHT}=0\\ var(\tau_{0})_{CCFHT}=\frac{1}{8\pi^{2}f_{0}^{2}}\Big{\{}\frac{1}{BSNR_{1}}+ \frac{1}{BSNR_{2}}+\frac{1}{BSNR_{1}.SNR_{2}}\Big{\}}\end{array} \tag{55}\]In general cases, supposing \(SNR_{1},\ SNR_{2}\) are significant, we have:  
\[var(\hat{\tau}_{0})_{CCFHT}\approx\frac{1}{8(\pi f_{0})^{2}}\bigg{\{}\tan^{2} \Theta_{0}\ +\frac{1}{N_{d}.\text{cos}^{2}\Theta_{0}}\bigg{(}\frac{1}{SNR_{1}}+\frac{1}{ SNR_{2}}\bigg{)}\bigg{\}} \tag{56}\]