## 2 Materials and Methods
### Bias and Variance Analysis with Conventional Cross-Correlation for TPE  
Similarly, in many works, the very conventional approach to time period or time delay estimation between sensors' output signals is using the crosscorrelation function which is defined as:  
\[R_{x_{1}x_{2}}(\tau)=\int_{0}^{T}x_{1}(\alpha)x_{2}(\tau-\alpha)da;for-T\leq \tau\leq T \tag{16}\]  
Thus, the time period is essentially estimated as:  
\[\tau_{0}=\arg\max_{-T\leq\tau\leq T}R_{x_{1}x_{2}}(\tau) \tag{17}\]  
Let \(y_{1}(t)=x_{1}(t)\); \(y_{2}(t)=x_{2}(-t)\), thus:  
\[y_{2}(t)=h_{2}(-t)\bigotimes\phi(-t-\tau_{0})+n_{2}(-t-\tau_{0}) \tag{18}\]  
and in the frequency domain:  
\[Y_{2}(f)=(H_{2}(-f)\cdot\Phi(-f)+N_{2}(-f))e^{j2\pi/\tau_{0}}=(H_{2}^{*}(f) \Phi^{*}(f)+N_{2}^{*}(f)) \tag{19}\]  
where \(C^{*}\) is the denotation of the conjugated complex number of a complex number \(C\). The cross-correlation function between the two signals \(x_{1}(t),\ x_{2}(t)\) is then rewritten as:  
\[R_{x_{1}x_{2}}(\tau)=\int_{0}^{T}x_{1}(\alpha)x_{2}(\alpha+\tau)d\alpha=\int _{0}^{T}x_{1}(\alpha)y_{2}(-\tau-\alpha)d\alpha=L(-\tau) \tag{20}\]  
\[\text{where}:\ L(\tau)=\int_{0}^{T}x_{1}(\alpha)y_{2}(\tau-\alpha)d\alpha \tag{21}\]  
Equation (22) drives the Fourier transform of the function \(L(\tau)\), which is defined as:  
\[ \begin{split} X_{1}(f)Y_{2}(f)&=(H_{1}(f)\Phi(f) +N_{1}(f))\cdot(H_{2}^{*}\Phi^{*}(f)+N_{2}^{*}(f))e^{j2\pi f\tau_{0}}\\ &=(D_{1}(f)+N_{1}(f))(D_{2}^{*}(f)+N_{2}^{*}(f))\cdot e^{j2\pi f \tau_{0}}\end{split}  \tag{22}\]  
Here, we denote the deterministic part of the output signal of each sensor in the frequency domain as:  
\[ \begin{split}& D_{1}(f)=H_{1}(f)\Phi(f);D_{2}(f)=H_{2}(f)\Phi(f)\\ &\text{and:}\ M(f)=(D_{1}(f)+N_{1}(f))(D_{2}^{*}(f)+N_{2}^{*}(f)) \end{split}  \tag{23}\]  
Therefore, \(L(\tau)\) is the inverse Fourier transform of the function \(M(f)\cdot e^{j2\pi f\tau_{0}}\):  
\[L(\tau)=\int_{0}^{T}x_{1}(\alpha)y_{2}(\tau-\alpha)d\alpha=\int_{-B}^{B}M(f) \cdot e^{j2\pi f(\tau+\tau_{0})}df \tag{24}\]Due to Equation (17), the following estimate is considered:  
\[\hat{\tau_{0}}=\text{argmax}(L(-\tau))\text{ or }\hat{\tau_{0}}=\text{arg}\bigg{(} \frac{d}{d\tau}L(-\tau)=0\bigg{)} \tag{25}\]  
Thus, the objective function is defined as \(J_{CCF}(\tau)=\frac{d}{d\tau}L(-\tau)\). Now, we proceed with some calculations to apply the analysis mentioned in Section 2.1 for bias and variance estimation. \[J_{CCF}(\tau)=-j2\pi\int_{-B}^{B}f.M(f).e^{j2\pi f(\tau_{0}-\tau)}df \tag{26}\]  
\[J_{CCF}(\tau_{0})=-j2\pi\int_{-B}^{B}f.M(f)df \tag{27}\]  
\[\mathbb{E}[J_{CCF}(\tau_{0})]=-j2\pi\int_{-B}^{B}f.\mathbb{E}[M(f)]df \tag{28}\]  
\[J_{CCF}(\tau_{0})^{2}=-4\pi^{2}\int_{-B}^{B}\int_{-B}^{B}f_{1}f_{2}.M(f_{1})M (f_{2})df_{1}df_{2} \tag{29}\]  
\[\mathbb{E}\Big{[}J_{CCF}(\tau_{0})^{2}\Big{]}=-4\pi^{2}\int_{-B}^{B}\int_{-B}^ {B}f_{1}f_{2}\mathbb{E}[M(f_{1})M(f_{2})]\ df_{1}df_{2} \tag{30}\]  
\[J_{CCF}^{\prime}(\tau)=\frac{d}{d\tau}J_{CCF}(\tau)=-4\pi^{2}\int_{-B}^{B}f^{ 2}.M(f)e^{j2\pi f(\tau_{0}-\tau)}df \tag{31}\]  
\[J_{CCF}^{\prime}(\tau_{0})=-4\pi^{2}\int_{-B}^{B}f^{2}.M(f)df \tag{32}\]  
\[\mathbb{E}\big{[}J_{CCF}^{\prime}(\tau_{0})\big{]}=-4\pi^{2}\int_{-B}^{B}f^{2}.\mathbb{E}\big{[}J_{CCF}^{\prime}(\tau_{0})\big{]}df \tag{33}\]  
Now, we use Equations (11)-(15) to explore the recent formulas, with the notation that \(D_{1}(f),\ D_{2}(f)\) is zero for any \(f\
eq\pm f_{0}\). To perform this, the term \(\Delta f\)--the frequency resolution--is introduced to manage the effect of the Dirac delta function \(\delta\) in an integral. The Dirac delta function is a mathematical idealization that assigns all energy at a single frequency, making direct integration problematic. To resolve this, \(\Delta f\) serves as a small frequency resolution element, ensuring the integral remains finite and meaningful in numerical and practical applications.