## 4 Signal Evaluation Approach under the Local Despreading Waveform Mismatch
### Formulation and Statistical Properties of Discriminator Output  
This subsection provides a derivation of the discriminator output, which is the foundation to estimate the error of pseudorange measurement. Based on the previously obtained correlation and I&D output Equation (10), the output of the early minus late power discriminator shown in Figure 1 can be expanded as:  
\[e(\varepsilon)=\left|\delta_{E}\right|^{2}-\left|\delta_{L}\right|^{2}\ =\ \left|c_{s 8}(\Delta/2)\right|^{2}-\left|c_{s8}(-\Delta/2)\right|^{2} \tag{17}\]  
Considering the random error caused by interference and Gaussian white noise, in stable tracking, the TOA estimation error is approximate to zero \(\varepsilon\approx 0\). By using Equation (17), the loop gain of \(e(\varepsilon)\) can be obtained after tedious derivation as:  
\[K=\left.\frac{\mathrm{d}e(\varepsilon)}{\mathrm{d}\varepsilon}\right|_{ \varepsilon\to 0}=8C_{s}^{2}\pi\int_{-\beta_{r}/2}^{\beta_{r}/2}\mathcal{X}_{s,s} (f)\mathrm{cos}(\pi f\Delta)df\ \times\int_{-\beta_{r}/2}^{\beta_{r}/2}f \mathcal{X}_{s,s}(f)\mathrm{sin}(\pi f\Delta)df \tag{18}\]  
Note that the cross-spectrum \(\mathcal{X}_{s,s}(f)\) is also used here. Then the variance of TOA estimation at the output of correlator is obtained as:  
\[ \begin{split}\mathrm{Var}(\varepsilon)=\frac{\mathrm{Var}\left\{e (\varepsilon)\right\}}{K^{2}}&=\frac{8}{K^{2}T}\left(\int_{- \beta_{r}/2}^{\beta_{r}/2}\mathcal{X}_{s,\hat{s}}(f)\mathrm{cos}(\pi f\Delta) df\right)^{2}\times\int_{-\beta_{r}/2}^{\beta_{r}/2}G_{w}(f)G_{S}(f)\mathrm{sin}^{2}( \pi f\Delta)df\\ &+\frac{2}{K^{2}T^{2}}\left[\left(\int_{-\beta_{r}/2}^{\beta_{r} /2}G_{w}(f)G_{S}(f)df\right)^{2}-\left(\int_{-\beta_{r}/2}^{\beta_{r}/2}G_{w}( f)G_{S}(f)\mathrm{cos}(2\pi f\Delta)df\right)^{2}\right]\end{split}  \tag{19}\]  
which is the open loop variance without being smoothed by the loop. Further, the variance of closed-loop TOA estimation error can be obtained with the relationship between the open-loop and the closed-loop result [11]:  
\[\sigma_{\mathrm{close}}^{2}(\varepsilon)=\mathrm{Var}(\varepsilon)\,2B_{L}T(1- 0.5B_{L}T) \tag{20}\]  
where \(B_{L}\) is the one-sided equivalent noise bandwidth of the tracking loop, \(T\) is the integration duration where \(0\leq B_{L}T\leq 0.5\) is satisfied, as previously defined in the model. Two assumptions are used here. First, let the early-late space \(\Delta\) tend to zero, which would obtain the lower bound of the code tracking error. The bound is strict in the sense that well-designed receiver can achieve competitive tracking accuracy [18]. Second, the noise and interference term \(w(t)\) is expanded into two terms as Equation (8), facilitating the study of the influence of noise and interference distinctly. Then Equation (20) becomes:  
\[ \begin{split}\sigma_{\Delta\to 0}^{2}(\varepsilon)&=\frac{B_{L}(1-0 5B_{L}T)}{\left(2\pi\right)^{2}}\left[\frac{\frac{\beta_{r}^{\beta_{r}/2}^{ \beta_{r}/2}f^{2}G_{S}(f)df}{\frac{G_{S}}{\beta_{0}}\left(\beta_{-\beta_{r}/2}^ {\beta_{r}/2}\mathcal{X}_{s,\hat{s}}(f)df\right)^{2}}+\frac{C_{s}}{\zeta_{s}} \frac{\int_{-\beta_{r}/2}^{\beta_{r}/2}f^{2}G_{S}(f)G_{S}(f)df}{\left(\beta_{- \beta_{r}/2}^{\beta_{r}/2}\mathcal{X}_{s,\hat{s}}(f)df\right)^{2}}\right]\\ &\times\left[1+\frac{\int_{-\beta_{r}/2}^{\beta_{r}/2}G_{S}(f)df}{ \frac{C_{s}}{\zeta_{0}}T\left(\beta_{-\beta_{r}/2}^{\beta_{r}/2}\mathcal{X}_{ s,\hat{s}}(f)df\right)^{2}}+\frac{C_{s}}{\zeta_{s}}\frac{\int_{-\beta_{r}/2}^{ \beta_{r}/2}G_{S}(f)G_{S}(f)df}{\left(\beta_{-\beta_{r}/2}^{\beta_{r}/2} \mathcal{X}_{s,\hat{s}}(f)df\right)^{2}}\right]\end{split}  \tag{21}\]  
Equation (21) provides a low bound for the code tracking error evaluation, which considers the effect of unmatched receiving by using the cross-spectrum term \(\mathcal{X}_{s,\hat{s}}\) in the denominator. Also, derivation shows that the local spectrum \(G_{S}\) appears in the numerator, rather than cross-spectrum \(\mathcal{X}_{s,\hat{s}}\), which is confused in the previous work [12].