# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 6 GNSS Meta-Signals
### AltBOC Spectral and Correlation Properties
Finally, define the autocorrelation function in terms of the PSD, we resort to the Wiener-Khintchine theorem [37] (Chapter 10), which states that the ACF and its PSD function are a Fourier transform pair defined as,  
\[ACF(t)=\int_{-\frac{Bt}{2}}^{\frac{Bt}{2}}G_{c}(f)e^{-j2\pi ft}df\, \tag{24}\]  
with \(Br\) the receiver bandwidth. The PSD of Galileo E5/E6 signals along with the GNSS meta-signal PSD defined by the AltBOC(50,10,5) and AltBOC(35,10,5) are illustrated in Figure 3 (left). Note from these results that the PSD of GNSS meta-signals slightly differ from the PSD of Galileo E5/E6. This is because the AltBOC modulation adds intermodulation products (refer to Equation (16)) to keep the constant envelope shape. Note also that generating those extra intermodulation products at the receiver replica is unnecessary since they were not generated at the transmission engine (Galileo E5 and E6 are transmitted with two independent HPA amplifiers). The corresponding ACFs are illustrated in Figure 3 (right), which shows that ACFs of GNSS meta-signals are narrower than those obtained for E5/E6. For completeness, we computed the ACF of the meta-signals _without intermodulation products_, but the difference with the ACF shown in Figure 3 is marginal, with no impact on the delay estimation accuracy. A summary of the different GNSS signals' main characteristics is given in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Signal** & **Modulation** & \(T_{\text{PRN }}\)**(ms)** & \(T_{\text{bit }}\)**(ms)** & **ACF Main Peak** \\ \hline GPS L1 C/A & BPSK(1) & 1 & 20 & \(\pm T_{c}\) \\ \hline GPS L1C Data & BOC(1,1) & 10 & 10 & \(\pm 0.33T_{c}\) \\ \hline GPS L1C Pilot & TMBOC & 10 & - & \(\pm 0.35T_{c}\) \\ \hline GPS L5-I & BPSK(10) & 1 & 10 & \(\pm 0.1T_{c}\) \\ \hline GPS L5-Q & BPSK(10) & 1 & - & \(\pm 0.1T_{c}\) \\ \hline Galileo E1B & CBOC \(+\) & 4 & 4 & \(\pm 0.36T_{c}\) \\ \hline Galileo E1C & CBOC \(-\) & 4 & - & \(\pm 0.34T_{c}\) \\ \hline Galileo E6B & BPSK(5) & 1 & 1 & \(\pm 0.2T_{c}\) \\ \hline Galileo E6C & BPSK(5) & 1 & - & \(\pm 0.2T_{c}\) \\ \hline Galileo E5b-I & BPSK(10) & 1 & 4 & \(\pm 0.1T_{c}\) \\ \hline Galileo E5a-I & BPSK(10) & 1 & 20 & \(\pm 0.1T_{c}\) \\ \hline Galileo E5a/B-Q & BPSK(10) & 1 & - & \(\pm 0.1T_{c}\) \\ \hline Galileo E5 & AltBOC(15,10) & 1 & 4 (E5b-I) & \(\pm 0.017T_{c}\) \\ \hline Meta-signal E5b-E6 BC & AltBOC(35,10,5) & 1 & 1 (E6B) & \(\pm 0.00725T_{c}\) \\ \hline Meta-signal E5a-E6 BC & AltBOC(50,10,5) & 1 & 1 (E6B) & \(\pm 0.005T_{c}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: GPS, Galileo, and meta-signals main characteristics. \(T_{c}\) is defined as the GPS L1 C/A chip period, i.e., \(1.023\times 10^{-6}\) s, and ACF main peak refers to the first zero-crossing of the ACF. Figure 3: **(Left)** Galileo E5/E6 and GNSS meta-signal PSD, and **(Right)** GPS L1 C/A, Galileo E6B, E5b-I, E5 and GNSS meta-signals ACFs, _with intermodulation products_. Notice the width of the E6B ACF in \([-\frac{T}{5},\frac{T}{5}]\) because it uses a BPSK(5), and the E5b-I ACF width in \([-\frac{T}{10},\frac{T}{10}]\) because it uses a BPSK(10).