# Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals
## 2 GNSS Signal Model, Meta-Signals, and HO-BOC Signals
### Generalized AltBOC  
The GNSS meta-signal is a basic concept that consists of processing two GNSS signals transmitted at two different carrier frequencies as a single GNSS signal [8; 9]. In order to process both signals jointly, the so-called alternate binary offset carrier (AltBOC) modulation or the alternate linear offset carrier (AltLOC) modulation [26] can be used, the goal being to express the two signals as a single one. The reader can refer to [27] and the previous references for a more exhaustive definition of the generalized alternate binary offset carrier (AltBOC) modulation and the corresponding spectral and ACF properties. The idea to process two GNSS signals located at two different bands jointly comes from the proposal made by the the Galileo Signal Task Force group in the year 2000. In the original proposal, an AltBOC signal was considered to transmit two independent signals in two separate bands using a unique high power amplifier (HPA). An AltLOC signal was also studied at the time, but it was quickly dropped because its envelope was not constant. Notice that a constant envelope signal at the transmitter is interesting in order to enhance the HPA efficiency. However, as was shown in [26], once the transmitter filters the intermodulation products and the harmonics of the AltBOC signal, the transmitted signal yields to an AltLOC modulation. Therefore, two different GNSS signals can be jointly processed at the receiver as a single AltLOC-modulated signal.  
The easiest form of AltLOC-modulated signal is the one where two independent PRN codes are multiplexed. Let us define the subcarrier with cosine and sine phasing as \(SC_{cos}(t)=\cos\left(2\pi F_{sub}t\right)\) and \(SC_{sin}(t)=\sin\left(2\pi F_{sub}t\right)\), respectively, where \(F_{sub}\) represents the subcarrier frequency. Then, we can build the single side band (SSB) subcarrier \(SC_{SSB}\) and its conjugate \(SC_{SSB}^{*}\) as,  
\[SC_{SSB}(t)=\frac{1}{\sqrt{2}}\left(SC_{cos}(t)\ +\ j\cdot SC_{sin}(t)\right)\ ;\ SC_{SSB}^{*}(t)=\frac{1}{\sqrt{2}}\left(SC_{cos}(t)-j\cdot SC_{sin}(t)\right) \tag{7}\]where \(j\) represents the imaginary part. The two-code AltLOC can be defined as,  
\[c(t)=c_{A}(t)\text{SC}^{*}_{SSB}(t)\ +\ c_{B}(t)\text{SC}_{SSB}(t)=\left[c_{A}(t)\ +\ c_{B}(t)\right]\text{SC}_{cos}(t)\ +\ j\cdot\left[c_{B}(t)-c_{A}(t)\right]\text{SC}_{sin}(t) \tag{8}\]  
where \(c_{A}(t)\) and \(c_{B}(t)\) represent the GNSS signals at the low (\(A\)) and high (\(B\)) frequency bands, respectively. Note from Equation (8) that the codes \(c_{A}(t)\) and \(c_{B}(t)\) are not required to have the same chip rate. We use the notation AltLOC(\(p,q,w,b\)) to represent the AltLOC receiver signal. Thus, given the fundamental frequency \(f_{0}=1.023\) MHz, \(p\) is the real number used to represent the subcarrier frequency \(F_{sub}\) through the equality \(F_{sub}=pf_{0}\); \(q\) is the real number used to represent the lower code chip rate \(f_{c,A}\) through the equality \(f_{c,A}=qf_{0}\); \(w\) is the real number used to represent the upper code chip rate \(f_{c,B}\) through the equality \(f_{c,B}=wf_{0}\); and \(b\) is the real number used to represent the receiver bandwidth \(BW\) through the equality \(BW=bf_{0}\).