# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 7 Results: Time-Delay Estimation Accuracy Limit for Some Representative GNSS Signals  
To assess the time-delay performance limit with the different GPS, Galileo and GNSS meta-signals introduced in the previous sections, we computed the closed-form CRB in Equation (6) and the corresponding MLE in Equation (3) for these GNSS representative signals:  
* the 1 ms of the GPS L1 C/A with PRN codes of length 1023 chips, BPSK(1);
* the 10 ms for the GPS L1C (data) with a PRN code of length 10,230 chips and a BOC(1,1) subcarrier;
* the 10 ms of the GPS L1C (pilot) with a PRN code of length 10,230 chips and a TMBOC subcarrier;
* the 4 ms of the Galileo E1B (data) with a PRN code of length 4096 chips and a CBOC(\(+\)) subcarrier;
* the 4 ms of the Galileo E1C (pilot) with a PRN code of length 4096 chips and a CBOC(\(-\)) subcarrier;
* the 1 ms of the GPS L5-I with PRN codes of length 10,230 chips, BPSK(10);
* the 1 ms of the Galileo E6B signal with PRN code of length 5115 chips, BPSK(5);
* the 1 ms of the Galileo E5b-I with PRN code of length 10,230 chips, BPSK(10);
* the 1 ms of the Galileo E5 AltBOC(15,10) with PRN codes of length 10230 chips;
* the 1 ms of the meta E5b + E6-BC AltBOC(35,10,5) signal, with PRNs of length 10,230 and 5115 chips; and
* the 1 ms of the meta E5a + E6-BC AltBOC(50,10,5) signal, with PRNs of length 10,230 and 5115 chips.  
The CRB in Equation (6) and the corresponding MLE in Equation (3) were computed considering \(\alpha=(1+j)\cdot\sqrt{SNR_{in}/2}\). The MLE was obtained from 1000 Monte Carlo runs. Section 7.1 shows the comparison of the legacy L1 C/A with the new GPS L1C and the Galileo E1 OS (E1B and E1C), the comparison with the other GPS civil signals at L2 and L5 is shown in Section 7.2, and the complete discussion with the other Galileo signals and meta-signals in Section 7.3. Notice that the \(\text{SNR}_{\text{out}}\) in the following results refers to the SNR at the output of the matched filter. The maximum SNR at the output of the ML matched filter is  
\[\text{SNR}_{\text{out}}=\frac{F_{s}\left|\alpha\right|^{2}\mathbf{c}^{H} \mathbf{c}}{\sigma_{n}^{2}}=\frac{C}{N_{0}}T_{\text{PRN}}L_{c}, \tag{25}\]  
where \(C/N_{0}\) (dB-Hz) is the carrier-to-noise density ratio, \(T_{\text{PRN}}\) is the single code duration (equal to 1 ms in the case of GPS L1 C/A), and \(L_{c}\) is the number of codes; therefore, \(T_{I}=T_{\text{PRN}}\times L_{c}\) is the coherent integration time. Then, we could verify that \(\text{SNR}_{\text{out}}=25\) dB and \(T_{I}=10\) ms imply a \(C/N_{0}=45\) dB-Hz, which is a nominal GNSS value. In addition, the MLE threshold, i.e., the point where the MLE reaches the optimal operation regime, was always found around \(\text{SNR}_{\text{out}}=15\) dB, which for \(T_{I}=1\) ms corresponds to a \(C/N_{0}=45\) dB-Hz, for \(T_{I}=10\) ms to a \(C/N_{0}=35\) dB-Hz, and for \(T_{I}=20\) ms to a \(C/N_{0}=32\) dB-Hz. The latter fixes the limit for the use of the GNSS signals in standard coherent integration architectures. To go below this \(C/N_{0}\) value, we have to resort to the so-called high-sensitivity GNSS (HS-GNSS) techniques, with combinations of coherent and non-coherent integrations due to the navigation data bits.