## 4 Performance Evaluation and Analysis
### Performance Analysis  
To analyze the performance of the improved algorithm proposed in this paper, we investigate it in terms of both bias and MSE [23]. Referring to Equation (13), the equation for bias is obtained as:  
\[bias(\hat{\tau}_{0})=E(\hat{\tau}_{0}-\tau_{0})=\frac{\partial^{2}\hat{\tau}_{0} }{\partial(E(\hat{\tau}_{0}))^{2}}\times var(\hat{\tau}_{0})+D \tag{23}\]  
Here, the first term on the right side of Equation (23) denotes random error, \(var(\cdot)\) indicates the variance, and \(\partial^{2}\hat{\tau}_{0}/\partial(E(\hat{\tau}_{0}))^{2}=\sum_{p=2}^{p}p(p- 1)a_{p}E^{p-2}(\hat{\tau}_{0})\). \(D\) accounts for the systematic bias, representing the \(\sum_{p=0}^{p}a_{p}\hat{\tau}^{p}\) misfit, which can be computed as  
\[D=\sum_{p=0}^{p}a_{p}\hat{\tau}_{0}^{p}-\tau_{0} \tag{24}\]  
If \(P\rightarrow+\infty\), then \(D\approx 0\). The MSE can be expressed as:  
\[MSE((\tilde{\tau}_{0})=E(\tilde{\tau}_{0}-\tau_{0})^{2}=var(\tilde{\tau}_{0}) +E^{2}(\tilde{\tau}_{0}-\tau_{0}) \tag{25}\]  
where \(var(\tilde{\tau}_{0})\) can be computed as:  
\[var(\tilde{\tau}_{0})=\left(\frac{\partial\tilde{\tau}_{0}}{\partial(E(\hat{ \tau}_{0}))}\right)^{2}\times var(\hat{\tau}_{0}) \tag{26}\]  
Here, \(\partial\hat{\tau}_{0}/\partial(E(\hat{\tau}_{0}))=\sum_{p=1}^{p}p\times a_{p} E^{p-1}(\hat{\tau}_{0})\). To facilitate the analysis of Equations (23) and (25), we combine Equation (7) to solve for \(var(\hat{\tau}_{0})\), as follows:  
\[ \begin{split} var(\hat{\tau}_{0})=&\frac{2\theta^ {2}}{(X-2Z+Y)^{4}}\times\\ &\bigg{[}(Y-Z)(Z-X)cov(x,y)+(X-Y)(Y-Z)cov(z,x)\\ &+(Z-X)(X-Y)cov(y,z)+\frac{(Y-Z)^{2}}{2}var(x)\\ &+\frac{(X-Z)^{2}}{2}var(y)+\frac{(Y-X)^{2}}{2}var(z)\bigg{]}. \end{split}  \tag{27}\]  
Here, \(cov(x,y)\) denotes the covariance of the random variables \(x\) and \(y\), \(X=E(\hat{R}_{x_{1}x_{2}}(\theta))\), \(Y=E(\hat{R}_{x_{1}x_{2}}(-\theta))\), \(Z=E(\hat{R}_{x_{1}x_{2}}(0))\), \(x=\hat{R}_{x_{1}x_{2}}(\theta),y=\hat{R}_{x_{1}x_{2}}(-\theta),z=\hat{R}_{x_{1 }x_{2}}(0)\). Considering the dependency of covariance on variance, we first address the expression for \(cov(\hat{R}_{x_{1}x_{2}}(\tau_{1}),\hat{R}_{x_{1}x_{2}}(\tau_{2}))\) as follows:  
\[ \begin{split} cov(\hat{R}_{x_{1}x_{2}}(\tau_{1}),\hat{R}_{x_{1}x_ {2}}(\tau_{2}))=& E(\hat{R}_{x_{1}x_{2}}(\tau_{1})\times\hat{R}_{x_ {1}x_{2}}(\tau_{2}))\\ &-E(\hat{R}_{x_{1}x_{2}}(\tau_{1}))E(\hat{R}_{x_{1}x_{2}}(\tau_{ 2}))\end{split}  \tag{28}\]  
We define \(x_{1}(kT)=x_{\tau 1}(kT)+i\times x_{\tau 1}(kT)\), \(x_{2}(kT+\tau T)=x_{\tau 2}(kT+\tau T)+i\times x_{\tau 2}(kT+\tau T)\); then, \(\Re(x_{1}^{*}(kT)x_{2}(kT+\tau T))=x_{\tau 1}(kT)\times x_{\tau 2}(kT+\tau T)+ xi1(kT)\times x_{\tau 2}(kT+\tau T)\). Thus, we obtain:  
\[E(\hat{R}_{x_{1}x_{2}}(\tau_{1})\times\hat{R}_{x_{1}x_{2}}(\tau_{2})) =\frac{1}{N^{2}}\sum_{k_{1}=0}^{N-1}\sum_{k_{2}=0}^{N-1}E\big{\{}[ \underbrace{x_{r1}(k_{1}T)\cdot x_{r2}(k_{1}T+\tau_{1}T)}_{\bigstar}\] \[+\underbrace{x_{i1}(k_{1}T)\cdot x_{i2}(k_{1}T+\tau_{1}T)}_{ \bigstar}\] \[\cdot[\underbrace{x_{r1}(k_{2}T)\cdot x_{r2}(k_{2}T+\tau_{2}T)}_{ \bigstar}\] \[+\underbrace{x_{i1}(k_{2}T)\cdot x_{i2}(k_{2}T+\tau_{2}T)}_{ \bigstar}]\big{\}} \tag{29}\]  
To simplify the analysis, we assume that the correlation between the signal \((s(kT))\), the real part \((s_{r}(kT))\), and the imaginary part \((s_{i}(kT))\) is approximately zero--i.e., \(E((s_{r}(kT)\cdot(s_{i}(kT))\approx 0,\mathbf{A}=1\).