# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 7 Results: Time-Delay Estimation Accuracy Limit for Some Representative GNSS Signals
### Performance Limits of Phase Estimation with AltBOC-Type GNSS Signals  
As already stated in the Introduction, the standard way to obtain GNSS precise positioning solutions is to exploit the signal phase information via RTK or PPP techniques [9]. The main problem with these techniques is that they have to deal with phase ambiguities, which imply long convergence \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SNR\({}_{\text{out}}\) (dB)** & \(T_{I}\) & **AltBOC(15,10)** & **AltBOC(35,10,5)** & **AltBOC(50,10,5)** \\ \hline
25 & 10 ms & 8 & 3.55 & 2.51 \\ \hline
30 & 30 ms & 4.47 & 2 & 1.41 \\ \hline
35 & 100 ms & 2.5 & 1.12 & 0.79 \\ \hline
40 & 300 ms & 1.41 & 0.63 & 0.45 \\ \hline
45 & 1 s & 0.8 & 0.35 & 0.25 \\ \hline
50 & 3 s & 0.45 & 0.2 & 0.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Galileo E5 and Meta-signals Time-delay Estimation Standard Deviation (cm). Results for Long Integration Times and \(C/N_{0}=45\) dB-Hz. times for PPP and complicated ambiguity resolution techniques for RTK, in addition to being limited to a low number of satellites to achieve high fixing success rates [9] (Chapter 23). From the CRB results in Section 3, while the time-delay estimation is related to the baseband signal resolution, as shown in the previous subsections for the different GPS and Galileo signals/meta-signals, phase estimation is mainly driven by the SNR at the output of the matched filter. Indeed, the phase estimation CRB is \(\text{CRB}_{\varphi|\mathbf{c}}\simeq\frac{1}{2\text{SNR}_{\text{out}}}\) (equality for real signals). Then, using fast codes such the ones in GPS L5 or Galileo E6, E5a, E5b, or AltBOC-type signals as in Galileo E5 and GNSS meta-signals, do not improve the phase estimation with respect to the legacy GPS L1 C/A signal. Then, the huge advantage of using these new signals in terms of time-delay estimation precision does not translate to the corresponding signal phase. This is an additional noteworthy result to further support the fact that AltBOC-type signals can be a promising solution for code-based precise positioning, because the advantage brought by joint delay/phase positioning using these signals may not be worth the inherent complexity and limitations of ambiguity resolution techniques. Let us recall that the MLE of time-delay and phase are given by  
\[\widehat{\tau}=\arg\max_{\tau}\left\{\left|\left(\mathbf{c}\left( \tau\right)^{H}\mathbf{c}\left(\tau\right)\right)^{-1}\mathbf{c}\left(\tau \right)^{H}\mathbf{x}\right|^{2}\right\}, \tag{26}\] \[\widehat{\varphi}\left(\widehat{\tau}\right)=\arg\left\{\left( \mathbf{c}\left(\widehat{\tau}\right)^{H}\mathbf{c}\left(\widehat{\tau} \right)\right)^{-1}\mathbf{c}\left(\widehat{\tau}\right)^{H}\mathbf{x}\right\}, \tag{27}\]  
then the phase is estimated as the argument of the cross-ambiguity function evaluated at the time-delay MLE. Therefore, we are only concerned by the receiver operation points where we obtain a correct delay estimate, that is, \(15<\text{SNR}_{\text{out}}<28\) dB (the upper value given by \(T_{I}=20\) ms at a nominal \(C/N_{0}=45\) dB-Hz, which is the \(T_{\text{bit}}\) coherent integration time limit in GPS L1 C/A signals without considering HS-GNSS architectures). In Table 3, we give some phase estimation precision results and the corresponding time-delay ones. Notice that \(\widehat{\varphi}\left(\widehat{\tau}\right)\) in meters was obtained as \(\frac{\lambda}{2\pi}\sqrt{\text{CRB}_{\varphi|\mathbf{c}}}\). The values for different wavelengths are almost the same (i.e., \(\lambda_{L1}=19.03\) cm, \(\lambda_{L5}=25.48\) cm, \(\lambda_{E5}=25.15\) cm, \(\lambda_{E5b-E6}=24.12\) cm and \(\lambda_{E5a-E6}=24.42\) cm), thus we use \(\lambda_{E5}\) in Table 3.  
From the GPS L1 C/A results shown in Table 3, we see that we obtain the typical values found in the GNSS literature, that is, code observable standard deviation \(\sigma_{\rho_{L1}}\simeq[2-5]\) m and phase observable standard deviation \(\sigma_{\varphi_{L1}}\simeq[1-3]\) mm.