# A Design Method of Code Correlation Reference Waveform in GNSS Based on Least-Squares Fitting
## 4 Performance Analysis
### Tracking Jitter  
The tracking jitter [9] quantifies the amount of noise transferred by a tracking loop from the input signal to the final delay estimate. If the thermal noise is only considered, the output of the non-coherent discriminator is given by:  
\[d_{nc}(\varepsilon) = (\sqrt{2C}R_{\hat{X}X}(\varepsilon)\cos(\varphi_{\epsilon})+N_{IX} )(\sqrt{2C}R_{\hat{X}W}(\varepsilon)\cos(\varphi_{\epsilon})+N_{I}) \tag{14}\] \[+ (\sqrt{2C}R_{\hat{X}X}(\varepsilon)\sin(\varphi_{\epsilon})+N_{ QX})(\sqrt{2C}R_{\hat{X}W}(\varepsilon)\sin(\varphi_{\epsilon})+N_{Q})\]  
Here, \(C\) stands for the signal power. \(N_{IX}\) and \(N_{QX}\) are the noise items of the in-phase and quadrature-phase branch during the prompt code correlation, with the variance of \(R_{XX}(0)N_{0}/T\). When the phase lock loop (PLL) is stable, \(\varphi_{\epsilon}\approx 0\). Then, the discriminator can be simplified as follows.  
\[d_{nc}(\varepsilon) = 2 CR_{\hat{X}X}(\varepsilon)R_{\hat{X}W}(\varepsilon)+\sqrt{2C}R_ {\hat{X}W}(\varepsilon)N_{IX} \tag{15}\] \[+ \sqrt{2C}R_{\hat{X}X}(\varepsilon)N_{I}+N_{IX}N_{I}+N_{QX}N_{Q}\]  
Using the linear model at \(\varepsilon=0\) for the code discriminator, it can be shown that the discriminator is as follows.  
\[d_{nc}(\varepsilon)=g_{0}\varepsilon+N \tag{16}\]  
Here, \(g_{0}\) represents the gain of the discriminator, and \(N\) means the Gaussian white noise with zero mean. Then, Equation (16) can be transformed into Equation (17).  
\[g_{0} = 2 CR_{\hat{X}X}(0)R^{\prime}_{\hat{X}W}(0)\] \[N = \sqrt{2C}R_{\hat{X}X}(0)N_{I}+\sqrt{2C}R_{\hat{X}W}(0)N_{IX}+N_{ IX}N_{I}+N_{QX}N_{Q} \tag{17}\]  
Here, we assume that the channel impulse response is undistorted and that the correlation function of the received signal between the local replica is symmetric, i.e., \(R^{\prime}_{\hat{X}X}(0)=0\). Furthermore, the designed reference waveform \(W(t)\) ensures \(R_{\hat{X}W}(0)=0\). The variance of \(N\) can be calculated as follows.  
\[\sigma_{N}^{2} = E[N^{2}] \tag{18}\] \[= 2 CR_{\hat{X}X}^{2}(0)E[N_{I}^{2}]+2 CR_{\hat{X}W}^{2}(0)E[N_{IX}^{ 2}]+E[N_{IX}^{2}N_{I}^{2}]+E[N_{QX}^{2}N_{Q}^{2}]\] \[= 2 CR_{\hat{X}X}^{2}(0)R_{WW}(0)N_{0}/T+2R_{XX}(0)R_{WW}(0)(N_{0}/T )^{2}\]  
From the above, the steady state normalized code error variance is given by [16]:  
\[\sigma_{N}^{2}=\frac{2B_{L}T\sigma_{N}^{2}}{g_{0}^{2}}=\frac{B_{ L}RWW(0)}{R^{\prime}_{\hat{X}W}(0)C/N_{0}}[1+\frac{1}{R^{2}_{\hat{X}X}(0)C/N_{0}T}] \tag{19}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Bandwidth** & **BOC(6,1)** & **BOC(7,1)** \\ \hline Infinite (chip\({}^{2}\)) & 0.0108 & 0.0087 \\
24.552 MHz (chip\({}^{2}\)) & 0.0119 & \\
20.46 MHz (chip\({}^{2}\)) & & 0.0093 \\
18.414 MHz (chip\({}^{2}\)) & 0.0119 & \\
16.368 MHz (chip\({}^{2}\)) & & 0.0093 \\ \hline \end{tabular}
\end{table}
Table 9: The multipath error envelope of the BOC(6,1) and BOC(7,1) signals.  
The output of coherent discriminator is given by:  
\[d_{c}(\varepsilon) = (\sqrt{2C}R_{\tilde{X}X}(\varepsilon)\cos(\varphi_{e})+N_{IX}) \tag{20}\] \[+ (\sqrt{2C}R_{\tilde{X}X}(\varepsilon)\sin(\varphi_{e})+N_{QX})\]  
Similarly, the corresponding steady state normalized code error variance can be given by:  
\[\sigma_{N}^{2}=\frac{2B_{L}R_{WW}(0)}{R_{XW}^{\prime 2}(0)C/N_{0}} \tag{21}\]  
In this case, the tracking jitters of the proposed discriminators are compared to that of the W2 and double-delta technique, assuming \(B_{L}=1\) Hz and \(T=1\) ms.