# Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals
## 4 GNSS Meta-Signal and HO-BOC Signals Delay/Phase Estimation Results  
In the problem of interest, the first step of the receiver (see Section 3.1) is the time-delay and phase ML estimation. Therefore, in order to assess the performance for the different GNSS meta-signals and HO-BOC signals in Section 2, we compared the delay and phase MLEs (9) and (10) with the corresponding CRBs in (11) and (12). In the rest of the article, we consider the following sampling frequencies:  
* Galileo E5 AltBOC(15,10)--\(F_{s}=60\) MHz. * L5Q AltLOC(25,10,1,75)--\(F_{s}=75\) MHz. * Galileo E5 BQ + E6B AltLOC(35,10,5,112)--\(F_{s}=112\) MHz. * BeiDou B2 AQ + B3Q AltLOC(45,10,10,120)--\(F_{s}=120\) MHz. * Galileo E5 AQ + E6B AltLOC(50,10,5,132)--\(F_{s}=133\) MHz. * BeiDou B2 + B3Q AltLOC(37.5,C\({}_{B2}\)(t),10,125)--\(F_{s}=125\) MHz. * Galileo E5 + E6B AltLOC(42.5,C\({}_{E5}\)(t),5,132)--\(F_{s}=135\) MHz. * M BOCsin(10,5)--\(F_{s}=30\) MHz. * Galileo E1 PRS BOCcos(15,2,5)--\(F_{s}=40\) MHz. * Galileo E6 PRS BOCcos(10,5)--\(F_{s}=30\) MHz. * BeiDou B1A BOCcos(14,2)--\(F_{s}=40\) MHz. The MLEs and CRBs were computed considering \(\alpha=(1\ +\ j)\cdot\sqrt{SNR_{in}/2}\), where \(j\) represents the imaginary number. The root mean squared error (RMSE) for the MLE was obtained from 1000 Monte Carlo runs. Notice that the \(\text{SNR}_{\text{out}}\) in the following results refers to the SNR at the output of the MLE, and the maximum is given by:  
\[\text{SNR}_{\text{out}}=\frac{F_{s}\left|\alpha\right|^{2}\text{c}^{H}\text{c}} {\sigma_{n}^{2}}=\frac{C}{N_{0}}T_{\text{PRN}}L_{c}, \tag{21}\]  
where \(C/N_{0}\) (dB-Hz) is the carrier-to-noise density ratio, \(T_{\text{PRN}}\) is the single code duration, \(L_{c}\) is the number of codes, and 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 implied a \(C/N_{0}=45\) dB-Hz, which was a nominal GNSS value. The results for the different meta-signals considered in this article are summarized in Figure 5.  
* First, notice from Subplot (a) that there was a difference of: (i) 2 dB between E5 and L2 + L5 and (ii) 1.44 dB between L2 + L5 and E5B + E6. Among the rest of the meta-signals, the maximum difference of 1.6 dB was exhibited between E5B + E6 and E5A + E6. Therefore, overall and after the convergence of the delay MLE, we had a maximum factor equal to three among meta-signals. * Considering \(\text{SNR}_{\text{out}}\) = 25 dB as a reference value, the time-delay standard deviation was (i.e., following the order in Figure 5a) roughly (results in cm): 12, 7.5, 5.4, 4.2, 3.7, 5.3, and 4.6. * What was more interesting was the impact of the subcarrier, i.e., the ACF, into the time-delay CRB values and the MLE convergence. Comparing the results in Figure 5 with the corresponding ACFs in Figure 2, it was clear that: (1) a narrower ACF main peak implied better time-delay estimation capabilities (see Subplots (a) and (b) in Figure 2), i.e., a lower CRB, (2) larger secondary peaks and closer to the main one directly impacted the convergence to the CRB and the ML threshold. This was directly related to possible false locks, which were more probable at lower SNR when the ACF secondary peaks were large and close to the main one of interest. For instance, for Galileo E5, E5 + E6 and B2 + B3, these secondary peaks (see Subplots (g) and (h) in Figure 2) only slightly affected the convergence to the CRB, which was reached for \(\text{SNR}_{\text{out}}\) = [16; 17; 18] dB. In contrast, larger secondary peaks in L2 + L5, B2A + B3, E5B + E6, and E5A + E6 led to a convergence in \(\text{SNR}_{\text{out}}\) = [20; 21; 22; 23; 24] dB. * Regarding the phase CRB and MLE, the latter converged only if the corresponding time-delay MLE did. This was because the phase MLE (10) was given by the argument evaluated at the time-delay MLE.