# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 6 GNSS Meta-Signals
### AltBOC Spectral and Correlation Properties  
If one wants to compute the AltBOC PSD, it is interesting to use the signal formulation introduced in [30]. According to [PERSON] [30], any chip within the PRN code can be seen as a number of equal-length deterministic segments with different amplitude levels (also known as Multilevel Coded Spreading (MCS) symbols). Then, the expression of the chip waveform is \(p_{chip}(t)=\sum_{n=0}^{N_{x}-1}a_{n}p_{sub_{chip}}(t-n\frac{T_{c}}{N_{x}})\), where \(N_{x}\) is the number of equal-length segments, i.e., sub-chips,and \(p_{sub_{chip}}\) represents the sub-chip shape. Note that in the GNSS framework \(p_{sub_{chip}}\) is usually assumed to be rectangular. Moreover, \(\frac{T_{c}}{N_{x}}\) can also be defined as the subcarrier period \(T_{sub}\). The transmitted signal is \(c(t)=\sum_{n=-\infty}^{+\infty}c_{n}p_{chip}(t-nT_{c})\), with \(c_{n}\) the amplitude of the PRN code sequence. Considering that the PRN code shows ideal statistical properties, the PSD of \(c(t)\) simplifies to \(G_{c}(f)=|C(f)|/T_{c}\)[30],  
\[G_{c}(f)=\frac{1}{T_{c}}\frac{\sin^{2}\left(\frac{\pi fT_{c}}{N_{x}}\right)}{ \left(\pi f\right)^{2}}\left|\left|\sum_{n=1}^{N_{x}}a_{n}e^{-2\pi j\frac{\pi f _{c}}{N_{x}}}\right|\right|^{2}. \tag{18}\]  
where the first term defines the PSD of a Binary Phase Shift Keying (BPSK) modulation with chip rate \(\frac{T_{c}}{N_{x}}\) and the second term determines the MCS modulation family, \(G_{c}(f)=G_{BPSK}(f)G_{mod}(f)\). Assuming that the AltBOC transmits codes with the same chip rate (symmetrical AltBOC), one can use the approach proposed in [36] to compute the symmetrical AltBOC PSD. Since GNSS meta-signals may be generated by non-symmetrical AltBOC waveforms, i.e., the chip rate of the upper and lower codes are not the same, we follow the approach [PERSON] et al. [33] to compute the GNSS meta-signals PSD. We use the notation AltBOC(p,q,w), where \(F_{sub}=pf_{0}\) is the subcarrier frequency with \(f_{0}=1.023\) MHz and \(f_{c,A}=qf_{0}\) and \(f_{c,B}=wf_{0}\) are the lower and upper codes' chip rate, respectively. The non-symmetrical AltBOC PSD expression is therefore [33],  
\[G_{AltBOC(p,q,w)}(f)=4 \begin{bmatrix}G_{BPSK,N_{A}}(f)G_{mod,N_{A}}(f)+G_{BPSK,N_{B}}(f)G_{mod,N_{B}}(f)\\ +G_{BPSK,N_{\overline{A}}}(f)G_{mod,N_{\overline{A}}}(f)+G_{BPSK,N_{\overline {B}}}(f)G_{mod,N_{\overline{B}}}(f)\end{bmatrix} , \tag{19}\]  
where  
\[G_{BPSK,N_{A}}(f) =f_{c,A}\frac{sin^{2}\left(\frac{\pi f}{N_{A/c,A}}\right)}{\left( \pi f\right)^{2}},\ G_{BPSK,N_{B}}(f)=f_{c,B}\frac{sin^{2}\left(\frac{\pi f}{ N_{B/c,B}}\right)}{\left(\pi f\right)^{2}}, \tag{20}\] \[G_{BPSK,N_{\overline{A}}}(f) =f_{c,\overline{A}}\frac{sin^{2}\left(\frac{\pi f}{N_{A/c, \overline{A}}}\right)}{\left(\pi f\right)^{2}}, \tag{21}\]  
with \(G_{BPSK,N_{\overline{A}}}(f)=G_{BPSK,N_{\overline{B}}}(f)\), \(N_{A}=4\left(2p/q\right)\), \(N_{B}=4\left(2p/w\right)\), \(N_{\overline{A}}=N_{\overline{B}}=4\left(2p/LCM(q,w)\right)\) and \(f_{c,\overline{A}}=LCM(f_{c,A},f_{c,B})\). Note that \(LCM\) stands for Least Common Multiple. In addition,  
\[G_{mod,N_{A}}(f) =\left|\left|\sum_{n=1}^{N_{sub}}a_{n_{A}}e^{-2\pi j\frac{\pi f}{ N_{A/c,A}}}\right|\right|^{2};\ G_{mod,N_{B}}(f)=\left|\left|\sum_{n=1}^{N_{sub}}a_{n_{B}}e^{-2 \pi j\frac{\pi f}{N_{B/c,B}}}\right|\right|^{2}, \tag{22}\] \[G_{mod,N_{\overline{A}}}(f) =\left|\left|\sum_{n=1}^{N_{sub}}a_{n_{\overline{A}}}e^{-2\pi j \frac{\pi f}{N_{A/c,\overline{A}}}}\right|\right|^{2};\ G_{mod,N_{\overline{B}}}(f)= \left|\left|\sum_{n=1}^{N_{sub}}a_{n_{B}}e^{-2\pi j\frac{\pi f}{N_{A/c, \overline{A}}}}\right|\right|^{2}, \tag{23}\]  
where \(a_{n_{A}}\), \(a_{n_{B}}\), \(a_{n_{\overline{A}}}\), and \(a_{n_{\overline{B}}}\) are the complex values derived from \(SC_{8,SSB}^{*}(t)\), \(SC_{8,SSB}(t)\), \(SC_{P,8,SSB}^{*}(t)\), and \(SC_{P,8,SSB}(t)\), respectively (refer to Equation (17)).