# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 6 GNSS Meta-Signals
### Generalized AltBOC  
The AltBOC modulation was original introduced as a method to combine Galileo signals within the E5a and E5b bands. This solution was rapidly accepted by the Galileo Signal Task Force (GSTF) since the AltBOC modulation: (i) provides a Constant Envelope Modulation (CEM), which avoids non-linear distortions at the output of the High Power Amplifier (HPA); and (ii) it provides a high level of isolation between two frequency bands [32]. The easiest form of AltBOC modulation is the one where two independent PRN codes are multiplexed. Let us define the BOC subcarrier with cosine and sine phasing as \(SC_{cos}(t)=sign\left(\cos\left(2\pi F_{sub}t\right)\right)\) and \(SC_{sin}(t)=sign\left(\sin\left(2\pi F_{sub}t\right)\right)\), respectively, where  
Figure 2: ACF of the GPS L1 C/A (**a**), GPS L5 (**b**), Galileo E6 (**c**), GPS L1C (**d**), Galileo E1 OS (**e**) and Galileo E5 (**f**). \(F_{sub}\) represents the subcarrier frequency. Then, we can built the Single Side Band (SSB) subcarrier \(SC_{SSB}\) and its conjugate \(SC_{SSB}^{*}\) as,  
\[SC_{4,SSB}(t)=\frac{1}{\sqrt{2}}\left(SC_{cos}(t)+j\cdot SC_{sin}(t)\right)\ ;\ SC_{4,SSB}^{*}(t)=\frac{1}{\sqrt{2}}\left(SC_{cos}(t)-j\cdot SC_{sin}(t) \right). \tag{12}\]  
Note that Equation (12) can take four values and it can be also derived as,  
\[\left\{ \begin{array}{lcl}SC_{SSB}(t)&=&e^{i\left(\frac{\pi}{4}+i\cdot\frac{ \pi}{2}\right)};\\ t&\mod T_{sub}&\in&\left[\frac{i\cdot T_{s}}{4},\frac{(i+1)\cdot T_{sub}}{4} \right],\end{array} \right. \tag{13}\]  
with \(i\in[0,1,2,3]\) and \(T_{sub}=1/F_{sub}\). Finally, the two-code AltBOC can be defined as,  
\[c(t)=c_{A}(t)SC_{4,SSB}^{*}(t)+c_{B}(t)SC_{4,SSB}(t)=\left[c_{A}(t)+c_{B}(t) \right]SC_{cos}(t)+j\cdot\left[c_{B}(t)-c_{A}(t)\right]SC_{sin}(t) \tag{14}\]  
where \(c_{A}(t)\) and \(c_{B}(t)\) represent binary PRN codes of the low (\(A\)) and high (\(B\)) band signals, respectively. Note from the previous equation that codes \(c_{A}(t)\) and \(c_{B}(t)\) are not required to have the same chip rate. Considering two independent codes (in-phase/quadrature) per each frequency band, a four-code AltBOC modulation can be expressed as,  
\[c(t)=\left[c_{A,I}(t)+j\cdot c_{A,Q}(t)\right]SC_{4,SSB}^{*}(t)+\left[c_{B,I}( t)+j\cdot c_{B,Q}(t)\right]SC_{4,SSB}(t)\, \tag{15}\]  
where \(c_{A,I}(t)\) and \(c_{A,Q}(t)\) represent binary in-phase/quadrature codes of the \(A\) band and \(c_{B,I}(t)\) and \(c_{B,Q}(t)\) represent binary in-phase/quadrature codes of the \(B\) band. The resulting constellation from Equation (15) was shown by [PERSON] et al. [32] to not a have constant envelope and consequently such modulation cannot be used with HPA working at the saturation point. In order to obtain a CEM, an intermodulation product [34] must be added within the composite signal, yielding Equation (15) to [35],  
\[c(t) =\left[c_{A,I}(t)+j\cdot c_{A,Q}(t)\right]SC_{8,SSB}^{*}(t)+ \left[c_{B,I}(t)+j\cdot c_{B,Q}(t)\right]SC_{8,SSB}(t)\] \[+\left[\overline{c_{A,I}(t)}+j\cdot\overline{c_{A,Q}(t)}\right]SC _{P,8,SSB}^{*}(t)+\left[\overline{c_{B,I}(t)}+j\cdot\overline{c_{B,Q}(t)} \right]SC_{P,8,SSB}(t) \tag{16}\]  
where \(\overline{c_{A,I}(t)}=c_{A,Q}(t)c_{B,I}(t)c_{B,Q}(t)\), \(\overline{c_{A,Q}(t)}=c_{A,I}(t)c_{B,I}(t)c_{B,Q}(t)\), \(\overline{c_{B,I}(t)}=c_{B,Q}(t)c_{A,I}(t)c_{A,Q}(t)\) and \(\overline{c_{B,Q}(t)}=c_{B,I}(t)c_{A,I}(t)c_{A,Q}(t)\), and \(SC_{8,SSB}(t)\) is the SSB \"single\" subcarrier of the four-code AltBOC. Moreover, \(SC_{P,8,SSB}(t)\) is defined as the \"product\" subcarrier. Both subcarriers can take eight different values [32],  
\[\left\{ \begin{array}{lcl}SC_{8,SSB}(t)&=&e^{i\left(\frac{\pi}{8}+i\cdot \frac{\pi}{4}\right)};\\ t&\mod T_{sub}&\in&\left[\frac{i\cdot T_{sub}}{8},\frac{(i+1)\cdot T_{s}}{8 }\right],\end{array} \right.,\ \ \left\{ \begin{array}{lcl}SC_{P,8,SSB}(t)&=&e^{i\left(\frac{\pi}{8}-i\cdot \frac{\pi}{4}\right)};\\ t&\mod T_{sub}&\in&\left[\frac{i\cdot T_{sub}}{8},\frac{(i+1)\cdot T_{s}}{8 }\right],\end{array} \right. \tag{17}\]  
with \(i\in[0,1,2,3,4,5,6,7]\).