# Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals
## 2 GNSS Signal Model, Meta-Signals, and HO-BOC Signals  
### Signal Model  
The GNSS signals broadcast by the different satellite constellations are typically built as a multilayer structure: (i) a low rate navigation message, encoded as binary phase-shift keying (BPSK) bits; (ii) a fast rate ranging code, so-called pseudo-random noise (PRN) code, with good autocorrelation and cross-correlation properties in order to allow individual satellite signals' processing (i.e., quasi orthogonality); (iii) a subcarrier that modulates the PRN code and shapes the autocorrelation function (ACF), i.e., no subcarrier is employed for the legacy GPS L1-C/A signal or binary offset carrier (BOC)-type subcarriers in modernized GPS and some Galileo signals; (iv) a carrier that is used to allocate the complete signal into the corresponding frequency. Notice that the signal may have data bits or not, depending on whether it belongs to a data component or a pilot component. In the sequel, and without loss of generality, we do not account for navigation data bits within the observation time. In general, the signal at the receiver antenna is the superposition of a set of signals at different frequency bands, plus environmental effects such as multipath and/or interferences. The latter two effects are out of the scope of this contribution, and because of the quasi orthogonal PRN code design, we can focus on one of these signals to define the model to be exploited. Therefore, we consider the transmission of a band-limited GNSS signal \(c(t)\) (bandwidth \(B\)), which contains both the PRN and subcarrier, over a carrier frequency \(f_{c}\) (\(\lambda_{c}=c/f_{c}\)), from a transmitter (satellite) \(T\) to a receiver \(R\). If \(T\) and \(R\) follow a relative uniform radial movement, then the \(T\)-to-\(R\) distance, \(\mathbf{p}_{TR}\left(t\right)\), can be approximated by a first-order distance-velocity model [13; 14; 15; 16],  
\[\left\|\mathbf{p}_{TR}\left(t\right)\right\|\triangleq\left\| \mathbf{p}_{R}\left(t\right)-\mathbf{p}_{T}\left(t-\tau\left(t\right)\right) \right\|=c\tau\left(t\right)\simeq d\ +\ vt,\] \[\tau\left(t\right)\simeq\tau\ +\ bt,\quad\tau=\frac{d}{c},\ b=\frac{v}{c}, \quad c=299,792,458\ \mathrm{m/s}, \tag{1}\]  
where \(d\) is the \(T\)-to-\(R\) relative radial distance and \(v\) is the \(T\)-to-\(R\) relative radial velocity. Note that the previous formula is characterized by a time-delay \(\left(\tau\right)\) due to the propagation path and a dilation \(\left(1-b\right)\) induced by the Doppler effect. The complex analytic signal at the output of the receiver's antenna can be written as:  
\[x_{A}(t)=\alpha_{R}c\left(\left(1-b\right)\left(t-\tau\right)\right)e^{j2\pi f _{c}\left(1-b\right)t}e^{-j2\pi f_{c}\tau}\ +\ n_{A}(t), \tag{2}\]  
with \(e\) the Euler number (exponential function), \(n_{A}(t)\) a zero-mean white complex Gaussian noise, and where the gain \(\alpha_{R}\) depends on the transmitted signal power, the transmitter/receiver antenna gains, polarization vectors, and the radial distance between T and R [17; 18]. Notice that the following the standard narrowband assumption (2) can be approximated as [19; 20]:  
\[x_{A}(t)\approx\alpha_{R}c\left(t-\tau\right)e^{j2\pi f_{c}\left(1-b\right)t} e^{-j2\pi f_{c}\tau}\ +\ n_{A}(t), \tag{3}\]and because we are only interested in the processing of time-delay and phase information, (3) can be further simplified considering that during the observation time (i.e., 1 ms), both \(T\) and \(R\) are static, that is their respective positions are constant \(\mathbf{p}_{T}(t)=\mathbf{p}_{T}\) and \(\mathbf{p}_{R}(t)=\mathbf{p}_{R}\).