## 4 Performance Evaluation and Analysis  
### Modified Performance Limits  
[PERSON] [22] derived the CRLB for the case of equal signal bandwidth and noise bandwidth; however, the signal bandwidth is smaller than the noise bandwidth in most scenarios. Therefore, CRLB expression needs to be adapted to make it suitable for more applications. The Cramer-Rao lower bound (CRLB) is computed as [22]:  
\[CRLB(\tau)=\min_{\tau}E\left(-\frac{1}{2}\frac{\hat{\sigma}^{2}(\int_{-\infty }^{+\infty}\hat{X}^{H}(f)Q(f)\hat{X}(f)\,df)}{\hat{\sigma}\tau^{2}}\right)^{-1} \tag{14}\]  
where \(H\) denotes conjugate transpose, \(\hat{X}(f)=[X_{1}(f),X_{2}(f)]^{T}\), which is the Fourier transform vector of \([x_{1}(kT),x_{2}(kT)]\). \(Q(f)\) is the inverse of the spectral density matrix, which can be expressed as:  
\[Q(f)=\frac{\begin{bmatrix}G_{x_{2}x_{2}}(f)&-G_{x_{1}x_{2}}(f)\\ -G_{x_{1}x_{2}}^{*}(f)&G_{x_{1}x_{1}}(f)\end{bmatrix}}{G_{x_{2}x_{2}}(f)G_{x_{ 1}x_{1}}(f)-|G_{x_{1}x_{2}}(f)|^{2}} \tag{15}\]Here, the \(G_{x1x1}(f),G_{x2x2}(f)\) in the above equation denote the power spectral densities (psds) of the signals \(x_{1}(kT),x_{2}(kT)\), respectively. \(G_{x1x2}(f)\) represents the cross-correlation power spectral density (psd) of the signals \(x_{1}(kT),x_{2}(kT)\), which can be formatted:  
\[\begin{cases}G_{x1x1}(f)=G_{\text{ss}}(f)+G_{n1n1}(f)\\ G_{x2x2}(f)=G_{\text{ss}}(f)+G_{n2n2}(f)\\ G_{x1x2}(f)=G_{\text{ss}}(f)e^{-2\pi f\tau}\end{cases} \tag{16}\]  
where \(G_{\text{ss}}(f),G_{n1n1}(f),G_{n2n2}(f)\) represent the power spectral densities of signals \(s(kT)\), \(n_{1}(kT)\), \(n_{2}(kT)\), respectively, and \(\tau\) indicates the time delay of signal \(x_{2}(kT)\) with regard to the reference signal.  
The derivation term of Equation (14) can be calculated as:  
\[\int_{-\infty}^{+\infty}\tilde{X}^{H}(f)Q(f)\tilde{X}(f)\,df=\int _{-\infty}^{+\infty}\frac{|X_{1}(f)|^{2}G_{x2x2}(f)+|X_{2}(f)|^{2}G_{x1x1}(f) }{G_{x2x2}(f)G_{x1x1}(f)-|G_{x1x2}(f)|^{2}}\,df\] \[+2 NT\int_{-\infty}^{+\infty}|G_{x1x2}(f)|\frac{G_{\text{ss}}(f)e ^{2\pi f(\tau-\tau_{0})}}{G_{x2x2}(f)G_{x1x1}(f)-|G_{x1x2}(f)|^{2}}\,df \tag{17}\]  
Then, we introduce the normalization factor of the cross-correlation psd \(C_{x1x2}(f)\):  
\[C_{x1x2}(f)=\frac{|G_{x1x2}(f)|^{2}}{G_{x1x1}(f)G_{x2x2}(f)}. \tag{18}\]  
To facilitate the analysis, assuming that the power spectra of signal is flat over the frequency range \([-B/2,B/2]\) and zero outside this range, Equation (14) is simplified as follows:  
\[CRLB(\tau)=\left(NT\int_{-\infty}^{+\infty}(2\pi f)^{2}\frac{C_{x1x2}(f)}{1-C _{x1x2}(f)}\,df\right)^{-1} \tag{19}\]  
As far as the noise and signal bandwidth are concerned, we set the noise bandwidth to \(B_{n}\)= \(1/T\) and the signal bandwidth to \(B\); thus, the power spectral density of \(x_{1}(kT)\) is obtained:  
\[\begin{cases}G_{x1x1}(f)=\frac{\sigma_{B}^{2}}{B_{n}}+\frac{\sigma_{n}^{2}}{B _{n}}&\text{if }|f|\leq\frac{B}{2}\\ G_{x1x1}(f)=\frac{\sigma_{n}^{2}}{B_{n}}&\text{if }\frac{B}{2}<|f|\leq\frac{B_{n}}{2}\\ G_{x1x1}(f)=0&\text{others}\end{cases} \tag{20}\]  
Then, we can obtain:  
\[\frac{C_{x1x2}(f)}{1-C_{x1x2}(f)}=\frac{\alpha^{2}SNR^{2}}{1+2\alpha SNR} \tag{21}\]  
where \(SNR=\sigma_{s}^{2}/\sigma_{n}^{2}\), \(\alpha=B_{n}/B\). By substituting Equation (21) into Equation (19), we can obtain:  
\[CRLB(\tau) =\left(NT\int_{-B/2}^{+B/2}(2\pi f)^{2}\frac{\alpha^{2}SNR^{2}}{ 1+2\alpha SNR}\,df\right)^{-1}\] \[=\frac{3}{\pi^{2}}\cdot\frac{1+2\alpha SNR}{\alpha^{2}SNR^{2}} \cdot\frac{1}{B^{3}NT} \tag{22}\]